Large deformation analysis of fibrous materials using rate constitutive equations

Rate constitutive equations (or hypoelastic constitutive equations) are frequently used in finite element software for finite strain analyses. For instance the material user subroutines of softwares such as ABAQUS® are based on such approaches in the case of large deformations. Among these approaches, most of them use the standard Green Naghdi or Jaumann objective derivatives. The mechanical behaviour at finite strains of fibrous media (fabrics, composite reinforcements, NCF, etc.) is specific because of the possible sliding between fibres. The rate constitutive equations can be used if the fibrous medium is considered as a continuum. In this case, some approaches use the standard Green Naghdi or Jaumann objective derivatives and the associated rotated frame. The constitutive matrix in the rotated frame is obtained from the one in the fibre frame, which is the only frame where the specificities of the fibrous medium are exhibited. An alternative consists in defining and using an objective derivative and a rotated frame from the rotation of the fibre. The aim of this paper is to show that only the second approach leads to satisfying results for some tests and to explain the reasons why this second approach is preferable. This approach is also used to analyse at mesoscopic scale, in-plane shear deformation of a woven unit cell.