The paper deals with an analysis of the stress and displacement fields in various cracked specimens (loaded in mode I as well as in mode II). The main goal is to determine an accurate approximation of the near-crack-tip fields which can be subsequently used e.g. in the case of quasi-brittle materials for estimation of the extent of a nonlinear zone at the crack tip in general, the fracture process zone in particular. Values of coefficients of the so-called higher-order terms of the power expansion by which the fields can be expressed are determined via regression from results of numerical simulations. The analysis is conducted using 2D numerical models and ANSYS FE computational system is employed. Various aspects regarding the description of the stress/displacement fields by means of the Williams expansion are discussed; especially the convergence of the coefficients of the first several terms of the series expansion, their absolute values and importance for an accurate stress approximation.