Rate-independent damage in thermo-viscoelastic materials with inertia
Transcript of Rate-independent damage in thermo-viscoelastic materials with inertia
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Rate-independent damagein thermo-viscoelastic materials with inertia
Giuliano Lazzaroni
SISSA, Trieste, Italy
STAMM 2014 at Poitiers
Joint work with R. Rossi (Brescia), M. Thomas (WIAS Berlin), and R. Toader (Udine)
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Damage in thermo-visco-elastodynamics
Aim: Existence result for evolutionary systems including • partial damageAim: Existence result for evolutionary systems including • viscosity & inertiaAim: Existence result for evolutionary systems including • thermal effects
8
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for u : damped equation of elastodynamics
for z : rate-independent, unidirectional flow rule
for ✓ : heat equation coupled in a consistent way
u = displacementz = damage (internal variable), z 2 [0, 1] ✓= temperature
⇢
z= 1 : sound materialz= 0 : most damaged state
Modelling: Frémond’s approach, Generalised Standard Materials
Difficulty: Interplay of rate-independent and rate-dependent phenomena (Roubícek)
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The model: 1. Thermo-viscoelasticity⌦⇢R3 bounded, Lipschitz, u(t, x)2R3, e(u)=ru+ruT
2 , z(t, x)2[0, 1], ✓(t, x)2R+
thermo-visco
elasticity: ⇢ u � div �(u) = F(t)
where �(u) =C e(u)
+ D e(u)� ✓B
(z)(z, ✓)
✓ � div�
K(z, ✓)r✓�
= |z|+⇥
D(z, ✓)e(u)�✓B⇤
: e(u) + H(t)
• Kelvin-Voigt rheology & inertia• Temperature changes produce additional stresses Thermal expansion term, coupling terms in the heat equation
Assumptions: C 2 C0,1(R;R3⇥3⇥3⇥3), D 2 C0(R⇥ R;R3⇥3⇥3⇥3)
Assumptions: C, D are bounded, symmetric, positive definite, uniformly in (z, ✓)Assumptions: C is nondecreasing: 8 z1 z2 C(z1)A : A C(z2)A : A
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The model: 1. Thermo-viscoelasticity⌦⇢R3 bounded, Lipschitz, u(t, x)2R3, e(u)=ru+ruT
2 , z(t, x)2[0, 1], ✓(t, x)2R+
thermo-
viscoelasticity: ⇢ u � div �(u) = F(t)
where �(u) =C e(u) + D e(u)
� ✓B
(z)(z, ✓)
✓ � div�
K(z, ✓)r✓�
= |z|+⇥
D(z, ✓)e(u)�✓B⇤
: e(u) + H(t)
• Kelvin-Voigt rheology & inertia
• Temperature changes produce additional stresses Thermal expansion term, coupling terms in the heat equation
Assumptions: C 2 C0,1(R;R3⇥3⇥3⇥3), D 2 C0(R⇥ R;R3⇥3⇥3⇥3)
Assumptions: C, D are bounded, symmetric, positive definite, uniformly in (z, ✓)Assumptions: C is nondecreasing: 8 z1 z2 C(z1)A : A C(z2)A : A
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The model: 1. Thermo-viscoelasticity⌦⇢R3 bounded, Lipschitz, u(t, x)2R3, e(u)=ru+ruT
2 , z(t, x)2[0, 1], ✓(t, x)2R+
thermo-
viscoelasticity: ⇢ u � div �(u) = F(t)
where �(u) =C(z)e(u) + D(z, ✓)e(u)
� ✓B✓ � div
�
K(z, ✓)r✓�
= |z|+⇥
D(z, ✓)e(u)�✓B⇤
: e(u) + H(t)
• Kelvin-Voigt rheology & inertia
• Temperature changes produce additional stresses Thermal expansion term, coupling terms in the heat equation
Assumptions: C 2 C0,1(R;R3⇥3⇥3⇥3), D 2 C0(R⇥ R;R3⇥3⇥3⇥3)
Assumptions: C, D are bounded, symmetric, positive definite, uniformly in (z, ✓)Assumptions: C is nondecreasing: 8 z1 z2 C(z1)A : A C(z2)A : A
• Uniform ellipticity of C: Damage is partial• Uniform ellipticity of D: Dynamics is damped• Monotonicity of C: Increase of damage reduces the stored elastic energy
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The model: 1. Thermo-viscoelasticity⌦⇢R3 bounded, Lipschitz, u(t, x)2R3, e(u)=ru+ruT
2 , z(t, x)2[0, 1], ✓(t, x)2R+
thermo-viscoelasticity: ⇢ u � div �(u) = F(t)
where �(u) =C(z)e(u) + D(z, ✓)e(u)� ✓B✓ � div
�
K(z, ✓)r✓�
= |z|+⇥
D(z, ✓)e(u)�✓B⇤
: e(u) + H(t)
• Kelvin-Voigt rheology & inertia• Temperature changes produce additional stresses Thermal expansion term, coupling terms in the heat equation
Assumptions: C 2 C0,1(R;R3⇥3⇥3⇥3), D 2 C0(R⇥ R;R3⇥3⇥3⇥3)
Assumptions: C, D are bounded, symmetric, positive definite, uniformly in (z, ✓)Assumptions: C is nondecreasing: 8 z1 z2 C(z1)A : A C(z2)A : AAssumptions: B 2 R3⇥3 constant, symmetricAssumptions: K 2 C0(R⇥ R;R3⇥3), symmetric, and 9 2 (1, 5
3 ) such thatAssumptions: 8 z, ✓, ⇠ : c1(|✓|+1) |⇠|2 K(z, ✓)⇠ · ⇠ c2(|✓|+1) |⇠|2
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Some remarks on the heat equation
cv(✓)| {z }
⌘1
✓ � div�
K(z, ✓)r✓�
= |z|+⇥
D(z, ✓)e(u)�✓B⇤
: e(u) + H(t)
• K = heat conductivity with subquadratic growth condition:
9 2 (1, 53 ) 8 z, ✓, ⇠ : c1(|✓|+1) |⇠|2 K(z, ✓)⇠ · ⇠ c2(|✓|+1) |⇠|2
? Borrowed from Rocca-Rossi, in the spirit of Feireisl-Petzeltová-Rocca? Needed in the proof of a-priori estimates? Applies to polymers such as PMMA
• cv(✓) = heat capacity, constant because of constitutive assumptions
? Allows us to avoid a so-called enthalpy transformation? Compatible with regimes where ✓ � ✓D (Debye temperature)? Indeed we show that temperature is > of a (tunable) threshold for suitable data
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Some remarks on the heat equation
cv(✓)| {z }
⌘1
✓ � div�
K(z, ✓)r✓�
= |z|+⇥
D(z, ✓)e(u)�✓B⇤
: e(u) + H(t)
• K = heat conductivity with subquadratic growth condition:
9 2 (1, 53 ) 8 z, ✓, ⇠ : c1(|✓|+1) |⇠|2 K(z, ✓)⇠ · ⇠ c2(|✓|+1) |⇠|2
? Borrowed from Rocca-Rossi, in the spirit of Feireisl-Petzeltová-Rocca? Needed in the proof of a-priori estimates? Applies to polymers such as PMMA
• cv(✓) = heat capacity, constant because of constitutive assumptions
? Allows us to avoid a so-called enthalpy transformation? Compatible with regimes where ✓ � ✓D (Debye temperature)? Indeed we show that temperature is > of a (tunable) threshold for suitable data
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The model: 2. Damage flow rule
@R1(z) + @zG(z,rz)� div�
D⇠G(z,rz)�
+ 12C
0(z)e(u) : e(u) 3 0
Assumptions: R1 is a 1-homogeneous dissipation potential of the form
R1(v) :=
(
|v| if v 0+1 otherwise
• Rate-independent: energy necessary to damage is independent of velocity• Enforces monotonicity (z 0): unidirectional damage, no healing
Assumptions: G 2 C0(R⇥R3;R[{1}), G(0, 0) = 0Assumptions: G(z, ⇠)<1 ) z2 [0, 1]Assumptions: G(z, ·) is convex and has q-growth from above and below, q > 1
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The model: 2. Damage flow rule
@R1(z) + @zG(z,rz)� div�
D⇠G(z,rz)�
+ 12C
0(z)e(u) : e(u) 3 0
Assumptions: R1 is a 1-homogeneous dissipation potential of the form
R1(v) :=
(
|v| if v 0+1 otherwise
Assumptions: G 2 C0(R⇥R3;R[{1}), G(0, 0) = 0Assumptions: G(z, ⇠)<1 ) z2 [0, 1]Assumptions: G(z, ·) is convex and has q-growth from above and below, q > 1
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The model: 2. Damage flow rule
@R1(z) + @zG(z,rz)� div�
D⇠G(z,rz)�
+ 12C
0(z)e(u) : e(u) 3 0
Assumptions: R1 is a 1-homogeneous dissipation potential of the form
R1(v) :=
(
|v| if v 0+1 otherwise
Assumptions: G 2 C0(R⇥R3;R[{1}), G(0, 0) = 0Assumptions: G(z, ⇠)<1 ) z2 [0, 1]Assumptions: G(z, ·) is convex and has q-growth from above and below, q > 1
Semistability
z(t) 2 argmin⇣
n
E(t, u(t), ⇣) +Z
⌦R1(⇣�z(t)) dx
o
where E(t, u, ⇣) :=Z
⌦
h
12C(⇣)e(u) : e(u) + G(⇣,r⇣)� F(t) u
i
dx
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Example: Ambrosio-Tortorelli modelPhase-field approximation
Regularisation of a sharp crack:The internal variable interpolates continuouslybetween sound and fractured material
Figure: "Phase field order parameter" by Cenna - Own work. Licensed under Public domain viaWikimedia Commons http://commons.wikimedia.org/wiki/File:Phase_field_order_parameter.jpg
Prototype of the time-discrete problem: given u, z0
, min0⇣z0
⇢
Z
⌦
12C(⇣)e(u) : e(u) dx +
Z
⌦G(⇣,r⇣) dx +
Z
⌦R1(⇣�z0) dx
�
Setting C(⇣) := (⇣2+�) I with � > 0, and G(⇣,r⇣) := |r⇣|2 + 12 (1+⇣2) + I[0,1](⇣)
, min0⇣z0
⇢
Z
⌦( 1
2 (⇣2+�) |e(u)|2 dx +
Z
⌦
12 (1�⇣)2 dx +
Z
⌦|r⇣|2 dx
�
Ambrosio-Tortorelli (without passage to brittle limit)
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The PDE system
Problem: Given F 2 H1(0, T; L2(⌦;R3)), H 2 L2(0, T; L2(⌦)), prove existence for8
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⇢ u � div⇣
D(z, ✓)e(u) + C(z)e(u)� ✓B⌘
= F(t)
@R1(z) + @zG(z,rz)� div�
D⇠G(z,rz)�
+ 12C
0(z)e(u) : e(u) 3 0
✓ � div�
K(z, ✓)r✓�
= R1(z) +⇥
D(z, ✓)e(u)�✓B⇤
: e(u) + H(t)
+ initial Cauchy conditons on u(0), u(0), z(0), ✓(0)
+ natural boundary conditions: Dirichlet/Neumann on u and Neumann on ✓
+ (all boundary conditions will be homogeneous in this talk)
Challenges: • Interplay of rate-independent| {z }
damage
and of rate-dependent| {z }
viscosity, heat
processes
Challenges: • Highly nonlinear coupling of the systemChallenges: • Dissipation rates as heat sources, L1 right-hand side in heat equation
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Literature
(Incomplete) List of related works:
• Roubícek: Coupling of rate-independent processes with viscosity, inertia, heat.We extend his ansatz to a unidirectional process, with D(z, ✓) nonconstant
• Feireisl-Petzeltová-Rocca and Rocca-Rossi: Treatment of the heat equation.They consider a rate-dependent setting for phase transitions and damage
• Larsen-Ortner-Süli: Dynamic evolution for unidirectional damage, isothermal.Ambrosio-Tortorelli setting with C(z) = D(z) = (z2 + �) eC(x)
• Vast literature on (fully) rate-independent processes, Mielke’s abstract theory.For damage: Mielke-Roubícek, Thomas-Mielke, Fiaschi-Knees-Stefanelli, . . .
• About rate-dependent damage (without inertia, with or without heat):Frémond, Bonetti, Bonfanti, Heinemann, Kraus, Nedjar, Schimperna, Segatti. . .
• Modelling and numerics, e.g. Miehe-Welschinger-Hofacker (inertia + heat)
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Literature
(Incomplete) List of related works:
• Roubícek: Coupling of rate-independent processes with viscosity, inertia, heat.We extend his ansatz to a unidirectional process, with D(z, ✓) nonconstant
• Feireisl-Petzeltová-Rocca and Rocca-Rossi: Treatment of the heat equation.They consider a rate-dependent setting for phase transitions and damage
• Larsen-Ortner-Süli: Dynamic evolution for unidirectional damage, isothermal.Ambrosio-Tortorelli setting with C(z) = D(z) = (z2 + �) eC(x)
• Vast literature on (fully) rate-independent processes, Mielke’s abstract theory.For damage: Mielke-Roubícek, Thomas-Mielke, Fiaschi-Knees-Stefanelli, . . .
• About rate-dependent damage (without inertia, with or without heat):Frémond, Bonetti, Bonfanti, Heinemann, Kraus, Nedjar, Schimperna, Segatti. . .
• Modelling and numerics, e.g. Miehe-Welschinger-Hofacker (inertia + heat)
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Existence result for the energetic formulationTheorem (L.-Rossi-Thomas-Toader):Given loading F , heat source H � 0 , and initial data u0 , u0 , z0 2 [0, 1] , ✓0 � ✓⇤ > 0 ,there exists an energetic solution (u, z, ✓) such that the initial conditions hold and
• unidirectionality and semistability: z(·, x) nonincreasing and for all t
8 ⇣ z(t) : E(t, u(t), z(t)) E(t, u(t), ⇣) +Z
⌦(z(t)�⇣) dx
where ⇣ 2 W1,q and z2 L1(W1,q)\ L1 \BV(L1)
• momentum equation in weak form for all t
• mechanical energy balance for all t
• heat equation in weak form for all t
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Existence result for the energetic formulationTheorem (L.-Rossi-Thomas-Toader):Given loading F , heat source H � 0 , and initial data u0 , u0 , z0 2 [0, 1] , ✓0 � ✓⇤ > 0 ,there exists an energetic solution (u, z, ✓) such that the initial conditions hold and
• unidirectionality and semistability: z(·, x) nonincreasing and for all t
8 ⇣ z(t) : E(t, u(t), z(t)) E(t, u(t), ⇣) +Z
⌦(z(t)�⇣) dx
• momentum equation in weak form for all t
� ⇢
Z t
0
Z
⌦u · v +
Z t
0
Z
⌦
⇣
D(z, ✓)e(u) + C(z)e(u)� ✓B⌘
: e(v)
= ⇢
Z
⌦u0 · v(0)� ⇢
Z
⌦u(t) · v(t) +
Z t
0
Z
⌦F v
8 v2 L2(H1)\W1,1(L2), where u2H1(H1D)\W1,1(L2)
• mechanical energy balance for all t
• heat equation in weak form for all t
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Existence result for the energetic formulationTheorem (L.-Rossi-Thomas-Toader):Given loading F , heat source H � 0 , and initial data u0 , u0 , z0 2 [0, 1] , ✓0 � ✓⇤ > 0 ,there exists an energetic solution (u, z, ✓) such that the initial conditions hold and
• unidirectionality and semistability: z(·, x) nonincreasing and for all t
8 ⇣ z(t) : E(t, u(t), z(t)) E(t, u(t), ⇣) +Z
⌦(z(t)�⇣) dx
• momentum equation in weak form for all t
• mechanical energy balance for all t
⇢2
Z
⌦|u(t)|2 +
Z
⌦(z0�z(t)) + E(t, u(t), z(t)) +
Z t
0
Z
⌦
⇥
D(z, ✓)e(u)�✓B⇤
: e(u)
= ⇢2
Z
⌦|u0|2 + E(0, u0, z(0))�
Z t
0
Z
⌦F u
• heat equation in weak form for all t
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Existence result for the energetic formulationTheorem (L.-Rossi-Thomas-Toader):Given loading F , heat source H � 0 , and initial data u0 , u0 , z0 2 [0, 1] , ✓0 � ✓⇤ > 0 ,there exists an energetic solution (u, z, ✓) such that the initial conditions hold and
• unidirectionality and semistability: z(·, x) nonincreasing and for all t
8 ⇣ z(t) : E(t, u(t), z(t)) E(t, u(t), ⇣) +Z
⌦(z(t)�⇣) dx
• momentum equation in weak form for all t
• mechanical energy balance for all t
• heat equation in weak form for all t
h✓(t), ⌘(t)i �Z
⌦✓0 ⌘(0)�
Z t
0
Z
⌦✓ ⌘ +
Z t
0
Z
⌦K(✓, z)r✓·r⌘
=
Z t
0
Z
⌦
⇥
D(z, ✓)e(u)�✓B⇤
: e(u) ⌘ +
Z t
0
Z
⌦⌘ |z|+
Z t
0
Z
⌦H ⌘
8 ⌘ 2H1(L2)\C0(W2,3+�), where ✓2 L2(H1)\ L1(L1)\BV((W2,3+�)⇤)
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Positivity of the temperature
• Under the previous assumptions (H � 0 , ✓0 � ✓⇤ > 0), there exists e✓ > 0 with
✓(t, x) � e✓ > 0 for a.a. (t, x)
• Furthermore, if in addition
9H⇤ > 0 : H(t, x) � H⇤ for a.a. (t, x) and ✓0(x) �p
H⇤/c for a.a. x
where c is a constant depending on B and D, then
✓(t, x) � maxn
e✓,p
H⇤/co
for a.a. (t, x)
We can tune the constant in such a way that ✓ � ✓D (Debye model)
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Strategy: 1. Time discretisation0 = t0
n < · · · < tnn = T with tk
n � tk�1n = T
n =: ⌧n
1. zkn 2 argmin
⇢
E(tkn, uk�1
n , z) +Z
⌦(zk�1
n �z) dx : z zk�1n
�
2. Solve the system in the unknowns (ukn, ✓
kn) :
⇢
Z
⌦
ukn�2uk�1
n +uk�2n
⌧ 2n
· v +Z
⌦⌧n|e(uk
n)|��2e(ukn) : e(v)
+
Z
⌦
⇣
D(zk�1n , ✓k�1
n )e⇣
ukn�uk�1
n⌧n
⌘
+ C(zkn)e(u
kn)� ✓k
n B⌘
: e(v) =Z
⌦Fk
n v
Z
⌦
✓kn�✓k�1
n⌧n
⌘ +
Z
⌦K(zk
n, ✓kn)r✓k
n ·r⌘ �Z
⌦
zk�1n �zk
n⌧n
⌘
=
Z
⌦
h
D(zk�1n , ✓k�1
n )e⇣
ukn�uk�1
n⌧n
⌘
� ✓kn B
i
: e⇣
ukn�uk�1
n⌧n
⌘
⌘ +
Z
⌦Hk
n ⌘
• Minimisation decoupled from the system of momentum & heat equation• Regularisation by �-Laplacian (� > 4), disappearing as n ! 1• Existence by direct method and theory of pseudomonotone operators
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Strategy: 2. Time-discrete to continuous
• Discrete energy inequality A priori estimates
• Enhanced estimates for ✓ by positivity (cf. Feireisl-Petzeltová-Rocca)
• Convergence by compactness and Helly’s selection principle
• Semistability in the limit via “mutual recovery sequence” (cf. Thomas-Mielke)
• Passage to the limit in the discrete energetic formulation
(cf. Mielke’s scheme for rate-independent processes(cf. and Roubícek’s one for coupling with rate-dependent effects)
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Vanishing viscosity & inertiaVanishing viscosity & inertia (cf. Roubícek, Dal Maso-Scala, isothermal)
In the limit we seek a rate-independent & temperature-independent model
Assumptions: Homogeneous Dirichlet problem, q-growth of G with q>3 , H" ' " ,
Assumptions: and K"(z, ✓) = 1"2 K(z, ✓) Heat conducted with infinite speed
8
>
>
>
<
>
>
>
:
"2⇢u" � div⇣
"D(z", ✓")e(u") + C(z")e(u")� ✓" B⌘
= F"
@R1(z") + @zG(z",rz")� div(D⇠G(z",rz")) + 12C
0(z")e(u") : e(u") 3 0
"✓" �
1"2
div(K"(z", ✓")r✓") = "R1(z") +⇥
"2D(z", ✓")e(u")� "✓" B⇤
: e(u") + H"
In the limit " ! 0 : Unidirectionality, semistability, � div�
C(z)e(u)�
= F(t) , and
E(t, u(t), z(t)) +Z
⌦(z(s)�z(t)) dx E(s, u(s), z(s)) +
Z t
s@tE(r, u(r), z(r)) dr
for all t and a.a. s 2 (0, t). Notion of local solution (Mielke, Roubícek, Stefanelli)
Moreover, ✓" * ✓? depending on time, constant in space
Giuliano Lazzaroni (SISSA) Rate-independent damage in thermo-viscoelastic materials 13 / 14
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Vanishing viscosity & inertiaVanishing viscosity & inertia (cf. Roubícek, Dal Maso-Scala, isothermal)
In the limit we seek a rate-independent & temperature-independent model
Assumptions: Homogeneous Dirichlet problem, q-growth of G with q>3 , H" ' " ,
Assumptions: and K"(z, ✓) = 1"2 K(z, ✓) Heat conducted with infinite speed
8
>
>
>
<
>
>
>
:
"2⇢u" � div⇣
"D(z", ✓")e(u") + C(z")e(u")� ✓" B⌘
= F"
@R1(z") + @zG(z",rz")� div(D⇠G(z",rz")) + 12C
0(z")e(u") : e(u") 3 0
"✓" � 1"2 div(K
"
(z", ✓")r✓") = "R1(z") +⇥
"2D(z", ✓")e(u")� "✓" B⇤
: e(u") + H"
In the limit " ! 0 : Unidirectionality, semistability, � div�
C(z)e(u)�
= F(t) , and
E(t, u(t), z(t)) +Z
⌦(z(s)�z(t)) dx E(s, u(s), z(s)) +
Z t
s@tE(r, u(r), z(r)) dr
for all t and a.a. s 2 (0, t). Notion of local solution (Mielke, Roubícek, Stefanelli)
Moreover, ✓" * ✓? depending on time, constant in space
Giuliano Lazzaroni (SISSA) Rate-independent damage in thermo-viscoelastic materials 13 / 14
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Vanishing viscosity & inertiaVanishing viscosity & inertia (cf. Roubícek, Dal Maso-Scala, isothermal)
In the limit we seek a rate-independent & temperature-independent model
Assumptions: Homogeneous Dirichlet problem, q-growth of G with q>3 , H" ' " ,
Assumptions: and K"(z, ✓) = 1"2 K(z, ✓) Heat conducted with infinite speed
8
>
>
>
<
>
>
>
:
"2⇢u" � div⇣
"D(z", ✓")e(u") + C(z")e(u")� ✓" B⌘
= F"
@R1(z") + @zG(z",rz")� div(D⇠G(z",rz")) + 12C
0(z")e(u") : e(u") 3 0
"✓" � 1"2 div(K
"
(z", ✓")r✓") = "R1(z") +⇥
"2D(z", ✓")e(u")� "✓" B⇤
: e(u") + H"
In the limit " ! 0 : Unidirectionality, semistability, � div�
C(z)e(u)�
= F(t) , and
E(t, u(t), z(t)) +Z
⌦(z(s)�z(t)) dx E(s, u(s), z(s)) +
Z t
s@tE(r, u(r), z(r)) dr
for all t and a.a. s 2 (0, t). Notion of local solution (Mielke, Roubícek, Stefanelli)
Moreover, ✓" * ✓? depending on time, constant in space
Giuliano Lazzaroni (SISSA) Rate-independent damage in thermo-viscoelastic materials 13 / 14
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Conclusion: Results and open problems
• Existence of evolutions for rate-independent, unidirectional, partial damagewith viscosity, inertia, and thermal effects basing on time-discretisation
• We can account for nonhomogeneous, time-dependent, Neumann conditions
• We cannot take nonhomogeneous, time-dependent, Dirichlet conditions for u(There are problems if D is not constant)
• We consider the limit for vanishing viscosity & inertia if K" ' "�2,obtaining a temperature-independent model
• Open problem: Vanishing viscosity & inertia without assuming K" ' "�2
• Open problem: Passage to the limit from Ambrosio-Tortorelli to a sharp crack
Giuliano Lazzaroni (SISSA) Rate-independent damage in thermo-viscoelastic materials 14 / 14