On Regularization of Singular Solutions of Orthotropic Elasticity Theory

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.