12 / 2021-02-18 17:18:30

Cauchy-Born hypothesis based numerical homogenization method

periodic unit cells,numerical homogenization,Cauchy-Born rule

Vibration of continuous systems

Asymptotic expansion based homogenization (AH) has been widely used to predict the effective macroscopic properties of periodic unit cells. In this work, we show that the homogenization process can be done in a much simpler manner for both continuum and discrete periodic unit cells 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 elastic tensor is quite easy and can rely entirely on commercial CAE software (e.g., ANSYS, ABAQUS, etc.) to complete 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 element of the elasticity tensor is then shown to be simply the specific 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 corresponding to extreme material property requirements, can be implemented in a straightforward way as well. Some numerical examples are tested, the results of which are compared with that in the literature. It is showed that the results of both the homogenization and inverse homogenization examples obtained by our methods agree very well with the ones in the literature, demonstrating the validity of the Cauchy-Born hypothesis based numerical homogenization method.