## Multiscale failure modelling of quasi-brittle materials

### Vinh Phu Nguyen

The conventional multiscale homogenization theory, which has mainly been applied to determine constitutive laws for bulk materials, suffers from two drawbacks when applied to softening materials. Firstly, the coarse scale response is sensitive to the finite element mesh used to discretize the coarse scale domain. Secondly, increasing the size of the fine scale model does not lead to a converged homogenized response. In other words, a Representative Volume Element (RVE) does not exist for softening materials in the classical sense of homogenization.

This work aims at devising a computational homogenization (CH) method, which is objective with respect to the (finite element) discretizations of the coarse scale/fine scale models and to the fine scale model size (size of a material sample upon which a fine scale model is applied). The method is applied to softening quasi-brittle materials. These materials have a random heterogeneous microstructure that undergoes localized damage such as concrete.

The concept of an RVE for quasi-brittle softening materials has been revisited in this work in which we developed a new averaging technique coined the failure zone averaging scheme. The basic idea of the scheme is to do the averaging over a propagating damaged zone, rather than over the entire fine scale domain. By doing so, we have been able to obtain homogenized initially rigid cohesive laws which are independent of fine scale model size (when this size is larger than a minimum value in order for the fine scale model to be independent of the microstructural randomness) which allows us to state that an RVE exists for softening materials when averaging towards a coarse scale cohesive law.

Two CH based multiscale cohesive crack frameworks are developed by which the two aforementioned drawbacks associated with conventional homogenization theories applied for softening materials are eliminated. According to the first framework, referred to as a discontinuous homogenization scheme, the coarse scale bulk material is assumed to be linear elastic with effective properties computed a-priori while the behavior of the coarse scale cohesive crack is coming from nested finite element computations performed on a fine scale model (in the spirit of FE 2 methods). The method provides an energetically equivalent means to upscale a fine scale localization band to a coarse scale cohesive crack. Due to the assumption on the elastic behaviour of the coarse scale bulk, the number of nested finite element models is limited e.g., equals the number of integration points locating on the coarse scale cracks thus reducing the computational expense. The second computational homogenization procedure that has been developed in this thesis is referred to as a continuous-discontinuous CH scheme. According to the second CH scheme, constitutive behaviour of the bulk material as well as that of the cohesive crack at the coarse scale are determined from nested fine scale finite element computations. Although computationally expensive, the continuous-discontinuous homogenization scheme is more accurate than the discontinuous CH scheme for problems in which the energy dissipated prior to fine scale localization is significant. The models are verified against direct numerical simulations (DNS) and numerical examples show that with CH models, significantly reduced computational expense has been obtained compared to DNS without compromising the accuracy.

Also presented in this thesis is a multiscale framework for modelling heterogeneous material layers. At the coarse scale, they are modelled by zero-thickness interface elements of which an initially elastic cohesive law is derived from a fine scale model. A new solution scheme to solve the coupled system of equations (the fine scale equilibrium equation and the fine/coarse scale transition equation) is presented.

A fully numerical framework to study the mechanical behavior of hardening cement pastes is developed in this dissertation. Based on this micromechanical model, a macro-meso-micro three scale model for concrete is proposed. In order to make multiscale simulations feasible, parallelization of the multiscale code is realized by solving the fine scale models in parallel.