On Boundary Conditions in Multiscale Second-Order Computational Homogenization Scheme

Abstract eng:
In recent years, a study of the influence of the microstructure on the material mechanical behavior has attracted a considerable attention, since almost all materials are micro heterogeneous due to their natural structure. Therefore, for the modeling of the macroscopic behavior of heterogeneous materials, the methods of modeling on several levels (multiscale methods) are increasingly developing, which go beyond limitations of continuum mechanics, allowing consideration of microstructure, and using the nonlocal theory [1]. To solve the boundary value problem at the micro level, the finite element method has been mostly applied. The paper presents a multiscale second-order computational homogenization scheme for small deformation theory and discusses some issues related to the application of the periodic boundary conditions on the representative volume element (RVE) at micro level presented in [2]. The comparison of the periodic boundary conditions with the displacement boundary conditions in the second-order computational homogenization is also given. The macro level is discretized by the C1 continuity two dimensional triangular finite element derived in [4], while the micro level is discretized by the C0 continuity quadrilateral finite element according to [2, 3]. Because of C1 to C0 continuity transition, an alternative relationship between the macro strain gradient and the micro strain is needed, which consequently results in an additional integral condition on micro level fluctuation field. For the rectangular RVE geometry and the periodic boundary conditions, the fluctuation integral condition is decomposed into two surface integrals, enforcing zero average fluctuations on the RVE boundaries. The aforementioned integral condition on micro level fluctuation field is implemented into the finite element software ABAQUS. A comparison of various numerical integration methods (Gauss, Simpson, trapezoidal rule) is demonstrated in a few simple examples. The results show that different integration methods satisfy the integral condition easily, but the deformed shape of RVE strongly depends on the method applied. The deformed configurations are compared with the results presented in [1]. It is shown that only trapezoidal rule gives an accurate deformed shape of RVE because of the favorable distribution of weight factors on the boundary nodes. In addition, in the corner nodes of RVE, the stress concentrations appears as a result of zero fluctuations in the RVE corners [1, 2] and the absence of the condition that volume integral of micro fluctuation field is set to zero [1].

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