Periodic Boundary Condition for Large Material Deformation

The use of periodic boundary conditions has been a common practice in numerical simulations for the study of constitutive response of materials. To consider material deformation, the common method of enforcing the periodic boundaries is to deform the computational domain with the material, which limits the strain of the material to order one. The method fails when strains are large or the deformation gradients are complicated, such as those containing large rotation or twist. For instance, it is quite difficulty to perform a simulation under a large pure shear deformation because of the distorted and elongated computational domain.

To avoid this difficulty, a different method has been developed by decomposing the velocity gradient matrix into the upper triangle part and the antisymmetric part representing a pure rotation. The antisymmetric part is then canceled by performing the simulation in a rotating reference frame and considering the associated inertial forces. Advantages of this method will be shown using examples.