The scope of this thesis is to analyse a number of fracture mechanics problems with cracks by means of mathematical methods and recently developed numerical techniques. Various physical models of cracks, in the framework of both continuous and discrete media, are considered. The main assumption that all models in this study have in common is concerned with the presence of a special region, often called process zone, located in front of the crack, separating the damaged zone and the undamaged part of the material. As opposed to the classic Griffith’s model of crack, where the observed effects are limited with elastic behaviour of the material, models with such regions allow non-linear effects. The primary goal of this research is to develop a model that describes the process of propagation of damage in solids, where the fracture is accompanied with a bridging or other complex processes. For this purpose, the analysis begins with the consideration of a problem of a static crack with process zone that lies between two different elastic media which is solved analytically. This problem is followed by a similar model with slightly modified conditions on the process zone and the solution is obtained numerically. Some limiting cases that complement the understanding of the problem are considered for both models. Note that both models mentioned above are introduced in the framework of continuum mechanics. Qualitative analysis allows to derive values of the parameters of process zone for which equilibrium cracks can exist. Also, as an addition, within the framework of the second model, the problem of crack propagation at a constant velocity is considered. The final task of this study is the problem of crack propagation in a discrete medium, where the body being damaged is represented by two parallel chains consisting of masses connected with elastic bonds. From mathematical point of view, the major challenge of the thesis is concerned with the factorisation of matrix-valued functions in the Wiener–Hopf method, which is an effective tool for solving boundary valued problems. Analytical solutions are obtained using widely known methods in the field of boundary value problems. To obtain an approximate solution, an effective numerical method is proposed, designed specifically for Wiener–Hopf type equations with matrix kernels. In a separate chapter, special cases of Wiener–Hopf matrix problems are considered, for which, under certain conditions, we find an exact solution. For more general cases, an iterative scheme for an approximate solution is proposed, which can serve as a tool for solving the above-mentioned discrete problem of fracture. Questions related to the presence of singularities and possible solutions are discussed. All calculations were performed using MATLAB. All expressions derived analytically are verified with Maple. The results are shown graphically and discussed at the end of each chapter, as well as in the concluding section.
Analysis of discrete and continuous problems for cracks with process zones with the Wiener–Hopf technique
Livasov, P. (Awdur). 2021
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