An edge dislocation near an anticrack in a confocal elliptical coating
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O R I G I NA L A RT I C L E
Xu Wang · Peter Schiavone
An edge dislocation near an anticrack in a confocal elliptical coating
Received: 21 August 2020 / Accepted: 23 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We present an analytical study of the problem associated with an edge dislocation near a completely coated finite anticrack (or rigid line inhomogeneity). The two foci of the elliptical coating-matrix interface are located at the two tips of the anticrack. In addition, the coating and the matrix have identical shear moduli but distinct Poisson’s ratios. By means of conformal mapping and analytic continuation, we obtain a closed-form representation of a specifically constructed auxiliary function defined in the entire image plane. This auxiliary function is then used to derive analytical expressions (in the image plane) of the two pairs of analytic functions which characterize the corresponding stress and displacement distributions. A closed-form expression representing the rigid-body rotation of the anticrack is presented by satisfying moment balance on a circular disk with sufficiently large radius. The mode I and mode II stress intensity factors at the two anticrack tips are determined explicitly. Keywords Anticrack · Confocal elliptical coating · Edge dislocation · Rigid-body rotation · Stress intensity factor · Analytic solution
1 Introduction The study of elastic fields resulting from the presence of rigid line inhomogeneities continues to attract increasing attention in the literature. These rigid line inhomogeneities are especially useful in the modeling of composites; for example, they can be used to represent thin hard fibers as part of a reinforcing phase in composite materials. Accordingly, researchers have considered material microsystems incorporating rigid line inhomogeneities in various different scenarios ranging from those involving the influence of rigid inhomogeneities placed near material interfaces [1–4], the interaction of dislocations with rigid line inhomogeneities [5] and the singular stress fields induced near the tips of rigid line inhomogeneities [6–9]. In all of the aforementioned studies, the rigid line inhomogeneity is either embedded in a homogeneous material [3,5–9] or lies at the interface of a bimaterial [1–4]. In the design of fibrous composites, a coating layer is usually inserted between the internal inhomogeneity (representing the fiber) and the surrounding matrix with the objective of Communicated by Andreas Öchsner. X. Wang (B) School of Mechanical and Power Engineering, East China University of Science and Technology, 130 Meilong Road, Shanghai 200237, China E-mail: [email protected] P. Schiavone (B) Department of Mechanical Engineering, University of Alberta, 10-203 Donadeo Innovation Centre for Engineering, Edmonton, AB T6G 1H9, Canada E-mail: [email protected]
X. Wang, P. Schiavone
improving the bonding between the inhomogeneity and the matrix and also to reduce the material mismatch induced stress concentration
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