Homogenization in periodically heterogeneous elastic bodies with multiple micro-contact

The aim of this contribution is to compute the effective in-plane tension and shear behaviour of textile-like elastic materials, that is, plates or shells with a periodic micro-structure composed of long woven or knitted fibres. The knitting or weaving results in multiple periodic contact between fibres. Mathematically the problem can be formulated as an in-plane elasticity problem defined in a heterogeneous domain with e-periodic micro-structure, including multiple micro-contact between the structural components, which is described by the Signorini and Tresca-friction contact conditions. The asymptotic analysis and homogenized limit for such problems was recently obtained by Damlamian et al. via periodic unfolding strategy. These results are briefly recalled in the paper. The obtained two-scale algorithm is implemented for some specific textile materials, which are macroscopic shells with a 3-D microstructure including contact. For each macroscopic deformation state, a contact problem in the periodicity cell or representative volume element is solved and the corresponding non-linear macroscopic stress-strain relation is obtained. The results are illustrated by the simulation of woven and knitted textiles.