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Mixed variational principle for shape memory solids

Vladimir A. Grachev and Yuriy S. Neustadt

Vol. 18 (2023), No. 5, 621–633
Abstract

The quasistatic deformation problem for shape memory solids is studied based on the measurements of displacements arising from force actions and temperature variations. The phenomenological approach relies on generalization curves for uniaxial tension and compression of specimens at different temperatures. Under proportional loading and at low temperature the alloy behaves as an ideal elastoplastic material; the residual strain is observed after unloading. If the deformed sample is heated to a certain temperature for each alloy, the initial shape is restored. The first curve of deformation can be described with the variational principle. Thus, it becomes clear how to explain “the reverse deformation” within the slightly modified theory of plasticity. It is necessary to replace the simply connected surface of loading with the doubly connected one, use the principle of orthogonality for thermodynamic forces and streams, and update the variational principle with two laws of thermodynamics.

Keywords
continuum mechanics, solids, variational principle, shape memory materials, thermal plasticity, space of bounded deformation, generalized solutions
Milestones
Received: 28 September 2021
Revised: 21 September 2022
Accepted: 14 June 2023
Published: 25 October 2023
Authors
Vladimir A. Grachev
Academy of Architecture and Building
Samara State Technical University
Samara
Russia
Yuriy S. Neustadt
Academy of Architecture and Building
Samara State Technical University
Samara
Russia