Existence, uniqueness and regularity of solutions for a thermomechanical model of shape memory alloys
This thesis is devoted to the study of a thermomechanical model describing the macroscopic behavior of shape memory alloys. The analyzed model takes into account the non-isothermal character of the phase transition, as well as the existence of the intrinsic dissipation. The first law of thermodynamics, the balance of momentum in its quasi-static form, the evolution equation for the internal variables (the volume fraction of martensite), together with the second principle of thermodynamics (the entropy inequality), lead to a partial differential equations system. In the circular cylindrical case the problem reduces to the following ordinary differential system: The unknown data are: the temperature θ at the surface of the body, the total fraction β of the martensite in the body, and the axial elongation ε of the sample in the Ox3 direction. The stress σ is supposed to be given. These all are real functions depending only on the time variable t > 0. The constants τ, Γ, L, C, E, g, p, q, T0, Ta, ΔT are all positive, T0 > Ta, Γ < L/C, and σ± := p(T0 - Ta + θ + βΔT) ± q. We prove uniqueness of solutions in a large class of functions spaces (abstract derivation structures), as well as existence and regularity in several such spaces.