Int J Fract DOI 10.1007/s10704-017-0190-6 ORIGINAL PAPER Virtual crack closure technique for an interface crack between two transversely isotropic materials Elad Farkash · Leslie Banks-Sills Received: 21 November 2016 / Accepted: 27 January 2017 © Springer Science+Business Media Dordrecht 2017 Abstract The virtual crack closure technique makes use of the forces ahead of the crack tip and the dis- placement jumps on the crack faces directly behind the crack tip to obtain the energy release rates G I and G II . The method was initially developed for cracks in lin- ear elastic, homogeneous and isotropic material and for four noded elements. The method was extended to eight noded and quarter-point elements, as well as bimate- rial cracks. For bimaterial cracks, it was shown that G I and G II depend upon the virtual crack extension Δa. Recently, equations were redeveloped for a crack along an interface between two dissimilar linear elastic, homogeneous and isotropic materials. The stress inten- sity factors were shown to be independent of Δa. For a better approximation of the Irwin crack closure inte- gral, use of many small elements as part of the virtual crack extension was suggested. In this investigation, the equations for an interface crack between two dissimilar linear elastic, homogeneous and transversely isotropic materials are derived. Auxiliary parameters are used to prescribe an optimal number of elements to be included in the virtual crack extension. In addition, in previous papers, use of elements smaller than the interpenetra- E. Farkash (B ) · L. Banks-Sills Dreszer Fracture Mechanics Laboratory, School of Mechanical Engineering, Tel Aviv University, 69978 Ramat Aviv, Israel e-mail: eladfarksh@mail.tau.ac.il L. Banks-Sills e-mail: banks@tau.ac.il tion zone were rejected. In this study, it is shown that these elements may, indeed, be used. Keywords Energy release rate · Finite element method · Interface crack · Interpenetration zone · Transversely isotropic · VCCT 1 Introduction Initial assumptions for the Virtual Crack Closure Tech- nique were made by Rybicki and Kanninen (1977). Using those assumptions, Raju (1987) presented a mathematical derivation for this method. While Rybicki and Kanninen (1977) proposed the method for four noded elements, Raju (1987) developed it for higher order elements, including eight noded and quarter-point elements. For the mathematical derivation, Raju (1987) used the Irwin (1958) crack closure integral which is given as G = lim Δa→0 1 Δa  Δa 0  σ yy (Δa − r )u y (r ) + σ yx (Δa − r )u x (r )  dr (1) in two dimensions. In Eq. (1), σ yy and σ yx are the tensile and shear stresses, respectively, ahead of the crack tip. The tensile stress σ yy is shown in Fig. 1a. The sliding and opening displacements are denoted as u x and u y , respectively; u y is shown in Fig. 1b. Note that r is the radial coordinate emanating from the crack tip whose length is a + Δa, as shown in Fig. 1b. The first and 123