A finite deformation isogeometric finite element approach to fibre-reinforced composites with fibre bending stiffness

Abstract

It is a common technique in many fields of engineering to reinforce materials with certain types of fibres in order to enhance the mechanical properties of the overall material. Specific simulation methods help to predict the behaviour of these composites in advance. In this regard, a widely established approach is the incorporation of the fibre direction vector as an additional argument of the energy function in order to capture the specific material properties in the fibre direction. While this model represents the transverse isotropy of a material, it cannot capture effects that result from a bending of the fibres and does not include any length scale that might allow the simulation of size effects. In this contribution, an enhanced approach is considered which relies on the introduction of higher-gradient contributions of the deformation map in the stored energy density function and which eventually allows accounting for fibre bending stiffness in simulations. The respective gradient fields are approximated by NURBS basis functions within an isogeometric finite element framework by taking advantage of their characteristic continuity properties. The isogeometric finite element approach that is presented in this contribution for fibre-reinforced composites with fibre bending stiffness accounts for finite deformations. It is shown that the proposed method is in accordance with semi-analytical solutions for a representative boundary value problem. In an additional example it is observed that the initial fibre orientation and the particular bending stiffness of the fibres influence the deformation as well as the stress response of the material.

On the incorporation of curvature effects into the isogeometric analysis of fibre‐reinforced solids

“…3. For further investigations of the influence of the fibre bending stiffness, the interested reader is referred to [2] and [5] where additional boundary value problems are analysed, including a comparison with an analytical solution for an isogeometric finite element framework with consideration of small as well as finite strains.…”

A general isogeometric finite element formulation for rotation‐free shells with in‐plane bending of embedded fibers

“…Further, the theory of Steigmann 17 and the implementation of Schulte et al 36 are restricted to two fiber families that are initially straight. Another IGA‐based finite element formulation for the gradient model of Spencer and Soldatos 14 has been presented recently by Witt et al 37 However, it is not a shell formulation and it is also restricted to initially straight fibers.…”