Mechanics of heterogeneous media and homogenization

In this theme, we focus on the behavior of heterogeneous media with the main objective of characterizing the macroscopic behavior of these media at a scale (often the engineering scale) where heterogeneities appear to be small. The research is mainly methodological, with the aim of developing the most general theories possible that can be applied to a wide range of materials. The challenge is to develop powerful numerical methods and tools that take into account the geometric characteristics of heterogeneities for highly nonlinear behaviors of constituents (e.g., elastoviscoplasticity, large deformations) with possible multiphysical couplings (such as thermomechanical). The areas of application include metallic materials (e.g., in the nuclear sector, particularly within the framework of the MISTRAL joint laboratory) and micro-architectured media (composite materials or media with periodic microstructures). The project related to this theme is divided into four areas, described below. The methodological approach adopted by the team focuses on developing techniques for changing scales and implementing them in calculation codes (mainly those developed by the team). Two areas of opportunity have also been identified, namely the use of machine learning techniques (AI) and the controlled design of micro-architectured environments.

Permanent Members : C. Bellis (CR), S. Bourgeois (MCF ECM), M. Garajeu (MCF AMU), N. Lahellec (PR AMU), H. Moulinec (IR CNRS)
Cross-team collaboration : B. Lombard (DR – équipe O&I)

Methods of Homogenization in Mean Fields

Micromechanical (or homogenization) approaches in “mean fields,” based on a limited but sufficient number of local state descriptors, make it possible to construct the macroscopic response of heterogeneous materials by taking into account the properties of the constituent phases and their microstructural morphology. Integrated at the integration point scale of a structural calculation, they can lead to significant savings in calculation time compared to “square finite element” methods, in which local interactions are described by extremely costly “full field” calculations. The application of these methods to nonlinear behaviors involves defining, through a linearization step of the constitutive equations of the phases, a linear comparison material (LCM) whose properties are obtained by efficient linear homogenization schemes. In the case of elastically dissipative behaviors (which couple reversible and irreversible phenomena), this LCM is linear viscoelastic.

The objective of this focus area is to advance these methods in three directions:

i) Development of a linearization method for fragile behavior (damage) using incremental variational principles developed in the laboratory, taking into account the statistics of the mechanical fields (first and second moments) present in the composite.

ii) Development of a method for solving the linear viscoelastic MLC to combine the advantages of direct methods based on the correspondence principle (simplicity of implementation and accuracy of the effective response) and variational methods based on approximations of microscopic fields (estimation of field statistics).

iii) Development of a method allowing the possible evolution of the geometry of the phases defining the MLC to take into account the presence of localized fields found when the behavior is highly nonlinear (e.g., damage).

Full-field homogenization methods (computational homogenization)

Our second area of study focuses on numerical aspects and scientific computing issues related to the homogenization methods described above. On the one hand, we are interested in questions of numerical analysis of algorithms with the aim of characterizing performance in terms of speed (convergence rate) and accuracy (via the development of a posteriori error estimators for nonlinear problems). On the other hand, we aim to improve the performance of these numerical methods through the development of fast optimization algorithms (within the framework of variational approaches) as well as adaptive computational tools dedicated to FFT approaches. These are at the heart of the techniques developed by the laboratory in this field, and we are continuing to develop them. Finally, these issues related to full-field computation on periodic microstructures are compounded by questions about the numerical generation of microstructures. The associated projects concern both theoretical developments (to follow up on work carried out on the generation of Voronoi tessellations) and software developments within the framework of the CraFT software. The previous topics mainly concern the static behavior of heterogeneous materials, but we are also interested in homogenization techniques in dynamics. Following preliminary work on the subject, notably in collaboration with the Waves & Imaging team, we are studying wave propagation problems in nonlinear heterogeneous media, particularly those with plastic behavior (deformation theory and then incremental theory), in order to characterize their macroscopic behavior. We also wish to continue studying the effective dynamic properties of media containing nonlinear interfaces, for which the transition from 2D to 3D poses new theoretical and numerical challenges.

Model reduction and machine learning

A first exploratory area focuses on model reduction methods for scale changes. We will first look at principal component analysis methods for linear problems, i.e., theoretical and numerical calculations of the eigenvalues of div(A grad) operators for periodic elastic problems. The main objective is to use model reduction approaches for nonlinear problems. It should be noted that these approaches are in line with methods historically developed by the team, such as the NTFA method. In the long term, we want to invest in artificial intelligence methods and apply them to micromechanics issues. Our goal is to evaluate both the ability of deep learning methods to identify the morphological parameters of the microstructure that drive the effective properties, for example based on simulation results generated using CraFT software on synthetic microstructures, and also the possibility of obtaining reduced behavior relationships through learning for highly nonlinear problems.

Design of micro-architectured environments

Finally, a second exploratory area has emerged from our activities and focuses on issues related to the controlled generation of microstructures. In the context of dynamic homogenization, an initial project involves systematizing the generation of heterogeneous materials through topological optimization. To do this, we need to develop both high-performance algorithms and digital tools that enable practical implementation. One example of interest that we will be working on is the design of microstructured thin films, an issue in which we already have expertise, with the aim of achieving target effective dynamic properties. A second project concerns the optimization of bistable microarchitectured structures. Continuing our work on tape measures, we will focus on the design of bistable microarchitectured structures with the aim of achieving target geometric configurations for stable configurations. In conclusion, it should be noted that these issues are related to the previous area of research, as we conjecture that the design of functional structures, via the determination of their local microscopic properties, could be achieved through the use of machine learning (AI) techniques on microstructures in conjunction with scaling methods.