Establishment of deformation simulation procedure by finite element method based on second-order homogenization using characteristic displacement function for macroscopic strain gradient

To evaluate effect of a relative scale of microstructure to macrostructure, a simulation procedure using second-order homogenization based finite element method was proposed. In this method, a microscopic characteristic displacement function for macroscopic strain gradient was added to the conventional first order homogenization method. Then, a procedure to solve a macroscopic boundary problem was established based on the principle of virtual work in macroscopic scale represented by the microscopic characteristic displacement function. To validate the proposed second-order homogenization method, computational simulations of deformation behavior of cavitated rubber (void) blended amorphous polymer were performed using the proposed second-order homogenization. From the result of bending deformation where tension or compression was given to upper side or lower side of the macroscopic model, the material containing larger void required a larger energy for the bending of the model. With decrease in the void size, the energy converged to that predicted by first-order homogenization method. Basically, the deformation behavior predicted by proposed homogenization model was qualitatively and quantitatively similar to that predicted by full scale model. The proposed model is expected to be applied for computational prediction of the scale-dependent deformation in various cases because the model does not limit the form of constitutive equation, shape of the unit cell and deformation mechanisms and structure of the material.