Optimization of functionally graded materials to make stress concentration vanish in a plate with circular hole

This paper is devoted to the minimization of the stress concentration factor in infinite plates with circular hole made of functionally graded materials and subjected to a far-field uniform uniaxial tension. Despite the vast literature on the versatility of these materials, the novelty of the results is that the material distribution is not limited to prefixed laws, as in many works available in the literature. Instead, it is assumed to be an unknown piecewise constant function, thus aiming to derive the material distribution by exploiting, at best, the inhomogeneity concept associated with functionally graded materials. After a brief review of the governing equations, the motivation, the statement and the mathematical formulation of the optimization problem are given under the hypothesis of axisymmetric material distribution. Still, the problem could not be solved analytically, therefore a direct transcription approach by the aid of finite difference method has been followed to convert it into a nonlinear programming problem, whose solution has been obtained numerically by dedicated gradient-based solvers. Numerical optimal solutions are reported in graphical forms, thoroughly discussed and validated by means of the finite element method. The developed numerical approach yields a material inhomogeneity obeying a sigmoid-like function and a uniform hoop stress along the radial direction, thus making the stress concentration factor at the rim of the circular hole vanish.