Stress-Strain State of a Viscoelastic Anisotropic Plate With a System of Elastic Inclusions

The linear viscoelasticity problem is solved for an infinite multiply connected anisotropic plate weakened by a system of elliptical and linear elasticinclusions. To obtain the solution, the small parameter method is applied, where the time variation of Poisson's ratios is chosen as the parameter, which allowed reducing the original time-dependent problem to a recurrence sequence of analogous boundary value problems in the theory of elasticity. The construction of the solution is based on the apparatus of complex potentials, conformal mapping methods, and the expansion of functions into Laurent series. To satisfy the boundary conditions on the contact contours, the generalized least squares method is used, which ensures high accuracy in determining unknown constants at any given moment of time. Analytical expressions for the bending moments in the plate, which explicitly contain the viscoelastic time operators, are presented in this work. For the case where the holes with inclusions degenerate into straight elastic lines, formulas for calculating the moment intensity factors (MIFs) are derived. The proposed approach allows for a correct description of the evolution of the singular behavior of moments and an evaluation of the mutual influence of the concentrator geometry layout and the rheological properties of materials on the stress-strain state of the plate. Numerical studies were carried out for plates made of materials with different relaxation properties (aluminum and epoxy resin) depending on the distance between the inclusions and their relative stiffness parameter. The time intervals of the most intense moment redistribution and when the stress state of the plate practically reaches a stationary value are established. It is proven that the concentration of moments nonlinearly depends on the inclusion stiffness: it is minimal at intermediate stiffness values and sharply increases in the cases of holes (absolutely soft inclusions) or absolutely rigid elements.