Comparison of prediction on effective elastic property and shape optimization of truss material with periodic microstructure

Abstract

This paper compares the representative volume element (RVE) method based on Dirichlet and Neumann boundary conditions with the homogenization method for predicting the effective elastic property of truss material with periodic microstructure. Numerical experiments show that, with increase of the number of the unit cell, n, the results of RVE method under the Dirichlet and Neumann boundary conditions converge towards those obtained with homogenization method from the above and below sides, respectively. For some specific types of the unit cell, RVE method gives the same results as those obtained with homogenization method, even if only one unit cell is included. For RVE method, a simple criterion for judging the existence of scale effects is whether the equilibrium of the boundary nodal forces is guaranteed under the Dirichlet boundary conditions, or whether the deformation compatibility at the unit cell boundaries is satisfied under the Neumann boundary conditions. We also discover that for a specific type of truss material, whose unit cell has no characteristic displacement defined in homogenization method, the volume average of members’ properties in the unit cell gives the exact prediction of the effective elastic properties. Finally, shape optimization technique is applied to find the optimal geometric shape of the unit cell for truss material with the maximum and minimum shear stiffness, and the numerical singularity involved is discussed as well.