Preferential stiffness and the crack-tip fields of an elastic porous solid based on the density-dependent moduli model

TL;DR

This study investigates how density-dependent material properties influence crack-tip fields and stiffness in elastic porous solids, revealing that the nonlinear model captures phenomena beyond classical elasticity, with implications for failure analysis.

Contribution

It introduces a density-dependent moduli model for porous solids, develops a solution algorithm using Newton's method and finite elements, and demonstrates the model's ability to describe complex crack behaviors.

Findings

Density-dependent moduli affect stiffness and failure modes.

Model parameter sign influences strength under different loadings.

Numerical results show phenomena beyond classical elasticity.

Abstract

In this paper, we study the preferential stiffness and the crack-tip fields for an elastic porous solid of which material properties are dependent upon the density. Such a description is necessary to describe the failure that can be caused by damaged pores in many porous bodies such as ceramics, concrete and human bones. To that end, we revisit a new class of implicit constitutive relations under the assumption of small deformation. Although the constitutive relationship \textit{appears linear} in both the Cauchy stress and linearized strain, the governing equation bestowed from the balance of linear momentum results in a quasi-linear partial differential equation (PDE) system. For the linearization and obtaining a sequence of elliptic PDEs, we propose the solution algorithm comprise a \textit{Newton's method} coupled with a bilinear continuous Galerkin-type finite elements for the discretization. Our algorithm exhibits an optimal rate of convergence for a manufactured solution. In the numerical experiments, we set the boundary value problems (BVPs) with edge crack {under different modes of loading (i.e., the pure mode-I, II, and the mixed-mode). From the numerical results, we find that the density-dependent moduli model describes diverse phenomena that are not captured within the framework of classical linearized elasticity. In particular,numerical solutions clearly indicate that the nonlinear \textit{modeling} parameter depending on its sign and magnitude can control preferential mechanical stiffness along with the change of volumetric strain; larger the parameter is in the positive value}, the responses are such that the strength of porous solid gets weaker against the tensile loading while stiffer against the in-plane shear (or compressive) loading, which is vice versa for the negative value of it.