Weight function approach to studying perfect and imperfect interfaces in anisotropic and piezoelectric bimaterials

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Student thesis: Doctoral ThesisDoctor of Philosophy

Original languageEnglish
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Award date08 Oct 2015
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Abstract

The focus of the thesis is interfacial crack problems in anisotropic and piezoelectric bimaterials. We seek to solve a variety of problems using weight function techniques and singular integral equations. We begin by studying a dynamic crack along a perfectly bonded interface in an anisotropic bimaterial. Using a weight function derived from a mirrored problem it is possible to derive important material parameters which govern the crack propagation. Following this a static crack is considered. However, in this case the materials are not bonded perfectly, an imperfect interface is present instead. A method is derived where singular integral equations for the imperfect interface problem are derived through use of perfect interface weight functions. The weight functions are then extended to fracture in piezoelectric bimaterials which allows equivalent integral equations to be derived relating the mechanical and electrical fields. In past literature a number of results have been found which can only be used when considering a symmetric load system on the crack faces. All of the problems considered here have asymmetric loading. Firstly, a steady-state formulation is used to derive asymptotic coefficients of the crack displacement and interfacial tractions for a dynamic crack along a perfect interface. The method can be used to find many asymptotic coefficients but the one of most importance here is the stress intensity factor which therefore enables the calculation of energy release rate at the crack tip. As an example an orthotropic bimaterial with two different loading configurations is used to examine the importance of crack speed and load asymmetry on the properties of the crack propagation. We proceed to study imperfect interface conditions for an anisotropic bimaterial. Usually when looking at such a problem it is necessary to derive new weight functions which correspond to the imperfect interface. An innovative method which makes use of the Betti formula and existing weight functions for the analogous perfect interface problem is derived. This procedure is used to obtain singular integral equations which relate the crack loading, which is assumed to be known, to the displacement jump over both the crack and interface and tractions along the bonded area between the materials. Examples of the results obtained through solving the integral equations numerically are given. Finally, we extend the weight functions used previously in the thesis to a piezoelectric setting. The general form of the weight function for any piezoelectric bimaterial is given before two specific examples are studied in depth. The examples are chosen in such a way to illustrate the effect that the poling direction of the bimaterial can have on both the mechanical and electrical fields. For both examples explicit expressions are derived for the weight functions which are then used to derive singular integral equations which can be used to study the effect of both mechanical loading and electrical charges being applied to the crack faces. To finish we present some examples for both poling directions to illustrate the use of the derived equations.