Heterogeneity and phase transformation in materials: Energy minimization, iterative methods and geometric nonlinearity

We consider the relationship between methods used to solve problems involving heterogeneities with a focus on phase transformations. We compare methods that solve for mechanical equilibrium based on iterative spectral techniques with those evolving a free energy using inertial and relaxational dynamics in terms of microstructure, convergence and efficiency for the two-dimensional versions of the cubic-to-tetragonal and hexagonal-to-orthorhombic transformations. We generalize the strain-based approach using kinematic compatibility, in conjunction with a Landau-based nonconvex energy functional, to the geometrically nonlinear case for computation using the iterative spectral as well energy-minimizing methods. Our approach uses the strain compatibility equations for geometrically nonlinear elasticity in two dimensions, the geometrically linear strain compatibility equations being recovered in the small strain limit. We show how for the two-dimensional version of the hexagonal-to-orthorhombic transformation with three variant distortions in the microstructure, the linear compatibility equations capture relatively large rotations with small to moderate stains. We propose a reconstruction that uses the mapping between displacement gradients and Lagrange strains to approximate the geometrically nonlinear result within the scope of the linear strain theory if the linear strains are substituted for the Lagrange strains. We illustrate this by evaluating the disclination angle associated with the unique microstructure for this transformation, which otherwise requires geometric nonlinear theory to correctly capture.