Abstract
Within the framework of Kirchhoff–Love plate theory, we analyze a variational model for elastic plates with rigid inclusions and interfacial cracks. The main feature of the model is a fully coupled nonpenetration condition that involves both the normal component of the longitudinal displacements and the normal derivative of the transverse deflection of the crack faces. Without making any artificial assumptions on the crack geometry and shape variation, we prove that the first-order shape derivative of the potential deformation energy is well defined and provide an explicit representation for it. The result is applied to derive the Griffith formula for the energy release rate associated with crack extension.
Highlights
Recent and ongoing advances in engineering and material science have increased the need for mathematical tools in order to design and optimize in an efficient way three-dimensional highly inhomogeneous thin structures.In this paper, we confine our attention to the model of a composite Kirchhoff– Love plate proposed in [17]
The motivation for such a study lies in the fact that this derivative can be used to calculate efficiently the energy release rates associated with variation of defects, which are utmost of importance to predict crack propagation [4,5,12,20]
The most important idea behind our technique originates from [40], where the Griffith formula for the energy release rate associated with crack extension was deduced in a rigorous way for two-dimensional elastic bodies with curvilinear cracks subjected to the Signorini conditions
Summary
Recent and ongoing advances in engineering and material science have increased the need for mathematical tools in order to design and optimize in an efficient way three-dimensional highly inhomogeneous thin structures. The most important idea behind our technique originates from [40], where the Griffith formula for the energy release rate associated with crack extension was deduced in a rigorous way for two-dimensional elastic bodies with curvilinear cracks subjected to the Signorini conditions Following this idea, in the case of Signorini-type constraints imposed on crack faces, general results on the shape differentiability of the potential deformation energy supported by explicit formulae for the first-order shape derivative were derived in [41] for linear elastic materials, in [32] for a Mindlin–Timoshenko elastic plate model, and in [42] for a Kirchhoff–Love elastic plate model. The paper closes with a short discussion of the case when the elastic properties of the plate are inhomogeneous and an application of the result to fracture mechanics: we derive the Griffith formula for the energy release rate associated with crack extension
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