A New High-order Deformation Theory and Solution Procedure Based on Homogenized Strain Energy Density

The classical continuum mechanics faces difficulties in solving problems involving highly inhomogeneous deformations. The proposed theory investigates the impact of higher-order microscopic deformation on modeling of material behaviors and provides a refined interpretation of strain gradients through the homogenized strain energy density. Only one scale parameter, i.e., the size of the Representative Volume Element (RVE), is required by the proposed theory. By employing the variational approach and the Augmented Lagrangian Method (ALM), the governing equations for deformation as well as the numerical solution procedure are derived. It is demonstrated that the homogenized energy theory offers plausible explanations and reasonable predictions for the problems yet unsolved by the classical theory such as the size effect of deformation. The concept of homogenized strain energy proves to be more suitable for describing the intricate mechanical behavior of materials. And higher order partial differential equations can be effectively solved by the ALM by introducing supplementary variables to lower the highest order of the equations.