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 fibres and instead allows the incorporation of the fibre bending stiffness into the material model. Amongst others, this new material parameter is associated with the gradient of the fibre direction which introduces a length scale to the model. As a special case, only the directional derivative in the fibre direction is considered to account for fibre curvature effects.
The incorporation of the gradient of the fibre direction in the stored energy density function causes the requirement of a generalised continuum theory. Several approaches to generalised continua have been examined in the past. In [
11,
21] strain gradient elasticity has been studied. Micromorphic and micropolar concepts have been addressed in [
10,
13,
15]. In the present contribution, the couple-stress theory is employed. The fundamental balance equations in this context have been derived in [
22]. A partial differential equation of fourth order with, in general, non-symmetric stresses and couple-stresses follows accordingly.
For the solution of the partial differential equation within a finite element framework, enhanced continuity requirements regarding the underlying basis functions need to be considered, cf. [
12,
35]. A multi-field approach using Lagrangian basis functions has been elaborated in [
2‐
4]. In [
23,
24], a formulation based on special types of elements, such as Hermite elements, is proposed to account for the higher continuity demand. In the present contribution, the isogeometric analysis, presented in [
6,
16], is employed. Therein, non-uniform rational B-Splines (NURBS) are used as basis functions. The fundamental theory of NURBS is provided in [
26], and it has been shown that these functions can fulfil the particular continuity requirements. The isogeometric approach has been applied to gradient elasticity problems in [
11,
32], and fibre-reinforced solids have been modelled within the isogeometric concept in [
28]. In the present contribution, the model of fibre-reinforced composites with fibre bending stiffness introduced in [
31] will be analysed within an isogeometric finite element framework. A major advantage of this approach is the possibility to discretise and solve the fourth-order partial differential equation directly as opposed to the multi-field method where the problem is split into two partial differential equations of second order. This leads to less degrees of freedom in the isogeometric approach which lowers the computational effort.
We will start with the presentation of the kinematics as well as the balance equations for the generalised continuum approach in Sect.
2 and recapitulate the constitutive relations for fibre-reinforced composites including fibre bending stiffness. In Sect.
3, a weak form of the governing equation is developed and the discretisation is performed within a finite element framework. Section
4 provides details on the isogeometric analysis including higher-order derivatives of the NURBS basis functions and the corresponding coordinate mappings. The main focus lies on the fulfilment of the continuity requirements that follow from the fourth-order partial differential equation and the weak form associated with it. The presented finite element method is applied to representative numerical examples in Sect.
5. A beam subject to a bending deformation is studied, and a cylindrical tube is analysed subject to a pure azimuthal shear deformation. The results from the isogeometric analysis are evaluated with special regard to the influence of the fibre properties, and, particularly for the cylindrical tube, they are compared against a semi-analytical solution that is provided in [
7].
Notation Let
\({\varvec{T}}\) and
\({\varvec{S}}\) denote tensor-valued quantities of arbitrary order in the Cartesian basis system. The single tensor contraction is introduced as
$$\begin{aligned} {\varvec{T}}\cdot {\varvec{S}} =&\, [T_{ij\ldots kl}\,{\varvec{e}}_i\otimes {\varvec{e}}_j\ldots \otimes {\varvec{e}}_k\otimes {\varvec{e}}_l] \cdot [S_{mn\ldots op}\,{\varvec{e}}_m\otimes {\varvec{e}}_n\ldots \otimes {\varvec{e}}_o\otimes {\varvec{e}}_p] \nonumber \\ =&\, T_{ij\ldots kl}\,S_{ln\ldots op}\,{\varvec{e}}_i\otimes {\varvec{e}}_j\ldots \otimes {\varvec{e}}_k\otimes {\varvec{e}}_n \ldots \otimes {\varvec{e}}_o\otimes {\varvec{e}}_p \end{aligned}$$
(1)
following Einstein’s summation convention. Analogously, the double and triple contractions are defined as
$$\begin{aligned} {\varvec{T}}:{\varvec{S}} =&\, [T_{ij\ldots kl}\,{\varvec{e}}_i\otimes {\varvec{e}}_j\ldots \otimes {\varvec{e}}_k\otimes {\varvec{e}}_l] :[S_{mn\ldots op}\,{\varvec{e}}_m\otimes {\varvec{e}}_n\ldots \otimes {\varvec{e}}_o\otimes {\varvec{e}}_p] \nonumber \\ =&\, T_{ij\ldots kl}\,S_{kl\ldots op}\,{\varvec{e}}_i\otimes {\varvec{e}}_j\ldots \otimes {\varvec{e}}_o\otimes {\varvec{e}}_p \end{aligned}$$
(2)
if
\({\varvec{T}}\) and
\({\varvec{S}}\) are of second or higher order and
$$\begin{aligned} {\varvec{T}}:\!\cdot \,{\varvec{S}} =&\, [T_{ij\ldots klm}\,{\varvec{e}}_i\otimes {\varvec{e}}_j\ldots \otimes {\varvec{e}}_k\otimes {\varvec{e}}_l\otimes {\varvec{e}}_m] :\!\cdot \,[S_{nop\ldots qr}\,{\varvec{e}}_n\otimes {\varvec{e}}_o\otimes {\varvec{e}}_p\ldots \otimes {\varvec{e}}_q\otimes {\varvec{e}}_r] \nonumber \\ =&\, T_{ij\ldots klm}\,S_{klm\ldots qr}\,{\varvec{e}}_i\otimes {\varvec{e}}_j\ldots \otimes {\varvec{e}}_q\otimes {\varvec{e}}_r \end{aligned}$$
(3)
if
\({\varvec{T}}\) and
\({\varvec{S}}\) are of at least third order.
For first-order tensors
\({\varvec{a}}\) and
\({\varvec{b}}\), the vector product is denoted under the use of the Levi-Civita tensor
\(\varvec{\epsilon }\) so that
$$\begin{aligned} {\varvec{a}}\times {\varvec{b}} = a_ib_j\epsilon _{ijk}\,{\varvec{e}}_k. \end{aligned}$$
(4)
For a second-order tensor
\({\varvec{T}}\), the product reads
$$\begin{aligned} {\varvec{T}}\times {\varvec{b}} = T_{ij}b_k\epsilon _{jkl}\,{\varvec{e}}_i\otimes {\varvec{e}}_l. \end{aligned}$$
(5)
The standard dyadic product is defined as
$$\begin{aligned} {\varvec{T}}\otimes {\varvec{S}} =&\, [T_{ij\ldots kl}\,{\varvec{e}}_i\otimes {\varvec{e}}_j\ldots \otimes {\varvec{e}}_k\otimes {\varvec{e}}_l] \otimes [S_{mn\ldots op}\,{\varvec{e}}_m\otimes {\varvec{e}}_n\ldots \otimes {\varvec{e}}_o\otimes {\varvec{e}}_p] \nonumber \\ =&\, T_{ij\ldots kl}\,S_{mn\ldots op}\,{\varvec{e}}_i\otimes {\varvec{e}}_j\ldots \otimes {\varvec{e}}_k\otimes {\varvec{e}}_l \otimes {\varvec{e}}_m\otimes {\varvec{e}}_n\ldots \otimes {\varvec{e}}_o\otimes {\varvec{e}}_p \end{aligned}$$
(6)
and two non-standard dyadic products are introduced in the form of
$$\begin{aligned} {\varvec{T}}\,{\overline{\otimes }}\,{\varvec{S}} =&\, [T_{ij\ldots kl}\,{\varvec{e}}_i\otimes {\varvec{e}}_j\ldots \otimes {\varvec{e}}_k\otimes {\varvec{e}}_l] \,{\overline{\otimes }}\,[S_{mn\ldots op}\,{\varvec{e}}_m\otimes {\varvec{e}}_n\ldots {\varvec{e}}_o\otimes {\varvec{e}}_p] \nonumber \\ =&\, T_{ij\ldots km}\,S_{ln\ldots op}\,{\varvec{e}}_i\otimes {\varvec{e}}_j\ldots \otimes {\varvec{e}}_k\otimes {\varvec{e}}_l \otimes {\varvec{e}}_m\otimes {\varvec{e}}_n\ldots \otimes {\varvec{e}}_o\otimes {\varvec{e}}_p \end{aligned}$$
(7)
and
$$\begin{aligned} {\varvec{T}}\,{\underline{\otimes }}\,{\varvec{S}} =&\, [T_{ij\ldots kl}\,{\varvec{e}}_i\otimes {\varvec{e}}_j\ldots \otimes {\varvec{e}}_k\otimes {\varvec{e}}_l] \,{\underline{\otimes }}\,[S_{mn\ldots op}\,{\varvec{e}}_m\otimes {\varvec{e}}_n\ldots {\varvec{e}}_o\otimes {\varvec{e}}_p] \nonumber \\ =&\, T_{ij\ldots kn}\,S_{lm\ldots op}\,{\varvec{e}}_i\otimes {\varvec{e}}_j\ldots \otimes {\varvec{e}}_k\otimes {\varvec{e}}_l \otimes {\varvec{e}}_m\otimes {\varvec{e}}_n\ldots \otimes {\varvec{e}}_o\otimes {\varvec{e}}_p. \end{aligned}$$
(8)
The application of the gradient operator to a tensor
\({\varvec{T}}\) of arbitrary order yields
$$\begin{aligned} \nabla _{\!\varvec{x}}{\varvec{T}} = \frac{\partial T_{ij\ldots kl}}{\partial x_m} \,{\varvec{e}}_i\otimes {\varvec{e}}_j\ldots \otimes {\varvec{e}}_k\otimes {\varvec{e}}_l\otimes {\varvec{e}}_m \end{aligned}$$
(9)
and applying the divergence operator results in
$$\begin{aligned} \nabla _{\!\varvec{x}}\cdot {\varvec{T}} = \nabla _{\!\varvec{x}}{\varvec{T}} : {\varvec{I}} = \frac{\partial T_{ij\ldots kl}}{\partial x_l} \,{\varvec{e}}_i\otimes {\varvec{e}}_j\ldots \otimes {\varvec{e}}_k, \end{aligned}$$
(10)
where
\({\varvec{I}}\) is the second-order identity tensor. The curl of a first-order tensor
\({\varvec{a}}\) is denoted as
$$\begin{aligned} \nabla _{\!\varvec{x}}\times {\varvec{a}} = \epsilon _{ijk} \frac{\partial a_{k}}{\partial x_j} \,{\varvec{e}}_i. \end{aligned}$$
(11)
Different kinds of transpositions of tensors are defined depending on the order of the tensor. For a second-order tensor
\({\varvec{T}}\), the transposition results in
$$\begin{aligned} {\varvec{T}}^\mathrm {t} = T_{ji} \,{\varvec{e}}_i\otimes {\varvec{e}}_j. \end{aligned}$$
(12)
For a fourth-order tensor
\({\varvec{S}}\), two different types of transpositions are used as this work proceeds, namely
$$\begin{aligned} {\varvec{S}}^\mathrm {t} = S_{jikl} \,{\varvec{e}}_i\otimes {\varvec{e}}_j\otimes {\varvec{e}}_k\otimes {\varvec{e}}_l\quad \hbox {and}\quad {\varvec{S}}^\mathrm {T} = S_{klij} \,{\varvec{e}}_i\otimes {\varvec{e}}_j\otimes {\varvec{e}}_k\otimes {\varvec{e}}_l. \end{aligned}$$
(13)
Similarly, for a third-order tensor
\({\varvec{R}}\),
$$\begin{aligned} {\varvec{R}}^\mathrm {t} = R_{jik} \,{\varvec{e}}_i\otimes {\varvec{e}}_j\otimes {\varvec{e}}_k \quad \hbox {and}\quad {\varvec{R}}^\mathrm {T} = R_{kij} \,{\varvec{e}}_i\otimes {\varvec{e}}_j\otimes {\varvec{e}}_k \end{aligned}$$
(14)
are employed.
The present contribution focusses on the simulation of fibre-reinforced composites where the fibres exhibit a certain bending stiffness. The material model has been developed in accordance with the derivations presented in [
31]. For a small strain setting, the stress and couple-stress tensor have been specified and the balance equations for a couple-stress theory have been derived. An isogeometric finite element framework has been developed for solving the resulting partial differential equation of fourth order because the particular continuity requirements can be fulfilled by the use of NURBS as basis functions.
With the isogeometric framework, two numerical examples have been examined. In the first example, a fibre-reinforced beam subject to a bending deformation has been simulated in order to study the influence of the fibre properties. It has been confirmed that an increasing fibre bending stiffness parameter leads to a stiffer response. In addition, the influence of the fibre orientation has been studied in detail. The minimum bending has been obtained for the fibres being aligned with the beam’s axis.
In the second example, a cylindrical tube under pure azimuthal shear has been analysed. Due to the assumption of radially aligned fibres, the equations for the displacement in radial and azimuthal direction become uncoupled. Under appropriate boundary conditions, the model has been simulated by the isogeometric finite element approach. In order to evaluate the respective simulation results, they have been compared against the semi-analytical solution provided in [
7]. It has been found that the isogeometric analysis yields highly accurate results in the modelling of fibre-reinforced composites including fibre bending stiffness. In particular for the symmetric stress components as well as for the couple-stress components, no significant difference to the semi-analytical results is observable when plotted over the radius of the cylindrical tube.
Moreover, it has been shown that the continuity properties of the basis functions have a significant impact on the simulation. By the knot removal procedure as well as the imposition of linear constraints for repeated control points, the continuity requirements for the analysis of materials including higher gradient contributions can be fulfilled globally within the isogeometric analysis.
In future research, the developed isogeometric finite element framework including higher gradients of the displacement field shall be extended to finite deformations by employing the respective constitutive relations specified in [
31]. In addition, more complex boundary value problems, particularly in the three-dimensional space, shall also be studied.
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