A Gradient-Based Constitutive Model for Shape Memory Alloys

Constitutive models are necessary to design shape memory alloy (SMA) components at nano- and micro-scales in NEMS and MEMS. The behavior of small-scale SMA structures deviates from that of the bulk material. Unfortunately, this response cannot be modeled using conventional constitutive models which lack an intrinsic length scale. At small scales, size effects are often observed along with large gradients in the stress or strain. Therefore, a gradient-based thermodynamically consistent constitutive framework is established. Generalized surface and body forces are assumed to contribute to the free energy as work conjugates to the martensite volume fraction, transformation strain tensor, and their spatial gradients. The rates of evolution of these variables are obtained by invoking the principal of maximum dissipation after assuming a transformation surface, which is a differential equation in space. This approach is compared to the theories that use a configurational force (microforce) balance law. The developed constitutive model includes energetic and dissipative length scales that can be calibrated experimentally. Boundary value problems, including pure bending of SMA beams and simple torsion of SMA cylindrical bars, are solved to demonstrate the capabilities of this model. These problems contain the differential equation for the transformation surface as well as the equilibrium equation and are solved analytically and numerically. The simplest version of the model, containing only the additional gradient of martensite volume fraction, predicts a response with greater transformation hardening for smaller structures.