Existence Theorem for a Thermomechanical Model of Shape Memory Alloys

Abstract

In this paper, we give an existence 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].…”

“…Since every solution of (T ) X(J) also satisfies (T ) C(J) , we deduce that (T ) X(J) is compatible if and only if the unique solution (β, θ) of (T ) C(J) (see [6], Th.3.1, p.543) satisfies β, θ ∈ X(J). Hence, for our problem, regularity of solutions (β, θ ∈ X(J) whenever σ ∈ X(J)) is equivalent to their existence.…”

“…The thermomechanical model studied in this paper will be published in [1], but a description of it can also be found in [2]. A variant which neglects the intrinsic dissipation was studied in [3] and [4].…”

“…If Γ0 = Γ, we also use the notations α+, γ+, U + r , V + r . We shall need the following three results proved in [6] (Lemma 2.1 and Prop.2.1 and 2.2).…”

A note on: “Monotone strain-stress models for shape memory alloys hysteresis loop and pseudoelastic behavior” [Z. Angew. Math. Phys. (ZAMP) 56 (2005), 304–356]

“…In [4] we established the uniqueness of solutions in a large class of spaces (the abstract derivation structures newly introduced in [4]). In [5] we proved the existence of solutions in the space of all continuous functions with respect to the derivative in the sense of distributions. Existence, uniqueness and regularity of solutions in various functions spaces were obtained in [6].…”