Abstract The quasi-static and unsteady growth for a plane strain mode I crack in power-law hardening materials is investigated under small-scale yielding conditions by means of the finite element method. In order to improve the quality of the computing program, some special and effective measures (for example, the prediction of the stiffness matrix) are proposed. Based on the asymptotic fields σ ij∼ ln A r α ̃ ), ϵ ij∼ ln A r α ̃ +1 , u i∼r ln A r α ̃ +1 (where α ̃ = 1 (n − 1) , n is the power-law hardening index), the numerical results are analysed. As a result, the important parameters in the asymptotic fields, which are undetermined by the asymptotic analysis or depend on the global solution, are determined. Moreover, some J-resistancecurves applicable to the engineering practice are given based on the fracture criterion of the critical opening displacement. Finally, the effect of different loading histories on the important parameters are discussed.