Scuola Internazionale Superiore di Studi...
Transcript of Scuola Internazionale Superiore di Studi...
Scuola Internazionale Superiore di Studi Avanzati
Ph.D. Course in Mathematical Analysis, Modelling, and Applications
On some mathematical problemsin fracture dynamics
Ph.D. Thesis
Candidate:Maicol Caponi
Advisor:Prof. Gianni Dal Maso
Academic Year 2018/2019
iii
Il presente lavoro costituisce la tesi presentata da Maicol Caponi, sotto la direzione delProf. Gianni Dal Maso, al fine di ottenere lâattestato di ricerca post-universitaria DoctorPhilosophiĂŠ presso la SISSA, Curriculum in Analisi Matematica, Modelli e Applicazioni.Ai sensi dellâart. 1, comma 4, dello Statuto della SISSA pubblicato sulla G.U. no. 36 del13.02.2012, il predetto attestato e equipollente al titolo di Dottore di Ricerca in Matematica.
Trieste, Anno Accademico 2018/2019
Ringraziamenti
Eccomi giunto alla fine di questa avventura durata 4 anni. E stata unâesperienza che nondimentichero mai, piena di bellissimi ricordi. Vorrei dunque utilizzare queste poche righe perringraziare tutti quelli che mi hanno accompagnato e aiutato durante questo percorso.
Voglio iniziare con il Prof. Gianni Dal Maso che con i suoi insegnamenti e consigli hapermesso la realizzazione di questa tesi. Inoltre, un sentito grazie va alla Prof. Patrizia Pucciche mi ha incoraggiato ed aiutato ad intraprendere questa avventura.
Un ringraziamento speciale va poi alla SISSA, per avermi accolto per questi 4 anni. Inparticolare, ringrazio Emanuele T. e Giovanni, che si sono rivelati essere i migliori colleghidi ufficio e amici con cui vivere questa esperienza. Inoltre, non posso non menzionare Mat-teo, Giulio e Daniele, nostri rivali nellâarte suprema del perdere tempo, o Filippo, Flavia eFrancesco, che hanno sopportato senza mai lamentarsi le mie intrusioni in ufficio per parlaredi matematica. Ringrazio poi Emanuele C., Veronica, Alessandro N. e Alessandro R., insiemeai quali mi sono divertito nel difficile (ma divertente) ruolo di rappresentante. Inoltre, ungrande grazie va a Giulia, Ornela, Enrico, Danka, Elio, Paolo e Michele, cosı come Ilaria,Carlo, Boris e Pavel, e tutte le persone della SISSA (e non) con cui ho condiviso questifantastici 4 anni.
Una menzione particolare va poi agli amici Emanuele G., Simone, Fabio, Bods, Andrea,Cristina, Francesca, e Franca, grazie ai quali i 5 anni di universita a Perugia sono volati.
Infine, ringrazio i miei genitori, Marcello e Marisa, che mi hanno sempre supportato (esopportato) in tutto, e senza i quali oggi non sarei quello che sono.
v
Contents
Introduction ix
1 Elastodynamics system in domains with growing cracks 11.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4 Continuous dependence on the data . . . . . . . . . . . . . . . . . . . . . . . 28
2 Dynamic energy-dissipation balance of a growing crack 372.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1.1 The change of variable approach . . . . . . . . . . . . . . . . . . . . . 382.2 Representation result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2.1 Local representation result . . . . . . . . . . . . . . . . . . . . . . . . 442.2.2 Global representation result . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3 Dynamic energy-dissipation balance . . . . . . . . . . . . . . . . . . . . . . . 53
3 A dynamic model for viscoelastic materials with growing cracks 633.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.2 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.3 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.4 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.5 An example of a growing crack . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4 A phase-field model of dynamic fracture 874.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2 The time discretization scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 964.3 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.4 The case without dissipative terms . . . . . . . . . . . . . . . . . . . . . . . . 114
Bibliography 121
vii
Introduction
This thesis is devoted to the study of several mathematical problems in fracture mechanicsfor brittle materials. The main ingredient to develop a reasonable model of these phenomenais Griffithâs criterion, originally formulated in [31] for the quasi-static setting, namely whenthe external data vary slowly compared to the elastic wave speed of the material. In thiscase, Griffith states there is an exact balance between the decrease of elastic energy duringthe evolution, and the energy used to increase the crack, which is assumed to be proportionalto its area.
In sharp-interface models, i.e., when the crack is identified with the discontinuity surfaceof the displacement in the reference configuration, this principle was turned into a precisedefinition by Francfort and Marigo in [26]. In the context of small-strain antiplane shear, thefollowing energy functional of Mumford-Shahâs type is considered:
1
2
â«Î©\Î|âu(x)|2 dx+Hdâ1(Î). (1)
Here, Ω â Rd (with d = 2 being the physically relevant case) is an open bounded set withLipschitz boundary, which represents the reference configuration of the elastic material, theclosed set Î â Ω describes the crack, and u â H1(Ω \ Î) is the antiplane displacement. Thefirst term represents the stored elastic energy of a homogeneous and isotropic material, whilethe second one, called surface energy, models the energy used to produce the crack. Here andhenceforth all physical constants are normalized to 1.
In this setting, given a time-dependent Dirichlet datum t 7â w(t) acting on âΩ, a quasi-static evolution is a pair t 7â (u(t),Ît) which at every time t minimizes (1) among all pairs(uâ,Îâ), where Îâ is a closed set with Îâ â Ît, and uâ â H1(Ω \ Îâ) with uâ = w(t) onâΩ \ Îâ. The minimum problem is complemented with the irreversibility condition Îs â Îtfor every s †t (meaning the crack can only increase in time), and with an energy-dissipationbalance for every time.
The first rigorous existence result for quasi-static evolutions was due to Dal Maso andToader in [22] in dimension d = 2, and with a restriction on the number of connected com-ponents of the crack set. Later, Francfort and Larsen in [27] removed these assumptions, bysetting the problem in the space SBV of special functions with bounded variation, introducedby De Giorgi and Ambrosio in [25]. More in general, in the context of linear elasticity, the dis-placement u : Ω\Îâ Rd is vector-valued, and the term |âu|2 in the elastic energy is replacedby CEu ·Eu, where Eu is the symmetric part of the gradient, namely Eu := 1
2(âu+âuT ),and C is the elastic tensor. Existence results for quasi-static evolutions in linear elasticity canbe found only in dimension d = 2, see [11] (which works under the same geometric restric-tions of [22]) and [29] (for the general case). For related results in dimension d > 2 we referto [12, 13]. A detailed analysis of variational models of quasi-static fracture can be foundin [5] and in the references therein.
In this thesis, we study several mathematical problems in fracture dynamics. In thissetting, the stationarity condition for the displacement has to be replaced by the fact thatu solves the elastodynamics system out of the crack, while the crack evolves according to a
ix
x Introduction
dynamic version of Griffithâs criterion, see [41, 28]. Therefore, any reasonable mathematicalmodel should follow the following principles:
(a) elastodynamics: away from the crack set, the displacement u evolves according to theelastodynamics system;
(b) irreversibility: the time-dependent crack t 7â Ît is increasing in time with respect toinclusion (Îs â Ît for every s †t);
(c) dynamic energy-dissipation balance: the work done by external forces is balanced bythe mechanical energy (sum of kinetic and elastic energy) and the energy dissipated tocreate a crack;
(d) maximal dissipation: if the crack can propagate while balancing energy, then it shouldpropagate.
The last condition, introduced in [35], is needed because a time-independent crack alwayssatisfies the first three principles. So far, it was not possible to prove the existence of asolution for a model satisfying (a)â(d) without stronger a priori regularity conditions on thecracks and their evolutions, which have not mechanical justifications. Some models for apeeling test in dimension 1, based on similar principles, have been recently analyzed in detailin [19, 47], obtaining existence and uniqueness results without a priori regularity assumptions.
The contents of the thesis are organized into four chapters.
Chapter 1: Elastodynamics system in domains with growing cracks
According to the principles stated before, a first step to study the dynamic crack propagationin an elastic material is to solve the elastodynamics system for a prescribed time-dependentcrack Îttâ[0,T ] satisfying the irreversibility condition. From the mathematical point of view,this means solving a hyperbolic-type system of the form
u(t, x)â div(C(t, x)Eu(t, x)) = f(t, x) t â [0, T ], x â Ω \ Ît, (2)
supplemented by boundary and initial conditions; the main difficulty in such a problem isthat the domain Ω \ Ît depends on time. In the literature, we can find several differentapproaches to hyperbolic systems in time-dependent domains. The first one is developedin [16] for the antiplane case, i.e., for the wave equation
u(t, x)ââu(t, x) = f(t, x) t â [0, T ], x â Ω \ Ît, (3)
with homogeneous Neumann conditions on the boundary of Ω\Ît. The existence of a solutionwith assigned initial data is proved by using a time-discrete approximation and passing tothe limit as the time step tends to zero. This construction leads to an existence result undervery weak conditions on the cracks Îttâ[0,T ]. A generalization of this construction to thevector-valued case can be found in the recent work [52]. Unfortunately, the uniqueness ofthese solutions is still an open problem in this setting.
In [43, 20] the authors use a different technique to study (3), which is based on a suitablechange of variables of class C2 that maps the domain (t, x) â [0, T ] à Ω : x â Ω \ Îtinto the cylinder [0, T ] Ă (Ω \ Î0). In this way, the wave equation (3) is transformed into anew hyperbolic equation in a fixed domain, with coefficients depending also on the change ofvariables. This method allows proving the existence and uniqueness of a solution, as well asa continuous dependence result. Nevertheless, such a change of variable can be constructedonly under very strong regularity assumptions on the cracks (see [20]).
Introduction xi
Finally, a third possible approach to the study of (3) can be found in a very recentpaper [15]. In this case, existence is proved by means on a suitable approximation of thewave equation via minima of convex functionals (see [48, 49]).
In Chapter 1 (which contains the results of [8]) we study the existence, uniqueness, andcontinuous dependence on the data for the solutions of the elastodynamics system (2). Onthe cracks Îttâ[0,T ] we assume the irreversibility condition and that they are contained in agiven C2 manifold Î of dimension dâ 1. The system (2) is supplemented by mixed Dirichlet-Neumann boundary conditions on âΩ and by homogeneous Neumann boundary conditionson the cracks Ît (traction free case).
These results are obtained by adapting the change of variable method introduced in [20]to the vector-valued case. Since the operator C is usually defined only on the subspace ofsymmetric matrices, it is convenient to introduce a new operator A defined on all RdĂd as
A(t, x)Ο := C(t, x)Οsym for every Ο â RdĂd,
where Οsym denotes the symmetric part of Ο. In this way, system (2) can be rephrased as
u(t, x)â div(A(t, x)âu(t, x)) = f(t, x) t â [0, T ], x â Ω \ Ît, (4)
and the change of variable approach leads to the transformed system
v(t, y)â div(B(t, y)âv(t, y))
+ p(t, y)âv(t, y) +âv(t, y)b(t, y) = g(t, y) t â [0, T ], y â Ω \ Î0,(5)
where the new coefficients B, p, b, and g are constructed starting from A and f . The boundaryconditions are also transformed by the change of variables and lead to mixed Dirichlet-Neumann boundary conditions on âΩ, and homogeneous Neumann boundary conditions onthe fixed crack Î0.
The main changes with respect to the paper [20] are in the treatment of the terms involv-ing B. Indeed, in linear elasticity, the natural ellipticity condition on A is the following:
A(t, x)Ο · Ο ℠λ0|Οsym|2 for every Ο â RdĂd, (6)
with λ0 positive constant. Unfortunately, this condition is not inherited by the transformedoperator B. To overcome this difficulty, we assume that B satisfies a weaker ellipticity assump-tion of integral type (see (1.2.1)), which always holds when A satisfies (6) and the velocityof the time-dependent diffeomorphisms used in the change of variables is sufficiently small(see (1.2.4)). Another difference with respect to [20] is that we consider also the case of non-homogeneous Neumann boundary conditions on the Neumann part of âΩ. This completesthe study of [20], including the case of traction forces acting on the boundary.
We first prove the existence and uniqueness of solutions to (5), with assigned initial andboundary conditions. Moreover, we prove an energy equality (see (1.3.2)), which is slightlydifferent from the one in [20], and takes into account the non-homogeneous boundary terms.This energy balance allows us to prove suitable continuity properties with respect to time forthe solutions v, which are important in the proof of the main existence result for (4).
Finally, in the last part, we prove the continuous dependence of the solutions on thecracks Îttâ[0,T ] and on the manifold Î. More precisely, given a sequence În, of manifoldsand a sequence Înt tâ[0,T ] of time-dependent cracks contained in În, we use the energyequality (1.3.2) to prove that, under appropriate convergence conditions, the solutions un
and vn to problems (4) and (5) corresponding to Înt tâ[0,T ] converge to the solutions u andv of the limit problems corresponding to Îttâ[0,T ].
These results have been used in [18] to prove an existence theorem for a model in fracturedynamics based on (a)â(d) with suitable conditions on the regularity of the cracks.
xii Introduction
Chapter 2: Dynamic energy-dissipation balance of a growing crack
Once we are able to solve the elastodynamics system on Ω \ Ît under suitable assumptionson Îttâ[0,T ], the next step towards the solution to the dynamic fracture problem, accordingto (a)â(d), is to select those cracks Îttâ[0,T ] such that the corresponding solutions u satisfythe dynamic energy-dissipation balance.
In Chapter 2 (which contains the results of [9], obtained in collaboration with I. Lu-cardesi and E. Tasso) we compute the mechanical energy (kinetic + elastic) of the solutioncorresponding to a sufficiently regular crack evolution Îttâ[0,T ] in the antiplane case. Weconsider as reference configuration a bounded open set Ω of R2 with Lipschitz boundary andwe assume that all the cracks Ît are contained in a fixed C3,1 curve Î â Ω with endpointson âΩ. In this case, Ît is determined at time t by the crack-tip position on Î, describedby the arc-length parameter s(t). Here we assume t 7â s(t) non-decreasing (irreversibilityassumption) and of class C3,1([0, T ]). Far from the crack set, the displacement u satisfies awave equation of the form
u(t, x)â div(A(x)âu(t, x)) = f(t, x) t â [0, T ], x â Ω \ Ît, (7)
where A is a suitable matrix field satisfying the usual ellipticity condition. The equation issupplemented by homogeneous mixed Dirichlet-Neumann boundary conditions on âΩ, homo-geneous Neumann boundary conditions on Ît, and initial conditions.
The mechanical energy associated with u at time t is given by
E(t) :=1
2
â«Î©\Ît|u(t, x)|2dx+
1
2
â«Î©\Ît
A(x)âu(t, x) · âu(t, x) dx. (8)
The difficulty of computing (8) is twofold: on one hand, the displacement has a singularbehavior near the crack-tip; moreover, the domain of u(t) contains a crack and varies intime. To handle the first issue, a representation result for u is in order: under suitableconditions on the initial data (see Theorems 2.2.4 and 2.2.10) we prove that for every time tthe displacement is of class H1 in a neighborhood of the tip of Ît and of class H2 far fromit, namely u(t) is of the form
u(t, x) = uR(t, x) + k(t)ζ(t, x)S(Ί(t, x)) x â Ω \ Ît, (9)
where uR(t) â H2(Ω \Ît), k(t) â R, ζ(t) is a cut-off function supported in a neighborhood ofthe moving tip of Ît, S â H1(R2 \ (Ï, 0) : Ï â€ 0), and Ί(t) is a diffeomorphism of Ω whichmaps the tip of Ît into the origin. Once fixed ζ, S, and Ί, the function uR and the constantk are uniquely determined. Actually, the coefficient k only depends on A, Î, and s. Inaddition, we provide another decomposition for u which is more explicit and better explainsthe behavior of the singular part (see Theorem 2.2.10).
The second issue is technical and we overcome it exploiting Geometric Measure Theorytechniques. The computation leads to the following formula:
E(t) +Ï
4
â« t
0k2(Ï)a(Ï)s(Ï) dÏ = E(0) +
â« t
0
â«Î©\ÎÏ
f(Ï, x)u(Ï, x) dx dÏ (10)
for every t, where a is a positive function which depends on A, Î, and s, and is equalto 1 when A is the identity matrix; see Theorem 2.3.7 for the proof of (10) when A = Id,and Remark 2.3.9 for the general case. We compare it with the dynamic energy-dissipationbalance, which in this case reads
E(t) +H1(Ît \ Î0) = E(0) +
â« t
0
â«Î©\ÎÏ
f(Ï, x)u(Ï, x) dx dÏ (11)
Introduction xiii
for every t. We deduce that (11) is satisfied if and only if at every time t in which the crack ismoving, namely when s(t) > 0, the function k(t), often called dynamic stress intensity factor,is equal to 2/
âÏa(t).
We mention that a similar result for a horizontal crack Ît := Ω â© (Ï, 0) â R2 : Ï â€ ctmoving with constant velocity c (with a suitable boundary datum) can be found in thepaper [17, Section 4]. The representation result stated in (9) extends the one of [43] valid forstraight cracks and A the identity matrix. Here we adapt their proof to the case of a curvedcrack and a constant (in time) matrix A, possibly depending on x; moreover, we remove arestrictive assumption on the acceleration s.
The main steps in the proof of (10) are the following: by performing four changes ofvariables, we reduce problem to a second order PDE of the form
v(t, x)â div(A(t, x)âv(t, x)) + l.o.t. = f(t, x) t â [0, T ], x â Ω \ Î0, (12)
with Ω Lipschitz planar domain and Î0 a C3,1 curve which is straight near its tip. Thematrix field A has time-dependent coefficients, but at the tip of Î0 it is constantly equalto the identity. Finally, the decomposition result for the solution v to (12), obtained viasemi-group theory, leads to (9) for u, the solution to the original problem.
Chapter 3: A dynamic model for viscoelastic materials with growing cracks
When we want to study the dynamic evolution of deformed materials with viscoelastic prop-erties, Kelvin-Voigtâs model is the most common one. If no crack is present, this leads in theantiplane case to the damped wave equation
u(t, x)ââu(t, x)ââu(t, x) = f(t, x) (t, x) â [0, T ]à Ω.
As it is well known, the solutions to this equation satisfy the dynamic energy-dissipationbalance
E(t) +
â« t
0
â«Î©|âu(Ï, x)|2dx dÏ = E(0) + work of external forces
for every t, where in this case E(t) := 12
â«Î© |u(t, x)|2dx+
â«Î© |âu(t, x)|2dx. If we consider also
the presence of a crack in the viscoelastic material, the damped wave equation becomes
u(t, x)ââu(t, x)ââu(t, x) = f(t, x) t â [0, T ], x â Ω \ Ît, (13)
and in this case, the dynamic energy-dissipation balance reads
E(t) +Hdâ1(Ît \ Î0) +
â« t
0
â«Î©\ÎÏ
|âu(Ï, x)|2dx dÏ = E(0) + work of external forces. (14)
For a prescribed crack evolution, this model was already considered by [16] in the antiplanecase, and more in general by [52] for the vector-valued case. As proved in the quoted papers,the solutions to (13) satisfy
E(t) +
â« t
0
â«Î©\ÎÏ
|âu(Ï, x)|2dx dÏ = E(0) + work of external forces
for every time t. This equality implies that (14) cannot be satisfied unless Ît = Î0 for every t,which means that the crack is not allowed to increase in time. This phenomenon was alreadywell known in mechanics as the viscoelastic paradox, see for instance [51, Chapter 7].
To overcome this problem, in Chapter 3 (which contains the results of [10], obtainedin collaboration with F. Sapio) we modify Kelvin-Voigtâs model by considering a possibly
xiv Introduction
degenerate viscosity term depending on t and x. More precisely, we study the followingequation
u(t, x)ââu(t, x)â div(Î2(t, x)âu(t, x)) = f(t, x) t â [0, T ], x â Ω \ Ît. (15)
On the function Î: (0, T )ĂΩâ R we only require some regularity assumptions; a particularlyinteresting case is when Î assumes the value zero on some points of Ω, which means that thematerial has no longer viscoelastic properties in such a zone.
The main result of Chapter 3 is Theorem 3.2.1 (see also Remark 3.3.4), in which we showthe existence of a solution to (15) and, more in general, to the analogous problem for thed-dimensional linear elasticity. To this aim, we first perform a time discretization in the samespirit of [16], and then we pass to the limit as the time step goes to zero by relying on energyestimates. As a byproduct, we derive an energy-dissipation inequality (see (3.3.4)), which isused to prove the validity of the initial conditions. By using the change of variables methodimplemented in [43, 20], we also prove a uniqueness result, but only in dimension d = 2 andwhen Î(t) vanishes on a neighborhood of the tip of Ît.
We complete the chapter by providing an example in d = 2 of a solution to (15) for whichthe crack can grow while balancing the energy. As we remarked before, this cannot happenfor the Kelvin-Voigtâs model. More precisely, when the crack Ît moves with constant speedalong the x1-axis and Î(t) is zero in a neighborhood of the crack-tip, we construct a functionu which solves (15) and satisfies
E(t) +H1(Ît \Î0) +
â« t
0
â«Î©\ÎÏ
|Î(Ï, x)âu(Ï, x)|2dx dÏ = E(0) + work of external forces (16)
for every time t. Notice that (16) is the natural formulation of the dynamic energy-dissipationbalance in this setting.
Chapter 4: A phase-field model of dynamic fracture
An alternative approach to the study of crack evolution is based on the so-called phase-fieldmodel, which relies on the Ambrosio-Tortorelliâs approximation of the energy functional (1).According to Ambrosio and Tortorelli [4], the (d â 1)-dimensional set Î is replaced by aphase-field variable vΔ : Ω â [0, 1] which is close to 0 in an Δ-neighborhood of Î, and closeto 1 away from it. Accordingly, the Griffithâs functional (1) is replaced by the Δ-dependentelliptic functionals
EΔ(u, v) + HΔ(v)
for u, v â H1(Ω), where
EΔ(u, v) :=1
2
â«Î©
[(v(x))2 + ηΔ] |âu(x)|2 dx,
HΔ(v) :=1
4Δ
â«Î©|1â v(x)|2 dx+ Δ
â«Î©|âv(x)|2 dx,
with 0 < ηΔ Δ. A minimum point (uΔ, vΔ) of EΔ + HΔ provides a good approximation ofa minimizer (u,Î) of (1) as Δ â 0+, in the sense that uΔ is close to u, vΔ is close to 0 nearÎ, and EΔ(uΔ, vΔ) + HΔ(vΔ) approximates the energy (1). For the corresponding quasi-staticevolution t 7â (uΔ(t), vΔ(t)), the minimality condition for (1) is replaced by
EΔ(uΔ(t), vΔ(t)) + HΔ(vΔ(t)) †EΔ(uâ, vâ) + HΔ(v
â) (17)
among every pair (uâ, vâ) with vâ †vΔ(t) and uâ = w(t) on âΩ. Notice that the inequalityvâ †vΔ(t) reflects the inclusion Îâ â Ît. As before, the minimum problem (17) is com-plemented with the irreversibility condition 0 †vΔ(t) †vΔ(s) †1 for every s †t, and
Introduction xv
with the energy-dissipation balance for every time; we refer to [30] for the convergence ofthis evolution, as Δ â 0+, toward the sharp-interface one described at the beginning of theintroduction.
In particular, a quasi-static phase-field evolution t 7â (uΔ(t), vΔ(t)) satisfies:
(Q1) for every t â [0, T ] the function uΔ(t) solves div([(vΔ(t))2 + ηΔ]âuΔ(t)) = 0 in Ω with
suitable boundary conditions;
(Q2) the map t 7â vΔ(t) is non-increasing (vΔ(t) †vΔ(s) for 0 †s †t †T ) and for everyt â [0, T ] the function vΔ(t) satisfies
EΔ(uΔ(t), vΔ(t)) + HΔ(vΔ(t)) †EΔ(uΔ(t), vâ) + HΔ(v
â)
for every vâ †vΔ(t);
(Q3) for every t â [0, T ] the pair (uΔ(t), vΔ(t)) satisfies the energy-dissipation balance
EΔ(uΔ(t), vΔ(t)) + HΔ(vΔ(t)) = EΔ(uΔ(0), vΔ(0)) + HΔ(vΔ(0)) + work of external data.
As explained before for the sharp interface model, in the dynamic case the first conditionis replaced by the wave equation, while in the energy balance we need to take into accountthe kinetic energy term. Developing these principles, in [6, 35, 36] the authors propose thefollowing phase-field model of dynamic crack propagation in linear elasticity:
(D1) uΔ solves uΔ â div([v2Δ + ηΔ]CEuΔ) = 0 in (0, T )à Ω with suitable boundary and initial
conditions;
(D2) the map t 7â vΔ(t) is non-increasing and for every t â [0, T ] the function vΔ(t) solves
EΔ(uΔ(t), vΔ(t)) + HΔ(vΔ(t)) †EΔ(uΔ(t), vâ) + HΔ(v
â) for every vâ †vΔ(t);
(D3) for every t â [0, T ] the pair (uΔ(t), vΔ(t)) satisfies the dynamic energy-dissipation balance
1
2
â«Î©|uΔ(t)|2dx+ EΔ(uΔ(t), vΔ(t)) + HΔ(vΔ(t))
=1
2
â«Î©|uΔ(0)|2dx+ EΔ(uΔ(0), vΔ(0)) + HΔ(vΔ(0)) + work of external data,
where EΔ(u, v) := 12
â«Î©[((v(x))2 + ηΔ]C(x)Eu(x) ·Eu(x) dx for u â H1(Ω;Rd) and v â H1(Ω).
A solution to this model is approximated by means of a time discretization with an alternatescheme: to pass from the previous time to the next one, one first solves the wave equation foru, keeping v fixed, and then a minimum problem for v, keeping u fixed. This method is usedin [36] to prove the existence of a pair (u, v) satisfying (D1)â(D3). For technical reasons, aviscoelastic dissipative term is added to (D1), which means that in [36] the following systemis considered
uΔ â div([v2Δ + ηΔ]C(EuΔ + EuΔ)) = 0 in (0, T )à Ω.
The disadvantage of this term appears when we consider the behavior of the solution asΔ â 0+, a problem which is out of the scope of this thesis. If we were able to prove theconvergence of the solution toward a dynamic sharp-interface evolution, then the dynamicenergy-dissipation balance for the damped wave equation in cracked domains of [16, 52] wouldimply that the limit crack does not depend on time, as explained above in the section aboutviscoelastic materials.
To avert this problem, in Chapter 4 (which contains the results of [7]) we propose adifferent model that avoids viscoelastic terms depending on the displacement and considerinstead a dissipative term related to the speed of the crack-tips. More precisely, given anatural number k â N âȘ 0, we consider a dynamic phase-field evolution t 7â (uΔ(t), vΔ(t))satisfying:
xvi Introduction
(D1) uΔ solves uΔ â div([(vΔ âš 0)2 + ηΔ]CEuΔ) = 0 in (0, T )à Ω with suitable boundary andinitial conditions;
(D2) the map t 7â vΔ(t) is non-increasing and for a.e. t â (0, T ) the function vΔ(t) solves thevariational inequality
EΔ(uΔ(t), vâ)â EΔ(uΔ(t), vΔ(t)) + HΔ(v
â)âHΔ(vΔ(t)) + (vΔ(t), vâ â vΔ(t))Hk(Ω) â„ 0
for every vâ †vΔ(t);
(D3) for every t â [0, T ] the pair (uΔ(t), vΔ(t)) satisfies the dynamic energy-dissipation balance
1
2
â«Î©|uΔ(t)|2dx+ EΔ(uΔ(t), vΔ(t)) + HΔ(vΔ(t)) +
â« t
0âvΔ(Ï)â2Hk(Ω)dÏ
=1
2
â«Î©|uΔ(0)|2dx+ EΔ(uΔ(0), vΔ(0)) + HΔ(vΔ(0)) + work of external data,
(18)
where in this case EΔ(u, v) := 12
â«Î©[(v(x) âš 0)2 + ηΔ]C(x)Eu(x) · Eu(x) dx. Notice that for
technical reasons the dissipative termâ« t
0âvΔ(Ï)â2Hk(Ω)
dÏ contains the norm in the Sobolev
space Hk(Ω), rather then the norm in L2(Ω), which is more frequently used in the literature.This choice guarantees more regularity in time for the phase-field function, more preciselythat vΔ â H1(0, T ;Hk(Ω)).
In the quasi-static setting, a condition similar to (D2) can be found in [42, 2], where itdefines a unilateral gradient flow evolution for the phase-field function vΔ. In sharp-interfacemodels, a crack-dependent term analogous to
â« t0âvΔ(Ï)â2
Hk(Ω)dÏ arises in the study of the
so-called vanishing viscosity evolutions, which are linked to the analysis of local minimizersof Griffithâs functional (1), see for example [46, 39]. We point out that a similar dissipativeterm also appears in [37] for a 1-dimensional debonding model.
By adapting the time discretization scheme of [6, 36], we show the existence of a dynamicphase-field evolution (uΔ, vΔ) which satisfies (D1)â(D3), provided that k > d/2, where d isthe dimension of the ambient space. This condition is crucial to obtain the validity of thedynamic energy-dissipation balance since in our case the viscoelastic dissipative term usedin [36] is not present.
We conclude Chapter 4 by analyzing the dynamic phase-field model (D1)â(D3) with noviscous terms. We show the existence of an evolution t 7â (uΔ(t), vΔ(t)) which satisfies (D1)and (D2), but only an energy-dissipation inequality (see (4.4.7)) instead of (D3).
Introduction xvii
Notation
Basic notation. The space of m Ă d matrices with real entries is denoted by RmĂd; incase m = d, the subspace of symmetric matrices is denoted by RdĂdsym, and the subspace of
orthogonal d Ă d matrices with determinant equal to 1 by SO(d). We denote by AT andAâ1, respectively, the transpose and the inverse of A â RdĂd, by AâT the transpose of theinverse, and by Asym the symmetric part, namely Asym := 1
2(A + AT ); we use Id to denotethe identity matrix in RdĂd. The Euclidian scalar product in Rd is denoted by · and thecorresponding Euclidian norm by | · |; the same notation is used also for RmĂd. We denoteby aâ b â RdĂd the tensor product between two vectors a, b â Rd, and by a b â RdĂdsym thesymmetrized tensor product, namely the symmetric part of aâ b.
The d-dimensional Lebesgue measure in Rd is denoted by Ld, and the (dâ1)-dimensionalHausdorff measure by Hdâ1. Given a bounded open set Ω with Lipschitz boundary, we denoteby Îœ the outer unit normal vector to âΩ, which is defined Hdâ1-a.e. on the boundary. We useBr(x) to denote the ball of radius r and center x in Rd, namely Br(x) := y â Rd : |yâx| < r,and id to denote the identity function in Rd, possibly restricted to a subset. Given twonumbers c1, c2 â R, we set c1 âš c2 := maxc1, c2 and c1 ⧠c2 := minc1, c2.
The partial derivatives with respect to the variable xi are denoted by âi or âxi . Givena function u : Rd â Rm, we denote its Jacobian matrix by âu, whose components are(âu)ij := âjui for i = 1, . . . ,m and j = 1, . . . , d. When u : Rd â Rd, we use Eu to de-note its symmetryzed gradient, namely Eu := 1
2(âu +âuT ). Given u : Rd â R, we use âu
to denote its Laplacian, which is defined as âu :=âd
i=1 â2i u. We set â2u := â(âu) and
â2u := â(âu), and we define inductively âku and âku for every k â N, with the conventionâ0u = â0u := u. For a tensor field T : Rd â RmĂd, by div T we mean its divergence withrespect to rows, namely (div T )i :=
âdj=1 âjTij for i = 1, . . . ,m.
Function spaces. Given two metric spaces X and Y , we use C0(X;Y ) and Lip(X;Y ) todenote, respectively, the space of continuous and Lipschitz functions from X to Y . Givenan open set Ω â Rd, we denote by Ck(Ω;Rm) the space of Rm-valued functions with k con-tinuous derivatives; we use Ckc (Ω;Rm) and Ck,1(Ω;Rm) to denote, respectively, the subspaceof functions with compact support in Ω, and of functions whose k-derivatives are Lipschitz.For every 1 †p †â we denote by Lp(Ω;Rm) the Lebesgue space of p-th power integrablefunctions, and by W k,p(Ω;Rm) the Sobolev space of functions with k derivatives; for p = 2we set Hk(Ω;Rm) := W k,2(Ω;Rm), and for m = 1 we omit Rm in the previous spaces. Theboundary values of a Sobolev function are always intended in the sense of traces. The scalarproduct in L2(Ω;Rm) is denoted by (·, ·)L2(Ω) and the norm in Lp(Ω;Rm) by â · âLp(Ω); asimilar notation is valid for the Sobolev spaces. For simplicity, we use â · âLâ(Ω) to denotealso the supremum norm of continuous and bounded functions.
The norm of a generic Banach space X is denoted by â · âX ; when X is a Hilbert space,we use (·, ·)X to denote its scalar product. We denote by X âČ the dual of X, and by ă·, ·ăXâČthe duality product between X âČ and X. Given two Banach spaces X1 and X2, the space oflinear and continuous maps from X1 to X2 is denoted by L (X1;X2); given A â L (X1;X2)and u â X1, we write Au â X2 to denote the image of u under A.
Given an open interval (a, b) â R and 1 †p †â, we denote by Lp(a, b;X) the spaceof Lp functions from (a, b) to X; we use W k,p(a, b;X) and Hk(a, b;X) (for p = 2) to denotethe Sobolev space of functions from (a, b) to X with k derivatives. Given u â W 1,p(a, b;X),we denote by u â Lp(a, b;X) its derivative in the sense distributions. The set of functionsfrom [a, b] to X with k continuous derivatives is denoted by Ck([a, b];X); we use Ckc (a, b;X)to denote the subspace of functions with compact support in (a, b). The space of absolutelycontinuous functions from [a, b] to X is denoted by AC([a, b];X); we use C0
w([a, b];X) to
xviii Introduction
denote the set of weakly continuous functions from [a, b] to X, namely
C0w([a, b];X) := u : [a, b]â X : t 7â ăxâČ, u(t)ăXâČ is continuous in [a, b] for every xâČ â X âČ.
When dealing with an element u â H1(a, b;X) we always assume u to be the continuousrepresentative of its class. In particular, it makes sense to consider the pointwise value u(t)for every t â [a, b].
Chapter 1
Elastodynamics system in domainswith growing cracks
In this chapter, we prove the existence, uniqueness, and continuous dependence results forthe elastodynamics system (4) via the change of variable approach of [43, 20].
The chapter is organized as follows. In Section 1.1 we list the main assumptions on theset Ω, on the geometry of the cracks Ît, and on the diffeomorphisms used for the changes ofvariables. Moreover, in Definitions 1.1.6 and 1.1.9 we specify the notion of weak solution toproblems (4) and (5). Section 1.2 deals with the study of the two problems (see Theorems 1.2.2and 1.2.3). We first show their equivalence (see Theorem 1.1.16), and then we prove anexistence and uniqueness result for (5) in a weaker sense (see Theorems 1.2.9). In Section 1.3we complete the proof of Theorems 1.2.2 and 1.2.3 by showing the energy equality (1.3.2),which ensures that the solution given by Theorem 1.2.9 is indeed a weak solution. Finally,Section 1.4 is devoted to the continuous dependence result, which is proved in Theorem 1.4.1.
The results contained in this chapter have been published in [8].
1.1 Preliminary results
Let T be a positive number, Ω â Rd be a bounded open set with Lipschitz boundary, âDΩbe a (possibly empty) Borel subset of âΩ, and âNΩ be its complement. Throughout thischapter we assume the following hypotheses on the geometry of the crack sets Îttâ[0,T ] andon the diffeomorphisms of Ω into itself mapping Î0 into Ît:
(H1) Î â Rd is a complete (d â 1)-dimensional C2 manifold with boundary âÎ such thatâΠ⩠Ω = â and Hdâ1(Î â© âΩ) = 0;
(H2) for every x â Îâ©Î© there exists an open neighborhood U of x in Rd such that (U â©Î©)\Îis the union of two disjoint open sets U+ and Uâ with Lipschitz boundary;
(H3) Îttâ[0,T ] is a family of (possibly irregular) closed subsets of Π⩠Ω satisfying Îs â Îtfor every 0 †s †t †T ;
(H4) Ί, Κ: [0, T ] à Ω â Ω are two continuous maps and the partial derivatives âtΊ, âtΚ,âiΊ, âiΚ, â2
ijΊ, â2ijΚ, âiâtΊ = âtâiΊ, âiâtΚ = âtâiΚ exist and are continuous for
i, j = 1, . . . , d;
(H5) Ί(t,Ω) = Ω, Ί(t,Îâ©Î©) = Îâ©Î©, Ί(t,Î0) = Ît, and Ί(t, y) = y for every t â [0, T ] andy in a neighborhood of âΩ;
(H6) Κ(t,Ί(t, y)) = y, Ί(t,Κ(t, x)) = x, and Ί(0, y) = y for every t â [0, T ] and x, y â Ω;
1
2 1.1. Preliminary results
(H7) âtΊ, âtΚ, âiΊ, âiΚ, â2ijΊ, â2
ijΚ, âiâtΊ, âiâtΚ belong to the space Lip([0, T ];C0(Ω;Rd))for i, j = 1, . . . , d;
(H8) there exists a constant L > 0 such that
|âiâtΊ(t, x)â âiâtΊ(t, y)| †L|xâ y|, |âiâtΚ(t, x)â âiâtΚ(t, y)| †L|xâ y|
for every t â [0, T ], x, y â Ω, and i = 1, . . . , d.
By using (H4) and (H6) we derive that detâΊ(t, y) 6= 0 and detâΚ(t, x) 6= 0 for everyt â [0, T ] and x, y â Ω. In particular, both determinants are positive, since âΊ(0, y) = Id forevery y â Ω.
Assumptions (H1) and (H2) imply the existence of the trace of Ï â H1(Ω\Î) on âΩ, andon Π⩠Ω from both sides. Indeed, we can find a finite number of open sets with Lipschitzboundary Uj â Ω \Î, j = 1, . . .m, such that ((Îâ©Î©)âȘâΩ) \ (Îâ©âΩ) â âȘmj=1âUj . Moreover,
since Hdâ1(Î â© âΩ) = 0, there exists a constant Ctr > 0, depending only on Ω and Î, suchthat
âÏâL2(âΩ) †CtrâÏâH1(Ω\Î) for every Ï â H1(Ω \ Î). (1.1.1)
In a similar way we obtain the embedding H1(Ω \ Î) â Lp(Ω) for every p â [1, 2â], where2â := 2n
nâ2 is the usual critical Sobolev exponent. In particular, there exists a constant Cp > 0,depending on Ω, Î, and p, such that
âÏâLp(Ω) †CpâÏâH1(Ω\Î) for every Ï â H1(Ω \ Î). (1.1.2)
Given a point y â Π⩠Ω, its trajectory in time is described by the time-dependent mapt 7â Ί(t, y) â Î. We infer that its velocity is tangential to the manifold Î at the point Ί(t, y),that is Ί(t, y) · Îœ(Ί(t, y)) = 0, where Îœ(x) is the unit normal vector to Î at x. By combiningthis equality with the relation
Îœ(Ί(t, y)) =âΊ(t, y)âT Îœ(y)
|âΊ(t, y)âT Îœ(y)|for y â Π⩠Ω,
we deduce
(âΊ(t, y)â1Ί(t, y)) · Îœ(y) = Κ(t,Ί(t, y)) · Îœ(y) = 0 for y â Π⩠Ω, (1.1.3)
or equivalentlyΊ(t,Κ(t, x)) · Îœ(x) = 0 for x â Π⩠Ω. (1.1.4)
In the following lemmas, we investigate some regularity properties of functions defined inΩ \Î, when composed with suitable diffeomorphisms of the domain into itself. Let us specifythe class of diffeomorphisms under study.
Definition 1.1.1. We say that Î: [0, T ] à Ω â Rd is admissible if it belongs to the spaceC1([0, T ] à Ω;Rd) and for every t â [0, T ] the function Î(t) is a C2 diffeomorphism of Ω initself such that Î(t,Ω) = Ω and Î(t,Π⩠Ω) = Π⩠Ω.
Notice that, according to (H4)â(H6), both Ί and Κ are admissible.
Lemma 1.1.2. Let f and fn, n â N, be elements of L2(Ω), and let Î and În, n â N, beadmissible diffeomorphisms. Assume there exist ÎŽ1, ÎŽ2 > 0 such that ÎŽ1 < detâÎn(t, x) < ÎŽ2
for every t â [0, T ], x â Ω, and n â N. Assume also that for every t â [0, T ]
În(t)â Î(t) in L2(Ω;Rd), fn â f in L2(Ω) as nââ.
Then, for every t â [0, T ] we have
fn(În(t))â f(Î(t)) in L2(Ω) as nââ.
Chapter 1. Elastodynamics system in domains with growing cracks 3
Proof. The proof of this result can be found in [20, Lemma A.7].
Lemma 1.1.3. Let Î be admissible. There exists a constant C > 0 such that for everyÏ â H1(Ω \ Î) we have
âÏ(Î(t))â Ï(Î(s))âL2(Ω\Î) †CââÏâL2(Ω\Î)|tâ s| for every 0 †s †t †T .
Proof. It is sufficient to repeat the proof of [20, Lemmas A.5], by approximating Ï â H1(Ω\Î)with a sequence of functions ÏΔ â Câ(Ω \ Î) â©H1(Ω \ Î) given by Meyers-Serrinâs theorem(see, e.g, [1, Theorem 3.16]), and integrating over Ω \ Î.
Lemma 1.1.4. Let Î be admissible and let t â [0, T ] be fixed. Then for every Ï â H1(Ω \Î)
1
h[Ï(Î(t+ h))â Ï(Î(t))]â âÏ(Î(t)) · Î(t) in L2(Ω \ Î) as hâ 0.
Proof. We argue again as in the proof of [20, Lemmas A.6], by approximating Ï with asequence of functions ÏΔ â Câ(Ω \ Î)â©H1(Ω \ Î) given by Meyers-Serrinâs theorem, and byintegrating over Ω \ Î. We only have to check that as hâ 0
Th(ÏΔ) :=1
h
â« h
0âÏΔ(Î(t+ Ï)) · Î(t+ Ï) dÏ â L(ÏΔ) := âÏΔ(Î(t)) · Î(t) in L2(Ω \ Î).
Since Î: [0, T ]ĂΩâ Rd is uniformly continuous, for every ÎŽ > 0 there exists Ï > 0 such that
|Î(t+ Ï, y)â Î(t, y)| < ÎŽ for every |Ï | < Ï and y â Ω. (1.1.5)
Similarly, fixed t â [0, T ], the map Îâ1(t) : Ωâ Rd is uniformly continuous, and so for everyη > 0 there exists ÎŽ > 0 such that
|Îâ1(t, y)â Îâ1(t, z)| < η for every y, z â Ω, with |y â z| < ÎŽ. (1.1.6)
By combining (1.1.5) and (1.1.6), we get that for every η > 0 there exists Ï > 0 such that
Î(t+ Ï,A) â Î(t, Iη(A)) for every set A â Ω and |Ï | < Ï,
where Iη(A) := x â Ω : dist(x,A) < η (we recall that dist(x,A) := infyâA |xâ y|).For every n â N we define Kn := x â Ω \ Î : dist(x, â(Ω \ Î)) â„ 1/n. The sets Kn are
compact, with Kn â Kn+1, and âȘân=1Kn = Ω \ Î. Fixed n â N, there exists η > 0 such thatIη(Kn) ââ Ω \ Î, which implies that Î(t, Iη(Kn)) ââ Ω \ Î. Therefore there exists Ï > 0such that for every |h| < Ï and y â Kn
|Th(ÏΔ)(y)| †1
h
â« h
0|âÏΔ(Î(t+ Ï, y)) · Î(t+ Ï, y)| dÏ â€ C,
for a constant C > 0 independent of h. Hence, by the dominated convergence theorem weconclude that âTh(ÏΔ) â L(ÏΔ)âL2(Kn) â 0 as h â 0, since Th(ÏΔ)(y) â L(ÏΔ)(y) for everyy â Ω \ Î. Similarly, there exists η > 0 such that Iη((Ω \ Î) \Kn+1) â (Ω \ Î) \Kn, and sowe can find Ï > 0 such that for every |h| < Ï
âTh(ÏΔ)â2L2((Ω\Î)\Kn+1) â€1
h
â«(Ω\Î)\Kn+1
â« h
0|âÏΔ(Î(t+ Ï, y)) · Î(t+ Ï, y)|2dÏ dy
†Câ«
(Ω\Î)\Kn|âÏΔ(Î(t, y))|2dy.
4 1.1. Preliminary results
Therefore, for every |h| < Ï
âTh(ÏΔ)â L(ÏΔ)âL2(Ω\Î)
†âTh(ÏΔ)â L(ÏΔ)âL2(Kn+1) + âTh(ÏΔ)âL2((Ω\Î)\Kn+1) + âL(ÏΔ)âL2((Ω\Î)\Kn+1)
†âTh(ÏΔ)â L(ÏΔ)âL2(Kn+1) + 2CââÏΔ(Î(t))âL2((Ω\Î)\Kn),
and consequently lim suphâ0 âTh(ÏΔ)âL(ÏΔ)âL2(Ω\Î) †2CââÏΔ(Î(t))âL2((Ω\Î)\Kn) for every
n â N. To conclude it is enough to observe that Ld((Ω \ Î) \ Kn) â 0 as n â â, andâÏΔ(Î(t)) â L2(Ω \ Î;Rd).
For every t â [0, T ] we introduce the space
H1D(Ω \ Ît;Rd) := Ï â H1(Ω \ Ît;Rd) : Ï = 0 on âDΩ, (1.1.7)
where the equality Ï = 0 on âDΩ refers to the trace of Ï on âΩ. We have that H1D(Ω\Ît;Rd)
is a Hilbert space endowed with the norm of H1(Ω \ Ît;Rd), and its dual is denoted byHâ1D (Ω \ Ît;Rd). The canonical isomorphism between H1
D(Ω \ Ît;Rd) and [H1D(Ω \ Ît)]
d
induces an isomorphism of Hâ1D (Ω \ Ît;Rd) into [Hâ1
D (Ω \ Ît)]d.
The transpose of the natural embedding H1D(Ω \Ît;Rd) â L2(Ω;Rd) induces the embed-
ding of L2(Ω;Rd) into Hâ1D (Ω \ Ît;Rd), which is defined by
ăg, ÏăHâ1D (Ω\Ît) := (g, Ï)L2(Ω) for g â L2(Ω;Rd) and Ï â H1
D(Ω \ Ît;Rd).
Given 0 †s †t †T , let Pst : Hâ1D (Ω \ Ît;Rd) â Hâ1
D (Ω \ Îs;Rd) be the transpose of thenatural embedding H1
D(Ω \ Îs;Rd) â H1D(Ω \ Ît;Rd), i.e.,
ăPst(g), ÏăHâ1D (Ω\Îs) := ăg, ÏăHâ1
D (Ω\Ît) for g â Hâ1D (Ω \ Ît;Rd) and Ï â H1
D(Ω \ Îs;Rd).
The operator Pst is continuous, with norm less than or equal to 1, but in general is notinjective, since H1
D(Ω \ Îs;Rd) is not dense in H1D(Ω \ Ît;Rd). Notice that Pst(g) = g for
every g â L2(Ω;Rd).Let C : [0, T ]à Ωâ L (RdĂdsym;RdĂdsym) be a time varying tensor field satisfying
C â Lip([0, T ];C0(Ω; L (RdĂdsym;RdĂdsym))),
C(t) â Lip(Ω; L (RdĂdsym;RdĂdsym)), ââC(t)âLâ(Ω) †C for every t â [0, T ],
(C(t, x)Ο1) · Ο2 = Ο1 · (C(t, x)Ο2) for every Ο1, Ο2 â RdĂdsym, t â [0, T ], x â Ω,
where C > 0 is a constant independent of t. Starting from the operator C(t, x) it is convienentto define a new operator A(t, x) â L (RdĂd,RdĂd) as:
A(t, x)Ο := C(t, x)Οsym for every Ο â RdĂd, t â [0, T ], x â Ω.
Clearly, A satisfies
A â Lip([0, T ];C0(Ω; L (RdĂd;RdĂd))), (1.1.8)
A(t) â Lip(Ω; L (RdĂd;RdĂd)), ââA(t)âLâ(Ω) †C for every t â [0, T ], (1.1.9)
(A(t, x)Ο1) · Ο2 = Ο1 · (A(t, x)Ο2) for every Ο1, Ο2 â RdĂd, t â [0, T ], x â Ω. (1.1.10)
Given
w â H2(0, T ;L2(Ω;Rd)) â©H1(0, T ;H1(Ω \ Î0;Rd)), (1.1.11)
f â L2(0, T ;L2(Ω;Rd)), F â H1(0, T ;L2(âNΩ;Rd)), (1.1.12)
Chapter 1. Elastodynamics system in domains with growing cracks 5
u0 â w(0) â H1D(Ω \ Î0;Rd), u1 â L2(Ω;Rd), (1.1.13)
we study the linear hyperbolic system
u(t)â div(A(t)âu(t)) = f(t) in Ω \ Ît, t â [0, T ], (1.1.14)
with boundary conditions formally written as
u(t) = w(t) on âDΩ, t â [0, T ], (1.1.15)
(A(t)âu(t))Îœ = F (t) on âNΩ, t â [0, T ], (1.1.16)
(A(t)âu(t))Îœ = 0 on Ît, t â [0, T ], (1.1.17)
and initial conditions
u(0) = u0, u(0) = u1 in Ω \ Î0. (1.1.18)
Remark 1.1.5. To give a precise meaning to (1.1.14)â(1.1.18), it is convenient to introducethe following notation. Given Ï â H1(Ω \ Ît;Rd), its gradient in the sense of distributionsis denoted by âÏ and it is an element of L2(Ω \ Ît;RdĂd). We extend it to a function inL2(Ω;RdĂd) by setting âÏ = 0 on Ît. Notice that this is not the gradient in the sense ofdistributions on Ω of the function Ï, considered as defined almost everywhere on Ω; indeedthe equality â«
ΩâÏ(x) · Ï(x) dx = â
â«Î©Ï(x) · divÏ(x) dx
holds for every Ï â Câc (Ω \Ît;RdĂd), but in general not for Ï â Câc (Ω;RdĂd). Similarly, weextend divÏ â L2(Ω \ Ît) to a function in L2(Ω) by setting divÏ = 0 on Ît.
We recall the notion of solution to (1.1.14)â(1.1.17) given in [20, Definition 2.4]. Weconsider functions u satisfying the following regularity assumptions:
u â C1([0, T ];L2(Ω;Rd)), (1.1.19)
u(t)â w(t) â H1D(Ω \ Ît;Rd) for every t â [0, T ], (1.1.20)
âu â C0([0, T ];L2(Ω;RdĂd)), (1.1.21)
u â AC([s, T ];Hâ1D (Ω \ Îs;Rd)) for every s â [0, T ), (1.1.22)
1
h[u(t+ h)â u(t)] u in Hâ1
D (Ω \ Ît;Rd) for a.e. t â (0, T ) as hâ 0, (1.1.23)
the function t 7â âu(t)âHâ1D (Ω\Ît) is integrable in (0, T ). (1.1.24)
The relationship between u and the distributional time derivative of u is explained in [20,Lemma 2.2], which shows that, under assumptions (1.1.19)â(1.1.24), the map t 7â Pst(u(t)) isthe distributional derivative of the function t 7â u(t) from (s, T ) to Hâ1
D (Ω\Îs;Rd). Moreover
u(t)â u(s) =
â« t
sPsÏ (u(Ï)) dÏ for every 0 †s †t †T .
Definition 1.1.6. Let A, w, f , and F be as in (1.1.8)â(1.1.12). We say that u is a weaksolution to the hyperbolic system (1.1.14) with boundary conditions (1.1.15)â(1.1.17) if usatisfies (1.1.19)â(1.1.24), and for a.e. t â (0, T ) we have
ău(t), ÏăHâ1D (Ω\Ît) + (A(t)âu(t),âÏ)L2(Ω) = (f(t), Ï)L2(Ω) + (F (t), Ï)L2(âNΩ) (1.1.25)
for every Ï â H1D(Ω \ Ît;Rd), where u(t) is defined in (1.1.23).
6 1.1. Preliminary results
Remark 1.1.7. Let us check that (1.1.25) makes sense for a.e. t â (0, T ). Thanks to (1.1.23)we have that u(t) â Hâ1
D (Ω \ Ît;Rd) for a.e. t â (0, T ), therefore it is in duality withÏ â H1
D(Ω \ Ît;Rd). Moreover, assumptions (1.1.8) and (1.1.21) implies that A(t)âu(t)belongs to L2(Ω;RdĂd) for every t â [0, T ]. Finally, thanks to (1.1.1) the last term of (1.1.25)is well defined for every t â [0, T ].
Remark 1.1.8. Notice that the Neumann boundary conditions (1.1.16) and (1.1.17) are ingeneral only formal. They can be obtained from (1.1.25), by using integration by parts inspace, only when u(t) and Ît are sufficiently regular.
Following [43], to prove the existence and uniqueness of a weak solution, we perform achange of variable. We denote by
v(t, y) := u(t,Ί(t, y)) t â [0, T ], y â Ω \ Î0, (1.1.26)
where Ί is the diffeomorphism introduced in (H4)â(H8), so that
u(t, x) = v(t,Κ(t, x)) t â [0, T ], x â Ω \ Ît. (1.1.27)
Notice that v(t) â H1(Ω \Î0;Rd) if and only if u(t) â H1(Ω \Ît;Rd) and that this change ofvariables maps the domain (t, x) : t â [0, T ], x â Ω \ Ît into the cylinder [0, T ] Ă (Ω \ Î0).The transformed system reads
v(t)â div(B(t)âv(t)) + p(t)âv(t)â 2âv(t)b(t) = g(t) in Ω \ Î0, t â [0, T ], (1.1.28)
where B(t, y) â L (RdĂd;RdĂd), p(t, y) â L (RdĂd;Rd), b(t, y) â Rd, and g(t, y) â Rd aredefined for t â [0, T ] and y â Ω as
B(t, y)Ο := (A(t,Ί(t, y))[ΟâΚ(t,Ί(t, y))])âΚ(t,Ί(t, y))T â Οb(t, y)â b(t, y), (1.1.29)
p(t, y)Ο := â[(B(t, y)Ο)â(detâΊ(t, y)) + ât(Οb(t, y) detâΊ(t, y))] detâΚ(t,Ί(t, y)), (1.1.30)
b(t, y) := âΚ(t,Ί(t, y)), (1.1.31)
g(t, y) := f(t,Ί(t, y)) (1.1.32)
for every Ο â RdĂd. The system is supplemented by boundary conditions formally written as
v(t) = w(t) on âDΩ, t â [0, T ], (1.1.33)
(B(t)âv(t))Îœ = F (t) on âNΩ, t â [0, T ], (1.1.34)
(B(t)âv(t))Îœ = 0 on Î0, t â [0, T ], (1.1.35)
and initial conditionsv(0) = v0, v(0) = v1 in Ω \ Î0, (1.1.36)
with initial datav0 := u0, v1 := u1 +âu0Ί(0). (1.1.37)
To give a precise meaning to the notion of solution to system (1.1.28) with boundary condi-tions (1.1.33)â(1.1.35), we consider functions v which satisfy the following regularity assump-tions:
v â C1([0, T ];L2(Ω;Rd)), (1.1.38)
v(t)â w(t) â H1D(Ω \ Î0;Rd) for every t â [0, T ], (1.1.39)
âv â C0([0, T ];L2(Ω;RdĂd)), (1.1.40)
v â AC([0, T ];Hâ1D (Ω \ Î0;Rd)). (1.1.41)
Chapter 1. Elastodynamics system in domains with growing cracks 7
Definition 1.1.9. Let A, w, f , and F be as in (1.1.8)â(1.1.12). Let B, p, b, and g bedefined according to (1.1.29)â(1.1.32). We say that v is a weak solution to the transformedsystem (1.1.28) with boundary conditions (1.1.33)â(1.1.35), if v satisfies (1.1.38)â(1.1.41) andfor a.e. t â (0, T ) we have
ăv(t), ÏăHâ1D (Ω\Î0) + (B(t)âv(t),âÏ)L2(Ω) + (p(t)âv(t), Ï)L2(Ω)
+ 2(v(t), div [Ïâ b(t)])L2(Ω) = (g(t), Ï)L2(Ω) + (F (t), Ï)L2(âNΩ)
(1.1.42)
for every Ï â H1D(Ω \ Î0;Rd).
Remark 1.1.10. Notice that (H5) and (1.1.3) imply b(t) = 0 on the boundary of Ω \ Î0.Hence, in the weak formulation of (1.1.28), it makes sense to pass from â2(âv(t)b(t), Ï)L2(Ω)
to 2(v(t),div[Ïâ b(t)])L2(Ω), which can be defined even for v(t) â L2(Ω;Rd).
Remark 1.1.11. Let v be a function which satisfies (1.1.38)â(1.1.41). Let us check thatthe scalar products in (1.1.42) are well defined for a.e. t â (0, T ). By (1.1.41) we havev(t) â Hâ1
D (Ω \ Î0;Rd) for a.e. t â (0, T ), therefore it is duality with Ï â H1D(Ω \ Î0;Rd).
In view of (1.1.38) and (1.1.40), for every t â [0, T ] the functions v(t) and âv(t) belong toL2(Ω;Rd) and L2(Ω;RdĂd), respectively. Hence, to ensure that the scalar products in theleft-hand side of (1.1.42) are well defined, we need to show that the coefficients B, p, b, anddiv b are essentially bounded in space for almost every time.
Thanks to (H4), (H7), (H8), (1.1.8), and (1.1.9), we derive that the maps t 7â A(t,Ί(t)),t 7â âΚ(t,Ί(t)), t 7â Κ(t,Ί(t)), and t 7â div(Κ(t,Ί(t))) are Lipschitz continuous from [0, T ]to Lâ(Ω; L (RdĂd;RdĂd)), Lâ(Ω;RdĂd), Lâ(Ω;Rd), and Lâ(Ω), respectively. Therefore, weget
B â Lip([0, T ];Lâ(Ω; L (RdĂd;RdĂd))), (1.1.43)
b â Lip([0, T ];Lâ(Ω;Rd)), div b â Lip([0, T ];Lâ(Ω)). (1.1.44)
We split the coefficient p defined in (1.1.30) into the sum p = p1 + p2, where the operatorsp1(t, y),p2(t, y) â L (RdĂd;Rd) are defined for t â [0, T ] and y â Ω as
p1(t, y)Ο := â[(B(t, y)Ο)â(detâΊ(t, y)) + Οb(t, y)ât(detâΊ(t, y))] detâΚ(t,Ί(t, y)),
p2(t, y)Ο := âΟb(t, y)
for every Ο â RdĂd. In view of the discussion above, p1 â Lip([0, T ];Lâ(Ω; L (RdĂd;Rd))),while p2 is an element of Lâ(0, T ;L2(Ω; L (RdĂd;Rd))), being b the distributional derivativeof a function in Lip([0, T ];Lâ(Ω;Rd)). Moreover, there exists a constant C > 0 such thatâp2(t)âLâ(Ω) †C for a.e. t â (0, T ). Finally, the function g defined in (1.1.32) belongs to
L2(0, T ;L2(Ω;Rd)), since f â L2(0, T ;L2(Ω;Rd)). Then the right-hand side of (1.1.42) iswell defined for a.e. t â (0, T ).
Remark 1.1.12. Thanks to (H1) and (H2), together with a partition of unity, we canintegrate by part in Ω \ Î and derive the following formula:â«
ΩâÏ(x) · h(x)Ï(x) dx = â
â«Î©Ï(x) div[h(x)Ï(x)] dx (1.1.45)
for every Ï, Ï â H1(Ω \ Î), and for every h â W 1,â(Ω;Rd), with h · Îœ = 0 on (Π⩠Ω) âȘ âΩ.Similarly, for every Ï âW 1,1(Ω \ Î) we haveâ«
ΩâÏ(x) · h(x) dx = â
â«Î©Ï(x) div h(x) dx. (1.1.46)
In particular, formulas (1.1.45) and (1.1.46) are satisfied if h is either Κ(t,Ί(t)) or Ί(t,Κ(t)),thanks to (1.1.3), (1.1.4), (H4), and (H5).
8 1.1. Preliminary results
Let us clarify the relation between problem (1.1.14) with boundary conditions (1.1.15)â(1.1.17), and problem (1.1.28) with boundary conditions (1.1.33)â(1.1.35). We start with thefollowing lemma.
Lemma 1.1.13. Suppose that u and v are related by (1.1.26) and (1.1.27). Then u satis-fies (1.1.19)â(1.1.24) if and only if v satisfies (1.1.38)â(1.1.41).
Proof. The proof is straightforward by applying Lemmas 2.8 and 2.11 of [20] to the com-ponents of u and v, which is possible thanks to our Lemmas 1.1.3 and 1.1.4, and for-mula (1.1.45).
By using the identification Hâ1D (Ω \ Ît;Rd) = [Hâ1
D (Ω \ Ît)]d and Lemmas 2.9 and 2.12
of [20] we derive the following two results.
Lemma 1.1.14. Assume that u satisfies (1.1.19)â(1.1.24). Then for a.e. t â (0, T ) we have
ăv(t), ÏăHâ1D (Ω\Î0)
= ău(t), Ï(Κ(t)) detâΚ(t)ăHâ1D (Ω\Ît) + (âu(t), ât[Ï(Κ(t))â Ί(t,Κ(t)) detâΚ(t)])L2(Ω)
+ (u(t), ât[Ï(Κ(t)) detâΚ(t)]â div[Ï(Κ(t))â Ί(t,Κ(t)) detâΚ(t)])L2(Ω) (1.1.47)
for every Ï â H1D(Ω \ Î0;Rd).
Lemma 1.1.15. Assume that v satisfies (1.1.38)â(1.1.41). Then for a.e. t â (0, T ) we have
ău(t), ÏăHâ1D (Ω\Ît)
= ăv(t), Ï(Ί(t)) detâΊ(t)ăHâ1D (Ω\Î0) + (âv(t), ât[Ï(Ί(t))â Κ(t,Ί(t)) detâΊ(t)])L2(Ω)
+ (v(t), ât[Ï(Ί(t)) detâΊ(t)]â div[Ï(Ί(t))â Κ(t,Ί(t)) detâΊ(t)])L2(Ω)
for every Ï â H1D(Ω \ Ît;Rd).
We can now specify the relation between the two problems.
Theorem 1.1.16. Under the assumptions of Definition 1.1.6, a function u is a weak solu-tion to problem (1.1.14) with boundary conditions (1.1.15)â(1.1.17), if and only if the cor-responding function v introduced in (1.1.26) is a weak solution to (1.1.28) with boundaryconditions (1.1.33)â(1.1.35).
Proof. Let us assume that u is a weak solution to problem (1.1.14) with boundary condi-tions (1.1.15)â(1.1.17). Thanks to Lemmas 1.1.13 and 1.1.14, the function v satisfies (1.1.38)â(1.1.41) and (1.1.47). Take an arbitrary test function Ï â H1
D(Ω\Î0;Rd). For every t â [0, T ]the function Ï(Κ(t)) detâΚ(t) is in H1
D(Ω \ Ît;Rd). Thus, by (1.1.25) we have
ău(t), Ï(Κ(t)) detâΚ(t)ăHâ1D (Ω\Ît)
= â(A(t)âu(t),â[Ï(Κ(t)) detâΚ(t)])L2(Ω) + (f(t), Ï(Κ(t)) detâΚ(t))L2(Ω)
+ (F (t), Ï(Κ(t)) detâΚ(t))L2(âNΩ).
By inserting this expression in (1.1.47) and using assumption (H4), we get
ăv(t), ÏăHâ1D (Ω\Î0)
= â(A(t)âu(t),â[Ï(Κ(t)) detâΚ(t)])L2(Ω) + (f(t), Ï(Κ(t)) detâΚ(t))L2(Ω)
+ (F (t), Ï)L2(âNΩ) + (âu(t), ât[Ï(Κ(t))â Ί(t,Κ(t)) detâΚ(t)])L2(Ω)
+ (u(t), ât[Ï(Κ(t)) detâΚ(t)]â div[Ï(Κ(t))â Ί(t,Κ(t)) detâΚ(t)])L2(Ω).
(1.1.48)
Chapter 1. Elastodynamics system in domains with growing cracks 9
Thanks to (1.1.27), by performing the same computations done in [20] we get
âu(t) = âv(t,Κ(t))âΚ(t), u(t) = v(t,Κ(t)) +âv(t,Κ(t))Κ(t). (1.1.49)
We insert this expression in (1.1.48) and we obtain that v satisfies (1.1.42). Notice thatthe boundary terms w and F remain the same through the change of variables since thediffeomorphisms Ί and Κ are the identity in a neighborhood of âΩ.
Similarly, by applying Lemmas 1.1.13 and 1.1.15, it is easy to check that if v is a weaksolution to problem (1.1.28) with boundary conditions (1.1.33)â(1.1.35), then u is a weaksolution to problem (1.1.14) with boundary conditions (1.1.15)â(1.1.17).
Remark 1.1.17. Given a weak solution u to (1.1.14)â(1.1.17), we can improve the integra-bility condition (1.1.24). Indeed, by (1.1.25), the Lipschitz regularity of A, the continuityproperty (1.1.21) of âu, and (1.1.1), we infer
âu(t)âHâ1D (Ω\Ît) †C(1 + âf(t)âL2(Ω) + âF (t)âL2(âNΩ)) for a.e. t â (0, T ),
where C > 0 is a constant independent of t. Therefore, the function t 7â âu(t)âHâ1D (Ω\Ît)
belongs to L2(0, T ), since f â L2(0, T ;L2(Ω;Rd)) and F â C0([0, T ];L2(âNΩ;Rd)). In ad-dition, if f â Lp(0, T ;L2(Ω;Rd)), with p â (2,â], then the function t 7â âu(t)âHâ1
D (Ω\Ît)belongs to Lp(0, T ). The same property is true also for a weak solution v of (1.1.28) withboundary conditions (1.1.33)â(1.1.35), by exploiting the regularity properties of v and âv,and the regularity of the coefficients (1.1.29)â(1.1.32) discussed in Remark 1.1.11.
1.2 Existence and uniqueness
To prove our existence and uniqueness results, for both problems (1.1.14) and (1.1.28), werequire an additional hypothesis on the operator B. We assume that there exist two constantsc0 > 0 and c1 â R such that for every t â [0, T ]
(B(t)âÏ,âÏ)L2(Ω) â„ c0âÏâ2H1(Ω\Î0) â c1âÏâ2L2(Ω) for every Ï â H1D(Ω \ Î0;Rd). (1.2.1)
Notice that assumption (1.2.1) is satisfied whenever the velocity of the diffeomorphism Ί issufficiently small and A satisfies the following standard ellipticity condition in linear elasticity:
(A(t, x)Ο) · Ο ℠λ0|Οsym|2 for every Ο â RdĂd, t â [0, T ], x â Ω, (1.2.2)
for a suitable constant λ0 > 0, independent of t and x. Indeed, thanks to assumptions (H1)and (H2) we can find a finite number of open sets Uj â Ω \ Î, j = 1, . . .m, with Lipschitzboundary, such that Ω \Î = âȘmj=1Uj . By using second Kornâs inequality in each Uj (see, e.g.,[44, Theorem 2.4]) and taking the sum over j, we can find a constant CK , depending only onΩ and Î, such that
ââÏâ2L2(Ω) †CK(âÏâ2L2(Ω) + âEÏâ2L2(Ω)
)for every Ï â H1(Ω \ Î;Rd), (1.2.3)
where EÏ is the symmetrized gradient of Ï, namely EÏ := 12(âÏ +âÏT ). Define
M := max(t,y)â[0,T ]ĂΩ
detâΊ(t, y), m := min(t,y)â[0,T ]ĂΩ
detâΊ(t, y).
For every t â [0, T ] and Ï â H1D(Ω \ Î0;Rd) we use the definition of B and the change of
variables formula, together with (1.1.10), (1.2.2), and (1.2.3), to derive
M
â«Î©B(t, y)âÏ(y) · âÏ(y) dy
10 1.2. Existence and uniqueness
â„â«
ΩB(t, y)âÏ(y) · âÏ(y) detâΊ(t, y) dy
=
â«Î©A(t, x)â(Ï(Κ(t, x))) · â(Ï(Κ(t, x))) dxâ
â«Î©|â(Ï(Κ(t, x)))Ί(t,Κ(t, x))|2dx
℠λ0
â«Î©|E(Ï(Κ(t, x)))|2dxâ
â«Î©|â(Ï(Κ(t, x)))Ί(t,Κ(t, x))|2dx
℠λ0
CK
â«Î©|â(Ï(Κ(t, x)))|2dxâ λ0
â«Î©|Ï(Κ(t, x))|2dxâ
â«Î©|â(Ï(Κ(t, x)))Ί(t,Κ(t, x))|2dx,
since Ï(Κ(t)) â H1(Ω \ Ît;Rd) â H1(Ω \ Î;Rd) for every t â [0, T ]. Hence, if we assume
|Ί(t, y)|2 < λ0
CKfor every t â [0, T ] and y â Ω, (1.2.4)
then by (H4) we obtain the existence of a constant ÎŽ > 0 such thatâ«Î©B(t, y)âÏ(y) · âÏ(y) dy â„ mÎŽ
M
â«Î©|âÏ(y)âΚ(t,Ί(t, y))|2dy â mλ0
M
â«Î©|Ï(y)|2dy,
which implies (1.2.1).
Remark 1.2.1. Assumption (1.2.4) imposes a condition on the velocity of the growing crackwhich depends on the geometry of the crack itself.
We have seen that problem (1.1.14) with boundary conditions (1.1.15)â(1.1.17), and prob-lem (1.1.28) with boundary conditions (1.1.33)â(1.1.35) are equivalent. We want to prove thefollowing existence theorem.
Theorem 1.2.2. Let be given A, w, f , F , u0, u1 as in (1.1.8)â(1.1.13). Let B, p, b, g,v0, v1 be defined according to (1.1.29)â(1.1.32) and (1.1.37), with B satisfying (1.2.1). Thenproblem (1.1.28) with boundary conditions (1.1.33)â(1.1.35) and initial conditions (1.1.36)admits a unique solution v, according to Definition 1.1.9.
The proof of Theorem 1.2.2 will be postponed at the end of Section 1.3 and it will obtainedas a consequence of Theorems 1.2.9 and 1.2.10 below and Proposition 1.3.1. Thanks toTheorem 1.1.16, as corollary we readily obtain the following result.
Theorem 1.2.3. Let be given A, w, f , F , u0, u1 as in (1.1.8)â(1.1.13). Assume that theoperator B defined in (1.1.29) satisfies (1.2.1). Then problem (1.1.14) with boundary condi-tions (1.1.15)â(1.1.17) and initial conditions (1.1.18) admits a unique solution u, accordingto Definition 1.1.6.
Proof. By using Theorems 1.1.16 and 1.2.2 there exists a solution u to (1.1.14)â(1.1.17).Moreover, the initial conditions (1.1.18) follows from the regularity conditions (1.1.19)â(1.1.24) of u and from the initial conditions of v. Finally, the solution is unique sinceevery solution u to (1.1.14) with boundary and initial conditions (1.1.15)â(1.1.18), givesa solution v to (1.1.28) with boundary and initial conditions (1.1.33)â(1.1.36), thanks toTheorem 1.1.16.
Remark 1.2.4. The existence and uniqueness results of this thesis improve the ones con-tained in [8]. Indeed, to prove Theorems 1.2.2 and 1.2.3 we only assume that w satis-fies (1.1.11), while in [8] is required
w â H2(0, T ;L2(Ω;Rd)) â©H1(0, T ;H1(Ω \ Î0;Rd)) â© L2(0, T ;H2(Ω \ Î0;Rd)).
Moreover, we remove the assumption
(A(t)âw(t))Îœ = 0 on âNΩ âȘ Ît, t â [0, T ],
which, on the contrary, is needed in [8].
Chapter 1. Elastodynamics system in domains with growing cracks 11
To prove Theorem 1.2.2, we introduce a notion of solution to (1.1.28) which is weaker thanthe one considered in Definition 1.1.9. Later, in Section 1.3, we will prove an energy equalitywhich ensure that this type of solution is more regular, namely it satisfies the regularityconditions (1.1.38)â(1.1.41).
Definition 1.2.5. Let A, w, f , and F be as in (1.1.8)â(1.1.13). Let B, p, b, and g bedefined according to (1.1.29)â(1.1.32). We say that v is a generalized solution to (1.1.28)with boundary conditions (1.1.33)â(1.1.35) if
v â Lâ(0, T ;H1(Ω \ Î0;Rd)) â©W 1,â(0, T ;L2(Ω;Rd)) â©H2(0, T ;Hâ1D (Ω \ Î0;Rd)),
v â w â Lâ(0, T ;H1D(Ω \ Î0;Rd)),
and for a.e. t â (0, T ) we have
ăv(t), ÏăHâ1D (Ω\Î0) + (B(t)âv(t),âÏ)L2(Ω) + (p(t)âv(t), Ï)L2(Ω)
+ 2(v(t), div[Ïâ b(t)])L2(Ω) = (g(t), Ï)L2(Ω) + (F (t), Ï)L2(âNΩ)
(1.2.5)
for every Ï â H1D(Ω \ Î0;Rd).
Remark 1.2.6. Since Câc (0, T ) â H1D(Ω \ Î0;Rd) is dense in L2(0, T ;H1
D(Ω \ Î0;Rd)), wecan recast equality (1.2.5) in the framework of the duality between L2(0, T ;Hâ1
D (Ω \Î0;Rd))and L2(0, T ;H1
D(Ω \ Î0;Rd)). Indeed, it is easy to see that (1.2.5) is equivalent toâ« T
0
[ăv(t), Ï(t)ăHâ1
D (Ω\Î0) + (B(t)âv(t),âÏ(t))L2(Ω) + (p(t)âv(t), Ï(t))L2(Ω)
]dt
+ 2
â« T
0(v(t), div[Ï(t)â b(t)])L2(Ω) dt =
â« T
0
[(g(t), Ï(t))L2(Ω) + (F (t), Ï(t))L2(âNΩ)
]dt
for every Ï â L2(0, T ;H1D(Ω \ Î0;Rd)).
Remark 1.2.7. Let us clarify the meaning of the initial conditions (1.1.36) for a generalizedsolution. We recall that if X,Y are two reflexive Banach spaces, with embedding X â Ycontinuous, then
C0w([0, T ];Y ) â© Lâ(0, T ;X) = C0
w([0, T ];X),
see for instance [24, Chapitre XVIII, §5, Lemme 6], where C0w([0, T ];X) and C0
w([0, T ];Y )denote the spaces of weakly continuous functions from [0, T ] to X and Y , respectively. Inparticular, we can apply this result to a generalized solution v. By taking X = H1(Ω\Î0;Rd)and Y = L2(Ω;Rd) and using
v â C0([0, T ];L2(Ω;Rd)) â© Lâ(0, T ;H1(Ω \ Î0;Rd)),
we have v â C0w([0, T ];H1(Ω \ Î0;Rd)). Therefore v(0) is an element of H1(Ω \ Î0;Rd).
Similarly, by taking X = L2(Ω;Rd) and Y = Hâ1D (Ω\Î0;Rd) we get v â C0
w([0, T ];L2(Ω;Rd)),since
v â C0([0, T ];Hâ1D (Ω \ Î0;Rd)) â© Lâ(0, T ;L2(Ω;Rd)).
Therefore v(0) is an element of L2(Ω;Rd). In particular, the initial conditions (1.1.36) arewell defined if v0 â H1(Ω\Î0;Rd) and v1 â L2(Ω;Rd). With a similar argument we also havev â w â C0
w([0, T ];H1D(Ω \ Î0;Rd)), which yields v(t) = w(t) on âDΩ in the sense of traces
for every t â [0, T ].
We recall an existence result for evolutionary problems of second order in time, whoseproof can be found for example in [24]. Let V , H be two separable Hilbert spaces, withembedding V â H continuous and dense, and for every t â [0, T ] let B(t; ·), A1(t; ·),A2(t; ·) : V Ă V â R be three families of continuous bilinear forms satisfying the follow-ing properties:
12 1.2. Existence and uniqueness
(i) the bilinear form B(t; ·) is symmetric for every t â [0, T ];
(ii) there exist c0 > 0, c1 â R such that B(t;Ï,Ï) â„ c0âÏâ2V â c1âÏâ2H for every t â [0, T ]and Ï â V ;
(iii) for every Ï, Ï â V the function t 7â B(t;Ï, Ï) is continuously differentiable in [0, T ];
(iv) there exists c2 > 0 such that |B(t;Ï, Ï)| †c2âÏâV âÏâV for every t â [0, T ] and Ï, Ï â V ;
(v) for every Ï, Ï â V the function t 7â A1(t;Ï, Ï) is continuous in [0, T ];
(vi) there exists c3 > 0 such that |A1(t;Ï, Ï)| †c3âÏâV âÏâH for every t â [0, T ] andÏ, Ï â V ;
(vii) for every Ï, Ï â V the function t 7â A2(t;Ï, Ï) is continuous in [0, T ];
(viii) there exists c4 > 0 such that |A2(t;Ï, Ï)| †c4âÏâV âÏâH for every t â [0, T ] andÏ, Ï â V ;
where t 7â B(t;Ï, Ï) denotes the derivative of t 7â B(t;Ï, Ï) for Ï, Ï â V .
Theorem 1.2.8. Let Δ > 0, v0 â V , v1 â H, g â L2(0, T ;V âČ), and B(t; ·), A1(t; ·), A2(t; ·),t â [0, T ], be three families of continuous bilinear forms over V Ă V satisfying assumptions(i)â(viii) above. Then there exists a function v â H1(0, T ;V )â©W 1,â(0, T ;H)â©H2(0, T ;V âČ)solution for a.e. t â (0, T ) to
ăv(t), ÏăV âČ + B(t; v(t), Ï) +A1(t; v(t), Ï) +A2(t; v(t), Ï) + Δ(v(t), Ï)V = ăg(t), ÏăV âČ (1.2.6)
for every Ï â V , with initial conditions v(0) = v0 and v(0) = v1.
Proof. See [24, Chapitre XVIII, §5, Theoreme 1 and Remarque 4].
We are now in a position to state the first existence result.
Theorem 1.2.9. Let A, w, f , F , u0, and u1 be as in (1.1.8)â(1.1.13). Let B, p, b, g, v0,and v1 be defined according to (1.1.29)â(1.1.32) and (1.1.37). Assume that B satisfies (1.2.1).Then there exists a generalized solution to (1.1.28) with boundary conditions (1.1.33)â(1.1.35)satisfying the initial conditions (1.1.36).
Proof. As in [20, Theorem 3.6], the proof is based on a perturbation argument, following thestandard procedure of [24]: we first fix Δ â (0, 1) and we study equation (1.2.5) with theadditional term
Δ(v(t), Ï)H1(Ω\Î0) for Ï â H1D(Ω \ Î0;Rd),
and then we let the viscosity parameter Δ tend to zero.Step 1: the perturbed problem. Let Δ â (0, 1) be fixed. We want to show the existence of
a function
vΔ â H1(0, T ;H1(Ω \ Î0;Rd)) â©W 1,â(0, T ;L2(Ω;Rd)) â©H2(0, T ;Hâ1D (Ω \ Î0;Rd)),
with vΔ â w â H1(0, T ;H1D(Ω \ Î0;Rd)), solution for a.e t â (0, T ) to the equation
ăvΔ(t), ÏăHâ1D (Ω\Î0) + (B(t)âvΔ(t),âÏ)L2(Ω) + (p(t)âvΔ(t), Ï)L2(Ω)
â 2(âvΔ(t)b(t), Ï)L2(Ω) + Δ(vΔ(t), Ï)H1(Ω\Î0) = (g(t), Ï)L2(Ω) + (F (t), Ï)L2(âNΩ)
(1.2.7)
for every Ï â H1D(Ω \ Î0;Rd), and which satisfies the initial conditions vΔ(0) = v0 and
vΔ(0) = v1. To this aim, we regularize our coefficient with respect to time by means of a
Chapter 1. Elastodynamics system in domains with growing cracks 13
sequence of mollifiers. Let Ïnn â Câc (R) be a sequence of functions such that Ïn â„ 0,supp(Ïn) â [â1/n, 1/n], and
â«R Ïn dx = 1 for every n â N, and let us extend our coefficients
B and p to all R, as done in [20, Theorem 3.6]. To deal with the boundary terms w and Fwe introduce the function g â L2(0, T ;Hâ1
D (Ω \ Î0;Rd)) defined for a.e. t â (0, T ) as
ăg(t), ÏăHâ1D (Ω\Î0)
:=â (w(t), Ï)L2(Ω) â ((ÏnâB)(t)âw(t),âÏ)L2(Ω)
â ((Ïnâp)(t)âw(t), Ï)L2(Ω) + 2(âw(t)b(t), Ï)L2(Ω)
â Δ(w(t), Ï)H1(Ω\Î0) + (g(t), Ï)L2(Ω) + (F (t), Ï)L2(âNΩ)
for Ï â H1D(Ω \ Î0;Rd); the regularity of g is a consequence of (1.1.1), (1.1.11), (1.1.12),
and Remark 1.1.11. We apply Theorem 1.2.8 with Hilbert spaces V = H1D(Ω \ Î0;Rd), H =
L2(Ω;Rd), bilinear forms B(t)(Ï, Ï) := ((ÏnâB)(t)âÏ,âÏ), A1(t)(Ï, Ï) := (Ïnâp)(t)âÏ, Ï),and A2(t)(Ï, Ï) := â2(âÏb(t), Ï) for Ï, Ï â V and t â [0, T ], forcing term g, and initialconditions v0 := v0 â w(0) and v1 := v1 â w(0). For every n â N this leads to a solutionvnΔ to (1.2.6) which satisfies the initial conditions v0 and v1. In particular, the functionvnΔ := vnΔ +w solves equation (1.2.7) with B(t) and p(t) replaced by (ÏnâB)(t) and (Ïnâp)(t),respectively, and initial conditions v0 and v1.
We fix t0 â (0, T ]. By taking vnΔ (t)â w(t) as test function in (1.2.7) and integrating over(0, t0), we getâ« t0
0
[ăvnΔ , vnΔ â wăHâ1
D (Ω\Î0) + ((ÏnâB)âvnΔ ,âvnΔ ââw)L2(Ω)
]dt
+
â« t0
0
[((Ïnâp)âvnΔ , vnΔ â w)L2(Ω) â 2(âvnΔ b, vnΔ â w)L2(Ω) + Δ(vnΔ , v
nΔ â w)H1(Ω\Î0)
]dt
=
â« t0
0
[(g, vnΔ â w)L2(Ω) + (F, vnΔ â w)L2(âNΩ)
]dt. (1.2.8)
For the first term we use the integration by parts formulaâ« t0
0ăvnΔ â w, vnΔ â wăHâ1
D (Ω\Î0) dt =1
2âvnΔ (t0)â w(t0)â2L2(Ω) â
1
2âv1 â w(0)â2L2(Ω)
to deduceâ« t0
0ăvnΔ , vnΔ â wăHâ1
D (Ω\Î0) dt =1
2âvnΔ (t0)â2L2(Ω) â
1
2âv1â2L2(Ω) â (vnΔ (t0), w(t0))L2(Ω)
+ (v1, w(0))L2(Ω) +
â« t0
0(vnΔ , w)L2(Ω) dt.
(1.2.9)
Similarly, for the second term we haveâ« t0
0((ÏnâB)âvnΔ ,âvnΔ ââw)L2(Ω) dt
=1
2((ÏnâB)(t0)âvnΔ (t0),âvnΔ (t0))L2(Ω) â
1
2((ÏnâB)(0)âv0,âv0)L2(Ω)
ââ« t0
0
[1
2(ât(ÏnâB)âvnΔ ,âvnΔ )L2(Ω) + ((ÏnâB)âvnΔ ,âw)L2(Ω)
]dt.
(1.2.10)
Since vnΔ (t) â w(t) â H1D(Ω \ Î0;Rd) for every t â [0, T ], by (1.2.1) we derive the following
estimate
((ÏnâB)(t0)[âvnΔ (t0)ââw(t0)],âvnΔ (t0)ââw(t0))L2(Ω) â„ c0âvnΔ (t0)â w(t0)â2H1(Ω\Î0)
â c1âvnΔ (t0)â w(t0)â2L2(Ω).
14 1.2. Existence and uniqueness
In particular, thanks to (1.1.11), (1.1.43), and Youngâs inequality, there exists a constantC > 0, independent of n, Δ, and t0, such that
c0
2âvnΔ (t0)â2H1(Ω\Î0) †((ÏnâB)(t0)âvnΔ (t0),âvnΔ (t0))L2(Ω) + C(1 + âvnΔ (t0)â2L2(Ω)). (1.2.11)
In addition, since vnΔ (t0) = v0 +⫠t0
0 vnΔ (t) dt, we get
âvnΔ (t0)â2L2(Ω) †2âv0â2L2(Ω) + 2T
â« t0
0âvnΔ â2L2(Ω) dt. (1.2.12)
By the regularity properties of B, p, g, and w we obtain another constant C > 0, independentof n, Δ, and t0, such thatâŁâŁâŁâŁâ« t0
0(vnΔ , w)L2(Ω) dt
âŁâŁâŁâŁ †C â« t0
0âvnΔ â2L2(Ω) dt, (1.2.13)âŁâŁâŁâŁâ« t0
0
1
2(ât(ÏnâB)âvnΔ ,âvnΔ )L2(Ω) dt
âŁâŁâŁâŁ †C â« t0
0âvnΔ â2H1(Ω\Î0) dt, (1.2.14)âŁâŁâŁâŁâ« t0
0((ÏnâB)âvnΔ ,âw)L2(Ω) dt
âŁâŁâŁâŁ †C â« t0
0âvnΔ â2H1(Ω\Î0) dt, (1.2.15)âŁâŁâŁâŁâ« t0
0((Ïnâp)âvnΔ , vnΔ â w)L2(Ω) dt
âŁâŁâŁâŁ †C â« t0
0
[âvnΔ â2H1(Ω\Î0) + âvnΔ â2L2(Ω)
]dt, (1.2.16)âŁâŁâŁâŁâ« t0
0(g, vnΔ â w)L2(Ω) dt
âŁâŁâŁâŁ †C (1 +
â« t0
0âvnΔ â2L2(Ω) dt
). (1.2.17)
Furthermore, we can use Youngâs inequality and (1.1.11) to find a further constant C > 0,independent of n, Δ, and t0, such thatâŁâŁ(vnΔ (t0), w(t0))L2(Ω) â (v1, w(0))L2(Ω)
âŁâŁ †1
4âvnΔ (t0)â2L2(Ω) + C. (1.2.18)
By formula (1.1.45) we derive
2
â« t0
0(âvnΔ b, w)L2(Ω) dt = â2
â« t0
0(vnΔ ,div[w â b])L2(Ω) dt,
which implies the existence of C > 0, independent of n, Δ, and t0, such thatâŁâŁâŁâŁ2 â« t0
0(âvnΔ b, w)L2(Ω) dt
âŁâŁâŁâŁ †C â« t0
0âvnΔ âL2(Ω) dt. (1.2.19)
Thanks to (1.1.46), for every Ï â H1D(Ω \ Î0;Rd) we have
2(âÏb(t), Ï)L2(Ω) =
â«Î©b(t, y) · â|Ï(y)|2dy = â
â«Î©
div b(t, y)|Ï(y)|2dy. (1.2.20)
Therefore, by (1.1.44) there exits a constant C > 0, independent of n, Δ, and t0, such thatâŁâŁâŁâŁ2â« t0
0(âvnΔ b, vnΔ )L2(Ω) dt
âŁâŁâŁâŁ †C â« t0
0âvnΔ â2L2(Ω) dt. (1.2.21)
Since F â H1(0, T ;L2(âNΩ;Rd)), we can integrate the last term in (1.2.8) by parts withrespect to time, and we obtainâ« t0
0(F, vnΔ â w)L2(âNΩ) dt = (F (t0), vnΔ (t0)â w(t0))L2(âNΩ) â (F (0), v0 â w(0))L2(âNΩ)
ââ« t0
0(F , vnΔ â w)L2(âNΩ) dt.
Chapter 1. Elastodynamics system in domains with growing cracks 15
We use the previous identity, together with (1.1.1) and Youngâs inequality, to deduce theexistence of a constant C > 0, independent of n, Δ, and t0, such thatâŁâŁâŁâŁâ« t0
0(F, vnΔ â w)L2(âNΩ) dt
âŁâŁâŁâŁ †c0
8âvnΔ (t0)â2H1(Ω\Î0) + C
(1 +
â« t0
0âvnΔ â2H1(Ω\Î0) dt
). (1.2.22)
Finally, again by Youngâs inequality
Δ
â« t0
0(vnΔ , v
nΔ â w)H1(Ω\Î0) dt ℠Δ
2
â« t0
0âvnΔ â2H1(Ω\Î0) dtâ 1
2
â« t0
0âwâ2H1(Ω\Î0) dt. (1.2.23)
By combining (1.2.8)â(1.2.19) with (1.2.21)â(1.2.23), we infer
1
4âvnΔ (t0)â2L2(Ω) +
c0
8âvnΔ (t0)â2H1(Ω\Î0) +
Δ
2
â« t0
0âvnΔ â2H1(Ω\Î0) dt
†C1 + C2
â« t0
0
[âvnΔ â2L2(Ω) + âvnΔ â2H1(Ω\Î0)
]dt,
for two constants C1 and C2 independent of n, Δ, and t0.Thanks to Gronwallâs lemma we conclude that there exists C > 0, independent of n, Δ,
and t0, such that
âvnΔ (t0)â2L2(Ω) + âvnΔ (t0)â2H1(Ω\Î0) †C for every t0 â [0, T ]. (1.2.24)
Hence, we have
vnΔ n is bounded in Lâ(0, T ;H1(Ω \ Î0;Rd)),vnΔ n is bounded in Lâ(0, T ;L2(Ω;Rd)),âΔvnΔ n is bounded in L2(0, T ;H1(Ω \ Î0;Rd)),
uniformly with respect to n and Δ. From these estimates, by using also the equation solvedby vnΔ and (1.2.20), we derive
vnΔ n is bounded in L2(0, T ;Hâ1D (Ω \ Î0;Rd)),
uniformly with respect to n and Δ. Therefore, up to a not relabeled subsequence, vnΔ con-verges weakly to a function vΔ in H1(0, T ;H1(Ω \ Î0;Rd)) and vnΔ converges weakly to vΔ inL2(0, T ;Hâ1
D (Ω \ Î0;Rd)) as n â â. Finally, we have vΔ â w â H1(0, T ;H1D(Ω \ Î0;Rd)),
since vnΔ â w â H1(0, T ;H1D(Ω \ Î0;Rd)) for every n â N.
Let us show that vΔ satisfies (1.2.7). Since B is symmetric, for every Ï â H1D(Ω \ Î0;Rd)
we have
((ÏnâB)(t)âvnΔ (t),âÏ)L2(Ω) = (âvnΔ (t), (ÏnâB)(t)âÏ)L2(Ω),
((Ïnâp)(t)âvnΔ (t), Ï)L2(Ω) = (âvnΔ (t), (Ïnâp?)(t)Ï)L2(Ω),
where p?(t, y) â L (Rd;RdĂd) is the transpose operator of p(t, y) â L (RdĂd;Rd), defined fort â [0, T ] and y â Ω by
(p?(t, y)a) · Ο = (p(t, y)Ο) · a for every a â Rd and Ο â RdĂd. (1.2.25)
By the regularity properties of B and p, as nââ we have
(ÏnâB)(t)âÏâ B(t)âÏ in L2(Ω;RdĂd) for every t â [0, T ],
(Ïnâp?)(t)Ïâ p?(t)Ï in L2(Ω;RdĂd) for a.e. t â (0, T ).
16 1.2. Existence and uniqueness
Thanks to the strong convergences above and the weak convergences of vnΔ , vnΔ and vnΔ , we canpass to the limit as n â â in the PDE solved by vnΔ and we obtain that the weak limit vΔsatisfies equation (1.2.7) (see Remark 1.2.6). Furthermore, the bound (1.2.24) and the weakconvergences of vnΔ , vnΔ and vnΔ , imply for every t â [0, T ]
vnΔ (t) vΔ(t) in H1(Ω \ Î0;Rd), vnΔ (t) vΔ(t) in L2(Ω;Rd) as nââ.
Hence vΔ satisfies the initial conditions vΔ(0) = v0 and vΔ(0) = v1.Step 2. Vanishing viscosity. As already done in Step 1 for vnΔ , we take vΔ â w as test
function in (1.2.7), and we integrate in (0, t0) to derive the energy equality
1
2âvΔ(t0)â2L2(Ω) +
1
2(B(t0)âvΔ(t0),âvΔ(t0))L2(Ω) + Δ
â« t0
0(vΔ, vΔ â w)H1(Ω\Î0) dt
=1
2âv1â2L2(Ω) +
1
2(B(0)âvΔ(0),âvΔ(0))L2(Ω)
+
â« t0
0
[1
2(BâvΔ,âvΔ)L2(Ω) + (BâvΔ,âw)L2(Ω) â (pâvΔ, vΔ â w)L2(Ω)
]dt
+
â« t0
0
[â(vΔ div b, vΔ) + 2(vΔ, div[w â b])L2(Ω) + (g, vnΔ â w)L2(Ω)
]dt
+
â« t0
0
[â(F , vΔ â w)L2(âNΩ) â (vΔ, w)L2(Ω)
]dt+ (F (t0), vΔ(t0)â w(t0))L2(âNΩ)
+ (vΔ(t0), w(t0))L2(Ω) â (F (0), v0 â w(0))L2(âNΩ) â (v1, w(0))L2(Ω).
(1.2.26)
By arguing as before and using the ellipticity condition (1.2.1) of B, we get the followingestimate:
1
4âvΔ(t0)â2L2(Ω) +
c0
8âvΔ(t0)â2H1(Ω\Î0) +
Δ
2
â« t0
0âvΔâ2H1(Ω\Î0) dt
†C1 + C2
â« t0
0
[âvΔâ2L2(Ω) + âvΔâ2H1(Ω\Î0)
]dt,
(1.2.27)
where C1 and C2 are two constants independent of Δ and t0. Therefore, Gronwallâs lemmayields
âvΔ(t0)â2L2(Ω) + âvΔ(t0)â2H1(Ω\Î0) †C for every t0 â [0, T ] (1.2.28)
for a constant C independent of Δ and t0. This implies that the sequence vΔΔ is uni-formly bounded in Lâ(0, T ;H1(Ω \ Î0;Rd)) and the sequence vΔΔ is uniformly bounded inLâ(0, T ;L2(Ω;Rd)). Moreover, by combining (1.2.27) and (1.2.28), we infer
Δ
â« T
0âvΔâ2H1(Ω\Î0) dt †C, (1.2.29)
for a constant C independent of Δ. By formula (1.1.45) for a.e. t â (0, T ) we have
(âvΔ(t)b(t), Ï)L2(Ω) = â(vΔ(t), div[Ïâ b(t)])L2(Ω) for every Ï â H1D(Ω \ Î0;Rd).
Thanks to (1.2.7) and the previous estimates we conclude that the sequence vΔΔ is uni-formly bounded in L2(0, T ;Hâ1
D (Ω \Î0;Rd)). Therefore, there exists a subsequence of Δ (notrelabeled) and a function
v â Lâ(0, T ;H1(Ω \ Î0;Rd)) â©W 1,â(0, T ;L2(Ω;Rd)) â©H2(0, T ;Hâ1D (Ω \ Î0;Rd))
such that the following convergences hold as Δâ 0+:
vΔ v in L2(0, T ;H1(Ω \ Î0;Rd)), (1.2.30)
Chapter 1. Elastodynamics system in domains with growing cracks 17
vΔ v in L2(0, T ;L2(Ω;Rd)), (1.2.31)
vΔ v in L2(0, T ;Hâ1D (Ω \ Î0;Rd)). (1.2.32)
Moreover, we have v âw â Lâ(0, T ;H1D(Ω \ Î0;Rd)). Notice that, a priori, the weak limit v
is not unique, but might depend on the particular subsequence chosen.Let us show that vΔ solves (1.2.5). We fix a test function Ï â L2(0, T ;H1
D(Ω \ Î0;Rd))for (1.2.5) (see Remark 1.2.6), and for every Δ â (0, 1) we haveâ« T
0
[ăvΔ, ÏăHâ1
D (Ω\Î0) + (BâvΔ,âÏ)L2(Ω) + (pâvΔ, Ï)L2(Ω) + 2(vΔ,div[Ïâ b])L2(Ω)
]dt
+ Δ
â« T
0(vΔ, Ï)H1(Ω\Î0) dt =
â« T
0[(g, Ï)L2(Ω) + (F,Ï)L2(âNΩ)] dt. (1.2.33)
Thanks to (1.2.29), as Δâ 0+ we getâŁâŁâŁâŁÎ” â« T
0(vΔ, Ï)H1(Ω\Î0) dt
âŁâŁâŁâŁ †âΔ⫠T
0
âΔâvΔâH1(Ω\Î0)âÏâH1(Ω\Î0) dt
â€âΔâÏâL2(0,T ;H1(Ω\Î0))
(â« T
0ΔâvΔâ2H1(Ω\Î0) dt
)1/2
â€âΔC â 0.
The last property, together with (1.2.30)â(1.2.32) and (1.2.33), gives that v solves (1.2.5).Finally, by arguing as in Step 1, for every t â [0, T ] we obtain
vΔ(t) v(t) in H1(Ω \ Î0;Rd), vΔ(t) v(t) in L2(Ω;Rd) as Δâ 0+. (1.2.34)
This gives the validity of the initial conditions for v.
The proof of uniqueness is similar to the one in [20] and relies on a standard techniquedue to Ladyzenskaya [34], which consists in taking as test function in (1.2.5) the primitive ofa solution.
Theorem 1.2.10. Under the assumptions of Theorem 1.2.9, there exists at most one gen-eralized solution to (1.1.28) with boundary conditions (1.1.33)â(1.1.35) satisfying the initialconditions (1.1.36).
Proof. By linearity, it is enough to show that the unique generalized solution v to prob-lem (1.1.28) with
g = F = w = v0 = v1 = 0
is u = 0. The proof is divided into two steps: first, we show the uniqueness in a small interval[0, t0]; then, by a continuity argument, we deduce the uniqueness in the all [0, T ].
Step 1. Let s â (0, T ] be fixed and let Ïs â L2(0, T ;H1D(Ω \ Î0;Rd)) be defined as
Ïs(t) :=
ââ« st v(Ï) dÏ if t â [0, s],
0 if t â [s, T ].
Notice that Ïs(s) = Ïs(T ) = 0. Moreover Ïs â L2(0, T ;H1D(Ω \ Î0;Rd)), indeed
Ïs(t) =
v(t) if t â [0, s),
0 if t â (s, T ].
By taking Ïs as test function in (1.2.5), we getâ« s
0
[ăv(t), Ïs(t)ăHâ1
D (Ω\Î0) + (B(t)âv(t),âÏs(t))L2(Ω)
]dt
+
â« s
0
[(p(t)âv(t), Ïs(t))L2(Ω) + 2(v(t),div[Ïs(t)â b(t)])L2(Ω)
]dt = 0.
(1.2.35)
18 1.2. Existence and uniqueness
We integrate the first term by parts with respect to time, and we obtainâ« s
0ăv(t), Ïs(t)ăHâ1
D (Ω\Î0) dt = ââ« s
0(v(t), v(t))L2(Ω) dt = â1
2âv(s)â2L2(Ω), (1.2.36)
since v0 = v1 = Ïs(s) = 0.Let us rewrite the second term involving B. By Definition 1.2.5 of generalized solution it
is easy to see that Ïs â Lip([0, T ];H1D(Ω \ Î0;Rd)). Therefore, thanks to (1.1.43), we have
that BâÏs â Lip([0, T ];L2(Ω;RdĂd)). We perform an integration by parts with respect totime and we use the fact that Ïs(s) = 0 in H1
D(Ω \ Î0;Rd) to deriveâ« s
0(B(t)âv(t),âÏs(t))L2(Ω) dt
= ââ« s
0(âÏs(t),B(t)âÏs(t))L2(Ω) dt
= â1
2(B(0)âÏs(0),âÏs(0))L2(Ω) â
1
2
â« s
0(B(t)âÏs(t),âÏs(t))L2(Ω) dt,
(1.2.37)
By combining (1.2.35)â(1.2.37) we get
1
2âv(s)â2L2(Ω) +
1
2(B(0)âÏs(0),âÏs(0))L2(Ω)
=
â« s
0
[â1
2(BâÏs,âÏs)L2(Ω) + (pâv, Ïs)L2(Ω) + 2(v,div[Ïs â b])L2(Ω)
]dt.
(1.2.38)
Let us bound from above the scalar products in the right-hand side of (1.2.38). By theLipschitz regularity of B there is C > 0 such that âB(t)âLâ(Ω) †C for a.e. t â (0, T ), and soâŁâŁâŁâŁ â« s
0(B(t)âÏs(t),âÏs(t))L2(Ω) dt
âŁâŁâŁâŁ †C â« s
0âÏs(t)â2H1(Ω\Î0) dt. (1.2.39)
For every t â [0, T ] we split div[Ïs(t) â b(t)] into the sum Ïs(t) div b(t) + âÏs(t)b(t).As already pointed in (1.1.44) we have div b â Lip([0, T ];Lâ(Ω)), therefore we can repeatthe same argument as before. By integrating by parts with respect to time and using theequalities v0 = Ïs(s) = 0, we obtainâ« s
0(v(t), Ïs(t) div b(t))L2(Ω) dt = â
â« s
0(v(t), v(t) div b(t) + Ïs(t) div b(t))L2(Ω) dt
†C⫠s
0
[âv(t)â2L2(Ω) + âÏs(t)â2H1(Ω\Î0)
]dt,
(1.2.40)
for some C > 0 independent of s. For the other term, we first perform an integration byparts with respect to time exploiting the assumptions v0 = Ïs(s) = 0, and then we useformula (1.1.46) and the regularity properties (1.1.44) of b to deduceâ« s
0(v(t),âÏs(t)b(t))L2(Ω) dt =
â« s
0
[1
2(v(t) div b(t), v(t))L2(Ω) â (v(t),âÏs(t)b(t))L2(Ω)
]dt
†C⫠s
0
[âv(t)â2L2(Ω) + âÏs(t)â2H1(Ω\Î0)
]dt, (1.2.41)
for a constant C > 0 independent of s.We now split p as p1 + p2, where p1(t, y), p2(t, y) â L (RdĂd;Rd) are defined for t â [0, T ]
and y â Ω as
p1(t, y)Ο := p1(t, y)Ο â p1(0, y)Ο = p1(t, y)Ο + Οb(0, y) div Ί(0, y),
p2(t, y)Ο := p2(t, y)Ο + p1(0, y)Ο = âΟ[b(t, y) + b(0, y) div Ί(0, y)].
Chapter 1. Elastodynamics system in domains with growing cracks 19
We have p1 â Lip([0, T ];Lâ(Ω; L (Rd;RdĂd))), therefore Ëp1 â Lâ(0, T ;L2(Ω; L (Rd;RdĂd)))and there exists C > 0 such that â Ëp1(t)âLâ(Ω) †C for a.e. t â (0, T ). By integrating byparts with respect to time and by exploiting the equalities Ïs(s) = p1(0) = 0, we getâ« s
0(p1(t)âv(t), Ïs(t))L2(Ω) dt =
â« s
0(âÏs(t), p?1(t)Ïs(t))L2(Ω) dt
= ââ« s
0(âÏs(t), Ëp?1(t)Ïs(t) + p?1(t)Ïs(t))L2(Ω) dt
= ââ« s
0( Ëp1(t)âÏs(t), Ïs(t) + v(t))L2(Ω) dt
†C⫠s
0
[âv(t)â2L2(Ω) + âÏs(t)â2H1(Ω\Î0)
]dt,
(1.2.42)
for a constant C > 0 independent of s, where p?1 is the transpose operator of p1, defined ina similar way to (1.2.25). On the other hand, by (1.1.44) we have div b â Lâ(0, T ;L2(Ω))and there exits C > 0 such that âdiv b(t)âLâ(Ω) †C for a.e. t â (0, T ). Furthermore,
b(0) div Ί(0) â Lip(Ω;Rd) thanks to (H8). Hence, by performing an integration by partswith respect to the space variable we obtainâ« s
0(p2(t)âv(t), Ïs(t))L2(Ω) dt = â
â« s
0(âv(t), Ïs(t)â (b(t) + b(0) div Ί(0)))L2(Ω) dt
=
â« s
0(v(t),div[Ïs(t)â (b(t) + b(0) div Ί(0))])L2(Ω) dt
†C⫠s
0
[âv(t)â2L2(Ω) + âÏs(t)â2H1(Ω\Î0)
]dt,
(1.2.43)
for a constant C > 0 independent of s. To derive (1.2.43) we have used formula (1.1.45) withh := b(t)+b(0) div Ί(0). This is possible since the function b(t)+b(0) div Ί(0) âW 1,â(Ω;Rd)for a.e. t â (0, T ) and satisfies (b(t)+b(0) div Ί(0)) ·Μ = 0 on (Îâ©Î©)âȘâΩ. Indeed b(t) ·Μ = 0on (Π⩠Ω) âȘ âΩ for every t â [0, T ] by (1.1.3) and (H5), and 1
h [b(t + h) â b(t)] â b(t) inC0(Ω;Rd) for a.e. t â (0, T ) by (H7).
Thanks to (1.2.38), the coercivity property (1.2.1) of B, the upper bounds (1.2.39)â(1.2.43), and
âÏs(0)â2L2(Ω) †Tâ« s
0âv(t)â2L2(Ω) dt,
we conclude
âv(s)â2L2(Ω) + c0âÏs(0)â2H1(Ω\Î0) †Câ« s
0
[âv(t)â2L2(Ω) + âÏs(t)â2H1(Ω\Î0)
]dt (1.2.44)
for a constant C independent of the parameter s chosen. Let us consider
ζ(t) :=
â« t
0v(Ï) dÏ t â [0, T ].
Then we can write Ïs(t) = ζ(t)â ζ(s) for every t â [0, s]; in particular
âÏs(0)âH1(Ω\Î0) = âζ(s)âH1(Ω\Î0), (1.2.45)â« s
0âÏs(t)â2H1(Ω\Î0) dt †2sâζ(s)â2H1(Ω\Î0) + 2
â« s
0âζ(t)â2H1(Ω\Î0) dt. (1.2.46)
By combining (1.2.44)â(1.2.46), we obtain
âv(s)â2L2(Ω) + (c0 â 2Cs)âζ(s)â2H1(Ω\Î0) †2C
â« s
0
[âv(t)â2L2(Ω) + âζ(t)â2H1(Ω\Î0)
]dt.
20 1.3. Energy balance
If s is small enough, e.g., s †t0 := c04C
, we can apply Gronwallâs lemma and obtain
v(s) = 0 for every s â [0, t0].
Step 2. Notice that the functions v : [0, T ] â L2(Ω;Rd) and v : [0, T ] â Hâ1D (Ω \ Î0;Rd)
are continuous, therefore we can define
tâ := supt â [0, T ] : v(s) = 0 for every s â [0, t].
Thanks to Step 1 and the continuity of v and v, we get that tâ â„ t0 > 0 and v(tâ) = v(tâ) = 0.Let us assume by contradiction that tâ < T . By repeating the strategy adopted in Step 1,with starting point tâ and initial set Ω \ Îtâ , we can find a point t1 > tâ such that v(s) = 0for every s â [tâ, t1], which leads to a contradiction. Therefore tâ = T and so v(s) = 0 forevery s â [0, T ].
Remark 1.2.11. Let v be the generalized solution to system (1.1.28) with boundary condi-tions (1.1.33)â(1.1.35) and initial conditions (1.1.36), and let vΔ be its viscous approximationobtained by solving (1.2.7). By using (1.2.28), (1.2.34), and the weak lower semicontinuityof the norm, there is a constant C > 0, independent of t, such that
âv(t)â2L2(Ω) + âv(t)â2H1(Ω\Î0) †C for every t â [0, T ]. (1.2.47)
If we consider u defined by (1.1.27), by using formulas (1.1.49) it is immediate to check thatfor another constant C > 0, independent of t, we have
âu(t)â2L2(Ω) + âu(t)â2H1(Ω\Ît) †C for every t â [0, T ].
1.3 Energy balance
In this section, following [20, Proposition 3.11], we prove an energy equality for the generalizedsolution v to (1.1.28). In order to state this result, we introduce the following definition forthe energy: given a function z â C0
w([0, T ];H1(Ω \ Î0;Rd)), with distributional derivativez â C0
w([0, T ];L2(Ω;Rd)), we set
EB(z; t) :=1
2âz(t)â2L2(Ω) +
1
2(B(t)âz(t),âz(t))L2(Ω) for t â [0, T ], (1.3.1)
where B is the operator defined in (1.1.29).
Proposition 1.3.1. Under the assumptions of Theorem 1.2.9, let v be the unique generalizedsolution to (1.1.28) with boundary conditions (1.1.33)â(1.1.35), satisfying the initial condi-tions (1.1.36). Then the energy t 7â EB(v; t) is a continuous function from [0, T ] to R andsatisfies
EB(v; t) = EB(v; 0) +R(v; t) for every t â [0, T ], (1.3.2)
where the remainder R is defined as
R(v; t) :=
â« t
0
[1
2(Bâv,âv)L2(Ω) + (Bâv,âw)L2(Ω) â (pâv, v â w)L2(Ω)
]dt
+
â« t
0
[â(v div b, v)L2(Ω) + 2(v,div[w â b])L2(Ω) + (g, v â w)L2(Ω)
]dt
+
â« t
0
[â(F , v â w)L2(âNΩ) â (v, w)L2(Ω)
]dt+ (F (t), v(t)â w(t))L2(âNΩ)
+ (v(t), w(t))L2(Ω) â (F (0), v(0)â w(0))L2(âNΩ) â (v(0), w(0))L2(Ω).
Chapter 1. Elastodynamics system in domains with growing cracks 21
Remark 1.3.2. If the solution v were smooth enough, then (1.3.2) would be straightforwardby taking v â w as test function in (1.2.5). In our case, we follow the proof of [20, Proposi-tion 3.11] by approximating v â w with H1
D(Ω \ Î0;Rd)-valued functions, in the same spiritof [40, Chapter 3, Lemma 8.3].
Proof of Proposition 1.3.1. The function t 7â (F (t), v(t))L2(âNΩ) is continuous from [0, T ]
to R, since F â C0([0, T ];L2(âNΩ;Rd)) and v â C0w([0, T ];L2(âNΩ;Rd)), thanks to (1.1.1).
Similarly, also the function t 7â (v(t), w(t))L2Ω) is continuous from [0, T ] to R. Then, to provethat t 7â EB(v; t) is continuous, it is enough to show that equality (1.3.2) holds.
For t = 0 equality (1.3.2) is trivial. Let t0 â (0, T ] be fixed and let Ξ0 denote thecharacteristic function of the time interval (0, t0). For every ÎŽ > 0, we call ΞΎ : R â Rthe function equals to 1 in [ÎŽ, t0 â ÎŽ], 0 outside [0, t0], and which is linear in [0, ÎŽ] and[t0 â ÎŽ, t0]; notice that ΞΎ â ÎŽ0 in L1(R) as ÎŽ â 0+. We also consider a sequence of mollifiersÏmm â Câc (R).
We want to approximate the function Ξ0(v â w) : [0, T ] â H1D(Ω \ Î0;Rd) by a suitable
sequence of functions in Câc (R;H1D(Ω \ Î0;Rd)). To this aim, we first extend the function
Ξ0(v â w) to all R be setting Ξ0(v â w) = 0 outside [0, T ]. In a similar way we extend everyfunction multiplied by either Ξ0 or ΞΎ.
For brevity, we set z := v â w. In view of the above definitions, for every m and ÎŽ fixedwe have
Ïmâ(ΞΎz) â Câc (R;H1D(Ω \ Î0;Rd)),
since ΞΎz â Lâ(R;H1D(Ω \ Î0;Rd)) has compact support, and the regularity follows from
dk
dtk(Ïmâ(ΞΎz)) =
(dkÏmdtk
)â(ΞΎz).
Moreover, we have Ïmâ(ΞΎ z) â Câc (R;L2(Ω;Rd)) and
Ïmâ(ΞΎ z) = Ïmâ(ΞΎz)â Ïmâ(ΞΎz), (1.3.3)
which implies Ïm â (ΞΎ z) â Câc (R;H1D(Ω \ Î0;Rd)). In a similar way, we can deduce that
Ïmâ(ΞΎ z) â Câc (R;L2(Ω;Rd)), since Ïmâ(ΞΎ z) â Câc (R;Hâ1D (Ω \ Î0;Rd)) and
Ïmâ(ΞΎ z) = Ïmâ(ΞΎ z)â Ïmâ(ΞΎ z). (1.3.4)
Since Ïmâ(ΞΎ z) â Câc (R;L2(Ω;Rd)), we deriveâ«R
d
dtâÏmâ(ΞΎ z)â2L2(Ω) dt = 0.
By differentiating the integrand and exploiting the properties of the convolution, we get
0 =
â«R
[(Ïmâ(ΞΎ z), Ïmâ(ΞΎ z))L2(Ω) + (Ïmâ(ΞΎ z), Ïmâ(ΞΎ z))L2(Ω)
]dt, (1.3.5)
being Ïmâ(ΞΎ z) well defined in L2(R;L2(Ω;Rd)). Let us study separately the behavior of eachterm in (1.3.5) as ÎŽ â 0+, keeping m fixed. For the first one we have
limÎŽâ0+
â«R
(Ïmâ(ΞΎ z), Ïmâ(ΞΎ z))L2(Ω) dt
= limÎŽâ0+
â«RΞΎ(z, ÏmâÏmâ(Ξ0z))L2(Ω) dt
= â(z(t0), (ÏmâÏmâ(Ξ0z))(t0))L2(Ω) + (z(0), (ÏmâÏmâ(Ξ0z))(0))L2(Ω).
(1.3.6)
22 1.3. Energy balance
To obtain (1.3.6), we have split ΞΎ as (ΞΎ â Ξ0) + Ξ0 and used the following facts:
Ïmâ(ΞΎ z)â Ïmâ(Ξ0z) in L2(R;L2(Ω;Rd)) as ÎŽ â 0+,
Ïmâ(ΞΎ z)ÎŽ is uniformly bounded in L2(R;L2(Ω;Rd)),s 7â (z(s), (ÏmâÏmâ(Ξ0z))(s))L2(Ω) is continuous on R.
The last property is a consequence of the fact that Ïm âÏm â(Ξ0z) â C0(R;L2(Ω;Rd)) andz â C0
w(R;L2(Ω;Rd)) (see Remark 1.2.7). The second term of (1.3.5) satisfies
limÎŽâ0+
â«R
(Ïmâ(ΞΎ z), Ïmâ(ΞΎ z))L2(Ω) dt =
â«R
(Ïmâ(Ξ0z), Ïmâ(Ξ0z))L2(Ω) dt. (1.3.7)
Indeed, the sequence Ïmâ(ΞΎ z)ÎŽ is uniformly bounded in L2(R;L2(Ω;Rd)) by (1.3.4) andÏmâ(ΞΎ z) converges strongly to Ïmâ(Ξ0z) in L2(R;Hâ1
D (Ω\Î0;Rd)) as ÎŽ â 0+. These two factsimply that Ïmâ(Ξ0z) is an element of L2(R;L2(Ω;Rd)) and Ïmâ(ΞΎ z) converges to Ïmâ(Ξ0z)weakly in L2(R;L2(Ω;Rd)) as ÎŽ â 0+. By combining (1.3.5)â(1.3.7), we getâ«
R(Ïmâ(Ξ0z), Ïmâ(Ξ0z))L2(Ω) dt
= (z(t0), (ÏmâÏmâ(Ξ0z))(t0))L2(Ω) â (z(0), (ÏmâÏmâ(Ξ0z))(0))L2(Ω).
In a similar way, we can proveâ«R
(Ïmâ(Ξ0w), Ïmâ(Ξ0w))L2(Ω) dt
= (w(t0), (ÏmâÏmâ(Ξ0w))(t0))L2(Ω) â (w(0), (ÏmâÏmâ(Ξ0w))(0))L2(Ω).
From the last two identities we deduceâ«R
(Ïmâ(Ξ0v), Ïmâ(Ξ0z))L2(Ω) dt
= (v(t0), (ÏmâÏmâ(Ξ0v))(t0))L2(Ω) â (v(0), (ÏmâÏmâ(Ξ0v))(0))L2(Ω)
â ((ÏmâÏmâ(Ξ0v))(t0), w(t0))L2(Ω) â (v(t0), (ÏmâÏmâ(Ξ0w))(t0))L2(Ω)
+ ((ÏmâÏmâ(Ξ0v))(0), w(0))L2(Ω) + (v(0), (ÏmâÏmâ(Ξ0w))(0))L2(Ω)
+
â«R
(Ïmâ(Ξ0v), Ïmâ(Ξ0w))L2(Ω) dt.
(1.3.8)
We apply the same argument to the function (BÏmâ(ΞΎâv), Ïmâ(ΞΎâv))L2(Ω) âW 1,â(R),which has compact support. Starting from the identityâ«
R
d
dt(BÏmâ(ΞΎâv), Ïmâ(ΞΎâv))L2(Ω) dt = 0,
we infer
0 =
â«R
[(BÏmâ(ΞΎâv), Ïmâ(ΞΎâv))L2(Ω) + 2(Ïmâ(ΞΎBâv), Ïmâ(ΞΎâv))L2(Ω)
]dt
+
â«R
2(Ïmâ(ΞΎBâv), Ïmâ(ΞΎâv))L2(Ω) dt
+
â«R
2(BÏmâ(ΞΎâv)â Ïmâ(ΞΎBâv), Ïmâ(ΞΎâv))L2(Ω) dt,
(1.3.9)
where Ïmâ(ΞΎâv) is well defined in L2(R;L2(Ω;RdĂd)) as
Ïmâ(ΞΎâv) := â(Ïmâ(ΞΎ v)) = Ïmâ(ΞΎâv)â Ïmâ(ΞΎâv). (1.3.10)
Chapter 1. Elastodynamics system in domains with growing cracks 23
Since Ïmâ(ΞΎ v) is uniformly bounded in L2(R;H1D(Ω \ Î0;Rd)) by (1.3.3), and it converges
strongly to Ïmâ(Ξ0v) in L2(R;L2(Ω;Rd)) as ÎŽ â 0+, we get
Ïmâ(ΞΎâv) Ïmâ(Ξ0âv) := â(Ïmâ(Ξ0v)) in L2(R;L2(Ω;RdĂd)) as ÎŽ â 0+. (1.3.11)
Moreover, the operator B â Lip(R;Lâ(Ω; L (RdĂd,RdĂd))), which yields that B is an elementof Lâ(R;L2(Ω; L (RdĂd,RdĂd))) and there exists C > 0 such that âB(s)âLâ(Ω) †C fora.e. s â R. By passing to the limit as ÎŽ â 0+ in (1.3.9), thanks to (1.3.11) and the properties:
BÏmâ(ΞΎâv)â BÏmâ(Ξ0âv) in L2(R;L2(Ω;RdĂd)) as ÎŽ â 0+,
BÏmâ(ΞΎâv)â BÏmâ(Ξ0âv) in L2(R;L2(Ω;RdĂd)) as ÎŽ â 0+,
Ïmâ(ΞΎBâv)â Ïmâ(Ξ0Bâv) in L2(R;L2(Ω;RdĂd)) as ÎŽ â 0+,
Ïmâ(ΞΎâv)â Ïmâ(Ξ0âv) in L2(R;L2(Ω;RdĂd)) as ÎŽ â 0+,
Ïmâ(ΞΎâv)â Ïmâ(Ξ0âv) in L2(R;L2(Ω;RdĂd)) as ÎŽ â 0+,
Ïmâ(ΞΎâv)ÎŽ is uniformly bounded in L2(R;L2(Ω;RdĂd)),s 7â ((ÏmâÏmâ(Ξ0Bâv))(s),âv(s))L2(Ω) is continuous on R,
we obtain the following identityâ«R
(Ïmâ(Ξ0Bâv), Ïmâ(Ξ0âz))L2(Ω) dt
= ((ÏmâÏmâ(Ξ0Bâv))(t0),âv(t0))L2(Ω) â ((ÏmâÏmâ(Ξ0Bâv))(0),âv(0))L2(Ω)
ââ«R
[1
2(BÏmâ(Ξ0âv), Ïmâ(Ξ0âv))L2(Ω) + (Ïmâ(Ξ0Bâv), Ïmâ(Ξ0âw))L2(Ω)
]dt
+
â«R
(Ïmâ(Ξ0Bâv)â BÏmâ(Ξ0âv), Ïmâ(Ξ0âv))L2(Ω) dt.
(1.3.12)
Let us consider the function (Ïmâ(ΞΎF ), Ïmâ(ΞΎz))L2(âNΩ) â Câc (R). We haveâ«R
d
dt(Ïmâ(ΞΎF ), Ïmâ(ΞΎz))L2(âNΩ) dt = 0,
which implies
0 =
â«R
[(Ïmâ(ΞΎF ), Ïmâ(ΞΎz))L2(âNΩ) + (Ïmâ(ΞΎF ), Ïmâ(ΞΎz))L2(âNΩ)
]dt
+
â«R
[(Ïmâ(ΞΎF ), Ïmâ(ΞΎz))L2(âNΩ) + (Ïmâ(ΞΎF ), Ïmâ(ΞΎ z))L2(âNΩ)
]dt,
being Ïmâ(ΞΎ z) well defined in L2(R;L2(âNΩ;Rd)) (see (1.1.1) and (1.3.3)). By the followingfacts:
Ïmâ(ΞΎF )â Ïmâ(Ξ0F ) in L2(R;L2(âNΩ;Rd)) as ÎŽ â 0+,
Ïmâ(ΞΎF )â Ïmâ(Ξ0F ) in L2(R;L2(âNΩ;Rd)) as ÎŽ â 0+,
Ïmâ(ΞΎz)â Ïmâ(Ξ0z) in L2(R;L2(âNΩ;Rd)) as ÎŽ â 0+,
Ïmâ(ΞΎ z) Ïmâ(Ξ0z) in L2(R;L2(âNΩ;Rd)) as ÎŽ â 0+,
Ïmâ(ΞΎF )ÎŽ is uniformly bounded in L2(R;L2(âNΩ;Rd)),Ïmâ(ΞΎz)ÎŽ is uniformly bounded in L2(R;L2(âNΩ;Rd)),s 7â (F (s), (ÏmâÏmâ(Ξ0z))(s))L2(âNΩ) is continuous on R,s 7â ((ÏmâÏmâ(Ξ0F ))(s), z(s))L2(âNΩ) is continuous on R,
24 1.3. Energy balance
as ÎŽ â 0+ we getâ«R
(Ïmâ(Ξ0F ), Ïmâ(Ξ0z))L2(âNΩ) dt
= ((ÏmâÏmâ(Ξ0F ))(t0), z(t0))L2(âNΩ) + (F (t0), (ÏmâÏmâ(Ξ0z))(t0))L2(âNΩ)
â ((ÏmâÏmâ(Ξ0F ))(0), z(0))L2(âNΩ) â (F (0), (ÏmâÏmâ(Ξ0z))(0))L2(âNΩ)
ââ«R
(Ïmâ(Ξ0F ), Ïmâ(Ξ0z))L2(âNΩ) dt.
(1.3.13)
We know that the function v solvesâ«R
[ăΞ0v, ÏăHâ1
D (Ω\Î0) + (Ξ0Bâv,âÏ)L2(Ω) + (Ξ0pâv, Ï)L2(Ω) + 2(Ξ0v,div[Ïâ b])L2(Ω)
]dt
=
â«R
[(Ξ0g, Ï)L2(Ω) + (Ξ0F,Ï)L2(âNΩ)
]dt
for every Ï â L2(R;H1D(Ω \ Î0;Rd)) (see Remark 1.2.6). In particular, by considering the
function Ï := Ïmâ(Ïmâ(Ξ0z)), which belongs to L2(R;H1D(Ω \Î0;Rd)) thanks to (1.3.3), and
exploiting the properties of the convolution, we obtainâ«R
[(Ïmâ(Ξ0v), Ïmâ(Ξ0z))L2(Ω) + (Ïmâ(Ξ0Bâv), Ïmâ(Ξ0âz))L2(Ω)
]dt
+
â«R
[(Ïmâ(Ξ0pâv), Ïmâ(Ξ0z))L2(Ω) + 2(Ïmâ(Ξ0v â b), Ïmâ(Ξ0âz))L2(Ω)
]dt
+
â«R
2(Ïmâ(Ξ0v div b), Ïmâ(Ξ0z))L2(Ω) dt
=
â«R
[(Ïmâ(Ξ0g), Ïmâ(Ξ0z))L2(Ω) + (Ïmâ(Ξ0F ), Ïmâ(Ξ0z))L2(âNΩ)
]dt.
(1.3.14)
We combine (1.3.8) and (1.3.12)â(1.3.14) to derive the following identity
(v(t0), (ÏmâÏmâ(Ξ0v))(t0))L2(Ω) + ((ÏmâÏmâ(Ξ0Bâv))(t0),âv(t0))L2(Ω)
â (v(0), (ÏmâÏmâ(Ξ0v))(0))L2(Ω) â ((ÏmâÏmâ(Ξ0Bâv))(0),âv(0))L2(Ω) = Rm(t0),(1.3.15)
where
Rm(t0) :=
â«R
[1
2(BÏmâ(Ξ0âv), Ïmâ(Ξ0âv))L2(Ω) + (Ïmâ(Ξ0Bâv), Ïmâ(Ξ0âw))L2(Ω)
]dt
+
â«R
(Ïmâ(Ξ0g)â Ïmâ(Ξ0pâv)â 2Ïmâ(Ξ0v div b), Ïmâ(Ξ0(v â w)))L2(Ω) dt
+
â«R
(BÏmâ(Ξ0âv)â Ïmâ(Ξ0Bâv), Ïmâ(Ξ0âv))L2(Ω) dt
ââ«R
2(Ïmâ(Ξ0v â b), Ïmâ(Ξ0(âv ââw)))L2(Ω) dt
ââ«R
[(Ïmâ(Ξ0F ), Ïmâ(Ξ0(v â w)))L2(âNΩ) + (Ïmâ(Ξ0v), Ïmâ(Ξ0w))L2(Ω)
]dt
+ ((ÏmâÏmâ(Ξ0F ))(t0), v(t0)â w(t0))L2(âNΩ) + ((ÏmâÏmâ(Ξ0v))(t0), w(t0))L2(Ω)
+ (F (t0), (ÏmâÏmâ(Ξ0(v â w)))(t0))L2(âNΩ) + (v(t0), (ÏmâÏmâ(Ξ0w))(t0))L2(Ω)
â ((ÏmâÏmâ(Ξ0F ))(0), v(0)â w(0))L2(âNΩ) â ((ÏmâÏmâ(Ξ0v))(0), w(0))L2(Ω)
â (F (0), (ÏmâÏmâ(Ξ0(v â w)))(0))L2(âNΩ) â (v(0), (ÏmâÏmâ(Ξ0w))(0))L2(Ω).
Chapter 1. Elastodynamics system in domains with growing cracks 25
Let us now perform the second passage to the limit: we let the index m associatedto the convolution Ïm tend to â. Let us study separately the asymptotic of the termsappearing (1.3.15). The left-hand side converges to
1
2âv(t0)â2L2(Ω) +
1
2(B(t0)âv(t0),âv(t0))L2(Ω) â
1
2âv(0)â2L2(Ω) â
1
2(B(0)âv(0),âv(0))L2(Ω).
Here we have used the weak continuity of v and âv (see Remark 1.2.7), the presence of Ξ0,and the fact that ÏmâÏm is still a smooth even mollifier. Similarly, the last eight terms ofRm(t0) converge to
(F (t0), v(t0))L2(âNΩ) + (v(t0), w(t0))L2(âNΩ) â (F (0), v0)L2(âNΩ) â (v1, w(0))L2(âNΩ).
By the strong approximation property of the convolution and the dominated convergencetheorem, it is easy check the following convergences:
limmââ
â«R
(BÏmâ(Ξ0âv), Ïmâ(Ξ0âv))L2(Ω) dt =
â« t0
0(Bâv,âv)L2(Ω) dt,
limmââ
â«R
(Ïmâ(Ξ0Bâv), Ïmâ(Ξ0âw))L2(Ω) dt =
â« t0
0(Bâv,âw)L2(Ω) dt,
limmââ
â«R
(Ïmâ(Ξ0g), Ïmâ(Ξ0(v â w))L2(Ω) dt =
â« t0
0(g, v â w)L2(Ω) dt,
limmââ
â«R
(Ïmâ(Ξ0pâv), Ïmâ(Ξ0(v â w)))L2(Ω) dt =
â« t0
0(pâv, v â w)L2(Ω) dt,
limmââ
â«R
(Ïmâ(Ξ0v div b), Ïmâ(Ξ0v))L2(Ω) dt =
â« t0
0(v div b, v â w)L2(Ω) dt,
limmââ
â«R
(Ïmâ(Ξ0v â b), Ïmâ(Ξ0âw))L2(Ω) dt =
â« t0
0(v â b,âw)L2(Ω) dt,
limmââ
â«R
(Ïmâ(Ξ0F ), Ïmâ(Ξ0(v â w)))L2(âNΩ) dt =
â« t0
0(F , v â w)L2(âNΩ) dt,
limmââ
â«R
(Ïmâ(Ξ0v), Ïmâ(Ξ0w))L2(Ω) dt =
â« t0
0(v, w)L2(Ω) dt.
For the remaining two terms of Rm(t0) we claim
limmââ
â«R
(Ïmâ(Ξ0v â b), Ïmâ(Ξ0âv))L2(Ω) dt = â1
2
â« t0
0(v div b, v)L2(Ω) dt, (1.3.16)
limmââ
â«R
(BÏmâ(Ξ0âv)â Ïmâ(Ξ0Bâv), Ïmâ(Ξ0âv))L2(Ω) dt = 0. (1.3.17)
Once proved the claim we are done: indeed, by combining the previous convergences withidentity (1.3.15) we get (1.3.2).
For simplicity we set
ζm := Ïmâ(Ξ0v â b)â Ïmâ(Ξ0v)â b, Ïm := Ïmâ(Ξ0v).
Hence, we may rephrase the integral in (1.3.16) asâ«R
(Ïmâ(Ξ0v â b), Ïmâ(Ξ0âv))L2(Ω) dt
=
â«R
[(Ïm â b,âÏm)L2(Ω) + (ζm,âÏm)L2(Ω)
]dt.
(1.3.18)
26 1.3. Energy balance
By integrating by parts (recall that b satisfies b · Îœ = 0 on âΩ âȘ Î0) we get
limmââ
â«R
(Ïm â b,âÏm)L2(Ω) = â1
2limmââ
â«R
(Ïm div b, Ïm)L2(Ω) = â1
2
â« t0
0(v div b, v)L2(Ω) dt,
since Ïm â Ξ0v in L2(R;L2(Ω;Rd)) as mââ. Therefore we have to prove that the secondterm in the right-hand side of (1.3.18) vanishes as mââ. Notice that (1.3.10) and (1.3.11)imply
âÏm(t) = (Ïmâ(Ξ0âv))(t) + Ïm(tâ t0)âv(t0)â Ïm(t)âv(0) for t â [0, T ].
Then, we may writeâ«R
(ζm(t),âÏm(t))L2(Ω) dt
= ââ«R
(ζm(t), (Ïmâ(Ξ0âv))(t))L2(Ω) dt+
â«R
(ζm(t), Ïm(tâ t0)âv(t0)â Ïm(t)âv(0))L2(Ω) dt.
Since Ïm and Ξ0 have compact support, for m large enough ζm and ζm are identically zerooutside the interval I = (â2T, 2T ). Clearly ζm â 0 in L2(I;L2(Ω;RdĂd)) as m â â, andlet us check that ζm converges to zero weakly in L2(I;L2(Ω;RdĂd)) as mââ. By (H7) weknow that b â Lip(I;Lâ(Ω;Rd)), so that b â Lâ(I;L2(Ω;Rd)) and there exists C > 0 suchthat âb(t)âLâ(Ω) †C for a.e. t â I. Therefore, for a.e. t â I, we may write
ζm(t) = (Ïmâ(Ξ0v â b))(t)â (Ïmâ(Ξ0v))(t)â b(t)â (Ïmâ(Ξ0v))(t)â b(t)
=
â«RÏm(tâ s)Ξ0(s)v(s)â [b(s)â b(t)] dsâ
â«RÏm(tâ s)Ξ0(s)v(s)â b(t)ds
=
â« t0
0Ïm(tâ s)v(s)â [b(s)â b(t)]
(sâ t)(sâ t) dsâ
â« t0
0Ïm(tâ s)v(s)â b(t) ds.
We use the Lâ-boundedness of t 7â âv(t)âL2(Ω), the previous properties of b, and the boundsâ«R|sÏm(s)| ds <â,
â«RÏm(s) ds <â,
to deduce that ζmm is uniformly bounded in L2(I;L2(Ω;RdĂd)). This gives that ζm 0in L2(I;L2(Ω;RdĂd)) as m â â, since ζm converges to 0 strongly in L2(I;L2(Ω;RdĂd)) asmââ. Hence
limmââ
â«R
(ζm, Ïmâ(Ξ0âv))L2(Ω) = 0.
Since the embedding H1(I;L2(Ω;RdĂd)) â C0(I;L2(Ω;RdĂd)) is continuous, the sequenceζmm is bounded in C0(I;L2(Ω;RdĂd)), and
âζm(t1)â ζm(t2)âL2(Ω) †âζmâL2(Ω)|t1 â t2|1/2 †C|t1 â t2|1/2 for every t1, t2 â I,
with C > 0 independent of t1 and t2. Then the sequence âζmâL2(Ω) : I â R, m â N,is equibounded and equicontinuos. By Ascoli-Arzelaâs theorem we get that ζm convergesstrongly to zero in C0(I;L2(Ω;RdĂd)), since âζmâL2(Ω) â 0 in L2(I). Notice that the function
t 7â Ïm(tâ t0)âv(t0)â Ïm(t)âv(0) is bounded in L1(I;L2(Ω;RdĂd)), hence
limmââ
â«R
(ζm(t), Ïm(tâ t0)âv(t0)â Ïm(t)âv(0))L2(Ω) dt = 0.
Similarly, by defining
Ïm := BÏmâ(Ξ0âv)â Ïmâ(Ξ0Bâv),
Chapter 1. Elastodynamics system in domains with growing cracks 27
we may writeâ«R
(BÏmâ(Ξ0âv)â Ïmâ(Ξ0Bâv), Ïmâ(Ξ0âv))L2(Ω) dt = ââ«R
(Ïm, Ïmâ(Ξ0âv))L2(Ω) dt.
As before, Ïm is identically zero outside of I, Ïm â 0 in L2(I;L2(Ω;RdĂd)), and the sequenceÏmm is bounded in L2(I;L2(Ω;RdĂd)) thanks to the Lipschitz regularity of B. HenceÏm 0 in L2(I;L2(Ω;RdĂd)) and (1.3.17) holds.
This concludes the proof of formula (1.3.2) and implies the desired continuity of the mapt 7â EB(v; t) in [0, T ].
We are now in a position to prove Theorem 1.2.2.
Proof of Theorem 1.2.2. In view of Theorems 1.2.9 and 1.2.10, we know that problem (1.1.28)with boundary and initial conditions (1.1.33)â(1.1.36) admits a unique generalized solution v(cf. Definition 1.2.5). Hence, to conclude the proof, it is enough to show that that everygeneralized solution v is indeed a weak solution (cf. Definition 1.1.9), more precisely itsatisfies (1.1.38)â(1.1.41).
As pointed out in Remark 1.2.7, v â C0([0, T ];L2(Ω;Rd)) â© C0w([0, T ];H1(Ω \ Î0;Rd)),
while v â C0([0, T ];Hâ1D (Ω \ Î0;Rd)) â© C0
w([0, T ];L2(Ω;Rd)). In addition, t 7â EB(v; t) is acontinuous function from [0, T ] to R, thanks to Proposition 1.3.1. Let us now prove thatt 7â âv(t) and t 7â v(t) are strongly continuous from [0, T ] to L2(Ω;RdĂd) and L2(Ω;Rd),respectively.
Let t0 â [0, T ] be fixed and let tkk â [0, T ] be a sequence of points converging to t0.Since v is weakly continuous, we have
âv(t0)â2L2(Ω) †lim infkââ
âv(tk)â2L2(Ω).
Moreover, condition (1.2.1) implies that (B(t0)âÏ,âÏ)L2(Ω)+c1âÏâ2L2(Ω), Ï â H1D(Ω\Î0;Rd),
is an equivalent norm on H1D(Ω\Î0;Rd). Hence, since z := vâw â C0
w([0, T ];H1D(Ω\Î0;Rd)),
we have
(B(t0)âz(t0),âz(t0))L2(Ω) + c1âz(t0)â2L2(Ω)
†lim infkââ
[(B(t0)âz(tk),âz(tk))L2(Ω) + c1âz(tk)â2L2(Ω)
]= lim inf
kââ(B(t0)âz(tk),âz(tk))L2(Ω) + c1âz(t0)â2L2(Ω),
thanks to the strong continuity and the weak continuity of z in L2(Ω;Rd) and H1(Ω\Î0;Rd),respectively. In particular, we can use (1.1.11) to derive
(B(t0)âv(t0),âv(t0))L2(Ω) †lim infkââ
(B(t0)âv(tk),âv(tk))L2(Ω).
Moreover, by the strong continuity of t 7â B(t) from [0, T ] to Lâ(Ω; L (RdĂd;RdĂd)) and thebound (1.2.47) we get
(B(t0)âv(t0),âv(t0))L2(Ω)
†lim infkââ
[(B(tk)âv(tk),âv(tk))L2(Ω) + ((B(t0)â B(tk))âv(tk),âv(tk))L2(Ω)
]†lim inf
kââ(B(tk)âv(tk),âv(tk))L2(Ω) + C lim
kâââB(t0)â B(tk)âLâ(Ω)
= lim infkââ
(B(tk)âv(tk),âv(tk))L2(Ω).
28 1.4. Continuous dependence on the data
Then
EB(v; t0) †1
2lim infkââ
âv(tk)â2L2(Ω) +1
2lim infkââ
(B(tk)âv(tk),âv(tk))L2(Ω)
†limkââ
EB(v; tk) = EB(v; t0),
which implies the continuity of t 7â âv(t)â2L2(Ω) and t 7â (B(t)âv(t),âv(t))L2(Ω) in t0 â [0, T ].
Thus v andâv are strongly continuous from [0, T ] to L2(Ω;Rd) and L2(Ω;RdĂd), respectively.Therefore, properties (1.1.38)â(1.1.40) are readily verified. Finally, since both v and v areelements of L2(0, T ;Hâ1
D (Ω \ Î0;Rd)), we infer that v â H1(0, T ;Hâ1D (Ω \ Î0;Rd)) which is
contained in AC([0, T ];Hâ1D (Ω \ Î0;Rd)). This gives (1.1.41) and concludes the proof.
1.4 Continuous dependence on the data
In this section, following the same procedure adopted in [20, Theorem 4.1], we use the en-ergy equality (1.3.2) to obtain a continuous dependence result on the data, both for prob-lem (1.1.14) with boundary and initial conditions (1.1.15)â(1.1.18), and problem (1.1.28) withboundary and initial conditions (1.1.33)â(1.1.36).
The initial crack Î0 is kept fixed. For every n â N we consider a family of closed setsÎnt tâ[0,T ] and a complete (d â 1)-dimensional C2 manifold În satisfying (H1)â(H8) withdiffeomorphisms Κn and Ίn, and we assume
În0 = Î0 for every n â N. (1.4.1)
Moreover, we consider a sequence An of tensor fields, fn of source terms, wn of Dirichletboundary data, Fn of Neumann boundary data, and (u0,n, u1,n) of initial data. The conver-gences of the corresponding solutions will be obtained under the assumptions detailed in thefollowing theorem.
Theorem 1.4.1. Assume that Î, Îttâ[0,T ], Ί, Κ satisfy (H1)â(H8). Let us consider a ten-sor field A which satisfies (1.1.8)â(1.1.10) and such that the transformed operator B satisfiesthe ellipticity condition (1.2.1). Let us also consider f , w, F , u0, and u1 satisfying (1.1.11)â(1.1.13). Assume that În, Înt tâ[0,T ], Ίn, Κn satisfying (H1)â(H8) and condition (1.4.1)for every n â N. Let us consider a sequence of tensor fields An which satisfy (1.1.8)â(1.1.10)for every n â N and such that the operators Bn, constructed starting from An, Ίn, and Κn,satisfy the ellipticity condition (1.2.1) with constants c0 and c1 independent of n. Let usconsider fn, wn, Fn, u0,n, and u1,n satisfying (1.1.11)â(1.1.13) for every n â N.
We assume there exists of a constant C > 0 such that every n â N and s, t â [0, T ]
âΊn(t)â Ίn(s)âLâ(Ω) †C|tâ s|, âΊn(t)â Ίn(s)âLâ(Ω) †C|tâ s|, (1.4.2)
ââiΊn(t)â âiΊn(s)âLâ(Ω) †C|tâ s|, ââ2ijΊ
n(t)âLâ(Ω) †C, (1.4.3)
âAn(t)â An(s)âLâ(Ω) †C|tâ s|, ââiAn(t)âLâ(Ω) †C, (1.4.4)
for every i, j = 1, . . . , d. Moreover, we assume the following convergences as nââ:
Ίn(t)â Ί(t) in L2(Ω;Rd), âiΊn(t)â âiΊ(t) in L2(Ω;Rd), (1.4.5)
Ίn(t)â Ί(t) in L2(Ω;Rd), â2ijΊ
n(t)â â2ijΊ(t) in L2(Ω;Rd), (1.4.6)
An(t)â A(t) in L2(Ω; L (RdĂd;RdĂd)), (1.4.7)
âiAn(t)â âiA(t) in L2(Ω; L (RdĂd;RdĂd)), (1.4.8)
An(t)â A(t) in L2(Ω; L (RdĂd;RdĂd)), (1.4.9)
wn â w in H2(0, T ;L2(Ω;Rd)) â©H1(0, T ;H1(Ω \ Î0;Rd)), (1.4.10)
Chapter 1. Elastodynamics system in domains with growing cracks 29
fn â f in L2(0, T ;L2(Ω;Rd)), Fn â F in H1(0, T ;L2(âNΩ;Rd)), (1.4.11)
u0,n â u0 in H1(Ω \ Î0;Rd), u1,n â u1 in L2(Ω;Rd) (1.4.12)
for a.e. t â (0, T ) and for every i, j = 1, . . . , d. Finally, we assume that (1.4.2), (1.4.3),(1.4.5), and (1.4.6) are true also for the sequence Κn with limit Κ.
For every n â N let un be the weak solution to problem (1.1.14) with growing crack Înt ,forcing term fn, boundary conditions (1.1.15)â(1.1.17) with w, F , and Ît replaced by wn,Fn and Înt , respectively, and initial data (u0,n, u1,n). Similarly, let vn be the weak solutionto (1.1.28) with boundary and conditions (1.1.33)â(1.1.36), where the coefficients (1.1.29)â(1.1.32) and the initial data (1.1.37) are constructed starting from Ίn,Κn,An, fn, u0,n, u1,n.Let u and v be the weak solutions to problem (1.1.14) with boundary and initial condi-tions (1.1.15)â(1.1.18) and problem (1.1.28) with boundary and initial conditions (1.1.33)â(1.1.36), respectively. Under the previous assumptions, for every t â [0, T ] as n â â wehave:
un(t)â u(t) in L2(Ω;Rd), âun(t)â âu(t) in L2(Ω;RdĂd), (1.4.13)
un(t)â u(t) in L2(Ω;Rd), (1.4.14)
vn(t)â v(t) in H1(Ω \ Î0;Rd), vn(t)â v(t) in L2(Ω;Rd). (1.4.15)
Remark 1.4.2. In the continuous dependence result of [8], both the initial crack and theDirichlet datum are fixed. In this thesis, we consider also the case of a sequence of Dirichletdata wn converging to w.
Remark 1.4.3. Since Ίn(0) = id for every n â N, assumption (1.4.5) implies
Ίn(t)â Ί(t), âiΊn(t)â âiΊ(t) in L2(Ω;Rd) as nââ (1.4.16)
for every t â [0, T ] and i = 1, . . . , d. Moreover, by (1.4.6)â(1.4.9) we also have
Ίn(t)â Ί(t) in L2(Ω;Rd), An(t)â A(t) in L2(Ω; L (RdĂd;RdĂd)) as nââ(1.4.17)
for every t â [0, T ]. Finally, we have â detâΊn(t)âLâ(Ω) †C and â detâΚn(t)âLâ(Ω) †C fora constant C > 0 independent of n and t. Thus there exists a constant ÎŽ0 > 0, independentof n, such that
detâΊn(t, y) â„ ÎŽ0, detâΚn(t, x) â„ ÎŽ0 for every t â [0, T ] and x, y â Ω. (1.4.18)
Proof of Theorem 1.4.1. We follow the lines of the proof of [20, Theorem 4.1]. As explained inthe quoted paper, the statement for the sequence unn is a consequence of the one for vnn.Indeed, let t â [0, T ] be fixed and let us assume that (1.4.15) is satisfied. By (1.1.27), (1.1.49),and the bounds (1.4.2) and (1.4.3) on the diffeomorphisms, we deduce that âun(t)n,un(t)n, and un(t)n are uniformly bounded in L2(Ω;RdĂd), L2(Ω;Rd), and L2(Ω;Rd),respectively. In particular, up to a subsequence, âun(t), un(t), and un(t) converge weakly inthese spaces. To determine the weak limits we fix a smooth function Ï â Câc (Ω;RdĂd). Bythe change of variable formula (1.1.49), we have
limnââ
(âun(t), Ï)L2(Ω) = limnââ
(âvn(t)âΚn(t,Ίn(t)), Ï(Ίn(t)) detâΊn(t))L2(Ω)
= (âv(t)âΚ(t,Ί(t)), Ï(Ί(t)) detâΊ(t))L2(Ω) = (âu(t), Ï)L2(Ω).
Hence âun(t) converges weakly to âu(t) in L2(Ω;RdĂd). Similarly, by using the conver-gences (1.4.15) and (1.4.16) we obtain that ââun(t)âL2(Ω) converges to ââu(t)âL2(Ω) as
nââ. Then âun(t)â âu(t) in L2(Ω;RdĂd), and the same argument applies to un(t) and
30 1.4. Continuous dependence on the data
un(t), which converge strongly in L2(Ω;Rd) to u(t) and u(t), respectively. This gives (1.4.13)and (1.4.14), since these limits do not depend on the subsequence.
Let us denote by Bn, pn, bn, gn, v0,n, and v1,n the coefficients of the system (1.1.28)constructed starting from Ίn, Κn, An, fn, u0,n, and u1,n. In view of (1.4.2)â(1.4.4) it is easyto check that for every n â N and i = 1, . . . , d
âBn(t)â Bn(s)âLâ(Ω) †C|tâ s|, âbn(t)â bn(s)âLâ(Ω) †C|tâ s| (1.4.19)
for every t, s â [0, T ], while
ââiBn(t)âLâ(Ω) †C, ââibn(t)âLâ(Ω) †C, âpn(t)âLâ(Ω) †C (1.4.20)
for a.e. t â (0, T ), where C is a constant independent of t, s, n, and i. Furthermore, theconvergences (1.4.5)â(1.4.9), the lower bounds (1.4.18), and Lemma 1.1.2 imply as nââ
Bn(t)â B(t) in L2(Ω; L (RdĂd;RdĂd)) for a.e. t â (0, T ), (1.4.21)
pn(t)â p(t) in L2(Ω; L (RdĂd;Rd)) for a.e. t â (0, T ), (1.4.22)
âibn(t)â âib(t) in L2(Ω) for a.e. t â (0, T ) (1.4.23)
for every i = 1, . . . , d. By using (1.4.19), (1.4.20), and Ascoli-Arzelaâs theorem, we also inferas nââ
Bn(t)â B(t) in C0(Ω; L (RdĂd;RdĂd)) for a.e. t â (0, T ), (1.4.24)
bn(t)â b(t) in C0(Ω;Rd) for a.e. t â (0, T ). (1.4.25)
Finally, by (1.4.11) and (1.4.12) we obtain as nââ
gn â g in L2(0, T ;L2(Ω;Rd)), (1.4.26)
v0,n â v0 in H1(Ω \ Î0;Rd), v1,n â v1 in L2(Ω;Rd). (1.4.27)
In order to prove the validity of (1.4.15), for every Δ â (0, 1) we consider the solution vΔ tothe perturbed problem (1.2.7) with coefficients B, p, b, g, w, F , v0, and v1, and the solutionvnΔ to the one with coefficients Bn, pn, bn, gn, wn, Fn, v0,n, and v1,n. For every t â [0, T ] asΔâ 0 we claim
vΔ(t)â v(t) in H1(Ω \ Î0;Rd), vΔ(t)â v(t) in L2(Ω;Rd). (1.4.28)
Moreover, we claim that there exists a sequence of parameters Δnn â (0, 1), converging to 0as nââ, such that for every t â [0, T ] as nââ
vnΔn(t)â vΔn(t)â 0 in H1(Ω \ Î0;Rd), vnΔn(t)â vΔn(t)â 0 in L2(Ω;Rd), (1.4.29)
vnΔn(t)â vn(t)â 0 in H1(Ω \ Î0;Rd), vnΔn(t)â vn(t)â 0 in L2(Ω;Rd). (1.4.30)
Notice that (1.4.28)â(1.4.30) imply (1.4.15). Indeed, by the triangle inequality, as nââ wehave
âvn(t)â v(t)âH1(Ω\Î0)
†âvn(t)â vnΔn(t)âH1(Ω\Î0) + âvnΔn(t)â vΔn(t)âH1(Ω\Î0) + âvΔn(t)â v(t)âH1(Ω\Î0) â 0
and the same holds true for âvn(t) â v(t)âL2(Ω). To prove (1.4.28)â(1.4.30) we divide theproof into several steps.
Chapter 1. Elastodynamics system in domains with growing cracks 31
Step 1. Strong convergence of vΔ. Let us define zΔ := vΔâv. By comparing the two energyequalities (1.2.26) and (1.3.2) we infer that zΔ satisfies
EB(zΔ; t) + Δ
â« t
0âvΔâ2H1(Ω\Î0)ds
=
â« t
0
[1
2(BâzΔ,âzΔ)L2(Ω) â (pâzΔ, zΔ)L2(Ω) â (zΔ div b, zΔ)L2(Ω)
]ds+RΔ(t),
(1.4.31)
where EB is defined in according to (1.3.1), and
RΔ(t) :=â (vΔ(t), v(t))L2(Ω) â (B(t)âvΔ(t),âv(t))L2(Ω) + âv1â2L2(Ω) + (B(0)âv0,âv0)L2(Ω)
+
â« t
0
[(BâvΔ,âv)L2(Ω) + (B(âvΔ +âv),âw)L2(Ω) â (pâvΔ, v)L2(Ω)
]ds
+
â« t
0
[â(pâv, vΔ)L2(Ω) + (p(âvΔ +âv), w)L2(Ω) â 2(v div b, vΔ)L2(Ω)
]ds
+
â« t
0
[2(vΔ + v,div[w â b])L2(Ω) + Δ(vΔ, w)H1(Ω\Î0) + (g, vΔ + v â 2w)L2(Ω)
]ds
+
â« t
0
[â(F , vΔ + v â 2w)L2(âNΩ) â (vΔ + v, w)L2(Ω)
]ds
+ (F (t), vΔ(t) + v(t)â 2w(t))L2(âNΩ) + (vΔ(t) + v(t), w(t))L2(Ω)
â 2(F (0), v0 â w(0))L2(âNΩ) â 2(v1, w(0))L2(Ω)
for every t â [0, T ]. Thanks to (1.2.28), as Δâ 0+ we haveâŁâŁâŁâŁÎ”â« t
0(vΔ, w)H1(Ω\Î0) ds
âŁâŁâŁâŁ †âΔ⫠T
0
âΔâvΔâH1(Ω\Î0)âwâH1(Ω\Î0) ds
â€âΔâwâL2(0,T ;H1(Ω\Î0))
(â« T
0ΔâvΔâ2H1(Ω\Î0) ds
)1/2
â€âΔC â 0.
Therefore, by using also the weak convergences (1.2.34) and the energy equality (1.3.2), wededuce that RΔ(t) â 0 as Δ â 0+. The uniform bounds on B, p, and div b, the ellipticitycondition (1.2.1), the estimate
âzΔ(t)â2L2(Ω) †Tâ« t
0âzΔ(s)â2L2(Ω) ds for every t â [0, T ], (1.4.32)
and the identity (1.4.31) imply
âzΔ(t)â2L2(Ω) + âzΔ(t)â2H1(Ω\Î0) †C(RΔ(t) +
â« t
0
[âzΔ(s)â2L2(Ω) + âzΔ(s)â2H1(Ω\Î0)
]ds
)for every t â [0, T ], with C > 0 independent of t and Δ. By applying Fatouâs lemma, for everyt â [0, T ] we have
lim supΔâ0+
[âzΔ(t)â2L2(Ω) + âzΔ(t)â2H1(Ω\Î0)
]†C lim sup
Δâ0+
(RΔ(t) +
â« t
0
[âzΔ(s)â2L2(Ω) + âzΔ(s)â2H1(Ω\Î0)
]ds
)†C
â« t
0lim supΔâ0+
[âzΔ(s)â2L2(Ω) + âzΔ(s)â2H1(Ω\Î0)
]ds.
32 1.4. Continuous dependence on the data
Thanks to Gronwallâs lemma we conclude
limΔâ0+
[âzΔ(t)â2L2(Ω) + âzΔ(t)â2H1(Ω\Î0)
]= 0 for every t â [0, T ],
which gives the convergences (1.4.28).
Step 2. Strong convergence of vnΔn â vΔn. Let Δnn â (0, 1) be a sequence of parametersto be fixed. The functions vnΔn and vΔn satisfy (1.2.7) with different coefficients, but withthe same viscosity Δn. By linearity, the function zn := (vnΔn â vΔn)â (wn â w) solves for a.e.t â (0, T )
ăzn(t), ÏăHâ1D (Ω\Î0) + (B(t)âzn(t),âÏ)L2(Ω) + (p(t)âzn(t), Ï)L2(Ω)
â 2(âzn(t)b(t), Ï)L2(Ω) + Δn(zn(t), Ï)H1(Ω\Î0) = ăqn(t), ÏăHâ1D (Ω\Î0)
(1.4.33)
for every Ï â H1D(Ω \ Î0;Rd), with initial data z0,n := (v0,n â v0) â (wn(0) â w(0)) and
z1,n := (v1,n â v1)â (wn(0)â w(0)), and right-hand side defined for t â [0, T ] as
ăqn(t), ÏăHâ1D (Ω\Î0)
:=â (wn(t)â w(t), Ï)L2(Ω) â (B(t)(âwn(t)ââw(t)),âÏ)L2(Ω)
â (p(t)(âwn(t)ââw(t)), Ï)L2(Ω) + 2((âwn(t)ââw(t))b(t), Ï)L2(Ω)
â Δn(wn(t)â w(t), Ï)H1(Ω\Î0) â ((Bn(t)â B(t))âvnΔn(t),âÏ)L2(Ω)
â ((pn(t)â p(t))âvnΔn(t), Ï)L2(Ω) â 2(vnΔn(t)â (bn(t)â b(t)),âÏ)L2(Ω)
â 2((div bn(t)â div b(t))vnΔn(t), Ï)L2(Ω) + (gn(t)â g(t), Ï)L2(Ω)
+ (Fn(t)â F (t), Ï)L2(Ω). (1.4.34)
In particular, the forcing term qn is an element of L2(0, T ;Hâ1D (Ω \ Î0;Rd)). Notice that we
have used the identity (1.1.46) for both bn and b to derive formula (1.4.33). By combiningthe energy equality (1.2.26) with (1.1.1), the uniform ellipticity condition (1.2.1) for Bn,the uniform bounds (1.4.19) and (1.4.20), and the convergences (1.4.10), (1.4.11), (1.4.26),and (1.4.27) we conclude that the sequences vnΔnn and vnΔnn are uniformly bounded withrespect to n in Lâ(0, T ;H1
D(Ω \ Î0;Rd)) and Lâ(0, T ;L2(Ω;Rd)), respectively. Moreover,these bounds do not depend on the sequence Δnn. By using (1.1.2), (1.4.19), (1.4.20),and (1.4.22)â(1.4.26) we conclude
qn â 0 in L2(0, T ;Hâ1D (Ω \ Î0;Rd)) as nââ, (1.4.35)
and the rate of this convergence is independent of the choice of Δnn â (0, 1). Notice that, topass to the limit in the first two terms in the right-hand side of (1.4.34), we have used (1.4.24)and (1.4.25).
Since zn â H1(0, T ;H1D(Ω \ Î0;Rd)), we can use zn as test function in (1.4.33), and by
integrating by parts in (0, t) for very t â (0, T ], we get
EB(zn; t) + Δn
â« t
0âznâ2H1(Ω\Î0) ds
= EB(zn; 0) +
â« t
0
[1
2(Bâzn,âzn)L2(Ω) â (pâzn, zn)L2(Ω) dsâ (zn div b, zn)L2(Ω)
]ds
+
â« t
0ăqn, znăHâ1
D (Ω\Î0) ds.
As in the previous step, the uniform bounds on B, p and div b, the ellipticity condition (1.2.1),
Chapter 1. Elastodynamics system in domains with growing cracks 33
and the estimate (1.2.12) applied to zn imply
1
2âzn(t)â2L2(Ω) +
c0
2âzn(t)â2H1(Ω\Î0) + Δn
â« t
0âznâ2H1(Ω\Î0) ds
†1
2âzn(0)â2L2(Ω) +
(c1 +
1
2
)âzn(0)â2H1(Ω\Î0) + C
â« t
0
[âznâ2L2(Ω) + âznâ2H1(Ω\Î0)
]ds
+
â« t
0|ăqn, znăHâ1
D (Ω\Î0)|ds
for a suitable constant C > 0 independent of n and t. We estimate from above the last termin the previous inequality asâ« t
0|ăqn, znăHâ1
D (Ω\Î0)|ds
†1
2âqnâL2(0,T ;Hâ1
D (Ω\Î0)) +1
2âqnâL2(0,T ;Hâ1
D (Ω\Î0))
â« t
0âznâ2H1(Ω\Î0) ds.
By choosing Δn â 0+ such that
Δn â1
2âqnâL2(0,T ;Hâ1
D (Ω\Î0)) â„ 0 for every n â N,
we obtain then following estimate
âzn(t)â2L2(Ω) + c0âzn(t)â2H1(Ω\Î0) †Cn + 2C
â« t
0
(âznâ2L2(Ω) + âznâ2H1(Ω\Î0)
)ds,
withCn := âzn(0)â2L2(Ω) + (2c1 + 1)âzn(0)â2H1(Ω\Î0) + âqnâL2(0,T ;Hâ1
D (Ω\Î0)).
The convergences (1.4.10), (1.4.27), and (1.4.35) yield that Cn â 0 as n â â. Therefore,thanks to Fatou and Gronwallâs lemmas, we derive
limnââ
[âzn(t)â2L2(Ω) + âzn(t)â2H1(Ω\Î0)
]= 0 for every t â [0, T ].
This fact, together with (1.4.10), proves (1.4.29).Step 3. Weak convergence of vn to v. For every n â N, the function vn satisfies for a.e.
t â (0, T )
ăvn(t), ÏăHâ1D (Ω\Î0) + (Bn(t)âvn(t),âÏ)L2(Ω) + (pn(t)âvn(t), Ï)L2(Ω)
+ 2(vn(t), div[Ïâ bn(t)])L2(Ω) = (gn(t), Ï)L2(Ω) + (Fn(t), Ï)L2(âNΩ)
(1.4.36)
for every Ï â H1D(Ω \ Î0;Rd). As shown in (1.2.47),there exists a constant C > 0 such that
âvn(t)â2L2(Ω) + âvn(t)â2H1(Ω\Î0) †C for every t â [0, T ]. (1.4.37)
In particular, the constant C can be chosen independent of n, thanks to (1.1.1), the uniform el-lipticity condition (1.2.1) for Bn, the bounds (1.4.19), (1.4.20), and the convergences (1.4.10),(1.4.11), (1.4.26), and (1.4.27). By using (1.4.36), we also infer that vnn is uniformlybounded in L2(0, T ;Hâ1
D (Ω \ Î0;Rd)). Hence, there exists a function
ζ â Lâ(0, T ;H1(Ω \ Î0;Rd)) â©W 1,â(0, T ;L2(Ω;Rd)) â©H2(0, T ;Hâ1D (Ω \ Î0;Rd))
such that, up to a subsequence (not relabeled), as nââ
vn ζ in L2(0, T ;H1(Ω \ Î0;Rd)), (1.4.38)
34 1.4. Continuous dependence on the data
vn ζ in L2(0, T ;L2(Ω;Rd)), (1.4.39)
vn ζ in L2(0, T ;Hâ1D (Ω \ Î0;Rd)). (1.4.40)
Moreover, thanks to (1.4.10), we have ζ â w â Lâ(0, T ;H1D(Ω \ Î0;Rd)). By combining
the strong convergences (1.4.11) and (1.4.22)â(1.4.27) with the weak convergences (1.4.38)â(1.4.40), we can pass to the limit as nââ in (1.4.36) and we derive that ζ is a generalizedsolution to the limit problem (1.2.5), with initial conditions v0 and v1. In view of Theo-rem 1.2.10 such solution is unique, therefore ζ = v. Since the result does not depend on thesubsequence, we conclude that the whole sequence vnn satisfies as nââ
vn v in L2(0, T ;H1(Ω \ Î0;Rd)),vn v in L2(0, T ;L2(Ω;Rd)),vn v in L2(0, T ;Hâ1
D (Ω \ Î0;Rd)).
These convergences, together with the bounds (1.4.37), for every t â [0, T ] imply
vn(t) v(t) in H1(Ω \ Î0;Rd), vn(t) v(t) in L2(Ω;Rd) as nââ. (1.4.41)
Step 4. Strong convergence of vnΔn â vn. For every n â N we define zn := vnΔn â v
n, whereΔn â (0, 1) are the parameters chosen in Step 1. Following the same procedure adopted inStep 1, we get
EB(zn; t) + Δn
â« t
0âvnΔnâ
2H1(Ω\Î0) ds
=
â« t
0
[1
2(Bnâzn,âzn)L2(Ω) â (pnâzn, zn)L2(Ω) â (zn div bn, zn)L2(Ω)
]ds+Rn(t),
with
Rn(t) :=â (vnΔn(t), vn(t))L2(Ω) â (Bn(t)âvnΔn(t),âvn(t))L2(Ω) + âv1,nâ2L2(Ω)
+ (Bn(0)âv0,n,âv0,n)L2(Ω) +
â« t
0(BnâvnΔn ,âv
n)L2(Ω) ds
+
â« t
0
[(Bn(âvnΔn +âvn),âwn)L2(Ω) â (pnâvnΔn , v
n)L2(Ω) â (pnâvn, vnΔn)L2(Ω)
]ds
+
â« t
0
[(pn(âvnΔn +âvn), wn)L2(Ω) + 2(vnΔn + vn,div[wn â bn])L2(Ω)
]ds
+
â« t
0
[â2(vn div bn, vnΔn)L2(Ω) + Δn(vnΔn , w
n)H1(Ω\Î0) + (gn, vnΔn + vn â 2wn)L2(Ω)
]ds
+
â« t
0
[â(Fn, vnΔn + vn â 2wn)L2(âNΩ) â (vnΔn + vn, wn)L2(Ω)
]ds
+ (Fn(t), vnΔn(t) + vn(t)â 2wn(t))L2(âNΩ) + (vnΔn(t) + vn(t), wn(t))L2(Ω)
â 2(Fn(0), v0,n â wn(0))L2(âNΩ) â 2(v1,n, wn(0))L2(Ω) (1.4.42)
for t â [0, T ]. By using the uniform bounds (1.4.19) and (1.4.20), the ellipticity condi-tion (1.2.1) for Bn and the estimate (1.4.32) for zn, we infer
âzn(t)â2L2(Ω) + âzn(t)â2H1(Ω\Î0) †C(Rn(t) +
â« t
0
[âznâ2L2(Ω) + âznâ2H1(Ω\Î0)
]ds
)(1.4.43)
for every t â [0, T ], where C > 0 is a constant independent of n and t.
Chapter 1. Elastodynamics system in domains with growing cracks 35
Let us show that Rn(t) â 0 as n â â. Thanks to Step 1 and 2, we know that vnΔn(t)converges to v(t) strongly in H1(Ω \ Î0;Rd) for every t â [0, T ] as n â â, while, vnΔn(t)converges to v(t) strongly in L2(Ω;Rd). We use now the weak convergences (1.4.41), togetherwith (1.4.16), (1.4.17), (1.4.27), and Lemma 1.1.2, to derive
limnââ
[(vnΔn(t), vn(t))L2(Ω) + (Bn(t)âvnΔn(t),âvn(t))L2(Ω)
]= âv(t)â2L2(Ω) + (B(t)âv(t),âv(t))L2(Ω),
limnââ
[âv1,nâ2L2(Ω) + (Bn(0)âv0,n,âv0,n)L2(Ω)
]= âv1â2L2(Ω) + (B(0)âv0,âv0)L2(Ω)
for every t â [0, T ]. Moreover, by arguing as in Step 3, it is easy to check
Δn
â« T
0âvnΔnâ
2H1(Ω\Î0)ds †C for every n â N
for a constant C > 0 independent of n. Therefore, for every t â [0, T ] as nâââŁâŁâŁâŁÎ”n â« t
0(vnΔn , w
n)H1(Ω\Î0) ds
âŁâŁâŁâŁ †âΔn â« T
0
âΔnâvnΔnâH1(Ω\Î0)âwnâH1(Ω\Î0) ds
â€âΔâwnâL2(0,T ;H1(Ω\Î0))
(â« T
0ΔnâvnΔnâ
2H1(Ω\Î0) ds
)1/2
â€âΔnC â 0.
since wn â w in L2(0, T ;H1(Ω \ Î0;Rd)). In view of the previous convergences, (1.4.10),(1.4.11), (1.4.19)â(1.4.27), and the dominated convergence theorem in the time variable, wecan pass to the limit as n â â in (1.4.42) and, by using the energy equality (1.3.2), weconclude that Rn(t) â 0 for every t â [0, T ] as n â â. Hence, we can apply Fatou andGronwallâs lemmas to (1.4.43) to derive
limnââ
[âzn(t)â2L2(Ω) + âzn(t)â2H1(Ω\Î0)
]= 0.
This convergence gives (1.4.30) and concludes the proof.
Chapter 2
Dynamic energy-dissipation balanceof a growing crack
In this chapter, we derive a formula for the mechanical energy (8) associated with the solutionsto the wave equation (7), and we derive necessary and sufficient conditions for the validity ofthe dynamic energy-dissipation balance (11).
The plan of the chapter is the following: in Section 2.1, we fix the standing assumptionson the crack set and the matrix A; moreover, we introduce the changes of variables whichtransform (7) into (12). Then, in Section 2.2, we prove the decomposition result (9), byadapting the proof of [43, Theorem 4.8] to our more general case, underlying the maindifferences. Finally, in Section 2.3, we prove the energy balance (10) from which we deducenecessary and sufficient conditions in order to get (11).
The results of this chapter, obtained in collaboration with I. Lucardesi and E. Tasso, arecontained in the submitted paper [9].
2.1 Preliminary results
We consider a bounded open set Ω â R2 with Lipschitz boundary âΩ, we take a Borel subsetâDΩ of âΩ (possibly empty), and we denote by âNΩ its complement. We fix a C3,1 curveÎł : [0, `] â Ω parametrized by arc-length, with endpoints on âΩ; namely, denoting by Î thesupport of Îł, we assume Îâ© âΩ = Îł(0)âȘ Îł(`). Let T be a positive number, s : [0, T ]â (0, `)be a non-decreasing function of class C3,1, and let us set
Ît := Îł(Ï) : 0 â€ Ï â€ s(t) for every t â [0, T ].
Ω
Îł(0)
Îł(s(0))
Îł(s(T ))Îł(`)
Figure 2.1: The endpoints of Î are Îł(0) and Îł(`) and belong to âΩ. We study the evolutionof the crack along Î from Îł(s(0)) to Îł(s(T )).
Let A â C2,1(Ω;R2Ă2sym) be a matrix field satisfying the ellipticity condition
(A(x)Ο) · Ο ℠λ0|Ο|2 for every Ο â R2 and x â Ω, (2.1.1)
37
38 2.1. Preliminary results
with λ0 > 0 independent of x. Given a function f â C0([0, T ];H1(Ω))â©Lip([0, T ];L2(Ω)) andsuitable initial data u0 and u1 (for their precise regularity, see Theorems 2.2.4 and 2.2.10),we consider the differential equation
u(t)â div(Aâu(t)) = f(t) in Ω \ Ît, t â [0, T ], (2.1.2)
with boundary conditions
u(t) = 0 on âDΩ, t â [0, T ], (2.1.3)
(Aâu(t)) · Îœ = 0 on âNΩ âȘ Ît, t â [0, T ], (2.1.4)
where Μ denotes the unit normal vector, and initial conditions
u(0) = u0, u(0) = u1 in Ω \ Î0. (2.1.5)
The equation (2.1.2) has to be intended in the following weak sense: for a.e. t â (0, T )
ău(t), ÏăHâ1D (Ω\Ît) + (Aâu(t),âÏ)L2(Ω) = (f(t), Ï)L2(Ω)
for every Ï â H1D(Ω \ Ît), where H1
D(Ω \ Ît) and Hâ1D (Ω \ Ît) are the spaces defined in
Chapter 1. We implicitly require u(t) to be in H1D(Ω \ Ît) and u(t) to be in Hâ1
D (Ω \ Ît) fora.e. t â (0, T ) (see also [20, Definition 2.4] and Definition 1.1.6).
We assume that the velocity of s is bounded as follows:
|s(t)|2 †λ0 â ÎŽ for every t â [0, T ], (2.1.6)
for a constant 0 < Ύ †λ0. This relation between s and the ellipticity constant λ0 of A is cru-cial in order to guarantee the resolvability of the problem (see also (2.1.14) in Lemma 2.1.1).(2.1.6) can be interpreted saying that the crack must evolve more slowly than the speed ofelastic waves.
2.1.1 The change of variable approach
We fix t0, t1 â [0, T ] such that 0 < t1 â t0 < Ï, with Ï sufficiently small. A comment on thevalue of Ï is postponed to Remark 2.1.3. In the following, we perform 4 changes of variables:first we act on the matrix A, transforming it into the identity on the crack set; then westraighten the crack in a neighborhood of Îł(s(t0)); then we recall the time-dependent changeof variables introduced in [20], that brings Ît into Ît0 for every t â [t0, t1]; finally, we performthe last change of variables in a neighborhood of the (fixed) crack-tip, in order to make theprincipal part of the transformed equation equal to the minus Laplacian. For the sake ofclarity, at each step, we use the superscript i = 1, . . . , 4, to denote the new objects: the
domain Ω(i), the crack set Î(i), and the time-dependent crack Î(i)t . We will also introduce the
matrix fields A(i), which characterize the leading part (with respect to the spatial variables)âdiv(A(i)âv) of the PDE (2.1.2) transformed.
Step 1. Thanks to the standing assumptions on A, we may find a matrix field Q of classC2,1(Ω;R2Ă2) such that
Q(x)A(x)Q(x)T = Id for every x â Ω, (2.1.7)
being Id the identity matrix. In particular we can choose Q(x) to be equal to the squareroot matrix of A(x)â1, namely Q(x) = Q(x)T and Q(x)2 = A(x)â1. It is easy to prove theexistence of a smooth diffeomorphism Ï â C3,1(Ω;R2) of Ω into itself which is the identity ina neighborhood of âΩ and satisfies âÏ(x) = Q(x) on Îâ©V , being V a suitable neighborhood
Chapter 2. Dynamic energy-dissipation balance of a growing crack 39
of Îł(s(t0)). Notice that the constraint âÏ = Q cannot be satisfied in the whole domain, sincethe rows of Q in general are not curl free. We set
Ω(1) := Ω, Î(1) := Ï(Î), Î(1)t := Ï(Ît) for t â [t0, t1],
A(1) := [âÏAâÏT ] Ïâ1.
Clearly, the matrix A(1) satisfies an ellipticity condition of type (2.1.1) for a suitable positiveconstant and it equals the identity matrix on Î(1). Moreover, we may easily write an arc-length parametrization Îł(1) of Î(1) exploiting that of Î, by setting
Îł(1) := Ï Îł ÎČ, ÎČâ1(Ï) :=
â« Ï
0
âŁâŁâŁâŁ d
dÏ(Ï Îł)(Ï)
âŁâŁâŁâŁ dÏ.
Accordingly, the time-dependent crack Î(1)t is parametrized by
Î(1)t = Îł(1)(s(1)(t)) for t â [t0, t1], s(1) := ÎČâ1 s.
The function s(1) is of class C3,1([t0, t1]) and, thanks to (2.1.7) and (2.1.6), satisfies thefollowing bound:
|s(1)(t)|2 =
âŁâŁâŁâŁdÎČâ1
ds(s(t))
âŁâŁâŁâŁ2 |s(t)|2 †max|Ο|=1, xâÎâ©V
|âÏ(x)Ο|2|s(t)|2 †1â c21 (2.1.8)
for every t â [t0, t1], where, for brevity, we have set c21 := ÎŽ
λ0. Moreover, for the sake of clarity,
we also fix a notation for the maximal acceleration: we set c2 as
c2 := maxtâ[t0,t1]
|s(1)(t)|. (2.1.9)
A direct computation proves that c2 is bounded and depends on λ0, ÎŽ, s, Îł, and â2Ï.Step 2. We now provide a change of variables Î of class C2,1 which straightens the crack
in a neighborhood of Îł(1)(s(1)(t0)). First, up to further compose Î with a rigid motion, we
may assume that the tip of Î(1)t0
is at the origin, and the tangent vector to Î(1) at the originis horizontal, namely
Îł(1)(s(1)(t0)) = 0, Îł(1)(s(1)(t0)) = e1 := (1, 0).
For brevity, we set Ï0 := s(1)(t0). We begin by transforming a tubular neighborhood U ofthe crack near 0 into a square: setting
U := Îł(1)(Ï0 + Ï) + ÏÎœ(1)(Ï0 + Ï) : Ï â (âΔ, Δ), Ï â (âΔ, Δ),
with Îœ(1) := (Îł(1))â„ and Δ > 0 such that U ââ Ω, we define Î: U â (âΔ, Δ)2 as the inverseof the function (Ï, Ï) 7â Îł(1)(Ï + Ï0) + ÏÎœ(1)(Ï + Ï0). The global diffeomorphism is obtainedby extending Î to the whole Ω. Accordingly, we set
Ω(2) := Î(Ω(1)), Î(2) := Î(Î(1)), Î(2)t := Î(Î
(1)t ) for t â [t0, t1],
A(2) := [âÎA(1)âÎT ] Îâ1.
The matrix field A(2) still satisfies an ellipticity condition of type (2.1.1), for a suitableconstant.
For x â Î(2) and in a neighborhood of the origin, setting y := Îâ1(x) â Î(1), we have
A(2)(x) = âÎ(y)A(1)(y)âÎ(y)T = âÎ(y)âÎ(y)T = [(â(Îâ1)(x))Tâ(Îâ1)(x)]â1 = Id.
40 2.1. Preliminary results
The last equality follows from
â(Îâ1)
âÏ(Ï, Ï) = Îł(1)(Ï0 + Ï) + Ï Îœ(1)(Ï0 + Ï),
â(Îâ1)
âÏ(Ï, Ï) = Îœ(1)(Ï0 + Ï), (2.1.10)
and the fact that here we consider x of the form x = (Ï, 0). In particular, we may be moreprecise on the ellipticity constant of A(2) restricted to a neighborhood of the origin: for everyΔ â (0, 1), there exists r > 0 such that
(A(2)(x)Ο) · Ο â„ (1â Δ)|Ο|2 for every Ο â R2 and |x| †r. (2.1.11)
Finally, we underline that if Ï := t1 â t0 is small enough (see also Remark 2.1.3), the
whole set Î(1)t1\ Î
(1)t0
is contained in U , so that the time-dependent crack Î(2)t satisfies
Î(2)t = Î
(2)t0âȘ (Ï, 0) â R2 : 0 â€ Ï â€ s(1)(t)â s(1)(t0) for every t â [t0, t1].
Step 3. Here we introduce a family of 1-parameter C2 diffeomorphisms Κ(t), t â [t0, t1],
which transform the time-dependent domain Ω(2) \ Î(2)t into Ω(2) \ Î
(2)t0
. All in all, we map
the domain (t, x) : t â [t1, t2], x â Ω(2) \ Î(2)t into the cylinder [t0, t1] Ă (Ω(2) \ Î
(2)t0
). Thisconstruction can be found in [43] and [20, Example 2.14], thus we limit ourselves to recall
the main properties: the diffeomorphism Κ: [t0, t1]à Ω(2) â Ω
(2)satisfies
Κ(t0) = id, Κ(t)|âΩ(2) = id, Κ(t,Î(2)t ) = Î
(2)t0
for t â [t0, t1].
The corresponding matrix field is
A(3)(t) := [âΚ(t)A(2)âΚ(t)T â Κ(t)â Κ(t)] Κâ1(t) for t â [t0, t1].
Notice that A(2) does not depend on time, while A(3) does. For t â [t0, t1] and x in aneighborhood of the origin we have
Κ(t, x) = xâ (s(1)(t)â s(1)(t0))e1, Κâ1(t, x) = x+ (s(1)(t)â s(1)(t0))e1, (2.1.12)
so that âΚ(t) = Id, Κ(t) = âs(1)(t)e1, and for x = (Ï, 0), with Ï small enough in modulus
A(3)(t, x) =
(1â |s(1)(t)|2 0
0 1
).
Step 4. In this last step we apply a change of variables P near the origin (namely the tip
of Î(2)t0
), in order to make the matrix field A(4), constructed as in the previous steps, satisfy
A(4)(t, 0) = Id for every t â [t0, t1]. To this aim, we recall the construction introduced in [43,Section 4].
We define α : [t0, t1]â [0,â) and d : [t0, t1]à Ω(2) â [0, c1] as
α(t) :=â
1â |s(1)(t)|2,
d(t, x) := α(t)kη(|x|) + (1â kη(|x|))c1,
where kη is the following cut-off function:
kη(Ï) :=
1 if 0 â€ Ï < η/2,(
2 Ïη â 2)2 (
4 Ïη â 1)
if η/2 â€ Ï < η,
0 if Ï â„ Î·.
(2.1.13)
Chapter 2. Dynamic energy-dissipation balance of a growing crack 41
Here η is a positive parameter, whose precise value will be specified later, small enough suchthat the ball Bη(0) â Ω(2). Eventually, we set
P (t, x) :=
(x1
d(t, x), x2
)t â [t0, t1], x â Ω(2).
For every t â [t0, t1] the map P (t) defines a diffeomorphism of Ω(2) into its dilation in thehorizontal direction
Ω(4) :=
(x1
c1, x2
): x â Ω(2)
,
which maps 0 in 0 and Î(2)t0
into a fixed set Î(4)t0
, horizontal near the origin. This chain of
transformations maps the Dirichlet part âDΩ into âDΩ(4) := (Î1(x)/c2,Î2(x)) : x â âDΩand the Neumann one âNΩ into âNΩ(4) := (Î1(x)/c2,Î2(x)) : x â âNΩ.
The matrix field A(4) associated to P reads
A(4)(t) := [âP (t)A(3)(t)âP (t)T â P (t)â P (t)â 2âP (t)Κ(t,Κâ1(t)) P (t)] Pâ1(t)
for t â [t0, t1]. The properties of A(4) are gathered in the following lemma.
Lemma 2.1.1. There exists a constant λ4 > 0 such that for every t â [t0, t1] and x â Ω(4)
(A(4)(t, x)Ο) · Ο ℠λ4|Ο|2 for every Ο â R2. (2.1.14)
Moreover, for every t â [t0, t1], there holds
A(4)(t, 0) = Id. (2.1.15)
Finally, there exists a vector field W : âNΩ(4) âȘ Î(4)t0â R2 such that for every t â [t0, t1] and
x â âNΩ(4) âȘ Î(4)t0
A(4)(t, x)T Μ(x) = W (x), (2.1.16)
and W (x) = Îœ(x) = e2 := (0, 1) in a neighborhood of the tip of Î(4)t0
.
Proof. Let t â [t0, t1] and x â Ω(4) be fixed. By setting y := Pâ1(t, x) â Ω(2), we distinguishthe three cases: |y| < η/2, η/2 †|y| †η, and |y| > η, where η is the constant introducedin (2.1.13). Without loss of generality, up to take η smaller, by recalling (2.1.12) we mayassume that if y â Bη(0)
âΚ(t,Κâ1(t, y)) = Id, Κ(t,Κâ1(t, y)) = âs(1)(t)e1,
so thatA(3)(t, Pâ1(t, x)) = A(3)(t, y) = A(2)(y)â |s(1)(t)|2e1 â e1.
Moreover, we take η < r, where r is the radius associated to Δ = c21/2 as in (2.1.11), so that
the ellipticity constant of A(2) in Bη(0) is 1â c21/2.
If |y| < η/2 we have
âP (t, y) =
( 1α(t) 0
0 1
), P (t, y) =
(ây1
α(t)α2(t)
0
),
thus
A(4)(t, x) =
( 1α(t) 0
0 1
)A(2)(y)
( 1α(t) 0
0 1
)â
(|s(1)(t)|2α(t)2 + y1
2s(1)(t)α(t)α3(t)
+ y21α2(t)α4(t)
0
0 0
).
42 2.1. Preliminary results
Since Pâ1(t, 0) = 0 and A(2)(0) = Id, we immediately get (2.1.15). For Ο arbitrary vectorof R2, we have
(A(4)(t, x)Ο) · Ο â„
(1â c2
1/2â |s(1)(t)|2
α2(t)â 2y1
s(1)(t)α(t)
α3(t)â y2
1
α2(t)
α4(t)
)Ο2
1 + (1â c21/2)Ο2
2 .
In view of the bounds (2.1.6), (2.1.9), and (2.1.8), we get
|α(t)| †c2
c1, c1 †|α(t)| †1,
in particular
(A(4)(t, x)Ο) · Ο â„(c2
1
2â 2η
c2
c41
â η2 c22
c61
)Ο2
1 +Ο2
2
2.
The coefficient of Ο1 is bounded from below, provided that η is small enough. This gives thestatement (2.1.14) for y â Bη/2(0).
Let now η/2 < |y| < η. In this case we have
âP (t, y) =1
d2(t, y)
(d(t, y)â y1â1d(t, y) ây1â2d(t, y)
0 d2(t, y)
),
P (t, y) =1
d2(t, y)
(ây1d(t, y)
0
).
Again, by exploiting the ellipticity of A(2) with constant (1â c21/2) â„ 1
2 and setting
m := y21 d(t, y)2 + 2y1s
(1)(t)d(t, y)(d(t, y)â y1â1d(t, y)),
p := d(t, y)â y1â1d(t, y), q := ây1â2d(t, y), d := d(t, y),
we get
(A(4)(t, x)Ο) · Ο ℠1
2|âP (t, y)T Ο|2 â m
d4Ο2
1
=1
2d4
[(p2 + q2 â 2m)Ο2
1 + 2qd2Ο1Ο2 + d4Ο22
]â„ 1
2
[p2 â
(1
Δâ 1
)q2 â 2|m|
]Ο2
1 +1
2(1â Δ)Ο2
2 ,
(2.1.17)
where in the last inequality we have used d †1 and Youngâs inequality with 0 < Δ < 1, whoseprecise value will be fixed later. Let us prove that, if η and Δ are well chosen, the coercivityof A(4) is guaranteed. The identities
âd(t, y) = (α(t)â c1)y
|y|kη(|y|), d(t, y) = â s
(1)(t)s(1)(t)kη(|y|)α(t)
,
together with the bounds
0 †kη(|y|) †1, c1 †d(t, y) †α(t) †1, â3
η†kη(|y|) †0,
give
1
d4â„ 1, p = d(t, y) +
y21
|y|(α(t)â c1)|kη(|y|)| â„ d(t, y) â„ c1,
q2 = (α(t)â c1)2 y21y
22
|y|2kη(|y|)2 †9(1â c1)2, |m| †42c2(1â c2
1)
c1η +
c22(1â c2
1)
c21
η2.
Chapter 2. Dynamic energy-dissipation balance of a growing crack 43
By inserting these estimates into (2.1.17), we infer
(A(4)(t, x)Ο) · Ο â„[c2
1
2â 9
2
(1
Δâ 1
)(1â c1)2 â 42c2(1â c2
1)
c1η â c2
2(1â c21)
c21
η2
]Ο2
1 +1â Δ
2Ο2
2 .
By taking
Δ :=9(1â c1)2
c21/2 + 9(1â c1)2
â (0, 1)
we have
c21
2â 9
2
(1
Δâ 1
)(1â c1)2 =
c21
4.
Thus, by choosing η small enough, we obtain the desired coercivity of A(4).
Finally, if |y| > η we have
âP (t, y) =
( 1c1
0
0 1
), P (t, y) = 0,
and condition (2.1.14) is readily satisfied in view of the ellipticity of A(3).
The assertion (2.1.16) is clearly verified for A(2): the matrix field does not depend ontime and equals to the identity on the crack, in a neighborhood of the origin. The lastdiffeomorphisms Κ and P both act in a neighborhood of the origin modifying the set onlyin the horizontal component; in particular they do not modify the normal to the crack in aneighborhood of the origin. As for the external boundary, Κ is the identity and P acts as aconstant dilation, so that
W (x) =
( 1c1
0
0 1
)A(2)(c1x1, x2)
( 1c1
0
0 1
)Îœ(x) on âNΩ(4).
This concludes the proof of the lemma.
Remark 2.1.2. The idea of the proof of Lemma 2.1.1 is taken from [43, Lemma 4.1]. Letus underline the main differences: in [43] the authors deal with the identity matrix as start-ing matrix field (here instead we have A(3)) and consider only the dynamics for which theacceleration of the crack-tip is bounded by a precise constant depending on c1 (in place ofour bound c2, not fixed a priori). We also point out that in [43] the study of the ellipticityof the transformed matrix field, in the annulus η/2 < |y| < η, is carried out forgetting thecoefficients out of the diagonal.
Remark 2.1.3. In our construction, a control on the maximal amplitude Ï of the timeinterval [t0, t1] is needed only in Step 2: roughly speaking, in order to straighten the set
Î(1)t1\ Î
(1)t0
and to remain inside Ω, we need to have enough room. A sufficient condition is
that the length of the set, which is at most Ïmaxtâ[0,T ] s(1)(t), has to be less than or equal
to the distance of the crack-tip Îł(1)(s(1)(t)) from the boundary âΩ, which is, thanks to the
assumption Î(1)T \ Î
(1)0 ââ Ω, bounded from below by a positive constant. Notice that if we
considered also a further diffeomorphism which is the identity in a neighborhood of Î(1)T \Î
(1)0
and stretches Ω near the boundary, then our results could be stated for every time t â [0, T ].
44 2.2. Representation result
2.2 Representation result
In this section we derive the decomposition (9) locally in time, namely in a time interval[t0, t1] small enough (see Section 2.1 and Remark 2.1.3). Finally, in Theorem 2.2.10 we givea global representation of u, valid in the whole time interval [0, T ].
Here we recall some classical facts of semigroup theory. Standard references on the subjectare the books [45] and [33]. Let X be a Banach space and A(t) : D(A(t)) â X â X adifferential operator. Consider the evolution problem
V (t) +A(t)V (t) = G(t), (2.2.1)
with initial condition V (0) = V0 and G forcing term (the boundary conditions are encodedin the function space X).
Definition 2.2.1. A triplet A;X,Y consisting of a family A = A(t), t â [0, T ] anda pair of real separable Banach spaces X and Y is called a constant domain system if thefollowing conditions hold:
(i) the space Y is embedded continuously and densely in X;
(ii) for every t the operator A(t) is linear and has constant domain D(A(t)) ⥠Y ;
(iii) the family A is a stable family of (negative) generators of strongly continuous semi-groups on X;
(iv) the operator A is essentially bounded from [0, T ] to the space of linear functionals fromY to X.
Theorem 2.2.2. Let A;X,Y form a constant domain system. Let us assume that V 0 â Yand G â Lip([0, T ];X). Then there exists a unique solution V â C0([0, T ];Y )â©C1([0, T ];X)to (2.2.1) with V (0) = V 0.
2.2.1 Local representation result
We fix t0, t1 â [0, T ] such that 0 < t1 â t0 < Ï. The chain of transformations introduced inSection 2.1 defines the family of time-dependent diffeomorphisms
Ί(t) := P (t) Κ(t) Î Ï, Ί(t) : Ωâ Ω(4), (2.2.2)
which map Î into Î(4), Ît into Î(4)t0
for every t â [t0, t1], âΩ into âΩ(4), the Dirchlet part âDΩ
into âDΩ(4), and the Neumann one âNΩ into âNΩ(4). For the sake of clarity, we denote by xthe variables in Ω and by y the new variables in Ω(4).
Looking for a solution u to (2.1.2) in [t0, t1] is equivalent to look for v := u Ίâ1 solutionto the equation
v(t)âdiv(A(4)(t)âv(t))+p(t) ·âv(t)â2b(t) ·âv(t) = g(t) in Ω(4) \Î(4)t0, t â [t0, t1], (2.2.3)
supplemented by the boundary conditions
v = 0 on âDΩ(4), t â [t0, t1], (2.2.4)
âW v = 0 on âNΩ(4) âȘ Î(4)t0, t â [t0, t1], (2.2.5)
and by suitable initial conditions v0 and v1 (see [20]). Here W is the vector field introduced
in (2.1.16) of Lemma 2.1.1, and for t â [t0, t1] and y â Ω(4)
p(t, y) := â[A(4)(t, y)â(detâΊâ1(t, y)) + ât(b(t, y) detâΊâ1(t, y))] detâΊ(t,Ίâ1(t, y)),
Chapter 2. Dynamic energy-dissipation balance of a growing crack 45
b(t, y) := âΊ(t,Ίâ1(t, y)),
g(t, y) := f(t,Ίâ1(t, y)).
The equation (2.2.3) has to be intended in the weak sense, namely valid for a.e. t â (0, T ) induality with an arbitrary test function in the space
H1D(Ω(4) \ Î
(4)t0
) := v â H1(Ω(4) \ Î(4)t0
) : v = 0 on âDΩ(4).
We implicitly require v(t) and v(t) to be in H1D(Ω(4) \ Î(4)(t0)), and v(t) to be in the dual
Hâ1D (Ω(4) \ Î(4)(t0)) for a.e. t â (0, T ).
The characterization of u will follow from that of v, slightly easier to be derived. Theadvantages in dealing with problem (2.2.3) are essentially 3: first of all, the domain is cylin-drical and constant in time; then, the fracture set is straight near the tip; finally, even if thecoefficients depend on space and time, the principal part of the spatial differential operatoris constant at the crack-tip. Before stating the result, we define
H := v â H2(Ω(4) \ Î(4)t0
) : (2.2.5) hold true â kζS : k â R,
where ζ is a cut-off function supported in neighborhood of the origin and
S(y) := Im(ây1 + iy2) =
y2â2â|y|+ y1
y â R2 \ (Ï, 0) : Ï â€ 0, (2.2.6)
with Im denoting the imaginary part of a complex number.
y
0f(Ï; 0) : Ï â€ 0g
Ξ 2 (Ï; Ï)
Ï = jyjS(Ï; Ξ) := Ï1
2 sin( Ξ2)
Figure 2.2: In polar coordinates, the function S reads S(Ï, Ξ) = Ï12 sin( Ξ2), where Ï is the dis-
tance from the origin and Ξ â (âÏ, Ï) is the angle which has a discontinuity on the horizontalhalf line (Ï, 0) : Ï â€ 0.
Proposition 2.2.3. Take v0 â H, v1 â H1D(Ω(4) \Î
(4)t0
), and g â Lip([t0, t1];L2(Ω(4))). Thenthere exists a unique solution v to (2.2.3)â(2.2.5) with v(t0) = v0, v(t0) = v1 in the class
v â C([t0, t1];H) â© C1([t0, t1];H1D(Ω(4) \ Î
(4)t0
)) ⩠C2([t0, t1];L2(Ω(4))).
Proof. Once we show that the triplet A;X;Y defined by
A(t) :=
(0 â1
âdiv(A(4)(t)â(·)) + p(t) · â(·) â2b(t) · â(·)
),
X := H1D(Ω(4) \ Î
(4)t0
)à L2(Ω(4)),
Y := HĂH1D(Ω(4) \ Î
(4)t0
),
is a constant domain system in [t0, t1] (cf. Definition 2.2.1), we are done. Indeed, we are ina position to apply Theorem 2.2.2 with
G(t) :=
(0g(t)
),
46 2.2. Representation result
and the searched v is the second component of the solution V to (2.2.1).The detailed proof of properties (i)â(iv) in Definition 2.2.1 can be found in [43, Theo-
rem 4.7], with the appropriate modifications (see Remark 2.1.2). Here we limit ourselvesto list the main ingredients. First of all, the domain of div(A(4)(t)â(·)) is constant intime: in view (2.1.15), its principal part, evaluated at the crack-tip, is the Laplace op-erator for every t, thus the domain of div(A(4)(t)â(·)) can be decomposed as the sum
v â H2(Ω(4) \ Î(4)t0
) : (2.2.5) holds true â kζS : k â R = H (cf. [32, Theorem 5.2.7]).Moreover, in view of (2.1.16), the boundary conditions (2.2.5) do not depend on time. Otherkey points are the equicoercivity in time of the bilinear form
(Ï0, Ï1) 7â (A(4)(t)âÏ0) · âÏ1 Ï0, Ï1 â H1D(Ω(4) \ Î
(4)t0
),
which is guaranteed by (2.1.14) and the propertyâ«Î©(4)\Î(4)
t0
Ï(y)âÏ(y) · b(t, y) dy = â1
2
â«Î©(4)\Î(4)
t0
Ï2(y) div b(t, y) dy,
valid for every Ï â H1D(Ω(4) \Î
(4)t0
). Finally, the needed continuity of the differential operatoris ensured by the following regularity properties of the coefficients: for every i, j = 1, 2 wehave
A(4)ij (t) â C0(Ω
(4)) for every t â [t0, t1],
A(4)ij , pi, bi â Lip([t0, t1];Lâ(Ω(4))),
ââA(4)ij (t)âLâ(Ω(4)) †C, âdiv b(t)âLâ(Ω(4)) †C for every t â [t0, t1],
for a suitable constant C > 0 independent of t.
We are now in a position to state the following representation result for u.
Theorem 2.2.4. Let f â C0([t0, t1];H1(Ω))â©Lip([t0, t1];L2(Ω)). Consider u0 and u1 of theform
u0 â k0ζS(Ί(t0)) â H2(Ω \ Ît0), (2.2.7)
u1 ââu0 ·(âΊâ1(t0,Ί(t0))Ί(t0)
)â H1(Ω \ Ît0), (2.2.8)
with u0 satisfying the boundary conditions (2.1.3) and (2.1.4), u1 = 0 on âDΩ, ζ cut-offfunction supported in neighborhood of Îł(s(t0)), and k0 â R. Then there exists a uniquesolution to (2.1.2)â(2.1.4) with initial conditions u(t0) = u0, u(t0) = u1 of the form
u(t, x) = uR(t, x) + k(t)ζ(t, x)S(Ί(t, x)) t â [t0, t1], x â Ω \ Ît, (2.2.9)
where ζ(t), t â [t0, t1], is a C2 (in time) family of cut-off functions supported in neighborhoodof Îł(s(t)), and k is a C2 function in [t0, t1] such that k(t0) = k0. Moreover, uR(t) â H2(Ω\Ît)for every t â [t0, t1], and
uR â C2([t0, t1];L2(Ω)), âuR â C1([t0, t1];L2(Ω;R2)), â2uR â C0([t0, t1];L2(Ω;R2Ă2)).
Remark 2.2.5. Notice that the equality u(t, x) = v(t,Ί(t, x)) implies
u0 = v0(Ί(t0)), u1 = v1(Ί(t0)) +âv0(Ί(t0)) · Ί(t0),
where the last term reads Ί(t0) = [P (t0,Κ(t0)) + âP (t0,Κ(t0))Κ(t0)] Î Ï. A priori,the function âv0 is just in L2 in a neighborhood of the origin and its gradient behaves
Chapter 2. Dynamic energy-dissipation balance of a growing crack 47
like |y|â3/2; nevertheless, since P (t, y) ⌠(y1, 0), we recover the L2 integrability of the gra-dient of âv0(Ί(t0)) · [P (t0,Κ(t0)) Î Ï]. The same reasoning does not apply for the termâv0(Ί(t0)) · [(âP (t0,Κ(t0))Κ(t0)) Î Ï], since the singularity of âv0 in a neighborhood ofthe orgin is not compensated by âP · Κ. Therefore we are not free to take u1 â H1
D(Ω \ Ît0)(as, on the contrary, is done in [43]).
Remark 2.2.6. Notice that the solution u to problem (2.1.2)â(2.1.5) displays a singularityonly at the crack-tip. Clearly, the fracture is responsible for this lack of regularity. Onthe other hand, the Dirichlet-Neumann boundary conditions do not produce any furthersingularity, due to the compatible initial data chosen.
2.2.2 Global representation result
We conclude the section by showing an alternative representation formula which can beexpressed for every time. This is done providing another expression for the singular function,as in [38], whose computation does not require to straighten the crack. To simplify thenotation we reduce ourselves to the case A = Id, so that the diffeomorphism Ï coincides withthe identity.
The chosen singular part of the solution to (2.1.2)â(2.1.5) is a suitable reparametrization ofthe function S introduced in (2.2.6). More precisely, fixed t0, t1 â [0, T ], with 0 < t1â t0 < Ï,for every t â [t0, t1] and x in a neighborhood of r(t) := Îł(s(t)), the singular part reads
S
(Î1(x)â (s(t)â s(t0))â
1â |s(t)|2,Î2(x)
). (2.2.10)
To compute (2.2.10) it is necessary to know the expression of Î, which is explicit only forsmall time and locally in space. We hence provide a more explicit formula for the singularpart, which has also the advantage of being defined for every time: for every t â [0, T ] we set
S(t, x) := Im
(â(xâ r(t)) · Îł(s(t))â
1â |s(t)|2+ i (xâ r(t)) · Îœ(s(t))
), (2.2.11)
where Îœ(Ï) â„ Îł(Ï) and S(t) is given by the unique continuous determination of the complexsquare function such that in x = r(t)+
â1â |s(t)|2Îł(s(t)) takes value 1 and its discontinuity
set lies on Ît. Roughly speaking, if we forget the termâ
1â |s(t)|2, the function (2.2.11) isthe determination of Im(
ây1 + iy2) in the orthonormal system with center Îł(s(t)) and axes
Îł(s(t)) and Îœ(s(t)).
Ît
r(t) := Îł(s(t)) _Îł(s(t))
Μ(s(t))x
Ï = jx r(t)j
Ξ
Ît Ξ
Ξ = 0
Ï1
2 sin( Ξ2)
Figure 2.3: A possible choice of determination of Im(ây1 + iy2), centered in r(t) = Îł(s(t))
with axes Îł(s(t)) and Îœ(s(t)), and with Ît as discontinuity set.
For every t â [0, T ] we consider the matrix R(t) â SO(2) that rotates the orthonormalsystem with axes Îł(s(t)) and Îœ(s(t)) in the one with axes e1 and e2. Thanks to our construc-tion of Î, and in particular to (2.1.10), the matrix R(t) coincides with âÎ(r(t)) in [t0, t1].
48 2.2. Representation result
By setting for t â [0, T ]
L(t) :=
(1â
1â|s(t)|20
0 1
), Ί(t, x) := L(t)R(t)(xâ r(t)) for x â Ω,
Ωt := Ί(t,Ω), Ît := Ί(t,Ît),
we may also write S(t, x) = S(t, Ί(t, x)) for t â [0, T ] and x â Ω \ Ît, where S(t) is given bythe continuous determination of Im(
ây1 + iy2) in Ωt \ Ît such that in y = (1, 0) takes the
value 1.
Lemma 2.2.7. Let ζ(t), t â [t0, t1], be a C2 (in time) family of cut-off functions with supportin a neighborhood of r(t). Let us define the function
w(t, x) := ζ(t, x)S(Ί(t, x))â S(t, x) for t â [t0, t1], x â Ω \ Ît . (2.2.12)
Then w(t) â H2(Ω \ Ît) for every t â [t0, t1].
Proof. Let us fix t â [t0, t1]. The function w(t) is of class C2 in Ω \ Ît and belongs to thespace H1(Ω \ Ît) â©H2((Ω \ Ît) \BΔ(r(t))) for every Δ > 0. Hence it remains to prove the L2
integrability of its second spatial derivatives in BΔ(r(t)). Let us choose Δ > 0 so small thatζ(t) = 1 on BΔ(r(t)). In BΔ(r(t)) \ Ît we have
â2jiw(t) =
dâh=1
[âhS(Ί(t))â2jiΊh(t)â âhS(t, Ί(t))â2
jiΊh(t)]
+
dâh,k=1
[â2hkS(Ί(t))âjΊk(t)âiΊh(t)â â2
hkS(t, Ί(t))âjΊk(t)âiΊh(t)]
=: I1(t) + I2(t)
for every i, j = 1, 2.
Notice that âS(Ί(t)),âS(t, Ί(t)) â L2(BΔ(r(t));R2), while â2Ί(t) and â2Ί(t) are uni-formly bounded in Ω. Therefore I1(t) â L2(BΔ(r(t))) and there exists a positive constant C,independent of t, such that
|I1(t, x)| †C|xâ r(t)|â12 for every x â BΔ(r(t)) \ Ît,
provided that Δ > 0 is small enough.
As for I2(t), we estimate it from above as
|I2(t)| â€dâ
h,k=1
|â2hkS(Ί(t))â â2
hkS(t, Ί(t))||âjΊk(t)||âiΊh(t)|
+
dâh,k=1
|â2hkS(Ί(t))||âjΊk(t)âiΊk(t)â âjΊk(t)âiΊh(t)|.
(2.2.13)
Let us study the right-hand side of (2.2.13). By choosing Δ small enough and using thedefinitions of Ί(t) and Ί(t), for every x â BΔ(r(t)) we deduce
|âjΊk(t, x)âiΊh(t, x)â âjΊk(t, x)âiΊh(t, x)|
†2
c21
ââÎâLâ(Ω(1))ââ2ÎâLâ(Ω(1))|xâ r(t)|,
(2.2.14)
Chapter 2. Dynamic energy-dissipation balance of a growing crack 49
since ââΊ(t)âLâ(Ω) †1c1ââÎâLâ(Ω(1)), ââΊ(t)âLâ(Ω) †1
c1ââÎâLâ(Ω(1)), and
|âΊ(t, x)ââΊ(t, x)| †1
c1|âÎ(x)âR(t)| †1
c1ââ2ÎâLâ(Ω(1))|xâ r(t)|.
Moreover, the function S satisfies |â2S(y)| †M |y|â32 for y â R2 \ (Ï, 0) : Ï â€ 0, with M
positive constant, while Î is invertible and |P (t, x)| â„ |x|. This allows us to conclude
|â2hkS(Ί(t, x))| â€MââÎâ1â
32
Lâ(Ω(2))|xâ r(t)|â
32 for every x â BΔ(r(t)) \ Ît. (2.2.15)
For the second term in the right-hand side of (2.2.13), we fix x â BΔ(r(t)) and we considerthe segment [Ί(t, x), Ί(t, x)] := λΊ(t, x) + (1 â λ)Ί(t, x) : λ â [0, 1] and the functiond(t, x) := dist([Ί(t, x), Ί(t, x)], 0). We claim that we can choose Δ > 0 so small that
d(t, x) â„ 1
2|xâ r(t)| for every x â BΔ(r(t)). (2.2.16)
Indeed let y â [Ί(t, x), Ί(t, x)] be such that |y| = d(t, x), then
|Ί(t, x)| †|y|+ |Ί(t, x)â y| †|y|+ |Ί(t, x)â Ί(t, x)|.
Since |P (t, x)| â„ |x| and R(t) is a rotation, for Δ small we deduce |Ί(t, x)| â„ |x â r(t)|. Onthe other hand, by Lagrangeâs theorem there exists z = z(t, x) â BΔ(r(t)) such that
Ί(t, x) = Ί(t, r(t)) +âΊ(t, r(t))(xâ r(t)) +â2Ί(t, z)(xâ r(t)) · (xâ r(t))= Ί(t, x) +â2Ί(t, z)(xâ r(t)) · (xâ r(t)).
Hence we derive the estimate
|Ί(t, x)â Ί(t, x)| †1
c1ââ2ÎâLâ(Ω(1))|xâ r(t)|
2 for every x â BΔ(r(t)), (2.2.17)
which implies
d(t, x) â„ |xâ r(t)| â 1
c1ââ2ÎâLâ(Ω(1))|xâ r(t)|
2 for every x â BΔ(r(t)).
In particular we obtain (2.2.16) by choosing Δ < c1/(2ââ2ÎâLâ(Ω(1))). Notice that Δ doesnot depend on t â [t0, t1].
Let us now fix x â BΔ(r(t)) \ Ît. Thanks to our construction of Ί and Ί, it is possibleto find two other determinations S±(t) of Im(
ây1 + iy2) in R2 such that their discontinuity
sets α(t) do not intersect the segment [Ί(t, x), Ί(t, x)], which is far way from 0. Moreover,we choose them in such a way that S+(t) is positive along (Ï, 0) : Ï â€ 0, while Sâ(t) isnegative, and S(Ί(t, x)) = S±(t,Ί(t, x)) if and only if S(t, Ί(t, x)) = S±(t, Ί(t, x)); notice
that |â3S±(t, y)| â€M |y|â52 for a positive constant M and for every y â R2 \α(t). By using
Lagrangeâs theorem, (2.2.16), and (2.2.17), we deduce
|â2hkS(Ί(t, x))â â2
hkS(t, Ί(t, x))| = |â2hkS
±(t,Ί(t, x))â â2hkS
±(t, Ί(t, x))|†|â3S±(t, z)||Ί(t, x)â Ί(t, x)|
†M
c1ââ2ÎâLâ(Ω(1))|d(t, x)|â
52 |xâ r(t)|2
†4â
2M
c1ââ2ÎâLâ(Ω(1))|xâ r(t)|
â 12 ,
(2.2.18)
50 2.2. Representation result
where z = z(t, x) â [Ί(t, x), Ί(t, x)]. Hence, by combining (2.2.13) with (2.2.14), (2.2.15),and (2.2.18), we obtain the existence of a positive constant C such that
|I2(t, x)| †C|xâ r(t)|â12 for every x â BΔ(r(t)) \ Ît.
In particular we get the following bound for â2w:
|â2w(t, x)| †C|xâ r(t)|â12 for every x â BΔ(r(t)) \ Ît, (2.2.19)
and consequently w(t) â H2(Ω \ Ît) for every t â [t0, t1].
In the following two lemmas, we investigate the regularity in time for w.
Lemma 2.2.8. Under the same assumptions of Lemma 2.2.7, the function w introducedin (2.2.12) is an element of C0([t0, t1];L2(Ω)). Moreover, âw â C0([t0, t1];L2(Ω;R2)) andâ2w â C0([t0, t1];L2(Ω;R2Ă2)).
Proof. The function ζ(S Ί) is continuous from [0, T ] to L2(Ω), since S belongs to the spaceC2(R2 \ (Ï, 0) : Ï â€ 0) â© L2
loc(R2) and Ί is continuous in [t0, t1] à Ω. We also claim that
S = S Ί â C0([t0, t1] Ă (Ω \ Î)) â© Lâ((t0, t1) à Ω). Indeed, let (tâ, xâ) â [t0, t1] Ă (Ω \ Î)and let (tj , xj)j â [t0, t1]Ă (Ω \Î) be a sequence of points converging to (tâ, xâ) as j ââ.Thanks to the convergence Ί(tj , xj) â Ί(tâ, xâ) â Ωtâ \ Îtâ as j â â, there exists j â Nsuch that
S(tj , Ί(tj , xj)) = S(tâ, Ί(tj , xj)) for every j â„ j.
This allows us to conclude that S(tj , xj) â S(tâ, xâ) as j â â, since the function S(tâ) is
continuous in Ωtâ \ Îtâ . Furthermore, there exists M > 0 such that |S(t, x)| â€M |Ί(t, x)|12 for
x â Ω \Î and t â [t0, t1], which yields that S is uniformly bounded in Ω \Î. We hence derivethe claim, which implies S â C0([t0, t1];L2(Ω)), by the dominated convergence theorem.
Arguing as before, we can easily derive that â(ζ(S Ί)) belongs to C0([t0, t1];L2(Ω;R2)),while âS = âΊT (âS Ί) â C0([t0, t1] Ă (Ω \ Î);R2). Therefore, thanks to the estimate
|âS(t, Ί(t, x))| †M |Ί(t, x)|â12 for x â Ω \ Î and t â [t0, t1], and the dominated converge
theorem, we conclude that âS â C0([t0, t1];L2(Ω;R2)).Finally, notice that the function â2w is continuous in [t0, t1] Ă (Ω \ Î). Let us now fix
tâ â [t0, t1] and let tjj be a sequence of points in [t0, t1] such that tj â tâ as j â â.Thanks to the estimate (2.2.19), we can find j â N and Δ > 0 such that
|â2w(tj , x)| †C|xâ r(tj)|â12 for every x â BΔ(r(tj)) \ Î and j â„ j,
with C independent of j. Here we have used the fact that the constant in (2.2.19) can bechosen uniform in time. Furthermore, the functions â2w(tj) are uniformly bounded withrespect to j outside the ball BΔ(r(tj)). We can hence apply the generalized dominatedconvergence theorem to deduce that â2w(tj) converges strongly to â2w(tâ) in L2(Ω;R2Ă2),which implies âw2 â C0([t0, t1];L2(Ω;R2Ă2)).
Lemma 2.2.9. Under the same assumptions of Lemma 2.2.7, the function w introducedin (2.2.12) is an element of C2([0, T ];L2(Ω)); moreover âw â C1([0, T ];L2(Ω;R2)).
Proof. For every x â Ω \ Î the function t 7â w(t, x) is differentiable in [t0, t1] and
w(t, x) = ζ(t, x)S(Ί(t, x)) + ζ(t, x)âS(Ί(t, x)) · Ί(t, x)ââS(t, Ί(t, x)) · ËΊ(t, x).
Indeed, fixed (tâ, xâ) â [t0, t1]Ă (Ω \ Î), we can find h > 0 such that for every |h| †h
S(tâ + h, Ί(tâ + h, xâ))â S(tâ, Ί(tâ, xâ))
h=S(tâ, Ί(tâ + h, xâ))â S(tâ, Ί(tâ, xâ))
h,
Chapter 2. Dynamic energy-dissipation balance of a growing crack 51
thanks to the fact that Ί(tâ + h, xâ) â Ί(tâ, xâ) â Ωtâ \ Îtâ for every xâ â Ω \ Î as h â 0.
In particular 1h [S(tâ+h, Ί(tâ+h, xâ))â S(tâ, Ί(tâ, xâ))]â âS(tâ, Ί(tâ, xâ)) · ËΊ(tâ, xâ), since
S(tâ) â C2(Ωtâ \ Îtâ). Hence, for every (t, x) â [t0, t1] Ă (Ω \ Î) and h â R such thatt+ h â [t0, t1] we may write
w(t+ h, x)â w(t, x)
h=
1
h
â« t+h
tw(Ï, x) dÏ.
By arguing as in the proof of the previous lemma we deduce that w â C0([t0, t1];L2(Ω)).Therefore we obtain that as hâ 0
1
h
â« t+h
tw(Ï) dÏ â w(t) in L2(Ω) for every t â [t0, t1],
and consequently 1h [w(t+ h)â w(t)]â w(t) in L2(Ω).
Similarly, for every x â Ω \ Î the map t 7â w(t, x) is differentiable in [t0, t1] and
w(t, x) = ζ(t, x)S(Ί(t, x)) + 2ζ(t, x)âS(Ί(t, x)) · Ί(t, x)
+ ζ(t, x)âS(Ί(t, x)) · Ί(t, x)ââS(t, Ί(t, x)) · šΊ(t, x)
+ ζ(t, x)â2S(Ί(t, x)) · [Ί(t, x)â Ί(t, x)â ËΊ(t, x)â ËΊ(t, x)]
+ [ζ(t, x)â2S(Ί(t, x))ââ2S(t, Ί(t, x))] ËΊ(t, x)â ËΊ(t, x).
We may find Δ > 0 so small that |Ί(t, x)â ËΊ(t, x)| †C|xâr(t)| in BΔ(r(t)) for every t â [t0, t1]and for a positive constant C. Therefore, we can proceed as in the proof of Lemma 2.2.7 toobtain that w(t) â L2(Ω) for every t â [t0, t1], with
|w(t, x)| †C|xâ r(t)|â12 for every x â BΔ(r(t)) \ Ît.
In particular, by arguing as in Lemma 2.2.8, this uniform estimate implies that w belongs toC0([t0, t1];L2(Ω)). We can hence repeat the same procedure adopted before for w to concludethat as hâ 0
w(t+ h)â w(t)
hâ w(t) in L2(Ω) for every t â [t0, t1],
which gives that w â C2([t0, t1];L2(Ω)).Finally, also the function t 7â âw(t, x) is differentiable in [t0, t1] for every x â Ω \Î, with
derivative
âw(t, x) = âζ(t, x)S(Ί(t, x)) +âζ(t, x)âS(Ί(t, x)) · Ί(t, x) + ζ(t, x)âΊ(t, x)TâS(Ί(t, x))
+ ζ(t, x)âΊ(t, x)TâS(Ί(t, x))ââ ËΊ(t, x)TâS(t, Ί(t, x))
+ [ζ(t, x)âΊ(t, x)T ââΊ(t, x)T ]â2S(Ί(t, x))Ί(t, x)
+ ζ(t, x)âΊ(t, x)Tâ2S(Ί(t, x))[Ί(t, x)â ËΊ(t, x)]
+âΊ(t, x)T [ζ(t, x)â2S(Ί(t, x))ââ2S(t, Ί(t, x))] ËΊ(t, x).
Moreover there exists Δ > 0 so small that for every t â [t0, t1]
|âw(t, x)| †C|xâ r(t)|â12 for every x â BΔ(r(t)) \ Ît,
which implies the continuity of the map t 7â âw(t) from [t0, t1] to L2(Ω;R2). Therefore, ashâ 0 we get
âw(t+ h)ââw(t)
hâ âw(t) in L2(Ω;R2) for every t â [t0, t1],
and in particular âw â C1([t0, t1];L2(Ω;R2)).
52 2.2. Representation result
Thanks to previous lemmas we derive the following decomposition result.
Theorem 2.2.10. Let f â C0([0, T ];H1(Ω))â© Lip([0, T ];L2(Ω)). Consider u0 and u1 of theform
u0 â k0S(0) â H2(Ω \ Î0),
u1 â k0 ËS(0) â H1(Ω \ Î0),
with u0 satisfying (2.1.3) and (2.1.4), u1 = 0 on âDΩ, and k0 â R. Then there exists aunique solution u to (2.1.2)â(2.1.4) with initial condition u(0) = u0 and u(0) = u1 of theform
u(t, x) = uR(t, x) + k(t)S(t, x) t â [0, T ], x â Ω \ Ît, (2.2.20)
where k â C2([0, T ]) and uR(t) â H2(Ω \ Ît) for every t â [0, T ]. Moreover
uR â C2([0, T ];L2(Ω)), âuR â C1([0, T ];L2(Ω;R2)), â2uR â C0([0, T ];L2(Ω;R2Ă2)).(2.2.21)
In particular the function k does not depend on the choice of Ί, but only on Πand s.
Proof. Thanks to our assumptions on f , u0 and u1 we can apply Theorem 2.2.4 with t0 = 0.Indeed, in view of the computations done before, we have
S(0)â ζ(0)S(Ί(0)) â H2(Ω \ Î0),
which gives (2.2.7). In particular
âu0 â k0ζ(0)âΊ(0)TâS(Ί(0)) â H1(Ω \ Î0),
from which we derive
k0 ËS(0)ââu0 ·
(âΊâ1(0,Ί(0))Ί(0)
)â H1(Ω \ Î0),
sinceËS(0)âζ(0)âΊ(0)TâS(Ί(0)) â H1(Ω\Î0), by arguing again as in the previous lemmas.
Therefore, also condition (2.2.8) is satisfied. This implies the representation formula (2.2.9)in [0, t1], with t1 < Ï. By combining (2.2.9) with Lemma 2.2.7, we deduce (2.2.20) in [0, t1].Indeed, we can write
u(t) = uR(t) + k(t)S(t) in Ω \ Ît, t â [0, t1],
where uR(t) := uR(t) + k(t)[ζ(t)S(Ί(t))â S(t)] â H2(Ω \ Ît).We can repeat this construction starting from t1 and we find a finite number of times
tini=0, with 0 =: t0 < t1 < · · · < tnâ1 < tn := T such that the solution u to (2.1.2)â(2.1.4)with initial conditions u(0) = u0 and u(0) = u1 can be written for i = 1, . . . , n as
u(t) = uRi (t) + ki(t)S(t) in Ω \ Ît, t â [tiâ1, ti].
Define k : [0, T ]â R and uR : [0, T ]â H2(Ω \Î) as k(t) := ki(t) and uR := uRi in [tiâ1, ti] forevery i = 1, . . . , n, respectively. The functions k and uR are well defined and do not dependon the particular choice of tini=0. Indeed, if we have
u(t) = uR1 (t) + k1(t)S(t) = uR2 (t) + k2(t)S(t) in Ω \ Ît
for a time t â [0, T ], then we derive
uR1 (t)â uR2 (t) = [k2(t)â k1(t)]S(t) in Ω \ Ît.
Since the left-hand side belongs to H2(Ω\Ît) while S(t) is an element of H1(Ω\Ît)\H2(Ω\Ît),such identity can be true if and only if k1(t) = k2(t) and uR1 (t) = uR2 (t). Hence, we deducethat k â C2([0, T ]) and that u satisfies the decomposition result (2.2.20) in [0, T ].
Finally, by combining the regularity in time of w, proved in Lemmas 2.2.8 and 2.2.9, withthe definition of uR, we conclude that uR satisfies (2.2.21).
Chapter 2. Dynamic energy-dissipation balance of a growing crack 53
Remark 2.2.11. When A 6= Id all the previous results are still true if we define
S(t, x) := Im
(â[A(r(t))â1 (xâ r(t))] · Îł(s(t))
cA,Îł(t)â
1â |cA,Îł(t)|2|s(t)|2+ i
(xâ r(t)) · Îœ(s(t))
cA,Μ(t)
), (2.2.22)
where cA,Îł(t) := |A(r(t))â1/2Îł(s(t))|, cA,Îœ(t) := |A(r(t))1/2Îœ(s(t))|, with A1/2 and Aâ1/2 the
square root matrices of A and Aâ1, respectively, and where the function S(t) is given bythe unique continuous determination of the complex square function such that in the pointx = r(t) +
â1/|cA,Îł(t)|2 â |s(t)|2Îł(s(t)) takes the value 1 and its discontinuity set lies on Ît.
Indeed, by exploiting the following identities in [t0, t1]
Îł(1)(s(1)(t)) =A(r(t))â1/2Îł(s(t))
|A(r(t))â1/2Îł(s(t))|, Îœ(1)(s(1)(t)) =
A(r(t))1/2Μ(s(t))
|A(r(t))1/2Μ(s(t))|,
s(1)(t) = |A(r(t))â1/2Îł(s(t))|s(t), âÏ(r(t)) = A(r(t))â1/2,
where Îł(1) and Îœ(1) are, respectively, the tangent and the unit normal vectors to the curveÎ(1) in the point Îł(1)(s(1)(t)), the function (2.2.22) can be rewritten as
Im
(â[âÏ(r(t)) (xâ r(t))] · Îł(1)(s(1)(t))â
1â |s(1)(t)|2+ i[âÏ(r(t)) (xâ r(t))] · Îœ(1)(s(1)(t))
).
In this case, it is enough to set Ί(t, x) := L(t)R(t)âÏ(r(t))(xâ r(t)) for t â [0, T ] and x â Ω,where L and R are constructed starting from Îł(1) and s(1), and we can proceed again as inLemmas 2.2.7â2.2.9, thanks to the fact that for every t â [t0, t1] and x â BΔ(r(t))
|Ί(t, x)â Ί(t, x)| †C|xâ r(t)|2, |âΊ(t, x)ââΊ(t, x)| †C|xâ r(t)|,
|Ί(t, x)â ËΊ(t, x)| †C|xâ r(t)|.
We hence obtain the decomposition result (2.2.20) with singular part (2.2.22). As a byprod-uct, arguing as in Theorem 2.2.10, we derive that the values of k do not depend on theparticular construction of Ί, but only on A, Î, and s.
We point out that the condition |s(t)|2 < 1/|cA,Îł(t)|2, which is necessary in order to
define S, is implied by (2.1.6). Indeed
1 = âÏ(r(t))A(r(t))âÏ(r(t))T Îł(s(t)) · Îł(s(t)) ℠λ0|A(r(t))â1/2Îł(s(t))|2 = λ0|cA,Îł(t)|2.
2.3 Dynamic energy-dissipation balance
In this section we derive formula (10) for the energy
E(t) :=1
2
â«Î©|u(t, x)|2dx+
1
2
â«Î©A(x)âu(t, x) · âu(t, x) dx t â [0, T ]
associated to u, solution to (2.1.2)â(2.1.4) with initial conditions u(0) = u0 and u(0) = u1.The computation is divided into three steps: first, in Proposition 2.3.5 we consider straight
cracks when A is the identity matrix; then, in Theorem 2.3.7 we adapt the techniques tocurved fractures; finally, in Remark 2.3.9 we generalize the former results to A 6= Id. To thisaim, some preliminaries are in order: first, in Remark 2.3.1 we compute the partial derivativesof u in a more convenient way, then in Lemmas 2.3.2 and 2.3.3 we provide two key results,based on Geometric Measure Theory. Once this is done, we deduce formula (10) in the timeinterval [t0, t1] where the decomposition (2.2.9) holds.
For brevity of notation, in this section we consider [t0, t1] = [0, 1]. All the results can beeasily extended to the general case. The global result in [0, T ] easily follows by iterating theprocedure a finite number of steps, and using both the additivity of the integrals and the factthat k depends only on A, Î, and s (see Theorem 2.2.10 and Remark 2.2.11).
54 2.3. Dynamic energy-dissipation balance
Remark 2.3.1. Let us focus our attention on a fracture which is straight in a neighborhoodof the tip. Without loss of generality, we may fix the origin so that for every t â [0, 1]
Ît \ Î0 = (Ï, 0) â R2 : 0 < Ï â€ s(t)â s(0).
The diffeomorphisms Ï and Î introduced in Section 2.1 can be both taken equal to theidentity, so that, in a neighborhood of the origin, the diffeomorphisms Ί(t) defined in (2.2.2)simply read
Ί(t, x) =
(x1 â (s(t)â s(0))â
1â |s(t)|, x2
)for t â [0, 1] and x â Ω.
Accordingly, the decomposition result in Theorem 2.2.4 states that the solution u to (2.1.2)â(2.1.5) with suitable initial conditions can be decomposed as
u(t, x) = uR(t, x) + k(t)ζ(t, x)S(t, x) for t â [0, 1] and x â Ω \ Ît,
where, for brevity, we have set S(t, x) := S(Ί(t, x)). We recall that uR(t) â H2(Ω \ Ît) forevery t â [0, 1] and S(y) = y2â
2â|y|+y1
for y â R2 \ (Ï, 0) : Ï â€ 0.Let us now compute the partial derivatives of u. For t â [0, 1] and x â Ω \ Ît we get
âu(t, x) = âuR(t, x) + k(t)âζ(t, x)S(t, x) + k(t)ζ(t, x)âS(t, x), (2.3.1)
u(t, x) = uR(t, x) + k(t)ζ(t, x)S(t, x) + k(t)ζ(t, x)S(t, x) + k(t)ζ(t, x) ËS(t, x). (2.3.2)
Since in R2 \ (Ï, 0) : Ï â€ 0 we have
â1S(y) = â y2
2â
2|y|â|y|+ y1
, â2S(y) =
â|y|+ y1
2â
2|y|,
â211S(y) =
2y1y2 + y2|y|4â
2|y|3â|y|+ y1
, â222S(y) = â 2y1y2 + y2|y|
4â
2|y|3â|y|+ y1
,
â212S(y) = â2
21S(y) =
â|y|+ y1(|y| â 2y1)
4â
2|y|3,
we claim
u(t)âu(t)â k2(t)ζ2(t) ËS(t)âS(t) âW 1,1(Ω \ Ît;R2) for every t â [0, 1].
Indeed, âuR(t), ζ(t)S(t), uR(t), ζ(t)S(t), and k(t)ζ(t)S(t) are functions in H1(Ω \ Ît) forevery t â [0, 1]; by the Sobolev embeddings theorem we deduce that they belong to Lp(Ω)
for every p â„ 1. By using also the explicit form of S(t) and ËS(t), one can check that thesefunctions are elements of W 1,4/3(Ω \ Ît). Therefore, we can easily conclude that the product
of each term in (2.3.1) with each term in (2.3.2), except for k2(t)ζ2(t) ËS(t)âS(t), is a functionin W 1,1(Ω \ Ît;R2) for every t â [0, 1].
Lemma 2.3.2. Let a, b â R, with a < 0 and b > 0, and define H+ := (x1, x2) â R2 : x2 â„ 0to be the upper half plane in R2. Let g : H+ â R be bounded, continuous at the origin, andcall Ï a modulus of continuity for g at x = 0. ThenâŁâŁâŁâŁ1Δ
⫠Δ
0
(â« b
ag(x1, x2)
x2
x21 + x2
2
dx1
)dx2 â Ïg(0, 0)
âŁâŁâŁâŁâ€ âgâLâ(H+)
(2Δ1/2|bâ a|+ Ξ(Δ)
)+ ÏÏ(Δ1/4),
(2.3.3)
Chapter 2. Dynamic energy-dissipation balance of a growing crack 55
where
Ξ(Δ) :=
âŁâŁâŁâŁÏ â â« 1
0
[arctan
(b
Δx2
)â arctan
(a
Δx2
)]dx2
âŁâŁâŁâŁ .In particular, for every g : H+ â R bounded and continuous at the origin, we have
limΔâ0+
1
Δ
⫠Δ
0
(â« b
ag(x1, x2)
x2
x21 + x2
2
dx1
)dx2 = Ïg(0, 0).
Proof. After a change of variable on the integral in (2.3.3), we can rewrite it asâ« 1
0
(â« b
ag(x1, Δx2)
Δx2
x21 + (Δx2)2
dx1
)dx2.
Notice thatâ« b
a
Δx2
x21 + (Δx2)2
dx1 =
â« b
aâ1 arctan
(x1
Δx2
)dx1 = arctan
(b
Δx2
)â arctan
(a
Δx2
),
thereforeâŁâŁâŁâŁâ« 1
0
(â« b
ag(x1, Δx2)
Δx2
x21 + (Δx2)2
dx1
)dx2 â Ïg(0, 0)
âŁâŁâŁâŁâ€âŁâŁâŁâŁâ« 1
0
(â« b
a[g(x1, Δx2)â g(0, 0)]
Δx2
x21 + (Δx2)2
dx1
)dx2
âŁâŁâŁâŁ+ g(0, 0)Ξ(Δ)
â€
âŁâŁâŁâŁâŁâ« 1
0
(â«(a,b)\(âΔ1/4,Δ1/4)
[g(x1, Δx2)â g(0, 0)]Δx2
x21 + (Δx2)2
dx1
)dx2
âŁâŁâŁâŁâŁ+ ÏÏ(Δ1/4) + g(0, 0)Ξ(Δ).
By using the estimate
supxâ[(a,b)\(âΔ1/4,Δ1/4)]Ă(0,1)
Δx2
(x21 + (Δx2)2)
†Δ1/2
1 + Δ3/2†Δ1/2,
valid for every Δ â (0, 1), we can continue the above chain of inequalities and we getâŁâŁâŁâŁâ« 1
0
(â« b
ag(x1, Δx2)
Δx2
x21 + (Δx2)2
dx1
)dx2 â Ïg(0, 0)
âŁâŁâŁâŁâ€ Δ1/2
â« 1
0
(â«(a,b)\(âΔ1/4,Δ1/4)
|g(x1, Δx2)â g(0, 0)|dx1
)dx2 + ÏÏ(Δ1/4) + g(0, 0)Ξ(Δ)
†2Δ1/2âgâLâ(H+)|bâ a|+ ÏÏ(Δ1/4) + g(0, 0)Ξ(Δ),
which is (2.3.3).
Lemma 2.3.3. Let Ω â R2 be open and let Îł : [0, `] â Ω be a Lipschitz curve. Let us set
Î := Îł(Ï) : Ï â [0, `], and for every Δ > 0 let us define ÏΔ(x) := dist(x,Î)Δ ⧠1 for x â Ω.
Then for every u âW 1,1(Ω \ Î) and for every v : Ωâ R bounded and satisfying
limxâx
v(x) = v(x) for every x â Î,
we have
limΔâ0+
â«dist+(x,Î)<Δ
u(x)v(x)|âÏΔ(x)| dx =
â«Îu+(x)v(x) dH1(x),
wheredist+(x,Î) < Δ :=
âÏâ[0,`]
(BΔ(Îł(Ï)) â© x â Ω : x · (Îł(Ï))â„ > 0
),
56 2.3. Dynamic energy-dissipation balance
and u+ is the trace on Î from dist+(x,Î) < Δ. Equivalently,
limΔâ0+
â«distâ(x,Î)<Δ
u(x)v(x)|âÏΔ(x)| dx =
â«Îuâ(y)v(y) dH1(y),
wheredistâ(x,Î) < Δ :=
âÏâ[0,`]
(BΔ(Îł(Ï)) â© x â Ω : x · (Îł(Ï))â„ < 0
),
and uâ is the trace on Î from distâ(x,Î) < Δ.
Proof. It is enough to apply the coarea formula to the Lipschitz maps ÏΔ.
Remark 2.3.4. In what follows we compute the energy balance in the case of homogeneousNeumann conditions on the whole âΩ. However, the same proof applies with no changes tothe case of Dirichlet boundary conditions. For example, to treat the homogeneous Dirichletcondition on âDΩ â âΩ, it is enough to check that the time derivative of the solution u(t)has still zero trace on âΩ, in such a way that it still remains an admissible test function. Butthis is simply because the incremental quotient in time 1
h [u(t + h) â u(t)] converges to u(t)as hâ 0, strongly in H1 in a sufficiently small neighborhood of âDΩ, so that u has still zerotrace on the Dirichlet part of the boundary.
Analogously, if we prescribe a regular enough non-homogeneous Dirichlet boundary condi-tion, we can rewrite the wave equation changing the forcing term f appearing in its right-handside, and turn the non-homogeneous Dirichlet condition into a homogeneous one. Also in thiscase, the computations follow unchanged.
Proposition 2.3.5. Let Ω â R2 be an open bounded set with Lipschitz boundary and letÎttâ[0,1] be a family of rectilinear cracks inside Ω of the form
Ît := Ω â© (Ï, 0) â R2 : Ï â€ s(t),
where s â C2([0, 1]) and s(t) â„ 0 for every t â [0, 1]. Suppose that a function u : [0, 1]ĂΩâ Rcan be decomposed as in (2.2.9) for t â [0, 1] and satisfies the wave equation
u(t)ââu(t) = f(t) in Ω \ Ît, t â [0, 1], (2.3.4)
with homogeneous Neumann boundary conditions on the boundary and on the cracks. Thenu satisfies the dynamic energy-dissipation balance (11) for every t â [0, 1] if and only if thestress intensity factor k is constantly equal to 2â
Ïin the set t â [0, 1] : s(t) > 0.
Proof. By hypothesis we can decompose the function u as u(t, x) = uR(t, x)+k(t)ζ(t, x)S(t, x)for t â [0, 1] and x â Ω \Ît, where uR(t) â H2(Ω \Ît), ζ(t) is a cut-off function supported ina neighborhood of the moving tip of Ît, and
S(t, x) := S
(x1 â (s(t)â s(0))â
1â |s(t)|2, x2
),
with S(y) = y2â2â|y|+y1
for y â R2 \ (Ï, 0) : Ï â€ 0.
Let us fix tâ â [0, 1] and for every Δ > 0 let us define ÏΔ(x) := dist(x,Îtâ\Î0)Δ ⧠1 for x â Ω.
Since ÏΔu(t) belongs to H1(Ω\Ît) for every t â [0, tâ], we can use it as test function in (2.3.4),and we get â« tâ
0ău(t), ÏΔu(t)ă(H1(Ω\Ît))âČ dt+
â« tâ
0(âu(t), ÏΔâu(t))L2(Ω) dt
+
â« tâ
0(âu(t),âÏΔu(t))L2(Ω) dt =
â« tâ
0(f(t), ÏΔu(t))L2(Ω) dt.
(2.3.5)
Chapter 2. Dynamic energy-dissipation balance of a growing crack 57
By using integration by parts with the fact that t 7â (u(t), ÏΔu(t))L2(Ω) is absolutely contin-uous, we obtainâ« tâ
0ău(t), ÏΔu(t)ă(H1(Ω\Ît))âČ dt =
1
2
â« tâ
0
d
dt(u(t), ÏΔu(t))L2(Ω) dt
=1
2(u(tâ), ÏΔu(tâ))L2(Ω) â
1
2(u(0), ÏΔu(0))L2(Ω),
and by passing to the limit as Δâ 0+ by the dominated convergence theorem, we have
limΔâ0+
â« tâ
0ău(t), ÏΔu(t)ă(H1(Ω\Ît))âČ dt =
1
2âu(tâ)â2L2(Ω) â
1
2âu(0)â2L2(Ω).
Analogously, we take the limit as Δâ 0+ in the right-hand side of (2.3.5) and in the secondterm in the left-hand side, and we get
limΔâ0+
â« tâ
0(âu(t),âu(t)ÏΔ)L2(Ω) dt =
1
2ââu(tâ)â2L2(Ω) â
1
2ââu(0)â2L2(Ω),
limΔâ0+
â« tâ
0(f(t), u(t)ÏΔ)L2(Ω) dt =
â« tâ
0(f(t), u(t))L2(Ω) dt.
The most delicate term is the third one in the left-hand side of (2.3.5). First of all, we writethe partial derivatives explicitly:
â[k(t)ζ(t, x)S(t, x)] = k(t)âζ(t, x)S(t, x) + k(t)ζ(t, x)âS(t, x),
ât[k(t)ζ(t, x)S(t, x)] = k(t)ζ(t, x)S(t, x) + k(t)ζ(t, x)S(t, x) + k(t)ζ(t, x) ËS(t, x).
Moreover, if we consider Ί1(t, x) = x1âs(t)â1â|s(t)|2
, we have
âS(t, x) =
(â1S(Ί1(t, x), x2)â
1â |s(t)|2, â2S(Ί1(t, x), x2)
)and
ËS(t, x) =âs(t)(1â |s(t)|2) + s(t)s(t)(x1 â (s(t)â s(0)))
(1â |s(t)|2)3/2â1S(Ί1(t, x), x2)
= Ί1(t, x)â
1â |s(t)|2â1S(t, x).
Thanks to Remark 2.3.1, the only contribution to the limit as Δâ 0+ is given byâ« tâ
0k2(t)(ζ2(t)âS(t),âÏΔ ËS(t))L2(Ω) dt.
Therefore, we need to compute
limΔâ0+
â« tâ
0
(â«dist(x,Ît\Î0)<Δ
k2(t)ζ2(t, x)âS(t, x) · âÏΔ(x) ËS(t, x) dx
)dt. (2.3.6)
To this aim, we set
IΔ(t) :=
â«dist(x,Ît\Î0)<Δ
k2(t)ζ2(t, x)âS(t, x) · âÏΔ(x) ËS(t, x) dx,
and we decompose IΔ as I+Δ + IâΔ , where I+
Δ is the integral restricted to the upper half planex â R2 : x2 > 0 and IâΔ is the integral restricted to the lower half plane x â R2 : x2 < 0.
58 2.3. Dynamic energy-dissipation balance
Let us focus on I+Δ (t). For brevity, we write r(t) := (s(t) â s(0), 0) for every t â [0, 1].
The gradient of ÏΔ reads
âÏΔ(x) =
e2Δ in x â R2 : 0 †x1 †s(tâ)â s(0), 0 †x2 < Δ,xΔ|x| in x â R2 : x â BΔ(0), x1 < 0, x2 â„ 0,xâr(tâ)Δ|xâr(tâ)| in x â R2 : x â BΔ(r(tâ)), x1 > s(tâ)â s(0), x2 â„ 0,0 otherwise.
Thus we get
I+Δ (t) =
1
Δ
â«[0,s(tâ)âs(0)]Ă(0,Δ]
k2(t)â
1â |s(t)|2ζ2(t, x)â2S(t, x)Ί1(t, x)â1S(t, x) dx
+1
Δ
â«BΔ(0)â©x1<0, x2â„0
k2(t)â
1â |s(t)|2ζ2(t, x)âS(t, x) · x|x|
Ί1(t, x)â1S(t, x) dx (2.3.7)
+1
Δ
â«BΔ(r(tâ))â©x1>s(tâ)âs(0), x2â„0
k2(t)â
1â |s(t)|2ζ2(t, x)âS(t, x) · xâ r(tâ)
|xâ r(tâ)|Ί1(t, x)â1S(t, x) dx.
We notice that the last two terms in (2.3.7) have integrands which are bounded on thedomains of integration, and so passing to the limit as Δ goes to 0 they do not give anycontribution. Thus we only have to analyze the first term of (2.3.7). By recalling that
S(t, x) = S(Ί1(t, x), x2), Ί1(t, x) = x1â(s(t)âs(0))â1â|s(t)|2
, and that ζ(t, x) = ζ(Ί1(t, x), x2), with ζ
cut-off function supported in a neighborhood of the origin, and making the change of variablexâČ1â
1â |s(t)|2 = x1 â (s(t)â s(0)), we rewrite the first term of (2.3.7) as
â k2(t)s(t)
Δ
⫠Δ
0
â« bt
at
ζ2(x1, x2)â1S(x1, x2)â2S(x1, x2) dx1 dx2
+k2(t)s(t)s(t)
Δâ
1â |s(t)|2
⫠Δ
0
â« bt
at
x1ζ2(x1, x2)â1S(x1, x2)â2S(x1, x2) dx1 dx2,
(2.3.8)
where the interval (at, bt) denotes the segment
(at, bt) :=
(s(0)â s(t)â
1â |s(t)|2,s(tâ)â s(t)â
1â |s(t)|2
).
Notice that
â k2(t)s(t)
Δ
⫠Δ
0
â« bt
at
ζ2(x1, x2)â1S(x1, x2)â2S(x1, x2) dx1 dx2
=k2(t)s(t)
Δ
⫠Δ
0
â« bt
at
ζ2(x1, x2)x2
8|x|2dx1 dx2,
and that the map (x1, x2) 7â ζ2(x1, x2) is bounded and continuous in (0, 0), with ζ(0, 0) = 1.Therefore we are in a position to apply Lemma 2.3.2, which gives
limΔâ0+
k2(t)s(t)
Δ
⫠Δ
0
â« bt
at
ζ2(x1, x2)x2
8|x|2dx1 dx2 =
Ï
8k2(t)s(t)ζ2(0, 0) =
Ï
8k2(t)s(t).
By arguing in the very same way, we can show that the limit as Δ â 0+ of the secondterm of (2.3.8), thanks to the presence of x1, is zero. This means that the limit of I+
Δ (t) is
limΔâ0+
I+Δ (t) =
Ï
8k2(t)s(t),
Chapter 2. Dynamic energy-dissipation balance of a growing crack 59
and, similarly,
limΔâ0+
IâΔ (t) =Ï
8k2(t)s(t).
All in all,
limΔâ0+
IΔ(t) = limΔâ0+
(I+Δ (t) + IâΔ (t)) =
Ï
4k2(t)s(t).
Thanks to the estimate in (2.3.3), we infer that the family of functions I+Δ (t)Δ>0 are domi-
nated on [0, 1] by a bounded function, and the same holds for IâΔ (t)Δ>0; by the dominatedconvergence theorem, we can pass the limit in (2.3.6) inside the integral in time, and we canwrite
limΔâ0+
â« tâ
0IΔ(t) dt =
â« tâ
0
Ï
4k2(t)s(t) dt.
So we deduce that the dynamic energy-dissipation balance (11) holds for every t â [0, 1] ifand only if the stress intensity factor k(t) is equal to 2â
Ïwhenever s(t) > 0.
Remark 2.3.6. We underline that our approach is different to that of Dal Maso, Larsen,and Toader in [17, Section 4]: in order to derive the energy balance associated to a horizontalcrack opening with constant velocity c, they prove that the mechanical energy of u(t) isconstant in the moving ellipse Er(t) := (x1, x2) â R2 : (x1âct)2/(1âc2)+x2
2 †r2 centeredat the crack-tip (ct, 0), for some small r > 0, and they make the explicit computation of theenergy in R2 \ Er(t).
We now generalize the previous result to non straight cracks.
Theorem 2.3.7. Let Ω â R2 be an open bounded set with Lipschitz boundary and letÎttâ[0,1] be a family of growing cracks inside Ω. Assume that there exists a bi-Lipschitzmap Î: Ωâ Ω with the following properties:
(i) Î(Ît \ Î0) = (Ï, 0) â R2 : 0 < Ï â€ s(t) â s(0), where s â C2([0, 1]) and s(t) â„ 0 forevery t â [0, 1],
(ii) H1 (Î(Ît \ Î0)) = H1 (Ît \ Î0) for every t â [0, 1];
(iii) limxâxâÎ(x) = âÎ(x) â SO(2)+, for every x â Î1 \ Î0.
Suppose that a function u : [0, 1] à Ω â R can be decomposed as in (2.2.9) for t â [0, 1] andsatisfies the wave equation
u(t)ââu(t) = f(t) in Ω \ Ît, t â [0, 1], (2.3.9)
with homogeneous Neumann boundary conditions on the boundary and on the cracks. Thenu satisfies the dynamic energy-dissipation balance (11) for every t â [0, 1] if and only if thestress intensity factor k is constantly equal to 2â
Ïin the set t â [0, 1] : s(t) > 0.
Proof. In view of (2.2.9), we have u(t, x) = uR(t, x) + k(t)ζ(t,Î(x))S(t,Î(x)), for t â [0, 1]and x â Ω\Ît, with uR(t) â H2(Ω\Ît), ζ(t)Î cut-off function supported in a neighborhoodof the moving tip of Ît, and
S(t,Î(x)) := S
(Î1(x)â (s(t)â s(0))â
1â |s(t)|2,Î2(x)
),
being S(y) = y2â2â|y|+y1
for y â R2 \ (Ï, 0) : Ï â€ 0.
60 2.3. Dynamic energy-dissipation balance
As in the proof of Proposition 2.3.5, we fix tâ â [0, 1] and for every Δ > 0 we define the
function ÏΔ(x) := dist(x,Îtâ\Î0)Δ ⧠1 for x â Ω. Since ÏΔu(t) â H1(Ω \ Ît), we can use it as test
function in (2.3.9), and we getâ« tâ
0ău(t), ÏΔu(t)ă(H1(Ω\Ît))âČ dt+
â« tâ
0(âu(t), ÏΔâu(t))L2(Ω) dt
+
â« tâ
0(âu(t),âÏΔu(t))L2(Ω) dt =
â« tâ
0(f(t), ÏΔu(t))L2(Ω) dt.
(2.3.10)
By integrating by parts, we easily obtain
limΔâ0+
â« tâ
0ău(t), ÏΔu(t)ă(H1(Ω\Ît))âČ dt =
1
2âu(tâ)â2L2(Ω) â
1
2âu(0)â2L2(Ω), (2.3.11)
limΔâ0+
â« tâ
0(âu(t), ÏΔâu(t))L2(Ω) dt =
1
2ââu(tâ)â2L2(Ω) â
1
2ââu(0)â2L2(Ω), (2.3.12)
limΔâ0+
â« tâ
0(f(t), ÏΔu(t))L2(Ω) dt =
â« tâ
0(f(t), u(t))L2(Ω) dt. (2.3.13)
The asymptotics as Δâ 0+ of the third term in the left-hand side of (2.3.10) is more delicateto handle. To simplify the notation, we set
ζ(t, x) := ζ(t,Î(x)), ÏΔ(x) := ÏΔ(Îâ1(x)) for t â [0, 1] and x â Ω.
By using Lemma 2.3.3 and Remark 2.3.1, as in the proof of the previous proposition for therectilinear case, we have that the only contribution to the limit as Δ â 0+ is given by thetermâ«
Ωk2(t)ζ
2(t, x)âS(t,Î(x)) · âÏΔ(x) ËS(t,Î(x)) dx
=
â«Î©k2(t)α(t)[âÎ(x)TâS(t,Î(x))] · âÏΔ(x)ζ
2(t, x)Ί1(t,Î(x))â1S(t,Î(x)) dx
=
â«Î©k2(t)α(t)[âÎ(Îâ1(x))TâS(t, x)] · âÏΔ(Îâ1(x))ζ2(t, x)Ί1(t, x)â1S(t, x)|JÎâ1(x)| dx
=
â«Î©k2(t)α(t)âS(t, x) · [B(Îâ1(x))âÏΔ(x)]ζ2(t, x)Ί1(t, x)â1S(t, x)|JÎâ1(x)| dx, (2.3.14)
where Ί1(t, x) := x1â(s(t)âs(0))â1â|s(t)|2
, B(x) := âÎ(x)âÎ(x)T , JÎâ1(x) := detâÎâ1(x), and
α(t) :=â
1â |s(t)|2 for t â [0, 1] and x â Ω. In the last equality we have used the coareaformula applied with the Lipschitz change of variables Îâ1.
Thanks to our construction of Î, for any x in a suitable small neighborhood of the tipof Î(Î1) we have
B(Îâ1(x)) =
(b11(x) 0
0 1
),
where b11 : R2 â R is a continuous function with b11(x1, 0) = 1. The last term in (2.3.14)can be split asâ«
Ωk2(t)α(t)b11(x)â1ÏΔ(x)ζ2(t, x)Ί1(t, x)(â1S(t, x))2|JÎâ1(x)| dx
+
â«Î©k2(t)α(t)â2S(t, x)â2ÏΔ(x)ζ2(t, x)Ί1(t, x)â1S(t, x)|JÎâ1(x)|dx.
Chapter 2. Dynamic energy-dissipation balance of a growing crack 61
By construction of Î, each line parallel to x â R2 : x2 = 0 is mapped by Îâ1 into a levelset of ÏΔ; more precisely ÏΔ(Î
â1(x â R2 : x2 = s)) = sΔ ⧠1, and this means that on the set
of points x â R2 : dist(x,Î(Î1)) †Δ, we have
âÏΔ(x) =
e2Δ in x â R2 : 0 †x1 †s(tâ)â s(0), 0 †x2 < Δ,xΔ|x| in x â R2 : x â BΔ(0), x1 < 0, x2 â„ 0,xâr(tâ)Δ|xâr(tâ)| in x â R2 : x â BΔ(r(tâ)), x1 > s(tâ)â s(0), x2 â„ 0,0 otherwise,
where, for brevity, we have set r(t) := (s(t)â s(0), 0) for every t â [0, 1].Since Î is a bi-Lipschitz map, JÎâ1 is bounded, thus by hypothesis (iii) we have
limxâ(s(t)âs(0),0)
|JÎâ1(x)| = 1,
for every t â [0, 1]. Moreover, in view of assumption (iii), we have that |JÎâ1| is continuouson the compact set Î1 \ Î0, hence uniformly continuous; therefore, proceeding exactly as inthe proof of Proposition 2.3.5, we can write
limΔâ0+
â«Î©k2(t)α(t)â2S(t, x)â2ÏΔ(x)ζ2(t, x)Ί1(t, x)â1S(t, x)|JÎâ1(x)|dx
=Ï
4k2(t)s(t).
(2.3.15)
Again by hypothesis (iii), we can apply estimate (2.3.3) and deduce that the sequence ofintegrands in (2.3.15) is dominated in t, so that we can apply the dominated convergencetheorem to deduce
limΔâ0+
â« tâ
0
(â«Î©k2(t)α(t)â2S(t, x)â2ÏΔ(x)ζ2(t, x)Ί1(t, x)â1S(t, x)|JÎâ1(x)|dx
)dt
=
â« tâ
0
Ï
4k2(t)s(t) dt.
(2.3.16)
By combining (2.3.10) with (2.3.11)â(2.3.13) and (2.3.16), we infer that
E(t)â E(0) +Ï
4
â« t
0k2(Ï)s(Ï) dÏ =
â« t
0(f(Ï), u(Ï))L2(Ω) dÏ for every t â [0, 1]. (2.3.17)
Hence, the dynamic energy-dissipation balance (11) is satisfied if and only ifâ« t
0
Ï
4k2(Ï)s(Ï) dÏ = H1(Ît \ Î0) = H1(Î(Ît \ Î0)) = s(t) for every t â [0, 1],
which is true if and only if k(t) is equal to 2âÏ
whenever s(t) > 0. This concludes the
proof.
Remark 2.3.8. Our approach is constructive and allows us to show the existence of time-dependent pairs t 7â (Ît, u(t)) satisfying the dynamic energy-dissipation balance (11). Un-der the standing assumptions on Ît, it is enough to take the forcing term f associated to
2âÏζ(Ί(t))S(Ί(t)) (which of course is a solution u(t)), where ζ is a suitable cut-off function
supported in a neighborhood of the origin. In order to ensure the homogeneous Neumanncondition on the fracture, we choose ζ satisfying â2ζ(y1, 0) = 0 for every y1 â R. This canbe achieved, e.g., by taking ζ(y1, y2) = Ï(y1)Ï(y2), where Ï â Câc (R) has compact supportcontained in (âΔ, Δ) and satisfies Ï = 1 in (âΔ/2, Δ/2), for some Δ > 0.
62 2.3. Dynamic energy-dissipation balance
Remark 2.3.9. When in equation (2.1.2) the matrix A is (possibly) not the identity, anenergy balance similar to (2.3.17) is still valid: for every t â [0, 1] there holds
E(t)â E(0) +Ï
4
â« t
0k2(Ï)a(Ï)s(Ï)dÏ =
â« t
0(f(Ï), u(Ï))L2(Ω)dÏ, (2.3.18)
where a is a function depending only on A, Î, and s, and it is given by
a(t) := |A(r(t))â1/2Îł(s(t))| · |A(r(t))1/2Îœ(s(t))| ·â
detA(r(t)).
Here A1/2 and Aâ1/2 denote the square root of the symmetric and positive definite matricesA and Aâ1, respectively, and Îł(s(t)) and Îœ(s(t)) are the tangent and unit normal vectorsto Î at the point r(t) := Îł(s(t)), respectively. In this case, the dynamic energy-dissipationbalance (11) holds true if and only if the stress intensity factor k(t) satisfies
k(t) =2âÏa(t)
during the crack opening, namely when s(t) > 0.In order to derive formula (2.3.18), we use the decomposition result (2.2.9) rewritten as
u(t, x) = uR(t, x) + k(t)ζ(t, x)S(t, Ï(x)),
where S(t, x) is the singular part of the solution relative to the transformed curve Î(1) = Ï(Î).Then we proceed as in the previous theorem and proposition: we test the PDE with ÏΔu(t),
where ÏΔ(x) := dist(x,Îtâ\Î0)Δ ⧠1 for x â Ω, and, as before, we notice that the only delicate
term is the one that converges to the integral in the left hand-side of (2.3.18):
limΔâ0+
â« tâ
0k2(t)
(â«Î©Î¶2(t, x)[A(x)âS(t, Ï(x))] · âÏΔ(x) ËS(t, Ï(x)) dx
)dt.
By applying the change of variables Ïâ1, we can rewrite the space integral in the previousexpression as follows:â«
Ωζ2(t, x)([âÏAâÏT ](Ïâ1(x))âS(t, x)) · âÏ(x)âTâÏΔ(x)(Ïâ1(x)) ËS(t, x)| detâÏâ1(x)|dx.
Finally, we work on the transformed curve Î(1), exactly as in the previous theorem, byusing the property of the singular part S(t, x) together with the following facts: by con-struction, [âÏAâÏT ] Ïâ1 is a continuous function which agrees with the identity on thepoints of Î(1); moreover âÏ(x)âTâÏΔ(Ïâ1(x)) is a continuous function which is equal to1Δ |A(r(t))1/2Îœ(s(t))|Îœ(1)(s(1)(t)) on the points Îł(1)(s(1)(t)) of Î(1); the velocity s(1) of the
curve Î(1) satisfies s(1)(t) = |A(r(t))â1/2Îł(s(t))|s(t); finally, |detâÏâ1(x)| is a continuousfunction equal to
âdetA(r(t)) on the points Îł(1)(s(1)(t)) of Î(1).
Remark 2.3.10. By combining Theorems 2.2.4 and 2.2.10 with Theorem 2.3.7 and Re-marks 2.3.4 and 2.3.9 we deduce that if f , u0, and u1 satisfy the assumptions of Theo-rem 2.2.10, then the unique solution u to (2.1.2)â(2.1.5) satisfies (10) for every t â [0, T ].This formula gives an important quantitative information on the functions k and s whichsatisfy the dynamic energy-dissipation balance (11): for every t â [0, T ](
2âÏa(t)
â k(t)
)s(t) = 0.
In particular, in the set t â [0, T ] : s(t) > 0 the stress intensity factor k coincides with thefunction 2/
âÏa.
Chapter 3
A dynamic model for viscoelasticmaterials with growing cracks
In this chapter we prove an existence and uniqueness result for equation (15) and the analo-gous problem in linear elasticity, that is system (3.1.12).
This chapter is organized as follows. In Section 3.1 we fix the notation adopted throughoutthe chapter, we list the main assumptions on the family of cracks Îttâ[0,T ] and on thefunction Î, and we specify the notion of solution to (3.1.12). In Section 3.2 we state ourmain existence result (Theorem 3.2.1), which is obtained by means of a time discretizationscheme. We conclude the proof of Theorem 3.2.1 in Section 3.3, where we show the validity ofthe initial conditions (3.1.21) and the energy-dissipation inequality (3.3.4). Section 3.4 dealswith the uniqueness problem. Under stronger regularity assumptions on the cracks sets, inTheorem 3.4.4 we prove the uniqueness, but only when the space dimension is d = 2. Tothis aim, we assume also that the function Î is zero in a neighborhood of the crack-tip. Weconclude with Section 3.5, where, in dimension d = 2 and for an antiplane evolution, we showan example of a moving crack which satisfies the dynamic energy-dissipation balance (16).
The results presented here are obtained in collaboration with F. Sapio and are containedin the submitted paper [10].
3.1 Preliminary results
Let T be a positive real number and let Ω â Rd be a bounded open set with Lipschitzboundary. Let âDΩ be a (possibly empty) Borel subset of âΩ and let âNΩ be its complement.We assume the following hypotheses on the geometry of the crack sets Îttâ[0,T ]:
(E1) Î â Ω is a closed set with Ld(Î) = 0 and Hdâ1(Î â© âΩ) = 0;
(E2) for every x â Î there exists an open neighborhood U of x in Rd such that (U ⩠Ω) \ Îis the union of two disjoint open sets U+ and Uâ with Lipschitz boundary;
(E3) Îttâ[0,T ] is a family of closed subsets of Î satisfying Îs â Ît for every 0 †s †t †T .
Remark 3.1.1. Assumptions (E1)â(E3) are a weaker version of assumptions (H1)â(H3) ofChapter 1. In particular, we do need Î to be a C2 manifold of dimension (d â 1), since weare not interested in define the trace of Ï â H1(Ω \ Î) on Î.
Thanks (E1)â(E3) the space L2(Ω \ Ît;Rm) coincides with L2(Ω;Rm) for every t â [0, T ]and m â N. In particular, we can extend a function Ï â L2(Ω \ Ît;Rm) to a functionin L2(Ω;Rm) by setting Ï = 0 on Ît. Moreover, by arguing as for (1.1.1), the trace of
63
64 3.1. Preliminary results
Ï â H1(Ω \ Î;Rd) on âΩ is well defined and there exists a constant Ctr > 0, depending onlyon Ω and Î, such that
âÏâL2(âΩ) †CtrâÏâH1(Ω\Î) for every Ï â H1(Ω \ Î;Rd). (3.1.1)
Hence, for every t â [0, T ] we can define the space H1D(Ω \ Ît;Rd) as done in (1.1.7), and
we denote its dual by Hâ1D (Ω \ Ît;Rd). Similarly, by proceeding as for (1.2.3), we can find a
constant CK , depending only on Ω and Î, such that
ââÏâ2L2(Ω) †CK(âÏâ2L2(Ω) + âEÏâ2L2(Ω)
)for every Ï â H1(Ω \ Î;Rd). (3.1.2)
Let C,D : Ωâ L (RdĂdsym;RdĂdsym) be two fourth-order tensors satisfying:
C,D â Lâ(Ω; L (RdĂdsym;RdĂdsym)), (3.1.3)
(C(x)Ο1) · Ο2 = Ο1 · (C(x)Ο2) for every Ο1, Ο2 â RdĂdsym and for a.e. x â Ω, (3.1.4)
(D(x)Ο1) · Ο2 = Ο1 · (D(x)Ο2) for every Ο1, Ο2 â RdĂdsym and for a.e. x â Ω. (3.1.5)
We require that C and D satisfy the following ellipticity condition, which is standard in linearelasticity:
C(x)Ο · Ο ℠λ1|Ο|2, D(x)Ο · Ο ℠λ2|Ο|2 for every Ο â RdĂdsym and for a.e. x â Ω, (3.1.6)
for two positive constants λ1, λ2 independent of x. By combining the ellipticity conditionfor C with the Kornâs inequality (3.1.2), we can find two constants c0 > 0 and c1 â R suchthat
(CEÏ,EÏ)L2(Ω) â„ c0âÏâ2H1(Ω\ÎT ) â c1âÏâ2L2(Ω) for every Ï â H1(Ω \ ÎT ;Rd). (3.1.7)
Let us consider a function Î: (0, T )à Ωâ R satisfying
Î â Lâ((0, T )à Ω), âÎ â Lâ((0, T )à Ω;Rd). (3.1.8)
Given
w â H2(0, T ;L2(Ω;Rd)) â©H1(0, T ;H1(Ω \ Î0;Rd)), (3.1.9)
f â L2(0, T ;L2(Ω;Rd)), F â H1(0, T ;L2(âNΩ;Rd)), (3.1.10)
u0 â w(0) â H1D(Ω \ Î0;Rd), u1 â L2(Ω;Rd), (3.1.11)
we want to find a solution to the viscoelastic dynamic system
u(t)â div(CEu(t))â div(Î2(t)DEu(t)) = f(t) in Ω \ Ît, t â [0, T ], (3.1.12)
satisfying the following boundary conditions
u(t) = w(t) on âDΩ, t â [0, T ], (3.1.13)
(CEu(t) + Î2(t)DEu(t))Îœ = F (t) on âNΩ, t â [0, T ], (3.1.14)
(CEu(t) + Î2(t)DEu(t))Îœ = 0 on Ît, t â [0, T ], (3.1.15)
and initial conditions
u(0) = u0, u(0) = u1 in Ω \ Î0. (3.1.16)
As pointed out in Chapter 1, the Neumann boundary conditions (3.1.14) and (3.1.15) areonly formal, and their meaning will be specified later in Definition 3.1.5.
Chapter 3. A dynamic model for viscoelastic materials with growing cracks 65
Throughout the chapter we always assume that the family of cracks Îttâ[0,T ] satisfies(E1)â(E3), as well as C, D, Î, f , w, F , u0, and u1 satisfy (3.1.3)â(3.1.11). In the following,we want to specify the notion of solution to problem (3.1.12)â(3.1.15). As pointed out inChapter 1, the main difficulty is to give a meaning to u(t). For this reason, we follow thedefinition given in [23, Definition 2.7], which does not require the second derivative of u intime. To simplify the notation, let us define the following three functional spaces:
V := Ï â L2(0, T ;H1(Ω \ ÎT ;Rd)) : Ï â L2(0, T ;L2(Ω;Rd)),Ï(t) â H1(Ω \ Ît;Rd) for a.e. t â (0, T ),
VD := Ï â V : Ï(t) â H1D(Ω \ Ît;Rd) for a.e. t â (0, T ),
W := u â V : Îu â L2(0, T ;H1(Ω \ ÎT ;Rd)),Î(t)u(t) â H1(Ω \ Ît;Rd) for a.e. t â (0, T ).
Remark 3.1.2. In the classical viscoelastic case, namely when Î is identically equal to 1,the solution u to system (3.1.12) has derivative u(t) â H1(Ω \ Ît;Rd) for a.e. t â (0, T ) withEu â L2(0, T ;L2(Ω;RdĂd)). For a generic Î we expect to have ÎEu â L2(0, T ;L2(Ω;RdĂd)).Therefore W is the natural setting where looking for a solution to (3.1.12). Indeed, from adistributional point of view we have
Î(t)Eu(t) = E(Î(t)u(t))ââÎ(t) u(t) in Ω \ Ît for a.e. t â (0, T ),
and E(Îu),âÎ u â L2(0, T ;L2(Ω;RdĂd)) if u â W, thanks to (3.1.8).
Remark 3.1.3. The setW coincides with the collection of functions u â H1(0, T ;L2(Ω;Rd))such that u(t) and Î(t)u(t) belong to H1(Ω \ Ît;Rd) for a.e. t â (0, T ) andâ« T
0âu(t)â2H1(Ω\Ît) + âÎ(t)u(t)â2H1(Ω\Ît) dt <â. (3.1.17)
Indeed the functions t 7â u(t) and t 7â Î(t)u(t) are strongly measurable from (0, T ) toH1(Ω \ ÎT ;Rd), which gives that (3.1.17) is well defined and u and Îu are elements ofL2(0, T ;H1(Ω \ ÎT ;Rd)). To prove the strong measurability of the two maps, it is enoughto observe that t 7â u(t) and t 7â Î(t)u(t) are weakly measurable from (0, T ) to L2(Ω;Rd),which is a separable Hilbert space. Moreover, t 7â Eu(t) and t 7â E(Î(t)u(t)) are weaklymeasurable from (0, T ) to L2(Ω;RdĂd), since for every Ï â Câc (Ω \ ÎT ) the maps
t 7ââ«
Ω\ÎTEu(t, x)Ï(x) dx = â
â«Î©\ÎT
u(t, x)âÏ(x) dx,
t 7ââ«
Ω\ÎTE(Κ(t, x)u(t, x))Ï(x) dx = â
â«Î©\ÎT
Κ(t, x)u(t, x)âÏ(x) dx
are measurable from (0, T ) to R, and Câc (Ω \ ÎT ) is dense in L2(Ω).
Lemma 3.1.4. The spaces V and W are Hilbert spaces with respect to the following norms:
âÏâV := (âÏâ2L2(0,T ;H1(Ω\ÎT )) + âÏâ2L2(0,T ;L2(Ω)))12 for Ï â V,
âuâW := (âuâ2V + âÎuâ2L2(0,T ;H1(Ω\ÎT )))12 for u â W.
Moreover, VD is a closed subspace of V.
Proof. It is clear that â·âV and â·âW are norms on V and W, respectively, induced by scalarproducts. We just have to check the completeness of such spaces with respect to these norms.
66 3.1. Preliminary results
Let Ïkk â V be a Cauchy sequence. Then, Ïkk and Ïkk are Cauchy sequencesin L2(0, T ;H1(Ω \ ÎT ;Rd)) and L2(0, T ;L2(Ω;Rd)), respectively, which are complete Hilbertspaces. Thus there exists Ï â L2(0, T ;H1(Ω\ÎT ;Rd)) with Ï â L2(0, T ;L2(Ω;Rd)) such thatÏk â Ï in L2(0, T ;H1(Ω \ ÎT ;Rd)) and Ïk â Ï in L2(0, T ;L2(Ω;Rd)). In particular thereexists a subsequence Ïkjj such that Ïkj (t) â Ï(t) in H1(Ω \ ÎT ;Rd) for a.e. t â (0, T ).
Since Ïkj (t) â H1(Ω \Ît;Rd) for a.e. t â (0, T ) we deduce that Ï(t) â H1(Ω \Ît;Rd) for a.e.t â (0, T ). Hence Ï â V and Ïk â Ï in V. With a similar argument, it is easy to check thatVD â V is a closed subspace.
Let us now consider a Cauchy sequence ukk â W. We have that ukk and Îukk areCauchy sequences in V and L2(0, T ;H1(Ω\ÎT ;Rd)), respectively, which are complete Hilbertspaces. Thus there exist u â V and z â L2(0, T ;H1(Ω \ ÎT ;Rd)) such that uk â u in V andÎuk â z in L2(0, T ;H1(Ω \ ÎT ;Rd)). Since uk â u in L2(0, T ;L2(Ω;Rd)) and Î belongsto Lâ((0, T ) à Ω), we derive that Îuk â Îu in L2(0, T ;L2(Ω;Rd)), which gives z = Îu.Finally let us prove that Î(t)u(t) â H1(Ω \ Ît;Rd) for a.e. t â (0, T ). Thanks to the factthat Îuk â Îu in L2(0, T ;H1(Ω \ ÎT ;Rd)), we can find a subsequence Îukjj such that
Î(t)ukj (t)â Î(t)u(t) in H1(Ω\ÎT ;Rd) for a.e. t â (0, T ). Since Î(t)ukj (t) â H1(Ω\Ît;Rd)for a.e. t â (0, T ), we deduce that Î(t)u(t) â H1(Ω \Ît;Rd) for a.e. t â (0, T ). Hence u â Wand uk â u in W.
We are in a position to define the notion of solution to (3.1.12)â(3.1.15).
Definition 3.1.5. We say that a function u â W is a solution to system (3.1.12) withboundary conditions (3.1.13)â(3.1.15) if uâ w â VD and
ââ« T
0(u(t), Ï(t))L2(Ω) dt+
â« T
0(CEu(t), EÏ(t))L2(Ω) dt
+
â« T
0(D[E(Î(t)u(t))â DâÎ(t) u(t)],Î(t)EÏ(t))L2(Ω) dt
=
â« T
0(f(t), Ï(t))L2(Ω) dt+
â« T
0(F (t), Ï(t))L2(âNΩ) dt
(3.1.18)
for every Ï â VD such that Ï(0) = Ï(T ) = 0.
Remark 3.1.6. When u is enough regular, for example u â L2(0, T ;H1(Ω \ ÎT ;Rd)) withu(t) â H1(Ω \ Ît;Rd) for a.e. t â (0, T ), we can write ÎEu = E(Îu) â âÎ u. There-fore (3.1.18) is coherent with the strong formulation (3.1.12). In particular, for a functionu â W we can define
ÎEu := E(Îu)ââÎ u â L2(0, T ;L2(Ω;RdĂd)), (3.1.19)
so that equation (3.1.18) can be written as
ââ« T
0(u(t), Ï(t))L2(Ω) dt+
â« T
0(CEu(t), EÏ(t))L2(Ω) dt
+
â« T
0(D[Î(t)Eu(t)],Î(t)EÏ(t))L2(Ω) dt
=
â« T
0(f(t), Ï(t))L2(Ω) dt+
â« T
0(F (t), Ï(t))L2(âNΩ) dt
(3.1.20)
for every Ï â VD such that Ï(0) = Ï(T ) = 0.
Remark 3.1.7. Notice that equation (3.1.20), which is formally obtained by integrating thePDE (3.1.12) in time and space and using the integration by part formula, can be rephrased
Chapter 3. A dynamic model for viscoelastic materials with growing cracks 67
pointwise in time, as done in the previous chapters (see, e.g., Definition 1.1.6). Indeed, byarguing as in [23, Theorem 2.17], it is possible to show that every solution u to (3.1.12)â(3.1.15) satisfies
ău(t), ÏăHâ1D (Ω\Ît) + (CEu(t), EÏ)L2(Ω) + (D[Î(t)Eu(t)],Î(t)EÏ)L2(Ω)
= (f(t), Ï)L2(Ω) + (F (t), Ï)L2(âNΩ)
for every Ï â H1D(Ω \ Ît;Rd), where the second derivative u is defined similarly to (1.1.23)
(see [23, Proposition 2.13]).
The notion of solution given in Definition 3.1.5 requires less regularity in time with respectto the ones given in the previous chapters. In particular, the functions u and u may not bedefined pointwise. Therefore, we define the initial conditions (3.1.16) as in [16].
Definition 3.1.8. We say that u â W satisfies the initial conditions (3.1.16) if
limhâ0+
1
h
â« h
0
(âu(t)â u0â2H1(Ω\Ît) + âu(t)â u1â2L2(Ω)
)dt = 0. (3.1.21)
3.2 Existence
We now state our main existence result, whose proof will be given at the end of Section 3.3.
Theorem 3.2.1. There exists a solution u â W to system (3.1.12)â(3.1.15), according toDefinition 3.1.5, which satisfies the initial conditions u(0) = u0 and u(0) = u1 in the senseof (3.1.21). Moreover, we have
u â C0w([0, T ];H1(Ω \ ÎT ;Rd)),
u â C0w([0, T ];L2(Ω;Rd)) â©H1(0, T ;Hâ1
D (Ω \ Î0)),
and as tâ 0+
u(t)â u0 in H1(Ω \ ÎT ;Rd), u(t)â u1 in L2(Ω;Rd).
To prove the existence of a solution to (3.1.12)â(3.1.15), we use a time discretizationscheme in the same spirit of [16]. Let us fix n â N and set
Ïn :=T
n, u0
n := u0, uâ1n := u0 â Ïnu1.
We define
F kn := F (kÏn), wkn := w(kÏn) for k = 0, . . . , n,
fkn :=1
Ïn
â« kÏn
(kâ1)Ïn
f(s) ds, Îkn :=
1
Ïn
â« kÏn
(kâ1)Ïn
Î(s) ds for k = 1, . . . , n,
ÎŽF kn :=F kn â F kâ1
n
Ïn, ÎŽwkn :=
wkn â wkâ1n
Ïnfor k = 1, . . . , n,
ÎŽw0n := w(0), ÎŽ2wkn :=
ÎŽwkn â ÎŽwkâ1n
Ïnfor k = 1, . . . , n.
For every k = 1, . . . , n let ukn â wkn â H1D(Ω \ ÎkÏn ;Rd), be the solution to
(ÎŽ2ukn, Ï)L2(Ω) + (CEukn, EÏ)L2(Ω) + (D[ÎknEÎŽu
kn],Îk
nEÏ)L2(Ω)
= (fkn , Ï)L2(Ω) + (F kn , Ï)L2(âNΩ)
(3.2.1)
68 3.2. Existence
for every Ï â H1D(Ω \ ÎkÏn ;Rd), where
ÎŽukn :=ukn â ukâ1
n
Ïnfor k = 0, . . . , n, ÎŽ2ukn :=
ÎŽukn â ÎŽukâ1n
Ïnfor k = 1, . . . , n.
The existence of a unique solution ukn to (3.2.1) is a consequence of Lax-Milgramâs theorem.
Remark 3.2.2. Since ÎŽukn â H1(Ω \ ÎkÏn ;Rd), we have ÎknEÎŽu
kn = E(Îk
nukn) â âÎk
n ukn.Hence, the discrete equation (3.2.1) is coherent with the weak formulation given in (3.1.18).
In the next lemma we show a uniform estimate for the family uknnk=1 with respect to nthat will be used later to pass to the limit in the discrete equation (3.2.1).
Lemma 3.2.3. There exists a constant C > 0, independent of n, such that
maxi=1,...,n
âÎŽuinâL2(Ω) + âuinâH1(Ω\ÎT )+nâi=1
ÏnâÎinEÎŽu
inâ2L2(Ω) †C. (3.2.2)
Proof. We fix n â N. To simplify the notation, for every Ï, Ï â H1(Ω \ ÎT ;Rd) we set
a(Ï, Ï) := (CEÏ,EÏ)L2(Ω), bkn(Ï, Ï) := (D[ÎknEÏ],Îk
nEÏ)L2(Ω) for k = 1, . . . , n.
By taking as test function Ï := Ïn(ÎŽukn â ÎŽwkn) â H1D(Ω \ ÎkÏn ;Rd) in (3.2.1), we obtain
âÎŽuknâ2L2(Ω) â (ÎŽukâ1n , ÎŽukn)L2(Ω) + a(ukn, u
kn)â a(ukn, u
kâ1n ) + Ïnb
kn(ÎŽukn, ÎŽu
kn) = ÏnL
kn
for k = 1, . . . , n, where
Lkn :=(fkn , ÎŽukn â ÎŽwkn)L2(Ω) + (F kn , ÎŽu
kn â ÎŽwkn)L2(âNΩ)
+ (Ύ2ukn, Ύwkn)L2(Ω) + a(ukn, Ύw
kn) + bkn(ÎŽukn, ÎŽw
kn).
Thanks to the following identities
âÎŽuknâ2L2(Ω) â (ÎŽukâ1n , ÎŽukn)L2(Ω) =
1
2âÎŽuknâ2L2(Ω) â
1
2âÎŽukâ1
n â2L2(Ω) +Ï2n
2âÎŽ2uknâ2L2(Ω),
a(ukn, ukn)â a(ukn, u
kâ1n ) =
1
2a(ukn, u
kn)â 1
2a(ukâ1
n , ukâ1n ) +
Ï2n
2a(ÎŽukn, ÎŽu
kn),
and by omitting the terms with Ï2n, which are non negative, we derive
1
2âÎŽuknâ2L2(Ω) â
1
2âÎŽukâ1
n â2L2(Ω) +1
2a(ukn, u
kn)â 1
2a(ukâ1
n , ukâ1n ) + Ïnb
kn(ÎŽukn, ÎŽu
kn) †ÏnLkn.
We fix i â 1, . . . , n and sum over k = 1, . . . , i to obtain the discrete energy inequality
1
2âÎŽuinâ2L2(Ω) +
1
2a(uin, u
in) +
iâk=1
Ïnbkn(ÎŽukn, ÎŽu
kn) †E0 +
iâk=1
ÏnLkn, (3.2.3)
where E0 := 12âu
1â2L2(Ω) + 12(CEu0, Eu0)L2(Ω). Let us now estimate the right-hand side
in (3.2.3) from above. By (3.1.1) and (3.1.3) we haveâŁâŁâŁâŁâŁiâ
k=1
Ïn(fkn , ÎŽukn)L2(Ω)
âŁâŁâŁâŁâŁ †1
2âfâ2L2(0,T ;L2(Ω)) +
1
2
iâk=1
ÏnâÎŽuknâ2L2(Ω), (3.2.4)âŁâŁâŁâŁâŁiâ
k=1
Ïn(fkn , ÎŽwkn)L2(Ω)
âŁâŁâŁâŁâŁ †1
2âfâ2L2(0,T ;L2(Ω)) +
1
2âwâ2L2(0,T ;L2(Ω)), (3.2.5)
Chapter 3. A dynamic model for viscoelastic materials with growing cracks 69
âŁâŁâŁâŁâŁiâ
k=1
Ïn(F kn , ÎŽwkn)L2(âNΩ)
âŁâŁâŁâŁâŁ †1
2âFâ2L2(0,T ;L2(âNΩ)) +
C2tr
2âwâ2L2(0,T ;H1(Ω\Î0)), (3.2.6)âŁâŁâŁâŁâŁ
iâk=1
Ïna(ukn, ÎŽwkn)
âŁâŁâŁâŁâŁ †1
2âCâLâ(Ω)
(âwâ2L2(0,T ;H1(Ω\Î0)) +
iâk=1
Ïnâuknâ2H1(Ω\ÎT )
). (3.2.7)
For the other term involving F kn , we perform the following discrete integration by parts
iâk=1
Ïn(F kn , ÎŽukn)L2(âNΩ) = (F in, u
in)L2(âNΩ) â (F (0), u0)L2(âNΩ)
âiâ
k=1
Ïn(ÎŽF kn , ukâ1n )L2(âNΩ).
(3.2.8)
Hence for every Δ â (0, 1), by using (3.1.1) and Youngâs inequality, we getâŁâŁâŁâŁâŁiâ
k=1
Ïn(F kn , ÎŽukn)L2(âNΩ)
âŁâŁâŁâŁâŁ †1
2ΔâF inâ2L2(âNΩ) +
Δ
2âuinâ2L2(âNΩ) + âF (0)âL2(âNΩ)âu0âL2(âNΩ)
+
iâk=1
ÏnâÎŽF knâL2(âNΩ)âukâ1n â2L2(âNΩ) (3.2.9)
†CΔ +ΔC2
tr
2âuinâ2H1(Ω\ÎT ) +
C2tr
2
iâk=1
Ïnâuknâ2H1(Ω\ÎT ),
where CΔ is a positive constant depending on Δ. Similarly to (3.2.8), we can say
iâk=1
Ïn(ÎŽ2ukn, ÎŽwkn)L2(Ω) = (ÎŽuin, ÎŽw
in)L2(Ω) â (ÎŽu0
n, Ύw0n)L2(Ω)
âiâ
k=1
Ïn(ÎŽukâ1n , ÎŽ2wkn)L2(Ω),
(3.2.10)
from which we deduce that for every Δ > 0âŁâŁâŁâŁâŁiâ
k=1
Ïn(ÎŽ2ukn, ÎŽwkn)L2(Ω)
âŁâŁâŁâŁâŁ †1
2ΔâÎŽwinâ2L2(Ω) +
Δ
2âÎŽuinâ2L2(Ω) + âu1âL2(Ω)âw(0)âL2(Ω)
+iâ
k=1
ÏnâÎŽukâ1n âL2(Ω)âÎŽ2wknâL2(Ω)
â€CΔ +Δ
2âÎŽuinâ2L2(Ω) +
1
2
iâk=1
ÏnâÎŽuknâ2L2(Ω),
(3.2.11)
where CΔ is a positive constant depending on Δ. We estimate from above the last term inright-hand side of (3.2.3) in the following way
iâk=1
Ïnbkn(ÎŽukn, ÎŽw
kn) â€
iâk=1
Ïn[bkn(ÎŽukn, ÎŽukn)]
12 [bkn(ÎŽwkn, ÎŽw
kn)]
12 (3.2.12)
†1
2
iâk=1
Ïnbkn(ÎŽukn, ÎŽu
kn) +
1
2âDâLâ(Ω)âÎâ2Lâ((0,T )ĂΩ)âwâ
2L2(0,T ;H1(Ω\Î0)).
70 3.2. Existence
By considering (3.2.3)â(3.2.12) and using (3.1.7) we obtain(1â Δ
2
)âÎŽuinâ2L2(Ω) +
c0 â ΔC2tr
2âuinâ2H1(Ω\Ît) â
c1
2âuinâ2L2(Ω) +
1
2
iâk=1
Ïnbkn(ÎŽukn, ÎŽu
kn)
†CΔ + Ciâ
k=1
Ïn
(âÎŽuknâ2L2(Ω) + âuknâ2H1(Ω\ÎT )
),
where CΔ and C are two positive constants, with CΔ depending on Δ. We can now choose
Δ < min
12 ,
c02C2
tr
and use the inequality
âuinâ2L2(Ω) â€
(âu0âL2(Ω) +
iâk=1
ÏnâÎŽuknâL2(Ω)
)2
†2âu0â2L2(Ω) + 2Tiâ
k=1
ÏnâÎŽuknâ2L2(Ω),
to derive the following estimate
1
4âÎŽuinâ2L2(Ω) +
1
4âuinâ2H1(Ω\ÎT ) +
1
2
iâk=1
Ïnbkn(ÎŽukn, ÎŽu
kn)
†C1 + C2
iâk=1
Ïn
(âÎŽuknâ2L2(Ω) + âuknâ2H1(Ω\ÎT )
),
(3.2.13)
where C1 and C2 are two positive constants depending only on u0, u1, f , F , and w. Thanksto a discrete version of Gronwallâs lemma (see, e.g., [3, Lemma 3.2.4]) we deduce the existenceof a constant C3 > 0, independent of i and n, such that
âÎŽuinâL2(Ω) + âuinâH1(Ω\ÎT ) †C3 for every i = 1, . . . , n and for every n â N.
By combining this last estimate with (3.2.13) and (3.1.6) we finally get (3.2.2) and we con-clude.
We now want to pass to the limit in (3.2.1) to obtain a solution to problem (3.1.12)â(3.1.15). To this aim, we define the piecewise affine interpolants un : [0, T ]â H1(Ω \ÎT ;Rd)of ujnnj=1 and uâČn : [0, T ]â L2(Ω;Rd) of ÎŽujnnj=1 as
un(t) := ujn + (tâ jÏn)ÎŽujn for t â [(j â 1)Ïn, jÏn], j = 1, . . . , n,
uâČn(t) := ÎŽujn + (tâ jÏn)ÎŽ2ujn for t â [(j â 1)Ïn, jÏn], j = 1, . . . , n.
We also define the backward interpolants un : [0, T ]â H1(Ω\ÎT ;Rd), uâČn : [0, T ]â L2(Ω;Rd)and the forward interpolants un : [0, T ] â H1(Ω \ ÎT ;Rd), uâČn : [0, T ] â L2(Ω;Rd) in thefollowing way:
un(t) := ujn for t â ((j â 1)Ïn, jÏn], j = 1, . . . , n, un(0) = u0n,
uâČn(t) := ÎŽujn for t â ((j â 1)Ïn, jÏn], j = 1, . . . , n, uâČn(0) = ÎŽu0n,
un(t) := ujâ1n for t â [(j â 1)Ïn, jÏn), j = 1, . . . , n, un(T ) = unn,
uâČn(t) := ÎŽujâ1n for t â [(j â 1)Ïn, jÏn), j = 1, . . . , n, uâČn(T ) = ÎŽunn.
Notice that un â H1(0, T ;L2(Ω;Rd)) with un(t) = ÎŽukn = uâČn(t) for t â ((k â 1)Ïn, kÏn) andk = 1, . . . , n. Let us also approximate Î and w by
În(t) := Îkn for t â ((k â 1)Ïn, kÏn], k = 1, . . . , n, În(0) := Î0
n,
wn(t) := wkn for t â ((k â 1)Ïn, kÏn], k = 1, . . . , n, wn(0) := w0n,
În(t) := Îkâ1n for t â [(k â 1)Ïn, kÏn), k = 1, . . . , n, În(T ) := În
n,
wn(t) := wkâ1n for t â [(k â 1)Ïn, kÏn), k = 1, . . . , n, wn(T ) := wnn.
Chapter 3. A dynamic model for viscoelastic materials with growing cracks 71
Lemma 3.2.4. There exists a function u â W, with u â w â VD, and a subsequence of n,not relabeled, such that the following convergences hold as nââ:
un u in L2(0, T ;H1(Ω \ ÎT ;Rd)), uâČn u in L2(0, T ;L2(Ω;Rd)), (3.2.14)
un u in L2(0, T ;H1(Ω \ ÎT ;Rd)), uâČn u in L2(0, T ;L2(Ω;Rd)), (3.2.15)
un u in L2(0, T ;H1(Ω \ ÎT ;Rd)), uâČn u in L2(0, T ;L2(Ω;Rd)), (3.2.16)
E(ÎnuâČn) E(Îu) in L2(0, T ;L2(Ω;RdĂd)), (3.2.17)
âÎn uâČn âÎ u in L2(0, T ;L2(Ω;RdĂd)). (3.2.18)
Proof. By Lemma 3.2.3 the sequences unn â Lâ(0, T ;H1(Ω\ÎT ;Rd))â©H1(0, T ;L2(Ω;Rd))and unn â Lâ(0, T ;H1(Ω \ ÎT ;Rd)) are uniformly bounded. Therefore, there exist twofunctions u â Lâ(0, T ;H1(Ω\ÎT ;Rd))â©H1(0, T ;L2(Ω;Rd)) and z â Lâ(0, T ;H1(Ω\ÎT ;Rd))such that, up to a not relabeled subsequence, as nââ
un u in L2(0, T ;H1(Ω \ ÎT ;Rd)), un u in L2(0, T ;L2(Ω;Rd)),un z in L2(0, T ;H1(Ω \ ÎT ;Rd)).
Moreover, we have u = z, since we can find a constant C > 0, independent of n, such that
âun â unâLâ(0,T ;L2(Ω;Rd)) †CÏn â 0 as nââ.
Since un(t) = un(tâÏn) for t â (Ïn, T ), uâČn(t) = un(t) for a.e. t â (0, T ), and uâČn(t) = uâČn(tâÏn)for t â (Ïn, T ), we deduce
un u in L2(0, T ;H1(Ω \ ÎT ;Rd)), uâČn u in L2(0, T ;L2(Ω;Rd)),uâČn u in L2(0, T ;L2(Ω;Rd)).
By using (3.2.2) we derive that the sequences E(ÎnuâČn)n â L2(0, T ;L2(Ω;RdĂd)) and
âÎnuâČnn â L2(0, T ;L2(Ω;RdĂd)) are uniformly bounded. Indeed, there exists a constantC > 0 independent of n such that
ââÎn uâČnâ2L2(0,T ;L2(Ω)) =nâk=1
â« kÏn
(kâ1)Ïn
ââÎkn ÎŽuknâ2L2(Ω) dt
†ââÎâ2Lâ((0,T )ĂΩ)
nâk=1
ÏnâÎŽuknâ2L2(Ω) †C,
âE(ÎnuâČn)â2L2(0,T ;L2(Ω)) =
nâk=1
â« kÏn
(kâ1)Ïn
âE(ÎknÎŽu
kn)â2L2(Ω) dt
=
nâk=1
ÏnâÎknEÎŽu
kn +âÎk
n ÎŽuknâ2L2(Ω)
†2nâk=1
ÏnâÎknEÎŽu
knâ2L2(Ω) + 2
nâk=1
ÏnââÎkn ÎŽuknâ2L2(Ω) †C.
Therefore, there exists z1, z2 â L2(0, T ;L2(Ω;RdĂd)) such that, up to a further not relabeledsubsequence, as nââ we have
âÎn uâČn z1 in L2(0, T ;L2(Ω;RdĂd)), E(ÎnuâČn) z2 in L2(0, T ;L2(Ω;RdĂd)).
72 3.2. Existence
We want to identify the limits z1 and z2. We fix a function Ï â L2(0, T ;L2(Ω;RdĂd)), andwe haveâ« T
0(âÎn uâČn, Ï)L2(Ω) dt =
1
2
â« T
0(uâČn, ÏâÎn)L2(Ω) dt+
1
2
â« T
0(uâČn, Ï
TâÎn)L2(Ω) dt
=
â« T
0(uâČn, Ï
symâÎn)L2(Ω) dt,
being Ïsym := Ï+ÏT
2 . Since uâČn u in L2(0, T ;L2(Ω;Rd)) and ÏsymâÎn â ÏsymâÎ inL2(0, T ;L2(Ω;Rd)) as nââ by dominated convergence theorem, we obtain
limnââ
â« T
0(âÎn uâČn, Ï)L2(Ω) dt =
â« T
0(u, ÏsymâÎ)L2(Ω) dt =
â« T
0(âÎ u, Ï)L2(Ω) dt,
and so z1 = âÎ u. Moreover, fixed Ï â L2(0, T ;L2(Ω;Rd)) we have
limnââ
â« T
0(Înu
âČn, Ï)L2(Ω) dt = lim
nââ
â« T
0(uâČn,ÎnÏ)L2(Ω) dt
=
â« T
0(u,ÎÏ)L2(Ω) dt =
â« T
0(Îu, Ï)L2(Ω) dt,
thanks to the fact that uâČn u in L2(0, T ;L2(Ω;Rd)) and ÎnÏâ ÎÏ in L2(0, T ;L2(Ω;Rd))as n â â, again by the dominated convergence theorem. Therefore Înu
âČn Îu in
L2(0, T ;L2(Ω;Rd)) as n â â, from which we get that E(ÎnuâČn) E(Îu) in the sense of
distributions as nââ, that gives z2 = E(Îu). In particular, Îu â L2(0, T ;H1(Ω\ÎT ;Rd)).Let us check that the limit point u is an element of W. To this aim we define the set
E := v â L2(0, T ;H1(Ω \ ÎT ;Rd)) : v(t) â H1(Ω \ Ît;Rd) for a.e. t â (0, T ).
Notice that E is a weakly closed subset of L2(0, T ;H1(Ω \ ÎT ;Rd)), since it is closed andconvex. Moreover, we have unn, Înu
âČnn â E. Indeed
un(t) = ukâ1n â H1(Ω \ Ît;Rd) for t â [(k â 1)Ïn, kÏn), k = 1, . . . , n,
În(t)uâČn(t) = Îkâ1n ÎŽukâ1
n â H1(Ω \ Ît;Rd) for t â [(k â 1)Ïn, kÏn), k = 1, . . . , n.
Since un u in L2(0, T ;H1(Ω \ ÎT ;Rd)) and ÎnuâČn Îu in L2(0, T ;H1(Ω \ ÎT ;Rd)), we
conclude that u,Îu â E. Finally, to show that uâ w â VD we observe
un(t)â wn(t) = ukâ1n â wkâ1
n â H1D(Ω \ Ît;Rd) for t â [(k â 1)Ïn, kÏn), k = 1, . . . , n.
Therefore
un â wnn â v â L2(0, T ;H1(Ω \ ÎT ;Rd)) : v(t) â H1D(Ω \ Ît;Rd) for a.e. t â (0, T ),
which is a closed convex subset of L2(0, T ;H1(Ω \ ÎT ;Rd)), and so it is weakly closed. Sinceun u in L2(0, T ;H1(Ω \ ÎT ;Rd)) and wn â w in L2(0, T ;H1(Ω \ Î0;Rd)) as n â â, weget that u(t)â w(t) â H1
D(Ω \ Ît;Rd) for a.e. t â (0, T ), which implies uâ w â VD.
We now use Lemma 3.2.4 to pass to the limit in the discrete equation (3.2.1).
Lemma 3.2.5. The function u â W given by Lemma 3.2.4 is a solution to (3.1.12)â(3.1.15),according to Definition 3.1.5.
Chapter 3. A dynamic model for viscoelastic materials with growing cracks 73
Proof. We only need to prove that u â W satisfies (3.1.18). Fixed n â N, we consider afunction Ï â C1
c (0, T ;H1(Ω \ ÎT ;Rd)) such that Ï(t) â H1D(Ω \ Ît;Rd) for every t â (0, T ),
and we set
Ïkn := Ï(kÏn) for k = 0, . . . , n, ÎŽÏkn :=Ïkn â Ïkâ1
n
Ïnfor k = 1, . . . , n.
By using ÏnÏkn â H1
D(Ω \ ÎkÏn ;Rd) as test function in (3.2.1) and summing over k = 1, . . . , nwe get
nâk=1
Ïn(ÎŽ2ukn, Ïkn)L2(Ω) +
nâk=1
Ïn(CEukn, EÏkn)L2(Ω) +nâk=1
Ïn(D[ÎknEÎŽu
kn],Îk
nEÏkn)L2(Ω)
=nâk=1
Ïn(fkn , Ïkn)L2(Ω) +
nâk=1
Ïn(F kn , Ïkn)L2(âNΩ). (3.2.19)
Let us define the approximating sequences
Ïn(t) := Ïkn, ÏâČn(t) := ÎŽÏkn for t â ((k â 1)Ïn, kÏn], k = 1, . . . , n.
Thanks to the identity
nâk=1
Ïn(ÎŽ2ukn, Ïkn)L2(Ω) = â
nâk=1
Ïn(ÎŽukâ1n , ÎŽÏkn)L2(Ω) = â
â« T
0(uâČn(t), ÏâČn(t))L2(Ω) dt,
from (3.2.19) we deduce the following equality
ââ« T
0(uâČn, Ï
âČn)L2(Ω) dt+
â« T
0(CEun, EÏn)L2(Ω) dt
+
â« T
0(D[E(Înu
âČn)ââÎn uâČn], EÏn)L2(Ω) dt
=
â« T
0(fn, Ïn)L2(Ω) dt+
â« T
0(Fn, Ïn)L2(âNΩ) dt,
(3.2.20)
where fn and Fn are the backward interpolants of fknnk=1 and F knnk=1, respectively. Noticethat as nââ we have
Ïn â Ï in L2(0, T ;H1(Ω \ ÎT ;Rd)), ÏâČn â Ï in L2(0, T ;L2(Ω;Rd)),fn â f in L2(0, T ;L2(Ω;Rd)), Fn â F in L2(0, T ;L2(âNΩ;Rd)).
By (3.2.14)â(3.2.18) and the above convergences we can pass to the limit as nââ in (3.2.20),and we get that u â W satisfies (3.1.18) for every Ï â C1
c (0, T ;H1(Ω \ ÎT ;Rd)) such thatÏ(t) â H1
D(Ω \ Ît;Rd) for every t â (0, T ). Finally, we can use a density argument (see [23,Remark 2.9]) to conclude that u â W is a solution to (3.1.12)â(3.1.15).
3.3 Initial conditions
To complete our existence result, it remains to prove that the solution u â W to (3.1.12)â(3.1.15) given by Lemma 3.2.4 satisfies the initial conditions (3.1.16) in the sense of (3.1.21).
We start by showing that the second distributional derivative u belongs to the spaceL2(0, T ;Hâ1
D (Ω\Î0;Rd)). By using the discrete equation (3.2.1), for every v â H1D(Ω\Î0;Rd)
with âvâH1(Ω\Î0) †1 we have
|(ÎŽ2ukn, v)L2(Ω)| †âCâLâ(Ω)âEuknâL2(Ω) + âDâLâ(Ω)âÎâLâ((0,T )ĂΩ)âÎknEÎŽu
knâL2(Ω)
+ âfknâL2(Ω) + CtrâF knâL2(âNΩ),
74 3.3. Initial conditions
and taking the supremum over v, we conclude
âÎŽ2uknâ2Hâ1D (Ω\Î0)
†C(âEuknâ2L2(Ω) + âÎknEÎŽu
knâ2L2(Ω) + âfknâ2L2(Ω) + âF knâ2L2(âNΩ))
for a positive constant C independent of n. We multiply this inequality by Ïn, we sum overk = 1, . . . , n, and we use (3.2.2) to obtain
nâk=1
ÏnâÎŽ2uknâ2Hâ1D (Ω\Î0)
†C for every n â N, (3.3.1)
where C is a positive constant independent of n. The above estimate implies that the sequenceuâČnn â H1(0, T ;Hâ1
D (Ω\Î0;Rd)) is uniformly bounded (uâČn(t) = ÎŽ2ukn for t â ((kâ1)Ïn, kÏn)and k = 1, . . . , n). Up to extract a further subsequence (not relabeled) from the one ofLemma 3.2.4, we deduce that there is z3 â H1(0, T ;Hâ1
D (Ω \ Î0;Rd)) such that
uâČn z3 in H1(0, T ;Hâ1D (Ω \ Î0;Rd)) as nââ. (3.3.2)
By using the following estimate
âuâČn â uâČnâL2(0,T ;Hâ1D (Ω\Î0)) †Ïnâu
âČnâL2(0,T ;Hâ1
D (Ω\Î0)) †CÏn â 0 as nââ,
we conclude that z3 = u, which gives u â L2(0, T ;Hâ1D (Ω \ Î0;Rd)).
The solution u â W given by Lemma 3.2.4 satisfies
u â Lâ(0, T ;H1(Ω \ ÎT ;Rd)), u â Lâ(0, T ;L2(Ω;Rd)),
and recalling Remark 1.2.7 we derive
u â C0w([0, T ];H1(Ω \ ÎT ;Rd)), u â C0
w([0, T ];L2(Ω;Rd)).
Therefore, by (3.2.14) and (3.3.2) for every t â [0, T ] we obtain
un(t) u(t) in L2(Ω;Rd), uâČn(t) u(t) in Hâ1D (Ω \ Î0;Rd) as nââ, (3.3.3)
so that u(0) = u0 and u(0) = u1, being un(0) = u0 and uâČn(0) = u1 for every n â N.
To prove
limhâ0+
1
h
â« h
0
(âu(t)â u0â2H1(Ω\Ît) + âu(t)â u1â2L2(Ω)
)dt = 0
we will actually show as tâ 0+
u(t)â u0 in H1(Ω \ ÎT ;Rd), u(t)â u1 in L2(Ω;Rd).
This is a consequence of an energy-dissipation inequality which holds for the solution u â Wto (3.1.12)â(3.1.15) given by Lemma 3.2.4. Let us define the mechanical energy of u as
E(t) :=1
2âu(t)â2L2(Ω) +
1
2(CEu(t), Eu(t))L2(Ω) for t â [0, T ].
Notice that E(t) is well defined for every t â [0, T ] since u â C0w([0, T ];H1(Ω \ ÎT ;Rd)) and
u â C0w([0, T ];L2(Ω;Rd)), and that E(0) = 1
2âu1â2L2(Ω) + 1
2(CEu0, Eu0)L2(Ω).
Chapter 3. A dynamic model for viscoelastic materials with growing cracks 75
Theorem 3.3.1. The solution u â W to (3.1.12)â(3.1.15) given by Lemma 3.2.4 satisfies forevery t â [0, T ] the following energy-dissipation inequality
E(t) +
â« t
0(D[ÎEu],ÎEu)L2(Ω) ds †E(0) +Wtot(t), (3.3.4)
where ÎEu is the function defined in (3.1.19) and Wtot(t) is the total work at time t â [0, T ],which is given by
Wtot(t) :=
â« t
0
[(f, uâ w)L2(Ω) + (CEu,Ew)L2(Ω) + (D[ÎEu],ÎEw)L2(Ω)
]ds
ââ« t
0
[(u, w)L2(Ω) + (F , uâ w)L2(âNΩ)
]ds+ (u(t), w(t))L2(Ω)
+ (F (t), u(t)â w(t))L2(âNΩ) â (u1, w(0))L2(Ω) â (F (0), u0 â w(0))L2(âNΩ).
Remark 3.3.2. The total work Wtot(t) is well defined for every t â [0, T ], since we haveF â C0([0, T ];L2(âNΩ;Rd)), w â C0([0, T ];L2(Ω;Rd)), u â C0
w([0, T ];H1(Ω \ ÎT ;Rd)), andu â C0
w([0, T ];L2(Ω;Rd)). In particular, the function t 7â Wtot(t) is continuous from [0, T ]to R.
Proof of Theorem 3.3.1. Fixed t â (0, T ], for every n â N there exists a unique j â 1, . . . , nsuch that t â ((j â 1)Ïn, jÏn]. After setting tn := jÏn, we can write (3.2.3) as
1
2âuâČn(t)â2L2(Ω) +
1
2(CEun(t), Eun(t))L2(Ω) +
â« tn
0(D[ÎnEu
âČn],ÎnEu
âČn)L2(Ω) ds
†E(0) +Wn(t),
(3.3.5)
where
Wn(t) :=
â« tn
0
[(fn, u
âČn â wâČn)L2(Ω) + (CEun, EwâČn)L2(Ω) + (D[ÎnEu
âČn],ÎnEw
âČn)L2(Ω)
]ds
+
â« tn
0
[(uâČn, w
âČn)L2(Ω) + (Fn, u
âČn â wâČn)L2(âNΩ)
]ds.
We want to pass to the limit as n â â in (3.3.5) and we start studying the left-hand side.Thanks to (3.2.2) and (3.3.1), as nââ we have
âun(t)â un(t)âL2(Ω) †ÏnâÎŽujnâL2(Ω) †CÏn â 0,
âuâČn(t)â uâČn(t)â2Hâ1D (Ω\Î0)
†Ï2nâÎŽ2ujnâ2Hâ1
D (Ω\Î0)†Ïn
nâk=1
ÏnâÎŽ2uknâ2Hâ1D (Ω\Î0)
†CÏn â 0.
The last two convergences, together (3.3.3), imply
un(t) u(t) in L2(Ω;Rd), uâČn(t) u(t) in Hâ1D (Ω \ Î0;Rd) as nââ,
and since âun(t)âH1(Ω\ÎT ) + âuâČn(t)âL2(Ω) †C for every n â N, we conclude
un(t) u(t) in H1(Ω \ ÎT ;Rd), uâČn(t) u(t) in L2(Ω;Rd) as nââ. (3.3.6)
Hence, we get
âu(t)â2L2(Ω) †lim infnââ
âuâČn(t)â2L2(Ω), (3.3.7)
(CEu(t), Eu(t))L2(Ω) †lim infnââ
(CEun(t), Eun(t))L2(Ω). (3.3.8)
76 3.3. Initial conditions
Thanks to Lemma 3.2.4 and (3.1.19), as nââ we obtain
ÎnEuâČn = E(Înu
âČn)ââÎn uâČn E(Îu)ââÎ u = ÎEu in L2(0, T ;L2(Ω;RdĂd)),
so that â« t
0(D[ÎEu],ÎEu)L2(Ω) ds †lim inf
nââ
â« t
0(D[ÎnEu
âČn],ÎnEu
âČn)L2(Ω) ds
†lim infnââ
â« tn
0(D[ÎnEu
âČn],ÎnEu
âČn)L2(Ω) ds,
(3.3.9)
since Ï 7ââ« t
0 (DEÏ(s), EÏ(s))L2(Ω) ds is lower semicontinuous on L2(0, T ;H1(Ω \ ÎT ;Rd))and t †tn.
Let us study the right-hand side of (3.3.5). By the following convergences as nââ:
fn â f in L2(0, T ;L2(Ω;Rd)), uâČn â wâČn uâ w in L2(0, T ;L2(Ω;Rd)),
we deduce
limnââ
â« tn
0(fn, u
âČn â wâČn)L2(Ω) ds =
â« t
0(f, uâ w)L2(Ω) ds. (3.3.10)
In a similar way, we can prove
limnââ
â« tn
0(CEun, EwâČn)L2(Ω) ds =
â« t
0(CEu,Ew)L2(Ω) ds, (3.3.11)
limnââ
â« tn
0(D[ÎnEu
âČn],ÎnEw
âČn)L2(Ω) ds =
â« t
0(D[ÎEu],ÎEw)L2(Ω) ds, (3.3.12)
since the following convergences hold as nââ:
EwâČn â Ew in L2(0, T ;L2(Ω;RdĂd)), CEun CEu in L2(0, T ;L2(Ω;RdĂd)),ÎnEw
âČn â ÎEw in L2(0, T ;L2(Ω;RdĂd)), ÎnEu
âČn ÎEu in L2(0, T ;L2(Ω;RdĂd)).
Now, we use formula (3.2.10) to deriveâ« tn
0(uâČn, w
âČn)L2(Ω) ds = (uâČn(t), wâČn(t))L2(Ω) â (u1, w(0))L2(Ω) â
â« tn
0(uâČn, w
âČn)L2(Ω) ds.
By arguing as before, we can deduce
limnââ
â« tn
0(uâČn, w
âČn)L2(Ω) ds = (u(t), w(t))L2(Ω) â (u1, w(0))L2(Ω) â
â« t
0(u, w)L2(Ω) ds, (3.3.13)
thanks to (3.3.6) and the following convergences as nââ
wâČn â w in L2(0, T ;L2(Ω;Rd)), uâČn u in L2(0, T ;L2(Ω;Rd)),
âwâČn(t)â w(t)âL2(Ω) â€1
Ïn
â« jÏn
(jâ1)Ïn
âw(s)â w(t)âL2(Ω) dsâ 0.
Notice that in the last convergence we have used w â C0([0, T ];L2(Ω;Rd)). Similarly we haveâ« tn
0(Fn, u
âČn â wâČn)L2(âNΩ) ds = (Fn(t), un(t)â wn(t))L2(âNΩ) â (F (0), u0 â w(0))L2(âNΩ)
ââ« tn
0(Fn, un â wn)L2(âNΩ) ds,
Chapter 3. A dynamic model for viscoelastic materials with growing cracks 77
and by using (3.1.1), (3.3.6), the continuity of F from [0, T ] in L2(âNΩ;Rd), and
Fn â F in L2(0, T ;L2(âNΩ;Rd)), un â wn uâ w in L2(0, T ;L2(âNΩ;Rd))
as nââ, we get
limnââ
â« tn
0(Fn, u
âČn â wâČn)L2(âNΩ) ds = (F (t), u(t)â w(t))L2(âNΩ) â (F (0), u0 â w(0))L2(âNΩ)
ââ« t
0(F , uâ w)L2(âNΩ) ds. (3.3.14)
By combining (3.3.7)â(3.3.14), we deduce the energy-dissipation inequality (3.3.4) for everyt â (0, T ]. Finally, for t = 0 the inequality trivially holds since u(0) = u0 and u(0) = u1.
Lemma 3.3.3. The solution u â W to (3.1.12)â(3.1.15) given by Lemma 3.2.4 satisfies
u(t)â u0 in H1(Ω \ ÎT ;Rd), u(t)â u1 in L2(Ω;Rd) as tâ 0+. (3.3.15)
In particular, u satisfies the initial conditions (3.1.16) in the sense of (3.1.21).
Proof. By sending tâ 0+ in the energy-dissipation inequality (3.3.4) and using the fact thatu â C0
w([0, T ];H1(Ω \ ÎT ;Rd)) and u â C0w([0, T ];L2(Ω;Rd)) we deduce
E(0) †lim inftâ0+
E(t) †lim suptâ0+
E(t) †E(0),
since the right-hand side of (3.3.4) is continuous in t, u(0) = u0, and u(0) = u1. There-fore there exists limtâ0+ E(t) = E(0). We combine this fact with the lower semicontinuityproperties of t 7â âu(t)â2L2(Ω) and t 7â (CEu(t), Eu(t))L2(Ω) to derive
limtâ0+
âu(t)â2L2(Ω) = âu1â2L2(Ω), limtâ0+
(CEu(t), Eu(t))L2(Ω) = (CEu0, Eu0)L2(Ω).
Finally, since we have
u(t) u1 in L2(Ω;Rd), Eu(t) Eu0 in L2(Ω;RdĂd) as tâ 0+,
we deduce (3.3.15). In particular, we derive that the functions u : [0, T ] â H1(Ω \ ÎT ;Rd)and u : [0, T ]â L2(Ω;Rd) are continuous at t = 0, which implies (3.1.21).
We are now in a position to prove Theorem 3.2.1.
Proof of Theorem 3.2.1. The proof is a consequence of Lemmas 3.2.5 and 3.3.3.
Remark 3.3.4. We have proved Theorem 3.2.1 for the d-dimensional linear elastic case,namely when the displacement u is a vector-valued function. The same result is true withidentical proofs in the antiplane case, that is when the displacement u is a scalar functionand satisfies (15).
3.4 Uniqueness
In this section we investigate the uniqueness properties of system (3.1.12) with boundaryand initial conditions (3.1.13)â(3.1.16). To this aim, we need to assume stronger regularityassumptions on the cracks Îttâ[0,T ] and the function Î. Moreover, we have to restrictour problem to the dimensional case d = 2, since in our proof we need to build a suitable
78 3.4. Uniqueness
family of diffeomorphisms which maps the time-dependent crack Ît into a fixed set, and thisconstruction is explicit only for d = 2 (see [20, Example 2.14]).
We proceed in two steps; first, in Lemma 3.4.2 we prove a uniqueness result in everydimension d, but when the cracks are not increasing, that is ÎT = Î0. Next, in Theorem 3.4.4we combine Lemma 3.4.2 with the uniqueness result of [23] and the finite speed propagationresult of [18] to prove the uniqueness in the case of a moving crack in d = 2.
Let us start with the following lemma, whose proof is analogous to the one of Proposi-tion 2.10 of [23].
Lemma 3.4.1. Let u â W be a solution to (3.1.12)â(3.1.15), according Definition 3.1.5,satisfying the initial condition u(0) = 0 in the following sense:
limhâ0+
1
h
â« h
0âu(t)â2L2(Ω) = 0.
Then u satisfies
ââ« T
0(u(t), Ï(t))L2(Ω) dt+
â« T
0(CEu(t), EÏ(t))L2(Ω) dt
+
â« T
0(D[Î(t)Eu(t)],Î(t)EÏ(t))L2(Ω) dt
=
â« T
0(f(t), Ï(t))L2(Ω) dt+
â« T
0(F (t), Ï(t))L2(âNΩ) dt
for every Ï â VD such that Ï(T ) = 0, where ÎEu is the function defined in (3.1.19).
Proof. We fix Ï â VD with Ï(T ) = 0 and for every Δ > 0 we define the function
ÏΔ(t) :=
tΔÏ(t) t â [0, Δ],
Ï(t) t â [Δ, T ].
We have that ÏΔ â VD and ÏΔ(0) = ÏΔ(T ) = 0, so we can use ÏΔ as test function in (3.1.18).By proceeding as in [23, Proposition 2.10] we obtain
limΔâ0+
â« T
0(u(t), ÏΔ(t))L2(Ω) dt =
â« T
0(u(t), Ï(t))L2(Ω) dt,
limΔâ0+
â« T
0(CEu(t), EÏΔ(t))L2(Ω) dt =
â« T
0(CEu(t), EÏ(t))L2(Ω) dt,
limΔâ0+
â« T
0(f(t), ÏΔ(t))L2(Ω) dt =
â« T
0(f(t), Ï(t))L2(Ω) dt.
It remains to consider the terms involving D and F . We haveâ« T
0(D[Î(t)Eu(t)],Î(t)EÏΔ(t))L2(Ω) dt
=
⫠Δ
0(D[Î(t)Eu(t)],
t
ΔÎ(t)EÏ(t))L2(Ω) dt+
â« T
Δ(D[Î(t)Eu(t)],Î(t)EÏ(t))L2(Ω) dt,
and by the dominated convergence theorem we get as Δâ 0+âŁâŁâŁâŁâ« Δ
0(D[Î(t)Eu(t)],
t
ΔÎ(t)EÏ(t))L2(Ω) dt
âŁâŁâŁâŁâ€ âDâLâ(Ω)âÎâLâ((0,T )ĂΩ)
⫠Δ
0âÎ(t)Eu(t)âL2(Ω)âEÏ(t)âL2(Ω) dtâ 0,
Chapter 3. A dynamic model for viscoelastic materials with growing cracks 79
â« T
Δ(D[Î(t)Eu(t)],Î(t)EÏ(t))L2(Ω) dtâ
â« T
0(D[Î(t)Eu(t)],Î(t)EÏ(t))L2(Ω) dt.
Similarly, we haveâ« T
0(F (t), ÏΔ(t))L2(âNΩ) dt =
⫠Δ
0(F (t),
t
ΔÏ(t))L2(âNΩ) dt+
â« T
Δ(F (t), Ï(t))L2(âNΩ) dt,
and as Δâ 0+âŁâŁâŁâŁâ« Δ
0(F (t),
t
ΔÏ(t))L2(âNΩ) dt
âŁâŁâŁâŁ †⫠Δ
0âF (t)âL2(âNΩ)âÏ(t)âL2(âNΩ) dtâ 0,â« T
Δ(F (t), Ï(t))L2(âNΩ) dtâ
â« T
0(F (t), Ï(t))L2(âNΩ) dt.
By combining together the previous convergences we get the thesis.
We prove now the uniqueness result in the case of a fixed domain, that is ÎT = Î0, byfollowing the same procedure adopted in Theorem 1.2.10 of Chapter 1. On the function Îwe assume
Î â Lip([0, T ]à Ω), âÎ â Lâ((0, T )à Ω;Rd), (3.4.1)
while on ÎT = Î0 we require only (E1)â(E3).
Lemma 3.4.2. Assume that Î satisfies (3.4.1) and ÎT = Î0. Then the viscoelastic dynamicsystem (3.1.12) with boundary conditions (3.1.13)â(3.1.15) has a unique solution, accordingto Definition 3.1.5, satifying u(0) = u0 and u(0) = u1 in the sense of (3.1.21).
Proof. Let u1, u2 â W be two solutions to (3.1.12)â(3.1.15) with initial conditions (3.1.16).The function u := u1 â u2 satisfies
limhâ0+
1
h
â« h
0(âu(t)â2H1(Ω\Ît) + âu(t)â2L2(Ω)) dt = 0, (3.4.2)
hence by Lemma 3.4.1 it solves
ââ« T
0(u(t), Ï(t))L2(Ω) dt+
â« T
0(CEu(t), EÏ(t))L2(Ω) dt
+
â« T
0(D[Î(t)Eu(t)],Î(t)EÏ(t))L2(Ω) dt = 0
(3.4.3)
for every Ï â VD such that Ï(T ) = 0. We fix s â (0, T ] and we consider the function
Ïs(t) :=
ââ« st u(Ï) dÏ t â [0, s],
0 t â [s, T ].
Since Ïs â VD and Ïs(T ) = 0, we can use it as test function in (3.4.3) to obtain
ââ« s
0(u(t), u(t))L2(Ω) dt+
â« s
0(CEÏs(t), EÏs(t))L2(Ω) dt
+
â« s
0(D[Î(t)Eu(t)],Î(t)EÏs(t))L2(Ω) dt = 0.
In particular we deduce
â 1
2
â« s
0
d
dtâu(t)â2L2(Ω) dt+
1
2
â« s
0
d
dt(CEÏs(t), EÏs(t))L2(Ω) dt
+
â« s
0(D[Î(t)Eu(t)],Î(t)EÏs(t))L2(Ω) dt = 0,
80 3.4. Uniqueness
which implies
1
2âu(s)â2L2(Ω) +
1
2(CEÏs(0), EÏs(0))L2(Ω) =
â« s
0(D[Î(t)Eu(t)],Î(t)EÏs(t))L2(Ω) dt, (3.4.4)
since u(0) = Ïs(s) = 0. From the distributional point of view, the following identity holds
d
dt(ÎEu) = ÎEu+ ÎEu â L2(0, T ;L2(Ω;RdĂd)). (3.4.5)
Indeed, for every Ï â Câc (0, T ;L2(Ω;RdĂd)) we haveâ« T
0(
d
dt(Î(t)Eu(t)), Ï(t))L2(Ω) dt
= ââ« T
0(Î(t)Eu(t), Ï(t))L2(Ω) dt
= ââ« T
0(E(Î(t)u(t))ââÎ(t) u(t), Ï(t))L2(Ω) dt
=
â« T
0(E(Î(t)u(t)) + E(Î(t)u(t))ââÎ(t) u(t)ââÎ(t) u(t), Ï(t))L2(Ω) dt
=
â« T
0(Î(t)Eu(t) + Î(t)Eu(t), Ï(t))L2(Ω) dt.
In particular ÎEu â H1(0, T ;L2(Ω;RdĂd)), so that by (3.4.2)
âÎ(0)Eu(0)â2L2(Ω) = limhâ0+
1
h
â« h
0âÎ(t)Eu(t)â2L2(Ω) dt
†âÎâ2Lâ((0,T )ĂΩ) limhâ0+
1
h
â« h
0âu(t)â2H1(Ω\Ît) dt = 0,
which yields Î(0)Eu(0) = 0. Thanks to (3.4.5) and to Îu â H1(0, T ;L2(Ω;Rd)), we deduce
d
dt(D[ÎEu],ÎEÏs)L2(Ω)
= 2(D[ÎEu], ÎEÏs)L2(Ω) + (D[ÎEu],ÎEÏs)L2(Ω) + (D[ÎEu],ÎEÏs)L2(Ω),
which impliesâ« s
0(D[ÎEu],ÎEÏs)L2(Ω) dt
=
â« s
0
[d
dt(D[ÎEu],ÎEÏs)L2(Ω) â 2(D[ÎEu], ÎEÏs)L2(Ω) â (D[ÎEu],ÎEÏs)L2(Ω)
]dt
†(D[Î(s)Eu(s)],Î(s)EÏs(s))L2(Ω) â (D[Î(0)Eu(0)],Î(0)EÏs(0))L2(Ω)
+
â« s
0
[2((D[ÎEu],ÎEu)L2(Ω)
) 12((D[ÎEÏs], ÎEÏs)L2(Ω)
) 12 â (D[ÎEu],ÎEÏs)L2(Ω)
]dt
â€â« s
0
[(D[ÎEu],ÎEu)L2(Ω) + (D[ÎEÏs], ÎEÏs)L2(Ω) â (D[ÎEu],ÎEÏs)L2(Ω)
]dt
†âDâLâ(Ω)âÎâ2Lâ((0,T )ĂΩ)
â« s
0âEÏsâ2L2(Ω)dt,
since EÏs(s) = Î(0)Eu(0) = 0 and EÏs = Eu in (0, s). By combining the previous inequalitywith (3.4.4) and using the coercivity of the tensor C, we derive
λ1
2âEÏs(0)â2L2(Ω) +
1
2âu(s)â2L2(Ω) â€
1
2(CEÏs(0), EÏs(0))L2(Ω) +
1
2âu(s)â2L2(Ω)
†âDâLâ(Ω)âÎâ2Lâ((0,T )ĂΩ)
â« s
0âEÏs(t)â2L2(Ω)dt.
Chapter 3. A dynamic model for viscoelastic materials with growing cracks 81
Let us now set ζ(t) :=⫠t
0 u(Ï) dÏ , then
âEÏs(0)â2L2(Ω) = âEζ(s)â2L2(Ω),
âEÏs(t)â2L2(Ω) = âEζ(t)â Eζ(s)â2L2(Ω) †2âEζ(t)â2L2(Ω) + 2âEζ(s)â2L2(Ω),
from which we deduce
λ1
2âEζ(s)â2L2(Ω) +
1
2âu(s)â2L2(Ω) †C
â« s
0âEζ(t)â2L2(Ω) dt+ CsâEζ(s)â2L2(Ω), (3.4.6)
where C := 2âDâLâ(Ω)âÎâ2Lâ((0,T )ĂΩ). Therefore, if we set s0 := λ14C , for all s †s0 we obtain
λ1
4âEζ(s)â2L2(Ω) â€
(λ1
2â Cs
)âEζ(s)â2L2(Ω) †C
â« s
0âEζ(t)â2L2(Ω) dt.
By Gronwallâs lemma the last inequality implies Eζ(s) = 0 for all s †s0. Hence, thanksto (3.4.6) we get âu(s)â2L2(Ω) †0 for all s †s0, which yields u(s) = 0 for all s †s0. Sinces0 depends only on C, D, and Î, we can repeat this argument starting from s0, and with afinite number of steps we obtain u = 0 on [0, T ].
We now are in a position to prove the uniqueness in the case of a moving crack. Weconsider the dimensional case d = 2, and we require the following assumptions:
(F1) there exists a C2,1 simple curve Î â Ω â R2, parametrized by arc-length Îł : [0, `]â Ω,such that Î â© âΩ = Îł(0) âȘ Îł(`) and Ω \ Î is the union of two disjoint open sets withLipschitz boundary;
(F2) Ît = Îł(Ï) : 0 â€ Ï â€ s(t), where s : [0, T ] â (0, `) is a non-decreasing function ofclass C1,1;
(F3) |s(t)|2 < λ1CK
for every t â [0, T ], where λ1 is the ellipticity constant of C and CK is theconstant that appears in Kornâs inequality (3.1.2).
Remark 3.4.3. Notice that hypotheses (F1) and (F2) imply (E1)â(E3). Moreover, by (F2)we have ÎT \ Î0 ââ Ω.
We also assume that Πsatisfies (3.4.1) and that there exists a constant Δ > 0, independentof t, such that
Î(t, x) = 0 for every t â [0, T ] and x â y â Ω : |y â Îł(s(t))| < Δ. (3.4.7)
Theorem 3.4.4. Assume d = 2, (F1)â(F3), and that Î satisfies (3.4.1) and (3.4.7). Thenthe system (3.1.12) with boundary conditions (3.1.13)â(3.1.15) has a unique solution u â W,according to Definition (3.1.5), satisfying u(0) = u0 and u(0) = u1 in the sense of (3.1.21).
Proof. As before let u1, u2 â W be two solutions to (3.1.12)â(3.1.15) with initial condi-tions (3.1.16). Then u := u1 â u2 satisfies (3.4.2) and (3.4.3) for every Ï â VD such thatÏ(T ) = 0. Let us define
t0 := supt â [0, T ] : u(s) = 0 for every s â [0, t],
and assume by contradiction that t0 < T . Consider first the case in which t0 > 0. By (F1),(F2), (3.4.1), and (3.4.7) we can find two open sets A1 and A2, with A1 ââ A2 ââ Ω, anda number ÎŽ > 0 such that for every t â [t0 â ÎŽ, t0 + ÎŽ] we have Îł(s(t)) â A1, Î(t, x) = 0 for
82 3.4. Uniqueness
every x â A2, and (A2 \A1)\Î is the union of two disjoint open sets with Lipschitz boundary.Let us define
V 1 := u â H1((A2 \A1) \ Ît0âÎŽ;R2) : u = 0 on âA1 âȘ âA2, H1 := L2(A2 \A1;R2).
Since every function in V 1 can be extended to a function in H1D(Ω \ Ît0âÎŽ;R2), by standard
results for linear hyperbolic equations (se, e.g., [24]) we deduce that u â L2(t0âÎŽ, t0+ÎŽ; (V 1)âČ)and u satisfies for a.e. t â (t0 â ÎŽ, t0 + ÎŽ)
ău(t), Ïă(V 1)âČ + (CEu(t), EÏ)H1 = 0 for every Ï â V 1.
Moreover, we have u(t0) = 0 as element of H1 and u(t0) = 0 as element of (V 1)âČ, sinceu(t) = 0 in [t0 â ÎŽ, t0), u â C0([t0 â ÎŽ, t0]; H1), and u â C0([t0 â ÎŽ, t0]; (V 1)âČ). We are nowin a position to apply the finite speed propagation result of [18, Theorem 6.1]. This theoremensures the existence of a third open set A3, with A1 ââ A3 ââ A2, such that, up to choosea smaller ÎŽ, we have u(t) = 0 on âA3 for every t â [t0, t0 + ÎŽ], and both (Ω \ A3) \ Î andA3 \ Î are union of two disjoint open sets with Lipschitz boundary.
In Ω \A3 the function u solves
ââ« t0+ÎŽ
t0âÎŽ
â«Î©\A3
u(t, x) · Ï(t, x) dx dt+
â« t0+ÎŽ
t0âÎŽ
â«Î©\A3
C(x)Eu(t, x) · EÏ(t, x) dx dt
+
â« t0+ÎŽ
t0âÎŽ
â«Î©\A3
D(x)Î(t, x)Eu(t, x) ·Î(t, x)EÏ(t, x) dx dt = 0
for every Ï â L2(t0â ÎŽ, t0 + ÎŽ; V 2)â©H1(t0â ÎŽ, t0 + ÎŽ; H2) such that Ï(t0â ÎŽ) = Ï(t0 + ÎŽ) = 0,where
V 2 := u â H1((Ω \A3) \ Ît0âÎŽ;R2) : u = 0 on âDΩ âȘ âA3, H2 := L2(Ω \A3;R2).
Since u(t) = 0 on âDΩâȘ âA3 for every t â [t0 â ÎŽ, t0 + ÎŽ] and u(t0 â ÎŽ) = u(t0 â ÎŽ) = 0 in thesense of (3.1.21) (we recall that u = 0 in [t0 â ÎŽ, t0)), we can apply Lemma 3.4.2 to deducethat u(t) = 0 in Ω \A3 for every t â [t0 â ÎŽ, t0 + ÎŽ].
On the other hand in A3, by setting
V 3t := u â H1(A3 \ Ît;R2) : u = 0 on âA3, H3 := L2(A3;R2),
we get that the function u solves
ââ« t0+ÎŽ
t0âÎŽ
â«A3
u(t, x) · Ï(t, x) dx dt+
â« t0+ÎŽ
t0âÎŽ
â«A3
C(x)Eu(t, x) · EÏ(t, x) dx dt = 0
for every function Ï â L2(t0âÎŽ, t0+ÎŽ; V 3t0+ÎŽ)â©H1(t0âÎŽ, t0+ÎŽ; H3) such that Ï(t) â V 3
t for a.e.t â (t0â ÎŽ, t0 + ÎŽ) and Ï(t0â ÎŽ) = Ï(t0 + ÎŽ) = 0. Here we would like to apply the uniquenessresult contained in [23, Theorem 4.3] (which is a slightly generalization of Theorem 1.2.10 ofChapter 1) for the spaces V 3
t tâ[t0âÎŽ,t0+ÎŽ] and H3, endowed with the usual norms, and forthe bilinear form
a(u, v) :=
â«A3
C(x)Eu(x) · Ev(x) dx for u, v â V 3t0+ÎŽ. (3.4.8)
As show in [20, Example 2.14] we can construct two maps Ί,Κ â C1,1([t0âÎŽ, t0 +ÎŽ]ĂA3;R2)such that for every t â [0, T ] the function Ί(t) : A3 â A3 is a diffeomorfisms of A3 in itselfwith inverse Κ(t) : A3 â A3. Moreover, Ί(0, y) = y for every y â A3, Ί(t,Î â© A3) = Î â© A3
and Ί(t,Ît0âÎŽâ©A3) = Îtâ©A3 for every t â [t0âÎŽ, t0+ÎŽ]. For every t â [t0âÎŽ, t0+ÎŽ], the maps
Chapter 3. A dynamic model for viscoelastic materials with growing cracks 83
(Qtu)(y) := u(Ί(t, y)), u â V 3t and y â A3, and (Rtv)(x) := v(Κ(t, x)), v â V 3
t0âÎŽ and x â A3,provide a family of linear and continuous operators which satisfies assumptions (U1)â(U8)of [23, Theorem 4.3]. The only condition to check is (U5), which is a ensured by (F3). Indeed,the bilinear form a satisfies the following ellipticity condition:
a(u, u) ℠λ1âEuâ2L2(A3) â„λ1
Ckâuâ2
V 3t0+ÎŽ
â λ1âuâ2H3 for every u â V 3t0+ÎŽ, (3.4.9)
where CK is the constant in Kornâs inequality in V 3t0+ÎŽ, namely
ââuâ2L2(A3) †CK(âuâ2L2(A3) + âEuâ2L2(A3)
)for every u â V 3
t0+ÎŽ.
Therefore have to show that Ί satisfies
|Ί(t, y)|2 < λ1
CKfor every t â [t0 â ÎŽ, t0 + ÎŽ] and y â A3,
which is analogous to the condition (1.2.4) appearing in Chapter 1. Thanks to (F3), we canconstruct the maps Ί and Κ in such a way that
|Ί(t, y)|2 < λ1
CKfor every t â [t0 â ÎŽ, t0 + ÎŽ] and y â A3,
as explained in [20, Example 3.1]. Moreover, every function in V 3t0+ÎŽ can be extended to a
function in H1(Ω\Î;Rd). Hence, we can use for Kornâs inequality in V 3t0+ÎŽ the same constant
CK of (3.1.2). This allows us to apply the uniqueness result [23, Theorem 4.3], which impliesu(t) = 0 in A3 for every t â [t0, t0 + ÎŽ]. In the case t0 = 0, it is enough to argue as before in[0, ÎŽ], by exploiting (3.4.2). Therefore u(t) = 0 in Ω for every t â [t0, t0 +ÎŽ], which contradictsthe maximality of t0. Hence t0 = T , that yields u(t) = 0 in Ω for every t â [0, T ].
Remark 3.4.5. Also Theorem 3.4.4 is true in the antiplane case, i.e, for (15), with identicalproof. Notice that, when the displacement is scalar, we do not need to use Kornâs inequalityin (3.4.9) to get the coercivity in V 3
t0+Ύ of the bilinear form a defined in (3.4.8). Therefore,in this case in (F3) it is enough to assume |s(t)|2 < λ1.
3.5 An example of a growing crack
We conclude this chapter with an example of a moving crack Îttâ[0,T ] and a solution tosystem (3.1.12)â(3.1.16) which satisfy the dynamic energy-dissipation balance (16), similarlyto the purely elastic case of [17].
In dimension d = 2 we consider an antiplane evolution, which means that the displacementu is scalar, and we take Ω := BR(0) â R2, with R > 0. We fix a constant 0 < c < 1 such thatcT < R, and we set
Ît := Ω â© (Ï, 0) â R2 : Ï â€ ct for every t â [0, T ].
Let us define the following function
S(x1, x2) := Im(âx1 + ix2) =
x2â2â|x|+ x1
x â R2 \ (Ï, 0) : Ï â€ 0,
where Im denotes the imaginary part of a complex number. Notice that the function Sbelongs to H1(Ω \ Î0) \H2(Ω \ Î0), and it is a solution to
âS = 0 in Ω \ Î0,
âS · Îœ = â2S = 0 on Î0.
84 3.5. An example of a growing crack
We consider the function
u(t, x) :=2âÏS
(x1 â ctâ
1â c2, x2
)t â [0, T ], x â Ω \ Ît,
and we define by w(t) its restriction to âΩ. Since u(t) has a singularity only at the crack-tip (ct, 0), the function w(t) can be seen as the trace on âΩ of a function belonging toH2(0, T ;L2(Ω)) â©H1(0, T ;H1(Ω \ Î0)), still denoted by w(t). It is easy to see that u solvesthe wave equation
u(t)ââu(t) = 0 in Ω \ Ît, t â [0, T ],
with boundary conditions
u(t) = w(t) on âΩ, t â [0, T ],
âu
âÎœ(t) = âu(t) · Îœ = 0 on Ît, t â [0, T ],
and initial data
u0(x1, x2) :=2âÏS
(x1â
1â c2, x2
)â H1(Ω \ Î0),
u1(x1, x2) := â 2âÏ
câ1â c2
â1S
(x1â
1â c2, x2
)â L2(Ω).
Let us consider a function Î satisfying the regularity assumptions (3.4.1) and condi-tion (3.4.7), namely
Î(t) = 0 on BΔ(t) := x â R2 : |xâ (ct, 0)| < Δ for every t â [0, T ],
with 0 < Δ < RâcT . In this case u is a solution, according to Definition 3.1.5, to the dampedwave equation
u(t)ââu(t)â div(Î2(t)âu(t)) = f(t) in Ω \ Ît, t â [0, T ],
with forcing term f given by
f := âdiv(Î2âu) = ââΠ· 2ÎâuâÎ2âu â L2(0, T ;L2(Ω)),
and boundary and initial conditions
u(t) = w(t) on âΩ, t â [0, T ],
âu
âÎœ(t) + Î2(t)
âu
âÎœ(t) = 0 on Ît, t â [0, T ],
u(0) = u0, u(0) = u1 in Ω \ Î0.
Notice that for the homogeneous Neumann boundary conditions on Ît we have used the factthat âu
âÎœ (t) = âu(t) · Îœ = â2u(t) = 0 on Ît. By the uniqueness result proved in the previoussection, the function u coincides with the solution given by Theorem 3.2.1. Thanks to thecomputations done in [17, Section 4], we know that u satisfies for every t â [0, T ] the followingdynamic energy-dissipation balance for the undamped equation, where ct coincides with thelength of Ît \ Î0:
1
2âu(t)â2L2(Ω) +
1
2ââu(t)â2L2(Ω) + ct
=1
2âu(0)â2L2(Ω) +
1
2ââu(0)â2L2(Ω) +
â« t
0(âu
âÎœ(s), w(s))L2(âΩ) ds.
(3.5.1)
Chapter 3. A dynamic model for viscoelastic materials with growing cracks 85
Moreover, we haveâ« t
0(âu
âÎœ(s), w(s))L2(âΩ) ds =
â« t
0(âu(s),âw(s))L2(Ω) dsâ
â« t
0(u(s), w(s))L2(Ω) ds
+ (u(t), w(t))L2(Ω) â (u(0), w(0))L2(Ω).
(3.5.2)
For every t â [0, T ] we compute
(f(t), u(t)â w(t))L2(Ω) = ââ«
(Ω\BΔ(t))\Îtdiv[Î2(t, x)âu(t, x)](u(t, x)â w(t, x)) dx
= ââ«
(Ω\BΔ(t))\Îtdiv[Î2(t, x)âu(t, x)(u(t, x)â w(t, x))] dx
+
â«(Ω\BΔ(t))\Ît
Î2(t, x)âu(t, x) · (âu(t, x)ââw(t, x)) dx.
If we denote by u+(t) and w+(t) the traces of u(t) and w(t) on Ît from above and by uâ(t)and wâ(t) the trace from below, thanks to the divergence theorem we haveâ«
(Ω\BΔ(t))\Îtdiv[Î2(t, x)âu(t, x)(u(t, x)â w(t, x))] dx
=
â«âΩ
Î2(t, x)âu
âÎœ(t, x)(u(t, x)â w(t, x)) dx+
â«âBΔ(t)
Î2(t, x)âu
âÎœ(t, x)(u(t, x)â w(t, x)) dx
ââ«
(Ω\BΔ(t))â©Ît
Î2(t, x)â2u+(t, x)(u+(t, x)â w+(t, x)) dH1(x)
+
â«(Ω\BΔ(t))â©Ît
Î2(t, x)â2uâ(t, x)(uâ(t, x)â wâ(t, x)) dH1(x) = 0,
since u(t) = w(t) on âΩ, Î(t) = 0 on âBΔ(t), and â2u(t) = 0 on Ît. Therefore for everyt â [0, T ] we get
(f(t), u(t)â w(t))L2(Ω) = âÎ(t)âu(t)â2L2(Ω) â (Î(t)âu(t),Î(t)âw(t))L2(Ω). (3.5.3)
By combining (3.5.1)â(3.5.3) we deduce that u satisfies for every t â [0, T ] the followingdynamic energy-dissipation balance
1
2âu(t)â2L2(Ω) +
1
2ââu(t)â2L2(Ω) + ct+
â« t
0âÎ(s)âu(s)â2L2(Ω) ds
=1
2âu(0)â2L2(Ω) +
1
2ââu(0)â2L2(Ω) +Wtot(t),
(3.5.4)
where in this case the total work takes the form
Wtot(t) :=
â« t
0
[(f(s), u(s)â w(s))L2(Ω) + (âu(s),âw(s))L2(Ω)
]ds
+
â« t
0
[(Î(s)âu(s),Î(s)âw(s))L2(Ω) â (u(s), w(s))L2(Ω)
]ds
+ (u(t), w(t))L2(Ω) â (u(0), w(0))L2(Ω).
Notice that equality (3.5.4) gives (16). This show that in this model the dynamic energy-dissipation balance can be satisfied by a moving crack, in contrast with the case Î = 1, whichalways leads to (14).
Chapter 4
A phase-field model of dynamicfracture
In this chapter we prove an existence result for the dynamic phase-field model of fracturewith a crack-dependent dissipation (D1)â(D3).
The chapter is organized as follows: in Section 4.1 we list the main assumptions on ourmodel, and in Theorem 4.1.5 we state our existence result. Section 4.2 is devoted to thestudy of the time discretization scheme. We construct an approximation of our evolutionby solving, with an alternate minimization procedure, problems (D1) and (D2). Next, weshow that this discrete evolution satisfies the estimate (4.2.17), which allows us to pass tothe limit as the time step tends to zero. For every k â N âȘ 0 we obtain the existence of adynamic evolution t 7â (u(t), v(t)) which satisfies (D1) and (D2), and the energy-dissipationinequality (4.2.32). We complete the proof of Theorem 4.1.5 in Section 4.3, where we provethat for k > d/2 our evolution is more regular in time, and it satisfies the dynamic energy-dissipation balance (18). Finally, in Section 4.4 we study the dynamic phase-field modelwithout dissipative terms (D1)â(D3).
The results contained in this chapter are the basis of the submitted paper [7].
4.1 Preliminary results
Let T be a positive number and let Ω â Rd be a bounded open set with Lipschitz boundary.We fix two (possibly empty) Borel subsets âD1Ω, âD2Ω of âΩ, and we denote by âN1Ω, âN2Ωtheir complements. We introduce the spaces
H1D1
(Ω;Rd) := Ï â H1(Ω;Rd) : Ï = 0 on âD1Ω,H1D2
(Ω) := Ï â H1(Ω) : Ï = 0 on âD2Ω,
and we denote by Hâ1D1
(Ω;Rd) the dual space of H1D1
(Ω;Rd). The transpose of the natural
embedding H1D1
(Ω;Rd) â L2(Ω;Rd) induces the embedding of L2(Ω;Rd) into Hâ1D1
(Ω;Rd),which is defined by
ăg, ÏăHâ1D1
(Ω):= (g, Ï)L2(Ω) for g â L2(Ω;Rd) and Ï â H1
D1(Ω;Rd).
Let C : Ωâ L (RdĂdsym;RdĂdsym) be a fourth-order tensor field satisfying the following naturalassumptions in linear elasticity:
C â Lâ(Ω; L (RdĂdsym;RdĂdsym)), (4.1.1)
(C(x)Ο1) · Ο2 = Ο1 · (C(x)Ο2) for a.e. x â Ω and for every Ο1, Ο2 â RdĂdsym, (4.1.2)
87
88 4.1. Preliminary results
C(x)Ο · Ο ℠λ0|Ο|2 for a.e. x â Ω and for every Ο â RdĂdsym, (4.1.3)
for a constant λ0 > 0. Thanks to second Kornâs inequality there exists a constant CK > 0,depending on Ω, such that
ââÏâL2(Ω) †CK(âÏâL2(Ω) + âEÏâL2(Ω)) for every Ï â H1(Ω;Rd).
By combining Kornâs inequality with (4.1.3), we obtain that C satisfies the following ellipticitycondition of integral type:
(CEÏ,EÏ)L2(Ω) â„ c0âÏâ2H1(Ω) â c1âÏâ2L2(Ω) for every Ï â H1(Ω;Rd), (4.1.4)
for two constants c0 > 0 and c1 â R.We fix Δ > 0 and we consider a map b : Râ [0,+â) satisfying
b â C1(R) is convex and non-decreasing, (4.1.5)
b(s) ℠η for every s â R and some η > 0. (4.1.6)
We define the functionals elastic energy E : H1(Ω;Rd)ĂH1(Ω)â [0,â] and surface energyH : H1(Ω)â [0,â) in the following way:
E (u, v) :=1
2
â«Î©b(v(x))C(x)Eu(x) · Eu(x) dx,
H (v) :=1
4Δ
â«Î©|1â v(x)|2 dx+ Δ
â«Î©|âv(x)|2 dx
for u â H1(Ω;Rd) and v â H1(Ω). We also define the kinetic energy K : L2(Ω;Rd)â [0,â)and the dissipative energy G : Hk(Ω)â [0,â) for every k â N âȘ 0 as
K (w) :=1
2
â«Î©|w(x)|2 dx, G (Ï) :=
kâi=0
αi
â«Î©|âiÏ(x)|2dx
for w â L2(Ω;Rd) and Ï â Hk(Ω), where αi, i = 0, . . . , k, are non negative numbers withα0, αk > 0 (we recall that H0(Ω) := L2(Ω)). Notice that, by [1, Corollary 4.16], the functionalG induces a norm on Hk(Ω) which is equivalent to the standard one. In particular, thereexist two constants ÎČ0, ÎČ1 > 0 such that
ÎČ0âÏâ2Hk(Ω) †G (Ï) †ÎČ1âÏâ2Hk(Ω) for every Ï â Hk(Ω).
Finally, we define the total energy F : H1(Ω;Rd)Ă L2(Ω;Rd)ĂH1(Ω)â [0,â] as
F (u,w, v) := K (w) + E (u, v) + H (v)
for u â H1(Ω;Rd), w â L2(Ω;Rd), and v â H1(Ω).Throughout the chapter we always assume that C and b satisfy (4.1.1)â(4.1.3), (4.1.5),
and (4.1.6), and that Δ is a fixed positive number. Given
w1 â H2(0, T ;L2(Ω;Rd)) â©H1(0, T ;H1(Ω;Rd)), (4.1.7)
w2 â H1(Ω) â©Hk(Ω) with w2 †1 on âD2Ω, (4.1.8)
f â L2(0, T ;L2(Ω;Rd)), g â H1(0, T ;Hâ1D1
(Ω;Rd)), (4.1.9)
u0 â w1(0) â H1D1
(Ω;Rd), u1 â L2(Ω;Rd), (4.1.10)
v0 â w2 â H1D2
(Ω) â©Hk(Ω) with v0 †1 in Ω, (4.1.11)
Chapter 4. A phase-field model of dynamic fracture 89
we search a pair (u, v) which solves the elastodynamics system
u(t)â div[b(v(t))CEu(t)] = f(t) + g(t) in Ω, t â [0, T ], (4.1.12)
with boundary conditions formally written as
u(t) = w1(t) on âD1Ω, t â [0, T ], (4.1.13)
v(t) = w2 on âD2Ω, t â [0, T ], (4.1.14)
(b(v(t))CEu(t))Îœ = 0 on âN1Ω, t â [0, T ], (4.1.15)
and initial conditions
u(0) = u0, u(0) = u1, v(0) = v0 in Ω. (4.1.16)
In addition, we require the irreversibility condition:
v(t) †v(s) in Ω for 0 †s †t †T , (4.1.17)
and for a.e. t â (0, T ) the following crack stability condition:
E (u(t), vâ)â E (u(t), v(t)) + H (vâ)âH (v(t))
+kâi=0
αi(âiv(t),âivâ ââiv(t))L2(Ω) â„ 0(4.1.18)
among all vâ âw2 â H1D2
(Ω) â©Hk(Ω) with vâ †v(t). Notice that the space H1(Ω) â©Hk(Ω)
coincides with either H1(Ω) (when k = 0) or Hk(Ω) (for k â„ 1). Finally, for every t â [0, T ]we ask the dynamic energy-dissipation balance:
F (u(t), u(t), v(t)) +
â« t
0G (v(s)) ds = F (u0, u1, v0) + Wtot(u, v; 0, t), (4.1.19)
where Wtot(u, v; t1, t2) is the total work over the time interval [t1, t2] â [0, T ], defined as
Wtot(u, v; t1, t2) :=
â« t2
t1
[(f(s), u(s)â w1(s))L2(Ω) + (b(v(s))CEu(s), Ew1(s))L2(Ω)
]ds
ââ« t2
t1
[(u(s), w1(s))L2(Ω) + ăg(s), u(s)â w1(s)ăHâ1
D1(Ω)
]ds
+ (u(t2), w1(t2))L2(Ω) + ăg(t2), u(t2)â w1(t2)ăHâ1D1
(Ω)
â (u(t1), w1(t1))L2(Ω) â ăg(t1), u(t1)â w1(t1)ăHâ1D1
(Ω).
Remark 4.1.1. A simple prototype for the function b is given by
b(s) := (s âš 0)2 + η for s â R.
In this case, the elastic energy becomes
E (u, v) =1
2
â«Î©
[(v(x) ⚠0)2 + η]C(x)Eu(x) · Eu(x) dx (4.1.20)
for u â H1(Ω;Rd) and v â H1(Ω), which corresponds to the phase-field model of dynamicfracture (D1)â(D3) considered in the introduction. Usually, in the phase-field setting, theelastic energy functional is defined as
1
2
â«Î©
[(v(x))2 + η]C(x)Eu(x) · Eu(x) dx
for u â H1(Ω;Rd) and v â H1(Ω), with v satisfying 0 †v †1. In our case, due to thepresence of the dissipative term introduced in (D2) and (D3), we need to consider phase-fieldfunctions v which may assume negative values. Therefore, we have to slightly modify theelastic energy functional by considering (4.1.20).
90 4.1. Preliminary results
Remark 4.1.2. We give an idea of the meaning of the term G (v) in the phase-field setting,by comparing it with a dissipation, in the sharp-interface case, which depends on the velocityof the crack-tips. We consider just an example in the particular case d = 2 and k = 0of a rectilinear crack Ît := (Ï, 0) : Ï â€ s(t), t â [0, T ], moving along the x1-axis, withs â C1([0, T ]), s(0) = 0, and s(t) â„ 0 for every t â [0, T ]. In view of the analysis done in [4],the sequence vΔ(t) which best approximate Ît takes the following form:
vΔ(t, x) := Κ
(dist(x,Ît)
Δ
)for (t, x) â [0, T ]Ă R2.
Here, Κ: R â [0, 1] is a C1 function satisfying Κ(s) = 0 for |s| †Ύ, with 0 < ÎŽ < 1, andΚ(s) = 1 for |s| â„ 1. The function vΔ â C1([0, T ]Ă R2) is constantly 0 in a ΔΎ-neighborhoodof Ît, and takes the value 1 outside a Δ-neighborhood of Ît. Moreover, its time derivativesatisfies
vΔ(t, x) = â s(t)Δâ1Ί
(xâ (s(t), 0)
Δ
)for (t, x) â [0, T ]Ă R2,
where Ί(y) := Κ(dist(y,Î0)) for y â R2. In particular for every t â [0, T ] we deduce
âvΔ(t)â2L2(Ω) =s(t)2
Δ2
â«R2
âŁâŁâŁâŁâ1Ί
(xâ (s(t), 0)
Δ
)âŁâŁâŁâŁ2 dx = s(t)2
â«R2
|â1Ί(y)|2 dy = CΊs(t)2.
Therefore, this term can be used to detect the dissipative effects due to the velocity of themoving crack. With similar computations, if there are m crack-tips with different velocitiessi(t), i = 1, . . . ,m, then the term âvΔ(t)â2L2(Ω) corresponds to a dissipation of the formâm
i=1Cis2i (t), with Ci positive constants.
To precise the notion of solution to problem (4.1.12)â(4.1.19), we consider a pair of func-tions (u, v) satisfying the following regularity assumptions:
u â C0([0, T ];H1(Ω;Rd)) â© C1([0, T ];L2(Ω;Rd)) â©H2(0, T ;Hâ1D1
(Ω;Rd)), (4.1.21)
u(t)â w1(t) â H1D1
(Ω;Rd) for every t â [0, T ], (4.1.22)
v â C0([0, T ];H1(Ω)) â©H1(0, T ;Hk(Ω)), (4.1.23)
v(t)â w2 â H1D2
(Ω) and v(t) †1 in Ω for every t â [0, T ]. (4.1.24)
Definition 4.1.3. Let w1, w2, f , and g be as in (4.1.7)â(4.1.9). We say that (u, v) is a weaksolution to the elastodynamics system (4.1.12) with boundary conditions (4.1.13)â(4.1.15), if(u, v) satisfies (4.1.21)â(4.1.24), and for a.e. t â (0, T ) we have
ău(t), ÏăHâ1D1
(Ω) + (b(v(t))CEu(t), EÏ)L2(Ω) = (f(t), Ï)L2(Ω) + ăg(t), ÏăHâ1D1
(Ω) (4.1.25)
for every Ï â H1D1
(Ω;Rd).
Remark 4.1.4. Since b satisfies (4.1.5) and (4.1.6), and v(t) †1 for every t â [0, T ], thefunction b(v(t)) belongs to Lâ(Ω) for every t â [0, T ]. Hence, equation (4.1.25) makes sensefor every Ï â H1
D1(Ω;Rd). Moreover, if (u, v) satisfies (4.1.21)â(4.1.24), then the function
(t1, t2) 7â Wtot(u, v; t1, t2) is well defined and continuous, thanks to the previous assumptionson C, b, w1, f , and g.
We state now our main result, whose proof will be given at the end of Section 4.3.
Theorem 4.1.5. Let k > d/2 and let w1, w2, f , g, u0, u1, and v0 be as in (4.1.7)â(4.1.11).Then there exists a weak solution (u, v) to problem (4.1.12) with boundary conditions (4.1.13)â(4.1.15) and initial conditions (4.1.16). Moreover, the pair (u, v) satisfies the irreversibilitycondition (4.1.17), the crack stability condition (4.1.18), and the dynamic energy-dissipationbalance (4.1.19).
Chapter 4. A phase-field model of dynamic fracture 91
Remark 4.1.6. According to Griffithâs dynamic criterion (see [41]), we expect the sum ofkinetic and elastic energy to be dissipated during the evolution, while it is balanced when wetake into account the surface energy associated to the phase-field function v. This happensin our case if we also consider
â« t0 G (v) ds. The presence of this term takes into account
the rate at which the function v is decreasing and it is a consequence of the crack stabilitycondition (4.1.18).
We need k > d/2 in order to obtain the energy equality (4.1.19). Indeed, in this casethe embedding Hk(Ω) â C0(Ω) is continuous and compact (see, e.g., [1, Theorem 6.2]),which implies that v(t) â C0(Ω) for a.e. t â (0, T ). This regularity is crucial, since weobtain (4.1.19) throughout another energy balance (see (4.3.20)), which is well defined onlywhen v(t) â Lâ(Ω).
Remark 4.1.7. In Theorem 4.1.5 we consider only the case of zero Neumann boundary data.Anyway, the previous result can be easily adapted to Neumann boundary conditions of theform
(b(v(t))CEu(t))Îœ = F (t) on âN1Ω, t â [0, T ], (4.1.26)
provided that F â H1(0, T ;L2(âN1Ω;Rd)). In this case a weak solution to problem (4.1.12)with Dirichlet boundary conditions (4.1.13) and (4.1.14), and Neumann boundary condi-tion (4.1.26) is a pair (u, v) satisfying (4.1.21)â(4.1.24) and for a.e. t â (0, T ) the equation
ău(t), ÏăHâ1D1
(Ω) + (b(v(t))CEu(t), EÏ)L2(Ω) = (f(t), Ï)L2(Ω) + ăg(t), ÏăHâ1D1
(Ω)
for every Ï â H1D1
(Ω;Rd), where the term g(t) â Hâ1D1
(Ω;Rd) is defined for t â [0, T ] as
ăg(t), ÏăHâ1D1
(Ω):= ăg(t), ÏăHâ1
D1(Ω) +
â«âN1
ΩF (t, x) · Ï(x)dH dâ1(x) for Ï â H1
D1(Ω;Rd).
Since g â H1(0, T ;Hâ1D1
(Ω;Rd)), we can apply Theorem 4.1.5 with g instead of g, and wederive the existence of a weak solution (u, v) to (4.1.12)â(4.1.14) with Neumann boundarycondition (4.1.26).
In the next lemma we show that for k > d/2 the dynamic energy-dissipation balance canbe rephrased in the following identity:
âvE (u(t), v(t))[v(t)] + âH (v(t))[v(t)] + G (v(t)) = 0 for a.e. t â (0, T ), (4.1.27)
where the derivatives âvE and âH take the form
âvE (u, v)[Ï] =1
2
â«Î©b(v)ÏCEu · Eudx for u â H1(Ω;Rd) and v, Ï â H1(Ω) â© Lâ(Ω),
âH (v)[Ï] =1
2Δ
â«Î©
(v â 1)Ïdx+ 2Δ
â«Î©âv · âÏdx for v, Ï â H1(Ω).
Lemma 4.1.8. Let k > d/2 and let w1, w2, f , g, u0, u1, and v0 be as in (4.1.7)â(4.1.11). As-sume that (u, v) is a weak solution to problem (4.1.12)â(4.1.15) with initial conditions (4.1.16).Then the dynamic energy-dissipation balance (4.1.19) is equivalent to identity (4.1.27).
Proof. We follow the same techniques of [21, Lemma 2.6]. Let us fix 0 < h < T and let usdefine the function
Ïh(t) :=u(t+ h)â u(t)
hâ w1(t+ h)â w1(t)
hfor t â [0, T â h].
92 4.1. Preliminary results
We use Ïh(t) as test function in (4.1.25) first at time t, and then at time t+ h. By summingthe two expressions and integrating in a fixed time interval [t1, t2] â [0, T âh], we obtain theidentityâ« t2
t1
ău(t+ h) + u(t), Ïh(t)ăHâ1D1
(Ω) dt
+
â« t2
t1
(b(v(t+ h))CEu(t+ h) + b(v(t))CEu(t), EÏh(t))L2(Ω) dt
=
â« t2
t1
(f(t+ h) + f(t), Ïh(t))L2(Ω) dt+
â« t2
t1
ăg(t+ h) + g(t), Ïh(t)ăHâ1D1
(Ω) dt.
(4.1.28)
We study these four terms separately. By performing an integration by parts, the first onebecomes â« t2
t1
ău(t+ h) + u(t), Ïh(t)ăHâ1D1
(Ω) dt
= ââ« t2
t1
(u(t+ h) + u(t), Ïh(t))L2(Ω) dt+ (u(t2 + h) + u(t2), Ïh(t2))L2(Ω)
â (u(t1 + h) + u(t1), Ïh(t1))L2(Ω)
= â1
h
â« t2+h
t2
âu(t)â2L2(Ω)dt+1
h
â« t1+h
t1
âu(t)â2L2(Ω)dt
+1
h
â« t2
t1
(u(t+ h) + u(t), w1(t+ h)â w1(t))L2(Ω) dt
+ (u(t2 + h) + u(t2), Ïh(t2))L2(Ω) â (u(t1 + h) + u(t1), Ïh(t1))L2(Ω).
Since u,w1 â C1([0, T ];L2(Ω;Rd)), by sending hâ 0+ we deduce
limhâ0+
[â1
h
â« t2+h
t2
âu(t)â2L2(Ω)dt+1
h
â« t1+h
t1
âu(t)â2L2(Ω)dt
]= ââu(t2)â2L2(Ω) + âu(t1)â2L2(Ω),
(4.1.29)
limhâ0+
[(u(t2 + h) + u(t2), Ïh(t2))L2(Ω) â (u(t1 + h) + u(t1), Ïh(t1))L2(Ω)
]= 2âu(t2)â2L2(Ω) â 2(u(t2), w1(t2))L2(Ω) â 2âu(t1)â2L2(Ω) + 2(u(t1), w1(t1))L2(Ω).
(4.1.30)
Notice that 1h [w1( ·+h)âw1] converges strongly to w1 in L2(t1, t2;L2(Ω;Rd)) as hâ 0+, since
w1 belongs to H1(0, T ;L2(Ω;Rd)). Therefore, there exist a sequence hm â 0+ as m â â,and a function Îș â L2(t1, t2) such that for a.e. t â (t1, t2)
1
hm(u(t+ hm) + u(t), w1(t+ hm)â w1(t))L2(Ω) â 2(u(t), w1(t))L2(Ω) as mââ,âŁâŁâŁâŁ 1
hm(u(t+ hm) + u(t), w1(t+ hm)â w1(t))L2(Ω)
âŁâŁâŁâŁ †2âuâLâ(0,T ;L2(Ω))Îș(t) for every m â N.
By the dominated convergence theorem we derive
limhâ0+
1
h
â« t2
t1
(u(t+ h) + u(t), w1(t+ h)â w1(t))L2(Ω) dt = 2
â« t2
t1
(u(t), w1(t))L2(Ω) dt, (4.1.31)
since the limit does not depend on the subsequence hmm. For the term involving f , weobserve that f( · + h) â f and Ïh â u â w1 in L2(t1, t2;L2(Ω;Rd)) as h â 0+. Hence, wehave
limhâ0+
â« t2
t1
(f(t+ h) + f(t), Ïh(t))L2(Ω) dt = 2
â« t2
t1
(f(t), u(t)â w1(t))L2(Ω) dt. (4.1.32)
Chapter 4. A phase-field model of dynamic fracture 93
By using the identityâ« t2
t1
ăg(t+ h) + g(t), Ïh(t)ăHâ1D1
(Ω) dt
=2
h
â« t2+h
t2
ăg(t), u(t)â w1(t)ăHâ1D1
(Ω) dtâ 2
h
â« t1+h
t1
ăg(t), u(t)â w1(t)ăHâ1D1
(Ω) dt
â 1
h
â« t2
t1
ăg(t+ h)â g(t), u(t+ h) + u(t)â w1(t+ h)â w1(t)ăHâ1D1
(Ω) dt,
and proceeding as before, we also deduce
limhâ0+
â« t2
t1
ăg(t+ h) + g(t), Ïh(t)ăHâ1D1
(Ω) dt
= 2ăg(t2), u(t2)â w1(t2)ăHâ1D1
(Ω) â 2ăg(t1), u(t1)â w1(t1)ăHâ1D1
(Ω)
â 2
â« t2
t1
ăg(t), u(t)â w1(t)ăHâ1D1
(Ω) dt.
(4.1.33)
It remains to study the last term, that can be rephrased in the following wayâ« t2
t1
(b(v(t+ h))CEu(t+ h) + b(v(t))CEu(t), EÏh(t))L2(Ω) dt
=1
h
â« t2+h
t2
(b(v(t))CEu(t), Eu(t))L2(Ω) dtâ 1
h
â« t1+h
t1
(b(v(t))CEu(t), Eu(t))L2(Ω) dt
â 1
h
â« t2
t1
([b(v(t+ h))â b(v(t))]CEu(t), Eu(t+ h))L2(Ω) dt
â 1
h
â« t2
t1
(b(v(t+ h))CEu(t+ h) + b(v(t))CEu(t), Ew1(t+ h)â Ew1(t))L2(Ω) dt.
Since Hk(Ω) â C0(Ω), we deduce that v belongs to the space C0([0, T ];C0(Ω)). Thisproperty, together with b â C1(R) and u â C0([0, T ];H1(Ω;Rd)), implies
limhâ0+
[1
h
â« t2+h
t2
(b(v(t))CEu(t), Eu(t))L2(Ω) dtâ 1
h
â« t1+h
t1
(b(v(t))CEu(t), Eu(t))L2(Ω) dt
]= (b(v(t2))CEu(t2), Eu(t2))L2(Ω) â (b(v(t1))CEu(t1), Eu(t1))L2(Ω). (4.1.34)
Moreover, the sequence 1h [v( · +h)âv] converges strongly to v in L2(t1, t2;C0(Ω)) as hâ 0+.
Therefore, there exist a subsequence hm â 0+ as mââ and a function Îș â L2(t1, t2) suchthat for a.e. t â (t1, t2)
v(t+ hm)â v(t)
hmâ v(t) in C0(Ω) as mââ,â„â„â„â„v(t+ hm)â v(t)
hm
â„â„â„â„Lâ(Ω)
†Îș(t) for every m â N.
Thanks to (4.1.5), we can apply Lagrangeâs theorem to derive for a.e. t â (t1, t2)
1
hm(b(v(t+ hm))â b(v(t))CEu(t), Eu(t+ hm))L2(Ω)
â (b(v(t))v(t)CEu(t), Eu(t))L2(Ω),
94 4.1. Preliminary results
as mââ, while for every m â NâŁâŁâŁâŁ 1
hm([b(v(t+ hm))â b(v(t))]CEu(t), Eu(t+ hm))L2(Ω)
âŁâŁâŁâŁâ€ b(1)âCâLâ(Ω)âEuâ2Lâ(0,T ;L2(Ω))Îș(t),
since u â C0([0, T ];H1(Ω;Rd)) and v(t) †1 for every t â [0, T ]. The dominated convergencetheorem yields
limhâ0+
1
h
â« t2
t1
([b(v(t+ h))â b(v(t))]CEu(t), Eu(t+ h))L2(Ω) dt
=
â« t2
t1
(b(v(t))v(t)CEu(t), Eu(t))L2(Ω) dt,
(4.1.35)
being the limit independent of the subsequence hmm. Finally, 1h [Ew1( · + h)â Ew1] con-
verges strongly to Ew1 in L2(t1, t2;L2(Ω;RdĂd)) as h â 0+. By arguing as in (4.1.31), thisfact gives
limhâ0+
1
h
â« t2
t1
(b(v(t+ h))CEu(t+ h) + b(v(t))CEu(t), Ew1(t+ h)â Ew1(t))L2(Ω) dt
= 2
â« t2
t1
(b(v(t))CEu(t), Ew1(t))L2(Ω) dt.
(4.1.36)
We combine together (4.1.28)â(4.1.36) to derive
K (u(t2)) + E (u(t2), v(t2))â 1
2
â« t2
t1
(b(v(t))v(t)CEu(t), Eu(t))L2(Ω) dt
= K (u(t1)) + E (u(t1), v(t1)) + Wtot(u, v; t1, t2)
for every t1, t2 â [0, T ) with t1 < t2. Since all terms in the previous equality are contin-uous with respect to t2, we deduce that a weak solution to (4.1.12)â(4.1.15) with initialconditions (4.1.16) satisfies the energy balance
K (u(t2)) + E (u(t2), v(t2))â 1
2
â« t2
t1
(b(v(t))v(t)CEu(t), Eu(t))L2(Ω) dt
= K (u(t1)) + E (u(t1), v(t1)) + Wtot(u, v; t1, t2)
(4.1.37)
for every t1, t2 â [0, T ] with t1 < t2.Let us assume now (4.1.27). Since v â H1(0, T ;Hk(Ω)), the function t 7â ζ(t) := H (v(t))
is absolutely continuous on [0, T ], with ζ(t) = âH (v(t))[v(t)] for a.e. t â (0, T ). By integrat-ing (4.1.27) over [t1, t2] â [0, T ], we obtain
â 1
2
â« t2
t1
(b(v(t))v(t)CEu(t), Eu(t))L2(Ω) dt = H (v(t2))âH (v(t1))+
â« t2
t1
G (v(t)) dt. (4.1.38)
This identity, together with (4.1.37), implies the dynamic energy-dissipation balance (4.1.19).On the other hand, if (4.1.19) is satisfied, by comparing it with (4.1.37) we deduce (4.1.38)for every interval [t1, t2] â [0, T ], from which (4.1.27) follows.
Remark 4.1.9. When k > d2 , the crack stability condition (4.1.18) is equivalent for a.e.
t â (0, T ) to the following variational inequality
âvE (u(t), v(t))[Ï] + âH (v(t))[Ï] +
kâi=0
αi(âiv(t),âiÏ)L2(Ω) â„ 0 (4.1.39)
Chapter 4. A phase-field model of dynamic fracture 95
among all Ï â H1D2
(Ω)â©Hk(Ω) with Ï â€ 0. Indeed, for every s â (0, 1] we can take v(t) + sÏas test function in (4.1.18). After some computations and by dividing by s, we deduce
E (u(t), v(t) + sÏ)â E (u(t), v(t))
s+ âH (v(t))[Ï] +
kâi=0
αi(âiv(t),âiÏ)L2(Ω)
+ s
[1
4ΔâÏâ2L2(Ω) + ΔââÏâ2L2(Ω)
]â„ 0.
(4.1.40)
Let us fix x â Ω. By Lagrangeâs theorem there exists zs(t, x) â [v(t, x) + sÏ(x), v(t, x)] suchthat
b(v(t, x) + sÏ(x))â b(v(x))
s= b(zs(t, x))Ï(x),
since b â C1(R). In particular, we have
limsâ0+
b(v(t, x) + sÏ(x))â b(v(x))
s= b(v(t, x))Ï(x),âŁâŁâŁâŁb(v(t, x) + sÏ(x))â b(v(x))
s
âŁâŁâŁâŁ †b(1)|Ï(x)|,
because b â C0(R) is non negative, non-decreasing, and zs(t, x) †v(t, x) †1. Then, thedominated convergence theorem yields
limsâ0+
E (u(t), v(t) + sÏ)â E (u(t), v(t))
s=
1
2
â«Î©b(v(t))ÏCEu(t)·Eu(t) dx = âvE (u(t), v(t))[Ï].
By sending s â 0+ in (4.1.40) we hence deduce (4.1.39). On the other hand, it is easy tocheck that (4.1.39) implies (4.1.18), by exploiting the convexity of vâ â E (u(t), vâ) + H (vâ)and taking Ï := vâ â v(t) for every vâ â w2 â H1
D2(Ω) â©Hk(Ω) with vâ †v(t).
The inequality (4.1.39) gives that for a.e. t â (0, T ) the distribution
â1
2b(v(t))CEu(t) · Eu(t)â 1
2Δ(v(t)â 1) + 2Δâv(t)â
kâi=0
αi(â1)iâiv(t)
is positive on Ω. Therefore it coincides with a positive Radon measure ”(t) on Ω, by Rieszâsrepresentation theorem. In particular, since Hk(Ω) â C0(Ω), for a.e. t â (0, T ) we deduce
ăζ(t), Ïă(Hk(Ω))âČ := âvE (u(t), v(t))[Ï]+âH (v(t))[Ï]+kâi=0
αi(âiv(t),âiÏ)L2(Ω) = ââ«
ΩÏd”(t)
for every function Ï â Hk(Ω) with compact support in Ω. We combine this fact withidentity (4.1.27) to derive for our model an analogous of the classical activation rule inGriffithâs criterion: for a.e. t â (0, T ) the positive measure ”(t) must vanish on the set ofpoints x â Ω where v(t, x) > 0. Indeed, let us consider a sequence Ïmm â Câc (Ω) such that0 †Ïm †Ïm+1 †1 in Ω for every m â N, and Ïm(x)â 1 for every x â Ω as mââ. Thefunction v(t) is admissible in (4.1.39) for a.e. t â (0, T ), since 1
h [v(t + h) â v(t)] â H1D2
(Ω)
converges strongly to v(t) in Hk(Ω) as h â 0+, and t 7â v(t) is non-decreasing in [0, T ].Therefore, thanks to (4.1.27) and (4.1.39), for a.e. t â (0, T ) we get
0 = ăζ(t), v(t)ă(Hk(Ω))âČ = ăζ(t), v(t)Ïmă(Hk(Ω))âČ + ăζ(t), v(t)(1â Ïm)ă(Hk(Ω))âČ
â„ ăζ(t), v(t)Ïmă(Hk(Ω))âČ = ââ«
Ωv(t)Ïmd”(t) â„ 0,
because v(t)Ïm â Hk(Ω) has compact support. Hence, for a.e. t â (0, T ) we have
0 = limmââ
â«Î©v(t)Ïmd”(t) =
â«Î©v(t) d”(t),
by the monotone convergence theorem, which implies our activation condition.
96 4.2. The time discretization scheme
4.2 The time discretization scheme
In this section we show some general results that are true for every k â N âȘ 0. In par-ticular, we prove that problem (4.1.12)â(4.1.16) admits a solution (u, v) (in a weaker sense)which satisfies the irreversibility condition (4.1.17) and the crack stability condition (4.1.18).Throughout this section, we always assume that the functions w1, w2, f , g, u0, u1, and v0
satisfy (4.1.7)â(4.1.11).We start by introducing the following notion of solution, which requires less regularity on
the time variable.
Definition 4.2.1. The pair (u, v) is a generalized solution to (4.1.12)â(4.1.15) if
u â Lâ(0, T ;H1(Ω;Rd)) â©W 1,â(0, T ;L2(Ω;Rd)) â©H2(0, T ;Hâ1D1
(Ω;Rd)), (4.2.1)
u(t)â w1(t) â H1D1
(Ω;Rd) for every t â [0, T ], (4.2.2)
v â Lâ(0, T ;H1(Ω)) â©H1(0, T ;Hk(Ω)), (4.2.3)
v(t)â w2 â H1D2
(Ω) and v(t) †1 in Ω for every t â [0, T ], (4.2.4)
and for a.e. t â (0, T ) equation (4.1.25) holds.
Remark 4.2.2. By arguing as in Remark 1.2.7, we derive that a generalized solution (u, v)to problem (4.1.12)â(4.1.15) satisfies u â C0
w([0, T ];H1(Ω;Rd)), u â C0w([0, T ];L2(Ω;Rd)),
and v â C0w([0, T ];H1(Ω)). Therefore, the initial conditions (4.1.16) makes sense, since the
functions u(t), u(t), and v(t) are uniquely defined for every t â [0, T ] as elements of H1(Ω;Rd),L2(Ω;Rd), and H1(Ω), respectively.
To show the existence of a generalized solution to (4.1.12)â(4.1.16), we approximate ourproblem by means of a time discretization with an alternate scheme, as done in [6, 36]. Wedivide the time interval [0, T ] by introducing n equispaced nodes, and in each of them wefirst solve the elastodynamics system (4.1.4), keeping v fixed, and then the crack stabilitycondition (4.1.18), keeping u fixed. Finally, we consider some interpolants of the discretesolutions and, thanks to an a priori estimate, we pass to the limit as nââ.
We fix n â N, and we set
Ïn =T
n, u0
n := u0, uâ1n := u0 â Ïnu1, v0
n := v0,
gjn := g(jÏn), wjn := w1(jÏn) for j = 0, . . . , n,
f jn :=1
Ïn
â« jÏn
(jâ1)Ïn
f(s) ds for j = 1, . . . , n.
For j = 1, . . . , n we consider the following two minimum problems:
(i) ujn â wjn â H1D1
(Ω;Rd) is the minimizer of
uâ 7â 1
2Ï2n
â„â„uâ â 2ujâ1n â ujâ2
n
â„â„2
L2(Ω)+ E (uâ, vjâ1
n )â (f jn, uâ)L2(Ω) â ăgjn, uâ â wjnăHâ1
D1(Ω)
among every uâ â wjn â H1D1
(Ω;Rd);
(ii) vjn â w2 â H1D2
(Ω) â©Hk(Ω) with vjn †vjâ1n is the minimizer of
vâ 7â E (ujn, vâ) + H (vâ) +
1
2ÏnG (vâ â vjâ1
n )
among every vâ â w2 â H1D2
(Ω) â©Hk(Ω) with vâ †vjâ1n .
Chapter 4. A phase-field model of dynamic fracture 97
Since C and b satisfy assumptions (4.1.1)â(4.1.3), (4.1.5), and (4.1.6), the discrete problems(i) and (ii) are well defined. In particular, for every j = 1, . . . , n there exists a unique pair(ujn, v
jn) â H1(Ω;Rd)Ă (H1(Ω) â©Hk(Ω)) solution to (i) and (ii).
Let us define
ÎŽujn :=ujn â ujâ1
n
Ïnfor j = 0, . . . , n,
ÎŽ2ujn :=ÎŽujn â ÎŽujâ1
n
Ïn, ÎŽvjn :=
vjn â vjâ1n
Ïnfor j = 1, . . . , n.
For j = 1, . . . , n the minimality of ujn implies
(ÎŽ2ujn, Ï)L2(Ω) + (b(vjâ1n )CEujn, EÏ)L2(Ω) = (f jn, Ï)L2(Ω) + ăgjn, ÏăHâ1
D1(Ω) (4.2.5)
for every Ï â H1D1
(Ω;Rd), which is the discrete counterpart of (4.1.25). Moreover, we can
characterize the function vjn in the following way.
Lemma 4.2.3. For j = 1, . . . , n the function vjn â w2 â H1D2
(Ω) â©Hk(Ω) with vjn †vjâ1n is
the unique solution to the variational inequality
E (ujn, vâ)â E (ujn, v
jn) + âH (vjn)[vâ â vjn] +
kâi=0
αi(âiÎŽvjn,âivâ ââivjn)L2(Ω) â„ 0 (4.2.6)
among all vâ â w2 â H1D2
(Ω) â© Hk(Ω) with vâ †vjâ1n . In particular, we have vjn †1 in Ω
andE (ujn, v
jn)â E (ujn, v
jâ1n )
Ïn+ âH (vjn)[ÎŽvjn] + G (ÎŽvjn) †0. (4.2.7)
Finally, if k = 0, w2 â„ 0 on âD2Ω, v0 â„ 0 in Ω, and b(s) = (s âš 0)2 + η for s â R, thenvjn â„ 0 in Ω for every j = 1, . . . , n.
Proof. Let vjn be the solution to (ii) and let vââw2 â H1D2
(Ω)â©Hk(Ω) be such that vâ †vjâ1n .
For every s â (0, 1] the function vjn + s(vââ vjn) is a competitor for (ii). Hence, by exploitingthe minimality of vjn and dividing by s, we deduce the following inequality
E (ujn, vjn + s(vâ â vjn))â E (ujn, v
jn)
s+ âH (vjn)[vâ â vjn] +
kâi=0
αi(âiÎŽvjn,âivâ ââivjn)L2(Ω)
+ s
[1
4Δâvâ â vjnâ2L2(Ω) + Δââvâ ââvjnâ2L2(Ω) +
1
2ÏnG (vâ â vjn)
]â„ 0. (4.2.8)
Notice that
E (ujn, vjn + s(vâ â vjn))â E (ujn, v
jn)
s†E (ujn, v
â)â E (ujn, vjn) for every s â (0, 1], (4.2.9)
since the difference quotients are non-decreasing in s â (0, 1], being b is convex. By combin-ing (4.2.8) with (4.2.9) and passing to the limit as s â 0+, we derive (4.2.6). On the otherhand, it is easy to see that every solution to (4.2.6) satisfies (ii), thanks to the convexity ofH and G . Finally, for every j = 1, . . . , n we have vjn †v0 †1 in Ω, and the inequality (4.2.7)is obtained by taking vâ = vjâ1
n in (4.2.6) and dividing by Ïn.Let us assume that k = 0, w2 â„ 0 on âD2Ω, v0 â„ 0 in Ω, and b(s) = (sâš0)2 + η for s â R.
The function v1n âš 0 is a competitor for (ii) and satisfies
E (u1n, v
1n âš 0) + H (v1
n âš 0) +1
2ÏnG ((v1
n âš 0)â v0) †E (u1n, v
1n) + H (v1
n) +1
2ÏnG (v1
n â v0),
98 4.2. The time discretization scheme
being E (u1n, v
1n âš 0) = E (u1
n, v1n), H (v1
n ⚠0) †H (v1n), and |(v1
n âš 0) â v0| †|v1n â v0| in Ω,
which is a consequence of v0 â„ 0. Hence, the function v1n âš 0 solves (ii). This fact implies
v1n = (v1
n âš 0) â„ 0 in Ω, since the minimum point is unique (the L2 norm is strictly convex).We now proceed by induction: if vjâ1
n ℠0 in Ω, we can argue as before to get
E (ujn, vjn âš 0) + H (vjn âš 0) +
1
2ÏnG ((vjn âš 0)â vjâ1
n ) †E (ujn, vjn) + H (vjn) +
1
2ÏnG (vjnâ vjâ1
n ),
which gives vjn = (vjn ⚠0) ℠0 in Ω for every j = 1 . . . , n.
As done in [36], we combine equation (4.2.5) with inequality (4.2.7) to derive a discreteenergy inequality for the family (ujn, vjn)nj=1.
Lemma 4.2.4. The family (ujn, vjn)nj=1, solution to problems (i) and (ii), satisfies for everyj = 1, . . . , n the discrete energy inequality
F (ujn, ÎŽujn, v
jn) +
jâl=1
ÏnG (ÎŽvln) +
jâl=1
Ï2nD
ln
†F (u0, u1, v0) +
jâl=1
Ïn
[(f ln, ÎŽu
ln â ÎŽwln)L2(Ω) + (b(vlâ1
n )CEuln, EΎwln)L2(Ω)
]
âjâl=1
Ïn
[(ÎŽulâ1
n , ÎŽ2wln)L2(Ω) â ăÎŽgln, ulâ1n â wiâ1
n ăHâ1D1
(Ω)
]+ (ÎŽujn, ÎŽw
jn)L2(Ω)
+ ăgjn, ujn â wjnăHâ1D1
(Ω) â (u1, w1(0))L2(Ω) â ăg(0), u0 â w1(0)ăHâ1D1
(Ω),
(4.2.10)
where ÎŽw0n := w1(0), ÎŽwjn := 1
Ïn(wjn â wjâ1
n ), ÎŽ2wjn := 1Ïn
(ÎŽwjn â ÎŽwjâ1n ), ÎŽgjn := 1
Ïn(gjn â gnâ1
n )
for j = 1, . . . , n, and the dissipation Djn are defined for j = 1, . . . , n as
Djn :=
1
2âÎŽ2ujnâ2L2(Ω) +
1
2(b(vjâ1
n )CEΎujn, EΎujn)L2(Ω) +1
4ΔâÎŽvjnâ2L2(Ω) + ΔââÎŽvjnâ2L2(Ω).
Proof. By using Ï := Ïn(ÎŽujn â ÎŽwjn) â H1D1
(Ω;Rd) as test function in (4.2.5), for everyj = 1, . . . , n we deduce the following identity
Ïn(ÎŽ2ujn, ÎŽujn)L2(Ω) + Ïn(b(vjâ1
n )CEujn, EΎujn)L2(Ω)
= Ïn(f jn, ÎŽujn â ÎŽwjn)L2(Ω) + Ïnăgjn, ÎŽujn â ÎŽwjnăHâ1
D1(Ω)
+ Ïn(ÎŽ2ujn, ÎŽwjn)L2(Ω) + Ïn(b(vjâ1
n )CEujn, EΎwjn)L2(Ω).
(4.2.11)
Thanks to the identity |a|2 â a · b = 12 |a|
2 â 12 |b|
2 + 12 |aâ b|
2 for a, b â Rd, we can write thefirst term as
Ïn(ÎŽ2ujn, ÎŽujn)L2(Ω) = âÎŽujnâ2L2(Ω) â (ÎŽujâ1
n , Ύujn)L2(Ω)
= K (ÎŽujn)âK (ÎŽujâ1n ) +
Ï2n
2âÎŽ2ujnâ2L2(Ω).
(4.2.12)
Similarly, we have
Ïn(b(vjâ1n )CEujn, EÎŽujn)L2(Ω) = E (ujn, v
jn)â E (ujâ1
n , vjâ1n ) +
Ï2n
2(b(vjâ1
n )CEΎujn, EΎujn)L2(Ω)
+1
2([b(vjâ1
n )â b(vjn)]CEujn, Eujn)L2(Ω). (4.2.13)
Chapter 4. A phase-field model of dynamic fracture 99
We use (4.2.7) to estimate from below the last term in the previous inequality as
1
2([b(vjâ1
n )â b(vjn)]CEujn, Eujn)L2(Ω)
â„ Ïn2Δ
(vjn â 1, ÎŽvjn)L2(Ω) + 2ΔÏn(âvjn,âÎŽvjn)L2(Ω) + ÏnG (ÎŽvjn)
= H (vjn)âH (vjâ1n ) + ÏnG (ÎŽvjn) +
Ï2n
4ΔâÎŽvjnâ2L2(Ω) + ΔÏ2
nââÎŽvjnâ2L2(Ω).
(4.2.14)
By combining (4.2.11)â(4.2.14), for every j = 1, . . . , n we obtain
F (ujn, ÎŽujn, v
jn)âF (ujâ1
n , ÎŽujâ1n , vjâ1
n ) + ÏnG (ÎŽvjn) + Ï2nD
jn
†Ïn(f jn, ÎŽujn â ÎŽwjn)L2(Ω) + Ïnăgjn, ÎŽujn â ÎŽwjnăHâ1
D1(Ω)
+ Ïn(ÎŽ2ujn, ÎŽwjn)L2(Ω) + Ïn(b(vjâ1
n )CEujn, EΎwjn)L2(Ω).
Finally, we sum over l = 1, . . . , j for every j â 1, . . . , n, and we use the identities
jâl=1
Ïnăgln, ÎŽuln â ÎŽwlnăHâ1D1
(Ω) = ăgjn, ujn â wjnăHâ1D1
(Ω) â ăg(0), u0 â w1(0)ăHâ1D1
(Ω)
âjâl=1
ÏnăÎŽgln, ulâ1n â wlâ1
n ăHâ1D1
(Ω),
(4.2.15)
jâl=1
Ïn(ÎŽ2uln, ÎŽwln)L2(Ω) = (ÎŽujn, ÎŽw
jn)L2(Ω) â (u1, w1(0))L2(Ω)
âjâl=1
Ïn(ÎŽulâ1n , ÎŽ2wln)L2(Ω),
(4.2.16)
to deduce the discrete energy inequality (4.2.10).
The first consequence of (4.2.10) is the following a priori estimate.
Lemma 4.2.5. There exists a constant C > 0, independent of n, such that
maxj=1,...,n
âÎŽujnâL2(Ω) + âujnâH1(Ω) + âvjnâH1(Ω)+nâj=1
ÏnâÎŽvjnâ2Hk(Ω) +nâj=1
Ï2nD
jn †C. (4.2.17)
Proof. Thanks to (4.1.4) and (4.1.6) we can estimate from below the left-hand side of (4.2.10)in the following way
F (ujn, ÎŽujn, v
jn) +
jâl=1
ÏnG (ÎŽvln) +
jâl=1
Ï2nD
ln
â„ 1
2âÎŽujnâ2L2(Ω) +
ηc0
2âujnâ2H1(Ω) â
ηc1
2âujnâ2L2(Ω)
(4.2.18)
for every j = 1, . . . , n. Let us now bound from above the right-hand side of (4.2.18). Wedefine
Ln := maxj=1,...,n
âÎŽujnâL2(Ω), Mn := maxj=1,...,n
âujnâH1(Ω),
and we use (4.1.7)â(4.1.11) to derive for every j = 1, . . . , n the following estimates:
jâl=1
Ïn(f ln, ÎŽuln â ÎŽwln)L2(Ω) †C1Ln + C2, (4.2.19)
100 4.2. The time discretization scheme
(ÎŽujn, ÎŽwjn)L2(Ω) â (u1, w1(0))L2(Ω) â
jâl=1
Ïn(ÎŽulâ1n , ÎŽ2wln)L2(Ω) †C1Ln + C2, (4.2.20)
ăgjn, ujn â wjnăHâ1D1
(Ω) â ăg(0), u0 â w1(0)ăHâ1D1
(Ω) âjâl=1
ÏnăÎŽgln, ulâ1n â wlâ1
n ăHâ1D1
(Ω)
†C1Mn + C2,
(4.2.21)
for two positive constants C1 and C2 independent of n. Moreover, since C belongs toLâ(Ω; L (RdĂd;RdĂd)), b is non-decreasing, and vjâ1
n †1, we get
jâl=1
Ïn(b(vlâ1n )CEuln, EÎŽwln)L2(Ω) †b(1)âCâLâ(Ω)
âTâEw1âL2(0,T ;L2(Ω))Mn (4.2.22)
for every j = 1, . . . , n. By combining (4.2.10) with (4.2.18)â(4.2.22) and the following estimate
âujnâL2(Ω) â€nâl=1
ÏnâÎŽulnâL2(Ω) + âu0âL2(Ω) †TLn + âu0âL2(Ω) for every j = 1, . . . , n,
we obtain the existence of two positive constants C1 and C2, independent of n, such that
(Ln +Mn)2 †C1(Ln +Mn) + C2 for every n â N.
This implies that Ln and Mn are uniformly bounded in n. In particular, there exists aconstant C > 0, independent of n, such that
K (ÎŽujn) + E (ujn, vjn) + H (vjn) +
jâl=1
ÏnG (ÎŽvln) +
jâl=1
Ï2nD
ln †C for every j = 1, . . . , n.
Finally, for j = 1, . . . , n we have
min
Δ,
1
4Δ
âvjn â 1â2H1(Ω) â€H (vjn) †C, ÎČ0
nâj=1
ÏnâÎŽvjnâ2Hk(Ω) â€nâj=1
ÏnG (ÎŽvjn) †C,
which gives the remaining estimates.
Remark 4.2.6. By combining together (4.2.5) and (4.2.17) we also obtain
nâj=1
ÏnâÎŽ2ujnâ2Hâ1D1
(Ω)+ maxj=1,...,n
âvjnâHk(Ω) †C
for a positive constant C independent of n. Indeed, by (4.2.5), for every j = 1, . . . , n we have
âÎŽ2ujnâHâ1D1
(Ω) = supÏâH1
D1(Ω;Rd), âÏâH1(Ω)â€1
âŁâŁ(ÎŽ2ujn, Ï)L2(Ω)
âŁâŁâ€ b(1)âCâLâ(Ω)âEujnâ2 + âf jnâL2(Ω) + âgjnâHâ1
D1(Ω).
Hence, thanks to (4.1.9) and (4.2.17), there exists a constant C > 0, independent of n, suchthat
nâj=1
ÏnâÎŽ2ujnâ2Hâ1D1
(Ω)†C(1 + âfâL2(0,T ;L2(Ω)) + âgâH1(0,T ;Hâ1
D1(Ω))).
Finally, also âvjnâHk(Ω) is uniformly bounded with respect to j and n, since
âvjnâHk(Ω) â€âT
(nâl=1
ÏnâÎŽvlnâ2Hk(Ω)
)1/2
+ âv0âHk(Ω) for every j = 1, . . . , n.
Chapter 4. A phase-field model of dynamic fracture 101
We now use the family (ujn, vjn)nj=1 to construct a generalized solution to (4.1.12)â
(4.1.18). As in Chapter 3, we denote by un : [0, T ] â H1(Ω;Rd) and uâČn : [0, T ] â L2(Ω;Rd)the piecewise affine interpolants of ujnnj=1 and ÎŽujnnj=1, respectively, which are defined inthe following way:
un(t) := ujn + (tâ jÏn)ÎŽujn for t â [(j â 1)Ïn, jÏn], j = 1, . . . , n,
uâČn(t) := ÎŽujn + (tâ jÏn)ÎŽ2ujn for t â [(j â 1)Ïn, jÏn], j = 1, . . . , n.
We also define the backward interpolants un : [0, T ]â H1(Ω;Rd) and uâČn : [0, T ]â L2(Ω;Rd),and the forward interpolants un : [0, T ]â H1(Ω;Rd) and uâČn : [0, T ]â L2(Ω;Rd) as:
un(t) := ujn for t â ((j â 1)Ïn, jÏn], j = 1, . . . , n, un(0) = u0n,
uâČn(t) := ÎŽujn for t â ((j â 1)Ïn, jÏn], j = 1, . . . , n, uâČn(0) = ÎŽu0n,
un(t) := ujâ1n for t â [(j â 1)Ïn, jÏn), j = 1, . . . , n, un(T ) = unn,
uâČn(t) := ÎŽujâ1n for t â [(j â 1)Ïn, jÏn), j = 1, . . . , n, uâČn(T ) = ÎŽunn.
In a similar way, we can define the piecewise affine interpolant vn : [0, T ] â H1(Ω) ofvjnnj=1, as well as the backward interpolant vn : [0, T ]â H1(Ω), and the forward interpolant
vn : [0, T ] â H1(Ω). Notice that un â H1(0, T ;L2(Ω;Rd)), uâČn â H1(0, T ;Hâ1D1
(Ω;Rd)),and vn â H1(0, T ;Hk(Ω)), with un(t) = uâČn(t) = ÎŽujn, uâČn(t) = ÎŽ2ujn, and vn(t) = ÎŽvjn fort â ((j â 1)Ïn, jÏn) and j = 1, . . . , n.
Lemma 4.2.7. There exist a subsequence of n, not relabeled, and two functions
u â Lâ(0, T ;H1(Ω;Rd)) â©W 1,â(0, T ;L2(Ω;Rd)) â©H2(0, T ;Hâ1D1
(Ω;Rd)),
v â Lâ(0, T ;H1(Ω)) â©H1(0, T ;Hk(Ω)),
such that the following convergences hold as nââ:
un u in H1(0, T ;L2(Ω;Rd)), uâČn u in H1(0, T ;Hâ1D1
(Ω;Rd)),
un â u in C0([0, T ];L2(Ω;Rd)), uâČn â u in C0([0, T ];Hâ1D1
(Ω;Rd)),
un u in L2(0, T ;H1(Ω;Rd)), uâČn u in L2(0, T ;L2(Ω;Rd)),un u in L2(0, T ;H1(Ω;Rd)), uâČn u in L2(0, T ;L2(Ω;Rd)),vn v in H1(0, T ;Hk(Ω)), vn â v in C0([0, T ];L2(Ω)),
vn v in L2(0, T ;H1(Ω)), vn v in L2(0, T ;H1(Ω)).
Proof. Thanks to (4.2.17) the sequence unn â Lâ(0, T ;H1(Ω;Rd))â©W 1,â(0, T ;L2(Ω;Rd))is uniformly bounded. Hence, by Aubin-Lionsâs lemma (see [50, Corollary 4]) there exist asubsequence of n, not relabeled, and a function
u â Lâ(0, T ;H1(Ω;Rd)) â©W 1,â(0, T ;L2(Ω;Rd)),
such that
un u in H1(0, T ;L2(Ω;Rd)), un â u in C0([0, T ];L2(Ω;Rd)) as nââ.
Moreover, the sequence unn â Lâ(0, T ;H1(Ω;Rd)) is uniformly bounded, and satisfies
âun(t)â un(t)âL2(Ω) †ÏnâunâLâ(0,T ;L2(Ω)) †CÏn for every t â [0, T ] and n â N, (4.2.23)
102 4.2. The time discretization scheme
where C is a positive constant independent of n and t. Therefore, there exists a furthersubsequence, not relabeled, such that
un u in L2(0, T ;H1(Ω;Rd)), un â u in Lâ(0, T ;L2(Ω;Rd)) as nââ.
Similarly, we have
un u in L2(0, T ;H1(Ω;Rd)), un â u in Lâ(0, T ;L2(Ω;Rd)) as nââ.
Let us now consider the sequence uâČnn â Lâ(0, T ;L2(Ω;Rd)) â© H1(0, T ;Hâ1D1
(Ω;Rd)).Since it is uniformly bounded with respect to n, we can apply again the Aubin-Lionsâs lemmaand we deduce the existence of
z â Lâ(0, T ;L2(Ω;Rd)) â©H1(0, T ;Hâ1D1
(Ω;Rd))
such that, up to a further (not relabeled) subsequence
uâČn z in H1(0, T ;Hâ1D1
(Ω;Rd)), uâČn â z in C0([0, T ];Hâ1D1
(Ω;Rd)) as nââ.
Furthermore, we have
âuâČn(t)â un(t)âHâ1D1
(Ω) = âuâČn(t)â uâČn(t)âHâ1D1
(Ω) â€âÏnâuâČnâL2(0,T ;Hâ1
D1(Ω)) †C
âÏn (4.2.24)
for every t â [0, T ] and n â N, with C > 0 independent of n and t. This fact implies thatz = u, and
uâČn u in L2(0, T ;L2(Ω;Rd)), uâČn â u in L2(0, T ;Hâ1D1
(Ω;Rd)) as nââ.
In a similar way, we get
uâČn u in L2(0, T ;L2(Ω;Rd)), uâČn â u in L2(0, T ;Hâ1D1
(Ω;Rd)) as nââ.
Finally, the thesis for the sequences vnn, vnn, and vnn is obtained as before, byusing (4.2.17) and the compactness of the embedding H1(Ω) â L2(Ω).
Remark 4.2.8. As already observed in Remark 4.2.2, we have u â C0w([0, T ];H1(Ω;Rd)),
u â C0w([0, T ];L2(Ω;Rd)), and v â C0
w([0, T ];H1(Ω)). By using the estimate (4.2.17), we get
âun(t)âH1(Ω) + âuâČn(t)âL2(Ω) †C for every t â [0, T ] and n â N
for a constant C > 0 independent of n and t. Hence, for every t â [0, T ] we derive
un(t) u(t) in H1(Ω;Rd), uâČn(t) u(t) in L2(Ω;Rd) as nââ,
thanks to the previous convergences. In particular, for every t â [0, T ] we can use (4.2.23)and (4.2.24) to obtain
un(t) u(t) in H1(Ω;Rd), uâČn(t) u(t) in L2(Ω;Rd) as nââ,un(t) u(t) in H1(Ω;Rd), uâČn(t) u(t) in L2(Ω;Rd) as nââ.
With a similar argument, for every t â [0, T ] we have
vn(t) v(t) in H1(Ω), vn(t) v(t) in H1(Ω), vn(t) v(t) in H1(Ω) as nââ.
We are now in a position to pass to the limit in the discrete problem (4.2.5).
Chapter 4. A phase-field model of dynamic fracture 103
Lemma 4.2.9. The pair (u, v) given by Lemma 4.2.7 is a generalized solution to prob-lem (4.1.12)â(4.1.15). Moreover, (u, v) satisfies the initial conditions (4.1.16) and the ir-reversibility condition (4.1.17). Finally, if k = 0, w2 â„ 0 on âD2Ω, v0 â„ 0 in Ω, andb(s) = (s âš 0)2 + η for s â R, then v(t) â„ 0 in Ω for every t â [0, T ].
Proof. The pair (u, v) given by Lemma 4.2.7 satisfies (4.2.1), (4.2.3), and the initial condi-tions (4.1.16), since u0 = un(0) u(0) in H1(Ω;Rd), u1 = uâČn(0) u(0) in L2(Ω;Rd), andv0 = vn(0) v(0) in H1(Ω) as n â â. If we consider the piecewise affine interpolant wnof wjnnj=1, for every t â [0, T ] we have un(t) â wn(t) â H1
D1(Ω;Rd) for every n â N and
wn(t)â w1(t) in H1(Ω;Rd) as nââ. Therefore, the function u satisfies (4.2.2). Similarly,vn(t)â w2 â H1
D2(Ω) and vn(t) †vn(s) †1 in Ω for every 0 †s †t †T and n â N, which
give (4.2.4) and (4.1.17). Finally, if k = 0, w2 â„ 0 on âD2Ω, v0 â„ 0 in Ω, and b(s) = (sâš0)2+ηfor s â R, then for every t â [0, T ] we deduce vn(t) â„ 0 in Ω, by Lemma 4.2.3, which impliesv(t) â„ 0 in Ω.
It remains to prove the validity of (4.1.25) for a.e. t â (0, T ). For every j = 1, . . . , nwe know that (ujn, v
jn) satisfies (4.2.5). In particular, by integrating it in [t1, t2] â [0, T ] and
using the previous notation, we deriveâ« t2
t1
ăuâČn(t), ÏăHâ1D1
(Ω) dt+
â« t2
t1
(b(vn(t))CEun(t), EÏ)L2(Ω) dt
=
â« t2
t1
(fn(t), Ï)L2(Ω) dt+
â« t2
t1
ăgn(t), ÏăHâ1D1
(Ω) dt
(4.2.25)
for every Ï â H1D1
(Ω;Rd), where fn and gn are the backward interpolants of f jnnj=1 and
gjnnj=1, respectively. We now pass to the limit as nââ in (4.2.25). For the first term wehave
limnââ
â« t2
t1
ăuâČn(t), ÏăHâ1D1
(Ω) dt =
â« t2
t1
ău(t), ÏăHâ1D1
(Ω) dt,
since uâČn u in L2(0, T ;Hâ1D1
(Ω;Rd)) as n â â. Moreover, as n â â it is easy to check
that fn converges strongly to f in L2(0, T ;L2(Ω;Rd)), while gn converges strongly to g inL2(0, T ;Hâ1
D1(Ω;Rd)), which implies
limnââ
[â« t2
t1
(fn(t), Ï)L2(Ω) dt+
â« t2
t1
ăgn(t), ÏăHâ1D1
(Ω) dt
]=
â« t2
t1
(f(t), Ï)L2(Ω) dt+
â« t2
t1
ăg(t), ÏăHâ1D1
(Ω) dt.
It remains to analyze the second term of (4.2.25). By the previous remark and using thecompactness of the embedding H1(Ω) â L2(Ω), we get that vn(t)â v(t) in L2(Ω) as nââfor every t â [0, T ]. Thanks to the estimate
|b(vn(t, x))C(x)EÏ(x)| †b(1)âCâLâ(Ω) |EÏ(x)| for every t â [0, T ] and a.e. x â Ω
and the dominated convergence theorem, we conclude that b(vn)CEÏ converges strongly tob(v)CEÏ in L2(0, T ;L2(Ω;RdĂd)). Hence, we obtain
limnââ
â« t2
t1
(b(vn(t))CEun(t), EÏ)L2(Ω) dt =
â« t2
t1
(b(v(t))CEu(t), EÏ)L2(Ω) dt,
since Eun Eu in L2(0, T ;L2(Ω;RdĂd)). Therefore, the pair (u, v) solvesâ« t2
t1
ău(t), ÏăHâ1D1
(Ω) dt+
â« t2
t1
(b(v(t))CEu(t), EÏ)L2(Ω) dt
=
â« t2
t1
(f(t), Ï)L2(Ω) dt+
â« t2
t1
ăg(t), ÏăHâ1D1
(Ω) dt
104 4.2. The time discretization scheme
for every Ï â H1D1
(Ω;Rd) and [t1, t2] â [0, T ]. We fix a countable dense set D â H1D1
(Ω;Rd).By Lebesgueâs differentiation theorem, we derive that the pair (u, v) solves (4.1.25) for a.e.t â (0, T ) and for every Ï â D . Finally, we use the density of D in H1
D1(Ω;Rd) to conclude
that the equation (4.1.25) is satisfied for every Ï â H1D1
(Ω;Rd).
In the next lemma we exploit the inequality (4.2.6) to prove (4.1.18).
Lemma 4.2.10. The pair (u, v) given by Lemma 4.2.7 satisfies for a.e. t â (0, T ) the crackstability condition (4.1.18).
Proof. For every j = 1, . . . , n the pair (ujn, vjn) satisfies the inequality (4.2.6), that can be
rephrased inE (un(t), vâ)â E (un(t), vn(t)) + âH (vn(t))[vâ â vn(t)]
+kâi=0
αi(âivn(t),âivâ ââivn(t))L2(Ω) â„ 0(4.2.26)
for a.e. t â (0, T ) and for every vâ â w2 â H1D2
(Ω) â© Hk(Ω) with vâ †vn(t). Given
Ï â H1D2
(Ω) â©Hk(Ω) with Ï â€ 0, the function Ï+ vn(t) is admissible for (4.2.26). After anintegration in [t1, t2] â [0, T ], we deduce the following inequalityâ« t2
t1
[E (un(t), Ï+ vn(t))â E (un(t), vn(t))] dt
+
â« t2
t1
âH (vn(t))[Ï] dt+
kâi=0
αi
â« t2
t1
(âivn(t),âiÏ)L2(Ω) dt â„ 0.
(4.2.27)
Let us send nââ. We have
limnââ
kâi=0
αi
â« t2
t1
(âivn(t),âiÏ)L2(Ω) dt =kâi=0
αi
â« t2
t1
(âiv(t),âiÏ)L2(Ω) dt, (4.2.28)
since vn v in L2(0, T ;Hk(Ω)). Moreover vn v in L2(0, T ;H1(Ω)), which implies
limnââ
â« t2
t1
âH (vn(t))[Ï] dt =
â« t2
t1
âH (v(t))[Ï] dt. (4.2.29)
The function Ï(x, y, Ο) := 12 [b(y) â b(Ï(x) + y)]C(x)Οsym · Οsym, (x, y, Ο) â Ω Ă R Ă RdĂd,
satisfies the assumptions of Ioffe-Olechâs theorem (see, e.g., [14, Theorem 3.4]). Thus, forevery t â [0, T ] we derive
E (u(t), v(t))â E (u(t), Ï+ v(t)) =
â«Î©Ï(x, v(t, x), Eu(t, x)) dx
†lim infnââ
â«Î©Ï(x, vn(t, x), Eun(t, x)) dx
= lim infnââ
[E (un(t), Ï+ vn(t))â E (un(t), vn(t))],
since vn(t) â v(t) in L2(Ω) and Eun(t) Eu(t) in L2(Ω;RdĂd) for every t â [0, T ]. ByFatouâs lemma, we concludeâ« t2
t1
[E (u(t), v(t))â E (u(t), Ï+ v(t))] dt
â€â« t2
t1
lim infnââ
[E (un(t), vn(t))â E (un(t), Ï+ vn(t))] dt
†lim infnââ
â« t2
t1
[E (un(t), vn(t))â E (un(t), Ï+ vn(t))] dt,
Chapter 4. A phase-field model of dynamic fracture 105
which gives â« t2
t1
[E (u(t), Ï+ v(t))â E (u(t), v(t))] dt
â„ lim supnââ
â« t2
t1
[E (un(t), Ï+ vn(t))â E (un(t), vn(t))] dt.
(4.2.30)
By combining (4.2.27)â(4.2.30) we obtain the following inequalityâ« t2
t1
[E (u(t), Ï+ v(t))â E (u(t), v(t)) + âH (v(t))[Ï] +
kâi=0
αi(âiv(t),âiÏ)L2(Ω)
]dt â„ 0.
We choose now a countable dense set D â Ï â H1D2
(Ω) â© Hk(Ω) : Ï â€ 0. Thanks toLebesgueâs differentiation theorem for a.e. t â (0, T ) we derive
E (u(t), Ï+ v(t))â E (u(t), v(t)) + âH (v(t))[Ï] +kâi=0
αi(âiv(t),âiÏ)L2(Ω) â„ 0 (4.2.31)
for every Ï â D . Finally, we use a density argument and the dominated convergence theoremto deduce that (4.2.31) is satisfied for every Ï â H1
D2(Ω) â©Hk(Ω) with Ï â€ 0. In particular,
for a.e. t â (0, T ) we get
E (u(t), vâ)â E (u(t), v(t)) + âH (v(t))[vâ â v(t)] +kâi=0
αi(âiv(t),âivâ ââiv(t))L2(Ω) â„ 0,
for every vâ â w2 â H1D2
(Ω) â©Hk(Ω) with vâ †v(t), by taking Ï := vâ â v(t). This impliesthe crack stability condition (4.1.18), since the map vâ 7âH (vâ) is convex.
We conclude this section by showing that the pair (u, v) given by Lemma 4.2.7 sat-isfies an energy-dissipation inequality. Notice that also for a generalized solution (u, v)the total work Wtot(u, v; t1, t2) is well defined for every t1, t2 â [0, T ]. Indeed, we haveu â C0
w([0, T ];H1(Ω;Rd)) and u â C0w([0, T ];H1(Ω;Rd)), which gives that u(t) â w1(t) and
u(t) are uniquely defined for every t â [0, T ] as elements of H1D1
(Ω;Rd) and L2(Ω;Rd), respec-tively. Moreover, by combining the weak continuity of u and u, with the strong continuity ofg, w1, and w1, it is easy to see that the function (t1, t2)â Wtot(t1, t2, u, v) is continuous.
Lemma 4.2.11. The pair (u, v) given by Lemma 4.2.7 satisfies for every t â [0, T ] theenergy-dissipation inequality
F (u(t), u(t), v(t)) +
â« t
0G (v(s)) ds †F (u0, u1, v0) + Wtot(u, v; 0, t). (4.2.32)
Proof. Let gn, wn, and wâČn be the piecewise affine interpolants of gjnnj=1, wjnnj=1, and
ÎŽwjnnj=1, respectively, and let wn, wâČn and wn, w
âČn be the backward and the forward inter-
polants of wjnnj=1 and ÎŽwjnnj=1, respectively.For t = 0 the inequality (4.2.32) trivially holds thanks to our initial conditions (4.1.16).
We fix t â (0, T ] and for every n â N we consider the unique j â 1, . . . , n such thatt â ((j â 1)Ïn, jÏn]. As done before, we use the previous interpolants and (4.2.10) to write
F (un(t), uâČn(t), vn(t)) +
â« tn
0G (vn) ds
†F (u0, u1, v0) +
â« tn
0[(fn, u
âČn â wâČn)L2(Ω) + (b(vn)CEun, EwâČn)L2(Ω)] ds
ââ« tn
0[ăgn, un â wnăHâ1
D1(Ω) + (uâČn, w
âČn)L2(Ω)] ds+ ăgn(t), un(t)â wn(t)ăHâ1
D1(Ω)
+ (uâČn(t), wâČn(t))L2(Ω) â ăg(0), u0 â w1(0)ăHâ1D1
(Ω) â (u1, w1(0))L2(Ω),
(4.2.33)
106 4.2. The time discretization scheme
where we have set tn := jÏn, and we have neglected the terms Djn, which are non negative.
It easy to see that the following convergences hold as nââ:
fn â f in L2(0, T ;L2(Ω;Rd)), gn â g in L2(0, T ;Hâ1D1
(Ω;Rd)),
wn â w1 in L2(0, T ;H1(Ω;Rd)), wâČn â w1 in L2(0, T ;H1(Ω;Rd)),wâČn â w1 in H1(0, T ;L2(Ω;Rd)).
By using also the ones of Lemma 4.2.7 and observing that tn â t as nââ, we deduce
limnââ
â« tn
0(fn(s), uâČn(s)â wâČn(s))L2(Ω) ds =
â« t
0(f(s), u(s)â w1(s))L2(Ω) ds, (4.2.34)
limnââ
â« tn
0ăgn(s), un(s)â wn(s)ăHâ1
D1(Ω) ds =
â« t
0ăg(s), u(s)â w1(s)ăHâ1
D1(Ω) ds, (4.2.35)
limnââ
â« tn
0(uâČn(s), wâČn(s))L2(Ω) ds =
â« t
0(u(s), w1(s))L2(Ω) ds. (4.2.36)
Moreover, the strong continuity of g, w1, and w1 in Hâ1D1
(Ω;Rd), H1(Ω;Rd), and L2(Ω;Rd),respectively, and the convergences of Remark 4.2.8, imply
limnââ
ăgn(t), un(t)â wn(t)ăHâ1D1
(Ω) = ăg(t), u(t)â w1(t)ăHâ1D1
(Ω), (4.2.37)
limnââ
(uâČn(t), wâČn(t))L2(Ω) = (u(t), w1(t))L2(Ω). (4.2.38)
It is easy to check that b(vn)CEwâČn â b(v)CEw1 in L2(0, T ;L2(Ω;RdĂd)), thanks to thedominated convergence theorem. By combining it with Eun Eu in L2(0, T ;L2(Ω;RdĂd)),we conclude
limnââ
â« tn
0(b(vn(s))CEun(s), EwâČn(s))L2(Ω) ds =
â« t
0(b(v(s))CEu(s), Ew1(s))L2(Ω) ds. (4.2.39)
If we now consider the left-hand side of (4.2.33), we get
K (u(t)) †lim infnââ
K (uâČn(t)), H (v(t)) †lim infnââ
H (vn(t)), (4.2.40)
since uâČn(t) u(t) in L2(Ω,Rd) and vn(t) v(t) in H1(Ω). Furthermore, we have vn v inL2(0, T ;Hk(Ω)) and t †tn, which givesâ« t
0G (v(s)) ds †lim inf
nââ
â« t
0G (vn(s)) ds †lim inf
nââ
â« tn
0G (vn(s)) ds. (4.2.41)
Finally, let us consider the function Ï(x, y, Ο) := 12b(y)C(x)Οsym ·Οsym, (x, y, Ο) â ΩĂRĂRdĂd.
As in the previous lemma, the function Ï satisfies the assumption of Ioffe-Olechâs theorem,while vn(t)â v(t) in L2(Ω) and Eun(t) Eu(t) in L2(Ω;RdĂd). Thus, we obtain
E (u(t), v(t)) =
â«Î©Ï(x, v(t, x), Eu(t, x)) dx
†lim infnââ
â«Î©Ï(x, vn(t, x), Eun(t, x)) dx = lim inf
nââE (un(t), vn(t)).
(4.2.42)
By combining (4.2.33) with (4.2.34)â(4.2.42) we deduce the inequality (4.2.32) for everyt â (0, T ].
Chapter 4. A phase-field model of dynamic fracture 107
4.3 Proof of the main result
In this section we show that for k > d/2 the generalized solution (u, v) given by Lemma 4.2.7is a weak solution and satisfies the identity (4.1.27). To this aim we need several lemmas: westart by proving that, given a function v â H1(0, T ;C0(Ω)) satisfying (4.1.17), there existsa unique solution u to equation (4.1.25). As a consequence, we deduce that the mechanicalenergy associated to u satisfies formula (4.3.20) for every t â [0, T ], which guarantees that thefunction u is more regular in time, namely u â C0([0, T ];H1(Ω;Rd)) â© C1([0, T ];L2(Ω;Rd)).Finally, we use the crack stability condition (4.1.18) and the energy-dissipation inequal-ity (4.2.32) to obtain (4.1.19) from (4.3.20).
Lemma 4.3.1. Let w1, f , g, u0, and u1 be as in (4.1.7), (4.1.9), and (4.1.10). Let usassume that Ï â H1(0, T ;C0(Ω)) satisfies (4.1.17). Then there exists a unique function zwhich satisfies (4.2.1), (4.2.2), the initial conditions z(0) = u0 and z(0) = u1, and whichsolves for a.e. t â (0, T ) the following equation:
ăz(t), ÏăHâ1D1
(Ω) + (b(Ï(t))CEz(t), EÏ)L2(Ω) = (f(t), Ï)L2(Ω) + ăg(t), ÏăHâ1D1
(Ω) (4.3.1)
for every Ï â H1D1
(Ω;Rd).
Proof. To prove the existence of a solution z to (4.3.1), we proceed as before. We fix n â Nand we define
Ïn :=T
n, z0
n := u0, zâ1n := u0 â Ïnu1, Ïjn := Ï(jÏn) for j = 0, . . . , n.
For j = 1, . . . , n we consider the unique solution zjn â wjn â H1D1
(Ω;Rd) to
(ÎŽ2zjn, Ï)L2(Ω) + (b(Ïjâ1n )CEzjn, EÏ)L2(Ω) = (f jn, Ï)L2(Ω) + ăgjn, ÏăHâ1
D1(Ω) (4.3.2)
for every Ï â H1D1
(Ω;Rd), where we have set ÎŽzjn := 1Ïn
(zjn â zjâ1n ) for j = 0, . . . , n, and
ÎŽ2zjn := 1Ïn
(ÎŽzjnâÎŽzjâ1n ) for j = 1, . . . , n. By using Ï := Ïn(ÎŽzjnâÎŽwjn) as test function in (4.3.2)
and proceeding as in Lemma 4.2.4, we get that the function zjn satisfies for j = 1, . . . , n
[K (ÎŽzjn) + E (zjn, Ïjn)]â [K (ÎŽzjâ1
n ) + E (zjâ1n , Ïjâ1
n )]â 1
2([b(Ïjn)â b(Ïjâ1
n )]CEzjn, Ezjn)L2(Ω)
†Ïn(f jn, ÎŽzjn â ÎŽwjn)L2(Ω) + Ïnăgjn, ÎŽzjn â ÎŽwjnăHâ1
D1(Ω)
+ Ïn(ÎŽ2zjn, ÎŽwjn)L2(Ω) + Ïn(b(vjâ1
n )CEzjn, EΎwjn)L2(Ω).
In particular, we can sum over l = 1, . . . , j for every j â 1, . . . , n and use the identi-ties (4.2.15) and (4.2.16) to derive the discrete energy inequality
K (ÎŽzjn) + E (zjn, Ïjn)â 1
2
jâl=1
([b(Ïln)â b(Ïlâ1n )]CEzln, Ezln)L2(Ω)
†K (u1) + E (u0, Ï(0)) +
jâl=1
Ïn[(f ln, ÎŽzln â ÎŽwln)L2(Ω) + (b(Ïlâ1
n )CEzln, EΎwln)L2(Ω)]
âjâl=1
Ïn[ăÎŽgln, zlâ1n â wlâ1
n ăHâ1D1
(Ω) + (ÎŽzlâ1n , ÎŽ2wln)L2(Ω)] + ăgjn, zjn â wjnăHâ1
D1(Ω)
+ (ÎŽzjn, ÎŽwjn)L2(Ω) â ăg(0), u0 â w1(0)ăHâ1
D1(Ω) â (u1, w1(0))L2(Ω).
(4.3.3)
108 4.3. Proof of the main result
Since Ïjn †Ïjâ1n and b is non-decreasing, the last term in the left-hand side is non negative.
Hence, by arguing as in Lemma 4.2.5 and in Remark 4.2.6, we can find a constant C > 0,independent of n, such that
maxj=1,...,n
[âÎŽzjnâL2(Ω) + âzjnâH1(Ω)
]+
nâj=1
ÏnâÎŽ2zjnâ2Hâ1D1
(Ω)†C.
Let zn, zâČn, zn, zâČn, and zn, zâČn be the piecewise affine, the backward, and the forward inter-polants of zjnnj=1 and ÎŽzjnnj=1, respectively. As in Lemma 4.2.7, the above estimate impliesthe existence of a subsequence of n, not relabeled, and function z satisfying (4.2.1), (4.2.2)and the initial conditions z(0) = u0 and z(0) = u1, such that the following convergences holdas nââ:
zn z in H1(0, T ;L2(Ω;Rd)), zâČn z in H1(0, T ;Hâ1D1
(Ω;Rd)),
zn â z in C0([0, T ];L2(Ω;Rd)), zâČn â z in C0([0, T ];Hâ1D1
(Ω;Rd)),
zn z in L2(0, T ;H1(Ω;Rd)), zâČn, z in L2(0, T ;L2(Ω;Rd)),zn z in L2(0, T ;H1(Ω;Rd)), zâČn z in L2(0, T ;L2(Ω;Rd)).
Let us now define the backward interpolant Ïn and the forward interpolant Ïn of Ïjnnj=1.By integrating the equation (4.3.2) in the time interval [t1, t2] â [0, T ], we obtainâ« t2
t1
ăzâČn(t), ÏăHâ1D1
(Ω) dt+
â« t2
t1
(b(Ïn(t))CEzn(t), EÏ)L2(Ω) dt
=
â« t2
t1
(fn(t), Ï)L2(Ω) dt+
â« t2
t1
ăgn(t), ÏăHâ1D1
(Ω) dt
for every Ï â H1D1
(Ω;Rd). Thanks to the fact that Ï â H1(0, T ;C0(Ω)) and the previousconvergences, we can pass to the limit as nââ as done in Lemma 4.2.9, and we deduceâ« t2
t1
ăz(t), ÏăHâ1D1
(Ω) dt+
â« t2
t1
(b(Ï(t))CEz(t), EÏ)L2(Ω) dt
=
â« t2
t1
(f(t), Ï)L2(Ω) dt+
â« t2
t1
ăg(t), ÏăHâ1D1
(Ω) dt
for every Ï â H1D1
(Ω;Rd). By Lebesgueâs differentiation theorem and a density argument we
can conclude that the function z solves (4.3.1) for a.e. t â (0, T ) and for every Ï â H1D1
(Ω;Rd).To prove the uniqueness, we use the same technique adopted in Theorem 1.2.10 and
Lemma 3.4.2. Let z1 and z2 be two solutions to (4.3.1) satisfying (4.2.1), (4.2.2), and theinitial conditions u0 and u1. The function z =: z1 â z2 belongs to the space
Lâ(0, T ;H1D1
(Ω;Rd)) â©W 1,â(0, T ;L2(Ω;Rd)) â©H2(0, T ;Hâ1D1
(Ω;Rd)),
and for a.e. t â (0, T ) solves
ăz(t), ÏăHâ1D1
(Ω) + (b(Ï(t))CEz(t), EÏ)L2(Ω) = 0 for every Ï â H1D1
(Ω;Rd),
with initial conditions z(0) = z(0) = 0. We fix s â (0, T ], and we consider the function
Ïs(t) =
ââ« st z(r) dr if t â [0, s],
0 if t â [s, T ].
Chapter 4. A phase-field model of dynamic fracture 109
Clearly, we have Ïs â C0([0, T ];H1D1
(Ω;Rd)) and Ïs(s) = 0. Moreover
Ïs(t) =
z(t) if t â [0, s),
0 if t â (s, T ],
which implies Ïs â Lâ(0, T ;H1D1
(Ω;Rd)). We use Ïs(t) as test function in (4.3.1) and weintegrate in [0, s] to deduceâ« s
0ăz(t), Ïs(t)ăHâ1
D1(Ω) dt+
â« s
0(b(Ï(t))CEz(t), EÏs(t))L2(Ω) dt = 0. (4.3.4)
By integration by parts, the first term becomesâ« s
0ăz(t), Ïs(t)ăHâ1
D1(Ω) dt = â
â« s
0(z(t), z(t))L2(Ω) dt = â1
2âz(s)â2L2(Ω),
since Ïs(s) = z(0) = z(0) = 0. Moreover, the function t 7â (b(Ï(t))CEÏs(t), EÏs(t))L2(Ω) is
absolutely continuous on [0, T ], because Ïs â H1(0, T ;H1D1
(Ω;Rd)) and Ï â H1(0, T ;C0(Ω)).Hence, we can integrate by parts the second terms of (4.3.4) to obtainâ« s
0(b(Ï(t))CE(z(t)), EÏs(t))L2(Ω) dt
= â1
2
â« s
0(b(Ï(t))Ï(t)CEÏs(t), EÏs(t))L2(Ω) dtâ 1
2(b(Ï(0))CEÏs(0), EÏs(0))L2(Ω),
since Ïs(s) = 0. These two identities imply that z and Ïs satisfy
âz(s)â2L2(Ω) + (b(Ï(0))CEÏs(0), EÏs(0))L2(Ω) = ââ« s
0(b(Ï(t))Ï(t)CEÏs(t), EÏs(t))L2(Ω) dt.
In particular, we get
âz(s)â2L2(Ω) + ηλ0âEÏs(0)â2L2(Ω)
†b(âÏâLâ(0,T ;C0(Ω)))âCâLâ(Ω)
â« s
0âÏ(t)âLâ(Ω)âEÏs(t)â2L2(Ω)dt,
since b is non-decreasing. Let us define ζ(t) :=⫠t
0 z(r) dr for t â [0, s]. Since Ïs(t) = ζ(t)âζ(s)for t â [0, s], we deduce that âEÏs(0)âL2(Ω) = âEζ(s)âL2(Ω) andâ« s
0âÏ(t)âLâ(Ω)âEÏs(t)â2L2(Ω)dt
†2âEζ(s)â2L2(Ω)
â« s
0âÏ(t)âLâ(Ω) dt+ 2
â« s
0âÏ(t)âLâ(Ω)âEζ(t)â2L2(Ω)dt
†2âsâÏâL2(0,T ;C0(Ω))âEζ(s)â2L2(Ω) + 2
â« s
0âÏ(t)âLâ(Ω)âEζ(t)â2L2(Ω)dt.
Hence, we have
âz(s)â2L2(Ω) +[ηλ0 â 2b(âÏâLâ(0,T ;C0(Ω)))âCâLâ(Ω)âÏâL2(0,T ;C0(Ω))
âs]âEζ(s)â2L2(Ω)
†2b(âÏâLâ(0,T ;C0(Ω)))âCâLâ(Ω)
â« s
0âÏ(t)âLâ(Ω)âEζ(t)â2L2(Ω)dt.
Let us set
t0 :=
(ηλ0
4b(âÏâLâ(0,T ;C0(Ω))âCâLâ(Ω)âÏâL2(0,T ;C0(Ω))
)2
.
110 4.3. Proof of the main result
By the previous estimate, for every s â [0, t0] we derive
âz(s)â2L2(Ω) +ηλ0
2âEζ(s)â2L2(Ω)
†2b(âÏâLâ(0,T ;C0(Ω)))âCâLâ(Ω)
â« s
0âÏ(t)âLâ(Ω)âEζ(t)â2L2(Ω)dt.
Thanks to Gronwallâs lemma (see, e.g., [24, Chapitre XVIII, §5, Lemme 1]), this inequalityimplies that z(s) = Eζ(s) = 0 for every s â [0, t0]. Since t0 depends only on C, b, and Ï, wecan repeat this procedure starting from t0 and, with a finite number of steps, we obtain thatz = 0 on the whole interval [0, T ].
Corollary 4.3.2. Let w1, f , g, u0, u1, and Ï be as in Lemma 4.3.1. Then the unique solutionz to (4.3.1) with initial conditions z(0) = u0 and z(0) = u1 satisfies for every t â [0, T ] thefollowing energy-dissipation inequality
K (z(t)) + E (z(t), Ï(t))â 1
2
â« t
0(b(Ï(s))Ï(s)CEz(s), Ez(s))L2(Ω) ds
†K (u1) + E (u0, Ï(0)) + Wtot(z, Ï; 0, t).
(4.3.5)
Proof. For t = 0 the inequality (4.3.5) is trivially true, thanks to the initial conditions of z.We fix t â (0, T ] and we write the inequality (4.3.3) as
K (zâČn(t)) + E (zn(t), Ïn(t))â 1
2Ïn
â« tn
0([b(Ïn)â b(Ïn)]CEzn, Ezn)L2(Ω) ds
†K (u1) + E (u0, Ï(0)) +
â« tn
0[(fn, z
âČn â wâČn)L2(Ω) + (b(Ïn)CEzn, EwâČn)L2(Ω)] ds
ââ« tn
0[ăgn, zn â wnăHâ1
D1(Ω) + (zâČn, w
âČn)L2(Ω)] ds+ ăgn(t), zn(t)â wn(t)ăHâ1
D1(Ω)
+ (zâČn(t), wâČn(t))L2(Ω) â ăg(0), u0 â w1(0)ăHâ1D1
(Ω) â (u1, w1(0))L2(Ω),
(4.3.6)
where tn := jÏn, and j is the unique element in 1, . . . , n for which t â ((jâ1)Ïn, jÏn]. To passto the limit as n â â in (4.3.6), we follow the same procedure adopted in Lemma 4.2.11.Notice that zn(t) z(t) in H1(Ω;Rd) and zâČn(t) z(t) in L2(Ω;Rd), by arguing as inRemark 4.2.8, while Ïn(t)â Ï(t) in C0(Ω). Hence, we derive
K (z(t)) †lim infnââ
K (zâČn(t)), E (z(t), Ï(t)) †lim infnââ
E (zn(t), Ïn(t)). (4.3.7)
Similarly, we combine the convergences given by the previous lemma, with Ïn(s) â Ï(s) inC0(Ω) for every s â [0, T ] and tn â t as nââ, to deduce
limnââ
â« tn
0(fn(s), zâČn(s)â wâČn(s))L2(Ω) ds =
â« t
0(f(s), z(s)â w1(s))L2(Ω) ds, (4.3.8)
limnââ
â« tn
0(b(Ïn(s))CEzn(s), EwâČn(s))L2(Ω)] ds =
â« t
0(b(Ï(s))CEz(s), Ew(s))L2(Ω) ds, (4.3.9)
limnââ
â« tn
0(zâČn(s), wâČn(s))L2(Ω) ds =
â« t
0(z(s), w1(s))L2(Ω) ds, (4.3.10)
limnââ
â« tn
0ăgn(s), zn(s)â wn(s)ăHâ1
D1(Ω) ds =
â« t
0ăg(s), z(s)â w1(s)ăHâ1
D1(Ω) ds, (4.3.11)
limnââ
(zâČn(t), wâČn(t))L2(Ω) = (z(t), w1(t))L2(Ω), (4.3.12)
limnââ
ăgn(t), zn(t)â wn(t)ăHâ1D1
(Ω) = ăg(t), z(t)â w1(t)ăHâ1D1
(Ω). (4.3.13)
Chapter 4. A phase-field model of dynamic fracture 111
Finally, for a.e. s â (0, T ) we haveâ„â„â„â„Ïn(s)â Ïn(s)
Ïnâ Ï(s)
â„â„â„â„Lâ(Ω)
†1
Ïn
â« s+Ïn
sâÏnâÏ(r)â Ï(s)âLâ(Ω) â 0 as nââ, (4.3.14)
since Ï â L2(0, T ;C0(Ω)). Let us fix s â (0, T ) for which (4.3.14) holds. By Lagrangeâstheorem for every x â Ω there exists a point rn(s, x) â [Ïn(s, x), Ïn(s, x)] such that
b(Ïn(s, x))â b(Ïn(s, x))
Ïn= b(rn(s, x))
Ïn(s, x)â Ïn(s, x)
Ïn.
Notice that rn(s, x)â Ï(s, x) as nââ for every x â Ω. Hence, for a.e. s â (0, T ) we get
limnââ
b(Ïn(s, x))â b(Ïn(s, x))
Ïn= b(Ï(s, x))Ï(s, x) for every x â Ω.
Furthermore, thanks to (4.3.14) there is a constant Cs > 0, which may depend on s, but it isindependent of n, such that for every x â ΩâŁâŁâŁâŁb(Ïn(s, x))â b(Ïn(s, x))
Ïn
âŁâŁâŁâŁ †b(âÏâLâ(0,T ;C0(Ω)))
â„â„â„â„Ïn(s)â Ïn(s)
Ïn
â„â„â„â„Lâ(Ω)
†b(âÏâLâ(0,T ;C0(Ω)))Cs.
Therefore, for a.e. s â (0, T ) we can apply the dominated convergence theorem to deduce
b(Ïn(s))â b(Ïn(s))
Ïnâ b(Ï(s))Ï(s) in L2(Ω) as nââ.
The function Ï(x, y, Ο) := 12 |y|C(x)Οsym · Οsym, (x, y, Ο) â ΩĂRĂRdĂd, satisfies the assump-
tions of Ioffe-Olechâs theorem, while Ezn(s) Ez(s) in L2(Ω;RdĂd) for every s â [0, T ].Then, we have
â 1
2(b(Ï(s))Ï(s)CEz(s), Ez(s))L2(Ω)
=
â«Î©Ï(x, b(Ï(s))Ï(s, x), Ez(s, x)) dx
†lim infnââ
â«Î©Ï
(x,b(Ïn(s, x))â b(Ïn(s, x))
Ïn, Ezn(s, x)
)dx
= lim infnââ
[â 1
2Ïn([b(Ïn(s))â b(Ïn(s))]CEzn(s), Ezn(s))L2(Ω)
]for a.e. s â (0, T ), being b(Ïn(s)) †b(Ïn(s)) in Ω. In particular, thanks to Fatouâs lemmawe get
â 1
2
â« t
0(b(Ï(s))Ï(s)CEz(s), Ez(s))L2(Ω) ds
â€â« t
0lim infnââ
[â 1
2Ïn([b(Ïn(s))â b(Ïn(s))]CEzn(s), Ezn(s))L2(Ω)
]ds
†lim infnââ
[â 1
2Ïn
â« tn
0([b(Ïn(s))â b(Ïn(s))]CEzn(s), Ezn(s))L2(Ω) ds
],
(4.3.15)
since t †tn. By combining (4.3.6)â(4.3.13) with (4.3.15) we deduce the inequality (4.3.5) forevery t â (0, T ].
112 4.3. Proof of the main result
The other inequality, at least for a.e. t â (0, T ), is a consequence of equation (4.3.1).
Lemma 4.3.3. Let w1, f , g, u0, u1, and Ï be as in Lemma 4.3.1. Then the unique solutionz to (4.3.1) with initial conditions z(0) = u0 and z(0) = u1 satisfies for a.e. t â (0, T )
K (z(t)) + E (z(t), Ï(t))â 1
2
â« t
0(b(Ï(s))Ï(s)CEz(s), Ez(s))L2(Ω) ds
â„ K (u1) + E (u0, Ï(0)) + Wtot(z, Ï; 0, t).
(4.3.16)
Proof. It is enough to proceed as done in Lemma 4.1.8, by using Lebesgueâs differentiationtheorem and the fact that z â C0
w([0, T ];H1(Ω;Rd)) and z â C0w([0, T ];L2(Ω;Rd)). This
ensures that z satisfies
K (z(t2)) + E (z(t2), Ï(t2))â 1
2
â« t2
t1
(b(Ï(s))Ï(s)CEz(s), Ez(s))L2(Ω) ds
= K (z(t1)) + E (z(t1), Ï(t1)) + Wtot(z, Ï; t1, t2)
for a.e. t1, t2 â (0, T ) with t1 < t2. Since the right-hand side is lower semicontinuous withrespect to t1, while the left-hand side is continuous, sending t1 â 0+ we deduce (4.3.16).
By combining the two previous results we obtain that the solution z to (4.3.1) satisfies
K (z(t)) + E (z(t), Ï(t))â 1
2
â« t
0(b(Ï(s))Ï(s)CEz(s), Ez(s))L2(Ω) ds
= K (u1) + E (u0, Ï(0)) + Wtot(z, Ï; 0, t).
(4.3.17)
for a.e. t â (0, T ). Actually, this is true for every time, as shown in the following lemma.
Lemma 4.3.4. Let w1, f , g, u0, u1, and Ï be as in Lemma 4.3.1. Then the unique solutionz to (4.3.1) with initial conditions z(0) = u0 and z(0) = u1 satisfies equality (4.3.17) forevery t â [0, T ]. In particular, the function t 7â K (z(t)) + E (z(t), Ï(t)) is continuous from[0, T ] to R and
z â C0([0, T ];H1(Ω;Rd)) â© C1([0, T ];L2(Ω;Rd)). (4.3.18)
Proof. We may assume that Ï, w1, f , and g are defined on [0, 2T ] and satisfy the hypotheses ofLemma 4.3.1 with T replaced by 2T . As for w1 and Ï, we can set w1(t) := 2w1(T )âw1(2Tât)and Ï(t) := Ï(T ) for t â (T, 2T ], respectively. By Lemma 4.3.1, the solution z on [0, T ] canbe extended to a solution on [0, 2T ] still denoted by z. Thanks to Corollary 4.3.2 andLemma 4.3.3, the function z satisfies equality (4.3.17) for a.e. t â (0, 2T ), and inequal-ity (4.3.5) for every t â [0, 2T ]. By contradiction assume the existence of a point t0 â [0, T ]such that
K (z(t0)) + E (z(t0), Ï(t0))â 1
2
â« t0
0(b(Ï(s))Ï(s)CEz(s), Ez(s))L2(Ω) ds
< K (u1) + E (u0, Ï(0)) + Wtot(z, Ï; 0, t0).
We have z(t0) â w(t0) â H1D1
(Ω;Rd) and z(t0) â L2(Ω;Rd), since z â C0w([0, T ];H1(Ω;Rd))
and z â C0w([0, T ];L2(Ω;Rd)). Then we can consider the solution z0 to (4.3.1) in [t0, 2T ] with
these initial conditions. The function defined by z in [0, t0] and z0 in [t0, 2T ] is still a solutionto (4.3.1) in [0, 2T ] and so, by uniqueness, we have z = z0 in [t0, 2T ]. Furthermore, in viewof (4.3.5) we deduce
K (z(t)) + E (z(t), Ï(t))â 1
2
â« t
t0
(b(Ï(s))Ï(s)CEz(s), Ez(s))L2(Ω) ds
†K (z(t0)) + E (z(t0), Ï(t0)) + Wtot(z, Ï; t0, t)
Chapter 4. A phase-field model of dynamic fracture 113
for every t â [t0, 2T ]. By combining the last two inequalities, we get
K (z(t)) + E (z(t), Ï(t))â 1
2
â« t
0(b(Ï(s))Ï(s)CEz(s), Ez(s))L2(Ω) ds
†K (z(t0)) + E (z(t0), Ï(t0)) + Wtot(z, Ï; t0, t)â1
2
â« t0
0(b(Ï(s))Ï(s)CEz(s), Ez(s))L2(Ω) ds
< K (u1) + E (u0, Ï(0)) + Wtot(z, Ï; 0, t0) + Wtot(z, Ï; t0, t)
= K (u1) + E (u0, Ï(0)) + Wtot(z, Ï; 0, t)
for every t â [t0, 2T ], which contradicts (4.3.17). Therefore, equality (4.3.17) holds for everyt â [0, T ], which implies the continuity of the function t 7â K (z(t))+E (z(t), Ï(t)) from [0, T ]to R.
Let us now prove (4.3.18). We fix t0 â [0, T ] and we consider a sequence of points tmmconverging to t0 as m â â. Since z â C0
w([0, T ];H1(Ω;Rd)) and z â C0w([0, T ];L2(Ω;Rd)),
we have
K (z(t0)) †lim infmââ
K (z(tm)), E (z(t0), Ï(t0)) †lim infmââ
E (z(tm), Ï(t0)).
Moreover, Ï â C0([0, T ];C0(Ω)) and b â C1(R), which implies as mââ
|E (z(tm), Ï(t0))â E (z(tm), Ï(tm))|
†1
2b(âÏâLâ(0,T ;C0(Ω)))âCâLâ(Ω)âEzâ2Lâ(0,T ;L2(Ω))âÏ(t0)â Ï(tm)âLâ(Ω) â 0.
In particular, we deduce
E (z(t0), Ï(t0)) †lim infmââ
E (z(tm), Ï(tm)).
The above inequalities and the continuity of t 7â K (z(t)) + E (z(t), Ï(t)) gives
K (z(t0)) + E (z(t0), Ï(t0)) †lim infmââ
K (z(tm)) + lim infmââ
E (z(tm), Ï(tm))
†limmââ
[K (z(tm)) + E (z(tm), Ï(tm))]
= K (z(t0)) + E (z(t0), Ï(t0)),
which implies the continuity of t 7â K (z(t)) and t 7â E (z(t), Ï(t)) in t0 â [0, T ]. In particular,we derive that the functions t 7â âz(t)âL2(Ω) and t 7â âz(t)âH1(Ω) are continuous from [0, T ]to R. By combining this fact with the weak continuity of z and z, we get (4.3.18).
We are now in a position to prove Theorem 4.1.5.
Proof of Theorem 4.1.5. By Lemmas 4.2.9 and 4.2.10, there exists a generalized solution(u, v) to (4.1.12)â(4.1.15) satisfying the initial conditions (4.1.16), the irreversibility con-dition (4.1.17), and the crack stability condition (4.1.18). Clearly, the function v satis-fies (4.1.23), since k â„ 1. Moreover, the function v = Ï is admissible in Lemmas 4.3.1and 4.3.4, since Hk(Ω) â C0(Ω). Therefore, u = z satisfies (4.1.21), which gives that (u, v)is a weak solution to (4.1.12)â(4.1.15).
It remains to prove that (u, v) satisfies the dynamic energy-dissipation balance (4.1.19).As observed in Remark 4.1.9, for k > d/2 the crack stability condition (4.1.18) is equivalentto the variational inequality (4.1.39) for a.e. t â (0, T ) and the function v(t) â Hk(Ω) isadmissible in (4.1.39). Therefore, we have
âvE (u(t), v(t))[v(t)] + âH (v(t))[v(t)] + G (v(t)) â„ 0 for a.e. t â (0, T ).
114 4.4. The case without dissipative terms
By integrating the above inequality in [0, t0] for every t0 â [0, T ], we getâ« t0
0âvE (u(t), v(t))[v(t)] dt+ H (v(t0))âH (v0) +
â« t0
0G (v(t)) dt â„ 0. (4.3.19)
Thanks to Lemma 4.3.4, for every t0 â [0, T ] the pair (u, v) satisfies
K (u(t0)) + E (u(t0), v(t0))â 1
2
â« t0
0(b(v(t))v(t)CEu(t), Eu(t))L2(Ω) dt
= K (u1) + E (u0, v0) + Wtot(u, v; 0, t0).
(4.3.20)
Hence, by combining (4.3.19) and (4.3.20), we deduce
F (u(t0), u(t0), v(t0)) +
â« t0
0G (v(t)) dt â„ F (u0, u1, v0) + Wtot(u, v; 0, t0)
for every t0 â [0, T ]. This inequality, together with (4.2.32), implies (4.1.19) and concludesthe proof.
4.4 The case without dissipative terms
We conclude the chapter by analyzing the dynamic phase-field model of crack propagationwithout dissipative terms. Given w1, w2, f , g, u0, u1, and v0 satisfying (4.1.7)â(4.1.11) and
v0 â arg minE (u0, vâ) + H (vâ) : vâ â w2 â H1D2
(Ω), vâ †v0 in Ω, (4.4.1)
we search a pair (u, v) which solves the elastodynamics system (4.1.12) with boundary andinitial conditions (4.1.13)â(4.1.16), the irreversibility condition (4.1.17), and the followingcrack stability condition for every t â [0, T ]
E (u(t), v(t)) + H (v(t)) †E (u(t), vâ) + H (vâ) (4.4.2)
among all vâ â w2 â H1D2
(Ω) with vâ †v(t).
Remark 4.4.1. We need the compatibility conditions (4.4.1) for the initial data (u0, v0),since we want that (4.4.2) is satisfied for every time. Notice that, given u0 â H1(Ω;Rd),an admissible v0 can be constructed by minimizing vâ 7â E (u0, vâ) + H (vâ) among allvâ â w2 â H1
D2(Ω) with vâ †1 in Ω.
In this section we consider the following notion of solution, which is a slightly modificationof Definition 4.2.1.
Definition 4.4.2. Let w1, w2, f , and g be as in (4.1.7)â(4.1.9). The pair (u, v) is a generalizedsolution to (4.1.12)â(4.1.15) if
u â Lâ(0, T ;H1(Ω;Rd)) â©W 1,â(0, T ;L2(Ω;Rd)) â©H2(0, T ;Hâ1D1
(Ω;Rd)), (4.4.3)
u(t)â w1(t) â H1D1
(Ω;Rd) for every t â [0, T ], (4.4.4)
v : [0, T ]â H1(Ω) with v â Lâ(0, T ;H1(Ω)), (4.4.5)
v(t)â w2 â H1D2
(Ω) and v(t) †1 in Ω for every t â [0, T ], (4.4.6)
and for a.e. t â (0, T ) equation (4.1.25) holds.
Remark 4.4.3. By exploiting the regularity properties (4.4.3), we deduce that u belongs toC0w([0, T ];H1(Ω;Rd)), while u is an element of C0
w([0, T ];L2(Ω;Rd)). Therefore, it makes senseto evaluate u and u at time 0. On the other hand, we require v to be defined pointwise for everyt â [0, T ], so that we can consider its precise value at 0. Hence, the initial condition (4.1.16)are well defined.
Chapter 4. A phase-field model of dynamic fracture 115
The main result of this section is the following theorem.
Theorem 4.4.4. Assume that w1, w2, f , g, u0, u1, and v0 satisfy (4.1.7)â(4.1.11) and(4.4.1). Then there exists a generalized solution (u, v) to problem (4.1.12)â(4.1.15) satisfyingthe initial condition (4.1.16), the irreversibility condition (4.1.17), and the crack stabilitycondition (4.4.2). Moreover, the pair (u, v) satisfies for every t â [0, T ] the following energy-dissipation inequality
F (u(t), u(t), v(t)) †F (u0, u1, v0) + Wtot(u, v; 0, t). (4.4.7)
Finally, if w2 â„ 0 on âD2Ω, v0 â„ 0 in Ω, and b(s) = (sâš 0)2 + η for s â R, then we can takev(t) â„ 0 in Ω for every t â [0, T ].
Remark 4.4.5. By choosing b(s) = (sâš0)+η for s â R we deduce the existence of a dynamicphase-field evolution (u, v) satisfying (D1) and (D2), since we can take v(t) â„ 0 in Ω andâ«
Ω[(v(x) âš 0)2 + η]C(x)Eu(x) · Eu(x) dx â€
â«Î©
[(v(x))2 + η]C(x)Eu(x) · Eu(x) dx
for every u â H1(Ω;Rd) and v â H1(Ω). Without adding a dissipative term to the model, weare not able to show the dynamic energy-dissipation balance (D3). However, we can alwaysselect a solution (u, v) which satisfies (4.4.7) for every t â [0, T ].
To prove Theorem 4.4.4 we perform a time discretization, as done in the previous sections.From now on we assume that w1, w2, f , g, u0, u1, and v0 satisfy (4.1.7)â(4.1.11) and (4.4.1).We fix n â N and for every j = 1, . . . , n we define inductively:
(i) ujn â wjn â H1D1
(Ω;Rd) is the minimizer of
uâ 7â 1
2Ï2n
â„â„uâ â 2ujâ1n â ujâ2
n
â„â„2
L2(Ω)+ E (uâ, vjâ1
n )â (f jn, uâ)L2(Ω) â ăgjn, uâ â wjnăHâ1
D1(Ω)
among every uâ â wjn â H1D1
(Ω;Rd);
(ii) vjn â w2 â H1D2
(Ω) with vjn †vjâ1n is the minimizer of
vâ 7â E (ujn, vâ) + H (vâ)
among every vâ â w2 â H1D2
(Ω) with vâ †vjâ1n .
As before, for every j = 1, . . . , n there exists a unique pair (ujn, vjn) â H1(Ω;Rd)ĂH1(Ω)
solution to problems (i) and (ii). Moreover, the function ujn solves (4.2.5), while the functionvjn satisfies
E (ujn, vâ)â E (ujn, v
jn) + âH (vjn)[vâ â vjn] â„ 0 (4.4.8)
among all vâ â w2 â H1D2
(Ω) with vâ †vjâ1n , arguing as in Lemma 4.2.3. In particular, if
w2 â„ 0 on âD2Ω, v0 â„ 0 in Ω, and b(s) = (s âš 0)2 + η for s â R, then for every j = 1, . . . , nwe can use vjn âš 0 â H1(Ω) as a competitor in (ii) to derive that vjn = vjn âš 0 â„ 0 in Ω.
Lemma 4.4.6. The family (ujn, vjn)nj=1, solution to (i) and (ii), satisfies for j = 1, . . . , nthe discrete energy inequality
F (ujn, ÎŽujn, v
jn) +
jâl=1
Ï2nD
ln
†F (u0, u1, v0) +
jâl=1
Ïn[(f ln, ÎŽuln â ÎŽwln)L2(Ω) + (b(vlâ1
n )CEuln, EΎwln)L2(Ω)]
âjâl=1
Ïn[(ÎŽulâ1n , ÎŽ2wln)L2(Ω) â ăÎŽgln, ulâ1
n â wiâ1n ăHâ1
D1(Ω)] + (Ύujn, Ύw
jn)L2(Ω)
+ ăgjn, ujn â wjnăHâ1D1
(Ω) â (u1, w1(0))L2(Ω) â ăg(0), u0 â w1(0)ăHâ1D1
(Ω).
116 4.4. The case without dissipative terms
In particular, there exists a constant C > 0, independent of n, such that
maxj=1,...,n
[âÎŽujnâL2(Ω) + âujnâH1(Ω) + âvjnâH1(Ω)
]+
nâj=1
ÏnâÎŽ2ujnâ2Hâ1D1
(Ω)+
nâj=1
Ï2nD
jn †C. (4.4.9)
Proof. It is enough to proceed as in Lemmas 4.2.4 and 4.2.5, and Remark 4.2.6.
As done in Section 4.2, we use the family (ujn, vjn)nj=1 and the estimate (4.4.9) to con-struct a generalized solution (u, v) to (4.1.12)â(4.1.16). Let un, uâČn, un, uâČn, and un, uâČn bethe piecewise affine, the backward, and the forward interpolants of ujnnj=1 and ÎŽujnnj=1,respectively. Moreover, we consider the backward interpolant vn and the forward interpolantvn of vjnnj=1.
Before passing to the limit as n â â, we recall the following Hellyâs type result forvector-valued functions.
Lemma 4.4.7. Let [a, b] â R and let Ïm : [a, b]â L2(Ω), m â N, be a sequence of functionssatisfying
Ïm(s) †Ïm(t) in Ω for every a †s †t †b and m â N.Assume there exists a constant C, independent of m, such that
âÏm(t)âL2(Ω) †C for every t â [a, b] and m â N.
Then there is a subsequence of m, not relabeled, and a function Ï : [a, b] â L2(Ω) such thatfor every t â [a, b]
Ïm(t) Ï(t) in L2(Ω) as mââ.Moreover, we have âÏ(t)âL2(Ω) †C for every t â [a, b] and
Ï(s) †Ï(t) in Ω for every a †s †t †b. (4.4.10)
Proof. Let us consider a countable dense set D â Ï â L2(Ω) : Ï â„ 0 and let us fix Ï â D .For every m â N the map t 7â
â«Î© Ïm(t, x)Ï(x) dx is non-decreasing and uniformly bounded
in [a, b], since âŁâŁâŁâŁâ«Î©Ïm(t, x)Ï(x) dx
âŁâŁâŁâŁ †CâÏâL2(Ω) for every t â [a, b]. (4.4.11)
By applying the Hellyâs theorem, we can find a subsequence of m, not relabeled, and afunction aÏ : [a, b]â R such that for every t â [a, b]â«
ΩÏm(t, x)Ï(x) dxâ aÏ(t) as mââ.
Moreover, thanks to a diagonal argument, the subsequence of m can be chosen independentof Ï â D .
We now fix t â [a, b] and Ï â L2(Ω) with Ï â„ 0. Given h > 0, there is Ïh â D such thatâÏ â ÏhâL2(Ω) < h. Moreover, thanks to the previous convergence we can find m â N suchthat for every m, l > mâŁâŁâŁâŁâ«
ΩÏm(t, x)Ïh(x) dxâ
â«Î©Ïj(t, x)Ïh(x) dx
âŁâŁâŁâŁ < h.
Therefore, the sequenceâ«
Ω Ïm(t, x)Ï(x) dx, m â N, is Cauchy in R. Indeed, for every h > 0there exists m â N such that for every m, l > mâŁâŁâŁâŁâ«
ΩÏm(t, x)Ï(x) dxâ
â«Î©Ïl(t, x)Ï(x) dx
âŁâŁâŁâŁâ€ 2CâÏâ ÏhâL2(Ω) +
âŁâŁâŁâŁâ«Î©Ïm(t, x)Ïh(x) dxâ
â«Î©Ïj(t, x)Ïh(x) dx
âŁâŁâŁâŁ< (2C + 1)h.
Chapter 4. A phase-field model of dynamic fracture 117
Hence, we can find an element aÏ(t) â R such thatâ«Î©Ïm(t, x)Ï(x) dxâ aÏ(t) as mââ.
In particular, for every t â [a, b] and Ï â L2(Ω) we have as mâââ«Î©Ïm(t, x)Ï(x) dx =
â«Î©Ïm(t, x)Ï+(x) dxâ
â«Î©Ïm(t, x)Ïâ(x) dx
â aÏ+(t)â aÏâ(t) =: aÏ(t),
where we have set Ï+ := Ï âš 0 and Ïâ := (âÏ) âš 0. For every t â [a, b] fixed, let us considerthe functional ζ(t) : L2(Ω)â R defined by
ζ(t)(Ï) := aÏ(t) for Ï â L2(Ω).
We have that ζ(t) linear and continuous on L2(Ω). Indeed, by (4.4.11) we deduce
|ζ(t)(Ï)| †CâÏâL2(Ω) for every Ï â L2(Ω).
Hence, Rieszâs representation theorem implies the existence of a function Ï(t) â L2(Ω) suchthat
aÏ(t) =
â«Î©Ï(t, x)Ï(x) dx for every Ï â L2(Ω).
In particular, for every t â [a, b] we deduce that Ïm(t) Ï(t) in L2(Ω) as m â â andâÏ(t)âL2(Ω) †C. Finally observe that Ï â L2(Ω) : Ï â„ 0 is a weakly closed subset ofL2(Ω). Therefore, we derive (4.4.10), since Ïm(t)âÏm(s) Ï(t)âÏ(s) in L2(Ω) as mââand Ïm(t)â Ïm(s) â Ï â L2(Ω) : Ï â„ 0 for every m â N and a †s †t †b.
Lemma 4.4.8. There exist a subsequence of n, not relabeled, and two functions
u â Lâ(0, T ;H1(Ω;Rd)) â©W 1,â(0, T ;L2(Ω;Rd)) â©H2(0, T ;Hâ1D1
(Ω;Rd)),v : [0, T ]â H1(Ω) with v â Lâ(0, T ;H1(Ω)),
such that as nââ
un u in H1(0, T ;L2(Ω;Rd)), uâČn u in H1(0, T ;Hâ1D1
(Ω;Rd)),
un â u in C0([0, T ];L2(Ω;Rd)), uâČn â u in C0([0, T ];Hâ1D1
(Ω;Rd)),
un u in L2(0, T ;H1(Ω;Rd)), uâČn u in L2(0, T ;L2(Ω;Rd)),un u in L2(0, T ;H1(Ω;Rd)), uâČn u in L2(0, T ;L2(Ω;Rd)),vn â v in L2(0, T ;L2(Ω)), vn v in L2(0, T ;H1(Ω)),
vn â v in L2(0, T ;L2(Ω)), vn v in L2(0, T ;H1(Ω)).
Moreover, for every t â [0, T ] as nââ we have
vn(t)â v(t) in L2(Ω), vn(t) v(t) in H1(Ω).
Proof. The existence of a limit point u and the related convergences can be obtained byarguing as in Lemma 4.2.7. Let us now consider the sequence vnn. For every n â N thefunctions vn : [0, T ]â L2(Ω) are non-increasing in [0, T ], that is
vn(t) †vn(s) in Ω for every 0 †s †t †T ,
118 4.4. The case without dissipative terms
and, in view of Lemma 4.4.6, there exists C > 0, independent of n, such that
âvn(t)âH1(Ω) †C for every t â [0, T ] and n â N. (4.4.12)
Therefore, we can apply Lemma 4.4.7. Up to extract a subsequence (not relabeled), we obtainthe existence of a non-increasing function v : [0, T ]â L2(Ω) such that as nââ
vn(t) v(t) in L2(Ω) for every t â [0, T ].
Moreover, by (4.4.12) for every t â [0, T ] we derive that v(t) â H1(Ω) and as nââ
vn(t) v(t) in H1(Ω), vn(t)â v(t) in L2(Ω),
thanks to Rellichâs theorem. Notice that the function v : [0, T ]â H1(Ω) is strongly measur-able. Indeed, it is weak measurable, since it is non-increasing, and with values in a separableHilbert space. In particular, we have v â Lâ(0, T ;H1(Ω)), since âv(t)âH1(Ω) †C for everyt â [0, T ]. By the dominated convergence theorem, as nââ we conclude
vn â v in L2(0, T ;L2(Ω)), vn v in L2(0, T ;H1(Ω)).
Finally, as nââ we have
vn â v in L2(0, T ;L2(Ω)), vn v in L2(0, T ;H1(Ω)),
since vn(t) = vn(tâ Ïn) for a.e. t â (Ïn, T ).
Remark 4.4.9. As pointed out in Remark 4.2.8, for every t â [0, T ] we have as nââ
un(t) u(t) in H1(Ω;Rd), uâČn(t) u(t) in L2(Ω;Rd),un(t) u(t) in H1(Ω;Rd), uâČn(t) u(t) in L2(Ω;Rd).
We are now in a position to prove Theorem 4.4.4.
Proof of Theorem 4.4.4. Thanks to the previous lemma there exists a pair (u, v) satisfy-ing (4.4.3)â(4.4.6), since un(t) â wn(t) â H1
D1(Ω;Rd) and vn(t) â w2 â H1
D2(Ω) for every
t â [0, T ] and n â N. Moreover, (u, v) satisfies the irreversibility condition (4.1.17) andthe initial conditions (4.1.16), thanks to (4.4.10) and the fact that u0 = un(0) u(0) inH1(Ω;Rd), u1 = uâČn(0) u(0) in L2(Ω;Rd), and v0 = vn(0) v(0) in H1(Ω) as nââ.
For every n â N and j = 1, . . . , n the pair (ujn, vjn) solves equation (4.2.5). In particular,
by integrating it over the time interval [t1, t2] â [0, T ], we deduceâ« t2
t1
ăuâČn(t), ÏăHâ1D1
(Ω) dt+
â« t2
t1
(b(vn(t))CEun(t), EÏ)L2(Ω) dt
=
â« t2
t1
(fn(t), Ï)L2(Ω) dt+
â« t2
t1
ăgn(t), ÏăHâ1D1
(Ω) dt
for every Ï â H1D1
(Ω;Rd). Let us pass to the limit as nââ. We have
limnââ
â« t2
t1
ăuâČn(t), ÏăHâ1D1
(Ω) dt =
â« t2
t1
ău(t), ÏăHâ1D1
(Ω) dt,
limnââ
â« t2
t1
(fn(t), Ï)L2(Ω) dt =
â« t2
t1
(f(t), Ï)L2(Ω) dt,
limnââ
â« t2
t1
ăgn(t), ÏăHâ1D1
(Ω) dt =
â« t2
t1
ăg(t), ÏăHâ1D1
(Ω) dt,
Chapter 4. A phase-field model of dynamic fracture 119
since uâČn u in L2(0, T ;Hâ1D1
(Ω;Rd)), gn â g in L2(0, T ;Hâ1D1
(Ω;Rd)), and fn â f in
L2(0, T ;L2(Ω;Rd)) as n â â. Moreover, the dominated convergence theorem yields thatb(vn)CEÏ â b(v)CEÏ in L2(0, T ;L2(Ω;RdĂd)) as nââ, being
|b(vn(t, x))C(x)EÏ(x)| †b(1)âCâLâ(Ω)|EÏ(x)| for every t â [0, T ] and a.e. x â Ω,
and vn â v in L2(0, T ;L2(Ω)). Therefore, we derive
limnââ
â« t2
t1
(b(vn(t))CEun(t), EÏ)L2(Ω) dt =
â« t2
t1
(b(v(t))CEu(t), EÏ)L2(Ω) dt,
because Eun Eu in L2(0, T ;L2(Ω;RdĂd)) as nââ. These facts imply that the pair (u, v)solves â« t2
t1
ău(t), ÏăHâ1D1
(Ω) dt+
â« t2
t1
(b(v(t))CEu(t), EÏ)L2(Ω) dt
=
â« t2
t1
(f(t), Ï)L2(Ω) dt+
â« t2
t1
ăg(t), ÏăHâ1D1
(Ω) dt
for every Ï â H1D1
(Ω;Rd) and [t1, t2] â [0, T ]. By Lebesgueâs differentiation theorem and adensity argument we hence obtain (4.1.25) for a.e. t â (0, T ).
For t = 0 the crack stability condition (4.4.2) trivially holds, since (u, v) satisfies theinitial conditions (4.1.16) and the compatibility condition (4.4.1). We fix t â (0, T ] and weuse the variational inequality (4.4.8) to derive
E (un(t), vâ)â E (un(t), vn(t)) + âH (vn(t))[vâ â vn(t)] â„ 0 (4.4.13)
among all vâ â w2 â H1D2
(Ω) with vâ †vn(tâ Ïn). Given Ï â H1D2
(Ω), with Ï â€ 0 in Ω, thefunction Ï+ vn(t) is admissible for (4.4.13). Hence, we have
E (un(t), Ï+ vn(t))â E (un(t), vn(t)) + âH (vn(t))[Ï] â„ 0.
Let us send nââ. Since vn(t) v(t) in H1(Ω), we deduce
limnââ
âH (vn(t))[Ï] = âH (v(t))[Ï].
Moreover, Eun(t) Eu(t) in L2(Ω;RdĂd) and vn(t) â v(t) in L2(Ω) as n â â, whichimplies
E (u(t), Ï+ v(t))â E (u(t), v(t)) â„ lim supnââ
[E (un(t), Ï+ vn(t))â E (un(t), vn(t))]
by Ioffe-Olechâs theorem, as in Lemma 4.2.10. If we combine these two results, for everyt â (0, T ] we get
E (u(t), Ï+ v(t))â E (u(t), v(t)) + âH (v(t))[Ï] â„ 0
for every Ï â H1D2
(Ω) with Ï â€ 0 in Ω. This implies (4.4.2), since the map vâ 7â H (vâ) isconvex.
It remains to prove the energy-dissipation inequality (4.4.7) for every t â [0, T ]. For t = 0we have actually the equality, thanks to the initial conditions (4.1.16). We now fix t â (0, T ]and we use (4.3.3) to write
F (un(t), uâČn(t), vn(t))
†F (u0, u1, v0) +
â« tn
0[(fn, u
âČn â wâČn)L2(Ω) + (b(vn)CEun, EwâČn)L2(Ω)] ds
ââ« tn
0[ăgn, un â wnăHâ1
D1(Ω) + (uâČn, w
âČn)L2(Ω)] ds+ ăgn(t), un(t)â wn(t)ăHâ1
D1(Ω)
+ (uâČn(t), wâČn(t))L2(Ω) â ăg(0), u0 â w1(0)ăHâ1D1
(Ω) â (u1, w1(0))L2(Ω)
120 4.4. The case without dissipative terms
for every n â N, where tn is the same number defined in Lemma 4.2.11. By using the factthat vn(t) v(t) in H1(Ω) as nââ, we deduce
H (v(t)) †lim infnââ
H (vn(t)).
Similarly, thanks to Ioffe-Olechâs theorem we derive
E (u(t), v(t)) †lim infnââ
E (un(t), vn(t)),
since vn(t)â v(t) in L2(Ω) and Eun(t) Eu(t) in L2(Ω;RdĂd). Finally, we can argue as inLemma 4.2.11 to derive that the remaining terms converge to Wtot(u, v; 0, t) as n â â. Bycombining the previous results, we deduce (4.4.7) for every t â (0, T ].
Finally, if w2 â„ 0 on âD2Ω, v0 â„ 0 in Ω, and b(s) = (s âš 0)2 + η for s â R, then we havevn(t) â„ 0 in Ω for every t â [0, T ], which implies v(t) â„ 0 in Ω.
Bibliography
[1] R.A. Adams: Sobolev spaces. Pure and Applied Mathematics, Vol. 65, Academic Press,New York, 1975.
[2] S. Almi, S. Belz, and M. Negri: Convergence of discrete and continuous unilat-eral flows for Ambrosio-Tortorelli energies and application to mechanics. ESAIM Math.Model. Numer. Anal. 53 (2019), 659â699.
[3] L. Ambrosio, N. Gigli, and G. Savare: Gradient flows in metric spaces and in thespace of probability measures. Lectures in Mathematics ETH Zurich. Birkhauser Verlag,Basel, 2005.
[4] L. Ambrosio and V.M. Tortorelli: Approximation of functionals depending onjumps by elliptic functionals via Î-convergence. Comm. Pure Appl. Math. 43 (1990),999â1036.
[5] B. Bourdin, G.A. Francfort, and J.J. Marigo: The variational approach to frac-ture. Reprinted from J. Elasticity 91 (2008), Springer, New York, 2008.
[6] B. Bourdin, C.J. Larsen, and C.L. Richardson: A time-discrete model for dy-namic fracture based on crack regularization. Int. J. Fracture 168 (2011), 133â143.
[7] M. Caponi: Existence of solutions to a phase-field model of dynamic fracturewith a crack-dependent dissipation. Submitted for publication (2018). Preprint SISSA06/2018/MATE.
[8] M. Caponi: Linear hyperbolic systems in domains with growing cracks, Milan J. Math.85 (2017), 149â185.
[9] M. Caponi, I. Lucardesi, and E. Tasso: Energy-dissipation balance of a smoothmoving crack. Submitted for publication (2018). Preprint SISSA 31/2018/MATE.
[10] M. Caponi and F. Sapio: A dynamic model for viscoelastic materials with prescribedgrowing cracks. Submitted for publication (2019). Preprint SISSA 12/2019/MATE.
[11] A. Chambolle: A density result in two-dimensional linearized elasticity, and applica-tions. Arch. Ration. Mech. Anal. 167 (2003), 211â233.
[12] A. Chambolle and V. Crismale: Compactness and lower semicontinuity in GSBD.To appear on J. Eur. Math. Soc. (JEMS) (2018). Preprint arXiv:1802.03302.
[13] A. Chambolle and V. Crismale: Existence of strong solutions to the Dirichlet prob-lem for the Griffith energy. Published online on Calc. Var. Partial Differential Equations(2019). DOI: https://doi.org/10.1007/s00526-019-1571-7.
[14] B. Dacorogna: Direct methods in the calculus of variations. Applied MathematicalSciences, Vol. 78, Springer-Verlag, Berlin, 1989.
121
122 Bibliography
[15] G. Dal Maso and L. De Luca: A minimization approach to the wave equa-tion on time-dependent domains. Published online on Adv. Calc. Var. (2019). DOI:https://doi.org/10.1515/acv-2018-0027.
[16] G. Dal Maso and C.J. Larsen: Existence for wave equations on domains with ar-bitrary growing cracks. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 22 (2011),387â408.
[17] G. Dal Maso, C.J. Larsen, and R. Toader: Existence for constrained dynamicGriffith fracture with a weak maximal dissipation condition. J. Mech. Phys. Solids 95(2016), 697â707.
[18] G. Dal Maso, C.J. Larsen, and R. Toader: Existence for elastodynamic Griffithfracture with a weak maximal dissipation condition. J. Math. Pures Appl. 127 (2019),160â191.
[19] G. Dal Maso, G. Lazzaroni, and L. Nardini: Existence and uniqueness of dynamicevolutions for a peeling test in dimension one. J. Differential Equations 261 (2016),4897â4923.
[20] G. Dal Maso and I. Lucardesi: The wave equation on domains with cracks growingon a prescribed path: existence, uniqueness, and continuous dependence on the data.Appl. Math. Res. Express 2017 (2017), 184â241.
[21] G. Dal Maso and R. Scala: Quasistatic evolution in perfect plasticity as limit ofdynamic processes. J. Dynam. Differential Equations 26 (2014), 915â954.
[22] G. Dal Maso and R. Toader: A model for the quasi-static growth of brittle fractures:existence and approximation results. Arch. Rat. Mech. Anal. 162 (2002), 101â135.
[23] G. Dal Maso and R. Toader: On the Cauchy problem for the wave equation ontime-dependent domains. J. Differential Equations 266 (2019), 3209-3246.
[24] R. Dautray and J.L. Lions: Analyse mathematique et calcul numerique pour lessciences et les techniques. Vol. 8. Evolution: semi-groupe, variationnel. Masson, Paris,1988.
[25] E. De Giorgi, L. Ambrosio: Un nuovo tipo di funzionale del calcolo delle variazioni.Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 82 (1988), 199â210.
[26] G.A. Francfort and J.J. Marigo: Revisiting brittle fracture as an energy mini-mization problem. J. Mech. Phys. Solids 46 (1998), 1319â1342.
[27] G.A. Francfort and C.J. Larsen: Existence and convergence for quasi-static evo-lution in brittle fracture. Comm. Pure Appl. Math. 56 (2003), 1465â1500.
[28] L.B. Freund: Dynamic fracture mechanics. Cambridge Monographs on Mechanics andApplied Mathematics, Cambridge University Press, Cambridge, 1990.
[29] M. Friedrich and F. Solombrino: Quasistatic crack growth in 2d-linearized elastic-ity. Ann. Inst. H. Poincare Anal. Non Lineaire 35 (2018), 28â64
[30] A. Giacomini: Ambrosio-Tortorelli aproximation of quasi-static evolution of brittlefractures. Calc. Var. Partial Differential Equations 22 (2005), 129â172.
[31] A.A. Griffith: The phenomena of rupture and flow in solids. Philos. Trans. Roy. Soc.London 221-A (1920), 163â198.
Bibliography 123
[32] P. Grisvard: Elliptic Problems in Nonsmooth Domains. Monographs and Studies inMath., vol. 24, Pitman, Boston, 1985.
[33] T. Kato: Abstract differential equations and nonlinear mixed problems. AccademiaNazionale dei Lincei, Scuola Normale Superiore, Lezione Fermiane, Pisa, 1985.
[34] O.A. Ladyzenskaya: On integral estimates, convergence, approximate methods, andsolution in functionals for elliptic operators. Vestnik Leningrad. Univ. 13 (1958), 60â69.
[35] C.J. Larsen: Models for dynamic fracture based on Griffithâs criterion. In: Hackl K.(eds.) âIUTAM Symposium on Variational Concepts with Applications to the Mechanicsof Materialsâ, IUTAM Bookseries, Vol 21, Springer, Dordrecht, 2010, 131â140.
[36] C.J. Larsen, C. Ortner, and E. Suli: Existence of solutions to a regularized modelof dynamic fracture. Math. Models Methods Appl. Sci. 20 (2010), 1021â1048.
[37] G. Lazzaroni and L. Nardini, Analysis of a dynamic peeling test with speed-dependent toughness. SIAM J. Appl. Math. 78 (2018), 1206â1227.
[38] G. Lazzaroni and R. Toader: Energy release rate and stress intensity factor inantiplane elasticity. J. Math. Pures Appl. 95 (2011), 565â584.
[39] G. Lazzaroni and R. Toader: A model for crack propagation based on viscousapproximation. Math. Models Methods Appl. Sci. 21 (2011), 2019â2047.
[40] J.L. Lions and E. Magenes: Non-homogeneous boundary value problems and applica-tions. Vol. I. Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972.
[41] N.F. Mott: Brittle fracture in mild steel plates. Engineering 165 (1948), 16â18.
[42] M. Negri: A unilateral L2-gradient flow and its quasi-static limit in phase-field fractureby an alternate minimizing movement. Adv. Calc. Var. 12 (2019), 1â29.
[43] S. Nicaise and A.M. Sandig: Dynamic crack propagation in a 2D elastic body: theout-of-plane case. J. Math. Anal. Appl. 329 (2007), 1â30.
[44] O.A. Oleinik, A.S. Shamaev, and G.A. Yosifian: Mathematical problems in elas-ticity and homogenization. Studies in Mathematics and its Applications, 26. North-Holland Publishing Co., Amsterdam, 1992.
[45] A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equa-tions. Appl. Math. Sci., vol. 44, Springer-Verlag, Berlin (1983)
[46] S. Racca: A viscosity-driven crack evolution. Adv. Calc. Var. 5 (2012), 433â483.
[47] F. Riva and L. Nardini: Existence and uniqueness of dynamic evolutions for a onedimensional debonding model with damping. Submitted for publication (2018). PreprintSISSA 28/2018/MATE.
[48] E. Serra and P. Tilli: Nonlinear wave equations as limits of convex minimizationproblems: proof of a conjecture by De Giorgi. Ann. of Math. 175 (2012), 1551â1574.
[49] E. Serra and P. Tilli: A minimization approach to hyperbolic Cauchy problems. J.Eur. Math. Soc. 18 (2016), 2019â2044.
[50] J. Simon: Compact sets in the space Lp(0, T ;B). Ann. Mat. Pura Appl. 146 (1987),65â96.
124 Bibliography
[51] L.I. Slepyan: Models and phenomena in fracture mechanics. Foundations of Engineer-ing Mechanics. Springer-Verlag, Berlin, 2002.
[52] E. Tasso: Weak formulation of elastodynamics in domains with growing cracks. Sub-mitted for publication (2018). Preprint SISSA 51/2018/MATE.