A multiscale continuum model of the grain-size dependence of the stress hysteresis in shape memory alloy polycrystals

In this paper, a multiscale continuum model is proposed to study the effect of grain size on the macroscopic dissipative response of shape memory alloy polycrystals during isothermal thermoelastic phase transition. In the simplest one-dimensional (1D) heterogeneous structural hierarchy, a series of non-local and non-convex double-well continuum elements are employed to model the micro-instability and the macroscopic stress hysteresis of the material under uniaxial quasi-static stretching. Three characteristic length scales (specimen size L, grain size l and intrinsic material length g) of a bulk polycrystal are imbedded in the 1D chain model and their important roles in the macroscopic dissipation are quantified. It is shown that the specific energy dissipation or the width of the stress hysteresis is governed by two non-dimensional ratios, N(=L/l  ) and l¯=(l/g). For a given specimen of size L, the hysteresis decreases rapidly at either very large or small values of l. In particular, it vanishes when the grain size is reduced to the nano-scale where the grain size and the interface thickness become comparable. The above results of the 1D model are reproduced in a two-dimensional (2D) non-local numerical experiment on the energy dissipation during multiple domain evolution in heterogeneous strips. The predictions of the two models agree well qualitatively with the recent experimental observations of the stress hysteresis in nano-grained superelastic NiTi polycrystals.