Design of 3D orthotropic materials with prescribed ratios for effective Young’s moduli

An orthotropic material is characterized by nine independent moduli. The ratios between the Young’s moduli in three directions are indicative of the level of orthotropy and the bulk modulus is indicative of the overall stiffness. In this paper we propose a method for designing the stiffest orthotropic material which has prescribed ratios for Young’s moduli. The material is modeled as a microstructure in a periodic unit cell. By using the homogenization method, the elasticity tensors are calculated and its compliance matrix is derived. A Lagrangian function is constructed to combine the objective and multiple equality constraints. To enable a bi-section search algorithm, the upper and lower bounds on those multipliers are derived by using a strain energy approach. The overall optimization is based on the bi-directional evolutionary structural optimization (BESO) method. Examples of various orthotropy ratios are investigated. The topology presents a constant pattern of material re-distributed along the strongest axis while the overall stiffness is maintained.