Optimization of axisymmetric elastic modulus distributions around a hole for increased strength

Holes in engineering structures cause stress concentrations that often lead to failure. In nature, however, blood vessel holes (foramina) in load-bearing bones are not normally involved in structural failures. It has been found that this behavior is linked to the material distribution near the hole. In the present paper, we have investigated the effectiveness of optimizing the radial distribution of the isotropic elastic modulus around a circular hole to increase load-carrying capacity. Bezier curves were used to describe the radial distribution of the elastic modulus. Since changing the elastic modulus usually affects the strength, the ratio of maximum principal stress to strength was chosen as the objective function for optimization. Using non-dimensional analysis of the 2-D elasticity equations, we identified three parameters that govern the optimum design and are applicable to a wide range of materials, loading, and geometries. The first is a material parameter that describes the relationship between the strength and elastic modulus, the second is the geometric parameter given by the ratio of the optimized field to the hole radius, and the third is the biaxial load ratio. The effect of failure criterion choice on the optimum elastic modulus distribution is also investigated. Optimum elastic modulus distributions for materials whose strength increases faster than the stiffness, as density and/or composition is varied, completely eliminated the effect of the hole by locally stiffening areas that experience high stresses. When the strength lagged behind the stiffness, optimum designs were similar to those found in bones, and relied on modulus distributions that direct the loads away from the hole.