Employing the Discrete Fourier Transform in the Analysis ...

International Journal for Multiscale Computational Engineering, 6(5)435–449(2008)

Employing the Discrete Fourier Transform in theAnalysis of Multiscale Problems

Michael RyvkinSchool of Mechanical Engineering, Tel Aviv University, Ramat Aviv 69978, Israel

ABSTRACT

The idea of employing the discrete Fourier transform casted as the representative cell methodfor the solution of multiscale problems is illustrated. Its application in combination withanalytical (structural mechanics methods, Wiener-Hopf method, integral transform methods)and numerical (finite element method, higher-order theory) methods is demonstrated. Bothcases of 1-D and 2-D translational symmetry are addressed. In particular, the problems forlayered, cellular, and perforated materials with and without flaws (cracks) are considered.The method is shown to be a convenient and universal analysis tool. Its numerical efficiencyallowed us to solve optimization problems characterized by multiple reanalysis.

KEYWORDS

discrete Fourier transform, periodic microstructure, representative cell, flaw, crack

*Address all correspondence to [email protected]

1543-1649/06/$35.00 © 2008 by Begell House, Inc. 435

436 MICHAEL RYVKIN

1. INTRODUCTION

In many cases, the multiscale problems arise in theanalysis of elastic systems possessing periodicity ingeometric and elastic properties. Typical examplesof such systems on a macroscale are large spacestructures or two-dimensional grids [1]. On a muchsmaller scale, there is a large family of compositematerials with periodic microstructure, includingfiber-reinforced and cellular materials [2]. It appearsthat in this situation, it is possible to avoid the ho-mogenization procedure and to get the result usingthe Discrete Fourier Transform (DFT). The employ-ing of this transform in the analysis of the periodicsystems has long been known. A rather comprehen-sive review of the works on this subject is given in[3]. It is found that most of the works are eitherfrom the field of dynamics or related to the buck-ling analysis in the framework of the Bloch waveapproach. Among the recent publications, it is per-tinent to note the works on the fracture in lattices(e.g., [4]), the Bloch wave study of pattern transfor-mations in periodic materials [5], and the analysis offinite repetitive structures [6].

In the present article, the application of the rep-resentative cell method suggested in [7] for the so-lution of multiscale problems is illustrated. Thismethod is based on the DFT and is found to bea very convenient and universal mathematical toolthat can be easily employed for the analysis of pe-riodic structures in combination with analytical aswell as numerical methods. In the next section, theidea of the method is illustrated by means of a triv-ial example. In Section 3, the applications for thestructures and the layered materials posessing 1-D translation symmetry are considered, whereas inSection 4, the case of 2-D symmetry is addressed. Inboth cases, special attention is given to the flaw ex-istence violating the periodicity. In the final section,several concluding remarks are drawn.

2. ILLUSTRATIVE EXAMPLE

It is convenient to present the idea of the representa-tive cell approach by means of the known structuralmechanics problem on bending of an infinite peri-odically supported beam (Fig. 1). The beam has uni-form bending stiffness EJ and, in the global coordi-nate system X,Y , occupies the region ∞ < X < ∞.The system periodicity in the geometric and elas-

tic properties is characterized by the vector b of 1-D translational symmetry defining the local scale ofthe problem. The beam is loaded by the two trans-verse forces Q,S acting in two neighboring spans atthe distances tQ and tS from the left support, respec-tively. Note that since the problem is linear, it wassufficient to consider only one force and then obtainthe multiple force solution by superposition. Thetwo forces are considered in the example simultane-ously to emphasize the fact that the loading in therepresentative cell problem, which will be obtainedlater, is complex valued.

The infinite domain is considered as an assem-blage of an infinite number of repetitive cells k =0,±1,±2, ... and local systems of coordinates −L ≤x ≤ L are introduced in each cell in an identicalmanner as shown in Fig. 1(b). Consequently, theproblem of deriving the bending deflections Y (X)in the global coordinates ∞ < X < ∞ is re-placed by the problem of displacements yk(x), k =0,±1,±2, ... in local coordinate systems, which isformulated as follows.

(i) Within the cell, the deflections obey the fourth-order differential equation

EIyIVk = Pk , k = 0,±1,±2, ... (1)

where

Pk = Qδk0δ(x− tQ) + Sδk1δ(x− tS) (2)

Here δij denotes the Kronecker symbol andδ(x) is the Delta function.

(ii) At the support point x = 0, which can be re-ferred to as the inner boundary Γ0, the bound-ary conditions are

yk(±0) = 0 (3)θk(+0) = θk(−0) , k = 0,±1,±2, ... (4)

Mk(+0) = Mk(−0) (5)

where θk(x) = y′k(x) is the rotation angle andMk(x) = −EJy′′k (x) is the bending moment.

(iii) At the outer boundaries Γ± : x = ±L, the conti-nuity conditions between the neighboring cellsmust be fulfilled. Defining the generalized dis-placement and force vectors as uk = yk,θk

International Journal for Multiscale Computational Engineering

EMPLOYING THE DISCRETE FOURIER TRANSFORM IN THE ANALYSIS OF MULTISCALE PROBLEMS 437

t Q

y

x

y

x

y

x

y

x

P*

*f +

*u−

y

x

Y

2L

X

Q S

t S

k=0 k=1 k=2 k=3k=−1k=−2

*f _

*u+

(a)

(b)

(c)

Fig. 1abc

FIGURE 1. (a) Infinite periodically supported beam subjected to local loading, (b) the representative cell scheme,and (c) the representative cell problem

and fk = Mk,Vk, respectively, where Vk =−EJy′′′k (x) is the shear force, we have

u+k = u−k+1 (6)

f+k = f−k+1 , k = 0,±1,±2, ... (7)

The superscript (±) denotes the correspondingboundary values, for example, u+

k ≡ yk(L), θ(L)and u−k ≡ yk(−L),θ(−L).

The formulated problem for the infinite beam isreduced to the problem for a finite beam of thelength 2L being the representative cell of the prob-lem (see Fig. 1(c)) by means of the DFT:

g∗(ϕ)=∞∑

k=−∞gk exp(iϕk) (8)

gk=12π

∫ π

−π

g∗ exp(−iϕk)dϕ, −π<ϕ<π (9)

Application of Eq. (8) to Eqs. (1)–(7) yields

EIyIV∗ =P∗, P∗=Qδ(x−tQ)+Sδ(x−tS)exp(iϕ) (10)

y∗(±0)=0 (11)θ∗(+0)=θ∗(−0) (12)

M∗(+0)=M∗(−0) (13)

Volume 6, Number 5, 2008

438 MICHAEL RYVKIN

u+∗ =exp(−iϕ)u−∗ (14)

f+∗ =exp(−iϕ)f−∗ (15)

It is important to note that the reduction of thescale of the consideration in the initial problem tothe scale of geometric periodicity is carried out in-dependently of the scale of the applied loading(the distance between the forces Q and S). Theobtained representative cell problem is formulatedwith respect to the complex-valued Fourier trans-form y∗(x,ϕ) of the actual deflections defined overthe actual geometric object—the representative cell.The opposite sides of the cell are connected by thespecific Born–Von Karman–type boundary condi-tions with a complex multiplier. However, the dif-ferential operator in Eq. (10) acting in the transformspace remains the same as in the initial bendingequation. Therefore the general solution has thestandard form (e.g., [8])

yj∗(x, ϕ)= Cj

1+Cj2x+Cj

3x2+Cj4x3

+16δ2j [Q(x−tQ)3+S(x−tS)3], j =1, 2

(16)

where the values of superscript j denote the left andright halves of the representative cell

y∗(x, ϕ) =

y1∗(x, ϕ) −L ≤ x ≤ 0

y2∗(x, ϕ) 0 ≤ x ≤ L

Eight constants of integration Cjm, j = 1, 2; m =

1, 2, 3, 4 are derived from the boundary conditions

(11)–(15) and, consequently, will represent the func-tions of the transform parameter ϕ. The actual de-flections in any cell are evaluated by the use of theinverse transformation formula (9):

yk(x) =12π

∫ π

−π

y∗(x,ϕ) exp(−ikϕ)dϕ (17)

3. SYSTEMS WITH 1-D TRANSLATIONALSYMMETRY

The idea illustrated in the previous section can beapplied to any continuous or discrete elastic systempossessing 1-D translational symmetry. An exam-ple of such an application is a problem on a peri-odic system of oblique stringers reinforcing an infi-nite elastic plate of thickness h solved in [9]. Thenonperiodic stress state is generated by the pointforce P applied to one of the stringers. In this case,the representative cell is an elastic domain—a stripof thickness 2a with a single stringer (see Fig. 2).However, the corresponding boundary value prob-lem preserves the general form (10)–(15) and con-sists of the following.

(i) The field equations within the cell with re-spect to the displacement transforms u∗(x, y) =u∗(x, y), v∗(x, y),

Lu∗ = 0 (18)

−Γ

οΓ

P

2a

xy

(a) (b)

Γ+

Fig. 2ab

FIGURE 2. (a) An infinite plate reinforced by a periodic system of stringers and (b) the representative cell of theproblem



where L is the Lame differential operator of theplane elasticity.

(ii) The boundary conditions at the inner boundaryΓ0 : (y = 0,−∞ < x < 0),

u∗(x,−0)=u∗(x, +0) (19)σy∗(x, +0)=σy∗(x,−0) (20)

D∂2u∗(x, 0)

∂x2=h[τ∗(x,−0)−τ∗(x, +0)] (21)

h

∫ ∞

0

[τ∗(x, +0)−τ∗(x,−0)]dx = P (22)

where σy∗, τ∗(x, y) ≡ τxy∗(x, y) are the trans-forms of the normal and shear stresses, re-spectively, and D is the axial stiffness of thestringers.

(iii) The Born–Von Karman–type boundary con-ditions for the displacements and tractionsf∗(x, y) ≡ σy∗(x, y), τxy∗(x, y) transformsat the corresponding points on the periodicboundaries Γ± : y = ±a:

u+∗ = exp(−iϕ)u−∗ (23)

f+∗ = exp(−iϕ)f−∗ (24)

The representative cell problem (18)–(24) is nottrivial; its analytical solution required the em-ployment of the Wiener-Hopf technique, buthaving it in hand, one easily obtains the com-ponents of the stress strain field in any point ofthe plate using the inverse DFT integration:

uk(x, y) =12π

∫ π

−π

u∗(x, y) exp(−iϕk)dϕ (25)

In several cases, the analytical solution for therepresentative cell domain having the form ofa strip can be obtained by the Muskhelishvilimethod. This fact allowed us to derive a closed-form solution of a problem of an incompletecontact of two dissimilar elastic half-planes[10].

Another example of an elastic system posess-ing 1-D translational symmetry is an infinite truss,shown in Fig. 3(a). The employment of the repre-sentative cell method in this case of a structure isdemonstrated in [11]. Here the representative cellconsists of eight rods (Fig. 3(b)). In the transformspace, the stresses in rods are defined by the dis-placements of six nodal points, indicated in the fig-ure u∗ = um∗, vm∗, m = 1, 2, .., 6, which can beseparated to the inner nodes u(in)

∗ for m = 1, 4, thenodes u−∗ for m = 2, 3 on the boundary Γ−, and thenodes u+

∗ for m = 5, 6 on the boundary Γ+. The rep-resentative cell problem in this case is also similarto the one presented in the illustrative example. Itincludes the following

(i) The cell equilibrium equation

Ku∗ = P ∗ + F ∗ + R∗ (26)

where K is the stiffness matrix of the cell cor-responding to the 12 nodal degrees of freedom,

1

2 3

4

65v

u

(a)

(b)

Fig 3abFIGURE 3. (a) An infinite truss and (b) its representative cell


440 MICHAEL RYVKIN

P ∗, R∗, and F ∗ ≡ F−∗ ,F +

∗ , are transforms ofthe external forces, the forces acting at the sup-port point, and the forces at the contact betweenthe neighboring cells, respectively.

(ii) The inner boundary conditions at the supportpoint

u1∗, v1∗ = 0, 0 (27)

(iii) The Born–Von Karman–type boundary condi-tions at the periodic boundaries relating the in-terface force and the displacements transforms

u+∗ = exp(−iϕ)u−∗ (28)

F +∗ = exp(−iϕ)F−

∗ (29)

After solving the cell problem by the standardstructural mechanics methods, one can, as inthe previous problems, return to the actualspace by the inverse transformation (9).

In the presented examples, the solution of therepresentative cell problem was obtained analyti-cally. This is possible for a rather limited class ofproblems; therefore the question of applicability ofthe approach in the case when only a numerical so-lution of the cell problem is available is of large im-portance. The positive answer to this question wasprovided by [12], where the representative cell tech-nique was employed in combination with the finiteelement method. In this work the harmonic prob-lem corresponding to the case of an antiplane defor-mation was addressed. The strong form of the rep-resentative cell problem for the displacement trans-form u∗(x, ϕ) is given by the equations

∆u∗(x, ϕ)=0, x∈Ω (30)u∗(x, ϕ)=U01∗(x), x∈Γ01 (31)

µ∇u∗(x, ϕ)·n=τ0∗(x), x∈Γ02 (32)exp(−iϕ)u∗(x−,ϕ)−u∗(x+, ϕ)=0, x±∈Γ± (33)

exp(−iϕ)∇u∗(x−, ϕ)·n−∇u∗(x+, ϕ)·n=0, x±∈Γ± (34)

Here Ω is the domain occupied by the representa-tive cell, x is the position vector in the local coordi-nate system, µ is the shear modulus of the material,Γ± are the periodic boundaries, Γ01 and Γ02 are theboundaries where the displacements U01∗ and thestress τ0∗ transforms are known, and vector n is thenormal to the corresponding boundary. In terms ofthe adopted notation for the equations forming the

representative cell problem, the harmonic Eq. (30)is of the type (i), the boundary conditions (31) and(32) are of the type (ii), and finally, the conditions atthe periodic boundaries (33) and (34) are of the type(iii). The latter conditions prevent the direct use of astandard commercial finite element code, and cer-tain adjustments are to be done. The weak formof this problem is formulated in a routine way, andthe only limitation to be introduced at the meshingstage is the identical discretization of the periodicboundaries, namely, there is one-to-one correspon-dence between the nodes at these boundaries andthe coordinates of each pair x−i , x+

j satisfy the rela-tion

x+j = x−i + b (35)

where b is the vector defining the system transla-tional symmetry. After repeating the solution proce-dure for several values of ϕ from the region (−π, π),one can find the sought function u(x) by the inversetransform numerical integration (9). The issue of therequired number of integration points was exam-ined in [3] where the finite element solutions for sev-eral more complicated elasticity problems were ob-tained. In particular, for the case of a plain problemfor an infinite strip, a comparison with the closedform analytical solution showed that it is sufficientto use nine Gauss integration points to provide 2%accuracy.

In many practically important cases, the multi-scale problem in periodically layered materials ap-pears as a result of the existence of a flaw (a crack)violating the material periodicity. For this kind ofproblem, the direct employment of the representa-tive cell method is impossible, and it can be usedin combination with the Green functions disloca-tion approach suggested in [13]. This combinedtechnique was employed for the first time in [14],where the problem on an antiplane deformation ofa bimaterial composite with a finite length inter-face crack subjected to uniform loading p was con-sidered (Figs. 4(a) and 4(b)). The body is dividedinto bilayered cells k = 0,±1,±2, .. and the crackis viewed as distributed dislocations with some un-known density f(t), 0 < t < a. Consequently, thedisplacements are presented by the use of the Greenfunction of the problem wk(x, y, t):

wk(x, y) =∫ a

0

wk(x, y, t)dt (36)



2a

h1

h2

µ 1

µ 2

(a) (b)

ln( )ln( )µ /µ 2 1

p

(c)

FIGURE 4. (a) A periodically layered composite with an interface crack, (b) its representative cell, and (c) the resultsfor the stress intensity factor

This function is given by the solution of an auxiliaryproblem on the same body without the crack andwith a single dislocation at the point x = t, y = 0 incell number 0:

∂

∂x[wk(x, +0)−wk(x,−0)]=f(t)δ(x−t)δok

0 < x < ∞, k = 0,±1,±2, ...

(37)

Since only the right hand part in the latter conditiondepends on k, the formulation of the representa-tive cell problem for the Green function transform isstraightforward. The final expression has the form

of a double integral of the Fourier and Laplace in-verse transforms. The dislocation density is definedthen from the numerical solution of the singular in-tegral equation expressing the stress conditions atthe crack faces. It should be emphasized that thesolution of the initial problem is obtained withoutany homogenization procedure. It provides all thedetails of the local stress distribution, in particular,the dependence of the stress intensity factor on theshear moduli ratio of the materials, which is de-picted in Fig. 4(c).


442 MICHAEL RYVKIN

Using the same strategy allowed us to solve alsomore complicated plane problems on finite cracksin periodically layered materials [15–17]. The ideaof further extension of the approach to the studyof semi-infinite cracks in periodically layered com-posites and composites of finite thickness was out-lined in [18] and implemented in [19] and [20]. Themethod hinges on the introduction of an unknownjump in the stresses or displacements along thesemi-infinite or infinite boundary, which then areadjusted to satisfy the given boundary conditions.

4. SYSTEMS WITH 2-D TRANSLATIONALSYMMETRY

4.1 Periodic Materials under Local Loading

The extension of the presented approach to the anal-ysis of elastic systems with 2-D translational sym-metry is straightforward. In this case, the represen-tative cell has the form of a parallelogram and con-tacts the neighboring ones through the two couplesof periodic boundaries. Consequently, the reductionof the modeling scale to the cell size is carried out bymeans of the double DFT:

g∗(ϕ1, ϕ2)=∑∞

k1=−∞∑∞

k2=−∞ gk1k2

×exp[i(k1ϕ1 + k2ϕ2)],

−π < ϕj < π, j = 1, 2

(38)

gk1k2 =1

4π2

∫ π

−π

∫ π

−π

g∗(ϕ1, ϕ2)

× exp[−i(k1ϕ1 + k2ϕ2)]dϕ1dϕ2

(39)

and the two groups of Born–Von Karman–typeboundary conditions, including the factors exp(iϕ1)and exp(iϕ2), are present.

The analysis of the infinite periodic grillage ar-rays presented in [21, 22] gives an example of em-ploying the approach for the case of 2-D periodicstructures rested on a periodic support array andsubjected to a transverse loading.

The next step in the approach development wasto consider 2-D periodic unsupported elastic sys-tems subjected to self-equilibrated loading. This al-lowed us to address the important class of problemsrelated to the local effects in periodic materials. Thefirst attempt was made by M. Ryvkin and E. Tad-mor (unpublished manuscript, 2001) who investi-gated the response of the truss material with trian-

gular cells to the action of force dipoles. Local ther-momechanical effects in continuous material–a uni-directional fiber reinforced composite–were studiedin [23]. An infinite elastic plane with the repetitivepattern appears at the cross section perpendicularto the fiber direction, and the representative cell isa rectangle −L/2 < x1 < L/2, −H/2 < x2 < H/2with a circular inclusion (see Fig. 5(a)). In particular,the problem on the stress state induced by the lo-cal heating was considered. The representative cellproblem formulation includes the following.

1. The field equations consisting of the equilib-rium equations

σkl,l∗ = 0 , k, l = 1, 2 (40)

and constitutive relations

σkl∗ = Cklmnεmn∗ − αkl∆T (41)

2. The Born–Von Karman–type conditions at twopairs of the periodic boundaries

uj∗(L/2, x2)=exp(−iϕ1)uj∗(−L/2, x2) (42)σ1j∗(L/2, x2)=exp(−iϕ1)σ1j∗(−L/2, x2) (43)

−H/2 < x2 < H/2uj∗(x1,H/2)=exp(−iϕ2)uj∗(x1,−H/2)

σ2j∗(x1,H/2)=exp(−iϕ2)σ2j∗(x1,−H/2) (44)−L/2 < x1 < L/2 (45)

where j = 1, 2.

Here uj∗, σkl∗, εmn∗ are the transforms of thedisplacements, stresses, and strains, respectively;Cklmn and αkl are the elements of the stiffness andthermal stress tensors of the materials occupying thecell; and ∆T is the transform of temperature devia-tion, which is assumed to be applied within the cell(0,0). In addition, the inner boundary conditions ex-pressing the perfect bonding between the fiber andthe matrix phases take place.

The obtained problem, clearly, allows only nu-merical solution. The finite element method is nota unique option, and in [23] the combination ofthe representative cell method and the higher-ordertheory was demonstarted. In accordance with thistheory, the elastic region is meshed into rectangu-lar subcells, and the displacements which are gov-erned by the continuum equations, are expanded



(0,-1) (0,0)

(1,-1) (1,0)

(-1,-1)(-1,0)

(0,1)

(-1,1)

(1,1)

s0

s0

X2, k2

X3, k3

L

H L

H

(k2, k3)

x1

x2

x ,k1 1

(a)

Fig.5a

0

-2 -1 210

-0.2

-0.1

0.1

s22(M

Pa)

x2/H

(b)

Fig.5b

FIGURE 5. (a) Cross-section of a unidirectional fiber reinforced composite with cell numbering and (b) the normalstress distribution induced by the local heating. The solid line corresponds to the case when the temperature jump isapplied at all the cell (0,0) regions and the dashed line applies to the case when just the fiber is heated

into second order in terms of local coordinates [24].The final results for the stresses appearing in theboron/epoxy composite with the volume fraction0.25 and H = L induced by the ∆T = 1°C temper-ature deviation are shown in Fig. 5(b). Two differ-

ent cases are considered. In the first case, the entiredomain of cell (0, 0) has been heated, while in thesecond case, the fiber only in this cell is heated. Thefigure well exhibits the locations of the fiber-matrixinterfaces. In addition, it can be observed that the


444 MICHAEL RYVKIN

heating of the single fiber barely affects the neigh-boring ones.

4.2 Fiber-Reinforced Materials with Flaws

Similar to the case of layered materials with 1-Dtranslational symmetry, for 2-D periodic materials,the local deviation from the periodic stress field mayappear not only as a result of a local thermome-chanical loading, but due to a flaw existence. Theapproach to the solution of multiscale problems ap-pearing in the latter case also hinges on the idea ofjumps applied along the flaw line. Consider, for ex-ample, the unidirectional fiber reinforced compositementioned in the previous section with a flaw as aresult of one missing fiber due to damage or pro-duction defect [25]. The stress state arising near theflaw as a result of the remote tensile loading per-pendicular to the fibers direction is to be derived.The solution is sought as a superposition of the fol-lowing ones for undamaged periodic composite: (1) theperiodic solution u0(x1, x2) corresponding to the re-mote loading, which is determined by the high fi-

delity generalized method of cells [26], and (2) NGreen function solutions un

k1k2(x1, x2), n = 1, 2, ..N ,

generated by unit stress jumps applied along the cir-cular flaw boundary, which are found by the repre-sentative cell method

uk1k2(x1, x2)=u0(x1, x2)+N∑

n=1

Cnunk1k2

(x1, x2) (46)

where Cn are some unknown coefficients. The lin-ear algebraic system of equations for their deriva-tion yields from the zero traction condition at themissed fiber boundary.

Cracks represent the most dangerous and, conse-quently, the most interesting type of flaws in brittlecomposite materials with periodic microstructure.The classical problem of fiber breakage in the 2-Dformulation [27] yields the problem on a layeredplane with a crack perpendicular to the interfaces(see the left part of Fig. 6(a)). The composite sub-jected to the remote tensile loading σ and the crackfaces are traction-free. To estimate the compositestrength, the exact stress distribution in the crack

matrix

fiber

x1

σ

σ22σ

(a)

(b)

x2

FIGURE 6. (a) A transverse crack in a fiber-reinforced composite and the normal stress distribution; (b) the repre-sentative cell employed in the analysis



vicinity is required. In this case, in view of the crackorientation, it is impossible to take the representa-tive cell in the form of an infinite strip, which allowsan analytical solution, and the finite rectangular rep-resentative cell supposing a numerical solution isused (Fig. 6(b)). It should be emphasized that thecell size is defined by the distance between the fibersonly in one direction, while the second cell dimen-sion can be chosen arbitrarily thanks to the transla-tional symmetry of the undamaged composite. Asin the case of the fiber loss problem, the solutionis derived by the use of the Green functions corre-sponding to the jumps imposed on the flaw (crack)line. These functions are determined by the com-bined use of the representative cell method and thehigher-order theory. A distinguishing feature of thisproblem is that the jumps in the displacements, andnot in the stress components, were considered. Eachunit jump was applied at the boundary of a subcell,the size of which was determined by the representa-tive cell discretization. The final results for the nor-mal stresses in the fiber direction are depicted in theright-hand part of Fig. 6(a). The stress singularitynear the crack tip and the stress concentration in thefirst intact fiber are well exhibited.

4.3 Cracks in Low-Density Materials

Many interesting multiscale problems emerge dur-ing the study of brittle fracture of the low-densitymaterials with microstructure generated by a pe-riodic system of voids. There are many examplesof such materials in nature, and they are mimickedin numerous engineering applications ranging fromperforated materials to honeycombs and foams. Ifthe failure is caused by a macrocrack propagation,the estimation of the material strength is based onthe fracture toughness concept, and it is necessary toconsider three different length scales. The first oneis the natural material scale Lm, defined by the sizeof the repetitive module of the material microstruc-ture; the second one is the crack length Lc; and fi-nally, the third is some macroscale Ls related to theoverall size of the model. Note that in the consid-ered materials, in contrast to the continuous ones,the stress singularity in the vicinity of the crack tip,which can be associated with the center of a void,is absent. Consequently, the fracture toughness canbe expressed in terms of the rupture modulus σfs

of the bulk parent material. The length scales in the

correct formulation of the problem on the fracturetoughness evaluation must fullfill the stipulation

Lm << Lc << Ls (47)

which causes a considerable computational chal-lenge in the framework of direct finite element mod-eling of a large domain with an embedded crack.The approach based on the DFT advocated in thepresent article confines the analysis to a single rep-resentative cell and allows us to overcome this ob-stacle.

Consider first the case of cellular material [28]when the relative density is very low and, con-sequently, the thin walls between the neighboringvoids can be adequately modeled by rigidly con-nected beam elements. The crack analysis of the ob-tained periodic beam lattice or honeycomb is basedon the assumption that the crack advance under re-mote tensile, say, loading σr takes place when theskin normal stress in the most loaded element in thecrack tip vicinity approaches the critical value σfs.A typical problem of this type for the kagome latticeis illustrated in Fig. 7(a).

The solution procedure can be outlined as fol-lows. The cracked lattice is viewed as the undam-aged periodic one, where the stresses in the ele-ments at the crack path are zero. Consequently, sim-ilar to Eq. (46), the stress state of the lattice is soughtin the form of the superposition of the solutions forthe pristine periodic structure

Uk1k2(x1, x2)=U0(x1, x2)+3N∑n=1

CnUnk1k2

(x1, x2) (48)

Here U0(x1, x2) is the periodic solution for the lat-tice subjected to the remote tensile loading, whichcan be obtained, for example, by the stress local-ization method presented in [29]; N is the num-ber of elements to be removed; and the solutionsUn

k1k2(x1, x2) are the specific unit force solutions. In-

stead of jumps, as in the fiber loss problem, these so-lutions are generated by self-equilibrated systems ofunit forces applied at the extremities of each elementto be abandoned. The unit force solutions are deter-mined by the representative cell method (the repre-sentative cell is depicted in Fig. 7(b)) and the systemof 3N linear algebraic equations for derivation of thecoefficients Cn yields from the condition of vanish-ing of the internal resultants in N beam elements.


446 MICHAEL RYVKIN

r

r

2a

(a)

L

t

(b)

FIGURE 7. (a) Kagome lattice with mode I crack, (b) its representative cell, and (c) the results of the crack lengthanalysis

Note that in the case of more simple string mass lat-tices, the preceding technique for structural degra-dation modeling by means of neutralizing forceswas employed in [4, 30].

For the given crack length 2a, the obtained solu-tion allows us to determine which remote loadingσr corresponds to the crack propagation condition,when the critical stress is reached in some elementin the crack tip vicinity. The possibility to derivefrom this result the crack length independent frac-ture toughness appears only in the case of a suffi-ciently long crack when the self-similar K-field inthe crack tip vicinity takes place. The linear elas-tic fracture mechanics dictate that in this situation,the remote stress must be inversely proportional to

the crack length. Therefore, to establish the propercrack length scale required for the fracture tough-ness evaluation, it is plausible to examine the prod-uct σr

√a. The results of such a crack length analy-

sis are presented in Fig. 7(c) for different values ofthe relative density parameter t/L (t is the elementthickness and L is their length). It is observed thatfor the considered parameter range, it is sufficient toemploy the cracks produced by 60 failed beam ele-ments.

In the case of low-density materials with moder-ate density (see Fig. 8(a)), the beam model for thecell walls becomes inapplicable. Therefore the onlyway to model fine material microstructure is to em-ploy some discrete (finite element) method. How-



FIGURE 8. (a) Low-density perforated material with optimal mode I fracture toughness, (b) the representative cellwith the finite element mesh, and (c) the principal stress distribution in the vicinity of a crack crossing 18 voids

ever, in this instance, employing the standard ap-proach for the fracture toughness evaluation, whichis required to mesh a sufficiently large domain tomeet the stipulation (47), is not practical due to ahuge number of the degrees of freedom. On theother hand, the use of the DFT helps to bridgethe different scales in this problem and leads tothe result [31]. The solution of the problem for along crack passing through many necks between thevoids is once more obtained as a superposition ofsome unit jump solutions. These solutions are de-rived by the representative cell method, and con-sequently, only the repetitive representative cell ismeshed, as shown in Fig. 8(b). The principal stressdistribution obtained in the vicinity of a macrocrack

passing through 18 voids is presented in Fig. 8(c).Using this result, one can easily calculate the frac-ture toughness. The numerical efficiency of the ap-proach allowed us to carry out a parametric studyand to optimize the crack arresting ability of the ma-terial. The layout presented in Fig. 8(a) provides themaximal fracture toughness for the relative densityρr = 0.5.

5. CONCLUDING REMARKS

The article presents, in a systematic manner, thedevelopment and applications of the idea to em-ploy the representative cell method in the multiscaleanalysis of periodic elastic systems ranging from


448 MICHAEL RYVKIN

structures to composite materials. The transition be-tween the macroscale, which is assumed to be in-finitely large for the considered case of translationalsymmetry, and the microscale, which is defined bythe system periodicity, is carried out by means of theDFT. Consequently, the approach does not includeany homogenization procedure, and the analysis isconfined to a representative cell. Of course, there isno free lunch, and this analysis is to be done manytimes for different values of the transform parame-ter. The applicability of the method in combinationto the analytical as well as the numerical methodsis demonstrated. Both cases of 1-D and 2-D trans-lational symmetry are considered (note that in therecent work of [32], it was shown that the methodworks in the case of 3-D translational symmetry aswell). It is demonstrated that the approach remainsapplicable also in the cases when the periodicity ofan elastic system is violated by a flaw.

ACKNOWLEDGMENT

The support of the ISF (Israel Science Foundation)under grant 838/06 is gratefully acknowledged.

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