Kirsch problem in classical and gradient elasticity, Part II: Isotropic and inhomogeneous bodies

Building upon Part I (Özer and Kröger, 2026) of this study for anisotropic and homogeneous bodies, Part II presents an analytical and theoretical investigation of the Kirsch problem, examining an isotropic but inhomogeneous plate with a circular hole of arbitrary radius under axial loading. Analytical solutions are explored for the stress and displacement fields around the hole, including the near- and far-field. The problem is addressed using both classical and gradient elasticity theories, and analytical solutions are investigated for both. Due to the absence of axial symmetry, the Airy potential stress function depends on two polar coordinates. Stress and displacement fields derived from Airy stress functions in both classical and gradient elasticity theories are obtained by solving partial differential equations. Solutions are independently analyzed for each theory, considering both classical and higher-order gradient boundary conditions and assuming a family of nonlinear variations in the radial coordinate of the inhomogeneous material. Our findings are derived using exact mathematical techniques like algebra and calculus, without approximations or simulations, while some results contain integrals, including special functions. Figures illustrating the analytical results are provided for each case, and explore the effect of the gradient parameter, in particular.