Perturbation analyses for slightly non-colinear and quasi-static crack extension were first performed by Banichuk (1970), and Goldstein and Salganik (1970, 1974) with the use of Muskhelishvili's complex potentials. Having obtained a rather simple expression of stress intensity factors, Cotterell and Rice (1980) examine the crack growth path of a semi-infinite crack in an infinitely extended plane. First order perturbation analyses are made by Sumi et al. (1981, 1986) for a straight crack in a finite body with the slightly branched and curved extension. The shape of branched and curved extension is approximated by a continuous function with three parameters, and the approximate stress intensity factors at the extended crack tip are obtained in terms of these shape parameters and the near tip stress field parameters ahead of the crack tip prior to its extension, where the effects of the geometry of the domain are also taken into account. In the present paper we extend our perturbation analysis from the first order to the second order with respect to the shape of the non-collinear crack path. An approximate description of the stress intensity factors is obtained at the kinked-curved crack tip, where the cracked body is subjected to an arbitrary far field boundary condition. A kind of a matched asymptotic expansion method is introduced in order to construct the solution, where the effects of the geometry of the domain are taken into account by alternately matching the far field asymptotic behavior and the near tip field asymptotic behavior in an ascending order of the square root of the crack extensional length. If we consider the smooth crack curving which confirms the continuity of the first derivative along the curved trajectory, the second order perturbation solution thus obtained gives the exact asymptotic property of the solution. The elastic energy release rate, which corresponds to the non-collinear crack growth, can be calculated by using Irwin's formula. As far as homogeneous materials are concerned, both the maximum energy release rate and the local symmetry critria predict the same crack growth direction within the second order approximation. Considering material inhomogeneity such as degradation zone, a crack may be branched and curved due to the spatial variation of the fracture toughness. The energy criterion is relevant to predict this type of behavior, which cannot be properly examined by simple stress or strain criteria.