Meshless Local Petrov-Galerkin Formulation for Problems in Composite Micromechanics

Abstract

In this paper we present the meshless local Petrov-Galerkin formulation for the generalized plane strain problem with specific emphasis on micromechanics of composite materials containing material discontinuities. The problem requires the introduction of an extra discrete degree of freedom, the out-of-plane uniform normal strain. The treatment of material discontinuity at the interface between the two phases of the composite is presented by means of direct imposition of interface boundary conditions. The meshless local Petrov-Galerkin method is used in the micromechanical model for predicting the elastic constants of the composite. To our knowledge, this is the first study in which the meshless local Petrov-Galerkin method is formulated for the so-called meshless local Petrov-Galerkin method based micromechanical analysis. Examples are presented to illustrate the effectiveness of the current method, and it is validated by comparing the results with available analytical and numerical solutions. The current method has the potential for use in micromechanics, especially for textile composites, where the meshing of the unit cell has been quite difficult. Nomenclature a = width, height, and depth of unit cell ax = vector of unknown parameters a j x b i = body force C = constant matrix of homogeneous composite c i = distance from node i to its third nearest neighboring node d i = distance from the sampling point x to the node x i E = Young's modulus I = identity matrix Jx = weighted discrete L 2 norm k = parameter in the Gaussian weight function (k 1) L s = part of local boundary over which no boundary conditions are specified N = total number of nodes n = number of points in the neighborhood of x for which wx x i > 0 n i = unit outward normal to the boundary P T x = vector of the complete monomial basis of order mm 3 r i = radius of the domain of influence of the weight function (r i 4c i ) r 0 = radius of the local domain r s = part of the local boundary located on the global boundary t i = prescribed traction on the boundary t u, v, w = u 1 , u 2 , u 3 u b = fictitious displacement on the interface u i = displacement field (trial function) u i = fictitious nodal value u i = prescribed displacement on the boundary ũ u i = actual displacement on the interface u h x = moving least-squares approximation V = volume of unit cell V f = fiber volume fraction v i = test function wx x i = weight function x 1 , x 2 , x 3 = Cartesian coordinates = penalty parameter ( 10 8 ) ij = Kronecker delta " = strain matrix computed from the test function " 0 = constant macroscopic direct strain in the z direction " M = macroscopic level strain matrix = Poisson's ratio ij = Cauchy stress tensor M = macroscopic level strain matrix x = shape function b , r = subsets of w.r.t. nodes on the interface and within the material = global domain s = local domain @ = boundary of the local domain s

Micromechanics of fibrous composites subjected to combined shear and thermal loading using a truly meshless method

“…Recently, MLPG method is employed for micromechanical analysis of the UD composite materials by Dang and Sankar [35,36]. They used the transformation approach for the enforcement of the continuity of displacement at the fiber-matrix interface as in the FEM.…”

“…Their model shows good agreement with the finite element results. However, the conventional MLPG formulation [35,36] with Gaussian weight functions and transformation technique is computationally expensive due to existence of the domain integrals in the weak form of equations.…”

A truly generalized plane strain meshless method for combined normal and shear loading of fibrous composites

“…Existing numerical solutions in the literature mainly include finite difference [11], finite element [12][13][14][15][16] and boundary element [17] methods. Recently, Dang and Sankar [18,19] employed the conventional MLPG method for micromechanical analysis of unidirectional composite. Dang and Sankar [18] used the penalty parameter method to enforce the essential boundary conditions to the RVE.…”

“…Recently, Dang and Sankar [18,19] employed the conventional MLPG method for micromechanical analysis of unidirectional composite. Dang and Sankar [18] used the penalty parameter method to enforce the essential boundary conditions to the RVE. They also used the transformation approach to enforce the continuity of displacement at the fiber-matrix interface.…”

A generalized plane strain meshless local Petrov–Galerkin method for the micromechanics of thermomechanical loading of composites

“…More recently, Dang and Sankar [2007] employed the conventional MLPG method for the micromechanical analysis of unidirectional composites. They used the penalty parameter method to enforce the essential boundary conditions on the RVE.…”