Issue34

Y. Sumi, Frattura ed Integrità Strutturale, 34 (2015) 43-59; DOI: 10.3221/IGF-ESIS.34.04 43 S OLUTIONS OF A CURVED CRACK he problem of a slightly non-collinear, quasi-static crack growth is considered. Problems of this kind have been treated by a first order perturbation method, in the context of Muskhelishvili’s complex potentials [18], by Banichuk [1] and Goldstein and Salganik [2,3]. Cotterell and Rice [4] employed the same method and obtained a rather simple first order expression for the stress intensity factors, which were used to examine the crack growth path of a semi-infinite crack in an infinitely extended domain. We shall consider the first and second order perturbation solutions which can take into account the effects of the geometry of the domain, i.e., finite outer boundaries as well as the finite crack length, Sumi et al . [5], Sumi [6,7]. For this purpose we first calculate the perturbation solution for a semi-infinite straight crack with a slightly kinked and curved extension, which will be used as the fundamental solution of the problem. We can establish the effect of the geometry of the domain by considering the far field behavior of the fundamental solution. Second Order Perturbation Solution Consider a homogeneous linearly elastic brittle solid containing a straight crack of length a . The stress tensor and the displacement vector  ij and u i are defined in the domain V occupied by the body. Surface tractions, t i , are prescribed on the part of the outer boundary, S t , and on the crack faces, S C ± , while surface displacements, v i , are prescribed on the remaining part of the outer boundary, S u (see Fig. 1). The Cartesian coordinate system ( x 1 , x 2 ) with the origin at the crack tip has its x 1 - axis along with the original crack line. In the following analysis this problem is referred to as the original problem.  Figure 1 : Straight crack with slightly kinked and curved extension. For a slightly non-symmetric loading system, we may have a slightly kinked and curved crack extension, whose projected length on the x 1 -axis is h , and whose deviation from the x 1 -axis is  ( h ) . To investigate the detailed distribution of stresses ahead of the extended crack tip, we consider the second order perturbation solution of the semi-infinite straight crack with slightly kinked and curved extension, whose near tip field solution gives the stress intensity factors. Then we account for the effect of the finite outer boundary by using the far field solution of the semi-infinite crack. In the following analysis we assume that the shape of the crack extension can be approximated by 3/2 2 ( ) , 1 1 1 1 x x x x        (1) disregarding higher order terms. Constants     and  are considered as shape parameters of the crack extension. This expression has the appropriate asymptotic form associated with the stress field ahead of a preexisting crack tip. The crack profile may be determined based on an appropriate crack path criterion which will be discussed in the following section. As is shown in Fig. 2, another linear orthogonal coordinate system (  1 ,  2 ) is introduced, , 1 1 x   (2) and the crack path deviation from the  1 -axis is measured by T