An improvement in the modeling of the second derivative of displacement field in structural analysis

The theory of locating error-minimal points for stress computation has been explained in some researches, and it is known that these points are the Gauss points of a quadrature rule one order less than that required for full integration of the element stiffness matrix in the finite element analysis. In the analysis of fracture mechanics of structures using 3D J -integral, second derivative of displacement field (SDD) is required, but it is not calculated in commercial finite element software. In this paper, it is shown that error-minimal points for stress computation are not suitable for SDD computation. Instead, they are most accurate at the Gauss points of a quadrature rule two order less than that required for full integration of the element stiffness matrix. For improving SDD evaluation, it is computed at the central Gauss point for all elements, and it is considered as the primary value for element nodes. The average of primary values of neighboring elements of each node is considered to be the final value at the node. SDD for the entire element is calculated by interpolation of its nodal values. It is shown that the results obtained using the proposed method, are more accurate than the results of differentiating the displacement field directly at the point of interest. The results are verified by comparison to simple analytical cases. The method is implemented in calculating the J -integral of a circular crack and the results are compared to well-known reference values. The proposed method shows great improvement over the conventional methods for calculation of SDD.