TY - JOUR
T1 - Variational bounds on the effective moduli of anisotropic composites
AU - Milton, Graeme W.
AU - Kohn, Robert V.
N1 - Funding Information:
We have been stimulated in this work by discussions with many colleagues, in particular G. Francfort, F. Murat and L. Tartar. GWM gratefully acknowledgess upport from the California Institute of Technology through a Weingart Fellowship, from the National Science Foundation through grant DMS-8312229, and from the Air Force Ohice of Scientific Research through grants AFOSR85-0017 and AFOSR86-0352. RVK gratefully acknowledges support from the National Science Foundation under grant DMS-83 12229f,r om the Office of Naval Research under grant NOOO14-83-0536a,n d from the Sloan Foundation. We are pleased to thank Denise Rosenthal for her help in preparing the manuscript and J. Berryman for his comments on the text.
PY - 1988
Y1 - 1988
N2 - The vritional inequalities of Hashin and Shtrikman are transformed to a simple and concise form. They are used to bound the effective conductivity tensor σ* of an anisotropic composite made from an arbitrary number of possibly anisotropic phases, and to bound the effective elasticity tensor C* of an anisotropic mixture of two well-ordered isotropic materials. The bounds depend on the conductivities and elastic moduli of the components and their respective volume fractions. When the components are isotropic the conductivity bounds, which constrain the eigenvalues of σ*, include those previously obtained by Hashin and Shtrikman, Murat and Tartar, and Lurie and Cherkaev. Our approach can also be used in the context of linear elasticity to derive bounds on C* for composites comprised of an arbitrary number of anisotropic phases. For two-component composites our bounds are tighter than those obtained by Kantor and Bergman and by Francfort and Murat, and are attained by sequentially layered laminate materials.
AB - The vritional inequalities of Hashin and Shtrikman are transformed to a simple and concise form. They are used to bound the effective conductivity tensor σ* of an anisotropic composite made from an arbitrary number of possibly anisotropic phases, and to bound the effective elasticity tensor C* of an anisotropic mixture of two well-ordered isotropic materials. The bounds depend on the conductivities and elastic moduli of the components and their respective volume fractions. When the components are isotropic the conductivity bounds, which constrain the eigenvalues of σ*, include those previously obtained by Hashin and Shtrikman, Murat and Tartar, and Lurie and Cherkaev. Our approach can also be used in the context of linear elasticity to derive bounds on C* for composites comprised of an arbitrary number of anisotropic phases. For two-component composites our bounds are tighter than those obtained by Kantor and Bergman and by Francfort and Murat, and are attained by sequentially layered laminate materials.
UR - http://www.scopus.com/inward/record.url?scp=45549116280&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=45549116280&partnerID=8YFLogxK
U2 - 10.1016/0022-5096(88)90001-4
DO - 10.1016/0022-5096(88)90001-4
M3 - Article
AN - SCOPUS:45549116280
SN - 0022-5096
VL - 36
SP - 597
EP - 629
JO - Journal of the Mechanics and Physics of Solids
JF - Journal of the Mechanics and Physics of Solids
IS - 6
ER -