Session: 16-01-01: Government Agency Student Poster Competition

Paper Number: 150819

150819 - The Nonlinear Eshelby Inclusion Problem and Its Isomorphic Limit 

In 1957, John D. Eshelby presented an exact linear solution for the deformation of an ellipsoidal inclusion embedded in an infinite matrix. His solution helped elucidate the effects of defects in metals and was instrumental in the development of the field of micromechanics. It has been applied and modified to characterize systems from stiff polymer composites to piezoelectric ceramics. However, as we work towards a better understanding of the micromechanics of soft materials and growth of biological systems, 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 has yet to be developed for ellipsoidal inclusions of arbitrary aspect ratios.

In this work, we use a semi-inverse method to attain an accurate nonlinear analytical solution for the stress and deformation fields around an elliptical inclusion undergoing isotropic growth within an infinite matrix. We restrict our attention to incompressible materials and, informed by Eshelby’s solution, we formulate a uniparameter mapping that takes one family of ellipsoids in the undeformed configuration, to a new family of ellipsoids in the current configuration; a more general method compared to previous studies, which consider confocal deformation fields that are fully prescribed kinematically. The configurational parameter is obtained in our formulation by using energy minimization to identify the corresponding configurational force and the value of the configurational parameter for which it vanishes. We apply this method to the growth of elliptical inclusions in 2D plane strain and to the growth of 3D spheroidal inclusions (both oblate and prolate) using a neo-Hookean material model. A striking agreement is shown between the resulting solutions of this almost entirely analytical method and the full deformation fields in the inclusion and matrix, as obtained in the linear limit via Eshelby’s solution, and using finite element analysis far into the nonlinear regime.

Additionally, we discover an important limit unique to nonlinear inclusion problems: the existence of a (non-spherical) asymptotic shape that the inclusion approaches as it grows. We call this limit the isomorphic limit and present an analytical solution for the isomorphic aspect ratios of spheroidal inclusions. Associated with this limit is an isomorphic pressure which approaches the well-known cavitation limit of (5/6)E when considering spherical inclusions, where E is the elastic modulus. We hypothesize that this pressure limit may have implications for the growth rate and survivability of tumors and embedded biofilms. We show that this limit depends only on the stress-free grown shape of the inclusion and is independent of its original ungrown shape.

Finally, we present an analysis of the applicability of the model to the growth of inclusions with properties different from the matrix in which they are embedded, commonly referred to as inhomogeneities. We demonstrate that the model provides a good approximation for this growth when the inclusion is softer than the matrix. Beyond the elucidation of incompatible growth morphologies of nonlinear inclusions and their application to various biological systems, this work paves the way for theoretical analysis of highly nonlinear mechanics problems and offers a potential path forward for the creation of homogenized nonlinear material models.

Presenting Author: Joseph Bonavia Massachusetts Institute of Technology

Presenting Author Biography: Joseph Bonavia is a third-year doctoral candidate at the Massachusetts Institute of Technology. His work focuses on the nonlinear mechanics of continua, specifically in problems related to growth and incompatibility.

Authors:

Joseph Bonavia Massachusetts Institute of Technology
Chockalingam Senthilnathan Massachusetts Institute of Technology
Tal Cohen Massachusetts Institute of Technology