Stiffness derivative finite element technique to determine nodal weight functions with singularity elements

Abstract

The use of the stiffness derivative technique coupled with “quarter-point” singular crack-tip elements permits very efficient finite element determination of both stress intensity factors and nodal weight functions. Two-dimensional results are presented in this paper to demonstrate that accurate stress intensity factors and nodal weight functions can be obtained from relatively coarse mesh models by coupling the stiffness derivative technique with singular elements. The principle of linear superposition implies that the calculation of stress intensity factors and nodal weight functions with crack-face loading, σ( r s ), is equivalent to loading the cracked body with remote loads, which produces σ( r s ) on the prospective crack face in the absence of crack. The verification of this equivalency is made numerically, using the virtual crack extension technique. Load independent nodal weight functions for two-dimensional crack geometry is demonstrated on various remote and crack-face loading conditions. The efficienct calculation of stress intensity factors with the use of the “uncracked” stress field and the crack-face nodal weight functions is also illustrated. In order to facilitate the utilization of the discretized crack-face nodal weight functions, an approach was developed for two-dimensional crack problems. Approximations of the crack-face nodal weight functions as a function of distance, ( r s ), from crack-tip has been sucessfully demonstrated by the following equation: h(a, r s,) = A(a) √r s + B(a) + C(a)√r s+ D(a)r s . Coefficients A(a), B(a), C(a) and D(a), which are functions of crack length ( a), can be obtained by least-squares fitting procedures. The crack-face nodal weight functions for a new crack geometry can be approximated using cubic spline interpolation of the coefficients A, B, C and D of varying crack lengths. This approach, demonstrated on the calculation of stress intensity factors for single edge crack geometry resulted in total loss of accuracy of less than 1%.