# Deformation induced loss of ellipticity in an anisotropic circular cylindrical tube

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## Abstract

When a transversely isotropic circular cylindrical tube is subject to axial extension and inflation, the governing equations of equilibrium can lose ellipticity under certain combinations of deformation and direction of transverse isotropy. In this paper, it is shown how the inclusion of an axial shear deformation moderates the loss of ellipticity condition. In particular, this condition is analysed for a material model consisting of an isotropic neo-Hookean matrix within which are embedded fibres whose properties are characterized by the addition to the strain-energy function of a reinforcing model depending on the local fibre direction.

## Keywords

Loss of ellipticity Axial shear Finite deformations Fibre kinking Fibre splitting## 1 Introduction

Motivated by instability phenomena in fibre-reinforced composite materials, this paper is concerned with an analysis of the loss of ellipticity of the equations governing the equilibrium of a transversely isotropic elastic material. The analysis is applied to a thick-walled circular cylindrical tube which is subject to the combination of axial extension, radial inflation and axial shear set in the context of the more general helical shear (which itself combines axial and azimuthal shear). In particular, the concern is with the transition from strong ellipticity to loss of ellipticity, which is associated with the emergence of surfaces of discontinuity, interpreted as relating to fibre kinking and fibre splitting. For a special material model consisting of an isotropic neo-Hookean matrix together with a standard reinforcing model depending on the stretch in the direction of transverse isotropy, this problem has been treated by El Hamdaoui et al. [1], and for some pointers to the literature on the helical shear problem, we refer to this paper. We mention below only the limited list of works concerned with the analysis of ellipticity and associated discontinuous solutions for various specializations of the tube problem considered here, and its variations.

Fosdick and MacSithigh [2] studied the helical shear problem for an incompressible isotropic elastic material with an emphasis on the structure of the energy function, with particular reference to its non-convexity and the related emergence of equilibrium states with discontinuous deformation gradients. The azimuthal shear problem for an incompressible isotropic elastic material was studied by Abeyaratne [3] with a focus on loss of ellipticity and the emergence of discontinuous solutions, while for the anti-plane shear problem Silling [4] considered numerically the passage from ellipticity to hyperbolicity of the governing equations resulting from deformation of an incompressible isotropic material containing a crack or a screw dislocation. For an incompressible *transversely* isotropic elastic material associated with a single family of fibre directions, the problem of loss of strong ellipticity for the azimuthal shear problem was first studied by Kassianidis et al. [5], who examined, in particular, the emergence and disappearance of non-uniqueness of solution. This was extended to the case of two symmetrically arranged fibre families by Dorfmann et al. [6] and El Hamdaoui and Merodio [7].

In the present work, while we adopt the neo-Hookean model for the matrix material, in contrast to the analysis in [1], we consider a different reinforcing model for the description of the anisotropy, which leads to quite different results in general compared with those in [1]. The following sections contain a general set-up of the problem in Sect. 2, including the description of the considered geometry and deformation, the constitutive equations and loss of ellipticity condition. Then, in Sect. 3, the constitutive law is specialized and the effect of axial shear on the loss of ellipticity is analysed in some detail for the separate cases of radial and axial transverse isotropy. Finally, some concluding remarks are provided in Sect. 4.

## 2 Basic equations

### 2.1 Geometry and kinematics

*A*,

*B*, respectively, are its internal and external radii and

*L*its length. Material points in the reference configuration are labelled by their position vector \(\mathbf {X}\), which is defined by \(\mathbf {X}=R\mathbf {E}_R+Z\mathbf {E}_Z\), where \((\mathbf {E}_R,\mathbf {E}_Z)\) are the unit basis vectors associated with (

*R*,

*Z*).

*r*(

*R*),

*g*(

*R*) and

*w*(

*R*) are unknown functions corresponding to radial deformation, azimuthal rotation and axial displacement, respectively. In principle, these can be determined from the solution of the equilibrium equations and boundary conditions. The corresponding position vector \(\mathbf {x}\) in the deformed configuration is given by \(\mathbf {x}=r\mathbf {e}_r+z\mathbf {e}_z\), where \((\mathbf {e}_r,\mathbf {e}_z)\) are the unit basis vectors associated with (

*r*,

*z*). In the following we use the notations

*R*. Then, by (4),

*r*(

*R*) is determined explicitly and the two functions

*g*(

*R*) and

*w*(

*R*) remain to be determined.

*preferred direction*, defined in the reference configuration and denoted here by the unit vector \(\mathbf {A}\), in general dependent on \(\mathbf {X}\). The (transversely isotropic) invariants are denoted here by \(I_4\) and \(I_5\) and defined by

### 2.2 Constitutive equations and equilibrium

*p*is a Lagrange multiplier associated with the constraint (4), \(\mathbf {I}\) is the identity tensor and \(\mathbf {F}^{-\mathrm {T}}=(\mathbf {F}^{-1})^{\mathrm {T}}=(\mathbf {F}^{\mathrm {T}})^{-1}\).

*W*depends on \(\mathbf {F}\) and \(\mathbf {A}\) through the combined invariants of \(\mathbf {C}\) and \(\mathbf {A}\) given in (9) and (10). Thus, \(W=W(I_1,I_2,I_4,I_5)\) and, from (11)\(_2\), it follows that

*p*. These restrictions were originally given in [8].

*p*. The azimuthal and axial equations are integrated immediately to give

### 2.3 Ellipticity

*strongly elliptic*if

*strong discontinuity*) or the deformation gradient is continuous but its gradient is discontinuous (a

*weak discontinuity*). The possible emergence of such a surface as the deformation proceeds is associated with the loss of (strong) ellipticity.

## 3 Application to \(I_5\) reinforcement

*W*is significant since, while \(I_4\) is the square of the stretch in the preferred direction, the invariant \(I_5\) involves not only the stretch but also shears in directions perpendicular to the preferred direction. The expression (13) for the Cauchy stress tensor then specializes to

*W*with respect to the deformation gradient \(\mathbf {F}\) in (19). This is given by

### 3.1 Radial fibres

Next, we analyse how the loss of ellipticity surface changes under an applied axial shear. For this purpose, we also illustrate this surface in the absence of any shear deformation \((\gamma _z=\gamma _\theta =0)\) for specific values of the reinforcing parameter \(\rho \), as a reference for comparing the solutions obtained when the shear deformation \(\gamma _z\) is involved.

*R*. Some comments on this non-homogeneous deformation are reserved for the concluding remarks in Sect. 4. Note that (36) and (38) are special cases of (39).

*r*,

*z*) plane, one with \(\chi > \pi /2\) and the other with \(\chi < \pi /2\). Also, a discontinuity surface parallel to the fibre with \(\chi =0\) (or \(\pi \)) is possible, which is interpreted as corresponding to fibre splitting, but, as already mentioned, loss of ellipticity with \(\chi =\pi /2\) always occurs before that with \(\chi =0\). Note that Fig. 4b is equivalent to Fig. 4a for \(\lambda _r\in [1/3,1]\) and shows \(\chi \) plotted against \(\lambda _r=\lambda _z^{-1}\) instead of \(\lambda _z\).

Figure 6 provides a selected alternative view of the results in Fig. 5 for \(\lambda _z=1.2\), and illustrates the symmetry \((\gamma _z,\chi )\leftrightarrow (-\gamma _z,\pi -\chi )\). For sufficiently large \(|\gamma _z|\) the deformation is strongly elliptic, but as the value of \(|\gamma _z|\) is reduced ellipticity is lost for two values of \(\chi \), one for positive \(\gamma _z\) and one for the negative value of equal magnitude, and the associated values of \(\chi \) are symmetric with respect to \(\chi =\pi /2\). It is clear that a non-zero \(\gamma _z\) delays the onset of ellipticity loss, with the fibre splitting mode (\(\chi =0\) or \(\chi =\pi \)) possible at a smaller value of \(|\gamma _z|\) than fibre kinking (\(\chi =\pi /2\)). This effect is shown in Fig. 5.

*r*,

*z*) plane, it can be shown that \(\gamma _\theta =0\) is a necessary consequence of (23) except for very special values of \(\chi \) and/or the deformation. This can be shown by expressing \(\bar{\mathbf {Q}}\) in the form

*f*is a known function of its arguments.

### 3.2 Axial fibres

In this subsection, we consider the fibres to be purely axial, so that \(\mathbf {A}=\mathbf {E}_Z\). Figure 7 illustrates the deformed configuration of a single fibre when the deformation (2) creates fibre contraction, again with \(\mathbf {m}\) and \(\mathbf {n}\) lying in the (\(\mathbf {e}_r,\mathbf {e}_z\)) plane.

In the absence of shear deformation \((\gamma _z =\gamma _\theta =0)\), the loss of ellipticity condition can be obtained from (39) by interchanging the roles of \(\lambda _r\) and \(\lambda _z\), and, bearing in mind the different ranges of values of the angle \(\chi \), plots equivalent to those in Figs. 3 and 4 can be obtained but are not included here.

*r*,

*z*) plane, it follows that \(\bar{\mathbf {Q}}\) can be written in the form (40), but in this case the coefficients \(\alpha \), \(\beta \) and \(\gamma \) are somewhat simpler. In particular, \(\beta \) is given by

## 4 Concluding remarks

The results in Sect. 3 serve to illustrate how axial shear affects the onset of loss of ellipticity when superimposed on axial extension and radial deformation in a fibre-reinforced elastic solid for the situations in which the fibre direction is either radial or axial and there is no azimuthal shear. Very similar results can be obtained for the case of azimuthal shear in the absence of axial shear with the vectors \(\mathbf {m}\) and \(\mathbf {n}\) in the \((r,\theta )\) plane instead of the (*r*, *z*) plane. For other fibre orientations, the resulting loss of ellipticity condition is quite complicated, and its analysis requires a separate and purely numerical approach.

The model for the fibre reinforcement adopted here was taken to depend on the invariant \(I_5\) and leads to results that are significantly different from those in [1], where an \(I_4\)-based model was used. In particular, while the loss of ellipticity associated with the \(I_4\) model admits only discontinuities related to fibre kinking, the \(I_5\) model may be related to both fibre kinking and fibre splitting, and, in the case of kinking, the discontinuities here develop differently from those in [1]. The main difference between the \(I_4\) and \(I_5\) invariants themselves, which influences the loss of ellipticity results through the strain-energy function, is that the former involves only fibre stretch while the latter relates to both fibre stretch and, in general, shearing, although it may be misleading to suggest that fibre splitting is directly related to shearing alone. Consideration of a strain-energy function which combines \(I_4\) and \(I_5\) would provide an obvious extension of the analysis considered here and in [1], although necessarily more complicated, and it would be of interest to investigate this further.

In the numerical examples in Sect. 3 we have assumed that \(\lambda _\theta =1\), i.e. \(r=R\), which means that, in the absence of shear, the deformation of the tube is homogeneous. When an axial shear is in place it depends on *r* and, depending on the constitutive law, it may result in the loss of ellipticity surface located at different values of \(r\in (a,b)\), with \(\gamma _z\) having different values on either side and part of the tube in the strongly elliptic regime. This effect is modified to some extent when \(\lambda _\theta \ne 1\) and the deformation is inhomogeneous in the absence of shear.

## Notes

### Acknowledgements

JM and RWO acknowledge the support by the Ministry of Economy in Spain, under the project reference DPI2014-58885-R.

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