| Summary: | This thesis consists of an introductory part and a review of the results obtained by the author and co-workers including discussion of related work by others. The thesis begins with an introduction to statistical physics of phase transitions and to percolation theory. This is followed by a presentation and discussion of the previous results in the field of rigidity percolation. Main body of the thesis consists of a study of rigidity percolation transition in two-dimensional random fibre networks. Finally, the author introduces a simplified model of a colliding membrane, and examines the role of connectivity in sedimented materials. Rigidity percolation is analysed in two-dimensional random fibre networks. A central force model consisting of only Hookean springs connecting the nearest neighbour crossing points is not rigid at any finite fibre density. The model can be made rigid with additional constraints. Three different rigidifying strategies were studied. All three transitions, corresponding to these strategies, were found to be continuous and in the same universality class as the two-dimensional central-force rigidity percolation in diluted lattices. This universality class is found to be different from the universality class of two-dimensional scalar percolation. Estimates for the transition thresholds and for the associated critical exponents are presented. At the percolation threshold the rigid backbone was found to break into rigid clusters, whose number diverges in the limit of infinite system size, when a critical bond is removed. A simplified model of an inflated closed membrane is used to demonstrate a rigidity transition which happens when impact energy is added to the membrane. The only relevant parameter in this transition is the ratio between the energy needed to collapse the membrane and the impact energy. The transition between the rigid and floppy states is discontinuous. Finally, in this thesis it is demonstrated that measures of connectivity together with the geometrical properties of the constituents determine the porosity of materials formed by random deposition.
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