Viscoelastic State of an Anisotropic Plate with a Single Elliptic Inclusion

Abstract

The linear viscoelasticity problem for an infinite anisotropic plate with an elliptical elastic inclusion under ideal mechanical contact conditions is solved. To obtain the solution, the small parameter method is applied, where the variation of Poisson's ratios over time is chosen as the parameter, effectively reducing the time-dependent problem to a sequence of analogous boundary value problems in the theory of elasticity. The construction of the solution is based on the complex potentials apparatus, conformal mapping methods, and Laurent series expansions. Boundary conditions at the contact interface are satisfied using the generalized least squares method, ensuring high accuracy of the unknown constants at any given moment. Analytical expressions for bending moments and shear forces in the plate are derived, explicitly incorporating viscoelastic time operators. For the case where the elliptical inclusion degenerates into a straight elastic line, formulas for calculating moment intensity factors at its endpoints are provided. The proposed approach allows for a correct description of the evolution of singular moment behavior and an evaluation of the material properties' influence on their temporal variation. Numerical studies were conducted for materials with various relaxation properties and different relative inclusion stiffnesses. It is established that the most intensive redistribution of moments occurs during the initial stage of the viscoelastic process, after which the stress state of the plate approaches a stationary phase. It is proven that moment concentration depends non-linearly on inclusion stiffness, being minimal at intermediate stiffness values and increasing sharply for holes or perfectly rigid inclusions. Isotropic plates are treated as a special case of anisotropic ones, allowing the results to be extended to a wide range of problems in composite mechanics and long-term strength prediction.