Asymptotic, second-order homogenization of linear elastic beam networks

Yang Ye (1) , Basile Audoly (1) , Claire Lestringant (2)

1 LMS - Laboratoire de mécanique des solides
2 DALEMBERT - Institut Jean Le Rond d'Alembert

We propose a general approach to the higher-order homogenization of discrete elastic networks made up of linear elastic beams or springs in dimension 2 or 3. The network may be nearly (rather than exactly) periodic: its elastic and geometric properties are allowed to vary slowly in space, in addition to being periodic at the scale of the unit cell. The reference configuration may be prestressed. A homogenized strain energy depending on both the macroscopic strain ɛ and its gradient ∇ɛ is obtained by means of a two-scale expansion. The homogenized energy is asymptotically exact two orders beyond that obtained by classical homogenization. The homogenization method is implemented in a symbolic calculation language and applied to various types of networks, such as a 2D honeycomb, a 2D Kagome lattice, a 3D truss and a 1D pantograph. It is validated by comparing the predictions of the microscopic displacement to that obtained by full, discrete simulations. This second-order method remains highly accurate even when the strain gradient effects are significant, such as near the lips of a crack tip or in regions where a gradient of pre-strain is imposed.

Mots clés: Elastic lattices; Asymptotic homogenization; Second-order homogenization; Energy methods

Domaines: Mécanique des solides [physics.class-ph]

Yang Ye, Basile Audoly, Claire Lestringant. Asymptotic, second-order homogenization of linear elastic beam networks. Journal of the Mechanics and Physics of Solids, inPress, ⟨10.1016/j.jmps.2024.105637⟩. ⟨hal-04549113v1⟩