A Semi-analytical Model for Nonlinear Elliptical Inclusions with Spherical Eigenstrains

Abstract

Motivated to understand the stresses induced by the formation of precipitates in metals, in 1957, John D. Eshelby provided a fully-analytical solution for the stress and deformation fields induced by an incompatible ellipsoidal inclusion embedded within an infinite matrix. Over the past six decades, his theory, which considers linearly elastic materials, has been essential in developing homogenized micromechanical models for metals and composites. However, as solid mechanics research increasingly focuses on soft materials such as biological tissues, a linear theory is no longer sufficient. Despite numerous potential applications ranging from medical diagnosis to industrial manufacturing processes, an accurate analytical or semi-analytical nonlinear extension of Eshelby’s theory of the elliptical inclusion problem has yet to be developed. This work presents a novel approach to solve the 2D elliptical inclusion problem, which satisfies incompressibility. It is shown to converge to the Eshelby solution in the linear limit for the case of isotropically growing inclusions. Moreover, this model matches almost identically to 2D finite element simulations for large incompatibilities, far beyond the linear range, while providing a complete description of the field through a single function. Finally, it is suggested that the simplified solution can enable the use of homogenization methods for future nonlinear micromechnical models and can help to elucidate various growth phenomena observed in nature.