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,
Heights.
’
NW, Arluntu.
GA 30318.5794,
USA
MI 48071, USA
East Lutzsing. MI 48824.1226,
USA
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-
* Corresponding
8944778.
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author. Tel.: + I-404-8946646;
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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.
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