Extending Esh3D Code to Solve Interacting Eshelby's Inhomogeneity Problems

Abstract

I present a new development of an open source code Esh3D in solving interacting Eshelby's inhomogeneity problems in whole, half, and finite spaces. The code resolves the interaction between multiple inhomogeneity bodies by assembling and solving a global linear system. For half or finite space problems, the code imposes Neumann and Dirichlet boundary conditions and resolves the surface‐body interaction by hybridizing Eshelby's analytical solutions with a numerical part. I demonstrate the impact of multibody interaction and surface‐body interaction by comparing the results of different problems in whole and finite spaces. This analytical‐numerical hybrid approach, compared to conventional finite element method, is inexpensive and flexible in modeling multiple and evolving inhomogeneity bodies.

Quartz Stressing and Fracturing by Pore Pressure Dropping Down to Negative Pressure

“…In particular, Eshelby studied how an applied stress, uniform at large distances, is disturbed by a cavity. From this early work, many analytical or closed-form analytical solutions have been proposed for spherical or spheroidal inclusions and considering isotropic or anisotropic materials. − The problem becomes more complicated for a spheroidal cavity hosted in an anisotropic solid. On the other hand, FEA has proved its reliability for solving mechanical problems especially in the case of linear elastic problems but also in the case of geometric nonlinearity and with nonlinear isotropic elastic materials .…”

“…In a first approach, the sample was modeled as a 2D axisymmetric solid containing an oblate spheroidal inclusion, and the finite-element-based results were compared with theoretical results obtained using the code recently developed by Meng et al − which gives quasianalytical strain/stress fields following Eshelby’s solution, with the elliptic integral approximated through a numerical routine. The purpose of this comparison was to determine the size of the solid surrounding the cavity to avoid boundary effects and define the meshing strategy to avoid numerical artifacts or errors and to better understand the stress field building around the inclusion.…”

“…The normalized radial coordinate is the radial coordinate divided by the half-length of the spheroid, r * = r / a . The stresses denoted “Esh” were calculated following Eshelby’s theory by applying script developed by Meng et al − and according to the superposition principle for linear elastic mechanics (top-left of the figure). Stresses that were noted by FEA were calculated using Comsol Multiphysics software.…”