Computational modeling of size-dependent superelasticity in shape memory alloys

Abstract

The superelastic effect in shape memory alloys (SMAs) is attributed to the stress-induced reversible austenitic-martensitic phase transformations. It is characterized by the development of significant strains which are fully recoverable upon unloading, and also characterized by the stress-hysteresis in the loading and unloading cycle which corresponds to the energy dissipated during phase transformations. Recently, experiments have revealed size-dependent effects in the superelastic responses of SMAs at micro- and nanoscales. For instance, the CuAlNi microwires and submicron pillars show a substantially higher capacity for the energy dissipation than that of bulk samples, which offers a significant promise for the applications in protective materials. In this thesis, a continuum model is developed in order to improve our understanding of size effects in SMAs at small scales. The modeling approach combines classic superelastic models, which use the volume fraction as an internal variable to represent the martensitic phase transformation, with strain gradient plasticity theories. Size effects are incorporated through two internal length scales, an energetic length scale and a dissipative length scale, which correspond to the martensitic volume fraction gradient and its time rate of change, respectively. Introducing the gradient of the martensitic volume fraction leads to coupled macro- and microforce balance equations, where the displacements and the martensitic volume fraction are both independent fields. A variational formulation for the temporally-discretized coupled macro- and microforce balance equations is proposed, as well as a computational framework based on this formulation. A robust and scalable parallel algorithm is implemented within this computational framework, which enables the large-scale three-dimensional study of size effects in SMAs with unprecedented resolution. This modeling and computational framework furnishes, in effect, a versatile tool to analyze a broad range of problems involving size effects in superelasticity with the potential to guide microstructure design and optimization. In particular, the model captures the increase of the stress hysteresis and strain hardening in bulk polycrystalline SMAs for decreasing grain size, as well as the increase of the residual strain for decreasing pillar size in NiTi pillars. The model confirms that constraints like grain boundaries and the surface Ti oxide layer are responsible for the size-dependent superelasticity in SMAs.