Numerical computing of elastic homogenized coefficients for periodic fibrous tissue

The homogenization theory in linear elasticity is applied to a periodic array of cylindrical inclusions in rectangular pattern extending to infinity in the inclusions axial direction, such that the deformation of tissue along this lastdirection is negligible. In the plane of deformation, the homogenization scheme is based on the average strain energy whereas in the third direction it is based on the average normal stress along this direction. Namely, these average quantities have to be the same on a Repeating Unit Cell (RUC) of heterogeneous and homogenized media when using a special form of boundary conditions forming by a periodic part and an affine part of displacement. It exists an infinity of RUCs generating the considered array. The computing procedure is tested with different choices of RUC to control that the results of the homogenization process are independent of the kind of RUC we employ. Then, the dependence of the homogenized coefficients on the microstructure can be studied. For instance, a special anisotropy and the role of the inclusion volume are investigated. In the second part of this work, mechanical traction tests are simulated. We consider two kinds of loading, applying a density of force or imposing a displacement. We test five samples of periodic array containing one, four, sixteen, sixty-four and one hundred ofRUCs. The evolution of mean stresses, strains and energy with the numbers of inclusions is studied. Evolutions depend on the kind of loading, but not their limits, which could be predicted by simulating traction test of the homogenized medium