2.1. Averaged mechanical response of a RVE The averaged mechanical properties of a containing the main microstructural features of the material, and solving the corresponding equilibrium problem. When a macroscopic point inside the RVE is given by where is the position vector within the RVE and heterogeneity of the material. Assuming that the macroscopic strain is the volume average of the fine-scale strain field resulting from the above equation, fluctuation, the Hill-Mandel condition (energy equivalence between the fine descriptions) implies that the macroscopic stress tensor is obtained as the volume average of the microstructural stress tensor. Considering the periodicity of next be identified based on the cell tying forces at nodes controlling the macroscopic loading as where the summation spans the nodes controlling the RVE loading. Any type of material behaviour can be postulated at the fine scale, and the periodicity of the microfluctuation field can be enforced by homogeneous linear connections between corresponding faces. controlling points (denoted 1 to 4 in Fig. deformation modes of the boundary of the RVE, provided identical meshes are used on faces of the RVE. The RVE equilibrium problem under the ma forces at the controlling points, which represent the action of the neighboring continuum on the RVE. The displacements of the controlling points, energetically conjugated to the imposed controlling forces, can be used to Figure 1. Control points for macroscopic quantities control on a RVE for upscaling principles 13th International Conference on Fracture June 16 -2- mechanical response of a RVE mechanical properties of a heterogeneous material can be deduced by loading equilibrium problem. When a macroscopic strain is applied to a RVE, the displacement of a . is a fluctuation field caused by the ulting from the above equation, and accounting for a periodic condition (energy equivalence between the fine-scale and macroscopic : 1 : he macroscopic stress tensor is obtained as the volume average of the microstructural the periodicity of the fluctuation field, the macroscopic stress tensor can 1 1 nodes controlling the RVE loading. Any type of material behaviour by homogeneous linear connections between corresponding faces. In a three-dimensional b points (denoted 1 to 4 in Fig. 1a) are used to apply the macroscopic stress or The RVE equilibrium problem under the macroscopic stress loading is then solved by imposing es, can be used to extract the macroscopic strain. . Control points for macroscopic quantities control on a RVE for upscaling principles June 16–21, 2013, Beijing, China material can be deduced by loading a RVE RVE, the displacement of a (1) and accounting for a periodic scale and macroscopic (2) , the macroscopic stress tensor can (3) dimensional body, four ) are used to apply the macroscopic stress or deformation modes of the boundary of the RVE, provided identical meshes are used on the opposite croscopic stress loading is then solved by imposing
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