Microstructures minimizing the energy of a two phase elastic composite in two space dimensions.

Abstract For modeling coherent phase transformations, and for applications to structural optimization, it is of interest to identify microstructures with minimal energy or maximal stiffness. The existence of a particularly simple microstructure with extremal elastic behavior, in the context of two-phase composites made from isotropic components in two space dimensions, has previously been shown. This “Vigdergauz microstructure” consists of a periodic array of appropriately shaped inclusions. We provide an alternative discussion of this microstructure and its properties. Our treatment includes an explicit formula for the shape of the inclusion, and an analysis of various limits. We also discuss the significance of this microstructure (i) for minimizing the maximum stress in a composite, and (ii) as a large volume fraction analog of Michell trusses in the theory of structural optimization.

[1]  W. C. Royster.  Theory of Functions of a Complex Variable. vol. 1. A. I. Markushevich. Translated from the Russian by Richard A. Silverman. Prentice-Hall, Englewood Cliffs, N. J., 1965. xvi + 459 pp. Illus. $16 , 1965 .

[2]  Martin P. Bendsøe,et al.  A Displacement-Based Topology Design Method with Self-Adaptive Layered Materials , 1993 .

[3]  S. B. Vigdergauz.  Optimality conditions in axisymmetric problems of elasticity theory , 1982 .

[4]  Zvi Hashin,et al.  The Elastic Moduli of Heterogeneous Materials , 1962 .

[5]  S. Vigdergauz.  Two-Dimensional Grained Composites of Extreme Rigidity , 1994 .

[6]  J. Cahn,et al.  A simple model for coherent equilibrium , 1984 .

[7]  Andrej Cherkaev,et al.  Effective Characteristics of Composite Materials and the Optimal Design of Structural Elements , 1997 .

[8]  T. Mura,et al.  Elastic fields of inclusions in anisotropic media , 1971 .

[9]  J. K. Lee,et al.  The strain energy of a coherent ellipsoidal precipitate , 1974 .

[10]  Graeme W. Milton,et al.  On characterizing the set of possible effective tensors of composites: The variational method and the translation method , 1990 .

[11]  Simona Socrate,et al.  Numerical determination of the elastic driving force for directional coarsening in Ni-superalloys , 1993 .

[12]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[13]  Andrej Cherkaev,et al.  Design of Composite Plates of Extremal Rigidity , 1997 .

[14]  Peter W Voorhees,et al.  THE EQUILIBRIUM SHAPE OF A MISFITTING PRECIPITATE , 1994 .

[15]  The inverse problem of the two-dimensional theory of elasticity in the hydrodynamic formulation , 1983 .

[16]  S. Vigdergauz.  Integral equation of the inverse problem of the plane theory of elasticity: PMM vol. 40, n≗ 3, 1976, pp. 566–569 , 1976 .

[17]  Gilles A. Francfort,et al.  Homogenization and optimal bounds in linear elasticity , 1986 .

[18]  Robert V. Kohn,et al.  Optimal bounds for the effective energy of a mixture of isotropic, incompressible, elastic materials , 1988 .

[19]  S. Shtrikman,et al.  A variational approach to the theory of the elastic behaviour of multiphase materials , 1963 .

[20]  Marco Avellaneda,et al.  Optimal bounds and microgeometries for elastic two-phase composites , 1987 .

[21]  George I. N. Rozvany,et al.  Structural Design via Optimality Criteria , 1989 .

[22]  L. Wheeler,et al.  On rigid inclusions of minimum stress concentration , 1986 .

[23]  G.I.N. Rozvany,et al.  OPTIMIZATION OF STRUCTURAL GEOMETRY , 1977 .

[24]  Piecewise-homogeneous plates of extremal stiffness☆ , 1989 .

[25]  W. Burnside.  Theory of Functions of a Complex Variable , 1893, Nature.

[26]  R. Kohn,et al.  Optimal design and relaxation of variational problems, III , 1986 .

[27]  R. Kohn,et al.  Variational bounds on the effective moduli of anisotropic composites , 1988 .

[28]  G. Allaire,et al.  Optimal design for minimum weight and compliance in plane stress using extremal microstructures , 1993 .

[29]  Robert V. Kohn,et al.  Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions , 1993 .

[30]  Graeme W. Milton,et al.  Modelling the Properties of Composites by Laminates , 1986 .

[31]  H. Aaronson,et al.  The elastic strain energy of coherent ellipsoidal precipitates in anisotropic crystalline solids , 1977 .

[32]  Robert V. Kohn,et al.  Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. , 1994 .

[33]  A. Pineau.  Influence of uniaxial stress on the morphology of coherent precipitates during coarsening—elastic energy considerations , 1976 .

[34]  L. Wheeler,et al.  On voids of minimum stress concentration , 1982 .

[35]  S. Vigdergauz.  On a case of the inverse problem of two-dimensional theory op elasticity: PMM vol. 41, n≗ 5, 1977, pp. 902–908 , 1977 .

[36]  G. Papanicolaou,et al.  Les méthodes de l'homogénéisation : théorie et applications en physique , 1985 .

[37]  Martin P. Bendsøe,et al.  Topology design of structures , 1993 .

[38]  Y. Grabovsky.  Bounds and extremal microstructures for two-component composites: a unified treatment based on the translation method , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[39]  R. A. Silverman,et al.  Theory of Functions of a Complex Variable , 1968 .

[40]  Robert V. Kohn,et al.  Optimal bounds on the effective behavior of a mixture of two well-ordered elastic materials , 1993 .

[41]  M. Bendsøe,et al.  Generating optimal topologies in structural design using a homogenization method , 1988 .

[42]  Robert Lipton,et al.  Inequalities for electric and elastic polarization tensors with applications to random composites , 1993 .

[43]  D. Kinderlehrer,et al.  Homogenization and effective moduli of materials and media , 1986 .

[44]  G. P. Cherepanov.  Inverse problems of the plane theory of elasticity: PMM vol. 38, n≗ 6, 1974, pp. 963–979 , 1974 .