Measurement and optimization of lattice materials – MOMAP

- an algorithm for optimizing cell geometry (use of invariants) and taking into account limitations (yield stress, buckling and manufacturing constraints).
- a specific mechanical test set-up for lattice materials. This device must allow any possible deformation tensor to be stressed, with minimal edge effects and allow X-ray tomography.
- a 3D imaging algorithm based on the fast and efficient VIC method adapted to cell metrology. We expect a better surface measurement (in 3D) than that provided by current reconstruction algorithms.
- a DIC/VIC algorithm allowing the measurement of the 3D deformation field of a specimen or structure. We expect a measurement with less noise than current methods.
- demonstrators (metallic and concrete) of optimized structures. We expect that these demonstrators will demonstrate the quality of the work as a whole and will demonstrate the interest of this new approach compared to current methods (topological optimization). The CEA will fabricate metallic lattices that will be measured by X-ray tomography and correlation of virtual images, in both free and loaded states. The company XtreeX will produce a concrete structure optimized with the proposed methods and which will be tested in the IFSTTAR laboratory.

The expected results are :
- a method for the optimization of quasi-periodic architectural materials within a structure, taking into account the manufacturing possibilities and the limit criteria of local or global buckling and elasticity
- a protocol for testing specimens made of an architectural material, resolving edge condition problems and allowing X-ray imaging and stress tensor testing of all components of the stress tensor.
- a 3D reconstruction method from raw X-ray tomography images, based on the virtual image correlation method, and showing a better reconstruction accuracy than current methods

Anglais (USA)
Glossaire

Prototype of the test fixture for triangular architectural material.

L. Calmettes, J. Réthoré et M.L.M François. 3D metrological shape measurements from X- ray radiographs, communication acceptée dans Photomechanics-iDiCs conference, 3-5 novembre 2021.

V. Jeanneau, C. Combescure et M.L.M. François. Tenseur d’élasticité et surface limite de linéarité d’un matériau architecturé 2D triangulaires sous état de déformation macroscopique, Congrès des jeunes chercheurs en Mécanique, en ligne, 24-27 aout 2021.

M.L.M François et Y. Lecieux. A 6-ring embedded strain tensor, proposé dans Meccanica

M.L.M. François, The Virtual Image Correlation Method: Principle and Uncertainty, dans iDIC, Portland, OR (USA), 14-18 octobre 2019

Lattice materials may induce a structural mass gain of magnitude 10 [Fleck, 2010]. However three scientific obstacles remain: - their fabrication cost - the computation of the optimal cell shape that remains long and difficult with the actual local methods [Laszczyk, 2011] - the quality control and the measurement of their mechanical properties which is quite impossible with classical mechanical testings [Bréchet, 2013] We will try to resolve the last two points. Topological optimisation is used to obtain the optimal shape of a bulk material. Here, given the global shape, we will search for the optimal local geometry of the cells (links widths, angles). The optimisation will be carried out by using the concept of homogeneous equivalent material concept, from its stiffness tensor, elastic and linearity limits and fabricability limits. The latter are associated to the 3D printing process of these materials. One participant [Olive, 2017] co-discovered the first set of stiffness tensor invariants. We will use some of these invariants and the orientation parameters within the optimisation process. The elasticity and the elasticity limits will be calculated analytically. In order to measure their actual value a testing device will be designed and realised. The boundary conditions will be imposed thanks to special links in order not to develop second gradient elasticity [Poncelet, 2018] and to obtain an homogenous strain field. In order to excite a sufficient number of strain states an hexapod (Stewart platform) device will be used. The strain filed will be measured thanks to 3D X-ray tomography. Tomography involves a reconstruction from a 3D volume from a collection of 2D radiographs. We will adapt the VIC (Virtual Image Correlation) method [Réthoré, 2013] for that. Using a parameteric description of the cells, it acts as an optimal filter, leading to better imaging resolution. This method will be used for the geometrical defects detection from tomographies at the cell scale. A coupled DIC (Digital Image Correlation) and VIC method will be developed and used at the structure scale for the specimen strain field measurement. CEA and Dorel company are partners of the projet. CEA will manufacture specimens and two structures of metallic lattices. These structures will be optimised thanks to the developed method and tested in order to verify their performances. Dorel will establish the specifications of a sketch structure (made of polymer) close to a baby equipment. One of them will be optimised with the new method and the other with present tools. They will be tested in order to estimate the gain of performance.

Fleck, N.A., Deshpande, V.S., Ashby, M.F. (2010). Micro-architectured materials: past, present and future. Proceedings of the royal society of London A. Vol. 466, No. 2121, pp. 2495-2516.
Laszczyk, L. (2011). Homogénéisation et optimisation topologique de panneaux architecturés, Thèse de Doctorat, Université de Grenoble.
Olive, M., Kolev, B., Auffray, N. (2017). A minimal integrity basis for the elasticity tensor. Archive for Rational Mechanics and Analysis, 226(1), 1-31.
Poncelet, M., Somera, A, Morel, C., Jailin, C, Auffray, N. (2019) An experimental evidence of the failure of Cauchy elasticity for the overall modeling of a non- centro-symmetric lattice under static loading. International Journal of Solids and Structures, 147 (15), 223-237
Réthoré, J. et François, M. (2013). Curve and boundaries measurement using B-splines and virtual images, Optics and Lasers in Engineering, 52, 145-155.