Convergent Spectral Approximations for the Thermomechanical Processes in Shape Memory Alloys

T. HERDMAN, P. MORIN AND R. SPIES

JOURNAL OF NONLINEAR ANALYSIS

Elsevier Science

Año: 2000 vol. 39 p. 11 - 32

0362-546X

In this article, discrete spectral approximations to the nonlinear partial differential equations that model the dynamics of thermomechanical martensitic transformations in one-dimensional shape memory alloys with non-convex Landau-Ginzburg potentials are developed. By using the theories of analytic semigroups and interpolation spaces and a generalization of Gronwall´s lemma for singular kernels, the convergence of the approximations is shown to hold not only in the state-space norm but also in a stronger delta-norm. Numerical experiments are performed using this scheme show that under different initial conditions and distributed external actions the model  is able to produce solutions whose qualitative behavior is found to be in close agreement with laboratory experiments performed on Shape Memory Alloys under similar conditions.equations that model the dynamics of thermomechanical martensitic transformations in one-dimensional shape memory alloys with non-convex Landau-Ginzburg potentials are developed. By using the theories of analytic semigroups and interpolation spaces and a generalization of Gronwall´s lemma for singular kernels, the convergence of the approximations is shown to hold not only in the state-space norm but also in a stronger delta-norm. Numerical experiments are performed using this scheme show that under different initial conditions and distributed external actions the model  is able to produce solutions whose qualitative behavior is found to be in close agreement with laboratory experiments performed on Shape Memory Alloys under similar conditions.