On Regularization of Singular Solutions of Orthotropic Elasticity Theory
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On Regularization of Singular Solutions of Orthotropic Elasticity Theory S. A. Lurie1* and D. B. Volkov-Bogorodskiy1** (Submitted by A. V. Lapin) 1
Institute of Applied Mechanics of Russian Academy of Sciences, Moscow, 125040 Russia Received March 29, 2020; revised April 5, 2020; accepted April 16, 2020
Abstract—The problem of the mechanics of cracks in an orthotropic plate is considered. A general solution form is proposed as a generalized Papkovich–Neuber representation for the plane problem of the theory of elasticity of an orthotropic body. This representation allows us to write the general solution in displacements through two vector potentials satisfying the generalized harmonic equations. The dependences between the vector potentials of Papkovich–Neuber and the stress function are presented. It is shown that there is a complex-valued solution form through four analytic functions of complex variables associated with coefficients that are the roots of the characteristic equation corresponding to the generalized biharmonic equation. It is proved the statement that the operator of the problem is written only through conjugate analytic functions of complex variables, and the general solution is written through arbitrary linear combinations of four functions of complex variables. A general form of solutions with a given singularities is presented, including representations for both cracks in Mode I and cracks in Mode II. The conditions are analyzed that make it possible to obtain generalized non-singular solutions for cracks of Mode I and II. Finally, we establish the conditions for the regularization of singular solutions through the solutions of the generalized Helmholtz equations that correspond to a particular version of the gradient theory of elasticity. DOI: 10.1134/S199508022010011X Keywords and phrases: generalized elasticity, orthotropic material, Papkovich–Neuber representation, nonsingular solutions, mechanics of cracks, radial multipliers.
1. INTRODUCTION The mechanics of linear elastic fracture is a very useful tool for assessing strength during design and therefore has been studied in detail as applied to various materials, isotropic or anisotropic [1]. The prediction of the occurrence and propagation of cracks is one of the main tasks in assessing strength and is extremely important for applications. As a rule, it is based on methods of fracture mechanics [2]. For anisotropic materials, fracture mechanics is based on fundamental results [3] and methods based on the complex analytic function theory [4, 5]. Note that the problems of fracture mechanics for anisotropic materials can be associated with many structural aspects. For example, fiber-reinforced polymer matrix materials or textile preforming fabric composites are typically modeled as anisotropic materials at the macroscopic level [6–8]. As a rule, the methods of linear fracture mechanics are used to describe stresses and displacements in bodies with cracks [9]. Frequently solved problems for cracks in orthotropic materials are of great practical
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