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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.

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A novel discretization method is proposed and developed for numerical solution of boundary value problems governed by partial differential equations. The spatial variation of field variables is approximated by using Lagrange finite elements for interpolation without discretization of the analysed domain into the mesh of finite elements. Only the net of nodal points is used for discrete degrees of freedom on the analysed domain and its boundary. The governing equations are considered at interior nodal points while the boundary conditions at nodal points on the boundary. The finite elements are created for each nodal point properly instead of using fixed finite elements like in standard Finite Element Method. In this way, we can eliminate interfaces between elements as well as the difficulties with continuity of derivatives of field variables on such interfaces. Both the strong and weak formulations are implemented for governing equations. The reliability (accuracy and efficiency) of the new method has been verified in numerical simulations for 2D problems of heat conduction in solids with possible continuous gradation of the heat conduction coefficient. Keywords: Lagrange finite element, bi-quadratic and bi-cubic approximation, strong and weak formulations.( 1 ) and the boundary conditions

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