Dislocation model of an asymmetric weak zone for problems of interaction between crack-like defects

Abstract

Crack-like defect models range from the well-known traction free Griffith-Irwin cracks, Barenblatt cuts containing small process zones at tips, to cuts subjected to cohesive forces over their entire length. This last type of defect may be regarded as a weak zone (WZ) in the solid which is normally closed but which can open progressively under sufficiently large remote external tensile forces. The fact that the WZ begins to open only when the remote force has reached a certain threshold level distinguishes the WZ from a real cohesive crack. The adhesive forces can be of very different physical origin -atomic, dislocational, localized porosity, etc. Healed cracks in glaciers and in the earth's crust are also WZs. The length of WZs can thus range from a few nanometers to hundreds of kilometers. It is interesting to investigate the fundamental behaviour of the WZs during their interaction with other defects. Note that the direct approach using some cohesive force-opening relationship leads to a complicated problem which is reduced to the solution of a system of nonlinear singular integral equations in [1]. Another approach was suggested in [2], whereby the cohesive forces were expressed in a series containing N terms with N free parameters. The free parameters were determined by imposing physically consistent conditions on the solution. This paper focuses on two topics. Firstly, guided by the results obtained in [2], a general approach to examining the behaviour of a WZ embedded in a very asymmetric external tensile stress field is developed by prescribing a priori the WZ asymmetric opening by a two-parameter basis function which meets all physical constraints. Secondly, this approach is exemplified on the problem of interaction between a long interface crack subjected to wedge opening forces and a short collinear WZ. The latter is separated from the main crack by a small strong microstructural feature of the material (i.e. by a small obstacle). The key questions that will be addressed are: (i) when does the WZ become the nucleus of a cohesive crack on its own without linking with the pre-existing long crack, and (ii) when does the WZ force the obstacle to rupture allowing the pre-existing crack to link with it. The critical applied load levels corresponding to these limiting situations will be determined. An asymmetric weak zone modelled as a special dislocationIn contrast to the direct method used in [2], we shall take an inverse approach when the WZ is located in an asymmetric tensile stress field and search for a basis function of asymmetric opening displacement of WZ in the formThe function G(X, η) depends on the parameter of asymmetry, η, i.e. the distance by which the maximum of w(X) is displaced from the centre of WZ, X = 0. It is assumed that η > 0, if the maximum is displaced to the left of centre.We shall seek the unknown function G(X, η) under the following mathematical constraints which result from obvious physical and geometrical considerations: (i) it is even with respect to a simultaneous change 1

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The interaction of a system of weakened zones at the interface of elastic media in a tensile stress field

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