Determination of Second-Order term Coefficients for the Inclined Crack in Orthotropic Plate using Singular Finite Elements

Abstract

The finite element method with quarter-point crack-tip elements is used and a simple formula for obtaining the coefficients of the second-order terms in the series expansion for near crack tip stresses in orthotropic materials under biaxial loading is presented. This formula is obtained by comparing the variation of the displacements along the crack tip element with the elastic field solution for the crack tip. Numerical examples are given for the validity of the present formulation. The results obtained are compared with the theoretical ones and a good agreement between the two solutions is obtained.Keywords: orthotropic plate, inclined crack, biaxial loading, stress intensity factor, second-order term coefficient, finite element analysis.1. Introduction. The importance of the second-order term of the series expansion for crack tip stresses and displacements is evident in the case of a flat crack under tensile loads in normal and tangential directions where the effect of the tangential stress appears only in the second term, and is thus completely omitted in the first-term formula. Omitting this term not only denies the physical presence of such a load, but also misleads one into thinking that load biaxiality does not affect the prediction of crack growth direction. The second-order term is therefore quantitatively significant for fracture mechanics analysis. Concerning the problem of cracked anisotropic materials, Yuan (2000a, 2000b) have investigated the effects of higher-order terms on the expression for crack-tip stresses. They derived the second and third terms of the crack tip stress field as additional parameters in characterizing the behavior of the crack. Lim et al (2001) have estimated the influence of second-order terms originated from load biaxiality on the stress and displacement fields in the vicinity of the crack tip under mode I in anisotropic materials and investigated their effects on the predicted crack growth direction. This analysis has been extended to the case of mixed-mode cracks by Lim and Sankar (2002), Carloni and Nobile (2002), Carloni et al (2003), Nobile et al (2004), Nobile and Carloni (2005) and Viola et al (2008).Numerical analysis for cracked orthotropic bodies was provided by Saouma and Sijiotis (1986), Boone et al (1987) and Garcia et al (2004), where they computed the coefficients of the first-order terms by employing the singular crack-tip element. However, to the best of the author's knowledge, little work has been done so far to obtain the coefficients of the secondorder terms for cracked orthotropic materials by using the finite element method(FEM). In this paper, the correct form of the second-order terms near a crack in orthotropic materials under biaxial loading is given. It is shown that the FEM with quarter-point singular crack-tip elements can be used to obtain the coefficients of the second-order terms. Special formulae

Mixed-Mode Crack Extension in Orthotropic Materials Under Biaxial Load

“…The expressions for the near crack tip stresses including the second order term can be found as (Lim 2011) . In these expressions is the applied stress, k is the horizontal load ratio, and is the crack inclination angle.…”