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 anin-plane elasticity problem defined in a heterogeneous domain with ε-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 periodicunfolding 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.