A numerical approach for the establishment of strain gradient constitutive relations in periodic heterogeneous materials

In this work, we developed a new numerical approach for the establishment of strain gradient constitutive equations in heterogeneous materials with periodic microstructure through homogenization over a unit cell. To this end, we considered a periodic unit cell of finite size, including different homogenous phases and subjected to gradient remote loading. The local elasticity problem was solved by combining the theoretical asymptotic analysis with the Fast Fourier Transform (FFT) method. The asymptotic analysis allows the establishment of differential equations of different orders. These equations were resolved by using the FFT method. By using the homogenization procedure proposed in Li (2011a,b), intrinsic strain gradient constitutive relations can be constructed. This approach fulfils the severe exigency on accuracy of the high-order homogenization. Numerical simulations on examples with 1D and 2D microstructures showed that this approach is highly accurate and efficient. Discussions were completed on the mechanical implications of the strain gradient constitutive laws through the obtained numerical results.