Architectured materials designed with higher-order homogenization – ArchiMatHOS

The initial scientific challenge was to develop a higher-order homogenization scheme applicable to continuous periodic microstructures. The second scientific challenge was to develop a topological optimization procedure for generating new microstructures with second-gradient properties on the basis of the homogenization scheme obtained. This procedure involved proving the existence and calculating the topological derivative, proposing relevant optimization functionals and implementing all these elements numerically. Finally, the challenge of the project was also to demonstrate that these materials derived from purely theoretical considerations are manufacturable, and that non-standard properties are indeed present.

We managed to synthesize a pantograph-type microstructure from topological optimization, to manufacture it, to test it and to compare experiments with the predictions of the proposed homogenization scheme. This type of microstructure had long been identified as a candidate for second gradient effects by one of the partners, and it is very remarkable that it emerged spontaneously from the optimization procedure. This is a major result, as there is really no study that covers each of these aspects (rigorous homogenization, optimization and experimental validation) in depth and in a consistent way, whereas the second gradient model has been the subject of many phenomenological and theoretical explorations before. It is the outcome of an efficient collaboration between mathematicians, researchers in theoretical mechanics, experimental mechanics and computational science.
The various spin-offs of this result include the identification of a biomedical application using an architectured deployable material for minimally invasive surgery, and the launch of an ANR project on architectured inflatable structures.

. This project has opened up several research topics. Firstly, the extension of the framework to finite deformations to synthesize deployable architectured materials. But also, the possibility of synthesizing architectured materials with other non-standard behaviors (such as enriched continua).
In addition, the biomedical application studied as a preliminary step in the project has enabled us to better identify the field of relevance of these architectured materials with internal mechanisms, and encourages us to extend our approach to non-linear geometry, possibly as part of a new ANR project to be submitted in October 2023.

The project resulted in the publication of 13 articles in peer-reviewed journals, some of them prestigious. In addition, two workshops were organized with internationally renowned keynote speakers to share the results of the project.

Recently, additive manufacturing and topology optimization allowed a rupture in material sciences called “material by design” where it becomes possible to design and fabricate materials with a given microstructure. Such materials are referred to as architectured materials because their effective properties arise not only from the constitutive material but also from the regular microstructure. These materials are raising high interest because they allow the fabrication of materials which may not be found in nature: meta-materials.
Multiscale modelling and homogenization play a central role in this science area since they are the keys for capturing the effective behavior of such materials. More precisely, it was mathematically proven that there is a wide variety of non-standard effective behaviors which may be reached form a simple mixture of constitutive materials. However, this theoretical result does not yield the corresponding microstructures.
The main goal of this project is to explore this variety of non-standard elastic behaviors in order to find and synthesize architectured materials with unprecedented properties: very large volume variation, high shock strength, high energy absorption, high formability. Architectured materials technologies will benefit from this project but also “smart materials” coupling different physical phenomena and driven by an actuation energy.
The first scientific challenge of this project is the derivation of higher-order homogenization schemes applied to periodic microstructures and based only on two-scale separation without ad hoc assumptions. This will be performed first in linear elasticity and then extended to large deformations. The second scientific challenge is the derivation and the implementation of a topology optimization procedure based on the new homogenization schemes in order to generate architectured materials with chosen non-standard properties. Mathematical assumptions made in the theoretical developments will be made with special care so that the obtained architectured materials will be indeed producible and testable.