Global Existence for Nonconvex Thermoelasticity

Marc Oliver Rieger and Johannes Zimmer

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Submission date: 19. Apr. 2004
published in: Advances in mathematical sciences and applications, 15 (2005) 2, p. 559-569 
Bibtex
MSC-Numbers: 35Q72, 74B20
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Abstract:
We prove global existence for a simplified model of one-dimensional thermoelasticity. The governing equations satisfy the balance of momentum and a modified energy balance. The application we wish to study by investigating this model are shape-memory alloys. They are a prominent example of solids undergoing structural phase transitions. A characteristic feature of these materials is that several crystalline variants are stable at low temperature. Consequently, the free energy considered here is nonconvex as a function of the deformation gradient for temperatures below a fixed threshold temperature. As a result of the nonconvexity of the free energy density, existence of weak solutions is not to be generally expected. We therefore show existence of a Young measure valued solution. The proof relies on vanishing capillarity.