Mathematical Modeling of Bending of Isotropic and Anisotropic Plates with Elliptical and Linear Inclusions

Abstract

The bending problem of an infinitely large thin anisotropic plate with an elliptical or linear elastic inclusion inserted into a hole without initial tension and under perfect mechanical contact with the plate matrix is solved. The plate at infinity is subjected to constant bending moments. The solution is obtained by employing the formalism of generalized complex potentials, expansions of functions into Laurent series and Faber polynomials, as well as conformal mapping techniques to transform the exterior of the unit circle into the exterior of an ellipse. An exact analytical solution for the case of an elliptical inclusion, providing expressions for bending moments and transverse forces both in the plate matrix and in the inclusion, is presented. For the case when the elliptical inclusion reduces to a line, formulas for calculating the moment intensity factors (MIF) at its ends are derived. This approach accurately captures the singular behavior of bending moments and identifies conditions under which MIF values are significant. Numerical studies were conducted for plates made of isotropic material (CAST–V) and anisotropic material (skew-wound glass-fiber-reinforced plastic) under various values of the inclusion’s relative stiffness and axis ratio. It was found that decreasing the inclusion’s stiffness leads to an increase in bending moments in certain contact zones, with higher moment concentrations in anisotropic plates compared to isotropic ones. For linear inclusions, significant MIF values arise only for substantially stiff or soft inclusions; when the stiffnesses of the plate and inclusion differ by less than a few times, MIF values are negligible, and it is inappropriate to discuss bending moment singularities. Isotropic plates are treated as a special case of anisotropic ones, enabling the extension of these results to a broad class of engineering problems involving composites and structures with embedded elements.