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.

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.

Electroelastic materials possess properties most attractive for the design of smart devices and systems such as actuators and sensors. Typical polymers show changes in shape under the action of an electric field, and vice versa, together with fast actuation times, high strain levels and low elastic moduli. This paper deals with an Ogden model inspired framework for large deformation electroelasticity which, as a special case, can also be reduced to the modelling of transversely isotropic elasticity. Extremal (local) states are elaborated based on a coaxiality analysis, i.e. extremal states of energy are considered at fixed deformation and changing direction of electric field, respectively, fixed electric field and changing principal directions of deformation. This analysis results in extremal states when stresses and strain commutate, respectively, dielectric displacements and electric field are aligned. In order to further elaborate electromechanical coupling properties, the sensitivity of stresses with respect to electric field is analysed. This sensitivity is represented by a third-order tensor which, in general, depends on deformation and electric field. To illustrate this third-order tensor, a decomposition into deviators is adopted. Related norms of these deviators, together with the electromechanical coupling contribution to the augmented energy, are investigated for different states under homogeneous deformation and changing electric field direction. The analysis is considered to contribute to a better understanding of electromechanical coupling properties and extremal states in large deformation electroelasticity and by that, as a long-term goal, may contribute to the improved design of related smart devices and systems. This article is part of the theme issue ‘The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity’.

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