Energy-Momentum Scheme with Drilling Degrees of Freedom for Composites with Curvature-Twist Stiffness

Abstract

This paper is motivated by the increasing application of 3D fiber-reinforced composites in rotating systems [1]. In 2D fiber-reinforced composites, single fibers with a diameter in the range of micrometers are embedded in a matrix material. But, these composites are prone to delamination damage, wherefore the development of 3D composites has been undertaken. Here, fiber bundles are woven, knitted, braided or stitched, in order to fix the fibers before they are surrounded by a matrix material. From a material modelling point of view, these two kind of composites make a huge difference, because a fiber bundle has to be considered as a beam-like structure with curvature-twist (bending as well as twisting) stiffness, in addition to the usual stretching stiffness (cf. [2]). The former is then responsible for the increasing strength-to-weight ratio of 3D fiber-reinforced composites for thin-walled lightweight structures. Therefore, 3D fiber-reinforced composites demand for a bespoke simulation technique. We have to consider a representative volume element, in which secondary effects as a micro inertia and a curvature-twist stiffness must be taken into account. We introduce these secondary effects in a continuum formulation by means of independent drilling degrees of freedom (cf. [3]). The resulting nonisothermal constrained micropolar continuum is derived by a mixed principle of virtual power (cf. [4]). This variational principle simultaneously generates in the discrete setting a mixed B-bar method and a Galerkin-based energy-momentum scheme of higher order. We also take into account viscoelastic material behaviour, which arises from a mixture of organic and inorganic fibers in a polymeric matrix material. Representative numerical examples demonstrate the twisting and bending stiffness of the fiber bundles.