Spring network models in elasticity and fracture of ... - Science Direct

COMPUTATIONAL MATERIALS SCIENCE Computational

Materials Science 7 ( 1996) 82-93

Spring network models in elasticity and fracture of composites and polycrystals M. Ostoja-Starzewski aT*,P.Y. Sheng b, K. Alzebdeh aInsritufe

vj’Puprr

Science und Technology.

und Georgiu

’ MGA Reseurch Corporution, ’ Depurtmmt

vj’Muteriul.s

Insritute

oj’Techn&~y.

900 Mm&line

Science und Mechmics.

Michigun

500 10th St.,

Street. Madison Stute University,

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’

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Abstract We review some recent advances in modeling of elastic composites and polycrystals by spring networks. In the first part, spring network models of anti-plane elasticity, planar classical elasticity, and planar micropolar elasticity are developed. In the second part, applications to progressive breakdown in elastic-brittle matrix-inclusion composites and aluminum sheets are discussed.

ture in this area has become

1. Introduction Spring network methods are based, in principle, on the atomic lattice models of materials. While it is unwieldy to work with the enormously large numbers of degrees of freedom that would be required in a true representation of a material specimen, a much cruder model requiring a very modest number of nodes per single heterogeneity (e.g., inclusion in a composite, or grain in a polycrystal) turns out to be sufficient for a number of applications. This method of studying micromechanics of materials has recently become quite popular but it is important to note that, in essence, it is a spin-off from the solid state physics where first studies were concerned with effective transport and breakdown properties of random media, see e.g., Refs. [l-4]. In fact, the litera-

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so extensive that it is impossible, in a regular paper, to give justice to all the works published on the subject. Thus we focus on several applications and extensions of spring networks of particular interest to us, and try to show connections with other, related studies. The paper is divided into two major parts: Spring Network Models and Applications to Effective Moduli and Fracture in Heterogeneous Materials. In the first part we outline the derivations of: an anti-plane elasticity (equivalently, in-plane conductivity) model, an in-plane anisotropic elasticity model (a generalization of the Kirkwood model), and an in-plane micropolar elasticity model. This part is complemented by a brief exposition of the so-called CLM transformation which allows a change of compliantes without a change of the stress field. In the second part we outline two applications of spring networks: anti-plane cracking in an elasticbrittle matrix-inclusion composite, and in-plane cracking in a thin aluminum (polycrystalline) sheet.

0 I996 Elsevier Science All rights reserved.

M. Ostoja-Starzewski

et al./Computationul

Both of these are simulated by a process in which the spring network bonds are progressively being taken out as they exceed the local strength criterion. The process of crack elimination represents thus the growth of a crack, or a field of cracks.

2. Spring network models 2.1. Basic idea of a spring network representation It is well known that the basic idea in setting up the spring network models is based on the equivalence of strain energy stored in a unit cell of a network of volume V

lb1

Econtinuum

= + c ( ,F ’ db)

f

/

u V”

i, j=

ai = CijEj,

1,2

(24

where ,a=(~,, u~,)(cT~, ai> and _E=(E,, E*) S @,, E&X Upon substitution into the momentum balance law gi,i = 0

(2.5)

’ ,E dV

(‘ij’,j),i

=

(2.2)

b in Eq. (2.21, stands for the bth spring (bond), and lb1 for the total number of bonds. Our discussion is

O

(2.6)

Henceforth, we are interested in approximations of locally homogeneous media, so that the governing equation becomes ciju,ij = 0

b

=

Of all the elasticity problems, the anti-plane one is the simplest on which to illustrate the spring network idea. Let us note that a number of problems are equivalent to it by virtue of mathematical analogies: elastic membrane, thermal conductivity, etc.; see also Ref. [5]. In the continuum setting we thus have the constitutive law

(2.1)

where the energies of the cell and its continuum equivalent, respectively, are

b

2.2. Anti-plane elasticity: square lattice

we obtain

Ecell = EcOntinuum

E cell= CE,

83

Materials Science 7 (1996) 82-93

(2.7)

In the special case of an isotropic medium Eq. (2.6) simplifies to a Laplace equation cu,;i = 0

(2.8)

set in the two dimensional (2D) setting so that, by a volume we actually mean an area of unit thickness. In the sequel we restrict ourselves to linear elastic springs and spatially linear displacement fields f! (i.e. uniform strain fields _E), so that Eq. (2.2) will become

a)

In Eq. (2.3) 2 is a generalized spring displacement and k its corresponding spring constant. The next step, that will depend on the particular topology of the unit element and on the particular model of interactions, will involve making a connection between of and ,E, and then deriving _Cfrom Eq. (2.11,. The corresponding procedures and resulting formulas are given below for several elasticity problems set in the square and triangular network geometries.

b) Fig. 1. (a) A triangular lattice with a hexagonal unit cell shown; (b) a square lattice with a square unit cell shown.

84

M. Ostoju-Starzewski et al./

Compututionul Muterids

We now discretize the material with a square lattice network, Fig. lb, whereby each node has one degree of freedom (anti-plane displacement u), and nearest neighbor nodes are connected by springs of constant k. It follows that the strain energy of a unit cell of such a lattice is (2.9) In the above we employed the uniform strain ,F= ( E , , is2). Also, I(‘) = ( Ejb’,lib’) is the vector of half-length of bond b. In view of Eq. (2.11, the stiffness tensor is obtained as i lyyb’, (2.10) i,j= 1,2 h= I where V = 4 if all the bonds are of unit length (Il(b)l = I). This leads to a relation between the bond s&ng constant k and the Cij tensor Cij = ;

c,, = c,, = 0

C,, =c,,=;,

(2.11)

In order to model an orthotropic medium, different bonds are applied in the X, and x2 directions: k(l) and kC2’.The strain energy of a unit cell is now 4 E

=

3

c h=

(2.12)

k(b)/jb)$bG;/ I

so that the stiffness tensor is

6 ,‘&b’l~b’l;b’

Cij = ;

(2.13)

h-l

which leads to relations k”’

c,, = -y>

k(2)

c,, = y,

c,, = c,, = 0 (2.14)

If one wants to model an anisotropic medium (i.e. with C,, # 01, one may either choose to rotate its principal axes to coincide with those of the square lattice and use the network model just described, or introduce diagonal bonds. In the latter case, the unit cell energy is given by the formula Eq. (2.12) with 161= 8, Fig. la. The expressions for Cij’s are C,, = ;

+ ,@‘,

c,, = T

c,, = c,, = kc5)- k’6’

+ k’6’ (2.15)

Science 7 (1996) 82-93

It will become clear in the next section how this model can be modified to a triangular spring network geometry. 2.3. In-plane elasticity: triangular lattice with central interactions

It is well known that, in the planar continuum setting the constitutive law is aiJ

=

‘ijkm

Ekm

i, j, k, m= 1, 2

3

(2.16)

which, upon substitution into the balance law (TlJ,j --0

(2.17)

results in the Navier’s equation for the displacement u,, that is, /LUi,,j +

KUj,j;

=

(2.18)

0

In Eq. (2.17) p is defined by o,2 = PB,~, which makes it the same as the classical three-dimensional shear modulus. On the other hand, K is the (planar) two-dimensional bulk modulus, that is defined by u,; = KEY,. See Appendix A for basic concepts of planar elasticity. As in the foregoing section, we are interested in approximations of locally homogeneous media. Consider a regular triangular network of Fig. la with central force interactions only, which are described, for each bond b, by F, = @l(ib)uj where

@II,“)= (y(‘)n\‘)njb)

(2.19)

Similar to the case of anti-plane elasticity, czcb) is the spring constant of half-lengths of such central (normal) interactions - i.e. of those parts of the springs that lie within the given unit cell. The unit vectors _ncb)at respective angles tYb) of the first three (Y springs are #I, = 0”

n(l)= 1, I

#) = _!

nI” = e/2,

2’

.f3)

=

_

.! 27

n$” = &i/2

e(Z) = 60”

n\” = 0, o(3)

=

120”

(2.20)

The other three springs (b = 4, 5, 6) must, by the requirement of symmetry with respect to the center of the unit cell, have the same properties as b = 1,2, 3, respectively. All the (Ysprings are of length 1, that is, the spacing of the triangular mesh is 21= s. The cell area is V= 2151~.

M. Ostoja-Starzewski

et al./ Computational Materials Science 7 (19%) 82-93

a3

Every node has two degrees of freedom, and it follows that the strain energy of a unit hexagonal cell of such a lattice, under conditions of uniform strain ,E= (E,,, e22r E,~), is

a4

a)

85

a2

4

i

a5

a1

a6

so that, again by Eq. (2.11, the stiffness tensor becomes

‘ijkm

(2.22)

=

b+l

In particular, taking all cz(‘) the same, we see that 9 c 1111=

3 c 1122 =

c2222 =

is"'

c,,,,

=

-a

86

3 C 1212

(2.23)

It is also observed that the condition C 1212

b)

=ir

=

~Wllll

-C1,22)

(2.24)

is satisfied, so that there are only two independent elastic moduli’, and the modeled continuum is isotropic. One might try to model anisotropy by considering three different (Y’S in Eq. (2.221, but this would be limited given the fact that only three of those can be varied - one needs to have six parameters in order to freely adjust any planar anisotropy which involves six independent Cijkm‘s. This can be achieved by introducing the additional angular springs as discussed below. In fact, angular springs are also the device to vary the Poisson’s ratio. All this is shown in Section 2.4. 2.4. In-plane elasticity: triangular lattice with central and angular interactions We continue with the triangular network, and introduce angular springs acting between the contiguous bonds incident onto the same node. These are assigned spring constants /3(@, and, again by the argument of symmetry with respect to the center of

Fig. 2. (a) Unit cell of a triangular lattice model; a,, . . , a6 are the normal spring constants, /3,, . . . . & are the angular spring constants; a, = q. a2 = Q~, a3 = CY~and j3, = &, & = &. & = &, in an anisotropic Kirkwood model; (b) details of the angular spring model.

the unit cell, only three of those can be independent. Thus, we arrive at six spring constants: (cr(‘), (Y(~), (Y(~),PC’), pc2), pc3)}. With reference to Fig. 2b, let A#*’ be the (infinitesimal) angle change of the bth spring orientation from the undeformed position. Noting that, p X ,u = lAt?, we obtain Aecb)= 8 ..E. n.n klJ JP 1 P

(2.25)

where ckij is the Levi-Civita permutation tensor. The angle change between two contiguous a springs (b and b + 1) is measured by A4= AI~(~+‘)-ABcb’, so that the energy stored in the spring pcb’ is

= ffl(b)( Ekijcjp( n$“+ ‘)n(pb+I) - nlb)n(pb)))’ (2.26) By superposing the energies of all the angular

86

M. Ostoju-Starzewski

et al./ Compututionul

bonds with the energy of Eq. (2.21), the elastic moduli are derived as [6,7]

Cijkm= f

,Cb),(,b),(~),Sp),Cb)

i

I

h=

J

Matrriuls Science 7 (1996) 82-93

The (Y and /3 constants are related to the planar bulk and shear moduli by K=

m

&ff),

I

(2.30)

-( p’b’+p

It is noted here that the angular springs have no effect on K, i.e. the presence of angular springs does not affect the dilatational response. The formula for a planar Poisson’s ratio is

(b- U)n;b)n$Hn:Wn(mh)

$$b+ I),Q) n Wnjb+ lk P (b)n(,b)n(j+ I)@+ lln(b)

-pb)S.

+P

v=-=

J

_p(b+jjkn~)n~)n~+ I,$++ 1)

K- p

Cl,,,

-

2c,2,*

(2.31)

C 1111

KfP

which, in view of Eq. (A.3), becomes

+P W@+

Un$Nn~Wn~+

1)

I

(2.27)

This provides the basis for a spring network representation of an anisotropic material; it is also a generalization of the Kirkwood model [8] of an isotropic material that is discussed below. By assigning the same (Yto all the normal and the same p to all the angular springs we recover the Kirkwood model whereby Eq. (2.27) simplifies to

1-g ZJ=

(2.32) 3+$

From Eq. (2.32) there follows the full range of Poisson’s ratio which can be covered with this model.lt has two limiting cases v=+

if;

Y= -1

_ 2n(b’n’,Mn~b’n’b’ J

m

~~~,r)~(j+

I),+b+

1 _

Ijn(b) m

+n(b)n(_b+:) W+:)@) 1

_

J

nk

aikn(b)n(,b)n(b+

+@+

PJP

m

ljn(b+ m

UnC+$On~+

1)

1)

(2.28)

It follows from the above that

(2.29) Again, condition Eq (2.24) is satisfied, so that there are only two indenendent elastic moduli.

P

+O(cw-model)

P if ; + = ( p - model)

(2.33)

For the Poisson’s ratio between - l/3 and l/3 one may also use a Keating model [9] which employs a different calculation of the energy stored in angular bonds. In order to model materials of the Poisson’s ratio above l/3 one can use a model developed in [lo] and [l 1I, which is based on a superposition of three honeycomb lattices, each with a different central force spring, resulting in a triangular lattice. Finally, it is interesting to note that Holnicki-Szulc and Rogula [12] used the spring network idea inversely, namely they simulated a discrete engineering structure by a continuum model. Their study gives a micromechanical basis for nonlocal and gradient-type elasticity models; see also Refs. [13,14]. 2.5. In-plane beam-type

elasticity: interactions

triangular

lattice

with

In the solid state physics literature the Kirkwood and Keating models are sometimes referred to as the

M. Ostoju-Starzewski et al. / Computational Materids Science 7 (1996) 82-93

‘beam-bending models’. This is a misnomer since there is no account taken in these models of the actual presence of moments and curvature change of spring bonds connecting the neighboring nodes. True beam bending was fully considered by Wozniak [ 151 and his coworkers, and, considering a limited access to this book, in the following we give a very brief account of the triangular lattice case. Let us focus on the deformations of a typical beam, its bending into a curved arch allowing the definition of its curvature, and a cut in a free body diagram specifying the normal force ,F, the shear force F, and the bending moment M, see Fig. 3. It follows that in 2D, the force field within the beam network is described by fields of force-stresses ok, and moment-stresses mk; note the additional presence of moment-stresses due to the beam-type interactions. This is called a micropolar elastic medium. The property of invariance of these stresses under an appropriate transformation in compliances is discussed in Appendix A. The kinematics of the network is now described by three functions u*( 5) 9

%(_x)*

cpw

ii(b)

I 4

b)

(2.34)

=

‘/,k

+

Elk

qy

Ki

=

‘P,i

(2.35)

-

(b)

t

which coincide with the actual displacements (u, , u,) and rotations (cp) at the fiber-fiber intersections. Within each triangular pore, these functions may be assumed to be linear. The local strain, -yk,, and curvature, K~, fields are related to u,, u2, and cp by Ykl

87

??

cl

where elk is the Ricci symbol. It follows from the geometric considerations that y’b’E

nk

(b) (b) n,

(2.36)

Yk/

is the average axial strain, with ~(~)-y(~)being its average axial length change. Similarly, -(b) = (b)-(b) Y - nk n, yk,

= n~bQi\bb4~,~k,- cp

(2.37)

is the difference between the rotation angle of the beam chord and the rotation angle of its end node. Finally, K(b) G ,,(kMKk

(2.38)

is the difference between the rotation angles of its ends. It follows from the beam theory that the mechani-

Fig. 3. The kinematics (a), curvature (b), and internal loads (c) in a single beam element; after Ref. [ 151.

cal (force-displacement and moment-rotation) sponse laws of each bond (Fig. 3) are given as

re-

M. Osto)ju-Stcrrze~vskiet d./

88

Compututionnl Materids

where A(‘) is the beam cross-sectional area, Zch’ is its centroidal moment of inertia with respect to an axis normal to the plane of the network, and .!Z”’ is the Young’s modulus of the beam’s material. All the beams are of length s = s(~), which is the spacing of the triangular mesh. Turning now to the continuum picture, the strain energy Eq. (2.3), is expressed as V V u cO”Il”““m = 2 YijCijkmYkm + I

Dij

=

KlDijK,

(2.40)

+ &Jn(b)R”(h) , m )

.;.b),~)S’b’

i

1 +I?/R E=3R 3 +l?/R 1 -R/R

i n\%jp)(frjh)n(,h)R(~J h= I

h=

which are seen to reduce to the formulas of Section 2.3 in the special case of flexural rigidity being absent. Furthermore, it follows from (A.3) that the effective Young’s modulus and Poisson’s ratio are

I

(2.47)

U=

from which we find Cijkm =

Science 7 (1996) 82-93

(2.41)

I

where 2E’b),_4’b’ R(b) =

3+2/R Regarding the Poisson’s ratio, we observe that the introduction of beam-type effects has a tendency to reduce it down from l/3. This is similar to the angular spring effects in the Kirkwood model, recall Eq. (2.33). However, noting that i/R = (w/s)‘, where w is the beam with, we see that this model does not admit v below N 0.2 and becomes questionable already at N 0.3 (i.e. w/s = 0.17). The case of short beams should be considered as a perforated plate problem.

s(“vT ’ (2.42) If we assume all the beams ( Rch) = R, etc.), we find C III1 = C,,,,

= 2(3R+R),

C 1122

=

C22l

I =

C 1221

=

CZll2

=

to be the same

C,,,,=;(R+3@

;(R-li) i(R-I?),

D,,=D,,=;S (2.43)

with all the other components of the stiffness tensors being zero. In other words, we have Cijkm = aij6,, B + Sik ajrnA + si, Sjkn Di’j”= 6,,r

(2.44)

3. Applications to effective moduii and fracture in heterogeneous materials 3.1. Fracture simulation Spring networks have been employed since the eighties to compute effective elastic moduli as well as to simulate crack formation in materials. The procedure is usually as follows: (i) Assignment of all the spring stiffnesses and strengths according to their placement in the body domain, i.e., depending on which phase does the given bond fallin. Any bond straddling the boundary between two phases (1 and 2) has its spring constant kc”) assigned according to a series spring system weighted by the partial lengths (1’ and r2) of the bond that belong to the respective domains, that is

in which E=n= r=

+(R-R”),

3s

$?,

I= 1g = 1’ + l2 (2.45)

The effective identified as K=

A=$(R+3R”)

bulk

p=i(R+R”)

and shear

moduli

are now

(2.46)

(3.1)

(ii) Loading of th e spring network by subjecting its external boundary to kinematic boundary conditions Ui = ZiiXj

(3.2)

hf. Ostoja-Starzewski

et al./Computational

where Eij is the macroscopic strain. An alternative approach to calculating effective properties, that is typically used in solid state physics, is based on the concept of a periodic window (with a heterogeneous microstructure of periodicity L or, equivalently, S), i.e., periodic boundary conditions Ui( 5) = z$( ,x + L) + ZijLj r&v) = -++g

(3.3)

Here &.= Lg, where ,e is the unit vector. Solution of the actual distribution of all the node displacements is typically obtained by a conjugate gradient method

M. (iii) In a fracture simulation one may increase the loading conditions through raising Eij by a small increment AEij, and then find the first bond(s) that exceeds the local fracture criterion. The latter one is formulated, in general, in terms of a force F in the given bond relative to its strength (3.4) F s F,, If Fq. (3.4) is met, the given bond is being removed from the lattice - thus representing a crack - and the macroscopic strain E is increased. The increase of Z by AZ is conducted by first unloading the entire lattice, and then reloading it by strain E + AZ. It is possible that more than one bond meets the fracture criterion at any given step, in which case all such bonds have to be removed at the same time. This process is continued until the lattice is completely cracked (cracked percolation). An alternative simulation method relies on the linear character of the entire body, even in the depleted state, and allows one to go right away to the most stressed bond without conducting many little steps. There are also two alternative, but fully equivalent, ways of formulating the fracture criterion: (i) in terms of the critical energy E, which may be stored in the bond, or (ii) in terms of the critical strain E,, (or elongation) of the bond. Indeed, these two options are preferred to Eq. (3.4) in actual simulations since the programs use conjugate gradient subroutines and are written in terms of the node displacements.

Materials Science 7 (1996) 82-93

89

crack formation in matrix-inclusion composites. Given the fact that both phases, matrix and inclusion, are elastic-brittle, the composite is specified by two dimensionless parameters: the stifiess ratio C’/C” and the strain-to-failure ratio e,!!/qT. In Fig. 3 we show the case of C’/C” = 2.0 and .$/L$ = 0.2, that is when the inclusions are stiffer by a factor of two, but are able to withstand only 20% of the strain level allowed by the matrix material. As shown by a sequence of plots in Fig. 4, damage forms initially in the denser regions of inclusions due to stress concentrations in their vicinity - stiff inclusions tend to form links carrying relatively more load. Microcracking then spreads (percolates) across the specimen whereby the random character of the damage pattern reflects the heterogeneity of the microstructure. A number of issues, such as basic classification of effective constitutive responses, geometric patterns of damage, varying degrees of randomness of the inclusions’ arrangements, and mesh resolution of continuum phases were investigated in Refs. [ 17,181. A number of studies were carried out by physicists on the related problem of breakdown of random

3.2. Fracture of a composite The spring network model discussed in Section 2.2 has been employed in Refs. [17,18] to simulate

Fig. 4. Evolution of a damage pattern in a sample of random composite at E’/E” = 0.2 and Cl/P = 2.0.

M. Ostoju-Stur:ewski

90

et ul./Compututiotml

Materids

Science

7 (1996)

82-93

(a)

Fig. 5. (a) An experimentally observed crack pattern in the aluminium sheet; (b) the damage/crack pattern from simulations spring network (N,, iV,) = (201, 169); after Ref. 171. A fine mesh seen in (b) is an artifact of the computer graphics.

of a triangular

M. Ostoja-Starzewski

et al./Computational

two-component networks without a disk composite microstructure; see e.g., Ref. [19]. These are typically known as the breakdown problems in conductivity of random media. 3.3. Fracture of a polycrystal The spring network model of Section 2.4 allows simulation of materials with local anisotropies. In fact, this model has been developed in order to complement experimental studies on fracture of thin aluminum sheets [7]. The 2D setup offered the possibility to observe the actual locations of cracks, and to study them as a function of the relative anisotropies of crystals in the sheet. Cracks were made to occur in the grain boundaries through the presence of gallium that was initially smeared onto the specimen. Also prior to the mechanical test, all the grain orientations in the sheet were measured through the Kikuchi surface backscattering technique and saved in a computer file. The specimen was then subjected to biaxial extension, whereby a crack pattern developed such as one shown in Fig. 5a. The scanned image of the polycrystalline specimen allowed the assignment of stiffness and strength properties to all the bonds of the spring-network according to which crystal they belonged to. This step relied on a classical transformation formula for a 4th rank tensor

CLnpy =

aniamjapka&ijkl~

n,

m,

P, 4 = 1,2,3

(3.5) where [a] were rotation matrices with respect to the reference orientation. Next, at every mesh node the in-plane (2D) portion of Cbmpcltensor, having six components, was identified and mapped one-to-one into the six spring constants (Y,, (Y*, (Ye, p,, &, & according to Eq. (2.26). Computer simulation of fracture, as described in Section 3.1 was next implemented in an attempt to reproduce the same failure patterns as observed experimentally. This was carried out on a rectangular (N, X IV,,) spring network, where N, and NY are the total numbers of mesh spacings in the x and y directions, respectively. In Fig. 5b we display the case (N,, NY)= (201, 169). It is seen that the cracking pattern matches the one obtained from the experiment very well.

Materials Science 7 (1996) 82-93

91

4. Spring networks versus finite elements In the linear elastic problems the spring networks are practically equivalent algebraically and allow the same level of resolution of small scale details as the finite elements (FE) providing the same number of nodes and linear interpolation functions are being used. The differences appear as one moves into more specialized applications. Advantages of spring networks: (i) The ability to easily simulate complex heterogeneous systems having very many degrees of freedom (up to 106-10’) as opposed to finite elements. The FE methods require meshes adjusted to the given microstructures, which may be costly preprocessing procedures. On the other hand, all the bond spring constants can be assigned according to their placement in the material in a much shorter time frame. This is an important consideration in simulations of many samples of a random medium (e.g., Ref. [20,301X (ii) No need to remesh or disconnect the finite elements; this is a time saving factor in fracture simulations using the spring networks. (iii) The ability to grasp spatially cooperative damage phenomena with very large numbers of cracks, as opposed to boundary elements and finite elements which are typically based on exact solvers and thus restricted to a smaller number of degrees of freedom and very few cracks. Advantages of FE: (i) Possibility of using higher order interpolation functions. (ii) Possibility of simulating non-linear elastic as well as inelastic, strain rate-dependent, etc., material behaviors, e.g., Ref. [21]. (iii) Possibility of simulating cracking accompanied by plastic flow, also in dynamic situations. Considering that the removal of spring bonds, or finite elements, as well as the remeshing of an FE mesh introduce artifacts into all the models, it has to be acknowledged that the basic problem of an accurate simulation of a progressive, non-rectilinear crack still remains an open challenge. One should also mention here the boundary element methods and the most recently developed element free approach [22]. A comprehensive discussion of the pros and cons of finite element versus spring network models in simu-

M. Ostoja-Starzrwski

92

et al./ Computational Materials Science 7 (1996182-93

lation of brittle fracture has been given by Jagota and Benison [23,24]. The major conclusion of these authors was that spring networks are naturally suited to mode1 materials whose topology corresponds to that of a chosen mesh of springs, e.g., a granular medium. Finally, on the subject of scaling of computing times with system size on parallel computers, we refer to a discussion by Plimpton [25], which, although set in the context of glass transitions via molecular dynamics simulations, is quite relevant for spring networks.

It is important to note here that v is seen to range from - 1 through + 1, in contradiction to v3D, which is bounded by - 1 I v3D5 0.5. For positive values, the two may be connected through v=

‘3D

’

-

(A.41

‘3D

This work was made possible through support by the National Science Foundation under grant MSS9202772.

A detailed discussion of relationships among this planar, the well-known plane stress, the well-known plane strain, and the 3D isotropic elasticity is given in that reference. However, a result of special interest studied there and in the companion paper [27] concerns a transformation of an original material with properties (K(X), p(_x)) into a new material with (barred) prope%es (Z(s), $5)). The CLM transformation (after the names Cherkaev, Lurie and Milton in this latter reference)

Appendix A

-_=-+2 K

Acknowledgements

1

1

The constitutive relations for a linear elastic isotropic 3D material are &II = &ii ’+ &I2

=

- ?D(%

+ %)I

‘3D

752

(A.1

>

3D

together with cyclic permutations 1 -+ 2 -+ 3, whereby E,, and v3D stand for the conventional 3D Young’s modulus and Poisson’s ratio. On the other hand, in 2D elasticity [26], there is no x3 direction, so that we have 1 E,, = - o,, E[

vu22

1)

1+v 812

=

~cT,~

E

(A.2)

with cyclic permutation 1 + 2. In Eq. (2.12), E and v stand for the 2D (or planar) Young’s modulus and Poisson’s ratio. It may be checked readily that the following relationships between E, v, K, and p (the latter two being the planar bulk and shear moduli) hold E

E

K= 2(1-

v) ’

4 1 -_=-+-

1

E

K

/.L’

p=

1 li’

1 -=

1 ---

L

CL

1

(A-5) f4

preserves the stress state; here A is an arbitrary constant restricted by the requirement that the compliances be nonnegative. Furthermore, the CLM theorem says that the effective properties of the original material are preserved under the same type of transformation 1

1

-3 2 eff

1

-+x7 K

1

1

--err P

CL

1 (~4.6)

-=--1

where A is the same constant as in Eq. (A.5). In fact, it was later established by Dundurs and Markenscoff [28] that the CLM theorem could be generalized to admit a linear transformation. In the case of a planar micropolar elastic material, one deals with four planar compliances 1

A=l=

A+$

K

M=-

SC-t

p=l P’

CY

1 Y+&

(A-7)

where A, CL,(Y, y, and E are the elastic moduli. As shown in Ref. [29] the CLM transformation can now be generalized to the following form

2(1 +v)

v= -K--P K+I1-

(A-3)

i=mM

(3.8)

M. Ostojadarzewski

et al./Computationd

where m is an arbitrary scalar. It can be shown that the CLM theorem for the composite holds here as well. Finally, we point out that the CLM transformation has an identical form in the case of locally anisotropic materials. One of the uses of the CLM theorem, in the context of spring networks modelling a heterogeneous linear elastic material, consists in a possibility of rapidly checking the entire computer code. References 111M. Sahimi and J.D. Goddard, Phys. Rev. B 33 (1986) 7848. 121B. Khan& G.G. Batrouni, S. Redner, L. de Arcangehs and H.J. Herrmann, Phys. Rev. B 37 (1988) 7625-7637. 131 H.J. Herrmann and S. Roux, eds., Statistical Models for the Fracture of Disordered Media (Elsevier Science Publishers, 1990). [41 A. Delaplaece, G. Pijaudier-Cabot and S. Roux, J. Mech. Phys. Solids 44 (19%) 99-136. 151 S. Toquato, Appl. Mech. Rev. 44 (1991) 37-76. l61P.Y. Sheng, Ph.D. thesis, Michigan State University (1995). [71 M. Grab, K. Alzebdeh, P.Y. Sheng, M.D. Vaudin, K.J. Bowman and M. Ostoja-Starzewski, Acta Mater. 40 (1996) 4003-4018. (81J.G. Kirkwood, J. Chem. Phys. 7 (1939) 506-509. [91 P.N. Keating, Phys. Rev. 145 (1966) 637-645. l101A.R. Day, K.A. Snyder, E.J. Garbcczi and M.E. Thorpe, J. Mech. Phys. Solids 40 (1992) 1031-1051. [Ill K.A. Snyder, E.J. Garboczi and A.R. Day, J. Appl. Phys. 72 (1992) 5948-5955. [I21 J. Holnicki-Szulc and D. Rogula, Arch. Mech. 31(6) (1979) 793-802.

Materials Science 7 (1996) 82-93

93

1131 K. Berglund, in: Mechanics of Micropolar Media, eds. 0. Brulin and R.K.T. Hsieh (World Scientific, 1982) pp. 35-86. iI41 A. Askar, Lattice Dynamical Foundations of Continuum Theories (World Scientific, 1985). [I51 C. Woiniak, Surface Lattice Structures (PWN-Polish Sci. Pub]., 1970) (in Polish). lt61 W.H. Press, W.T. Vetterling, S.A. Teukolsky and B.P. Flannery, Numerical Recipes (Cambridge University Press, 1990). [171 M. Ostoja-Starzewski, P.Y. Sheng and I. Jasiuk, ASME J. Eng. Mater. Tech. 116 (1994) 384-391. 1181M. Ostoja-Starzewski, P.Y. Sheng and I. Jasiuk, Eng. Fract. Mech. (1996), to be published. [I91 P.M. Duxbury, P.D. BeaIe and C. Moukarzel, Phys. Rev. B 51 (1995) 3476-3488. La M. Ostoja-Statzewski, Appl. Mech. Rev. 47( I, Part 2) ( 1994) S221-S230. 1211 S. Schmauder, ed., Comp. Mater. Sci. 4 (1993) special issue. WI T. Belytschko, Y.Y. Lu and L. Gu, Eng. Fract. Mech. 51 (1995) 295-315. [23]

A. Jagota and S.J. Bennison, in: Breakdown and Non-hnearity in Soft Condensed Matter, eds. K.K. Bardhart, B.K. Chakrabarti and A. Hansen (Springer-Verlag, 1993). [24] A. Jagota and S.J. Bennison, Model]. Simul. Mater. Sci. Eng. 3 (1995) 485-501. 1251 S. Plimpton. Comp. Mater. Sci. 4 (1995) 361-364. [26] M.E. Thorpe and 1. Jasiuk, Proc. R. Sot. London A 438 (1992) 531-544. [27] A.V. Cherkaev, K.A. LuBe and G.W. Milton, Proc. R. Sot. London A 438 (1992) 519-529. [28] J. Dundurs and X. Markenscoff, Proc. R. See. London A 443 ( 1993) 289-300. [29] M. Ostoja-Starzewski and I. Jasiuk, Proc. R. Sot. London A 45 1 ( 1995) 453-470. M. Ostoja-Starzewski and J. Schuhe, Phys. Rev. B 54 ( 1%) 278-285.