The Relaxation of Two-well Energies with Possibly Unequal Moduli

The elastic energy of a multiphase solid is a function of its microstructure. Determining the infimum of the energy of such a solid and characterizing the associated optimal microstructures is an important problem that arises in the modeling of the shape memory effect, microstructure evolution, and optimal design. Mathematically, the problem is to determine the relaxation under fixed phase fraction of a multiwell energy. This paper addresses two such problems in the geometrically linear setting. First, in two dimensions, we compute the relaxation under fixed phase fraction for a two-well elastic energy with arbitrary elastic moduli and transformation strains, and provide a characterization of the optimal microstructures and the associated strain. Second, in three dimensions, we compute the relaxation under fixed phase fraction for a two-well elastic energy when either (1) both elastic moduli are isotropic, or (2) the elastic moduli are well ordered and the smaller elastic modulus is isotropic. In both cases we impose no restrictions on the transformation strains. We provide a characterization of the optimal microstructures and the associated strain. We also compute a lower bound that is optimal except possibly in one regime when either (1) both elastic moduli are cubic, or (2) the elastic moduli are well ordered and the smaller elastic modulus is cubic; for moduli with arbitrary symmetry we obtain a lower bound that is sometimes optimal. In all these cases we impose no restrictions on the transformation strains and whenever the bound is optimal we provide a characterization of the optimal microstructures and the associated strain. In both two and three dimensions the quasiconvex envelope of the energy can be obtained by minimizing over the phase fraction. We also characterize optimal microstructures under applied stress.

[1]  Olivier Pironneau,et al.  Optimal Shape Design , 2000 .

[2]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[3]  Ingram Olkin,et al.  Inequalities: Theory of Majorization and Its Application , 1979 .

[4]  利久 亀井,et al.  California Institute of Technology , 1958, Nature.

[5]  Robert V. Kohn,et al.  Optimal lower bounds on the elastic energy of a composite made from two non-well-ordered isotropic materials , 1994 .

[6]  S. Childress,et al.  Macroscopic Properties of Disordered Media , 1982 .

[7]  H. Weinert.  Ekeland, I. / Temam, R., Convex Analysis and Variational Problems. Amsterdam‐Oxford. North‐Holland Publ. Company. 1976. IX, 402 S., Dfl. 85.00. US $ 29.50 (SMAA 1) , 1979 .

[8]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[9]  R. Kohn,et al.  Structural design optimization, homogenization and relaxation of variational problems , 1982 .

[10]  G. Milton.  The Theory of Composites , 2002 .

[11]  G. Allaire,et al.  Minimizers for a double-well problem with affine boundary conditions , 1999, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[12]  Björn Birnir,et al.  Complex hill's equation and the complex periodic korteweg‐de vries equations , 1986 .

[13]  The structure of rank-one convex quadratic forms on linear elastic strains , 2003, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[14]  Paolo Marcellini,et al.  Semicontinuity problems in the calculus of variations , 1980 .

[15]  Gérard A. Maugin,et al.  Continuum models and discrete systems , 1990 .

[16]  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.

[17]  K. Lurie,et al.  Exact estimates of the conductivity of a binary mixture of isotropic materials , 1986, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[18]  Andrej Cherkaev,et al.  Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion , 1984, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[19]  Robert V. Kohn,et al.  The relaxation of a double-well energy , 1991 .

[20]  A. Roytburd.  Elastic domains and polydomain phases in solids , 1993 .

[21]  Luc Tartar,et al.  Compensated compactness and applications to partial differential equations , 1979 .

[22]  Robert V. Kohn,et al.  Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. , 1995 .

[23]  David Kinderlehrer,et al.  Equilibrium configurations of crystals , 1988 .

[24]  K. Bhattacharya.  Microstructure of martensite : why it forms and how it gives rise to the shape-memory effect , 2003 .

[25]  Andrej Cherkaev,et al.  Microstructures of Composites of Extremal Rigidity and Exact Bounds on the Associated Energy Density , 1997 .

[26]  L. Gibiansky,et al.  The exact coupled bounds for effective tensors of electrical and magnetic properties of two-component two-dimensional composites , 1992, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

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

[28]  V. P. Smyshlyaev,et al.  On the relation of a three-well energy , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

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

[30]  G. D. Maso,et al.  An Introduction to-convergence , 1993 .

[31]  I. Chenchiah.  Energy-minimizing microstructures in multiphase elastic solids , 2004 .

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

[33]  L. Tartar.  Optimal Shape Design, Lecture Notes in Maths. 1740, A. Cellina & A. Ornelas eds, 47{156, Springer, 2000. An Introduction to the Homogenization Method in Optimal Design , 2000 .

[34]  Pablo Pedregal,et al.  Characterizations of young measures generated by gradients , 1991 .

[35]  Sanjay Govindjee,et al.  The free energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis , 2003 .

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

[37]  Robert V. Kohn,et al.  Explicit relaxation of a variational problem in optimal design , 1983 .

[38]  R. Kohn,et al.  Geometric parameters and the relaxation of multiwell energies , 1993 .

[39]  Ines Gloeckner,et al.  Variational Methods for Structural Optimization , 2002 .

[40]  Andrej Cherkaev,et al.  G-closure of some particular sets of admissible material characteristics for the problem of bending of thin elastic plates , 1984 .

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

[42]  A twodimensional variational model for the equilibrium configuration of an incompressible, elastic body with a three-well elastic potential , 1999 .

[43]  L. Cesari,et al.  Contributions to modern calculus of variations , 1987 .

[44]  Nikan B. Firoozye.  Optimal use of the translation method and relaxations of variational problems , 1991 .

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

[46]  J. Ball,et al.  Fine phase mixtures as minimizers of energy , 1987 .

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

[48]  Andrej Cherkaev,et al.  On the effective conductivity of polycrystals and a three‐dimensional phase‐interchange inequality , 1988 .

[49]  A. Pipkin,et al.  ELASTIC MATERIALS WITH TWO PREFERRED STATES , 1991 .

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

[51]  G. A. Seregin.  The uniqueness of solutions of some variational problems of the theory of phase equilibrium in solid bodies , 1996 .

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

[53]  Bernard Dacorogna,et al.  Quasiconvexity and relaxation of nonconvex problems in the calculus of variations , 1982 .