ELSEVIER Engineering Analysis with Boundary Elements 20 (1997) 287-298 © 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain Plh S 0 9 5 5 - 7 9 9 7 ( 9 7 ) 0 0 0 7 0 - 2 0955-7997/97/$17.00 Three-dimensional fracture analysis in transversely isotropic solids A. Sfiez, M. P. Ariza & J. Dominguez* Escuela Superior de Ingenieros, Universidad de Sevilla, Av. Reina Mercedes s/n, 41012-Sevilla, Spain (Received 21 August 1997; accepted 5 September 1997) In this paper a boundary element formulation for three-dimensional crack problems in transversely isotropic bodies is presented. Quarter-point and singular quarter-point elements are implemented in a quadratic isoparametric element context. The point load fundamental solution for transversely isotropic media is implemented. Numerical solutions to several three-dimensional crack problems are obtained. The accuracy and robustness of the present approach for the analysis of fracture mechanics problems in transversely isotropic bodies are shown by comparison of some of the results obtained with existing analytical solutions. The approach is shown to be a simple and useful tool for the evaluation of stress intensity factors in transversely isotropic media. © 1998 Elsevier Science Ltd. All rights reserved Keywords: Cracks, fracture mechanics, stress intensity factor, boundary elements, transversely isotropic bodies. 1 INTRODUCTION Transversely isotropic elastic materials are those with an axis of symmetry such that all directions perpendicular to this axis are equivalent. In other words, any plane perpen- dicular to the axis is a plane of isotropy. Crystals of the hexagonal system are transversely isotropic solids. There is also a significant number of fiber-reinforced composites which show this kind of behavior from a macroscopic point of view. They are composite solids made of unidirectionally oriented fibers, the fiber diameter and the fiber spacing being much smaller than the dimensions of the body. The basic fracture mechanics concepts for transversely isotropic solids were set 30 years ago by Kassir and Sih. I They showed that the stress singularity of the order ~r near the periphery of three-dimensional cracks, well known for isotropic materials, remains in the case of transversely iso- tropic solids. The magnitude of the local stresses may also be described in this case in terms of stress intensity factors (SIFs). Kassir and Sih obtained expressions of stresses and displacements near the crack front for cracks of arbitrary shape in a plane perpendicular to the material axis of symmetry. Sih et al. 2 had obtained expressions for stress and displacement fields near the tips of cracks in two- *To whom correspondence should be addressed. 287 dimensional anisotropic bodies 3 years previously. More recently, Rajiyah and Atluri 3 generalized the analytical solution for the SIF of a flat elliptical crack, to arbitrary crack face loading. The previous solution obtained by Kassir and Sih I was limited to the case of constant and linear variations of tractions on the crack face. Rajiyah and Atluri 3 employed their generalized solution with the finite element alternating technique to analyze embedded and surface elliptical cracks in transversely isotropic bodies of finite dimensions where the crack plane is perpen- dicular to the material axis. The number of boundary element papers dedicated to the analysis of fracture mechanics problems in non-isotropic materials is rather small. Among these papers, the early work of Snyder and Cruse 4 and the more recent one of Chan and Cruse 5 should be mentioned. Cruse and co-work- ers developed a boundary element approach for the analysis of two-dimensional crack problems in fully anisotropic bodies. They employed a special fundamental solution which incorporates the presence of a traction-free crack in an infinite anisotropic body. The use of such fundamental solution makes it unnecessary to represent the crack surface in the boundary element discretization. The stress intensity factors are computed using a path independent integral once the boundary displacements and tractions have been obtained. In 1992, Tan and Gao 6 presented the quadratic
Three-dimensional fracture analysis in transversely isotropic solids
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