Determination of stresses near the cracks in plates on the basis of integral equations relatively the displacement jumps
Journal Title: Вісник Тернопільського національного технічного університету - Year 2015, Vol 78, Issue 2
Abstract
A calculation algorithm of stresses in the plates with cracks, based on integral equations written relative to the displacement jumps of their faces, is presented. Singular integral equations are constructed by the Muskhelishvili method. To solve the integral equations the method of mechanical quadratures is used, where the quadrature formulas of Gauss type are used for regular integrals and integrals with Cauchy kernels. As a result the problem is reduced to solution a system of linear algebraic equations with respect to the displacement jumps of the crack faces at the chosen nodes. The stress intensity factors (SIF) are expressed in terms of the values of these displacements by means of interpolation formula or polynomial constructed by the least quadrature method. The cases when this approach has advantages over the known algorithms in literature, based on integral equations constructed relative to the derivatives from the displacement jumps, are considered. In particular, the calculations of SIF are carried out for the plates with rectilinear and curvilinear cracks under the concentrated forces acting in the crack vicinity. It is shown that the obtained results for the case of rectilinear cracks coincide practically with solution found analytically. It is shown that the proposed approach can be used to study SIF in the cases when loads are described by discontinuous functions (the problems of such class arise when considering plastic deformation in the crack tip vicinity according to Dugdeila model or when considering the cohesive force). The problem arising when considering the plate with curvilinear crack extending at infinity is studied in detail. It is assumed that on the crack extension the cohesive forces arise. The normal component of cohesive forces is assumed to be given. The tangential of component of cohesive forces and the sizes of the region of their action are found from the condition of stress boundedness. In special cases obtained results of calculations are coordinated with the known in the literature solutions For curved crack the region action sizes of forces of cohesion and their tangential component depending on the applied efforts and the form of crack are investigated.
Authors and Affiliations
Tetyana Solyar
Keywords
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