Nonlinear differential equation approach for the two-way shape memory effects of one-dimensional shape memory alloy structures

Abstract

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.