A unified definition for stress intensity factors of interface corners and cracks

Abstract

Based upon linear fracture mechanics, it is well known that the singular order of stresses near the crack tip in homogeneous materials is a constant value À1/2, which is nothing to do with the material properties. For the interface cracks between two dissimilar materials, the near tip stresses are oscillatory due to the order of singularity being À1/2 ± ie and À1/2. The oscillation index e is a constant related to the elastic properties of both materials. While for the general interface corners, their singular orders depend on the corner angle as well as the elastic properties of the materials. Owing to the difference of the singular orders of homogeneous cracks, interface cracks and interface corners, their associated stress intensity factors are usually defined separately and even not compatibly. Since homogenous cracks and interface cracks are just special cases of interface corners, in order to build a direct connection among them a unified definition for their stress intensity factors is proposed in this paper. Based upon the analytical solutions obtained previously for the multibonded anisotropic wedges, the near tip solutions for the general interface corners have been divided into five different categories depending on whether the singular order is distinct or repeated, real or complex. To provide a stable and efficient computing approach for the general mixed-mode stress intensity factors, the path-independent H-integral based on reciprocal theorem of Betti and Rayleigh is established in this paper. The complementary solutions needed for calculation of H-integral are also provided in this paper. To illustrate our results, several different kinds of examples are shown such as cracks in homogenous isotropic or anisotropic materials, central or edge notches in isotropic materials, interface cracks and interface corners between two dissimilar materials.

Path-independent H-integral for interface corners under thermal loadings

“…Since the singular order δ may be distinct or repeated, real or complex, combination of all the possible solutions associated with the most critical singular order whose real part δ R is maximal leads, as in [Hwu and Kuo 2007;Hwu and Ikeda 2008], from (6) to…”

“…In (7) the angular brackets < < > > stand for a diagonal matrix in which each component is varied according to the subscript α, for example, < < z α > > = diag[z 1 , z 2 , z 3 ]; δ R and α are, respectively, the real and imaginary parts of the most critical singular order δ c determined by (5a) with 0 < Re(δ) < 1; l is a length parameter which may be chosen arbitrarily as long as it is held fixed when specimens of a given material pair are compared; V (θ) and (θ ) are eigenfunction matrices related to E(θ )w 0 ; = (0), detailed expressions for which can be found in [Hwu and Kuo 2007;Kuo and Hwu 2010]; and k is a vector containing different modes of stress intensity factors and is defined by…”

“…To overcome this problem, several path-independent integrals have been proposed for crack problems such as the J-integral [Rice 1968], the L-integral [Choi and Earmme 1992], the M-integral [Im and Kim 2000], and the H-integral [Bueckner 1973;Stern 1976;Sinclair et al 1984;Chen 1985]. For corner problems that are usually in the status of mixed-mode intensity, the H-integral was suggested by [Hwu and Kuo 2007] for two-dimensional interface corners, and modified by [Kuo and Hwu 2010] for three-dimensional interface corners, which are valid for pure mechanical loading conditions. For interface corners under thermal loadings, the H-integral was modified by [Banks-Sills and Ishbir 2004;Nomura et al 2009].…”

“…Since the relation between the stress intensity factors k and the H-integral is obtained from the results of the H-integral passing through the singular field, it can now be written by referring to the relation obtained for the pure mechanical loading problem [Hwu and Kuo 2007], that is,…”

“…By this formalism, analytical closed-form solutions for the orders of heat flux/stress singularity and near-tip solutions of multimaterial anisotropic wedges under thermal loadings have been obtained in [Hwu and Lee 2004]. To understand the fracture behavior of interface corners, a unified definition of stress intensity factors for connecting cracks, corners, interface cracks, and interface corners was proposed in [Hwu and Kuo 2007]. In that work, in order to avoid the oscillatory singular problems of interface corners a path-independent H-integral [Bueckner 1973;Stern 1976] was suggested to calculate the stress intensity factors.…”