Loss of ellipticity in the combined helical, axial and radial elastic deformations of a fibre-reinforced circular cylindrical tube

Accepted Manuscript Loss of ellipticity in the combined helical, axial and radial elastic deformations of a fibre-reinforced circular cylindrical tube Mustapha El Hamdaoui, José Merodio, Ray W. Ogden PII: DOI: Reference:

S0020-7683(15)00097-9 http://dx.doi.org/10.1016/j.ijsolstr.2015.02.043 SAS 8684

To appear in:

International Journal of Solids and Structures

Received Date: Revised Date:

26 June 2014 19 February 2015

Please cite this article as: Hamdaoui, M.E., Merodio, J., Ogden, R.W., Loss of ellipticity in the combined helical, axial and radial elastic deformations of a fibre-reinforced circular cylindrical tube, International Journal of Solids and Structures (2015), doi: http://dx.doi.org/10.1016/j.ijsolstr.2015.02.043

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Loss of ellipticity in the combined helical, axial and radial elastic deformations of a fibre-reinforced circular cylindrical tube 1

Mustapha El Hamdaoui, 1 Jos´e Merodio, 2 Ray W. Ogden 1

Department of Continuum Mechanics and Structures E.T.S. Ingenieros Caminos, Canales y Puertos

Universidad Polit´ecnica de Madrid, 28040 Madrid, Spain 2

School of Mathematics and Statistics, University of Glasgow Glasgow G12 8QW, UK

Abstract In this paper we consider theoretically the finite deformation of a circular cylindrical tube of a transversely isotropic elastic material, specifically the combined axial stretch, inflation and helical shear deformation, with particular reference to the failure of ellipticity. For a simple form of strain-energy function specific examples involving axial and radial directions of transverse isotropy are then considered, leading to different predictions of the onset of ellipticity failure.

1

1

Introduction

The aim of this work is to analyze the equilibrium configurations of a tube of transversely isotropic hyperelastic material subject to combined axial compression (or extension), inflation and helical shear deformations for which the governing differential equation varies in type locally from strongly elliptic to non-elliptic, or vice versa, as the deformation proceeds. This change is associated with the possible emergence of surfaces of weak discontinuity in the deformation, i.e. surfaces on which certain second derivatives of the deformation are discontinuous, sometimes referred to as weak solutions, or strong discontinuity where first derivatives of the deformation are discontinuous. For definiteness in this paper we shall use the terminology ‘weak solutions’ in referring to the discontinuities. This analysis has been motivated by instability phenomena in fibre-reinforced composite materials. In particular, the material under consideration is an isotropic neo-Hookean base (or matrix) material augmented by an energy function that accounts for the existence of fibre reinforcement, and in this work we will deal in particular with the so-called standard model of reinforcement. The loss of ellipticity of the governing differential equations for the considered material is interpreted in terms of fibre failure. The helical shear problem has been studied by many authors from several points of view in the case of an isotropic material, starting from the pioneering work of Rivlin (1949). These include the study of combined axial and azimuthal shear of a circular cylindrical tube of incompressible isotropic elastic material by Ogden et al. (1973), in which some universal relations between the stress components were provided, and the works of Beatty and Jiang (1999) and Kirkinis and Ogden (2003), which were concerned with compressible materials capable of supporting helical shear. Horgan and Saccomandi (2003) investigated different constitutive models that account for hardening at large deformations in the case of a circular cylindrical tube composed of an incompressible hyperelastic material. None

2

of these papers were concerned with the loss of ellipticity, but, by contrast, Fosdick and MacSithigh (1983) provided a detailed study of helical shear with emphasis on the structure of the energy function and its convexity, with particular reference to a non-convex energy function and the emergence of equilibrium configurations with discontinuous deformation gradients. In the case of an anisotropic material the problem of helical shear has barely been studied, although Jiang and Beatty (2001) derived a necessary and sufficient condition for the strain-energy function to admit helical shear deformations for a compressible, anisotropic hyperelastic circular tube, considering transverse isotropy as a special case. Again, loss of ellipticity was not considered. In terms of loss of ellipticity, Abeyaratne (1981) investigated the emergence of solutions involving discontinuous deformation gradients associated with loss of ellipticity in the finite twisting of an incompressible isotropic elastic tube, while more recently Kassianidis et al. (2008) and Gao and Ogden (2008), from different perspectives, have analyzed the problem of azimuthal shear of a circular cylindrical tube of incompressible transversely isotropic elastic material where loss of strong ellipticity and the emergence of discontinuous (or non-smooth) solutions was examined. Dorfmann et al. (2010) similarly studied the azimuthal shear problem, but with the anisotropy associated with two symmetrically disposed preferred material directions. The problem of a tube of transversely isotropic elastic material subject to radial and axial deformations coupled with torsion was examined recently by El Hamdaoui et al. (2014) but without reference to loss of ellipticity. The discussion above relates to deformations of a thick-walled tube. For azimuthal shear, in particular, which is a plane strain deformation, the emergence, development and disappearance of weak surfaces is a feature of the analysis in Kassianidis et al. (2008), Gao and Ogden (2008) and Dorfmann et al. (2010). This is also the case for the threedimensional helical shear problem which is analyzed herein. 3

Similar phenomena also arise for the problem of rectilinear shear, which has been studied by Merodio et al. (2007), Destrade et al. (2009) and Baek and Pence (2010). In particular, Merodio et al. (2007) examined the existence of discontinuous solutions associated with fibre kinking for different fibre orientations for the (non-homogeneous) rectilinear shear of a finite thickness slab of transversely isotropic elastic material between two rigid plates. For the standard reinforcing model they obtained a closed-form expression for the amount of shear as a function of the through-thickness coordinate and they showed that fibres were subject to contraction on both sides of a singularity (a kink surface). Destrade et al. (2009) studied the same problem but with two distinct families of fibres with the shear direction bisecting the directions of the two fibre families. They showed that if the two fibre families have the same mechanical properties then no singularities can arise, but that singularities can develop when one fibre family is significantly stiffer than the other. The paper by Baek and Pence (2010) is concerned with a transversely isotropic material based on the standard reinforcing model, first under simple shear (homogeneous) and then subject to rectilinear shear (inhomogeneous). For simple shear they analyzed in detail the effect of fibre orientation on the emergence, development and disappearance of singular surfaces (kink surfaces) as the shear stress is applied. They went on to extend their analysis to the rectilinear shear problem and found, in particular, that pairs of singular surfaces were nucleated at a critical value of the shear stress and then annihilated at a second critical value, this being associated with a non-monotonic shear stress shear amount of shear response. In Section 2 we provide a summary of the basic ingredients of the kinematics and nonlinear elasticity theory, with particular reference to transversely isotropic materials and the loss of ellipticity condition. This is then applied, in Section 3, to a reduced form of the transversely isotropic constitutive law involving one isotropic and one transversely isotropic invariant with two examples of fibre distributions – axial and radial – and it is 4

shown how the emergence or disappearance of singular surfaces depends of the geometrical parameters, the deformation and the strength of the anisotropy. Finally, in Section 4 some concluding remarks are made.

2

Problem formulation

2.1

Kinematics and constitutive laws

We consider a circular cylindrical tube with an undeformed and stress-free reference configuration defined by

A ≤ R ≤ B,

0 ≤ Θ ≤ 2π,

0 ≤ Z ≤ L,

(1)

where (R, Θ, Z) are cylindrical polar coordinates with associated unit basis vectors (ER , EΘ , EZ ). The position vector, denoted X, of a material point in this configuration is given by X = RER + ZEZ relative to an origin on the tube axis. The deformation of the cylinder is described by the equations

r = r(R),

θ = Θ + g(R),

z = λz Z + w(R),

(2)

where (r, θ, z) are cylindrical coordinates with unit basis vectors (er , eθ , ez ) and X becomes x = rer + zez (with the same origin). The constant λz is the axial stretch of the cylinder, and g(R) and w(R) are unknown azimuthal and axial displacement functions to be determined from the solution of the equilibrium equations and boundary conditions. The deformation gradient tensor is denoted F and given by Gradx, where Grad is the gradient operator with respect to X. We assume that the material is incompressible, so

5

that the constraint det F = 1

(3)

is satisfied. For the considered deformation F is given by

F=

∂x 1 ∂x ∂x ⊗ ER + ⊗ EΘ + ⊗ EZ , ∂R R ∂Θ ∂Z

(4)

and is calculated explicitly as

F = (λr er + γθ eθ + γz ez ) ⊗ ER + λθ eθ ⊗ EΘ + λz ez ⊗ EZ ,

(5)

where λr = r0 (R) is the radial stretch, λθ = r/R is the azimuthal stretch, γθ = rg 0 (R) and γz = w0 (R), the prime indicating differentiation with respect to R. Then, by incompressibility, λr = (λz λθ )−1 ,

(6)

2 2 r2 = a2 + λ−1 z (R − A ),

(7)

and

where a = r(A) is the deformed inner radius of the tube, and we adopt the notation b = r(B) for the outer deformed radius. The right Cauchy–Green deformation tensor C = FT F is given by C = (γz2 + γθ2 + λ2r )ER ⊗ ER + λ2θ EΘ ⊗ EΘ + λ2z EZ ⊗ EZ + γθ λθ (ER ⊗ EΘ + EΘ ⊗ ER ) + γz λz (ER ⊗ EZ + EZ ⊗ ER ),

6

(8)

and the left Cauchy–Green deformation tensor FFT by B = λ2r er ⊗ er + (γθ2 + λ2θ ) eθ ⊗ eθ + (λ2z + γz2 ) ez ⊗ ez + γθ λr (er ⊗ eθ + eθ ⊗ er ) + γz λr (er ⊗ ez + ez ⊗ er ) + γθ γz (eθ ⊗ ez + ez ⊗ eθ ).

(9)

We consider an incompressible elastic material with strain-energy function W (F) per unit volume, and by objectivity it depends on F only through C. The nominal and Cauchy stress tensors S and σ are given by

S=

∂W − pF−1 , ∂F

σ=F

∂W − pI, ∂F

(10)

where p is a Lagrange multiplier associated with the incompressibility constraint and I is the identity tensor. In this paper we are concerned with a transversely isotropic material with the direction of transverse isotropy denoted by the unit vector A in the reference configuration. This can be thought of an isotropic matrix material reinforced by a single family of fibres (with A the local fibre direction), although this is not essential. For such a material W can be expressed in terms of four invariants in the incompressible case, and in standard notation these are typically taken to be

I1 = trC,

1 I2 = [I12 − tr(C2 )], 2

I4 = A · (CA),

I5 = A · (C2 A),

(11)

where (by incompressibility) I3 ≡ det C = 1 has been omitted. In general, A depends on position X. With W = W (I1 , I2 , I4 , I5 ) the Cauchy stress (10)2 expands out in the standard general form σ = 2W1 B + 2W2 (I1 B − B2 ) + 2W4 a ⊗ a + 2W5 (a ⊗ Ba + Ba ⊗ a) − pI, 7

(12)

where a = FA, which, in general, is not a unit vector, and Wi = ∂W/∂Ii , i = 1, 2, 4, 5. We shall be using specific examples of this constitutive law in the body of this paper. For background on the formulation of equations for fibre-reinforced materials we refer to the work of Spencer (1972, 1984), Holzapfel (2001) and Ogden (2003), for example. For the considered deformation, we have I1 = γθ2 + γz2 + λ2r + λ2θ + λ2z ,

(13)

a = AR λr er + (AR γθ + AΘ λθ )eθ + (AR γz + AZ λz )ez

(14)

I4 = a · a = (γθ AR + λθ AΘ )2 + (γz AR + λz AZ )2 + λ2r A2R ,

(15)

and

where AR , AΘ , AZ are the components of A with respect to the basis (ER , EΘ , EZ ). Expressions for I2 and I5 are more lengthy and are not needed in the following. In the reference configuration, we have I1 = I2 = 3 and I4 = I5 = 1. Then, assuming the reference configuration is stress free, from (12) the restrictions (Merodio and Ogden, 2002) 2W1 + 4W2 − p = 0,

W4 + 2W5 = 0,

(16)

are obtained, all terms of which are evaluated in the reference configuration.

2.2

Equilibrium and ellipticity

In the absence of body forces the equilibrium equation can be expressed in the two equivalent forms DivS = 0,

divσ = 0,

8

(17)

where Div and div are the divergence operators with respect to X and x, respectively. We are interested in situations where the character of the equilibrium equation changes from strongly elliptic to non-elliptic. For this purpose we write the first of these equations in (Cartesian) component form as ∂ ∂Xα



∂W −1 − pFαi ∂Fiα



≡ Aαiβj xj,αβ − p,i = 0,

(18)

where Roman indices are associated with the deformed configuration (x has Cartesian components xi , i = 1, 2, 3) and Greek indices with the reference configuration (X has Cartesian components Xα , α = 1, 2, 3), a subscript following a comma stands for differentiation with −1 respect to the relevant coordinate, Fαi is defined as (F−1 )αi , we have used the identity −1 (Fαi ),α = 0, and Aαiβj are the components of the elasticity tensor defined by

Aαiβj =

∂ 2W . ∂Fiα ∂Fjβ

(19)

In these differentiations the components Fiα are regarded as independent, the incompressibility being taken care of by the Lagrange multiplier. The usual summation convention applies for repeated indices. Associated with Aαiβj is the so-called acoustic tensor Q(n), the components of which are defined by Qij = Fpα Fqβ

∂ 2W np nq . ∂Fjβ Fiα

(20)

Evaluation of the elliptic character of the equations is determined by the properties of Q(n). First, we note that the strong ellipticity condition has the form

[Q(n)m] · m > 0 for all non–zero m and n such that m · n = 0,

9

(21)

the latter restriction arising from incompressibility. Typically, m and n can be taken to be unit vectors. For some background discussion on the notion of strong ellipticity we refer to section 6.4 of Ogden (1997). Strong ellipticity is lost if there are (non-zero) real m and n satisfying the eigenvalue problem [Q(n) − n ⊗ Q(n)n]m = 0 and hence [Q(n)m] · m = 0.

(22)

Note that ellipticity requires only [Q(n)m] · m 6= 0 and that strong ellipticity is a special case of this. In the literature loss of ellipticity (for example, in Qiu and Pence, 1997b) is sometimes referred to as loss of ordinary ellipticity. However, since we will assume that the transition is from strong ellipticity we refer only to loss of ellipticity. We note that Q(n) − n ⊗ Q(n)n is not in general symmetric. However, because of incompressibility, the eigenvalue problem can be cast as a two-dimensional problem by ¯ = ¯IQ¯I, the projection of Q on to the plane normal to n, where ¯I = I − n ⊗ n defining Q is the projection operator. This yields the two-dimensional eigenvalue problem ¯ Q(n)m =0

(23)

¯ in the plane normal to n, where Q(n) is symmetric. This equation identifies admissible values of n and m. The unit normal n is a solution (not necessarily unique) of the 2 × 2 determinantal equation ¯ = 0, det Q

(24)

and once n is determined (when it exists) m is found from (23) subject, of course, to m · n = 0. At this point we note that for the considered deformation the equilibrium equation

10

(17)2 reduces to the three scalar equations d 1 (σrr ) + (σrr − σθθ ) = 0, dr r

d 2 (r σrθ ) = 0, dr

d (rσrz ) = 0. dr

(25)

The first of these is used to essentially determine the radial stress σrr (equivalently the Lagrange multiplier p) when suitable boundary conditions are specified on the lateral boundaries of the tube. The second and third equations can be integrated as σrθ = c1 /r2 ,

σrz = c2 /r,

(26)

where c1 and c2 are constants related to the shear stresses on the lateral boundaries. These solutions will be used in the next section. Explicit expressions for σrr , σθθ and σzz can be obtained by specializing (12), but are not needed in this paper. They are just required to maintain the specified deformation.

3

Application to I4 reinforcement

We now specialize the constitutive law so that the strain energy depends only on I1 and I4 , i.e. W = W (I1 , I4 ). The Cauchy stress (12) then simplifies to

σ = 2W1 B + 2W4 a ⊗ a − pI,

(27)

while the restrictions (16) give, in the reference configuration,

2W1 − p = 0,

11

W4 = 0.

(28)

Next we specialize the fibre direction to lie locally in the (R, Z) plane, so that

A = AR ER + AZ EZ ,

where A2R + A2Z = 1,

(29)

and the expression (15) for I4 reduces to I4 = (γθ2 + λ2r )A2R + (γz AR + λz AZ )2 .

(30)

We note, in particular, that for the considered specialization σrθ and σrz are given by σrθ = 2(W1 + W4 A2R )λr γθ ,

σrz = 2[(W1 + W4 A2R )γz + W4 AR AZ λz ]λr ,

(31)

and in conjunction with (26) provide two coupled nonlinear algebraic equations that in principle can be used to determine γθ and γz . In general these solutions may not be unique, as, for example, in the azimuthal shear problem discussed in Kassianidis et al. (2008) and Gao and Ogden (2008).

3.1

Loss of Ellipticity

The second derivative of W with respect to the deformation gradient (19) has a relatively simple form since the invariants I2 and I5 are absent. It is given by ∂ 2W = 2W1 δij δαβ + 2W4 δij δrβ Ar As + 4W44 Fir Fjs Ar Aα As Aβ , ∂Fjβ Fiα

(32)

and, by use of (20), we obtain the corresponding acoustic tensor in the form Q = 2W1 [(Bn) · n] I + 2W4 (a · n)2 I + 4W44 (a · n)2 a ⊗ a,

12

(33)

where W44 = ∂ 2 W/∂I42 . Then ¯ = α¯I + β¯ Q a⊗¯ a,

(34)

where α = 2W1 n · (Bn) + 2W4 (a · n)2 ,

β = 4W44 (a · n)2 ,

(35)

¯ a = ¯Ia, and we note that m · a = m · ¯ a and n · ¯ a = 0. Only values of n satisfying (24) are admissible, and in respect of (34) this equation reduces to α(α + β¯ a·¯ a) = 0.

(36)

Thus, either α = 0 or α + β¯ a·¯ a = 0, and these two possibilities will in general determine different values of n. From equation (23), by taking the scalar product with m and m × n in turn, we obtain two scalar equations, both of which must hold: α + β(m · a)2 = 0,

β(m · a)[(m × n) · a] = 0.

(37)

First, if α = 0 then it follows from (37) that

m · a = 0 or β = 0.

(38)

Second, if α + β¯ a·¯ a = 0 then either β = 0 or (m · a)2 = ¯ a·¯ a and (m · a)[(m × n) · a] = 0.

(39)

Note that β = W44 (n · a)2 vanishes if either W44 = 0 or n · a = 0. We exclude the first of these by setting W44 > 0, as will be justified shortly in considering a particular form 13

of strain-energy function. Essentially, this condition is sufficient to ensure that increasing extension in the direction of transverse isotropy is accompanied by an increasing uniaxial stress in the same direction. To illustrate these results we consider two special examples of the direction of transverse isotropy, axial (A = EZ ) and radial (A = ER ). 3.1.1

Example 1: fibres with A = EZ

Consider now the special case in which the fibres are aligned initially along the axis of the tube, so that AR = 0, AZ = 1. Then a = λz ez and, considering m and n to lie in the (r, θ) plane, we may write

m = cos χ er − sin χ ez ,

n = sin χ er + cos χ ez ,

(40)

where the angle χ ∈ [−π/2, π/2], when it exists, defines (locally, i.e. as a function of r) a curve of weak discontinuity on which ellipticity has failed. The invariant I4 in (30) reduces to I4 = a · a = λ2z ,

(41)

and the fibres are not therefore affected by either the helical shear or radial deformation. Also, we have

¯ a = a − (n · a)n = −λz sin χm,

m · a = −λz sin χ,

n · a = λz cos χ.

(42)

In this case the shear stresses (31) reduce to

σrθ = 2W1 λr γθ ,

σrz = 2W1 λr γz ,

14

(43)

which are indistinguishable from those arising for an isotropic material when W1 is independent of I4 , i.e. the axial fibres do not influence the shear response. By combining (43) with (26) we obtain γθ /γz = c1 /c2 r,

(44)

which is the same as that arising for pure helical shear of an isotropic elastic material (Ogden et al., 1973), for which λz = 1 and r = R. Thus, as in the case of pure helical shear, γθ and γz are not independent. If α = 0 it follows from (38) that either χ = 0 or χ = π/2 (independently of r), which may be interpreted, for example, as related to fibre breaking (the line of discontinuity is perpendicular to the direction of transverse isotropy) or splitting (parallel to the direction of transverse isotropy). On the other hand, if α + β¯ a·¯ a = 0 the equations in (39) are automatically satisfied and this condition can be written explicitly as W1 [λ2r s2 + (λ2z + γz2 )c2 + 2λr γz sc] + W4 λ2z c2 + 2W44 λ4z s2 c2 = 0,

(45)

where, for compactness, we have introduced the notations s = sin χ, c = cos χ. Now, for purposes of illustration, we specialize to a prototype model consisting of an isotropic neo-Hookean base (matrix) material augmented with the standard reinforcing model (see, for example, Qiu and Pence, 1997a and references therein). The strain-energy function is given by 1 W = µ[(I1 − 3) + ρ(I4 − 1)2 ], 2

(46)

in which the constant µ (> 0) is the shear modulus of the base material and the constant ρ (> 0) is a measure of the strength of the anisotropy introduced by the preferred direction. In general both µ and ρ could depend on X, but here we assume that they are constants.

15

It follows that W4 = µρ(I4 − 1),

2W1 = µ,

W44 = µρ.

(47)

In particular, we note that W1 > 0, W44 > 0, but W4 can be positive or negative, depending on whether the fibres are extended or contracted. In this problem this corresponds to λz > 1 or λz < 1. Now, from (43) and (26) and the definitions γθ = rg 0 (R) and γz = w0 (R) we obtain the equations rg 0 (R) =

c1 λz λθ , µr2

w0 (R) =

c2 λz λθ , µr

(48)

which are now decoupled equations from which we may determine g(R) and w(R) separately. By using (7) and λθ = r/R these equations can be integrated to give, assuming g(A) = w(A) = 0,

g(R) =

c1 λ2z log(λa /λθ ), µ(λz a2 − A2 )

w(R) =

c2 λ z log(R/A), µ

(49)

where λa = a/A. For the pure helical shear problem, for which λz = 1 and r = R, these reduce to (Ogden et al., 1973) c1 g(r) = 2µ



1 1 − 2 2 a r



,

w(r) =

c2 log(r/a), µ

(50)

the first requiring taking of the limit a → A after setting λz = 1. On use of (47) the loss of ellipticity condition (45) reduces to λ2r s2 + (λ2z + γz2 )c2 + 2λr γz sc + 2ρ(λ2z − 1)λ2z c2 + 4ρλ4z s2 c2 = 0,

(51)

which, in particular, is quadratic in γz . This defines the deformation, through λθ , λz −1 (λr = λ−1 θ λz ) and γz , the angle χ and the anisotropy parameter ρ for which a curve (and

16

a surface of revolution) of discontinuity can exist for fibres that are initially distributed locally in the (R, Z)−plane parallel to the EZ direction. This is a local condition and we note that in general λr , γz and χ depend on r. This leads to a curve of discontinuity which is locally parallel to the unit vector m and has normal n in the considered plane. Note, in particular, that the possible appearance of weak solutions does not depend on the azimuthal shear γθ with this distribution of fibres. It is worth mentioning here that if (γz , sin χ) is a solution of (51), then so is (−γz , − sin χ), not surprisingly since there is symmetry about the axial direction. Without reference to dependence on r we now consider some properties of equation (51). If λθ = λz = 1, so the deformation is pure helical shear, (51) reduces to γz2 c2 + 2γz sc + 4ρs2 c2 + 1 = 0,

(52)

which has no real solution for γz . Thus, the pure helical shear deformation does not admit weak solutions (at least for the considered simple model). In general equation (51) admits real solutions for γz if and only if s2 ≤

−2 2ρ(λ−2 z − 1) − λz , 4ρ

which requires λz < 1 and also λz ≤

r

1−

1 . 2ρ

(53)

(54)

The latter recovers the restriction on λz found by Merodio and Pence (2001) as a necessary condition for the appearance of an orthogonal weak solution (χ = 0). Note that this is independent of the azimuthal stretch λθ . Satisfaction of the equality (54) requires that ρ > 0.5, as also does (51) in the case of pure compressive loading, in agreement with Qiu and Pence (1997b) in which it was shown

17

that, under plane strain, ρ > 0.5 is a necessary condition for the loss of ellipticity for a that, under plane strain, ρ > 0.5 is a necessary condition for the loss of ellipticity for a that, under plane strain, ρ > 0.5 is a necessary condition for the loss of ellipticity for a reinforced neo-Hookean material. Figure 1 provides a plot of the boundary of the region reinforced neo-Hookean material. Figure 1 provides a plot of the boundary of the region reinforced neo-Hookean material. Figure 1 provides a plot of the boundary of the region defined in (54). Strong ellipticity holds in the region above the curve. defined in (54). Strong ellipticity holds in the region above the curve. defined in (54). Strong ellipticity holds in the region above the curve. 1.0 1.0 0.8

λz0.8 λz0.6 0.6

0.4 0.4 0.2 0.2 0.0 0.00 0

2 2

4 4

6 6

ρ ρ

8 8

10 10

12 12

Figure in in (54). z zversus Figure1.1: Plot Plotofofthe thecurve curveofofλλ versusρ ρcorresponding correspondingtotoequality equality (54). Figure 1: Plot of the curve of λz versus ρ corresponding to equality in (54).

In Fig. 2 and Fig. 3, respectively, we illustrate the curves corresponding to loss of

(a) in the (λ , χ) and (γ , χ) planes based on(b) ellipticity equation (51). In Fig. 2(a) and Fig. z z 1.0 1.0

2(b) the plots in the (λz , χ) plane correspond to γz = 0 and γz = 1, respectively, with the0.5following values of ρ: 1, 3, 9, 30, corresponding 0.5 to relatively weak to relatively strong χ χ reinforcement. In Fig. 3 the plots are in the (γz , χ) plane for the representative value ρ = 9 0.0

0.0

with λz = 0.6, 0.7, 0.85, 0.95. In each case λθ has been set at 1 because λθ does not have a -0.5

-0.5 significant influence on the loss of ellipticity condition for the considered fibre distribution,

in the sense that the shapes of the curves are-1.0 essentially unaffected by a change in the -1.0

value0.0 of λθ . 0.2 In these0.4figures0.6and subsequently expressed 0.8 1.0 χ is 0.0 0.2 in radians. 0.4 0.6 0.8 1.0 λz λz In Fig. 4, for ρ = 9, we illustrate how the loss of ellipticity limit in the (λz , χ) plane

Figure 2: Plots of positive the curves toofsolutions of (51): χasversus γz = 0 z for changes with both andcorresponding negative values γz . In particular, notedλin the (a) caption Figure of theλcurves corresponding to solutions of (51): χ versus λ for (a) γ =0 and (b)2:γzPlots = 1 with = 1 and ρ = 1, 3, 9, 30, respectively the continuous, dashed, z z thick θ and 1 with = 12(b), andCurves ρ =curves 1, for 3, 9,negative 30,negative respectively thejust continuous, dashed, thick continuous and dotted γz are just the reflections of (in those of Fig.(b)2, γin respect ofλFig. the for γz are the reflections thefor z = θ curves. continuous and positive γz in thedotted axis χcurves. = 0. Curves for negative γz are just the reflections of those for χpositive = 0 axis) of the the axis corresponding curves for positive γz . Clearly, as the magnitude of γz γz in χ = 0. In Fig. 2 and Fig. 3, respectively, we illustrate the curves corresponding to loss of 18 18

0.0 0

2

4

6

8

ρ

10

12

Figure 1: Plot of the curve of λz versus ρ corresponding to equality in (54).

(a)

(b)

1.0 1.0 0.5

0.5

χ

χ 0.0

0.0

-0.5

-0.5

-1.0 -1.0 0.0

0.2

0.4

λz

0.6

0.8

1.0

0.0

0.2

0.4

λz

0.6

0.8

1.0

Figure 2. Plots of the curves corresponding to solutions of (51): χ versus λz for (a) γz = 0 Figure of theλcurves corresponding of (51): versus λz for (a) γz thick =0 and (b)2:γzPlots = 1 with ρ = 1, 3, 9,to 30,solutions respectively the χcontinuous, dashed, θ = 1 and and (b) γ = 1 with λ = 1 and ρ = 1, 3, 9, 30, respectively the continuous, dashed, thick continuousz and dottedθ curves. Curves for negative γz are just the reflections of those for continuous and positive γz in thedotted axis χcurves. = 0. Curves for negative γz are just the reflections of those for positive γz in the axis χ = 0. 1.0

18

0.5

χ 0.0 -0.5 -1.0 -4

-2

0

2

4

γz Figure 3. Plots of the curves corresponding to solutions of (51) with χ versus γz for λz = 0.6, 0.7, 0.85, 0.95, corresponding to the continuous, dashed, thick continuous and thick dotted curves, respectively. increases the initiation of weak solutions requires increased axial compression. For each value of ρ used here the weak solutions begin to appear starting from a value of λz reducing from 1 which satisfies equality in (54) and corresponding to χ = 0. It can be 19

(a)

(b)

1.5

1.5

1.0

χ

χ

1.0

0.5

0.5

0.0

0.0

-0.5

-0.5

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-1.5 0.0

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λz

0.6

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λz

(c)

(d)

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χ

χ

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χ

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-1.5 0.0

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λz

0.6

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λz

Figure 4. Plots of the curves corresponding to solutions of (51) with χ versus λz for ρ = 9 Figure of following the curvesvalues corresponding solutions of (51)(c) with χ versus λz for(e) ρ= 9 and λθ =4:1Plots for the of γz : (a)to±0.2; (b) ±0.7; ±1.2; (d) ±1.8; ±2.5; and λ = 1 for the following values of γ : (a) ±0.2; (b) ±0.7; (c) ±1.2; (d) ±1.8; (e) ±2.5; (f) ±5.θ The continuous (dashed) curvesz correspond to positive (negative) values of γz . (f) ±5. The continuous (dashed) curves correspond to positive (negative) values of γz .

20 20

seen from Fig. 2(a) that for larger values of ρ the region enclosed by the curve bounding the strongly elliptic region (which is outside the enclosed region) is more significant than for smaller values, i.e. the stronger the anisotropy the sooner weak solutions appear. Thus, for small values of ρ the mechanical properties of the material are dominated by the isotropic base material, in which case larger compressions are required for the initiation of weak solutions, while for larger values of ρ the anisotropic component dominates the mechanical behaviour of the composite and for weak solutions to appear a smaller compression is needed. We now consider the r dependence of equation (51). For this purpose we note that from equations (26)2 , with the constant c2 replaced by τ b, and (43)2 we have µγz = τ b/rλr , where τ is the axial shear stress applied on the outer boundary r = b of the tube. Also, by making use of the definition λθ = r/R and the incompressibility condition (6), equation (7) can be arranged in the form λ2r

=

λ−1 z



2 a2 − λ−1 z A 1− r2



.

(55)

By substituting for γz and λr into equation (51) we then have an equation from which we can elicit the dependence of χ on r for different choices of the parameter ρ, the axial stretch λz , the tube cross-sectional geometry (reference and deformed) and the axial shear stress. Note that there are no real solutions of (51) for λz ≥ 1. For numerical purposes we define the dimensionless quantities

η = B/A,

a ¯ = a/A,

¯b = b/A,

r¯ = r/A,

τ¯ = τ /µ,

(56)

noting that 2 ¯b2 = a ¯2 + λ−1 z (η − 1).

The range of values of r¯ is then [¯ a, ¯b].

21

(57)

(a)

(b)

0.5

0.5

χ

χ

0.0

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-0.5

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r¯

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r¯

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(d)

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χ

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r¯

3.0

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Figure5.5:Plots Plotsofofχχversus versusr¯r¯based basedon on equation equation (51) (51) with with ρρ = = 12, Figure 12, λ λzz = = 0.8, 0.8, ηη = = B/A B/A==3,3, 1.5and andτ¯ τ¯==0,0,0.7, 0.7,1.5, 1.5,2.1 2.1inin(a), (a),(b), (b), (c), (c), (d), (d), respectively. respectively. a ¯ a¯==1.5 Fig. 5,5,for forrepresentative representative values values of of these these quantities, quantities, we we illustrate InInFig. illustrate how how χχ depends depends Wetake takeρ ρ==12, 12,λλz==0.8, 0.8, ηη == 3, 3, a = 1.5 1.5 and and aa series series of ononr¯.r¯.We ¯a¯ = of values values of of τ¯τ¯,, specifically specifically z 0.7, 1.5,2.1. 2.1.The Thegeneral generalnature natureofofthe the solutions solutions follows follows the the same 0,0, 0.7, 1.5, same trend trend for for other other values values of these quantities. of these quantities. For τ¯ = 0, Fig. 5(a) shows two curves, symmetrically disposed with respect to χ = For τ¯ = 0, Fig. 5(a) shows two curves, symmetrically disposed with respect to χ = 0. These curves (actually straight lines in this case) correspond to two possible, but 0. These curves (actually straight lines in this case) correspond to two possible, but equivalent (because z and −z may be interchanged), curves with different orientations (at equivalent (because z and −z may be interchanged), curves with different orientations (at approximately ±30◦ to the radial direction). Figures 5(b)–(d) show how the curves change approximately ±30◦ to the radial direction). Figures 5(b)–(d) show how the curves change as γz is increased from 0. The symmetry is broken and the two curves evolve differently, as γz is increased from 0. The symmetry is broken and the two curves evolve differently, the discontinuities disappear from the inner part of the tube and eventually disappear from the discontinuities disappear from the inner part of the tube and eventually disappear from the tube altogether. 22 22

(a)

(b)

(c)

(d)

Figure 6: 6. Depiction Depiction of Figure of the the curves curves of of discontinuity discontinuity as as functions functionsofofthe thedeformed deformedtube tuberadius radius r for the following values of τ ¯ : (a) 0; (b) 0.7; (c) 1.5; (d) 2.1. In each figure a longitudinal r for the following values of τ¯: (a) 0; (b) 0.7; (c) 1.5; (d) 2.1. In each figure a longitudinal section of of the the tube section tube is is shown shown with with the the inner inner radius radius on on the the left left and andthe theouter outerradius radiuson onthe the right; the discontinuity curve is based on the centre of the tube section. right; the discontinuity curve is based on the centre of the tube section. the tube tube altogether. altogether. the For the the same For same values values of of the the quantities quantities used used for for Fig. Fig. 5,5, Fig. Fig. 66 shows shows aa schematic schematicofof the upper discontinuity curves themselves through the tube thickness as τ¯ increases. Note the upper discontinuity curves themselves through the tube thickness as τ¯ increases. Note that in Fig. 6(c) and (d) the orientation of the curves changes as χ changes from negative that in Fig. 6(c) and (d) the orientation of the curves changes as χ changes from negative to positive. By contrast, the corresponding curves for the lower branches in Fig. 5 (not to positive. By contrast, the corresponding curves for the lower branches in Fig. 5 (not shown) maintain negative values of χ for all r ∈ [a, b] as τ¯ increases. Results for negative shown) maintain negative values of χ for all r ∈ [a, b] as τ¯ increases. Results for negative values of τ¯ are merely obtained from those for positive values by reversing the sign of χ. values of τ¯ are merely obtained from those for positive values by reversing the sign of χ. Of course, the curves, which lie locally in the (r, z) plane, are cross sections of rotationally Of course, the curves, which lie locally in the (r, z) plane, are cross sections of rotationally symmetric surfaces of discontinuity. symmetric surfaces of discontinuity. 3.1.2 3.1.2

Example 2: fibres with A = ER Example 2: fibres with A = ER

We now consider the fibres to be purely radial, so that A = ER . We again define χ as the We now consider the fibres to be purely radial, so that A = ER . We again define χ as the angle between the vector m and the er direction, but because of the different orientation angle between the vector m and the er direction, but because of the different orientation

23 23

of the preferred direction m and n are given by of the preferred direction m and n are given by

m = cos χ er + sin χ ez , m = cos χ er + sin χ ez ,

n = − sin χ er + cos χ ez

n = − sin χ er + cos χ ez

(58) (58)

as distinct from (40).

as distinct from (40).

The invariant I4 in this case is given by The invariant I4 in this case is given by

−2 2 2 θ λ−2 z) , II44==γγz2 z++γθ2γθ++(λ(λ θ λz ) ,

(59) (59)

pp 2 2 γ 2 and and λθθ in in Fig. Fig. 77 for fortwo twodifferent differentvalues values γ = γθ2 + γθγ+ three andisisplotted plotted against against λ of of γ = z three z and

values In the the absence absenceofofhelical helicalshear shear = 0), I4 1 1 θ λzλ> 1),which which means means that that the thethe radial direction andand the the loss loss of of (λ(λ the fibres fibresare arecontracted contractedin in radial direction r r<<1), ellipticitymay may occur occur before before inflation farfar as as I4 is γθ and ellipticity inflationofofthe thetube tubebegins. begins.AsAs I4 concerned, is concerned, γθ and haveequivalent equivalent effects. effects. γzγzhave 2.5

2.0

I4 1.5

1.0

0.5

0.5

1.0

1.5

2.0

λθ

2.5

3.0

p p Figure7:7. Plot Plot of of II44 versus versus λλθθ for 0.8 γ = 0, 1, γ =γ =γθ2 +γγ2z2+ . γ 2. Figure for λλzz ==1.2, 1.2,1.0, 1.0, 0.8and and γ = 0, where 1, where z θ Thedashed, dashed,dotted dotted and and continuous 1.2, 1, 0.8, respectively. TheThe upper The continuouscurves curvesare areforforλzλ= = 1.2, 1, 0.8, respectively. upper z threecurves curvesare are for for γγ = = 00 and 1. 1. three and the thelower lowerthree threeforforγ γ= =

24

24

In this case the shear stresses (31) reduce to

σrθ = 2(W1 + W4 )λr γθ ,

σrz = 2(W1 + W4 )λr γz ,

(60)

which, in contrast to those in (43), exhibit dependence on W4 and hence are affected by the presence of the fibres. However, it is worth noting that the connection (44) also holds in this case, but the equations do not admit simple exact solutions (for the model (46) a cubic equation for γz is obtained as a function of r, for example). For the model (46), the loss of ellipticity condition (22)2 in this case may be written b4 γz4 + b3 γz3 + b2 γz2 + b1 γz + b0 = 0,

(61)

where we have defined the coefficients b4 = 2ρc2 (1 + 2s2 ),

b3 = 4ρsc(1 − 4s2 )λr ,

b2 = c2 [1 + 2ρ(γθ2 − 1)] + 6ρ(1 − 4s2 c2 )λ2r , b1 = −2sc[1 + 2ρ(γθ2 − 1) + 2ρ(3 − 4s2 )λ2r ]λr , b0 = s2 [1 + 2ρ(γθ2 − 1)]λ2r + 2ρs2 (1 + 2c2 )λ4r + c2 λ2z ,

(62)

and we have again used the notations s = sin χ, c = cos χ. In the absence of helical shear equation (61) reduces to b0 = 0, which, for χ = π/2, yields λθ =

λ−1 z

r

2ρ . 2ρ − 1

(63)

For a given axial extension λz , this is the limiting value of λθ for the loss of ellipticity corresponding to the orthogonal weak solution χ = ±π/2. It should also be noticed that ρ > 0.5 is required for loss of ellipticity in this case (Qiu and Pence, 1997b). 25

In equality (63) onon ρ for λz λ=z 1. As As InFig. Fig. 88we weillustrate illustratethe thedependence dependenceofofλθλsatisfying equality (63) ρ for = 1. θ satisfying mentioned larger ρ)ρ) thethe smaller thethe deformation mentionedbefore, before,the thestronger strongerthe theanisotropy anisotropy(i.e. (i.e.the the larger smaller deformation required, deformation required forfor lossloss of of required,while whilethe theweaker weakerthe theanisotropy anisotropythe thelarger largeris isthe the deformation required −1 ellipticity. λr λ=r = λ−1 λz λ=z 1) is is ellipticity. The Theregion regionofofstrong strongellipticity ellipticitylies liesbelow belowthe thecurve. curve.If If (for = 1) θ λθ(for

plotted in in Fig. 1. 1. plottedinstead insteadofofλλθ θthen thenthe thecurve curvetakes takesononthe theform formshown shown Fig. 1.6 1.5

λθ 1.4 1.3 1.2 1.1 1.0 5

10

ρ

15

20

Figure 8. Plot of λθ against ρ based on equation (63) for λz = 1. Strong ellipticity holds Figure 8: Plot of λθ against ρ based on equation (63) for λz = 1. Strong ellipticity holds below the curve. below the curve. Similarly to the case discussed in Section 3.1.1 we now present the curves corresponding Similarly to the case discussed in Section 3.1.1 we now present the curves corresponding to loss of ellipticity, with χ plotted against λr for a series of values of ρ in Fig. 9(a) with to loss of ellipticity, with χ plotted against λr for a series of values of ρ in Fig. 9(a) with γz = 0 and against γz for several values of λr with ρ = 9 in Fig. 9(b). In each case we γz = 0 and against γz for several values of λr with ρ = 9 in Fig. 9(b). In each case we have set λz = 1 and γθ = 0. We note that γθ was taken to be zero and λz was fixed at 1 have set λz = 1 and γθ = 0. We note that γθ was taken to be zero and λz was fixed at 1 because they do not have a significant influence on the qualitative nature of the results. because they do not have a significant influence on the qualitative nature of the results. Figure 9(a) shows plots of χ against the radial stretch λr for γz = 0, λz = 1 and Figure 9(a) shows plots of χ against the radial stretch λr for γz = 0, λz = 1 and different values of ρ, and it is similar to Fig. 2(a). When γz is changed the curves in 9(a) different values of ρ, and it is similar to Fig. 2(a). When γz is changed the curves in 9(a) change in a similar way to those shown in Fig. 2(b) so we do not show plots for different change in a similar way to those shown in Fig. 2(b) so we do not show plots for different values of γz is this case. For γz = 0 the only weak solution captured is the orthogonal values of γz is this case. For γz = 0 the only weak solution captured is the orthogonal solution corresponding to χ = ±π/2 and as the magnitude of γz is increased the weak solution corresponding to χ = ±π/2 and as the magnitude of γz is increased the weak solution becomes non-orthogonal, is delayed with respect to decreasing λr and is phased solution becomes non-orthogonal, is delayed with respect to decreasing λr and is phased 26 26

(a)

(b)

3.0

3.0

2.5

2.5

χ

χ

2.0

2.0

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0.0 0.0

0.2

0.4

0.6

0.8

0.0 -1.0

1.0

λr

-0.5

0.0

0.5

1.0

γz

Figure 9. Plots of χ versus (a) λr and (b) γz based of equation (61) with λz = 1, γθ = 0. Figure Plots of χ versus (a) λr and (b) γz basedto of the equation (61) with λz dashed, = 1, γθ = 0. In (a) γ9: corresponding thin continuous, thick z = 0 and ρ = 1, 3, 9, 30, In (a) γ = 0 and ρ = 1, 3, 9, 30, corresponding to the thin continuous, dashed, thick z continuous, and thick dotted curves, respectively. In (b) ρ = 9 and λr = 0.6, 0.7, 0.85, 0.95, continuous, and dotted curves, respectively. In (b) ρ = 9 andand λr =thick 0.6, 0.7, 0.85,curves, 0.95, corresponding tothick the thin continuous, dashed, thick continuous dotted corresponding to the thin continuous, dashed, thick continuous and thick dotted curves, respectively. respectively. out for sufficiently large values of γz . Figure 9(b) shows plots of χ against γz for ρ = 9, the curves are not affected by the change (χ, γθ ) → (−χ, −γθ ). The different orientations λz = 1 and several values of λr . This has features similar to those shown in Fig. 3, but of the closed curves in Fig. 9(b) compared with those in Fig. 3 are merely associated with the range of values of γz involved is significantly smaller. Note that, similarly to Fig. 3, the different ranges of values of χ used in the two cases. Changing the value of λz from the curves are not affected by the change (χ, γθ ) → (−χ, −γθ ). The different orientations 1 has only a marginal effect, but changing γθ from 0 to positive or negative values again of the closed curves in Fig. 9(b) compared with those in Fig. 3 are merely associated with delays the onset of loss of ellipticity with respect to decreasing λr and for large enough γθ the different ranges of values of χ used in the two cases. Changing the value of λz from loss of ellipticity is excluded altogether. 1 has only a marginal effect, but changing γθ from 0 to positive or negative values again delays the onset of loss of ellipticity with respect to decreasing λr and for large enough γθ

4

Concluding remarks

loss of ellipticity is excluded altogether. In this paper we have analyzed the emergence of loss of ellipticity for a specific boundaryvalue problem involving non-homogeneous deformations of a circular cylindrical tube, one of the few such problems for which this type of analysis has been conducted for transversely isotropic elastic materials. Specifically, we have considered helical shear superimposed on

27 27

4

Concluding remarks

In this paper we have analyzed the emergence of loss of ellipticity for a specific boundaryvalue problem involving non-homogeneous deformations of a circular cylindrical tube, one of the few such problems for which this type of analysis has been conducted for transversely isotropic elastic materials. Specifically, we have considered helical shear superimposed on the axial and radial deformation of the tube with the direction of transverse isotropy either axial or radial, and for a specific form of the constitutive law we have obtained explicit expressions that show how the loss of ellipticity depends on the various contributions to the deformation and the strength of the anisotropy. The loss of ellipticity presages, for example, the emergence of non-uniqueness of solutions involving weak or strong discontinuities, leading to non-smooth solutions such as fibre kinking in a fibre-reinforced material and possible material failure (see, for example, Budiansky and Fleck, 1993; Kyriakides et al., 1995; Lee et al., 2000; Merodio and Ogden, 2002, 2003, and references therein). This type of failure might, for example, be induced by small imperfections in the material that arise during the manufacturing process.

Acknowledgements The authors acknowledge support from the Ministerio de Ciencia, Spain, under the project number DPI2011-26167. Mustapha El Hamdaoui also thanks the Ministerio de Econom´ıa y Competitividad, Spain, for funding under grant DPI2008-03769.

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Baek, S. and Pence, T. J. (2010). Emergence and disappearance of load unduced fiber kinking surfaces in transversely isotropic hyperelastic materials. Z. Angew. Math. Phys., 61:745–772. Beatty, M.F. and Jiang, Q. (1999). On compressible materials capable of sustaining axisymmetric shear deformations. Part 3: Helical shear of isotropic hyperelastic materials. Q. Appl. Math., 57:681–697. Budiansky, B. and Fleck, N. A. (1993). Compressive failure of fiber composites. J. Mech. Phys. Solids, 41:183–211. Destrade, M., Saccomandi, G., and Sgura, I. (2009). Inhomogeneous shear of orthotropic non-linearly elastic solids: Singular solutions and biomechanical interpretation. Int. J. Eng. Sci., 47:1170–1181. Dorfmann, A., Merodio, J., and Ogden, R. W. (2010). Non-smooth solutions in the azimuthal shear of an anisotropic nonlinearly elastic material. J. Eng. Math., 68:27–36. El Hamdaoui, M., Merodio, J., Ogden, R.W., and Rodriguez, J. (2014). Finite elastic deformations of transversely isotropic circular cylindrical tubes. Int. J. Solids Structures, 51:1188–1196. Fosdick, R.L. and MacSithigh, G. (1983). Helical shear of an elastic, circular tube with a non-convex stored energy. Arch. Rat. Mech. Anal., 84:31–53. Gao, D.Y. and Ogden, R.W. (2008). Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem. Z. Angew. Math. Phys., 59:498–517. Holzapfel, G.A. (2001) Nonlinear Solid Mechanics: a Continuum Approach for Engineering, 2nd edn. John Wiley & Sons, Chichester.

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Horgan, C.O. and Saccomandi, G. (2003). Helical shear for hardening generalized neoHookean elastic materials. Math. Mech. Solids, 8:539–559. Jiang, Q. and Beatty, M.F. (2001). On compressible materials capable of sustaining axisymmetric shear deformations. Part 4: Helical shear of anisotropic hyperelastic materials. J. Elasticity, 62:47–83. Kassianidis, F., Ogden, R.W., Merodio, J., and Pence, T.J. (2008). Azimuthal shear of a transversely isotropic elastic solid. Math. Mech. Solids, 13:690–724. Kirkinis, E. and Ogden, R.W. (2003). On helical shear of a compressible elastic circular cylindrical tube. Q. J. Mech. Appl. Math., 56:105–122. Kyriakides, S., Arseculerane, R., Perry, E., and Liechti, K. M. (1995). On the compressive failure of fiber reinforced composites. Int. J. Solids Structures, 32:689–738. Lee, S. H., Yerramalli, C. S., and Waas, A. M. (2000). Compressive splitting response of glass-fiber reinforced unidirectional composites. Compos. Sci. Technol., 60:2957–2966. Merodio, J. and Ogden, R.W. (2002). Material instabilities in fiber-reinforced nonlinearly elastic solids under plane deformation. Arch. Mech., 54:525–552. Merodio, J. and Ogden, R.W. (2003). Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation. Int. J. Solids Structures, 40:4707–4727. Merodio, J. and Pence, T.J. (2001). Kink surfaces in a directionally reinforced neo-Hookean material under plane deformation: I. mechanical equilibrium. J. Elasticity, 62:119–144. Merodio, J., Saccomandi, G., and Sgura, I. (2007). The rectilinear shear of fiber-reinforced incompressible non-linearly elastic solids. Int. J. Non-Lin. Mech., 42:342–354.

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Ogden, R.W. (1997). Non-linear Elastic Deformations. Dover Publications, New York. Ogden, R.W. (2003). Nonlinear Elasticity with Application to Material Modelling. Centre of Excellence for Advanced Materials and Structures, Lecture Notes 6, Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw. Ogden, R.W., Chadwick, P., and Haddon, E.W. (1973). Combined axial and torsional shear of a tube of incompressible isotropic elastic material. Q. J. Mech. Appl. Math., 26:23–41. Qiu, G.Y. and Pence, T.J. (1997a). Remarks on the behavior of simple directionally reinforced incompressible nonlinearly elastic solids. J. Elasticity, 49:1–30. Qiu, G.Y. and Pence, T.J. (1997b). Loss of ellipticity in plane deformation of a simple directionally reinforced incompressible nonlinearly elastic solid. J. Elasticity, 49:31–63. Rivlin, R.S. (1949). Large elastic deformations of isotropic materials. VI. Further results in the theory of torsion, shear and flexure. Phil. Trans. R. Soc. Lond A, 242:173–195. Spencer, A.J.M. (1972). Deformations of Fibre-reinforced Materials. Oxford University Press. Spencer, A.J.M. (1984). Constitutive theory for strongly anisotropic solids, in: Continuum Theory of the Mechanics of Fibre-reinforced Composites, (Spencer, A.J.M., Ed.), pp.1– 32, CISM Courses and Lectures, no. 282, Springer, Wien 1984.

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Finite elastic deformations of transversely isotropic circular cylindrical tubes Highlights x x x x

Extension, inflation, torsion of a tube of transversely isotropic material Pressure, axial load and moment formulas Admissible directions of transverse isotropy for controllability and universality Illustrations for a prototype strain-energy function