Fast solution to determine the elastic behavior of 3D reconstructed biopolymer cellular structures

Abstract : The “fast Fourier transform method”, originally introduced by Moulinec and Suquet (1994), is a breakthrough in numerical algorithms used to estimate the linear and non-linear behavior of heterogeneous media. The method consists of a fixed-point algorithm based on the Lippman-Schwinger equations which is directly applied to microstructure images, thus requiring no meshing. Iterations are carried out in the real space and Fourier domain, where the local behavior of the material and the fields admissibility conditions, respectively, are accounted for. In the classical framework of linear elasticity, the local strain is admissible when it derives from a displacement field, whereas the stress is divergence-free. Eyre and Milton (1998) and Michel et al. (2001) proposed refined algorithms with much better convergence properties in cases of highly (or even infinitely) contrasted media, i.e. porous or rigidly-reinforced composites with arbitrary microstructures, which are used in this work. The Fast Fourier transform method also allows for large-size computations, where the the microstructure is more accurately discretized than in traditional finite element methods. Recently, 3D computations on materials with large representative volume elements have been successfully carried out on systems that contain up to 500^3 discretization points (Willot and Jeulin, 2008). The numerical FFT method is applied to segmented images of a biopolymer cellular material made of starch and containing a large volume fraction of voids (up to 90%). These images are crops of real structures acquired using X-ray tomography (Babin et al, 2007). Under a small deformation assumption, the local behaviour of starch is approximated by an isotropic, linear elastic law, and incompressibility is assumed. Such assumption is in accordance with previously published works dealing with both the cellular and the intrinsic material (Guessasma et al, 2009; Babin et al, 2007). The material is subjected to either hydrostatic or shear strain loading, and its macroscopic as well as local linear elastic response is computed with the FFT method, on images of sizes 200^3 to 300^3, where each voxel has a physical dimension in the range 10–40 µm depending on the sample, more exactly on the cellular structure. As expected, the material behavior is found to be macroscopically compressible due to the presence of voids, and according to the numerical computations, roughly isotropic. This homogenization procedure is compared to experimental data obtained for two sets of microstructures.
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Poster
3D Imaging, Analysis, Modeling and Simulation of Macroscopic Properties, Apr 2010, Fontainebleau, France. 2010
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François Willot, Sofiane Guessasma, Dominique Jeulin, Guy Della Valle, Anne-Laure Reguerre. Fast solution to determine the elastic behavior of 3D reconstructed biopolymer cellular structures. 3D Imaging, Analysis, Modeling and Simulation of Macroscopic Properties, Apr 2010, Fontainebleau, France. 2010. 〈hal-01713400〉

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