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Analysis of a thermomechanical model of shape memory alloys, pp. 487-536 $100.00
Authors:  Aida Timofte and Vlad Timofte
This chapter is devoted to the mathematical study of a thermomechanical model
describing the macroscopic behavior of shape memory alloys (SMA). Various experiments
(slow uniaxial traction-compression tests on cylindrical SMA samples) emphasized
the temperature changes during hysteresis loops and relaxation and creep phenomena.
They also established the absence of permanent deformations at the end of
a loading-unloading cycle, the so-called pseudoelastic behavior of SMA. The macroscopic
phenomena observed during the experiments can be predicted by using a classical
(Gibbs) thermomechanical model founded on a free energy, which is convex with
respect to the strain and to the martensitic volume fraction, and concave with respect
to the temperature. The 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 quasistatic form, together with
the evolution equation for the internal variables, form a partial differential equation
system, whose solutions fulfill the second principle of thermodynamics (the entropy
inequality). In the circular cylindrical case with non-negligible radius, this system
reduces to an ordinary differential system. In this case, we prove the uniqueness of
solutions in a large class of spaces (abstract differential structures). Existence and
regularity of solutions are then established in various functions spaces endowed with
natural derivatives. At the end of the chapter, for a particular traction-compression test
we present the exact solution, together with the associated hysteresis loop. 

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Analysis of a thermomechanical model of shape memory alloys, pp. 487-536