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Stick-slip interface motion as a singular limit of the viscosity-capillarity model. (English) Zbl 1057.74028
It is known that materials undergoing stress-induced martensitic phase transformations, when subjected to cyclic loading, exhibit hysteresis loops in load-displacement curves which are often serrated. A common approach, the idea of which was introduced by Ericksen, explains these phenomena within the framework of elasticity theory and involves minimization of a nonconvex elastic energy for the material.
This paper is a follow-up on a study by the author and P. Rosakis [J. Nonlinear Sci. 9, No. 6, 697–719 (1999; Zbl 0989.74054)] of interface dynamics and hysteresis in materials with solid-solid phase transformations. The dynamics of a viscoelastic Ericksen’s bar with a nonconvex double-well energy density, subjected to time-dependent displacement-controlled loading, are studied. The model investigates the relation between the viscoelastic Ericksen’s bar models with and without interfacial energy modeled by a strain-gradient capillarity term.
Numerically, it is predicted the hysteretic behavior in the overall load-elongation diagram. The author shows that, for given loadings, a sufficiently small interfacial energy coefficient results in the end-load serrations, due to which one can always find its small enough value at which the strain in the bar grows to adjust to the loading, while the phase boundary moves very slowly. The system goes away from the minimum-energy two-phase branch with both potential energy and end load increasing. Eventually the strain enters the spinodal region (or the region, where the elastic energy density is concave) in a portion of the bar, and the instability causes the system to lower the energy and adjust the location of the phase boundary.
These results suggest that the stick-slip interface motion and serrated hysteresis loop predicted earlier in the absence of interfacial energy are the singular limit of the viscosity-capillarity model as the capillarity coefficient tends to zero.

MSC:
74N20 Dynamics of phase boundaries in solids
74N30 Problems involving hysteresis in solids
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