Дослідження напружено-деформованого стану біля кутової точки межі поділу матеріалів, з якої виходить міжфазна тріщина

Abstract

В умовах плоскої деформації за допомогою перетворення Мелліна виконано розрахунок полів напружень і переміщень біля вершини півнескінченної міжфазної тріщини, яка виходить з кутової точки ламаної межі поділу двох різних матеріалів. Досліджено вплив кута зламу і пружних параметрів з'єднаних матеріалів на показник сингулярності в околі вершини. Знайдено коефіцієнти інтенсивності напружень біля вершини тріщини при навантаженні її берегів заданим напруженням.

Under the plane strain we consider the problem of the stress-strain state of piece-homogeneous body at the semi-infinite interfacial crack tip threshold, initiated from the angular point of the broken interface of two different isotropic elastic media. Part of the crack lips is loaded by given tangential and tensile normal stresses. The ideal joint is expected to be on the interface. Having applied the Mellin integral transformation to equations of equilibrium, condition of compatibility of strains, the Hooke’s law and boundary conditions, we reduce the formulated static boundary problem of the elastic theory to the system of algebraic equations, which determines the transforms of stresses and displacement at the crack tip threshold. By inverse Mellin transformation of the found transforms we deduce the formulas for the asymptotic field of stresses and displacements at the crack tip threshold as an expansion in terms of value of distance to the tip. These powers are the roots of some transcendent algebraic equation, obtained during the solution of the problem. The numerical analysis of roots of this characteristic equation of the problem for some elastic parameters of the joined materials and the interfacial angles shows the presence of values with negative real part among them, which causes the stress concentration at approaching to the crack tip. It is discovered that the singular behaviour of stresses at the crack tip threshold differs from the root. Physically incorrect spatial oscillations caused by the presence of imaginary part in the index of singularity superimpose on the power singularity of stresses in some interval of interfacial angles depended on the elastic parameters of the joined materials. It is supposed that the formation of area of contact of crack lips corresponds to this situation. The expression for the crack opening at the tip threshold was found and the possible contact zone sizes were estimated. On the basis of this expression the condition of correctness of the found problem solution was formulated. By comparing the obtained expansion of stresses with the solution of analogical canonical problem the similar formulas for the coefficients of intensity were deduced. The necessity of introduction of the two coefficients of intensity due to the absence of symmetry in the structure of piece-homogeneous body with the broken interface was substantiated. The obtained formulas were applied in the particular cases of the constant stresses action on the part of the crack lips or the concentrated forces. The obtained results can be used for the estimation of the fracture toughness of piece-homogeneous body with an interfacial crack for the calculations of the pre-fracture zone at the crack tip threshold in particular.