The Simply Laminated Microstructure in Martensitic Crystals that Undergo a Cubic‐to‐Orthorhombic Phase Transformation

Abstract. We study simply laminated microstructures of a martensitic crystal capable of undergoing a cubic‐to‐orthorhombic transformation of type ${\mathcal P}^{(432)} \to {\mathcal P}^{(222)'}$. The free energy density modeling such a crystal is minimized on six energy wells that are pairwise rank‐one connected. We consider the energy minimization problem with Dirichlet boundary data compatible with an arbitrary but fixed simple laminate. We first show that for all but a few isolated values of transformation strains, this problem has a unique Young measure solution solely characterized by the boundary data that represents the simply laminated microstructure. We then present a theory of stability for such a microstructure, and apply it to the conforming finite element approximation to obtain the corresponding error estimates for the finite element energy minimizers.

[1]  C. B. Morrey.  Multiple Integrals in the Calculus of Variations , 1966 .

[2]  J. Ericksen.  Stable equilibrium configurations of elastic crystals , 1986 .

[3]  D. Kinderlehrer,et al.  Numerical analysis of oscillations in multiple well problems , 1995 .

[4]  Richard D. James,et al.  Kinetics of materials with wiggly energies: Theory and application to the evolution of twinning microstructures in a Cu-Al-Ni shape memory alloy , 1996 .

[5]  Kewei Zhang.  A construction of quasiconvex functions with linear growth at infinity , 1992 .

[6]  R. Clifton,et al.  On pressure-shear plate impact for studying the kinetics of stress-induced phase transformations , 1992 .

[7]  K. Bhattacharya.  THEORY OF MARTENSITIC MICROSTRUCTURE AND THE SHAPE-MEMORY EFFECT , 2022 .

[8]  Mitchell Luskin,et al.  Analysis of the finite element approximation of microstructure in micromagnetics , 1992 .

[9]  L. Schetky.  Shape-memory alloys , 1979 .

[10]  B. Dacorogna.  Direct methods in the calculus of variations , 1989 .

[11]  J. Ball.  Convexity conditions and existence theorems in nonlinear elasticity , 1976 .

[12]  R. Kohn,et al.  Modeling of Branched Needle Microstructures at the Edge of a Martensite Laminate , 1995 .

[13]  S. Müller.  A sharp version of Zhang's theorem on truncating sequences of gradients , 1999 .

[14]  J. Ericksen,et al.  Nilpotent energies in liquid crystal theory , 1962 .

[15]  K. Bhattacharya.  Self-accommodation in martensite , 1992 .

[16]  J. Ball,et al.  Hysteresis During Stress-Induced Variant Rearrangement , 1995 .

[17]  B. Li,et al.  Finite Element Analysis of Microstructure for the Cubic to Tetragonal Transformation , 1998 .

[18]  R. D. James,et al.  Proposed experimental tests of a theory of fine microstructure and the two-well problem , 1992, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

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

[20]  J. Ericksen,et al.  Constitutive theory for some constrained elastic crystals , 1986 .

[21]  K. Shimizu,et al.  Morphology and Crystallography of Thermoelastic γ' Cu-Al-Ni Martensite , 1969 .

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

[23]  Yu. G. Reshetnyak.  On the stability of conformal mappings in multidimensional spaces , 1967 .

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

[25]  David Kinderlehrer,et al.  Theory of diffusionless phase transitions , 1989 .

[26]  J. Ball.  A version of the fundamental theorem for young measures , 1989 .

[27]  PDEs and Continuum Models of Phase Transitions , 1989 .

[28]  Mitchell Luskin,et al.  Approximation of a Martensitic Laminate with Varying Volume Fractions , 1999 .

[29]  Huibin Xu,et al.  Observations on a CuAlNi single crystal , 1990 .

[30]  J. Wloka,et al.  Partial differential equations , 1987 .

[31]  Bo Li,et al.  Nonconforming finite element approximation of crystalline microstructure , 1998, Math. Comput..

[32]  V. Sverák.  Lower-Semicontinuity of Variational Integrals and Compensated Compactness , 1995 .

[33]  J. C. Currie,et al.  Weak continuity and variational problems of arbitrary order , 1981 .

[34]  Kaushik Bhattacharya,et al.  Wedge-like microstructure in martensites , 1991 .

[35]  Mitchell Luskin,et al.  On the computation of crystalline microstructure , 1996, Acta Numerica.

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

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

[38]  Mitchell Luskin,et al.  Approximation of a laminated microstructure for a rotationally invariant, double well energy density , 1996 .