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Thermal control of the Souza-Auricchio model for shape memory alloys. (English) Zbl 1352.74095

Summary: We address the thermal control of the quasi-static evolution of a polycrystalline shape memory alloy specimen. The thermomechanical evolution of the body is described by means of the phenomenological Souza-Auricchio model [L. Souza et al., Eur. J. Mech. A/Solids 17, No. 5, 789–806 (1998; Zbl 0921.73024) and F. Auricchio and L. Petrini [Int. J. Numer. Methods Eng. 55, No. 11, 1255–1284 (2002; Zbl 1062.74580)]. By assuming to be able to control the temperature of the body in time we determine the corresponding quasi-static evolution in the energetic sense. By recovering in this context a result by F. Rindler [SIAM J. Control Optim. 47, No. 6, 2773–2794 (2008; Zbl 1176.49005); SIAM J. Numer. Anal. 47, No. 5, 3884–3909 (2009; Zbl 1206.65173)] we prove the existence of optimal controls for a suitably large class of cost functionals and comment on their possible approximation.

MSC:

74F05 Thermal effects in solid mechanics
49J45 Methods involving semicontinuity and convergence; relaxation
74E15 Crystalline structure
74G65 Energy minimization in equilibrium problems in solid mechanics
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