Numerical modelling of cracks in metals and experimental and numerical modelling of notches in polymer materials are presented in this thesis. Thick-walled metal pipes with semielliptical cracks of various dimensions were subjected to internal pressure. On the basis of extensive numerical analysis of the case, expressions for a more accurate assessment of the Jintegral and limit pressure in comparison to the previously recommended solutions are proposed. A new loading function is recommended instead of limit pressure in the expressions for reference stress and the J-integral. As a result, the results for the J-integral obtained by using the new reference stress method deviate insignificantly from the results obtained from the incremental theory of plasticity. This is illustrated by the examples of the thick-walled metal pipes of the VVER 1000 steam generators. The results obtained by using the proposed expressions for limit pressure in thick-walled metal pipes with cracks were compared with the results available in literature. An analysis of the plastic collapse of the pipes in steam generators has shown that the results obtained by using expressions for plastic collapse pressure from literature may indeed be very unreliable. The impact of the selection of the flow stress factor on the estimation of the plastic collapse pressure in pipes was investigated. Instead of the flow stress factor which depends only on the type of material, a new factor, which depends both on the material and on the crack dimensions, is proposed for the plastic collapse pressure expression. This provides more exact modelling not only for shallow cracks but also for deep cracks in pipes when compared to the existing solutions. Furthermore, a very detailed investigation was conducted into the use of the sub-modelling technique when calculating fracture mechanics parameters in thick-walled metal pipes. Optimal sizes of the sub-model related to the length of the crack are proposed for calculating the parameters of the linear elastic and the elastic-plastic fracture mechanics. Regarding the research into polymers, a numerical algorithm for modelling the behaviour of medium-density polyethylene for the case of creeping, including viscoelastic and viscoplastic effects, was derived on the basis of experimental results available in literature and of formulations similar to those proposed for metal materials. The new numerical algorithm proposed is intended for analysing the mechanisms in primary and secondary creep and fracture during the slow crack growth in polyethylene. Modelling the real behaviour of materials allows a more accurate calculation of parameters of fracture mechanics. The results of numerical tests were compared to the available experimental and numerical solutions. In addition, the paper includes the conducted experimental research into the fracture and the time-dependent behaviour of PE100 polyethylene at various temperatures and deformation rates, which has not been reported in literature. Particular material parameters for a primarysecondary creep law were determined. This brought the numerical modelling close to the real behaviour of materials. The results of the numerical analysis for notched action in pipes made of PE100 polyethylene were compared to experimental results in literature.