The effect of the lattice parameter of functionally graded materials on the dynamic stress field near crack tips
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The effect of the lattice parameter of functionally graded materials on the dynamic stress field near crack tips Jun Liang Æ Zhen Gong Zhou
Received: 18 January 2006 / Accepted: 6 March 2006 / Published online: 13 July 2006 Springer Science+Business Media B.V. 2006
Abstract In this paper, the effect of the lattice parameter of functionally graded materials on the dynamic stress fields near crack tips subjected to the harmonic anti-plane shear waves is investigated by means of non-local theory. By use of the Fourier transform, the problem can be solved with the help of a pair of dual integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the dual integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularities are present near crack tips. The non-local elastic solution yields a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion in functionally graded materials. Keywords Crack Æ Non-local theory Æ Functionally graded materials Æ Lattice parameter
Introduction In recent years, functionally graded materials (FGMs) have been widely introduced and J. Liang (&) Æ Z. G. Zhou Center for Composite Materials, Harbin Institute of Technology, 1247, Harbin 150001, P. R. China e-mail: [email protected]
applied to the development of thermal and structural components due to its ability to not only reduce the residual and thermal stresses but to increase the bonding strength and toughness as well. To help the development of such materials, many analytical and theoretical studies in fracture mechanics have been widely done. In an attempt to address the issues pertaining to the fracture analysis of bonded media with such transitional interfacial properties, a series of solutions to certain crack problems were obtained by Erdogan and his associates (Erdogan and Wu 1997; Delale and Erdogan 1988). However, it is found that all the solutions in references (Erdogan and Wu 1997; Delale and Erdogan 1988; Bao and Cai 1997; Shbeeb and Binienda 1999) contain the stress singularities near the crack tips. This phenomenon is not reasonable according to the physical nature. As a result of this, beginning with Griffith, all fracture criteria in practice today based on other considerations, e.g. energy, the J-integral (Rice 1968) and the strain gradient theory (Xia and Hutchinson 1996). To overcome the stress singularities at the crack tips in the classical elastic fracture theory, Eringen (Eringen Speziale and Kim 1977; Eringen 1978; Eringen 1979) used non-local theory to discuss the stress near the tip of a sharp line crack in an isotropic elastic plate subject to uniform tension, shear and anti-plane shear, and the resulting solutions did not contain
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any stress singularities at the crack tips. This allows us to use the maximum stress as a fracture criterion. Recently, some fracture problems (Zhou et al. 2003; Z
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