Keywords: phase transformations, nanoscale, shape memory effects, Ginzburg-Landau theory, nonlinear thermoelasticity.

Phase transformations are ubiquitous in many problems of science and engineering where we have to analyze the evolution of complex nonlinear systems. In this contribution, we focus on one class of such problems that are brought about by smart materials and structures technologies where materials with shape memory effect have found numerous applications at different spatial scales. We provide a brief historical context to the subject, classifications of phase transformations, and their applications. We consider a general three-dimensional model of dynamic nonlinear thermoelasticity, based on a coupled system of partial differential equations derived within the Landau-Ginzburg-Devonshire framework, which we apply to study the dynamics of materials with shape memory and associated phase transformations. We provide details of key mathematical difficulties in analyzing this model numerically. We briefly discuss the most efficient numerical techniques in this context, including conservative numerical approximations based on the modified integro-interpolational methodology where in addition to the interpolation of the solution with respect to independent variables, we also perform the Steklov averaging of nonlinear terms. Our focus is on a mathematical model and its numerical discretization which we construct to analyze the wave propagation in materials with shape memory. From a mathematical point of view, the result is a system of coupled nonlinear time-dependent partial differential equations, known as the Ginzburg-Landau-Devonshire system. The effect of internal friction on wave propagation patterns is analyzed under shock loadings implemented using stress boundary conditions. For practical numerical simulations the constructed model of coupled nonlinear system of partial differential equations can often be reduced to a system of differential-algebraic equations, where the Chebyshev collocation method can be employed for the spatial discretization, while backward differentiation can be used for the integration with respect to time.

In the last part of this paper, we discuss a relatively simple and computationally inexpensive model to study phase transformations in finite nanostructures with our major focus given here to nanowires of finite length. We show that in the latter case, the models describing shape memory effects at the mesoscopic level can be reduced to a two-dimensional case and we demonstrate our results on the example of the cubic-to-tetragonal transformations (approximated by square-to-rectangular transformations in the two-dimensional case). Our results were obtained under the conditions of the full coupling between thermal and mechanical fields. This new feature of our model extends recently reported phase-field-based models for studying microstructures and shape memory effects at the nanoscale level where thermal field coupling was neglected. We demonstrated the existence of a critical dimension for finite length nanowires exhibiting shape memory effects. Representative examples of modelling were given for nanowires of different widths showing the importance of geometrical constraints in studying the properties of nanowires.

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