Modelling the deep drawing of a 3D woven fabric with a second gradient model

Abstract

Experimental testing on dry woven fabrics exhibits a complex set of evidences that are difficult to completely describe using classical continuum models. The aim of this paper is to show how the introduction of energy terms related to the micro-deformation mechanisms of the fabric, in particular to the bending stiffness of the yarns, helps in the modelling of the mechanical behaviour of this kind of materials. To this aim, a second gradient, hyperelastic, initially orthotropic continuum theory is proposed to model fibrous composite interlocks at finite strains. In particular, the present work explores the relationship between the onset of wrinkling during the simulation of the deep drawing of a woven fabric and the use of a second gradient model. It is shown that the introduction of second gradient terms accounting for the description of in-plane and out-of-plane bending rigidities decreases the onset of wrinkles during the simulation of deep drawing.

In this work, a quadratic energy, roughly proportional to the square of the curvature of the fibres, is presented and implemented in the simulations. This simple constitutive assumption allows the effects of the second gradient energy on both the wrinkling description and the numerical stability of the model to be clearly shown. The results obtained in second gradient simulations are descriptive of the experimental evidence of deep drawing whose description is targeted in this work. The present paper provides additional evidence of the fact that first gradient continuum theories alone cannot be considered fully descriptive of the behaviour of dry woven composite reinforcements. On the other hand, the proposed second gradient model for fibrous composite reinforcements opens the way both to the more accurate simulation of complex forming processes and to the possibility of controlling the onset of wrinkles.