A new methodology to derive the linear effective constitutive law for a group of composites with random microstructure of a special kind is described as an extension of the methodology proposed in Warren and Kraynik (1988) and of the methodology used in polycrystal theory. The results are expressed in the form of specific bounds on effective elastic constants. Practical importance is in the specific bounds when the methodology is applied to cellular solids. Several examples are shown and compared with other published results. The new contribution of this paper lies in the presentation of the methodology, derivation of new specific bounds in two dimensions, and comments related to already published works on cellular solids.

1.
Avellaneda
M.
, and
Milton
G. W.
,
1989
, “
Optimal Bounds on the Effective Bulk Modulus of Polycrystals
,”
SIAM Journal of Applied Mathematics
, Vol.
49
, pp.
824
837
.
2.
Bakhvalov, N., and Panasenko, G., 1989, Homogenization: Averaging Processes in Periodic Media, Kluwer, Dordrecht, The Netherlands.
3.
Bensoussan, A., Lions, J. L., and Papanicolau, G., 1978, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam.
4.
Christensen
R. M.
,
1994
, “
Heterogeneous Material Mechanics at Various Scales
,”
ASME Applied Mechanics Reviews
, Vol.
47
, pp.
S20–S33
S20–S33
.
5.
Dimitmvova´, Z., 1997, “Mechanical Properties of Cellular Materials,” Ph.D. dissertation, Instituto Superior Te´cnico, Lisbon, Portugal.
6.
Dimitmvova´
Z.
, and
Faria
L.
,
1999
, “
New Methodology to Establish Bounds on Effective Properties of Cellular Solids
,”
Mechanics of Composite Materials and Structures
, Vol.
6
, pp.
331
346
.
7.
Duvaut, G., 1976, “Homogeneization et Materiaux Composite,” Theoretical and Applied Mechanics, ed. P. Ciarlet and M. Rouseau, eds. North-Holland, Amsterdam.
8.
Ferrari
M.
, and
Johnson
G. C.
,
1988
, “
The Equilibrium Properties of a 6 mm Polycrystal Exhibiting Transverse Isotmpy
,”
Journal of Applied Physics
, Vol.
63
, pp.
4460
4468
.
9.
Gent
A. N.
, and
Thomas
A. G.
,
1959
, “
The Deformation of Foamed Elastic Materials
,”
Journal of Applied Polymer Science
, Vol.
1
, pp.
107
113
.
10.
Gibson, L. J., and Ashby, M. F., 1988, Cellular Solids. Structure and Properties, Pergamon Press, Oxford.
11.
Guedes, J. M., 1990, “Nonlinear Computational Models for Composite Materials Using Homogenization,” Ph.D. dissertation, The University of Michigan, Ann Arbor, MI.
12.
Hashin
Z.
,
1965
, “
On Elastic Behavior of Fibre Reinforced Materials of Arbitrary Transverse Phase Geometry
,”
Journal of the Mechanics and Physics of Solids
, Vol.
13
, pp.
119
134
.
13.
Hashin, Z., 1970, “Theory of Composite Materials,” Mechanics of Composite Materials, F. W. Wendt, H. Liebowitz, and N. Perrone, eds., Pergamon Press, Oxford.
14.
Hashin
Z.
,
1983
, “
Analysis of Composite Materials—A Survey
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
50
, pp.
481
505
.
15.
Hashin
Z.
, and
Shtrikman
S.
,
1962
a, “
On Some Variational Principles in Anisotropic and Nonhomogeneous Elasticity
,”
Journal of the Mechanics and Physics of Solids
, Vol.
10
, pp.
335
342
.
16.
Hashin
Z.
, and
Shtrikman
S.
,
1962
b, “
A Variational Approach to the Theory of the Elastic Behavior of Polycrystals
,”
Journal of the Mechanics and Physics of Solids
, Vol.
10
, pp.
343
352
.
17.
Hashin
Z.
, and
Shtrikman
S.
,
1963
, “
A Variational Approach to the Theory of the Elastic Behavior of Multiphase Materials
,”
Journal of the Mechanics and Physics of Solids
, Vol.
11
, pp.
127
140
.
18.
Lagzdins, A., Tamuzs, V., Teters, G., and Kregers, A., 1992, Orientational Averaging in Mechanics of Solids, Longman Group, London.
19.
Lekhnitskii, S. G., 1981, Theory of Elasticity of an Anisotropic Body, Mir, Moscow.
20.
Lu
B.
, and
Torquato
S.
,
1990
, “
n-Point Probability Functions for a Lattice Model of Heterogeneous Media
,”
Physical Review B
, Vol.
42
, pp.
4453
4459
.
21.
Meister
R.
, and
Peselnick
L.
,
1966
, “
Variational Method of Determining Effective Moduli of Polycrystals with Tetragonal Symmetry
,”
Journal of Applied Physics
, Vol.
37
, pp.
4121
4125
.
22.
Nemat-Nasser, S., and Hori, M., 1993, Micromechanics: Overall Properties of Heterogeneous Materials (North-Holland Series in Applied Mathematics and Mechanics), Vol. 37, ed. J. D. Achenbach, B. Budiansky, H. A. Lauwerier, P. G. Saffman, L. Van Wijngaarden, and J. R. Willis, eds., North-Holland Amsterdam.
23.
Ostoja-Starzewski
M.
,
1993
, “
Micromechanics as a Basis of Stochastic Finite Elements and Differences: An Overview
,”
ASME Applied Mechanics Reviews
, Vol.
46
, pp.
S136–S147
S136–S147
.
24.
Ostoja-Starzewski
M.
,
1994
, “
Micromechanics as a Basis of Continuum Random Fields
,”
ASME Applied Mechanics Reviews
, Vol.
47
, pp.
S221–S230
S221–S230
.
25.
Ostoja-Starzewski
M.
,
1998
, “
Random Field Models of Heterogeneous Materials
,”
International Journal of Solids and Structures
, Vol.
35
, pp.
2429
2455
.
26.
Peselnick
L.
, and
Meister
R.
,
1965
, “
Variational Method of Determining Effective Moduli of Polycrystals: (A) Hexagonal Symetry, (B) Trigonal Symmetry
,”
Journal of Applied Physics
, Vol.
36
, pp.
2879
2884
.
27.
Schulgasser
K.
,
1983
, “
Sphere Assemblage Model for Polycrystals and Symmetric Materials
,”
Journal of Applied Physics
, Vol.
54
, pp.
1380
1382
.
28.
Suquet, P. M., 1985a, “Elements of Homogenization for Inelastic Solid Mechanics,” Homogenization Techniques for Composite Media, Proceedings, Udine, Italy, E. Sanchez-Palencia, and A. Zaoui, eds., Lecture Notes in Physics, 272, Springer-Verlag, New York, pp. 193–278.
29.
Suquet, P. M., 1985b, “Local and Global Aspects in the Mathematical Theory of Plasticity,” Plasticity Today—Modeling Methods and Applications, A. Sawczuk, and G. Bianchi, Elsevier, London, pp. 279–310.
30.
Torquato
S.
,
1991
, “
Random Heterogeneous Media: Microstructure and Improved Bounds on Effective Properties
,”
ASME Applied Mechanics Reviews
, Vol.
44
, pp.
37
76
.
31.
Torquato
S.
,
1994
, “
Macroscopic Behavior of Random Media from the Microstructure
,”
ASME Applied Mechanics Reviews
, Vol.
47
, pp.
S29–S37
S29–S37
.
32.
Torquato
S.
,
1998
, “
Morphology and Effective Properties of Disordered Heterogeneous Media
,”
International Journal of Solids and Structures
, Vol.
35
, pp.
2385
2406
.
33.
Torquato
S.
, and
Stell
G.
,
1982
, “
Microstructure of Two-Phase Random Media. I. The n-Point Probability Functions
,”
Journal of Chemical Physics
, Vol.
77
, pp.
2071
2077
.
34.
Warren
W. E.
, and
Kraynik
A. M.
,
1988
, “
The Linear Elastic Properties of Open-Cell Foams
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
55
, pp.
341
346
.
35.
Watt
J. P.
,
1979
, “
Hashin-Shtrikman Bounds on the Effective Elastic Moduli of Polycrystals With Orthorhombic Symmetry
,”
Journal of Applied Physics
, Vol.
50
, pp.
6290
6295
.
36.
Watt
J. P.
,
1980
, “
Hashin-Shtrikman Bounds on the Effective Elastic Moduli of Polycrystals with Monoclinic Symmetry
,”
Journal of Applied Physics
, Vol.
51
, pp.
1520
1524
.
37.
Watt
J. P.
, and
Peselnick
L.
,
1980
, “
Clarification of the Hashin-Shtrikman Bounds on the Effective Elastic Moduli of Polycrystals with Hexagonal, Trigonal, and Totragonal Symmetries
,”
Journal of Applied Physics
, Vol.
51
, pp.
1525
1531
.
38.
Werner
E.
,
Siegmund
T.
,
Weinhandl
H.
, and
Fischer
F. D.
,
1994
, “
Properties of Random Polycrystalline Two-Phase Materials
,”
ASME Applied Mechanics Reviews
, Vol.
47
, pp.
S231–S240
S231–S240
.
This content is only available via PDF.
You do not currently have access to this content.