Seminar professor Ganghoffer

The talk aims to provide a classification of generalized continua constructed by a micromechanical approach, relying on an extension of the celebrated Hill macrohomogeneity condition for the micromorphic effective medium. One key aspect of the method relies on the identification of the homogeneous displacement obtained from energetic arguments, which exactly represents the effective micromorphic medium. It is corrected by a fluctuation measuring its deviation from the true response of the initially heterogeneous periodic medium. The homogenized micromorphic constitutive law is derived based on variational principles and the solutions of the unit cell BVP’s. The proposed homogenization framework is illustrated by numerical examples in the domain of composites and architected materials.

The ranking of the different effective media arising from micromorphic continuum versus the small-scale parameter (ratio of unit cell size to a macroscopic dimension) is established, using a projection of the micro-deformation into reduced higher order kinematic variables.

Based on a surface formulation of the extended Hill macrohomogeneity condition, edge effects in the response of lattice structures are quantified thanks to the higher-order terms of the constitutive law. This approach is exemplified in the case of the bending of an architected beam modeled in the framework of Cosserat mechanics. The energetic formulation of a second strain gradient continuum allows revisiting the notion of anisotropic surface energy, thereby providing a generalization of Mindlin’s model of surface energy.

We conclude by highligthing the prominent role of topology optimization methods as a rationale to synthesize microstructures prone to either classical or higher-order behaviors.

References

Alavi et al. Construction of micromorphic continua by homogenization based on variational principles. J. Mech. Phys. Solids, 153, 104278 (2021).
Reda, et al, 2021. Homogenization towards chiral Cosserat continua and applications to enhanced Timoshenko beam theories. Mechanics of Materials, 2021.
Alavi et al., 2022. Continualization method of lattice materials and analysis of size effects revisited based on Cosserat models. Int. J. Solids Struct., 254–255 (2022) 111894.
Mawassi et al., 2022. Analysis of surface effects based on first and second strain gradient mechanics. Mech. Mat., 175(13): 104462.
R. D. Mindlin, “Micro-structure in linear elasticity,” Archive for Rational Mechanics andAnalysis, vol. 16, no. 1, pp. 51–78, Jan. 1964, doi: 10.1007/BF00248490.

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