The main idea of this work is part of an approach that attempts to represent the quasi-brittle behavior of heterogeneous materials such as cementitious ones. The guideline followed fits in a sequenced multi-scale framework for which descriptions of the material are selected at a thin scale (mesoscopic or microscopic) and information is transferred to a larger scale (macroscopic). It shows how the explicit representation of heterogeneities offers interesting prospects on identification, understanding and modeling of macroscopic behaviors. In practice, from a simple description of each phases and interfaces behavior, a structural effect that leads to more complex macroscopic behavior is observed.
This work is therefore focusing on two main axes. On the one hand, the morphological representation of the heterogeneities is handle using the excursion sets theory. Randomly shaped inclusions, which geometrical and topological characteristics are analytically controlled, are produced by applying a threshold on realizations of correlated Random Fields.
On the other hand, the FE implementation of both heterogeneity and local degradation behavior (micro-cracking) are dealt with by a double kinematics enhancement (weak and strong discontinuity) using the Embedded Finite Element Method. Finally, combining both axes of the problematic, the resulting model is tested by modeling cementitious materials at the meso-scale under uniaxial loadings mainly. It reveals an emergent macroscopic response that exhibits several features such as asymmetry of the tension-compression stress-strain relationship, cracks' patterns or historical-dependency, which are typical of concrete-like materials.
We propose an analytical model to predict critical probability of percolation. It is based on the excursion set theory which allows us to consider N-dimensional bounded regions. Details are given for the 3D case and statistically Representative Volume Elements are calculated. Finally generalisation to the N-dimensional case is made.
The work proposes a FE2 multiscale approach to computational modeling of material failure in concrete-like structures, made of cement/aggregate-type composite materials. Keeping the approach in a classical homogenization setting, a multiscale model is proposed, which naturally provides a microscopic lengthscale to be exported to the macrostructure. There, this length scale is used as regularization parameter in the context of the Continuum Strong Discontinuity Approach to material failure, and finite elements with embedded strong discontinuities (E-FEM). The resulting technique allows robust modeling of crack propagation at the structural scale, accounting for the mesostructure morphology, supplies proper energy dissipation and solutions independent of the finite element and RVE sizes. Application to a number of examples, in the range from light-aggregate concrete to regular concrete, shows the potentiality of the method.
Stabilization technics are used in order to address the commun numerical issues of highly localized strain fields. A double strain-injection scheme of weak and strong discontinuities along with a crack path field method [Oliver 2014] is used at the macroscopic scale.
The reduced order part of the work addresses the problematic of reducing the computational time of multi-scale framework by solving, at each point of the structural scale, the Representative Volume Element (RVE) equilibrium problem using Reduce-Basis approximations. A specific methodology is proposed in order to tackle non-smooth fields. Therefore, the framework can find its applications into quasi-brittle heterogeneous materials, such as cementitious ones, and aims at performing crack propagation computations at a structural level.
The model is based on the Finite Element Method at both scales linked by weak coupling (FE2). The macro-strain acts as a loading parameter on the RVE whereas the macro-stress is the resulting homogenization of the micro-stress. The degradation mechanisms are modeled by the Continuum Strong Discontinuity Approach (CSDA) which lead to kinematic of the macroscopic crack defined in a small dimensional space.
The basis used for the reduction are constructed with a partitioned version of the Proper Orthogonal Decomposition of a set of a pre-computed snapshots, allowing us to consider distinctly the different regime of the solution (elastic, hardening or softening). Then, in addition to the projection of the solution (strains) onto a reduced basis, the interpolation of the off-line terms (stresses) is also made in a reduced-order space. To avoid the resulting ill-posed formulation, an expansion of the approximation space is made. Finally, based on both precision and stability considerations, a point-selecting algorithm (greedy) is perform in order to reduce the number of integration points.
The specificity of the method in regards with the potentially non-smooth solutions comes from a decomposition of the fields between elastic and inelastic domains of the RVE thus increasing the efficiency of the greedy algorithm. Such a decomposition is possible thanks to the CSDA that allows us to fully express the RVE formulation in terms of strains.