### Meshless Local Petrov-Galerkin Formulation for Problems in Composite Micromechanics

Thi D. Dang, Bhavani V. Sankar
2007 AIAA Journal
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
more » ... ions. 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