13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- On Symmetric Crack Formation in Plates under Central Bending Ilya Nickolaevich Dashevskiy A. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 119526, Russia dash@ipmnet.ru Abstract The energy approach is used to propose a model of brittle fracture of a thin plate (and a wedge) under bending by a point indenter, which permits studying some possible mechanisms determining the number of sectors into which the plate breaks. Since the energy necessary to form new cracks and the total elastic bending energy of the n triangular sectors-beams arising under bending vary in opposite directions with variation in both the crack length L and n, it follows that the total energy required to form n sectors has a minimum depending on L and n, and it is this minimum that determines the number n of the arising sectors. In the simplest scheme, the number of developing cracks turns out to be independent of the plate physical-mechanical characteristics, and its thickness and varies from 2 to 4 as the wedge opening angle varies from 0 to 2π. An analysis is performed and a qualitative interpretation of the obtained results is given. Possible refinements of the proposed model in various directions are discussed. Keywords Crack Formation, Plate, Bending, Model 1. Introduction. Statement of the Model. Energy Relations When studying the interaction of ice fields with icebreakers, ice-resistant structure footings, and other objects and in several other cases (fracture of glass and other brittle materials), there arise problems leading to the scheme of fracture of a plate made of a brittle material by a point indenter or by a lumped force in which several cracks begin to develop under the indenter and cut out the corresponding number of sectors in the plate [1,2], being different in different cases. For the theoretical estimate of the number of sectors arising in crack formation in a plate under the action of an indenter, we assume that: (1) The plate is loaded by a point indenter. (2) As the plate strength is exhausted, fracture occurs instantaneously with the formation of a symmetric system of radial cracks. (3) One can neglect the irreversible (nonelastic, thermal, etc.) losses (i.e., the plate behavior is quasibrittle) and the possible dynamics (vibrations and waves). (4) The main contribution to the energy balance equation is made by the energy of formation of new surfaces (cracks) and by the elastic bending energy of the arising sectors. In this case, for simplicity, we assume that the strain of the undisturbed (and hence preserving the former rigidity) peripheral part of the plate is small and its contribution to the energy balance equation can be neglected. Thus, in fact, it becomes an unstrained foundation for the arising sectors, which are rigidly fixed to it by their bases. (5) The minimum-energy-consuming fracture scheme is realized; i.e., the total energy is minimal in this case. First, consider the case in which the load is applied at the plate center. Under the assumption that the arising sectors are equal to each other, we can write W = nLhγ + nU, (1) where W = W (n, L) is a function of the total energy expenditure in the crack formation, n is the
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