A Screw Dislocation in a Monoclinic Tri-Material
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A Screw Dislocation in a Monoclinic Tri-Material Xu Wang1 · Peter Schiavone2
Received: 20 August 2020 / Accepted: 28 September 2020 © Springer Nature B.V. 2020
Abstract Employing primarily analytic continuation, we derive analytical solutions to the anti-plane problem associated with a screw dislocation located anywhere inside an anisotropic tri-material composed of an intermediate anisotropic elastic layer of finite thickness sandwiched between two semi-infinite anisotropic elastic media. All three phases of the tri-material are monoclinic with the symmetry plane at x3 = 0 in a Cartesian coordinate system. We obtain explicit expressions for each one of the three analytic functions defined in its respective anisotropic elastic phase. This allows for the complete determination of the stresses and displacement in the tri-material. In addition, we present simple yet concise expressions for the image force acting on the screw dislocation. An interfacial Zener-Stroh crack in the tri-material is discussed using the analytical solution developed for an interfacial screw dislocation. Keywords Screw dislocation · Monoclinic tri-material · Analytical solution · Analytic continuation · Image force Mathematics Subject Classification 30B40 · 30E25 · 35Q74 · 45E05 · 74B05 · 74M25 · 74R10 · 74S70
1 Introduction Using, respectively, the image method and an approach based on intuition and analogy, Chou [1] and Lin and Chou [2] derived analytical solutions to the problem of a screw dislocation
B X. Wang
[email protected]
B P. Schiavone
[email protected]
1
School of Mechanical and Power Engineering, East China University of Science and Technology, 130 Meilong Road, Shanghai 200237, China
2
Department of Mechanical Engineering, University of Alberta, 10-203 Donadeo Innovation Centre for Engineering, Edmonton, Alberta T6G 1H9, Canada
X. Wang, P. Schiavone
in isotropic and orthotropic tri-materials composed of an intermediate layer sandwiched between two semi-infinite phases. Even for the case of an orthotropic tri-material, Lin and Chou [2] further admitted that “Like the isotropic case, a formal analysis of this problem appears to be intractable.” Ma and Lu [3] similarly acknowledged that “It is impossible to use the conventional image method to obtain the solution of anisotropic multilayered problem directly.” On the other hand, the method of analytic continuation has been successfully employed to derive Green’s functions for isotropic and anisotropic bimaterials by Suo [4, 5] and for multilayered isotropic laminates by Wang and Schiavone [6]. We are very much interested in further exploiting the potential offered by analytic continuation in deriving Green’s function solutions for anisotropic multilayered structures subjected to a line dislocation and/or a line force. The current study is focused on presenting a rigorous analysis of the problem of a screw dislocation in an unbounded anisotropic tri-material. All three phases of the tri-material are monoclinic with the symmetry plane at x3 = 0. Using analytic con
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