Static and dynamic behavior of porous elastic materials based on micro-dilatation theory: A numerical study using the MLPG method

Abstract

Available online xxx Keyword: Porous material Cowin-Nunziato model Local integral equations Moving least square method Micro-dilatation Stress intensity factor (SIF) a b s t r a c t A meshless local Petrov-Galerkin (MLPG) model of porous elastic materials based on micro-dilatation theory by Cowin and Nunziato (1983) is developed. . This theory describes properties of homogeneous elastic materials with voids free of fluid. The primal fields (mechanical displacements, and change in matrix volume fraction which is also called micro-dilatation) are coupled in the constitutive equations. The governing differential equations are satisfied in the weak form on small circular subdomains for 2D problems. Only one node is lying at the center of each subdomain spread on the analyzed domain. A Heaviside step function is applied as test functions in the weak-form to derive local integral equations on subdomains. The spatial variation of the displacements and micro-dilatation are approximated by the moving least-squares (MLS) scheme. After performing the spatial integrations, a system of ordinary differential equations for certain nodal unknowns is obtained.

Bending problems in the theory of elastic materials with voids and surface effects

“…To eliminate the rigid displacements of the beam, we prohibit the vertical displacements of the beam's point located at the origin of coordinate system (see Figure 2). Rigid rotations of the beam are eliminated owing to symmetry conditions (27), (28).…”

“…Similar to the pure bending problem, only one-quarter of the beam is considered (Figure 2). Symmetry conditions (27) and (28) are used. To eliminate rigid displacements of the beam in vertical direction, the point at the origin of the Figure 2.…”

“…Based on the experimental data, the relation between the two non-classical parameters (coupling modulus and void stiffness parameter) was first found by Jeong et al [23]. Lakes [28], Ciarletta et al [31] and Rueger and Lakes [32] showed that there are scale effects in bending tests and torsion tests for polyurethane foams. It was noted that these results are in contradiction with the theoretical predictions that follow from the micro-dilatation theory [1].…”

“…In Thurieau [27], the authors modified the boundary element method to gain a numerical solution of the micro-dilatational theory as well as to study the problems of uniform and non-uniform deformations of the hollow thick cylinder. In a recent paper [28], the authors used the meshless local Petrov-Galerkin method [29] to solve the different 2D problems in micro-dilatational theory, including the cantilever beam bending problem.…”

On deformation of porous plates

“…For instance, in [4] there constructed Green-Lame, Boussinesq-Papkovitch-Neuber and Cauchy-Kovalevski-Somigliana solutions. A meshless local Petrov-Galerkin (MLPG) model for a porous elastic material on the basis of Cowin-Nunziato model is constructed in [18]. In [17] authors studied the process for distribution of plane waves in a porous media with the account of temperature.…”