We consider a strongly nonlinear PDE system describing solid–solid phase transitions in shape memory alloys. The system accounts for the evolution of an order parameter χ (related to different symmetries of the crystal lattice in the phase configurations), of the stress (and the displacement u), and of the absolute temperature ϑ. The resulting equations present several technical difficulties to be tackled; in particular, we emphasize the presence of nonlinear coupling terms, higher order dissipative contributions, possibly multivalued operators. As for the evolution of temperature, a highly nonlinear parabolic equation has to be solved for a right hand side that is controlled only in L1. We prove the existence of a solution for a regularized version by use of a time discretization technique. Then, we perform suitable a priori estimates which allow us pass to the limit and find a weak global-in-time solution to the system.
Existence of Solutions for a Mathematical Model Related to Solid–Solid Phase Transitions in Shape Memory Alloys / E. Bonetti, P. Colli, M. Fabrizio, G. Gilardi. - In: ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS. - ISSN 0003-9527. - 219:1(2016 Jan), pp. 203-254. [10.1007/s00205-015-0896-4]
Existence of Solutions for a Mathematical Model Related to Solid–Solid Phase Transitions in Shape Memory Alloys
E. BonettiPrimo
;
2016
Abstract
We consider a strongly nonlinear PDE system describing solid–solid phase transitions in shape memory alloys. The system accounts for the evolution of an order parameter χ (related to different symmetries of the crystal lattice in the phase configurations), of the stress (and the displacement u), and of the absolute temperature ϑ. The resulting equations present several technical difficulties to be tackled; in particular, we emphasize the presence of nonlinear coupling terms, higher order dissipative contributions, possibly multivalued operators. As for the evolution of temperature, a highly nonlinear parabolic equation has to be solved for a right hand side that is controlled only in L1. We prove the existence of a solution for a regularized version by use of a time discretization technique. Then, we perform suitable a priori estimates which allow us pass to the limit and find a weak global-in-time solution to the system.File | Dimensione | Formato | |
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