Identifiability of the Landau-Ginzburg potential in a mathematical model of shape memory alloys
The authors consider an initial boundary value problem (IBVP) that involves a pair of coupled nonlinear partial differential equations. These equations arise from conservation laws of linear momentum and energy, and describe structural phase transitions in one-dimensional shape memory solids with non-convex Landau-Ginzburg potentials. As in previous work by one of the authors [\textit{R. D. Spies}, J. Math. Anal. Appl. 190, 58-100 (1995; Zbl 0829.35132)], the IBVP is first expressed as a semilinear Cauchy problem of the form \[ \dot z(t)= A(q)z(t) +F(q,t,z), \quad z(0) =z_0, \tag{1} \] set in an appropriate Hilbert space \(Z_q\), where \(q\) is a vector of admissible parameters. In this abstract formulation, \(A(q)\) is a differential operator associated with the linear part of the coupled partial differential equations, while \(F(q,t,z)\) corresponds to the nonlinear part of the system. The theories of analytic semigroups and real interpolation spaces for maximal accretive operators are then used to show that solutions of (1) depend continuously on the admissible parameters. In addition, the non-physical parameters that define the free energy are shown to be identifiable from the model.
equations of thermo-viscoelasto-plasticity
scientific article; zbMATH DE number 1053565
scientific article