Homogenization in a simpler way: analysis and optimization of periodic unit cells with Cauchy–Born hypothesis

Asymptotic expansion based homogenization has been widely used to predict the effective macroscopic properties of periodic unit cells (PUCs). In this work, we show that the homogenization process can be done in a much more elegant manner for both continuum and discrete PUCs by taking advantage of the Cauchy–Born’s hypothesis, which is a widely used rule in the area of solid physics to relate the position of the atoms in a crystal lattice and the overall strain of the medium. It is shown that in the proposed method, the derivation process of the effective elasticity tensor is quite easy and can rely entirely on commercial CAE software (e.g., ANSYS, ABAQUS, etc.) to accomplish the homogenization task. In detail, after the discretization of the unit cell with finite elements, one only needs to apply affine boundary conditions at the exterior boundaries of the unit cell and then call the FEA solver to find the static displacement field under such affine boundary conditions. The entries of the elasticity tensor can then be expressed using the stain energy of the unit cell. After deriving the sensitivity information of the Cauchy–Born hypothesis based homogenization process, the inverse homogenization process, which attempts to find the optimal layout exhibiting pre-determined desirable material properties, can be implemented in a straightforward way as well. Some numerical examples are tested and compared with the results in the literature. It is showed that the results of both the homogenization and inverse homogenization examples obtained by our method agree very well with the ones in the literature, demonstrating the validity of the Cauchy–Born hypothesis based numerical homogenization method.