On the nonlinear analysis of pre-strained shape memory alloy beams

We present a theoretical and numerical investigation of the transverse dynamic response of monolithic Shape Memory Alloy (SMA) beams under the effect of axial pre-strains. The SMA beam is taken as benchmark structure to evaluate the effect of (axial) pre-strain on either the free or the forced dynamics. The combination of the static stress, produced by the pre -loading conditions, with the dynamic stress, associated with the vibratory response, affects the occurrence of the stress-induced SMA phase transformation hence providing a mechanism to passively tune the dynamic response. Particular attention is given to the effect that different pre -strain levels have on the overall ability to dissipate mechanical energy, that is on the effective damping. The numerical model of the SMA beam accounts for both material and geometric nonlinear behaviors. The nonlinear material behavior is due to the SMA phase-transformation and it is modeled via the one-dimensional improved Brinson's model (including tension- compression asymmetry), while the geometric nonlinear behavior is due to large displacements occurring during the phase transformation and it is accounted for via von Karman assumptions. The resulting model is solved numerically via the finite element method and used to evaluate both the free and the forced response of the beam. The free response analysis suggests the existence of optimal pre-strain levels to achieve maximum damping capacity, while the forced response highlights the occurrence of nonlinear dynamic features, such as bifurcation. In both cases, it is found that the level of pre-strain can significantly affect the dynamic response as well as the effective damping. The results presented in this work provide useful guidelines to understand the dynamics of continuous SMA structures under pre-strain and the ability to tune either their resulting dynamics or dissipation properties.