Tabesh, Majid (2015-08). A Gradient-based Constitutive Model to Predict Size Effects in the Response of Shape Memory Alloys. Doctoral Dissertation.
Shape memory alloys (SMAs) show size effect in their response because the behavior of small-scale SMA structures deviates from that of the bulk material. Ni-Fe-Ga ferromagnetic SMA micropillars, for example, demonstrated a significantly increased hardening in their compressive stress-strain response as their diameter approached micron and submicron scales. This response cannot be modeled using conventional theories that lack an intrinsic length scale in their constitutive models. Constitutive models, however, are crucial for the design and simulation of SMA components at nano and micron scales as in NEMS and MEMS. Therefore, to capture such a size effect, a gradient-based thermodynamically consistent constitutive framework is established. We assume the existence of generalized surface and body forces that contribute to the free energy as work conjugates to the generalized variables of martensite volume fraction, transformation strain tensor, and their spatial gradients. The rates of evolution of the generalized variables are obtained by invoking the principal of maximum dissipation after assuming a transformation surface. This approach is compared to the theories that use a configurational force balance law. The developed constitutive model includes various energetic and dissipative length scales that can be calibrated experimentally. To demonstrate the capabilities of this model, a series of boundary value problems are solved. The boundary value problems contain the differential equation for the transformation surface as well as the equilibrium equation and are solved analytically and numerically. Example problems include pure bending of SMA beams, simple torsion of SMA cylindrical bars, and compression of SMA micro/nanopillars. The simplest version of the model, containing only the additional gradient of martensite volume fraction, predicts a response with greater hardening for smaller structures. Also once calibrated, the model can qualitatively predict the experimentally observed response of Ni-Fe-Ga micropillars under compression.