Local vs. nonlocal integral elasticity-based phase field models including surface tension and simulations of single and two variant martensitic transformations and twinning

In this paper, single and two variant martensitic transformations including twinning are studied using the phase field approach based on local elasticity and nonlocal integral elasticity including the surface tension. In contrast to the well-known two-phase (TP) kernel, the compensated TP (CTP) kernel fulfills the normalization condition for the boundary region and removes boundary effects. The coupled local/nonlocal integral elasticity and Ginzburg–Landau equations are solved using the FEM and the verification of numerical procedure is presented. The local and nonlocal models with both kernels are compared for various phase transformation examples. For a single variant in a homogeneous sample under uniform applied stress, the TP kernel resolves an unphysically heterogeneous growth and consequently, the surface tension appears. In contrast, the CTP kernel and local model predict a similar homogeneous growth with zero surface tension since no interface appears. Including a boundary layer with a lower Young’s modulus than the bulk region results in a heterogeneous growth for all the models. For the creation of a stationary austenite–martensite interface, both kernels show the same solution different than that of the local model. For the formation of a martensite–martensite interface, all the models show the same solution, where the angle between the martensitic planes is in a very good agreement with existing MD simulations and analytical solution. For the embryo growth of both martensitic variants, a similar evolution occurs for the local model and the CTP kernel. Generally, the CTP kernel gives a maximum stress larger than the TP kernel and lower than the local model. However, the kernel can predict larger stresses than the local model if their morphologies are not similar at the same time. For the creation of twin structures, both kernels lead to a larger number of twins since they resolve lower stresses than the local model and their twin structures are the same since they are not affected by the boundaries due to their large distance. The ratio of the characteristic length to the martensite–martensite interface width is also found as a key parameter in resolving the twin structure. For the two variant martensitic transformation in the presence of a crack, despite a similar evolution, the TP kernel and the local model show the fastest and slowest transformations, as well as the lowest and highest stress concentrations at the crack tip, respectively. However, the local model shows a lower stress concentration when austenite appears at the crack tip. In all the given examples, the surface tension is found generally much smaller than the total stress; thus, it shows no practical effect on the PT.