Nonlocal integral elasticity based phase field modelling and simulations of nanoscale thermal- and stress-induced martensitic transformations using a boundary effect compensation kernel

Abstract

In this paper, a nonlocal elasticity based phase field model is presented for thermal and stress induced martensitic phase transformations (PTs) at the nanoscale. The main drawbacks of previous kernels are resolved by introducing the compensated two-phase (CTP) kernel as a combination of the modified and the two-phase (TP) kernels which compensates the boundary effects. The finite element method and COMSOL code are used to solve‌ the coupled Ginzburg-Landau and local/nonlocal elasticity equations. Several important PT examples are presented and the results of the local elasticity and nonlocal elasticity with different kernels are compared. The thermal induced interface propagation is found almost similar for all cases. In contrast, at the equilibrium temperature, the local elasticity creates a stationary vertical interface while both kernels result in a similar 450–rotated interface and a different stress field with lower values. For the nucleus growth, similar morphology and growth rate are found for all cases except during the initial growth where the CTP kernel leads to the growth rate smaller than that of the local elasticity and larger than that of the TP kernel. For the reverse PT, with nonhomogeneous initial conditions, the austenitic embryo grows and the interface propagates until the entire sample transforms back to austenite. The role of the CTP kernel is pronounced with homogeneous initial conditions where the TP kernel represents a nonhomogeneous reverse PT while both the local elasticity and CTP kernel show a similar homogenous reverse PT with no interface. For the precipitate induced PT, in contrast to the TP kernel for which the entire sample transforms to martensite, there remains some noncomplete martensite near the precipitate for the local elasticity and CTP kernel. The CTP kernel also predicts the initial growth rate and stress concentrations between the local elasticity and TP kernel. Note that when boundary periodicity is included, the precipitate induced growth is suppressed. In the presence of a nanobubble which represents internal surfaces, the growth starts from its surface and proceeds if the boundary periodicity is removed and both kernels show a smaller growth rate than the local elasticity. The current study allows for a better understanding of the nonlocal elasticity theory and its application within the kinetics modeling of martensitic PTs and similar nanoscale phenomena.