Dynamics of martensite phase transitions in shape memory beams under buckling and postbuckling conditions

Abstract

The numerical simulation of phase transitions in shape memory alloy beams under combined mechanical loads and temperature fields is performed on the background of the once coupled Movchan’s theory of thermoelastic phase transforms and the finite element approach. The completely incremental formulation of the model of thermoelastic behavior of shape memory alloys is constructed and implemented into the finite element code. The geometrically and physically nonlinear solid finite element model is used to investigate the dynamics of the phase transitions in prismatic beams being axially compressed and cooled through the temperature range of direct martensite transforms. The trivial equilibrium state of a beam is perturbed by applying small initial deflections to study the beam buckling and postbuckling behavior in terms of Lyapunov’s concept. It is shown that the clamped-clamped shape memory alloy beam buckles after the initiation of the direct martensite transition at compression forces about 11-14% of the critical forces obtained analytically and numerically and corresponding to the minimum elasticity modulus of the entirely martensite phase constitution. The presented numerical results are consistent with the physical test data and very close to the analytical estimations based on the assumption of the ”supplementary phase transform occurring everywhere” advanced by A. Movchan and L. Silchenko. The distribution of the martensite volume ratio over the beam cross-section remains almost linear due to the beam deflection as well at a bifurcation point as in the postbuckling state. This heterogeneity acts as a supplementary perturbation which results in the buckling at very low compression forces. Thus, the assumption of the decisive effect of phase transitions due to the combined compression and cooling on the beam instability is vindicated by the numerical simulation, and the concept of the ”supplementary phase transition occurring everywhere” being an extension of Shenley’s buckling concept is validated for the practice.