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Numerical Analysis of a Relaxed Variational Model of Hysteresis in Two-Phase Solids

Published online by Cambridge University Press:  15 April 2002

Carsten Carstensen
Institute for Applied Mathematics and Numerical Analysis, Vienna University of Technology, Wiedner Hauptstraße 8-10/115, A-1040 Wien, Austria. (
Petr Plecháč
Mathematical Institute, University of Warwick, Coventry, UK. (
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This paper presents the numerical analysis for a variational formulation of rate-independent phase transformations in elastic solids due to Mielke et al. The new model itself suggests an implicit time-discretization which is combined with the finite element method in space. A priori error estimates are established for the quasioptimal spatial approximation of the stress field within one time-step. A posteriori error estimates motivate an adaptive mesh-refining algorithm for efficient discretization. The proposed scheme enables numerical simulations which show that the model allows for hysteresis.

Research Article
© EDP Sciences, SMAI, 2001

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Ball, J.M. and James, R.D., Fine phase mixtures as minimisers of energy. Arch. Rational Mech. Anal. 100 (1987) 13-52. CrossRef
Ball, J.M. and James, R.D., Proposed experimental tests of the theory of fine microstructure and the two-well problem. Philos. Trans. Roy. Soc. London Ser. A 338 (1992) 389-450. CrossRef
Bildhauer, M., Fuchs, M. and Seregin, G., Local regularity of solutions of variational problems for the equilibrium configuration of an incompressible, multiphase elastic body. Nonlin. Diff. Equations Appl. 8 (2001) 53-81. CrossRef
S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, in Texts in Applied Mathematics 15 . Springer-Verlag, New York (1994).
Carstensen, C. and Funken, S. A., Constants in Clément-interpolation error and residual based a posteriori estimates in finite element methods. East-West J. Numer. Math. 8 (2000) 153-175.
Carstensen, C. and Funken, S. A., Fully reliable localised error control in the FEM. SIAM J. Sci. Comput. 21 (2000) 1465-1484. CrossRef
C. Carstensen and S. Müller, Local stress regularity in scalar non-convex variational problems. In preparation.
Carstensen, C. and Plechác, P., Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp. 66 (1997) 997-1026. CrossRef
Carstensen, C. and Petr Plechác, Numerical analysis of compatible phase transitions in elastic solids. SIAM J. Numer. Anal. 37 (2000) 2061-2081. CrossRef
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978).
Le Dret, H. and Raoult, A., Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results. Arch. Rational Mech. Anal. 154 (1999) 101-134. CrossRef
K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations, in Acta Numerica, A. Iserles, Ed., Cambridge University Press, Cambridge (1995) 105-158.
Fonseca, I. and Müller, S., A-quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal. 30 (1999) 1355-1390. CrossRef
Goodman, J., Kohn, R.V. and Reyna, L., Numerical study of a relaxed variational problem from optimal design. Comput. Methods Appl. Mech. Engrg. 57 (1986) 107-127. CrossRef
A. G. Khachaturyan, Theory of Structural Transformations in Solids. John Wiley & Sons, New York (1983).
M.S. Kuczma, A. Mielke and E. Stein, Modelling of hysteresis in two-phase systems. Solid Mechanics Conference (1999); Arch. Mech. 51 (1999) 693-715.
Kohn, R.V., The relaxation of a double-well energy. Contin. Mech. Thermodyn. 3 (1991) 193-236. CrossRef
M. Luskin, On the computation of crystalline microstructure, in Acta Numerica, A. Iserles, Ed., Cambridge University Press, Cambridge (1996) 191-257.
A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis, in Workshop of Continuum Mechanics in Analysis and Engineering, H.-D. Alber, D. Bateau and R. Farwig, Eds. , Shaker-Verlag, Aachen (1999) 117-129.
A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle. Submitted to Arch. Rational Mech. Anal.
A. L. Roitburd, Martensitic transformation as a typical phase transformation in solids, in Solid State Physics 33 , Academic Press, New York (1978) 317-390.
Seregin, G.A., The regularity properties of solutions of variational problems in the theory of phase transitions in an elastic body. St. Petersbg. Math. J. 7 (1996) 979-1003, English translation from Algebra Anal. 7 (1995) 153-187.
R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques, in Wiley-Teubner Series Advances in Numerical Mathematics, John Wiley & Sons, Chichester; Teubner, Stuttgart (1996).