We discuss a relatively simple and computationally inexpensive model that has recently been developed to study phase
transformations and shape memory effects in finite nanostructures. Our major focus is given to nanowires of finite length
and other nanostructures where size effects are pronounced. The main tool used here is based on mesoscopic models
developed with the phase-field approach which we and other authors have applied before to study ferroelectrics at the
nanoscale. We study the cubic-to-tetragonal transformations in which case the 2D analogue of the model describes the
square-to-rectangle phase transformations. The actual model is based on a coupled system of partial differential equations
and is solved with a combination of the Chebyshev collocation method and the extended proper orthogonal decomposition.
The developed model and its numerical implementation allow us to study properties of nanostructures and several
representative examples of mechanical behavior of nanostructures are discussed.
In the current paper, a macroscopic differential model is constructed for the modeling of two-way shape memory effects
in one-dimensional shape memory alloy (SMA) structures. The model is based on the phenomenological theory of
thermoelastic phase transformations in SMAs. Hysteresis loops in both mechanical and thermal fields are treated as
macroscopic illustrations of martensite transformations and martensite variant re-orientations. A non-convex free energy
function is constructed such that each of its local equilibriums can be used to characterize one of the phases involved in
the transformations. System states (strain) can be transformed upon external loadings (mechanical or thermal) from one
stable equilibrium to another, thus the dynamics of phase transformations can be modeled by investigating the system
state transformations. Governing equations for the transformation dynamics are formulated by employing the Lagrange's
equation, and are expressed as nonlinear differential equations. Numerical examples of thermal and mechanical
hysteresis loops associated with the transformations caused by thermal and mechanical loadings are presented. Two way
shape memory effects and pseudo-elastic effects are successfully modeled.