Acta mater. 48 (2000) 4893–4900
www.elsevier.com/locate/actamat
EFFECTS OF CELL IRREGULARITY ON THE ELASTIC
PROPERTIES OF OPEN-CELL FOAMS
H. X. ZHU1, J. R. HOBDELL2 and A. H. WINDLE1*
1
Department of Materials Science and Metallurgy, Cambridge University, Pembroke Street, Cambridge
CB2 3QZ, UK and 2Huntsman Corporation, Everslaan 45, B-3078 Everberg, Belgium
( Received 5 November 1999; received in revised form 18 July 2000; accepted 18 July 2000 )
Abstract—Foams are more and more widely used because of their high mechanical properties relative to
their low density. Most available mechanical models are based on idealised unit-cell structures. A significant
disadvantage of the unit-cell modelling approach is that it does not account for the natural variations in
microstructure that are typical for most foam structures. Our objective has been to investigate how the cell
irregularity affects the elastic properties of open-cell foams. We generated periodic, three-dimensional (3D),
random samples with different degrees of irregularity, and used finite element analysis (FEA) to determine
the effective elastic properties. The geometrical properties were investigated for 3D random open-cell foams
and related to the elastic properties. The results indicate that the more irregular the foams, the larger will be
their effective Young’s modulus and shear modulus at constant overall relative density. On the other hand,
the bulk modulus reduces with increasing degree of cell irregularity, while the Poisson’s ratio is largely
independent.  2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
Keywords: Foams; Elastic; Computer simulation; Finite elements; Microstructure
1. INTRODUCTION
Foams are widely used in different areas, and their
mechanical behaviour is determined by the cell structure and the mechanical properties of the solid
material. The strong connections between mechanical
behaviour and cell structure of foams are generally
acknowledged, but poorly understood.
Mechanical models of foams are usually based on
models of cell structure. The simplest approach to
mechanical properties is to use dimensional analysis
[1], which gives the dependence of the foam properties on the relative density but not on cell geometry;
the constants relating to cell geometry need to be
determined by fitting experimental data. The second
method is to analyse a repeating unit cell, such as a
tetrakaidecahedron, using finite elements or structural
mechanics [2–8]. As long as the mechanical properties of the solid are known, this method can give the
full response of the foam subjected to a stress or
strain. The third approach is to model the real foam
structure by the random Voronoi technique followed
by finite element analysis (FEA) [9–11]. The advan-
* To whom all correspondence should be addressed. Tel.:
0044 01223 334321; fax: 0044 01223 335637.
E-mail address: Ahw1@cus.cam.ac.uk (A.H. Windle)
tage of this approach is that it gives a better representation of the cell geometry of foams.
Van der Burg et al. [9] have modelled foam structure by random Voronoi cells using finite element
analysis. They started from regular body-centred
cubic (bcc) and face-centred cubic (fcc) lattice nuclei
distributions, subsequently gave the nuclei positions
an increasing random offset and constructed the random structure using the Voronoi procedure. However,
all of the struts at a boundary are normal to the
boundary face, giving a much stiffer structure (up to
six times stiffer at high strain compression [10]).
Also, as their model is not periodic, they could not
have been able to apply periodic boundary conditions
in the finite element analysis. As is well known,
boundary conditions can greatly influence the mechanical response of a structure, and even mixed boundary conditions tend to underestimate the effective
Young’s modulus of foams [12, 13].
The aim of this work was to determine the influence of disorder in foam cell size and shape on the
linear elastic response of three-dimensional (3D)
foams. We have constructed 3D periodic random
structures with different degrees of irregularity, and
applied finite element analysis to determine the effective mechanical properties. The geometrical properties of random foams have also been investigated and
related to the mechanical properties.
1359-6454/00/$20.00  2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
PII: S 1 3 5 9 - 6 4 5 4 ( 0 0 ) 0 0 2 8 2 - 2
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ZHU et al.: ELASTIC PROPERTIES OF FOAMS
2. THE DEFINITION OF REGULARITY
2.1. The construction of periodic random Voronoi
foams
Cellular solids are usually formed by the nucleation
and growth of cells. If all cells nucleate randomly in
space at the same time and grow at the same linear
rate, the resulting structure is a Voronoi foam. The
central field V0, which is surrounded by 26 equivalent
boxes, is taken to be a cube in the present statistical
study. An orthogonal coordinate system is chosen,
with the origin at one corner of the cube, and
nucleation points are created in the cube by generating x, y and z coordinates independently from the
pseudo-random numbers between 0 and 1. After the
first point is specified, each subsequent random point
is accepted only if it is greater than a minimum allowable distance d from any existing point, until n nuclei
are seeded in the cube. The periodic Voronoi foam
sample used in the FEA model is the “unit” cell of
an infinite periodic foam [Fig. 1(a)]. A dedicated code
has been developed to construct the periodic random
Voronoi structures [14].
2.2. The definition of regularity
The fully ordered limit of the 3D Voronoi tessellation is effectively a cubic lattice of tetrakaidecahedral cells. Such cells have a surface area very close to
the minimum for a given cell volume.
To construct a regular lattice with n identical
tetrakaidecahedral cells in the volume V0, the minimum distance d0 between any two adjacent nuclei is
given by
d0 ⫽
冉 冊
√6 V0 1/3
.
2 √2 n
(1)
To construct a random Voronoi tessellation with n
cells in the volume V0, the maximum d (the minimum
distance between any two nuclei) should be less than
d0; otherwise, it is impossible to obtain n cells. The
regularity of a 3D Voronoi tessellation can be measured by [15]:
a⫽
d
.
d0
(2)
For a regular lattice with tetrakaidecahedral cells, d
equals d0 and a is 1. For a completely random
Voronoi tessellation, d equals 0 and thus a is 0.
3. COMPUTATIONAL ASPECTS AND RESULTS
3.1. General methodology
All struts in the foam are represented mechanically
by beams that are rigidly connected in vertices. In
real foams, the cross-section of the struts is a plateau
border and the area of the cross-section is variable
along the strut length, thickening continuously as the
vertices are approached. For simplicity of the model,
all of the struts are assumed to have the same and
constant plateau border cross-section with area A.
Consequently, the foam relative density r is determined by the cross-section area A and specified by
A
r⫽
Fig. 1. (a) An undeformed random Voronoi foam with 27 complete cells; (b) the deformed structure with periodic boundary
conditions.
冘
N
i⫽1
V0
li
,
(3)
where li are the cell strut lengths and N is the total
number of cell struts.
An ABAQUS standard program was used in the
FEA analysis. Each strut was modelled, depending on
its length, with one to five Timoshenko beam
elements (ABAQUS element type B32). The Young’s
modulus of the struts, Es, was set to 1.0 and the Poisson’s ratio to 0.3. To determine the Young’s moduli
and the Poisson’s ratios related to orthogonal directions, an effective compressive foam strain of ⫺0.001
was imposed in the x, y and z directions indepen-
ZHU et al.: ELASTIC PROPERTIES OF FOAMS
dently in separate analyses. To determine the shear
moduli, an effective compressive foam strain of
⫺0.001 in the y direction and an effective tensile
strain of 0.001 in the x or z direction were imposed
simultaneously. Twenty samples were analysed for
each varying degree of regularity, and the mean
values were taken as the results.
3.2. Boundary conditions
Before performing any calculations, one must
decide upon the most suitable boundary conditions in
the finite element analysis. The best boundary conditions should lead to the average global behaviour
of the 3D foams, and avoid any localised deformation
near the boundaries of the mesh.
Three types of boundary conditions—(1) mixed
boundary conditions, (2) prescribed displacement
boundary conditions and (3) periodic boundary conditions—are usually used in finite element analysis.
The mixed boundary conditions enforce the normal
displacement and eliminate the tangential force and
the bending moment at nodes on the boundaries. The
prescribed displacement boundary conditions impose
very strong restrictions and are usually used in problems related to plastic deformation. The periodic
boundary conditions assume that the corresponding
nodes on the opposite struts of the mesh have the
same expansion in the normal directions, the same
displacements in the other directions, and the same
rotations in all directions [see Fig. 1(b)].
To compare the mixed boundary conditions with
the periodic boundary conditions, some cubic samples
with 128 complete cells were taken from an infinite
perfect foam having tetrakaidecahedral cells and a
relative density of 0.01. Finite element analysis was
performed for each sample to derive the reduced
Young’s modulus (the foam’s Young’s modulus E, an
effective modulus, divided by Esr2), using the mixed
boundary conditions and the periodic boundary conditions, respectively. The reduced Young’s moduli
derived by periodic boundary conditions are almost
the same and very close to the theoretical result of
0.99 [4] for each of the samples. However, the
reduced Young’s moduli derived by the mixed boundary conditions gleaned a wide range of values from
0.7 to 0.86, and all were much lower than the theoretical one. The above test suggests that the mixed
boundary conditions tend to underestimate the
Young’s modulus. Therefore, only periodic boundary
conditions are used from this point onwards.
4895
ratios as well as their standard deviations are given
in Fig. 2(a) and (b). It is shown that the mean of the
non-dimensional Young’s moduli is almost the same
for samples with varying number of cells, but generally, the fewer the number of cells, the larger will be
the standard deviation. When N ⫽ 125, the standard
deviation is about 5% of the mean. Thus the finite
element results given below are the mean results over
20 random Voronoi foam samples having a total number of cells fixed at N ⫽ 125.
3.4. Isotropic properties
Table 1 lists the reduced Young’s moduli, E∗1 , E∗2
and E∗3 , the reduced shear moduli, G∗12, and the Poisson’s ratios, n∗21 and n∗23, of 20 samples with a regularity parameter a ⫽ 0.7. Although the ratios of the
Young’s moduli in three directions (E∗2 /E∗1 and
E∗2 /E∗3 ) varied over a considerable range, the models
were, on average, isotropic.
3.5. Effects of cell regularity on the elastic properties
Taking the strut bending, twisting and stretching as
the deformation mechanisms, Zhu et al. [4] derived
the closed-form results of the effective bulk modulus,
3.3. Mesh sensitivity
A mesh sensitivity study was performed by changing the total number N of cells for samples having a
regularity parameter a ⫽ 0.5. Twenty random
samples were investigated for each different number
of cells: N ⫽ 27, 64, 125, 343 and 512. Each Voronoi
foam was generated by using a different list of random numbers, and had the same relative density of
0.01. The reduced Young’s moduli and the Poisson’s
Fig. 2. Effects of the number of cells on (a) the reduced
Young’s modulus and (b) the Poisson’s ratio of Voronoi foams
having a regularity parameter a ⫽ 0.5 and a relative density
r ⫽ 0.01.
4896
ZHU et al.: ELASTIC PROPERTIES OF FOAMS
Table 1. Non-dimensional Young’s moduli, shear moduli and Poisson’s ratios of 20 isotropic, periodic Voronoi foams with relative density r ⫽
0.01 and regularity parameter a ⫽ 0.7, as determined by FEA
E∗1
E∗2
E∗2
n∗21
n∗23
G∗12
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1.6915
1.5049
1.3569
1.3367
1.5681
1.4122
1.6535
1.4562
1.2560
1.4516
1.4584
1.3868
1.3631
1.5293
1.2940
1.5350
1.4151
1.3788
1.5137
1.5255
1.5495
1.5727
1.3258
1.3423
1.3450
1.3323
1.5556
1.4397
1.3002
1.4371
1.4258
1.4280
1.2980
1.5578
1.3719
1.4695
1.4550
1.3366
1.4137
1.3925
1.6543
1.5281
1.4092
1.2997
1.4675
1.3941
1.5606
1.2987
1.2759
1.5480
1.4798
1.4082
1.3410
1.5715
1.3864
1.4624
1.5529
1.3562
1.3193
1.5349
0.45962
0.48202
0.49577
0.46609
0.44962
0.46757
0.44811
0.41242
0.48658
0.50925
0.48507
0.48241
0.47211
0.48507
0.51283
0.45206
0.52165
0.47021
0.40762
0.48344
0.48298
0.46820
0.46004
0.49355
0.50414
0.48779
0.50265
0.53923
0.46991
0.44617
0.47004
0.47064
0.48902
0.46171
0.44210
0.50093
0.42870
0.48826
0.54547
0.47317
0.5479
0.5230
0.4467
0.4571
0.4916
0.4633
0.5490
0.5117
0.4324
0.4777
0.4838
0.4770
0.4485
0.5213
0.4451
0.5139
0.4738
0.4595
0.5150
0.4849
Mean
Standard deviation
1.4544
0.11010
1.4175
0.08641
1.4424
0.10644
0.47248
0.02859
0.48124
0.02839
0.4799
0.02984
Model no.
Young’s modulus, shear modulus and the Poisson’s
ratio for a perfect foam with tetrakaidecahedral cells,
and found that the regular foam is almost isotropic.
If the cross-section of the strut is a plateau border,
the effective Young’s modulus is given by
E⫽
1.009Esr2
,
1 ⫹ 1.514r
(4)
the effective shear modulus by
G⫽
0.32Esr2
,
1 ⫹ 0.96r
Esr
9
(5)
(6)
and the Poisson’s ratio by
n∗ ⫽ 0.5
1⫺1.514r
,
1 ⫹ 1.514r
(8)
the reduced shear modulus by
G∗ ⫽
G
0.32
⫽
,
2
Esr
1 ⫹ 0.96r
(9)
and the reduced bulk modulus by
the bulk modulus by
K⫽
E
1.009
⫽
,
2
Esr
1 ⫹ 1.514r
E∗ ⫽
(7)
where r is the relative density of the foam and Es is
the Young’s modulus of the solid from which the
foam is made.
For low-density foams (r smaller than 0.03), strut
bending and twisting are the dominant deformation
mechanisms. The reduced Young’s modulus of opencell foams can be expressed by
K∗ ⫽
K
1
⫽ .
Esr 9
(10)
For random Voronoi foams, the effects of cell regularity on reduced Young’s modulus, shear modulus,
bulk modulus and Poisson’s ratio of foams with a
constant relative density of 0.01 are shown in Figs.
3–6. For highly irregular (a smaller than 0.5) and
very low-density (r ⫽ 0.01) foams, the Young’s
modulus and the shear modulus are about 50% larger
than those of a perfect regular foam, and the bulk
modulus is about 20% smaller than that of a perfect
regular foam. With increasing cell regularity, the
trend is towards the values for the perfect tetrakaidecahedronic foam, with the Young’s modulus and the
shear modulus reducing slightly, the bulk modulus
increasing significantly, but the Poisson’s ratio constant at the perfect foam value [equation (7)].
3.6. Effects of foam relative density
Figures 7 and 8 present the reduced Young’s moduli and the Poisson’s ratios as functions of relative
density for random Voronoi foams with different
ZHU et al.: ELASTIC PROPERTIES OF FOAMS
Fig. 3. Effects of cell regularity on the reduced Young’s modulus of random Voronoi foams having a constant relative density r ⫽ 0.01. The diamond points on the curve are the results
predicted by equation (14).
Fig. 4. Effects of cell regularity on the reduced shear modulus
of random Voronoi foams having a constant relative density
r ⫽ 0.01.
4897
Fig. 5. Effects of cell regularity on the reduced bulk modulus
of random Voronoi foams having a constant relative density
r ⫽ 0.01.
Fig. 6. Effects of cell regularity on the Poisson’s ratio of random Voronoi foams having a constant relative density r ⫽
0.01.
values of cell regularity parameter a. For varying
degrees of cell regularity parameter a, both the
reduced Young’s modulus and the Poisson’s ratio
decrease with increasing relative density. The theoretical results [equations (7) and (8)] and the computational results for a perfect foam are also included
for comparison.
4. DISCUSSION
Gibson and Ashby have done very extensive studies of the mechanical properties of foams. Taking the
strut bending as the only deformation mechanism, and
using dimensional analysis and data fitting, they [1]
derived the Young’s modulus, the shear modulus and
the Poisson’s ratio for low-density foams:
E ⫽ Esr2,
(11)
Fig. 7. Effects of relative density on the reduced Young’s
modulus of random Voronoi foams with varying degrees of
regularity parameter a.
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ZHU et al.: ELASTIC PROPERTIES OF FOAMS
Table 2. The relations between the compressive strain and the tangential
(Young’s) modulus of a perfect low-density (r less than 0.01) foam
Compressive strain
Reduced tangential modulus
0.0
0.01
0.02
0.03
0.04
0.05
Fig. 8. Effects of relative density on the Poisson’s ratio of random Voronoi foams with varying degrees of regularity parameter a.
3
G ⫽ Esr2
8
(12)
1
n∗ ⫽ .
3
(13)
and
The struts were treated as beams; hence, their
model and results apply for low-density foams.
Gibson and Ashby’s reduced Young’s modulus is
a constant independent of relative density and very
similar to our theoretical prediction for a perfect foam
[equation (4)]. However, a real foam is a random
foam, and the strut cross-section is a plateau border.
Figure 7 would suggest that for real foams the
reduced Young’s modulus is really rather dependent
on the relative density, suggesting that a curve convex
to the relative Young’s modulus would be a more
approximate fit to the comprehensive data set of Fig.
5.9, p. 131 of Ref. [1] Furthermore, with such a comprehensive data set, gleaned from a wide range of
sources, the question of experimental boundary conditions and sample size must always be an issue. To
measure the Young’s modulus, a foam is usually
compressed to a strain of about 0.02. According to
the theoretical analysis [5], the relationships between
the compressive strain and the reduced tangential
(Young’s) modulus of a perfect low-density (r less
than 0.01) foam are given in Table 2. If we measure
the Young’s modulus at a compressive strain of 0.02,
the result obtained is about 6.5% smaller even for
very low-density and periodic foams under periodic
boundary conditions.
Our computational Poisson’s ratios (Fig. 6) for
low-density (r less than 0.01) random foams are
almost the same as the theoretical prediction [equ-
1.003
0.9687
0.93736
0.90699
0.87929
0.85399
ation (7)] for a perfect regular foam. However, Gibson and Ashby’s experimentally measured Poisson’s
ratios for random foams are 1/3 [equation (13)],
which is much smaller. The reasons are the same as
in measuring the Young’s modulus. Figure 8 shows
that the Poisson’s ratios reduce with increasing relative density for foams having varying degrees of
regularity. Also, according to the theoretical analysis
[5], the Poisson’s ratio of a foam reduces with compressive strain. Consequently, the experimentally
measured Poisson’s ratio should be smaller than the
theoretical one.
Our computational results indicate that, when the
relative density r is less than 0.04, the highly irregular foams have a much larger Young’s modulus and
shear modulus than a perfect foam, which is very
similar to two-dimensional foams [12]. To explain
this qualitatively, we assume that the individual random cells in a foam can be treated as springs in parallel. In a foam, each strut is shared by three cells and
hence the relative density of a cell is one-third of the
total strut length of the cell, multiplied by the area of
the strut cross-section and divided by the cell volume.
Based on the analysis of 106 cells of varying degrees
of regularity, Fig. 9 shows the normalised cell density
(the density of a cell divided by the overall density
of the Voronoi foam) distributions for 3D Voronoi
tessellations. Assuming that the power law [equation
(11)] can be used for the individual cells of a random
Voronoi foam having a low relative density, the
reduced effective Young’s modulus of the foam
will be
冘
N
E
⫽
d2P (d ),
Esr2 i ⫽ 1 i v i
(14)
where di is the normalised cell density (the cell density divided by the overall foam density) and Pv(di)
is the volume probability of the cells having a normalised cell density di.
Using the cell density distributions (Fig. 9), equation (14) gives a reduced Young’s moduli for random Voronoi foams with varying degrees of regularity, as shown in Fig. 3. Actually, there are some
four-faced cells in highly irregular foams and those
cells are much stiffer than the power law [equation
(11)] describes. Many other different geometrical
ZHU et al.: ELASTIC PROPERTIES OF FOAMS
4899
Fig. 9. The normalised density distributions of cells in random Voronoi foams having different values of the
regularity parameters: (a) a ⫽ 0.1, (b) a ⫽ 0.3, (c) a ⫽ 0.5 and (d) a ⫽ 0.7. The data were grouped in equal
intervals of width 0.1.
properties also affect the elastic properties of random
Voronoi foams. Equation (14) is just an attempt to
explain our FEA results.
The difference between the computational result
and the theoretical prediction [equation (8)] for the
Young’s modulus of a perfect foam increases with
the foam relative density. In the finite element analysis, the Timoshenko beam elements considered the
shear deformation mechanism which was ignored in
our theoretical analysis [4]; hence, equation (8) overestimated the Young’s modulus of a perfect foam
with a large relative density. However, the beam
theory was used in both the theoretical analysis [4]
and the FEA simulations, and the struts should be thin
and long. If the cross-section of the struts is an equilateral triangle of side t in a perfect foam, the relations
between the foam relative density r and the ratio of
the strut length l to the strut cross-section dimension
t are listed in Table 3.
To use the beam theory correctly, l/t should be
Table 3. The relationship between the foam relative density r and the
ratio of the strut length l to the strut cross-section dimension t
l/t
r
3
4
5
6
7
0.051
0.0287
0.0183
0.0128
0.0094
larger than 4 and the foam relative density should be
smaller than 0.0287. However, in real foams, the strut
cross-section is a plateau border that has a larger
dimension than an equilateral triangle. That makes r
smaller. Thus, there are errors in the computational
results when r is larger than 0.025.
Van der Burg et al. [9] have modelled foam structure by random Voronoi cells using finite element
simulation. They treated the struts as beam elements
with the same and constant circular cross-section.
4900
ZHU et al.: ELASTIC PROPERTIES OF FOAMS
Although they found that the more irregular the
foams, the larger the Young’s modulus, they claimed
that the Young’s modulus of a highly irregular foam
agrees with Gibson and Ashby’s result. However, we
do not favour their conclusion. According to the
theoretical analysis [4], if the cross-section of the
struts is a circle, the Young’s modulus of a perfect
foam is
E⫽
0.60021Esr2
.
1 ⫹ 0.9003r
(15)
It is about 40% smaller than the regular foam whose
strut cross-section is a plateau border [see equation
(4)]. Owing to the facts that all of the struts at the
boundaries are normal to the boundary faces in their
models, and that they did not use periodic boundary
conditions, it is very difficult to compare their results
with ours.
5. CONCLUSIONS
Finite element models of random periodic Voronoi
foams were developed to investigate the effects of
cell regularity on the elastic properties. All of the
struts were treated as beam elements with the same
and constant cross-section; consequently, the analyses
were limited to models having low relative density.
The results indicate that, for low-density foams, the
highly irregular foams have a larger Young’s modulus and shear modulus, and a smaller bulk modulus,
than a perfect foam. The Poisson’s ratio does not
change with the cell regularity parameter a, but
reduces gradually with increasing relative density r.
Although the Young’s modulus varies in different
directions for each sample, the models are, on average, isotropic, and the relationship G∗ ⫽ E∗/[2(1 ⫹
n∗)] holds for foams with different values of regularity parameter a.
Acknowledgements—The authors wish to thank Dr. Anthony
Cunningham of Huntsman Corporation for his continued support of, and intellectual contributions to this work.
REFERENCES
1. Gibson, L. J. and Ashby, M. F., Cellular Solids: Structure
and Properties. Cambridge University Press, Cambridge,
1988.
2. Warren, W. E. and Kraynik, A. M., J. Appl. Mech., 1988,
55, 341.
3. Warren, W. E. and Kraynik, A. M., J. Appl. Mech., 1997,
64, 787.
4. Zhu, H. X., Knott, J. F. and Mills, N. J., J. Mech. Phys.
Solids. 1997, 45, 319.
5. Zhu, H. X., Mills, N. J. and Knott, J. F., J. Mech. Phys.
Solids. 1997, 45, 1875.
6. Zhu, H. X. and Mills, N. J., J. Mech. Phys. Solids. 1999,
47, 1437.
7. Zhu, H. X. and Mills, N. J., Int. J. Solids Struct., 2000,
37, 1931.
8. Mills, N. J. and Zhu, H. X., J. Mech. Phys. Solids. 1999,
47, 669.
9. Van der Burg, M. W. D., Shulmeister, V., Van der Giessen,
E. and Marissen, V., J. Cell. Plast., 1997, 33, 31.
10. Shulmeister, V., Van der Burg, M. W. D., Van der Giessen,
E. and Marissen, V., Mech. Mater., 1998, 30, 125.
11. Silva, M. J., Hayes, W. C. and Gibson, L. J., Int. J. Mech.
Sci., 1995, 37, 1161.
12. Zhu, H. X., Hobdell, J. R. and Windle, A. H., J. Mech.
Phys. Solids., 2000, submitted for publication.
13. Chen, C., Lu, T. J. and Fleck, N. A., J. Mech. Phys. Solids.
1999, 47, 2235.
14. Zhu, H. X. and Windle, A. H., Phil. Mag., 2000, submitted
for publication.
15. Zhu, H. X., Thorpe, S. M. and Windle, A. H., Phil. Mag.,
2000, submitted for publication.