DOI 10.1007/s10704-005-1922-6 International Journal of Fracture (2005) 133:1–41 © Springer 2005 A through interface crack between a ±45 ◦ transversely isotropic pair of materials YUVAL FREED ∗ and LESLIE BANKS-SILLS The Dreszer Fracture Mechanics Laboratory, Department of Solid Mechanics, Materials and Systems, The Fleischman Faculty of Engineering, Tel Aviv University, Ramat Aviv 69978, Israel ∗ Author for correspondence (E-mail:yuval@eng.tau.ac.il) Received 1 August 2004; accepted in revised form 8 February 2005 Abstract. In this study, the first term of the asymptotic displacement and stress fields is determined analytically for a straight through crack along the interface between a ±45 ◦ transversely isotropic pair of materials. Since with this configuration, there is full coupling between the modes, this prob- lem requires a three-dimensional treatment. To calculate stress intensity factors, a three-dimensional M-integral is derived using the asymptotic fields as auxiliary solutions. The displacement extrapolation method is derived as well, and used to check the results obtained by the M-integral. Two numeri- cal test cases are employed to examine the accuracy of both methods. Results obtained for other mechanical problems are presented as well. Key words: Fiber reinforced material, interface crack, stress intensity factors, three-dimensional J -inte- gral, three-dimensional M-integral, transversely isotropic material. 1. Introduction The use of fiber reinforced composites has increased recently in various applications such as sport and medical equipment, the automotive industry and aerospace struc- tures. Therefore, it is important to understand the behavior of delaminations or inter- face cracks in composites and predict their propagation. These materials may be treated effectively as anisotropic. Although interface cracks in anisotropic media have been widely investigated in the past few years (see for example Ting, 1986, 1995; Suo, 1990; Hwu, 1993; Banks-Sills and Boniface, 2000), only few studies, Qu and Xue (1998) for instance, have concentrated on three-dimensional problems. This is a result of the complexity of such problems and the extremely large computational effort required in numerical analysis. This paper focuses on the behavior of a crack along the interface between a ±45 ◦ transversely isotropic pair of materials. Asymptotic stress and displacement fields, as well as fracture mechanics parameters, such as Griffith’s energy release rate, crack face displacements, and stresses along the interface are obtained. Based on these ana- lytical expressions, two methods for calculating stress intensity factors are derived. Since a ±45 ◦ transversely isotropic pair causes in-plane and out-of-plane coupling, all modes are involved for all displacements and stresses. Therefore, the crack should be treated in three dimensions. Guided by the results of Hartranft and Sih (1969) for a three-dimensional through crack in a homogeneous material however, conditions of