Constitutive modeling of Cu-Al-Ni shape memory alloys

Abstract

Certain alloys can exist in multiple phases, which, in the context of solids, essentially mean multiple crystallographic structures. For example, at certain compositions, a Cu-Al-Ni alloy can exist as a cubic lattice (austenite), an orthorhombic lattice ([beta]'1-martensite) or a monoclinic lattice (([beta]'1-martensite). The material changes from one phase to another under various conditions of thermal and/or mechanical loading. Under certain loads, multiple phases can coexist and when this happens, a sharp interface separates any two phases. As the stress or temperature changes, the interface propagates through the material and particles transform from one phase to the other as they cross the moving phase boundary. (A martensitic phase can exist in the form of many "variants", and an interface between co-existing variants is a twin boundary.) The constitutive modeling of such materials is made difficult by the inherent anisotropic nature of such materials and by the non monotonicity of the stress-strain curves. We develop a systematic method by which we can calculate the free-energy of such a material based on its symmetry. The velocity with which interfaces propagates controls the rate of phase transformations (i.e. the "kinetics"). It is well known that classical balance laws are insufficient for a complete description of the behavior of materials undergoing phase transformations. The classical continuum theory describes the bulk regions (regions away from the interfaces) in a satisfactory manner but leaves a gap in the information concerning the interface. This lacuna has been filled by either including nucleation and kinetic criteria that are consistent with the second law of thermodynamics, or by regularizing the continuum theory in some consistent manner. The above treatments seek to provide information on the boundaries between the phases. However, they suffer from the drawback that even though they are meant to be continuum scale descriptions of microscale phenomena which take place on the transformation front they do not model the physics of the transformations. A more natural way of obtaining the relevant information would be to directly study the transformation process at a microscale and then perform an appropriate homogenization so that the resulting law is applicable at a continuum scale. Such an approach would facilitate a deeper understanding of the transformation process as well as enable the continuum theory to reflect the micromechanical processes that govern the transformation. We develop a lattice model of twin and phase boundaries that accounts for microstructural effects. The model incorporates the effect of ledges in the interface. A quasi continuum model is obtained by approximating the resulting difference-differential equation of motion of the ledge, but retaining leading discreteness effects. The quasicontinuum model now models the interface at a continuum scale but incorporates lattice effects. The kinetic relation obtained from such a model explains the experimentally observed difference in the stress required for moving boundaries between different variants of martensite. The kinetic relation obtained for phase boundaries has the feature that the hysteresis loops do not decrease in size to zero for vanishing loading rates.