Martensitic phase transformations in shape -memory alloys: Constitutive modeling and numerical simulation

A continuum-level constitutive model for martensitic phase transformations in shape-memory alloys (SMAs) is developed in the framework of irreversible thermodynamics with internal variables. The thermodynamic framework is used to gain insight into the effect of evolving internal variables (which account for the state of the phase transformation) on the driving force for the phase changes and, consequently, on the overall macroscopic material stress-strain response. As an important class of active materials, SMAs derive unique properties from martensitic phase transformations and are used in a wide variety of applications including sensors, actuators, and medical devices such as stents. These unique properties, such as pseudoelasticity and the shape-memory effect, are a result of a solid-to-solid phase transformation, which occurs on the level of the crystal lattice. Although the underlying mechanism for the phase transformation is microscopic, a macroscopic continuum-level description is the ideal choice to model SMAs for use in applications on the size scale of devices. Therefore, a constitutive model is required that couples the microscopic phenomena to the macroscopic response. A finite-element implementation incorporating the SMA material model (an Abaqus UMAT) has been developed to perform numerical simulations of the stress-strain response of SMA single crystals. A backward Euler integration with Newton iteration of the nonlinear rate equations is employed to integrate the constitutive equations at each of the finite-element integration points to determine the current state of the transformation during three-dimensional, non-linear, finite-element simulations. The derivation of an analytical expression for the material stiffness tensor required as a UMAT output is also discussed. Utilizing an implicit integration scheme with analytical derivatives yields a computationally efficient algorithm. From the numerical simulations the effect that the evolving elastic properties have on the driving force is shown to significantly affect the overall stress-strain response of single crystal SMAs. Additionally, the effect of ambient temperature the tension-compression asymmetry associated with the pseudoelastic stress-strain response (and its dependence on the orientation of the loading axis relative to various crystallographic axes), as well as the behavior of SMAs around a stress concentration are also discussed