An isogeometric finite element approach to fibre-reinforced composites with fibre bending stiffness
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O R I G I NA L
Carina Witt
· Tobias Kaiser
· Andreas Menzel
An isogeometric finite element approach to fibre-reinforced composites with fibre bending stiffness
Received: 15 May 2020 / Accepted: 14 August 2020 © The Author(s) 2020
Abstract In the modelling of fibre-reinforced composites, it is well established to consider the fibre direction in the stored energy in order to account for the transverse isotropy of the overall material, induced by a single family of fibres. However, this approach does not include any length scale and therefore lacks in the prediction of size effects that may occur from the fibre diameter or spacing. By making use of a generalised continuum model including non-symmetric stresses and couple-stresses, the gradient of the fibre direction vector can be taken into account as an additional parameter of the stored energy density function. As a consequence, the enhanced model considers the bending stiffness of the fibres and includes information on the material length scale. Along with additional material parameters, increased continuity requirements on the basis functions follow in the finite element analysis. The isogeometric finite element method provides a framework which can fulfil these requirements of the corresponding weak formulation. In the present contribution, the method is applied to two representative numerical examples. At first, the bending deformation of a cantilever beam is studied in order to analyse the influence of the fibre properties. An increasingly stiff response is observed as the fibre bending stiffness increases and as the fibre orientation aligns with the beam’s axis. Secondly, a fibre-reinforced cylindrical tube under a pure azimuthal shear deformation is considered. The corresponding simulation results are compared against a semi-analytical solution. It is shown that the isogeometric analysis yields highly accurate results for the boundary value problem under consideration. Keywords Isogeometric analysis · Fibre bending stiffness · Fibre curvature · Generalised continuum · Gradient elasticity
1 Introduction In a wide range of applications, materials are reinforced with fibres in order to improve their mechanical behaviour, cf. [14,17,18,27]. Electrical and thermal properties of a material can also be manipulated by the reinforcement with certain types of fibres, see [1,34]. When fibre-reinforced composites are modelled numerically, the fibres are, in most cases, characterised by a direction vector only. This leads to a classic structural tensor approach under the assumption of perfectly flexible fibres, cf. [5,19,20,29,30,36]. Although this concept yields accurate results for many applications, it cannot account for size effects that follow from particular fibre properties such as their diameter or spacing. On these grounds, a new approach has been presented in [31]. It drops the assumption of perfectly flexible C. Witt (B) · T. Kaiser · A. Menzel Institute of Mechanics, TU Dortmund, Leonhard-Euler-Strasse 5, 44227 Dortmund, Germany E-mail: carina.witt@tu-d
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