Unit cell model

Hi Gouse,

You have to use homogenization to compute the macroscopic homogenized stress-strain relation from unit cell computations. The procedure is given in details in the works of Kounetzsova, Eindhoven University of Technology that can be easily found by googling. Briefly, the procedure is that, for a given macroscopic strain epsilon_M, the prescribed boundary displacements imposed on three corner nodes x_I of the unit cell are given by epsilon_M \cdot x_I. Periodic boundary conditions are imposed for opposite edges. Then the unit cell problem is solved. The converged linear system of the unit cell reads K fu = d f. This can be condensed to K* du* = d f* (1) for the dofs of the three corner nodes on which the prescribed displacements have been imposed. Using averaing theorem, you can write

macroscopic stress = inverse of unit cell area * integration of micro stress over unit cell domain

                           = inverse of unit cell area * sum_{i}^{3} f_i \otimes x_i

Then, the variation of macroscopic stress is given by

d sigma_M =  inverse of unit cell area * sum_{i}^{3} d f_i \otimes x_i

                = inverse of unit cell area * sum_{i}^{3} K* d u*_i \otimes x_i ( (1) used)

                = inverse of unit cell area * sum_{i}^{3} K* \cdot (d epsilon_M) \cdot x_i \otimes x_i ( (1) used)

From which you can compute the macroscopic tangent.

Hope that it helped. If you need further help, just ask :-)

I've found the three part review by Hassini and Hinton quite helpful.

 Here's a link to one of them: http://dx.doi.org/10.1016/S0045-7949(98)00132-1

 -Nachiket