Improved Numerical Determination of the Effective Stress within the Framework of Local Approach Cleavage Models

For the determination of the Weibull stress or analogous effective stress measures in the assessment of the cleavage failure probability based on finite element results, robust and numerically stable procedures are necessary. For the quantitative assessment of the numerical errors in the approximate integration procedures for the determination of the Weibull stress or corresponding stress measures, an appropriate error indicator in analogy to adaptive finite element meshing methods was derived. The mathematical considerations reveal that an evaluation of the cleavage probability by means on the classical Beremin (1983) model using the standard one-point Gauss integration scheme may result in distinct numerical errors. Therefore, an enhanced numerical integration formula was derived and implemented. This formula facilitates a numerical evaluation of the integrals with improved accuracy.