Elastic strain energy of inhomogeneous solids

A two-phase elastically inhomogeneous coherent mixture of two phases with the transformation-induced crystal lattice misfit is considered under the loaded and unloaded conditions. It is assumed that the constituent phases particles, composing the mixture, have the structural pattern of an arbitrary morphology but the system is macroscopically homogeneous. It is shown that the microstructure-dependent part of the strain energy in the loaded and unloaded conditions is described by the same equations with a minor modification of constants. This allows us to use the theory for a characterization of coarsening under applied stress. The theory that formulates the phase transformation-induced strain energy in terms of the interaction between finite elements of the constituent phases is proposed. It is shown that the strain energy of the system can be presented as a sum of multiparticle interactions between finite elements of the constituent phases, pairwise, triplet, quadruplet, and so on, the n-particle interaction energy being related to the (n-2)th order term in the Taylor expansion of the Green function with respect to the elastic modulus misfit. Particularly, the zeroth order term in such an expansion corresponding to the homogeneous modulus case is related to the pairwise interaction between finite elements. The applicability and accuracy of the effective elastic media approximation are discussed.