Measurement and optimization of lattice materials – MOMAP

1 Calculation of the linearity limit surface

The linearity limit surface we're looking for corresponds to the intersection of three domains: the elastic transformation domain (absence of plasticity, damage, fracture), the local buckling-free domain (absence of finite-wavelength buckling patterns), and the global buckling-free domain (absence of infinite-wavelength buckling modes).

For the latter, an original method has been developed. At the scale of a single cell, boundary conditions incorporating a buckling pattern were introduced, in a context of large transformations. The load factor was then found by canceling the determinant of the stiffness matrix.

2 Polynomial approximant of the linearity boundary surface

Following work on invariants in other collaborations (in particular the previous MatSyMat project), a threshold surface based on 4th-order polynomial invariants has been developed. The number of monomials was reduced by considering the symmetry class to which the threshold surface (inherited from the microstructure) is supposed to belong. A distance minimization method with measurement points has also been developed.

3 Topological optimization of architectural materials

Classical topological optimization methods have been redesigned to take account of the new threshold surface formulation.

4 Mechanical test set-up for measuring elasticity and linearity boundary surface of architectural materials

The new test set-up introduces boundary conditions that respect the so-called Cauchy-Born periodicity conditions, so as not to generate the buckling (bifurcation) we're looking for. The original triangular shape of the test specimen allows the application of displacement boundary conditions, also imposed by the balance of the set-up, and the architectural bars are held at mid-length by pivots.

5 Shape measurement using X-ray tomography, 3D-VIC method

Reconstruction of the shape of an architectural material specimen is carried out by image correlation between actual tomograph projections (2D radiographs) and those simulated digitally (from CAD and tomograph physics). This takes advantage of knowledge of the theoretical shape (the initial CAD) of the object. There are as many 2D calculations as there are radiographs, but no costly 3D calculations. The measurement consists of a CAD of the object, i.e. an analytical, parameterized shape (not a set of voxels). This information provides information on deviations from the prescribed CAD, i.e. the shape defects or deformation sought. The correlation between real and simulated radiographs is that of VIC (2D), itself close to image correlation methods (DIC).

I Calculation of the linearity limit surface

The published method was developed in 2D and limited to triangular architectures, which are therefore reserved for rigid, structural applications. However, the method can be used for any straight-bar architecture.

The software, initially developed as part of V. Jeanneau's thesis, was fine-tuned by an intern (R. Picard). It is now available as Open-Source on the Zenodo platform.

2 Polynomial approximant of the linearity limit surface

The new polynomial limit surface for triangular mesh architectures was finalized during N. Kesmia's PhD thesis. Already shown at scientific conferences, the final publication is currently being written.

3 Topological optimization of architectural materials

This point was not finalized. The task proved too ambitious and required finalization of the work on threshold surfaces. However, progress has been made and is currently being written up. As a result, the demonstrator planned for Xtreee (an optimized plate) could not be produced.

4 Mechanical test set-up for architectural materials and measurement of linearity limit surfaces

The principle of the test fixture was validated during Mr. Gaudeul's internship at Calcul-Méca. The concept then evolved progressively, with the simulation of many different kinematics and various dimensions, until the definition drawings were produced in June 2024. The assembly was delivered in September 2024. It is the first in the world to be capable of generating a deformation state of any (tensorial) direction in 2D, while imposing regular (Cauchy-Born) boundary conditions on the edges. Publication of the assembly concept is in progress.

5 Shape measurement using 3D-VIC X-ray tomography

Developed during L. Calmettes' PhD thesis from the 2D VIC versions and some previous developments (by us and other authors) in 3D, the 3D-VIC method, using correlation on projections and RBF shape functions, has been finalized and published. The program is now operational on Cuda and available as Open-Source. New features are that i) the measurement obtained is of sub-pixel precision and ii) requires only a few (4 or 5...) radiographs. This compares with the precision of several voxels and the use of around 1,000 radiographs for existing reconstruction methods. Finally, the problem of a priori measurement error assessment has been solved (and published) in 2D.

Outstanding feature : see above

Future prospect :

I Calculation of the linearity boundary surface

Extending the method from 2D to 3D is perfectly feasible, requiring only a few technical developments. On the other hand, the method remains rather slow: it takes several minutes to determine the threshold surface for a given mesh. It is likely that the program can be speeded up by working on both the programming and the theoretical problem. This work is already in progress.

2 Polynomial approximation of the linearity boundary surface

Identifying the optimal parameters to describe a load surface defined by a set of points (using the previous method or another) is still slow. An automatic process would be necessary. It would also be possible, using a model reduction method, to extrapolate load surfaces for any mesh geometry from the knowledge of a finite number of them.

3 Topological optimization of architectural materials

This task is the least advanced in the project, with much work still to be done to optimize the shape of the part (macro scale) and the cell (meso scale). The intermediate solution envisaged is to impose homogeneous macroscopic geometry (the same mesh everywhere) in order to focus on optimizing cell shape, a theme close to known work on optimizing reinforcement placement in composite parts.

4 Mechanical test set-up for architectural materials and linearity boundary surface measurement

The set-up has been completed and the measurement protocol is being finalized. An optical VIC-2D measurement method is to be developed by M. François and V. Rey, which uses known bifurcated modes as the basis for displacements, in order to identify their appearance with sub-pixel precision. A preliminary measurement and validation using a VIC code with an existing bar model (J. Réthoré) is planned. Finally, a local buckling detection algorithm using 2D FFT, developed by F. Amiot (Femto), is planned.

Ultimately, this set-up and measurement method should enable us to measure the 2D elasticity and limit surface of a triangular 2D structure. This will enable us not only to validate theoretical approaches, but also to carry out measurements on the actual performance of the architectures, taking into account shape and material defects. Fatigue measurements could also be envisaged.

5 3D-VIC X-ray tomography shape measurement

The code is now operational, but still needs to be enhanced by taking tomograph defects into account, in order to improve accuracy. The limits of the method - number of cells imaged, influence of noise, measurement on more diffracting materials (metal) - remain to be tested. The integration of 3D-VIC measurement and 3D mechanical testing (still rudimentary) remains to be done.

Jeanneau, V.; Combescure, C.; Franc¸ois, M. Homogenized elasticity and domain of linear elasticity of 2D architecture materials. International Journal of Solids and Structures. 2023, 269(1), 112185.

Combescure, C. Selecting generalized continuum theories for nonlinear periodic solids based on the instabilities of the underlying microstructure. 19th US National Congress of Theoretical and Applied Mechanics, Jun 2022, Austin, United States.

Calmettes, L.; Franc¸ois, M.; Re´thore´, J. Shape measurements of lattice materials from few X-ray radiographs using the 3D virtual image correlation method. International Journal for Numerical Methods in Engineering. 2023.

Franc¸ois, M. Uncertainty of the virtual image correlation method. International Journal for Numerical Methods in Engineering. 2022, 123(18), 4367–4390.

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.