Kirsch problem in classical and gradient elasticity. Part I: Anisotropic and homogeneous bodies

In this first part of our study, we examine the Kirsch problem analytically for a body composed of a homogeneous but anisotropic material from the perspectives of both classical and gradient elasticity theories. As a continuation, the second part will address the Kirsch problem for a body that is isotropic but inhomogeneous. In the model, the plane is assumed anisotropic and, with positive ϵ_ij≪1 characterizing the degree of anisotropy, we consider a weakly anisotropic material for which the six elastic coefficients deviate slightly from their values in the equivalent isotropic material. The Airy stress function is used to obtain analytical solutions for stress fields. Similarly, a gradient Airy stress function notation is employed to solve the Kirsch problem in gradient elasticity theory. The stress and displacement fields for the anisotropic Kirsch problem are determined analytically within both classical and gradient elasticity frameworks. The analytical solutions from isotropic and classical elasticity are obtained and compared with existing literature. In addition to the classical boundary conditions, the higher-order gradient boundary conditions are also included in the stress field calculations. The differences that emerge within the scope of classical and gradient elasticity theories are also examined, along with a comparative analysis of the graphical representations of the analytical solutions obtained and the size effects in the gradient elasticity theory. Furthermore, based on both classical and gradient elasticity theories, the comparative presentation of the influence of anisotropic material qualities on the solutions is provided. In this study, we represent the analytical solutions for the homogeneous but anisotropic Kirsch problem, using both classical elasticity and gradient elasticity theory for the first time in the literature.