Model reduction for the forming process of fibrous composites structures via second gradient enriched continuum models

Abstract

This paper describes phenomena associated to the microstructure, such as the local bending stiffness of the yarns, with a second gradient parameter associated to terms in the energy with higher order derivatives. The results obtained with this enriched continuous model are presented and it is shown how the main deformation mode can be controlled to reproduce experimental evidence. In particular, increasing the second gradient parameter, the deformation mode switches from shear to one with a constant curvature along the specimen. Two fibrous composite reinforcements with two different geometries are modeled: composite airframe for aeronautic industry and an automotive leaf-spring for automotive industry. The two application cases are described as homogeneous second gradient continua with an energy depending on the in-plane and out-of-plane curvature of the yarns. This addition allows to take into account the effects of the local bending stiffness of the yarns at the mesoscale in a homogenized way.

Influence of In-Plane Bending Behaviour on Textile Composite Reinforcement Forming

“…Nevertheless, it is necessary to highlight that these two deformation modes can be decoupled when writing internal virtual work. Similar hypothesis is done in second gradient models where the hyperelastic second gradient strain energy density is supposed to be a function of Cauchy-Green strain gradient only (whereas the first gradient energy is only a function of Cauchy-Green strain) [67,68,70]. The in-plane bending behaviour is part of the nature of textile reinforcement: even if there is only one yarn direction (in this case shear angle definition is not straightforward), it is still possible to define in-plane bending field as the gradient of fiber direction.…”

“…In-plane bending of fabrics was also observed during forming process and related to the occurrence of yarn buckling [62,63,64]. Second gradient methods have been proposed for the modelling of this specific phenomenon during BET [65,66,67,68,69] and were applied to actual forming [70]. However, the implementation of second gradient methods in finite element codes and the determination of associated material parameters are not direct.…”

Modeling and analysis of in-plane bending in fibrous reinforcements with rotation-free shell finite elements.

“…The above-mentioned second gradient models are the only ones up to now that can also take into account the bending of the fibers in the plan. They were recently used to simulate standard forming cases with finite element (Barbagallo et al, 2017(Barbagallo et al, , 2019 but their use is still cost-effective due to high order shape functions.…”