Finite strain phase-field microelasticity theory for modeling microstructural evolution

Abstract

Implementing the phase-field model at finite strains is usually considered unattainable through the spectral method, largely because of the nonlinearity in the transformation-induced microelasticity. Here we present a phase-field microelasticity (PFM) theory at finite strains with a representation in the reference configuration, allowing the spectral method to be readily incorporated. Following the spirit of Khachaturyan’s PFM theory at small strains, the elastic energy is formulated as a functional of microstructure (order parameters) solely, which should automatically satisfy the mechanical equilibrium. Thermodynamic consistency of the current theory under multiplicative decomposition of the total deformation gradient (into elastic and inelastic parts) and in conjunction with hyperelasticity and the time-dependent Ginzburg-Landau equation is shown rigorously. The new theory is first applied to the classical Eshelby’s inclusion problem, where shear-dilation coupling due to geometric nonlinearity is shown and a convergence study between small strain and finite strain theories is also carried out. The effects of geometric nonlinearity on the co-evolution of micromechanics and microstructure is further studied through modeling the growth of {101¯2}〈1¯011〉 deformation twins in magnesium. The simulation results suggest significant differences in terms of the shape of and the stress field around the deformation twin. In particular, the current finite strain PFM theory predicts a deviation of the twin boundary plane from the theoretical K1 plane, which is not captured in the small strain theory nor in the crystallographic theory. A parametric study further reveals that the observed deviation is caused by the tip effect of the finite-sized twin plate when the aspect ratio is relatively small. The symmetry of the stress field distribution around the twin tip is also found to be drastically different between the small strain and finite strain based phase-field modeling. The sharp twin tip observed in experiments is also shown to be likely related to the anisotropy in twin/matrix interface mobility.