Computational Homogenization of Transient Problems - Application to Porous Media

Doctoral thesis, 2012

This thesis is devoted to the development and analysis of a homogenization-based computational strategy for simulating the transient and nonlinear coupled consolidation problem in micro-heterogeneous porous media. A variational framework and the pertinent finite element (FE) strategy, based on a generalized macro-homogeneity condition, are proposed for the spatial homogenization of a class of transient problems, whereby the classical assumption of first order homogenization is adopted. The homogenization is carried out on Representative Volume Elements (RVEs), which in the discrete setting are introduced in quadrature points of the macroscale elements in the spatial domain. The corresponding algorithm is thus of a nested character, often referred to as a ”Finite Element Square” (FE2) implementation. Along with the classical averages, a higher order conservation quantity is obtained. Dirichlet and Neumann boundary conditions are imposed on the RVEs as prolongation conditions that connect the macro- and subscales. The corresponding algorithmic tangent tensors for fully nonlinear material response are derived. The accuracy of homogenization relies on various features of the RVEs, including size, geometrical structure, subscale constitutive models, prolongation conditions and FEdiscretization. In particular, it is important to choose the proper prolongation conditions in order to keep the RVE size small and the computation efficient. From the saddle-point properties of an appropriately defined RVE-potential it is possible to establish bounds on the homogenized response within a given time increment, and the bounds are obtained for combinations of Dirichlet and Neumann boundary conditions on the coupled fields. The theoretical bounds are verified numerically for the special cases of drained and undrained response. For the case of very stiff and low-permeable particles in a soft and permeable matrix, which is pertinent to asphalt concrete, it was observed numerically that the most favourable choice is Neumann condition on the displacement and Dirichlet condition on the pore pressure field. The target application of the proposed computational strategy is the transient response of water-saturated asphalt concrete, which has a random substructure and viscous material behaviour. However, the numerical results in the thesis are obtained for a simplified micro-structure and elastic response of both ballast and mastic.