Homogenization of a Cracked Poroelastic Medium - Numerical calculation of the coefficients and damage

This work is devoted to the use of double scale asymptotic expansions and
the posterior use of numerical tools in order to get the coe cients for an homogeneous
macroscopic model starting from heterogeneous microscopic description.
The particular problem of this project is a cracked poroelastic medium, that is a
Biot-type problem both in the grains and in the cracks. In order to solve the expanded
problem resulting from the double scale asymptotic expansions a nite
element code is used. Numerical simulations allow one to obtain the homogenized
coe cients for a macroscale description. These results are not isotropic
despite having isotropic microscale coe cients due to the crack geometry and
orientation. In a second part of the project damage is introduced in the cracks.
Damage depends on the crack's opening, this makes the problem nonlinear and
makes necessary the use of appropriate numerical tools to nd the solution. The
commercial nite element code used in the rst part can not reproduce the damage
problem, therefore a Matlab code is developed. This code consists of a FEM
implementation in the porous part, with the appropriate linking conditions for
the cracks to create the macrograin in the REV. In order to nd the solution
for the di erent nonlinearities of the problem iterative procedures such as the
secant method are introduced. The resulting model is able to reproduce any
time-history of strain, and an evolution of the homogenized coe cients can be
obtained from this time-history, unlike in the linear case, bifurcation phenomena
can be observed for the damage problem.