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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
20/2004

Global Existence for Nonconvex Thermoelasticity

Marc Oliver Rieger and Johannes Zimmer

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.

Received:
Apr 19, 2004
Published:
Apr 19, 2004
MSC Codes:
35Q72, 74B20

Related publications

inJournal
2005 Repository Open Access
Marc Oliver Rieger and Johannes Zimmer

Global existence for nonconvex thermoelasticity

In: Advances in mathematical sciences and applications, 15 (2005) 2, pp. 559-569