Uniqueness Theorem for a Thermomechanical Model of Shape Memory Alloys

Abstract

In this paper, we give a uniqueness theorem concerning a thermomechanical model which describes the behavior of shape memory alloys and takes into account the nonisothermal character of the phase transformations, as well as the existence of the intrinsic dissipation.

Existence, Uniqueness and Regularity of Solutions for a Thermomechanical Model of Shape Memory Alloys

“…In the present work we prove existence, uniqueness and regularity of solutions in various functions spaces for the circular cylindrical case. Uniqueness in a large class of spaces (abstract derivation structures), as well as existence in the space of continuous functions were established in [5] and [6]. Some of the prerequisites can be found in [5] and [6].…”

“…r (see [5], Prop.6.1, p.460). For simplicity, we will write α, γ, Ur, Vr instead of α0, γ0, U 0 r , V 0 r , when no confusion can arise or when distinction is not important.…”

“…In the circular cylindrical case, the given axial stress σ, the martensitic volume fraction β, the variation θ of the absolute temperature of the sample with respect to T0 (temperature of the room), and the axial strain ε depend only on the time variable t ∈ J (J interval with min J = 0) and satisfy the following system (see [3] and [5] for details): where σ ± := p(T0−Ta+θ+β∆T )±q. The constants τ, Γ, L, C, E, g, p, q, T0, Ta, ∆T are all positive and T0 > Ta, Γ < L/C.…”

“…In order to remove the derivatives of β from (E), we introduced in [5] a new notion. A point t ∈ J0 (J0 an interval) is said to be an increment point for u ∈ C(J0), if and only if for every neighborhood V of t, we have t1 < t2 and u(t1) < u(t2) for some t1, t2 ∈ V ∩ J0.…”

“…The system (T ) may be considered for any of the functions spaces and derivatives listed below (see [5] for the definition of an abstract derivation structure X(J) and the corresponding system (T ) X(J) ). …”