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The plane-stress equations of linear elasticity are used in conjunction with those of the boundary element method to develop a novel curved, quadratic boundary element applicable to structures composed of anisotropic materials in a state of plane stress or plane strain. The curved boundary element is developed to solve two-dimensional, elastostatic problems of arbitrary shape, connectivity, and material type. As a result of the anisotropy, complex variables are employed in the fundamental solution derivations for a concentrated unit-magnitude force in an infinite elastic anisotropic medium. Once known, the fundamental solutions are evaluated numerically by using the known displacement and traction boundary values in an integral formulation with Gaussian quadrature. All the integral equations of the boundary element method are evaluated using one of two methods: either regular Gaussian quadrature or a combination of regular and logarithmic Gaussian quadrature. The regular Gaussian quadrature is used to evaluate most of the integrals along the boundary, and the combined scheme is employed for integrals that are singular. Individual element contributions are assembled into the global matrices of the standard boundary element method, manipulated to form a system of linear equations, and the resulting system is solved. The interior displacements and stresses are found through a separate set of auxiliary equations that are derived using an Airy-type stress function in terms of complex variables. The capabilities and accuracy of this method are demonstrated for a laminated-composite plate with a central, elliptical cutout that is subjected to uniform tension along one of the straight edges of the plate. Comparison of the boundary element results for this problem with corresponding results from an analytical model show a difference of less than 1%. Stanley S. Smeltzer III, Mechanics and Durability Branch, 8 W. Taylor St., NASA Langley Research Center, Hampton, VA 23681-2199. Eric C. Klang, Mechanical and Aerospace Engineering Department, Box 7910, Raleigh, NC 27695-7910. INTRODUCTION Composite materials continue to see increased usage in space, aircraft, industrial, and recreational markets around the world. This upward trend is reflected in the recent advancements made for finite element methods; however, relatively few advances for boundary element methods (BEM) have been made for composite structures [1,2]. Boundary element advancements for composite structures are important, since boundary elements offer increased efficiency and accuracy over finite elements for infinite and semi-infinite regions, regions with large response gradients, and boundaries with complex geometry as well as reduced pre-processing time for discretizing a problem. The initial application of the BEM to bodies composed of anisotropic materials was made by Rizzo [3]. Cruse presented a further advancement of the BEM in 1971 that included a derivation of the traction and displacement fundamental solutions for an anisotropic plate as well as determining stress concentrations for anisotropic plates with circular and elliptical cutouts [4]. Recent texts written about the BEM have focused on basic problems associated with isotropic structures as well as advanced topics in the areas of fracture mechanics [5], acoustics, and the evaluation of complex integrals for body forces and threedimensional volumes [6]. The primary contribution of the curved, quadratic boundary element developed in the present paper is to provide a capability for modeling anisotropic regions that have highly curved boundaries. Instead of modeling a circular or elliptical boundary by using many straight elements to approximate the geometry, a relatively few curved elements can be used to model the geometry exactly with no loss in solution accuracy. An important aspect of the curved, quadratic boundary element resulting from the material anisotropy is the treatment of complicated singular integrals. The integrand of the singular integrals is an elaborate expression in terms of complex variables that results from the anisotropic material behavior, quadratic shape functions, and Gaussian quadrature evaluation points. All integrals are evaluated using the quadratic shape functions and Gaussian quadrature, which provides very accurate modeling of curvilinear boundaries. Thus, by careful manipulation of the singular integrals a sophisticated scheme has been developed for analyzing structures that are composed of anisotropic materials, have complex curved boundaries, and are in a state of plane strain or plane stress. The objective of the present paper is to demonstrate a numerical method for evaluating elaborate, complex singular integrals that are used in the development of a curved, quadratic boundary element for structures composed of anisotropic materials that are in a state of plane stress or plane strain (referred to herein as plane anisotropic structures). This objective is accomplished by presenting the basic equations that govern a two-dimensional, linear elastostatic problem, the development of the boundary integral equation, the boundary element solution process, and the treatment of singular integrals using the developed method. A brief description of the modeling characteristics and analysis results for a classic structural mechanics problem is then presented to demonstrate the accuracy and computational efficiency of the curved boundary element for use with complex geometry. THEORY Boundary Integral Equation (BIE) Formulation First, the equations that govern two-dimensional linear elastostatics as given by Brebbia are stated [7]. The equations of equilibrium are written using indicial notation as, b 0 kl,l l σ + = (k,l = 1,2) in Ω (1) where σkl are the components of the stress tensor, bl are the components of the body-force vector, and Ω is the problem domain enclosed by a boundary denoted by Γ. The equations derived in this paper follow the rules of indicial notation; specifically, partial differentiation is represented by terms containing a comma and repeated indices indicate terms that are summed. The kinematic relations are 1 (u u ) 2 ij i,j j,i ε = + (i, j = 1,2) (2) where εij are the components of the strain tensor and ui represents the components of the displacement vector. An additional equation that the strain components must satisfy when the displacement field is not the primary dependent variable is the compatibility equation. This equation in two dimensions is stated as 2 11,22 22,11 12,12 ε + ε = ε (3) Eq. 3 provides a necessary and sufficient condition for specified strain components to give displacements that are single-valued and continuous for simply connected regions. Since the problems of interest in the present paper are two-dimensional in nature, the constitutive equation for a state of plane stress is given for a plane anisotropic structure as 11 12 16 12 22 26 16 26 66 S S S S S S 2 S S S 11 11