In this thesis, crack growth in steel, polyethylene and polyethylene composites is simulated by means of the cohesive zone model in the finite-element-program ABAQUS. The cohesive zone model is a fracture mechanical approach that is often used to simulate crack growth. Two parameters determine the model, the cohesive energy and the cohesive strength. To check the applicability and the correctness of the approach, crack growth is first simulated for a steel where experimental results are available from a previous investigation. The results of the simulation fit well to the experimental data, however, some problems appeared regarding the initiation of crack growth during the optimization stage of the modelling. The problems were caused by strong deformation near the crack tip, which did not occur in the simulations with polyethylene. The reason of the different behaviour lies in the different ratio between the cohesive strength and the yield strength in the two different material groups. In contrast to polyethylene, very high strains are required to reach the cohesive strength for steel. It was possible to solve the problem by refining the cohesive elements. The two types of polyethylene investigated, PE1 and PE2, differ in the stress strain behaviour, the cohesive energy and the cohesive strength. The results of the crack growth simulations show that the resistance against the initiation of crack growth is higher in PE1. This is caused primarily due to the higher cohesive energy of PE1. To analyze crack growth in polyethylene composites, a bi-material model was generated. The interface between the two materials is perpendicular to the direction of crack growth and is assumed to be perfect. If the initial crack lies in PE1 and grows towards the PE2-interface, a strong, abrupt crack extension step occurs under constant load. If the crack grows from PE2 to PE1, crack growth is initiated earlier and the crack temporarily stops at the interface. The investigations show the strong link between crack growth behaviour and the different arrangements of the two types of polyethylene. This thesis reveals the intense influence of the stress strain behaviour and the cohesive zone parameters of the material on the requirements of the modelling and the results of the numerical simulation.