Multi-parameter approximation of the stress field in a cracked body in the more distant surroundings of the crack tip

• Stress field in more distant surroundings from crack tip is approximated by truncated Williams series.
• Coefficients of terms up to 12th order are determined via regression of FEM results.
• Backward reconstruction of the crack-tip stress field is performed and analysed.
• Example of wedge splitting test specimen with various crack lengths is investigated.
• Hints towards correct approximation of crack-tip fields are given.

The accuracy of the multi-parameter approximation of the stress fields in a cracked body is studied in the paper. This approximation, which uses the Williams power series expansion (WE), is intended to be used to estimate the extent of the nonlinear zone, which plays a significant role within tensile failure and the fatigue assessment of non-brittle materials. The characteristics of this zone could be potentially incorporated into methods of determining the true values of fracture parameters and the fatigue behaviour descriptors of materials exhibiting nonlinear failure. Considering the fact that in the case of elastic–plastic and especially quasi-brittle materials the size of this zone is substantial in comparison to specimen dimensions, it is necessary to consider a large region around the crack tip for this task. An automatic routine implemented as a Java application was created to determine the values of coefficients of the higher order terms of the WE that describe crack-tip fields. These values are calculated using the Over-Deterministic Method (ODM), which is applied to the results of the finite element (FE) analysis of an arbitrary mode I test geometry. Furthermore, another Java application developed by the authors provides an analytical reconstruction of the crack-tip stress field by means of the truncated WE, and enables detailed analysis of the crack-tip stress field approximation. The developed procedures simplify the analysis of the description of mechanical fields at a greater distance from the crack tip considerably. The presented research is focused on the optimisation of the selection of FE nodal results entering the ODM procedure used to determine the values of coefficients of the higher order terms of the WE. The aim is to improve the accuracy of the approximation.