The paper describes a second-order two-scale computational homogenization procedure for modeling of heterogeneous materials at small strains. The Aifantis theory of linear elasticity has been described and implemented into the two dimensional C1 continuity triangular finite element formulation. The element has been verified on several patch tests and the computational efficiency of numerical integration of the element stiffness matrix has been tested as well. Furthermore, the C1 two dimensional triangular finite element based on full second gradient continuum is formulated and used for the macrolevel discretization in the frame of a multiscale scheme, where the RVE is discretized by the C0 quadrilateral finite element. The application of generalized periodic boundary conditions and the microfluctuation integral condition on RVE has been investigated. The presented numerical algorithms have been implemented into FE software ABAQUS via user subroutines and verified on a pure bending problem. The comparability of RVE size to the length scale parameter of gradient elasticity has been proved, and elastoplastic behavior of heterogeneous material has been also considered. The results obtained show good numerical efficiency of the proposed algorithms.
Bubble-enhanced quadrilateral finite element formulation for nonlinear analysis of geotechnical problems