Approximation of a Martensitic Laminate with Varying Volume Fractions

We give results for the approximation of a laminate with varying volume fractions for multi-well energy minimization problems modeling martensitic crystals that can undergo either an orthorhombic to monoclinic or a cubic to tetragonal transformation. We construct energy minimizing sequences of deformations which satisfy the corresponding boundary condition, and we establish a series of error bounds in terms of the elastic energy for the approximation of the limiting macroscopic deformation and the simply laminated microstructure. Finally, we give results for the corresponding nite element approximation of the laminate with varying volume fractions. R esum e. Nous consid erons des probl emes de minimisation d' energie avec de multiples puits de po- tentiel. De tels probl emes mod elisent, pour des cristaux martensitiques, des transitions de phase d'un r eseau orthorhombique a monoclinique, ou d'un r eseau cubique a t etragonal, par exemple. Des r esultats d'approximation des structures laminaires correspondantes, avec fractions volumiques variables, sont donn es. Des suites minimisantes, avec d eformations compatibles avec les conditions au bord, sont construites et permettent l'obtention de plusieurs estimations d'erreur concernant l'approxi- mation de la d eformation macroscopique limite en fonction de l' energie elastique. Finalement, nous d ecrivons des r esultats concernant l'approximation par el ements nis de la structure laminaire avec fractions volumiques variables.

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

[2]  Pablo Pedregal,et al.  On the numerical analysis of non-convex variational problems , 1996 .

[3]  Michel Chipot,et al.  Numerical approximations in variational problems with potential wells , 1992 .

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

[5]  Donald A. French,et al.  On the convergence of finite-element approximations of a relaxed variational problem , 1990 .

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

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

[8]  David Kinderlehrer,et al.  Twinning of Crystals (II) , 1987 .

[9]  J. Christian,et al.  Experiments on the martensitic transformation in single crystals of indium-thallium alloys , 1954 .

[10]  Carsten Carstensen,et al.  Numerical solution of the scalar double-well problem allowing microstructure , 1997, Math. Comput..

[11]  Ingo Müller,et al.  Metastability and incompletely posed problems , 1987 .

[12]  Mitchell Luskin,et al.  Numerical approximation of the solution of variational problem with a double well potential , 1991 .

[13]  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.

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

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

[16]  Willard Miller,et al.  The IMA volumes in mathematics and its applications , 1986 .

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

[18]  Michel Chipot,et al.  Numerical analysis of oscillations in nonconvex problems , 1991 .

[19]  Tomáš Roubíček,et al.  Numerical approximation of relaxed variational problems. , 1996 .

[20]  T. Roub.  Numerical Approximation of Relaxed Variational Problems , 1996 .

[21]  Mitchell Luskin,et al.  Optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem , 1991 .

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

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

[24]  Pierre-Alain Gremaud,et al.  Numerical analysis of a nonconvex variational problem related to solid-solid phase transitions , 1994 .

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

[26]  Pablo Pedregal,et al.  Numerical approximation of parametrized measures , 1995 .

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

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

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

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

[31]  R. A. Nicolaides,et al.  Strong convergence of numerical solutions to degenerate variational problems , 1995 .

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