has been studied recently in respect of pure torsion by various authors (see, for example, [START_REF] Beatty | Topics in finite elasticity: hyperelasticity of rubber, elastomers and biological tissuesðwith examples[END_REF], [START_REF] Beatty | Introduction to nonlinear elasticity[END_REF] and [START_REF] Carroll | Finite strain solutions in compressible isotropic elasticity[END_REF]). Loss of ellipticity for this material model during a pure torsional deformation was examined by Horgan and Polignone [START_REF] Horgan | A note on pure torsion of a circular cylinder for a compressible nonlinearly elastic material with nonconvex strain-energy[END_REF].

The work that is closest in character to the present article is that by Polignone and Horgan [START_REF] Polignone | Pure torsion of compressible nonlinearly elastic circular cylinders[END_REF], in which the authors derive a necessary and sufficient condition on the strain energy for pure torsion to be sustainable, but without imposing zero-traction conditions on the lateral surface of the cylinder. They proposed and examined several material models expressed in terms of the principal invariants of the CauchyïGreen deformation tensors.

In the present article we first summarize, in section 2, the necessary kinematics. The required general form of constitutive law for a compressible elastic material is then given in section 3, where we also derive the equilibrium equations appropriate for the combination of torsion with a uniform extension of a circular cylinder. In section 4, we specialize the problem by requiring that the deformation is isochoric and that no change in radius accompanies the torsion (i.e., we consider SXUH WRUVLRQ ). New necessary and sufficient conditions on the strainenergy function for the material to sustain pure torsion ZLWK ]HUR WUDFWLRQ RQ WKH ODWHUDO VXUIDFH RI WKH F\OLQGHU are obtained in terms of the principal stretches. It is shown how these relate to the condition derived in [START_REF] Polignone | Pure torsion of compressible nonlinearly elastic circular cylinders[END_REF], which did not impose the zero-traction requirement. We also show that, for any material satisfying these conditions a simple connection between the axial load on the cylinder and the torque required to achieve the torsion holds, a relation found previously for a special BlatzïKo material by Beatty [START_REF] Beatty | Topics in finite elasticity: hyperelasticity of rubber, elastomers and biological tissuesðwith examples[END_REF].

In section 5 we then focus on analysing the necessary and sufficient conditions when expressed in terms of the principal invariants L 4 > L 5 > L 6 of the CauchyïGreen deformation tensor. We note that a special form of the BlatzïKo material satisfying these condition was shown by Polignone and Horgan [START_REF] Polignone | Pure torsion of compressible nonlinearly elastic circular cylinders[END_REF] to meet the requirements considered. We also observe that a class of materials considered by Jiang and Ogden [START_REF] Jiang | Some new solutions for the axial shear of a circular cylindrical tube of compressible elastic material[END_REF] in the context of axial shear of a circular cylindrical tube satisfies the same requirements. Several specific forms of strainenergy function are examined. We emphasize that for a strain-energy function that does not satisfy the necessary and sufficient conditions mentioned above but does satisfy the condition given in [START_REF] Polignone | Pure torsion of compressible nonlinearly elastic circular cylinders[END_REF], a distribution of surface tractions is required on the lateral surface in order to maintain the pure torsional deformation. These tractions are calculated for some particular energy functions.

In section 6 we examine the modifications to the theory needed for consideration of energy functions expressed in terms of the principal invariants of the stretch tensors, and again some particular examples are studied.

In spirit this paper follows the work of Jiang and Ogden [START_REF] Jiang | Some new solutions for the axial shear of a circular cylindrical tube of compressible elastic material[END_REF][START_REF] Jiang | On azimuthal shear of a circular cylindrical tube of compressible elastic material. 4 -0HFK[END_REF] in that an isochoric form of deformation is used to place restrictions on the form of the strain-energy function by means of the radial equation of equilibrium. This general approach (which need not be restricted to consideration of isochoric deformations) has provided a systematic way of deriving new solutions to several boundary-value problems.

TORSION COMBINED WITH UNIFORM AXIAL EXTENSION

We consider a compressible non-linearly elastic circular cylinder whose cross-section in its natural (unstressed) configuration is defined by

3 5> 3 ] O> (1)
where U> > ]> are cylindrical polar coordinates.

In the deformed configuration the cylinder is subjected to a uniform axial stretch [ , which is held fixed while a torsional deformation is superimposed. The plane face } @ [ O is then rotated, while that at } @ 3 is fixed, in such a way that plane sections of the cylinder normal to its axis remain plane and the radius turns through an angle [ ], where 3 represents the angular twist per unit length of the deformed cylinder. The resulting deformation is then defined by

u @ u+U,> @ . [ ]> } @ [ ]> (2)
where u> > } are the cylindrical polar coordinates of a material point in the deformed configuration.

The matrix representation, ) say, of the deformation gradient tensor I with respect to the two sets of cylindrical polar coordinates has the form

) @ 3 C b u 3 3 3 u@U [ u 3 3 [ 4 D > (3)
where the dot signifies differentiation with respect to U. The left CauchyïGreen strain tensor E @ II W has the matrix representation

% @ 3 C b u 5 3 3 3 +u@U, 5 . 5 [ 5 u 5 5 [ u 3 5 [ u 5 [ 4 D = (4)
Let y +J, , l @ 4> 5> 6, be the unit Eulerian principal axes associated with this deformationðthat is, the principal axes of E. We see that the radial unit vector h S is the Eulerian principal axis associated with the principal stretch

4 @ b u= (5)
We may express the remaining two principal directions in terms of the cylindrical polar axes h r , h [ . Thus, we write ) where ! defines the orientation of the axes y +5, > y +6, relative to h r > h [ = The polar decomposition theorem guarantees the existence and uniqueness of the (positive definite, symmetric) left stretch tensor Y through

y +5, @ frv ! h r . vlq ! h [ > y +6, @ vlq ! h r . frv ! h [ > (6
I @ YU> ( 7
)
where U is a proper orthogonal tensor. The spectral decomposition of Y has the form

Y @ 6 [ J@4 J y +J, y +J, > (8)
which identifies the (positive) principal stretches 4 > 5 > 6 of the deformation. By defining the rotation matrix

3 @ 3 C 4 3 3 3 frv ! vlq ! 3 vlq ! frv ! 4 D (9)
and comparing the entries of % and 9 (the matrix representation of Y) through % @ 39 5 3 W , we obtain the connections 5 5 frv 5 ! . 5 6 vlq 5 ! @ u 5 U 5 . 5 [ 5 u 5 > (10)

5 5 vlq 5 ! . 5 6 frv 5 ! @ 5 [ > (11) + 5 5 5 6 , vlq ! frv ! @ 5 [ u> (12)
from which it may be deduced that

5 5 . 5 6 @ u 5 U 5 . 5 [ 5 u 5 . 5 [ > (13)
+ 5 5 5 [ ,+ 5 [ 5 6 , @ 7 [ 5 u 5 > (14)
and

5 6 @ [ u U = (15)
Further, we obtain the explicit expression for ! in the form frv 5! @ 5 5 . 5

Counterparts of the above formulae for the Lagrangian principal axes can be found in ( [START_REF] Ogden | 1RQ/LQHDU (ODVWLF 'HIRUPDWLRQV[END_REF], section 5.2.5).

EQUILIBRIUM AND STRESSïDEFORMATION EQUATIONS

For an isotropic elastic material, which we are considering here, the Cauchy stress tensor j, is coaxial with the left CauchyïGreen strain tensor E. The non-zero components of j in the current configuration can therefore be expressed in terms of its principal stresses 4 > 5 > 6 through SS @ 4 > r[ @ + 5 6 , frv ! vlq ! > (17) rr @ 5 frv 5 ! . 6 vlq 5 ! >

[[ @ 5 vlq 5 ! . 6 frv 5 ! =

The connection

5 [ u+ rr [[ , @ +u 5 @U 5 . 5 [ 5 u 5 5 [ , r[ (19)
(see also Holzapfel HW DO [21]) may be obtained from ( 17) and ( 18) on use of [START_REF] Horgan | A note on pure torsion of a circular cylinder for a compressible nonlinearly elastic material with nonconvex strain-energy[END_REF]. Note that this is not a universal relation since u+U, depends on the solution of the equilibrium equation and hence, in general, on the form of the strain-energy function. This contrasts with the situation for incompressible materials, for which the counterpart of ( 19) LV a universal relation. For a general discussion of universal relations we refer the reader to the recent review by Saccomandi [START_REF] Saccomandi | Universal results in finite elasticity[END_REF].

Since we are considering compressible materials, the principal stresses are given by

4 5 6 J @ J CZ C J > l5 i4> 5> 6j +qr vxppdwlrq ryhu l,> (20)
where Z @ Z+ 4 > 5 > 6 , is the strain energy per unit reference volume, which is a symmetric function of the principal stretches. For consistency with the classical theory Z should satisfy

Z+4> 4> 4, @ 3> Z J +4> 4> 4, @ 3> l@ 4> 5> 6> (21)
Z JK +4> 4> 4, @ 5 6 > = l 9 @ m> Z JJ +4> 4> 4, @ . 7 6 > l @ 4> 5> 6> [START_REF] Saccomandi | Universal results in finite elasticity[END_REF] where, in the latter, no summation is implied by the repetition of the index l, and the notation Z J @ CZ@C J , Z JK @ C 5 Z@C J C K is adopted. In [START_REF] Saccomandi | Universal results in finite elasticity[END_REF], +A3, denotes the shear modulus and +A3, the bulk modulus in the natural configuration. The principal invariants L 4 > L 5 > L 6 of E are given in terms of the principal stretches by L 4 @ 5 4 . 5 5 . 5 6 > L 5 @ 5 5 5 6 . 5 6 5 4 . 5 4 5 5 > L 6 @ 5 4 5 5 5 6 >

and we write Z to represent the energy function when expressed in terms of L 4 > L 5 > L 6 . Similarly, we write Z for dependence on the principal invariants l 4 > l 5 > l 6 of Y, these being defined by l 4 @ 4 . 5 . 6 > l 5 @ 5 6 . 6 4 . 4 5 > l 6 @ 4 5 6 =

We also note the connections

L 4 @ l 5 4 5l 5 > L 5 @ l 5 5 5l 4 l 6 > L 6 @ l 5 6 = (25)
Thus, we have

Z+ 4 > 5 > 6 , @ Z+L 4 > L 5 > L 6 , @ Z+l 4 > l 5 > l 6 ,= (26)
The analogues of ( 21) and ( 22) for Z and Z, respectively, are

Z+6> 6> 4, = 3> Z 4 . Z 5 @ + Z 5 . Z 6 , @ 5 > (27) Z 44 . 7 Z 45 . 7 Z 55 . 5 Z 46 . 7 Z 56 . Z 66 @ 7 . 6 > (28)
where the derivatives are evaluated for L 4 @ 6> L 5 @ 6> L 6 @ 4, and

Z+6> 6> 4, = 3> Z 4 . Z 5 @ + Z 5 . Z 6 , @ 5> (29) Z 44 . 7 Z 45 . 7 Z 55 . 5 Z 46 . 7 Z 56 . Z 66 @ . 7 6 > (30)
with the derivatives evaluated for l 4 @ 6> l 5 @ 6> l 6 @ 4. In each case the subscripts 4> 5> 6 indicate differentiation with respect to the relevant arguments, i.e., L 4 > L 5 > L 6 or l 4 > l 5 > l 6 respectively. Since the deformation gradient depends only on the radial coordinate, the equilibrium equation div j @ 3 (in the absence of body forces) reduces to the radial equation

g SS gu . 4 u + SS rr , @ 3= (31)
In terms of the principal stretches equation (31) can be written in the form This equation is identical to the one derived in [START_REF] Ogden | 1RQ/LQHDU (ODVWLF 'HIRUPDWLRQV[END_REF] based on a Lagrangian formulation of the problem.

Note that for the problems considered here there is just the one equilibrium equation. This contrasts with the situation for the (isochoric) azimuthal shear and axial shear problems for a circular cylindrical tube considered by Jiang and Ogden [START_REF] Jiang | On azimuthal shear of a circular cylindrical tube of compressible elastic material. 4 -0HFK[END_REF][START_REF] Jiang | Some new solutions for the axial shear of a circular cylindrical tube of compressible elastic material[END_REF] and others. In each of these problem there are two equilibrium equations to be satisfied for the (unknown) radially dependent deformation field. Compatibility of these two equations leads to restrictions on the form of strain-energy function. Appropriate shear and radial tractions are required on the lateral surfaces of the tube in order to maintain the deformation. In the current problem the equilibrium equation serves to determine u +U, for any given form of (compressible isotropic) strain-energy function. On the lateral surface of the cylinder there will in general be radial tractions, but we may also wish to consider the possibility that the lateral surface is traction free. In this case the boundary condition to be satisfied is

SS @ 4 @ 3 rq U @ D> (33)
and the deformation is then maintained by applying an axial load and torque.

ISOCHORIC SPECIALIZATION

We now specialize the deformation so that it is isochoric, comprising an isochoric simple tension and an isochoric torsion. We therefore have u @ 45 [ U and equations ( 5), ( 15) and ( 13) respectively reduce to

4 @ 45 [ > 5 6 @ 45 [ > 5 5 . 5 6 @ 5 [ . 4 [ . [ 5 U 5 = (34)
Since the deformation is isochoric L 6 @ 4 and the principal invariants L 4 > L 5 are given by

L 4 @ 5 [ . 5 4 [ . [ 5 U 5 > L 5 @ 5 [ . 5 [ . 5 U 5 = (35)
Corresponding expressions may be written down for l 4 > l 5 but we do not need them here.

In terms of the stretches, the equations ( 17) and ( 18) for the stress components may now be given the explicit forms while the equilibrium equation (32) specializes accordingly. On use of (34) we may write the equilibrium equation explicitly in terms of the stretches. After some manipulation this leads to in which the derivatives of Z are evaluated for (34). This provides a necessary and sufficient condition for the energy function to admit the deformation considered and generalizes the result given in [START_REF] Polignone | Pure torsion of compressible nonlinearly elastic circular cylinders[END_REF] for [ @ 4 to the case [ 9 @ 4.

SS @ 45 [ Z 4 > r[ @ s + 5 5 5 [ ,+ 5
When [ @ 4 equation (39) reduces to

+ Z 45 4 Z 46 , . + . 4 ,Z 4 @ 5 Z 5 . 5 Z 6 > (40)
where is defined by

@ 4 @ U> (41)
and we have set 5 @ > 6 @ 4 . In this special case the deformation is locally a simple shear of magnitude in the (h r > h [ ) plane and is referred to as SXUH WRUVLRQ. Equation ( 40) is equivalent to the result given in [START_REF] Polignone | Pure torsion of compressible nonlinearly elastic circular cylinders[END_REF] except that a factor of has been removed from the latter. As pointed out in [START_REF] Polignone | Pure torsion of compressible nonlinearly elastic circular cylinders[END_REF], satisfaction of (40) does not guarantee that the zero-traction boundary condition on the lateral surface of the cylinder is satisfied and therefore, in general, appropriate radial tractions need to be supplied in order to maintain the deformation.

We now obtain necessary and sufficient conditions on the energy function that ensure that the deformation is sustainable and that, in addition, the zero-traction boundary condition is met.

Consider the stress SS given by (36) 4 . Since this is evaluated for the deformation given by (34) and [ is constant, SS depends on the deformation and the radial coordinate only through the combination U. We therefore write SS +U, @ +U, to represent this dependence. On the lateral surface we then have SS +D, @ +D,. If we require the lateral traction to vanish then, from (33), SS +D, @ +D, @ 3 iru doo 3=

Hence

g g SS +D, @ 3 + D,D @ 3 iru doo A 3> (43)
and it then follows that, for any fixed A 3, g gU SS +U, @ 3 +U, @ 3 iru doo U A 3

(although we only require it to hold for 3 ? U ? D). Thus, SS +U, is constant, and since it vanishes for U @ D, SS 3. From (31) we deduce that rr 3 also. The conditions on the strain-energy function that ensure these identities are, from (36) and (37),

Z 4 @ 3> 5 + 5 5 5 [ ,Z 5 6 + 5 6 5 [ ,Z 6 @ 3> (45)
in which the terms are evaluated for the stretches given by (34). The conditions (45) are QHFHVVDU\ DQG VXIILFLHQW for the strain-energy function to admit the combined isochoric torsion and uniform extension ZLWK ]HUR WUDFWLRQV on the lateral surface of the cylinder. It is worth noting that, since the tractions vanish, these conditions also ensure that the deformation is supported by a circular cylindrical tube. Note that derivation of the second equation in (45) requires the assumption that 5 9 @ 6 . This is justified since it is easy to show from (34) that 5 @ 6 only in the trivial situation @ 3> [ @ 4 with 5 @ 6 @ 4.

When [ @ 4 equations (45) reduce to

Z 4 @ 3> 5 Z 5 . 5 Z 6 @ 3> (46)
evaluated for 5 @ > 6 @ 4 > 4 @ U.

Beatty [START_REF] Beatty | Topics in finite elasticity: hyperelasticity of rubber, elastomers and biological tissuesðwith examples[END_REF], in considering a special BlatzïKo form of strain-energy function, showed that for pure torsion ( [ @ 4) the resultant axial load Q on any cross-section of the cylinder and the resultant moment P are related by

Q @ P= (47)
See also the discussion in [START_REF] Polignone | Pure torsion of compressible nonlinearly elastic circular cylinders[END_REF]. We now show that this holds for any strain-energy function satisfying the necessary and sufficient conditions derived above.

For the more general situation with [ 9 @ 4 we have

Q @ 5 4 [ " ] 3
[[ U gU> P @ 5 65 [ " ] 3 r[ U 5 gU> (48)
where the integrals (which are independent of ]) are taken over any cross-section of cylinder. On specializing the relation [START_REF] Jiang | On azimuthal shear of a circular cylindrical tube of compressible elastic material. 4 -0HFK[END_REF] to the deformation (34) and setting rr @ 3, we obtain

Q @ P . 5 4 + 45 [ :5 [ , " ] 3 r[ gU> (49)
so (47) follows when [ @ 4. The formulae (48) and ( 49), with appropriate changes in the limits of the integrals, also apply for a tube. When [ @ 4 and pure torsion is considered equation [START_REF] Jiang | On azimuthal shear of a circular cylindrical tube of compressible elastic material. 4 -0HFK[END_REF] reduces to rr [[ @ r[ . Thus, when rr @ 3 we see that the connection [[ @ r[ , and hence (47), LV D QHFHVVDU\

FRQGLWLRQ IRU SXUH WRUVLRQ WR EH DGPLWWHG LQ WKH DEVHQFH RI VXUIDFH WUDFWLRQV RQ WKH ODWHUDO VXUIDFH.

A similar statement can be made for the case [ 9 @ 4.

ENERGY FUNCTIONS IN TERMS OF

L 4 > L 5 > L 6
When expressed in terms of the strain energy Z+L 4 > L 5 > L 6 ,, the Cauchy stress components (36)ï(38) take the forms [ , . 5

SS @ 5 4 [ ^ Z 4 . Z 5 + 5 5 . 5 6 , . Z 6 [ `> (50) r[ @ 5 Z 4 . Z 5
Z 5 + 5 5 . 5 6 , 4 [ . 5 Z 6 > (52) [[ @ 5 Z 4 5 [ . 7 Z 5 [ . 5 Z 6 = (53)
The combination of the identities SS @ rr 3 then leads to Z 4 @ 3> Z 5 + 5 5 . as given by Polignone and Horgan [START_REF] Polignone | Pure torsion of compressible nonlinearly elastic circular cylinders[END_REF], where L @ L 4 @ L 5 @ 6 . 5 > L 6 @ 4> @ U. In [17], L was left as 6 . 5 U 5 and coefficients of terms involving 5 U 5 were set to zero in order to find restrictions on the energy function that ensured satisfaction of (55). This approach is more restrictive than the one we adopt in section 5.1. For the case [ @ 4, it is easy to show by differentiation of (54) and use of ( 35) that (55) follows from (54). For [ 9 @ 4 a similar deduction can be made. When equations (54) hold it follows from (51) and (53) that

r[ @ 5 4 [ Z 5 s + 5 5 5 [ ,+ 5 [ 5 6 , (56)
and

[[ @ 5 4 [ Z 5 + 5 5 . 5 6 5 5 [ ,> (57)
and when [ @ 4 these reduce to

r[ @ 5 Z 5 > [[ @ 5 5 Z 5 = (58)
Since L 6 @ 4 the first equation in (54) shows that Z+L 4 > L 5 > 4, LV LQGHSHQGHQW RI L 4 SURYLGHG WKDW L 4 DQG L 5 DUH LQGHSHQGHQW. Note, however, that in the second equation in (54) the term Z 6 +L 4 > L 5 > 4, may depend on L 4 . In this case 5 5 . 5 6 @ L 4 4

[

@ [ L 5 5 [ > (59)
can be used (in principle) to determine [ in terms of L 4 and L 5 and the second equation in (54) can then be written as an ordinary differential equation with L 5 as the independent variable. In general this is impractical if not impossible. To illustrate the procedure, therefore, we shall restrict attention to [ @ 4 in what follows.

3XUH WRUVLRQ

If [ @ 4 we have L 5 @ L 4 , and (54) 4 does QRW then imply that Z is independent of L 4 since it applies only for the restricted manifold L 5 @ L 4 > L 6 @ 4 in (L 4 > L 5 > L 6 )-space. If, however, Z LV independent of L 4 , then (54) 4 is automatically satisfied.

-2 PDWHULDOV

An example of a class of strain-energy functions independent of L 4 is that introduced by Jiang and Ogden [START_REF] Jiang | Some new solutions for the axial shear of a circular cylindrical tube of compressible elastic material[END_REF]. We write this as Z @ Z MR5 +L 5 > L 6 ,, where Z MR5 +L 5 > L 6 , @ j+L 5 ,k 4 +L 6 , . k 5 +L 6 ,>

j> k 4 > k 5 are functions to be determined and in the subscript JO2 the 2 is used to distinguish (60) from a class of energy functions introduced by Jiang and Ogden [START_REF] Jiang | On azimuthal shear of a circular cylindrical tube of compressible elastic material. 4 -0HFK[END_REF], for which the subscript JO1 will be used later. Substitution of (60) into (54) 5 leads to an expression for the function j, and hence (60) may be given the explicit form

Z MR5 +L 5 > L 6 , @ 5 L n +L 5 4, L k 4 +L 6 , . k 5 +L 6 ,> (61)
where n 9 @ 3 is a disposable parameter and, for consistency with ( 27) and ( 28), k 4 > k 5 must satisfy

k 4 +4, @ 4> k 3 4 +4, @ n> k 5 +4, @ n > k 3 5 +4, @ 3> (62)
and

k 33 5 +4, . n k 33 4 +4, @ 7 . 6 . +n . 4,= (63)
In the case n @ 3 equation ( 61) is replaced by

Z MR5 +L 5 > L 6 , @ orj+L 5 4,k 4 +L 6 , . k 5 +L 6 ,> (64)
and ( 62) and ( 63) by (iii) n ? 4 5 : r[ has a maximum and r[ $ 3 as $ 4.

k 4 +4, @ 4> k 3 4 +4, @ 3> k 5 +4, @ orj 5> k 3 5 +4, @ > (65)
Of the three cases (i)ï(iii), only (i) can be regarded as yielding a stable solution for all values of the applied twist A 3. But even in this case stability may be lost at some critical value of if there is an energetic preference for departure from the pure torsion. Analysis of stability is beyond the scope of the present paper, but will be examined elsewhere.

Finally in this section we calculate the resulting torque P using (48) 5 with [ @ 4 and (67). This gives P @ D 6 5 L 4 n+n . 4, 6 " ^+5 . 5 " , L +n 5 " 5, . 5 L.4 `> n9 @ 3> 4> (69)

where " @ D. Corresponding results may be obtained for n @ 3 and n @ 4 but we omit them here. We give results for two specific values of n. For n @ 4 equation (69) gives the standard result

P @ 4 5 D 7 (70)
obtainable for the neo-Hookean strain energy in the incompressible theory and also for the special BlatzïKo material considered in [START_REF] Polignone | Pure torsion of compressible nonlinearly elastic circular cylinders[END_REF]. For n @ 5 the corresponding specialization of (69) is

P @ 4 5 D 7 4 . 4 6 5 D 5 = ( 71
)
Since P is positive for all A 3 in these examples it follows from (47) that Q ? 3. Thus, a compressive axial load is required to maintain [ @ 4, suggesting that in the absence of axial load the cylinder would lengthen. This conclusion also applies for all other n since it is easy to show from (69) that P A 3 for all A 3.

%ODW]±.R FRQVLGHUDWLRQV

We now consider a modification of the JO2 material class (60) of the form Z @ j+L 5 @L 6 ,k 4 +L 6 , .

k 5 +L 6 ,= (72)
This is motivated by reference to the special BlatzïKo material of the form Z @ 5

L 5 L 6 4 4 . 4 5 L w +4 5w , 6 > (73)
where is Poissonòs ratio, given by @ 6 5 5+6 . ,

in terms of the bulk modulus and shear modulus .

The procedure used to determine the function j in respect of the JO2 materials leads, in this case, to Z having the form Z @ 5n h{s^n+L 5 @L 6 6,`k 4 +L 6 , .

k 5 +L 6 ,> (75)
provided that n 9 @ 3, where

k 4 +4, @ 4> k 3 4 +4, @ n> k 5 +4, @ 5n > k 3 5 +4, @ 3 (76)
and

k 33 5 +4, . 5n k 33 4 +4, @ 7 . n 5 5 6 = (77)
These should be contrasted with the corresponding expressions (61)ï(63) for JO2 materials.

If n @ 3 equations ( 75)ï(77) are replaced by Z @ 5

L 5 L 6 k 4 +L 6 , . k 5 +L 6 ,> (78)
k 4 +4, = 4> k 3 4 +4, @ 3> k 5 +4, @ 6 5 > k 3 5 +4, @ 5 (79)
and

k 33 5 +4, . 6 5 k 33 4 +4, @ 7
5 6 > (80)
respectively. For the special case in which k 4 +L 6 , 4, equations (78)ï(80) reduce to Z @ 5

L 5 L 6 . k 5 +L 6 ,> (81)
k 5 +4, @ 6 5 > k 3 5 +4, @ 5 > k 33 5 +4, @ 7
5 6 > (82)
which provides a slight generalization of the BlatzïKo model (73) in that k 5 +L 6 , need not be of the specific form required by (73).

In respect of ( 75) and ( 78) the shear stress is given by r[ @ h{s+n 5 ,, and this is a monotonically increasing function of for all if and only if n 3. The resultant torque on the ends of the cylinder is calculated as

P @ D 6 n 5 6 " ^4 h{s+n 5 " , . n 5 " h{s+n 5 " ,`> n9 @ 3> (83)
while for n @ 3 it is again given by (70). In each case P A 3 and Q ? 3, as for the JO2 materials.

*HQHUDWLRQ RI RWKHU HQHUJ\ IXQFWLRQV

One possible approach to finding energy functions that can support pure torsion is to start with a relatively simple energy function and then extend it by the addition of a term which can then be determined by application of the necessary and sufficient conditions. We illustrate this method with just one example. Consider the energy function containing a term linear in L 5 and given by Z @ d 5 ^L5 k 4 +L 6 , . j+L 5 ,k 5 +L 6 , .

k 6 +L 6 ,`> (84)
where d is a constant and the function j is to be determined, while the functions k 4 > k 5 > k 6 are to be consistent with the requirements ( 27) and [START_REF] Beatty | On compressible materials capable of sustaining axisymmetric shear deformations. III. Helical shear of isotropic hyperelastic materials[END_REF]. Two separate possibilities arise when (84) is substituted into (54) 5 (for [ @ 4).

In the first case j is quadratic in L 5 and (84) can be written as

Z @ d 5 ^L5 k 4 +L 6 , . 4 5 +n . 4,L 5 5 k 5 +L 6 , . k 6 +L 6 ,`> (85)
where

d @ 4@+6n . 7,> (86)
k 4 +4, @ k 5 +4, @ 4> k 3 4 +4, @ n> k 3 5 +4, @ 5> (87) k 6 +4, @ 6 5 +6n . 8,> k 3 6 +4, @ 4> (88)
k 33 6 +4, . < 5 +n . 4,k 33 5 +4, . 6k 33 4 +4, @ +6n .

and n is a disposable constant.

In the second case (84) becomes

Z @ d 5 ^L5 k 4 +L 6 , +d 4, pd5 N 4 +L 5 4, N k 5 +L 6 , . k 6 +L 6 ,`> (90)
where

k 4 +4, @ k 5 +4, @ 4> k 3 4 +4, @ 4> k 3 5 +4, @ p> (91)
k 6 +4, @ 4 pd ^5 . +6p 5,d`> k 3 6 +4, @ 4> (92)
Here, both d and p are disposable constants, but in the special case p @ 5, equation ( 85) is not recovered since n @ 4 and the energy is then linear in L 5 . Thus, equation ( 85) is not a special case of (90).

Restrictions on the values of n> d> p may be determined by considering the monotonicity of r[ as in sections 5.1.1 and 5.1.2, but we omit the details here.

Further forms of energy function can be generated by replacing the linear term in L 5 in (84) by other specific functions of L 5 and in this way an extensive catalogue of energy functions that can support pure torsion with zero lateral traction can be built up.

(QHUJ\ IXQFWLRQV ZLWK SS 9 @ 3 If it is not required to satisfy the zero-traction condition on U @ D, then in order for an energy function to support pure torsion it is only necessary that equation (55) be satisfied. For this purpose we examine the class of strain-energy functions Z @ Z MR4 +L 4 > L 6 , introduced by Jiang and Ogden [START_REF] Jiang | On azimuthal shear of a circular cylindrical tube of compressible elastic material. 4 -0HFK[END_REF]. These have the form

Z MR4 +L 4 > L 6 , @ i +L 4 ,k 4 +L 6 , . k 5 +L 6 ,= (94)
Substitution of ( 94) into (55) with L 6 @ 4 leads to

Z MR4 +L 4 > L 6 , @ 5n h{s^n+L 4 6,`k 4 +L 6 , . k 5 +L 6 ,> (95)
where n 9 @ 3 is a disposable constant and

k 4 +4, = 4> k 3 4 +4, @ 4 5 n> k 5 +4, @ 5n > k 3 5 +4, @ 7n > (96)
k 33 5 +4, . 5n k 33 4 +4, @ 7 . 9 +6n 4,=

The resulting shear stress is r[ @ h{s+n 5 ,, which is the same as that discussed in respect of (81). From (50) the radial stress is calculated as

SS @ 5n ^h{s+n 5 , 4`= (98)
It follows that SS A 3 for 9 @ 3 for all n. This is consistent with the discussion in section 5.1.1 where we noted that an axial compression was required to prevent the cylinder extending during torsion. In this case, however, it follows from (50) and ( 53) that [[ @ SS and hence that the axial load is positive, in contrast to the situation in section 5.1.1. Thus, the tendency for the axial load to extend the cylinder is counterbalanced by the tensile radial traction (which tends to shorten the cylinder).

In a similar way to that in section 5.1.3 we may generate classes of strain-energy functions, except that now this is done using (55) rather than (54). We consider the energy function Z+L 4 > L 5 > L 6 , @ i +L 4 ,k 4 +L 6 , . j+L 5 ,k 5 +L 6 , .

k 6 +L 6 ,> (99)
where i and j are to be determined and k 4 > k 5 > k 6 play the same roles as previously. This can be regarded as a generalization of the BlatzïKo material Z+L 4 > L 5 > L

where (102) hold with p @ 4 and ( 103) is replaced by 6d 5nd +5n 4,d orj 5 . 5e @ = (105)

In each case the stresses r[ > SS > [[ may be calculated but no new features arise so we omit the details. By taking specific values of the constants several of the particular examples discussed previously can be recovered. A further class of energy functions can be generated by replacing j+L 5 , by j+L 5 @L 6 , in (99) in a similar way to that in section 5.1.2.

ENERGY FUNCTIONS IN TERMS OF

l 4 > l 5 > l 6
Here we consider an alternative formulation of the pure torsion problem based on the principal invariants l 4 > l 5 > l 6 defined in [START_REF] Horgan | Equilibrium solutions for compressible nonlinear elasticity[END_REF]. The strain energy is written as Z @ Z+l 4 > l 5 > l 6 ,, as in [START_REF] Haughton | Circular shearing of compressible elastic cylinders. 4 -0HFK[END_REF]

when evaluated for l @ l 4 @ l 5 @ .

We recall that the subscripts on Z denote derivatives with respect to l 4 > l 5 > l 6 . When SS @ rr @ 3 we obtain, after some rearrangement, l Z 4 .

Z 5 @ 3> +l 5 l 4,

Z 4 Z 6 @ 3> (112)
again evaluated for (110). These are the counterparts and equivalents of (54). In general, application of (112) leads to different classes of energy function from those generated on the basis of (54).

To be specific we consider energy functions of the form Z+l 4 > l 5 > l 6 , @ i +l 4 ,k 4 +l 6 , . j+l 5 ,k 5 +l 6 , . k 6 +l 6 ,>

which includes many well-known energy functions as special cases: for example, the class I, class II and class III materials considered by Carroll [START_REF] Carroll | Finite strain solutions in compressible isotropic elasticity[END_REF], which are, respectively, linear in +l 5 > l 6 ,, +l 4 > l 6 , and +l 4 > l 5 ,; see also Carroll and Horgan [START_REF] Carroll | Finite strain solutions for a compressible elastic solid. 4[END_REF] and the recent review by Horgan [START_REF] Horgan | Equilibrium solutions for compressible nonlinear elasticity[END_REF]. For materials of class III the torsion problem was studied in [START_REF] Polignone | Pure torsion of compressible nonlinearly elastic circular cylinders[END_REF], and it was found that pure torsion can be sustained only in the presence of a (uniformly distributed) tensile loading on the lateral surface of the cylinder. Clearly, class III materials cannot satisfy (112) 4 except trivially. It was also shown in [START_REF] Polignone | Pure torsion of compressible nonlinearly elastic circular cylinders[END_REF] that class II materials cannot sustain pure torsion and that (in general) neither can class I materials.

On substitution of (113) into (112) we obtain li 3 +l, . j 3 +l, @ 3> +l 5 l 4, i 3 +l, n i +l, pj+l, k 3 6 +4, @ 3>

where, on application of (29), we have k 4 +4, @ k 5 +4, @ 4> k 3 4 +4, @ n> k 3 5 +4, @ p> (115) k 6 +4, @ i +6, j+6,> i 3 +6, . j 3 +6, @ 5> (116) j 3 +6, . n i +6, . pj+6, . k 3 6 +4, @ 5=

We have not written down the specialization of (30) explicitly. Equations (114) may be solved to give i and j. In general the solution involves logarithms and we omit the details. There is one special case in which logarithms do not arise, and we therefore focus on this for purposes of illustration. This corresponds to n @ 4> p @ 5 in (115) and leads to Z @ l 4 k 4 +l 6 , . 5 l 5 5 k 5 +l 6 , . k 6 +l 6 ,>

with

k 6 +4, @ 6 5 > k 3 6 +4, @ > k 33 6 +4, . < 5 k 33 5 +4, 6k 33 4 +4, @ . 8; 6 = (119)
From the specialization of (25) for l 6 @ 4 we note that L 4 @ l 5 4 5l 5 > L 5 @ l 5 5 5l 4 , and hence it is clear that (118) is not independent of L 4 when regarded as a function of L 4 and L 5 .

Note that for the material (118) we obtain r[ @ > [[ @ 5 , which are the same results as are obtained for (78) with n @ 3. This means, in particular, that it is not possible to distinguish between the two different strain-energy functions on the basis of pure torsion alone.

Finally in this section, we consider the compressible Varga material defined by Z @ d+l 4 6, . e+l 5 6, . k+l 6 ,>

where d and e are constants. This corresponds to the class III materials introduced by Carroll [START_REF] Carroll | Finite strain solutions in compressible isotropic elasticity[END_REF] (see also Haughton [25]). As already noted, these materials can support pure torsion, but require a radial traction on the lateral surface of the cylinder in order to do so. For (120) to support pure torsion it is necessary and sufficient that e @ d. The energy function ( 120) is interesting because it can also support pure azimuthal shear [START_REF] Haughton | Circular shearing of compressible elastic cylinders. 4 -0HFK[END_REF] and pure axial shear [START_REF] Jiang | Some new solutions for the axial shear of a circular cylindrical tube of compressible elastic material[END_REF]. The present authors have also shown that the same is true for helical shear [START_REF] Kirkinis | On helical shear of an isotropic compressible elastic tube[END_REF] and have obtained a closed-form solution to the helical shear problem in this case (see Beatty and Jiang [START_REF] Beatty | On compressible materials capable of sustaining axisymmetric shear deformations. III. Helical shear of isotropic hyperelastic materials[END_REF] for a discussion of helical shear).

It is evident that the procedure discussed in the above sections can be used to derive further classes of energy function and to derive material models that best fit experimental data.

INCOMPRESSIBLE MATERIALS

In [START_REF] Jiang | On azimuthal shear of a circular cylindrical tube of compressible elastic material. 4 -0HFK[END_REF] and [START_REF] Jiang | Some new solutions for the axial shear of a circular cylindrical tube of compressible elastic material[END_REF] we discussed the procedure for generating strain-energy functions and corresponding solutions for incompressible materials in respect of pure azimuthal shear and pure axial shear of a circular cylindrical tube. A similar procedure can be adopted in respect of the pure torsion considered here.

For this purpose, we define the strain-energy functions z+L 4 > L 5 , @ Z+L 4 > L 5 > 4,> z+l 4 > l 5 , @ Z+l 4 > l 5 > 4,>

where Z+L 4 > L 5 > L 6 ,, or Z+l 4 > l 5 > l 6 ,, corresponds to one of the models derived in the foregoing sections. The Cauchy stress tensor j for an incompressible material is written as j @ 5 z 4 E 5 z 5 E 4 sL> (122

)
where s is an arbitrary hydrostatic pressure to be determined by the equilibrium equations and the boundary conditions. Equivalently, it may be written as

j @ z 4 Y z 5 Y 4 sL> ( 123
)
where V is the left stretch tensor and, in general, s in (123) differs from that in (122). Thus, results obtained for pure torsion for the considered compressible materials apply also to incompressible materials. Instead of imposing restrictions on the strain-energy function, the equilibrium equation serves to determine s. The procedure of obtaining results for an incompressible problem using corresponding results from the isochoric compressible case was illustrated in [START_REF] Jiang | On azimuthal shear of a circular cylindrical tube of compressible elastic material. 4 -0HFK[END_REF] and [START_REF] Jiang | Some new solutions for the axial shear of a circular cylindrical tube of compressible elastic material[END_REF].

$FNQRZOHGJPHQWV 7KH ZRUN RI ( .LUNLQLV ZDV VXSSRUWHG E\ WKH 8. (QJLQHHULQJ DQG 3K\VLFDO 6FLHQFHV 5HVHDUFK &RXQFLO DQG WKH 8QLYHUVLW\ RI *ODVJRZ

4 [

 4

  is a constant and is Poissonòs ratio, as discussed earlier. For purposes of illustration we set i +L 4 , @ dL 4 , where d is a constant.Equation (55) then leads to an equation for j+L ,, the solution of which reveals two separate cases. Firstly, we obtain Z @ dL 4 k 4 +L 6 , . +4,, d> e> n> p and and . We omit the details. The second case corresponds to p @ 4 and gives Z @ dL 4 k 4 +L 6 , . +L 6 , . k 6 +L 6 ,>

	6 , @	5	d	L 4 4		4	.	4 5	L w +4 5w , 6
	.	5	+4 d,	L 5 L 6			4		4	.	4 5	L 6 w +4 5w ,	>	(100)
	where d e	+L 5 4, N 5 N	.		d 5	5n 4 p 4	L 5	k 5 +L 6 , . k 6 +L 6 ,>	(101)
	for p 9 @ 4, where
	k 4 +4, @ 4>	k 5 +4, @ 4>		k 3 4 +4, @ n>	k 3 5 +4, @ p=	(102)
	Three of the constants d> e> n> p are independent. They are related by
		d	5p . 5n 6 p 4	. ep @ >	(103)
	while k 6 +4,> k 3 6 +4, are given in terms of the constants through (27). Equation (28) provides a connection between k 33 4 +4,> k 33 5 +4,> k 33 6 k eL 5 l d 5 +5n 4,+L 5 4, orj+L 5 4, k 5

  . The Cauchy stress components (50)ï(53) become

	SS @	Z 4 . +l 4, Z 5 .	Z 6 >	(106)
	rr @	4 l 4	Z 4 + 5 . 5, . +l 5,+l . 4, Z 5 . +l 4, Z 6 >	(107)
	r[ @	l 4	Z 4 .	Z 5 >		(108)
	[[ @	4 l 4	5 Z 4 . +l . 4, Z 5 . +l 4, Z 6