Rotation of cylinders of special compressible materials

0020-7225/92 $5.00+ 0.00 Copyright 0 1992Pergamon Press plc

Int. J. Engng Sci. Vol. 30, No. 2, pp. 213-221, 1992 Printed in Great Britain. All rights reserved

ROTATION OF CYLINDERS OF SPECIAL COMPRESSIBLE MATERIALS JEREMIAH Department

G. MURPHY?

of Mechancial Engineering,

and FRANCIS

J. ROONEY

University of California at Berkeley, Berkeley, CA 94720, U.S.A.

Abstract-In this paper we seek closed form solutions to the boundary value problems associated with the rotation of solid cylinders of finitely-deforming compressible elastic materials. Two exact solutions to the equations of motion will be obtained and some of the qualitative features of these solutions will be studied. The strain energy functions used are quadratic in one stretch invariant and linear in another.

1. INTRODUCTION Green and Shield [l] have shown that shape-preserving deformations of a solid circular cylinder resulting from steady rotation about the axis of symmetry are controllable deformations for homogeneous isotropic incompressible materials. The qualitative features of boundary value problems associated with the rotation of solid cylinders of specific incompressible materials have been studied by Green and Zerna [2] and by Chadwick, Creasy and Hart [3]. Aspects of the bifurcation and the stability of finitely deformed rotating elastic cylinders were considered by Haughton and Ogden [4] and by Hill [5]. The rotation of cylinders is not a controllable deformation for homogeneous isotropic compressible elastic materials. Additionally, Carroll [6] has shown that rotation of cylinders is not a controllable deformation for three specific (though still quite general) compressible materials, materials which he termed materials of types I, II and III. The materials treated were such that the strain energy function, expressed as a function of the principal invariants of the stretch tensors, has the separable form:

W =fW + g(4 + G),

(1.1)

where f, g and h are twice continuously differentiable functions. The three special classes of material considered in [6] have the following form: for each class, two of the functions f, g and h are linear functions and the third is an arbitrary function. Since the rotation of cylinders was found not to be controllable for these materials, the form of the deformation field describing such rotations is dependent on the form of the strain energy functions of these materials. It turns out that for some specific forms of the appropriate arbitrary function, closed form solutions of the equations of motion can be obtained. Some such solutions will be obtained in this paper and these will form the basis of the discussion of the qualitative features of the behaviour of rotating cylinders of compressible materials. We will study strain energy functions which have a particularly simple form: the strain energy functions will be quadratic in one stretch invariant and linear in another. These materials we will term the standard materials of types I, II and III, as appropriate. The standard materials of types I and III have obvious closed form solutions to the equations of motion whereas the standard material of type II does not. The emphasis of this paper will be on closed form solutions to boundary value problems associated with the rotation of cylinders of compressible materials. Accordingly, we will concentrate on the standard material of type I as the associated boundary value problems are especially amenable to analysis. t Author to whom correspondenceshould be addressed and presently at the Department of Mathematical University College Cork, Cork, Eire. 213

Physics,

214

J. G. MURPHY 2.

The response function

and F. J. ROUNEY

PRELIMINARIES

of an elastic body is described

completely

by the form of its strain energy

W = e(F),

where F is the deformation

(2.11

gradient tensor satisfying det F > 0.

(2.2)

F has the polar decompositions F=RU=VR,

(2.3)

where the rotation R is a proper orthogonal tensor and the stretch tensors U, V are positive definite and symmetric. Invariance under superposed rigid-body motions leads to W = W(U).

(2.41

The assumption of material isotropy further leads to W = Wii(i,, i2, i3),

(2.5)

where il, iz and i3 are the principal invariants of the stretch tensor I_J (and of V, since U, V have identical invariants). The stress response equations

P=$

T = i;‘PFT,

(2.6)

where P and T are the Piala and Cauchy stress tensors, then Iead to a representation (2.71 on application of the Cayley-Hamilton theorem. Carroll [6] introduced the following form of the strain energy function W =fGr) + g(G) + W3),

(2.8)

where f, g and h are twice continuously differentiable functions of the appropriate the stretch tensor. Use of (2.8) in the stress-strain relation (2.7) leads to: T = (ia’

+g’fi&tr

V-"11+ i;ifr(ii)V

- g’ff‘z)V-l.

invariant of

f2.o)

The conditions that the strain energy and the stress in a material with a strain energy function of the separable form (2.8) vanish in the reference configuration are easily seen to be: f(3) f g(3) + W) = 0,

f’(3) + 2g’(3) -I-h’(l) = 0.

(2.10)

Since the finite defo~at~on theory includes the linear theory as a limiting case, one restriction on the separable form (2.8) is immediately obvious: on restriction to infi~tesimal deformations, the bulk and shear moduli should be positive. This condition is equivalent to: f”(3) + 4g”(3) “+h”(1) +g’(3) + h’(1) = A,

g’(3) + h’(1) = -2/A,

(2.11)

where A, p are the Lam& constants for the material satisfying F>O,

3A+2~>0.

(2.12)

The simplest form of the separable strain energy function (2.8) satisfying conditions (2.10) and (2.11) will be quadratic in one stretch invariant and linear in another. Such materials we will term standard materials and we now introduce standard materials of types I, II and III.

215

Rotation of cylinders of special compressible materials

Standard material of type I R++2p)i: This material was introduced sector of this material [B].

(2.13)

-(3d+2p)i,-2&+;1+3p

by Sensenig [7] who later considered the bending of a cylindrical

Standard material of type II (2.14) Standard material of type III (2.15) We will now consider the steady rotation of cylinders of compressible

3. STEADY

ROTATION

Motions having cylindrical coordinate

materials.

OF CYLINDERS

representation

r = P(R),

8 = 0 + wt,

z = yz,

(3.1)

with dr/dR > 0 and y > 0 and where (R, 0, Z) and (r, 8, z) are coordinates of a particle before and after deformation, describe steady rotation, with angular frequency w, about the Z-axis, along with constant radial deformation and uniform axial stretch. The physical components of the deformation and stretch tensors are: F=V=diag(&,i,

y).

(3.2)

From (3.2) we can calculate the principal invariants: .

dr

r

r dr il= yEz.

iz =

ll=G+R+Y,

(3.3)

For an elastic material with strain energy function of the separable form (2.8), substitution from equations (3.2) and (3.3) in the stress response equation (2.9) gives the principal stresses as:

T, =:f’(&)+ Tee =i$f

(~+~)g’(i.)+h’(i,),

‘(iI) + ($+t)g’(iJ

+ h’(i3),

(3.4) (3.5)

and (3.6) The azimuthal and axial equations of motion are satisfied identically and the radial equation is:

-$T, + f (T, -

Tee)= -poi;‘w2r,

(3.7)

where p,, is the mass density in the reference configuration. Substitution from (3.4) and (3.5) into (3.7) yields: (3.8) ES 30:2-G

216

J. G. MURPHY and F. J. ROONEY

Substitution of the appropriate form of the strain energy function of each of the standard materials of types I, II and 111 into (3.8) will yield an equation to determine the form of the radial deformation field. We will next attempt to determine this field for each of the standard materials.

4. THE

STANDARD

MATERIAL

OF

Substitution of (2.13) into (3.8) yields the following equation radial deformation field P = P(R): ,, d’r dr R-dR’+Rz+r

t&%, ----R~-l ( A+2@

)

TYPE

I

for the determination

_=()

of the

(3.1)

Let

Now (4.1) is seen to be Bessel’s equation of order 1 and has the solution: (4.3)

where A, B are constants. Consideration of solid cylinders requires that we set B equal to zero and therefore we obtain the following motion which describes the steady rotation of a cylinder of the standard material of type I: 6, = 0 + Ol,

r=AJ,(fRj,

We suppose that the plane surfaces Z = f (A) The curved (B) There is an

undeformed

con~guration

z = yz.

(4.4)

is a solid cylinder of radius R,,. bounded by the conditions to impose is:

L. One set of natural boundary

surface is traction free. axial force of resultant T applied to the faces 2 = zt:L.. Accordingly

we set

i(Rt,) T=2Jr These boundary

!-$1

T,,rdr,

Z=rfiL.

(4.5)

conditions yield after a routine calculation: A=

TA - 2~(3A + 2@rR;

(4.6)

7rR,,{J,(X)[2A2 + 2dc(, + 4~7 - XJ,,(X)[A + 2~1% ’

and y = 3A.+ 2~ + [Td - 2~(3A + 211).?cR~1[2~.MX)- (A + 2HX-MX)I A. nAR;{(2k2 -I-2h,u +4$)5,(X) - (A +2&X&,(X)}

’

where X = oR,/s > 0. The qualitative features of the above motion can now be studied but the analysis, although straightforward, is tedious. An examination of such qualitative features is greatly facilitated by setting il = ~1and we will proceed under this assumption. We obtain: A

T, - 10

R,=

U,c,X) - 9X-&,(X) ’

y = 5 + $ (U,(X) - 3XMX)). where T, is the non-dimensionalised

(4.8) (4.9)

force given by: T, = -

T

~R;P.

(4.10)

Rotation of cylinders of special compressible materials

217

standard harmonic material

physically %3liStiC

region

non-dim. applied force Fig. 1. Physically realistic region for type I materials.

For

the above solution to be physically realistic, it must satisfy the following constraints: r > 0,

An examination have that:

of (4.8) reveals that

$0,

y>0.

(4.11)

A = 0 (and consequently r = 0)for T,= 10. Thus we must T,< 10.

(4.12)

X < 1.8,

(4.13)

Similarly requiring that drldR f 0 yields:

since, from tables, J;(X)lx=1.8 = 0. Now for 0 0, 0 < R G R. and drldR>O, OsRsR,. It remains to satisfy the kinematical constraint that y > 0. The curve y = 0 in the T,-X plane must be determined numerically, noting a result due to Sensenig [7] who found that for the special material considered here, a finite total load of amount T,= -5/2 would collapse a non-rotating cylinder down to zero height and finite radius. The region bounded by the curves y = 0, T,= 10 and X = 1.8 is given in Fig. 1. This region constitutes the domain of physically realistic behaviour (in the sense that inequalities (4.11) are satisfied) and henceforth we will assume that the applied loads and angular velocities are such that we stay within this region.

5. SOME QUALITATIVE

FEATURES

We first consider the volume change associated with the axial loading of a rotating cylinder. We define the relative volume change, vol,, in the obvious way: vol

=

c

d% - Jcr(&)*yL=lRR;L

yR;,

(5. I)

218

J. G. MURPHY

and F. J. ROONEY

standard harmonic material

-4-

-6 -

-8 -

\ t

I

\

_:

i -10 -

-12

0

\ t=8.0 \ ;\ 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1

2

non-dim. angular velocity Fig. 2. Relative volume change for type I materials.

where R, is the radial stretch of the cylindrical surface, r(R,)/Ro. In Fig. 2 we plot relative volume change vs non-dimensionalised angular velocity for five different non-dimensionalised axial loads: T,,(=t) = -2.0, 0, 2.0, 5.0, 8.0. The plots are truncated where we leave the physically realistic region of Fig. 1. In Fig. 2 we see that there are three distinct modes of material behaviour: monotonically increasing relative volume change with respect to angular velocity (t = -2.0) plots with turning points (t = 0, 2.0) and monotonically decreasing behaviour (t = 5.0, 8.0). We note that relative volume change is insensitive to angular velocity until we approach the boundary of the region of physically realistic behaviour. Thus for moderate angular velocities, we would not be able to distinguish a rotating cylinder of the standard material from a cylinder of an incompressible material from consideration of the relative volume change alone. We now consider the radial stretch of the cylindrical surface, R,. Now from (4.4) and (4.8) we conclude that T, - 10

q(x) Therefore

- 9X&(X)

J&O

(5.2)

for fixed T,, 9(T, - 10) dR d-~ = (ar1(x) _ gxJ,(x))z

PIJO - X(J ?I+ J:C

(5.3)

It is easy to show that 2I,(X)J,(X) - X(Jo(X)2 + Jl(X)‘) < 0, for X Z=1. From tables we find that W,J, - X(J$ + J:) < 0, for 0 0, we have that

dRs dy>o.

(5.4)

In particular (5.4) holds for all physically allowable X and Tn. Haughton and Ogden [4] have considered the same problem for a special incompressible material. They found that there are no turning points in the plot of non-dimensionalized angular velocity vs radial stretch, as in (5.4), for rotating cylinders under compressive axial loads. However for T,, > 0, the relationship

219

Rotation of cylinders of special compressible materials

between angular velocity and radial stretch is not unique. The axial stretch, y, is determined from (4.8) and (4.9) to be: y = 5 + (T” - 10) Therefore

U,(X) - 3XJo(X) t 8&(X) - 9X.&(X) I

(5.5)

for fixed T,, 6(T, - 10)

dy

&Ij= (arI(x)

_ gxJo(x))2

{-WJO

+ xv

(5.6)

; +m

Using the same reasoning as before, we conclude that

dy

-gp’

(5.7)

for all physically allowable X and Tn. We note from (5.2) and (5.4) that as T,+ 10, we get that y+ 5 and R,-t 0 for all allowable X. Thus a finite total load of amount T, = 10 would collapse a rotating cylinder down to zero radius and finite height. We now plot both radial and axial stretches versus non-dimensionalised angular velocity for the five axial loads considered previously: T,( =t) = -2.0, 0, 2.0, 5.0, 8.0. The plots are again truncated where we leave the physically realistic region of Fig. 1. As shown below in Figs 3 and 4, all plots are monotonic and we see that the axial stretch goes to zero for t = -2.0, 0, 2.0. The solution to the corresponding problem of plane strain is easily obtained. Setting y = 1 in (4.9), we obtain from (4.8) that A 4 E = 3X&(X) - W,(X) * Consequently

(5.8)

the radial stretch, R,, is given by: 4JdX) Rs = 3XJ,(X) - U,(X) ’

(5.9)

The satisfaction of the kinematical inequalities r > 0,

$0,

(5.10)

requires that X < 1.8, as before. A plot of R, vs angular velocity for this case is given in Fig. 5. _.._.._. t m8.0 5-

r 0 5 L iiJ ?? E

d

4-

-..-..

5

t -3.0

= -___---

;A8 t-0 t w-2.0

-

321

I _.___._.._ 1 I

.._.. -..-. 1 I I

1

1.0

1.5

0.5 Angular

1

1 2.0

velocity

Fig. 3. Radial stretch for type I materials.

0.5 Angular

1.0

1.5

2.0

velocity

Fig. 4. Axial stretch for type I materials.

220

J. G. MURPHY

and

F. J. ROONEY

4.0 3.5

-

3.0

-..-..-.

Type (

!

TypeIll

!

i

-

i ‘IO

2.5

-

i

/

5

‘j

2.0

3 5 m a:

-

.’ ./’

1.5-

1. _.H

0.5

-

I

I

I

I

0.5

Fig. 5. Plane

THE

I

1.0

Angular

6.

./’

,/-

1

, 2.0

velocity

strain

STANDARD

I 1.5

radial

stretch.

MATERIAL

OF TYPE

Substitution of (2.14) into (3.8) yields the following equation radial displacement field for the standard material of type II:

II

for the determination

;(*+2&(;+9&((;++$fy;)++=0.

(6. I)

This equation does not have an obvious closed form solution and therefore further with the analysis. We have included this section for completeness.

7. THE

Substitution

STANDARD

MATERIAL

OF TYPE

we will proceed no

III

of (2.15) into (3.8) yields:

(A+ This equation

of the

has the following

--r dr WY-& RdR 1

I

; P~wzR-o Y

(7.1)

’

solution: r2= CUR’-

P”fjJ2 4v + WY2

(7.2)

R4+/3,

where CY,/3 are arbitrary constants. Thus the following deformation of solid cylinders of the standard type III material: 2 ,.2=R2 (ypa@ 8 = 0 + WC, R2 , 1 4(A + 2P)Y2 I

field describes the rotation

z = yz.

(7.3)

Application of the boundary conditions associated with the rotation of axially loaded cylinders yields the following system of coupled non-linear equations to determine (Yand y (in contrast to the unique solution obtained for the standard material of type I), where the notation of Section 4 has been used and we have again assumed that A = p for clarity: 64y2 - (10~ - 6ay2 + 3X2)2(4cuy2 - X2) = 0, 12y4$ - 2czy2(3X2 + 10~) + X2(5y + X2) + 4y3(2 - T,) = 0. This system of equations can now be solved numerically kinematical

constraints

(4.11)

are satisfied.

The analysis

(7.4)

to obtain values of CV,y such that the is lengthy and tedious. A comparison

Rotation of cylinders of special compressible materials

221

of the qualitative features of (7.3) with the solution for the standard type I material can be immediately obtained on consideration of the corresponding plane strain problem. We will proceed therefore setting y equal to 1 in (7.4), and we obtain the following cubic equation to determine LY: 144~~ - a?(480 + 180X’) + a(400 + 360X’ + 72X4) - 100X* - 60X4 - 9X6 - 64 = 0

(7.5)

We choose the unique solution branch which has the value 1 when X equals 0. To ensure that LY is real we must restrict X so that 0 G X < 0.54. For this range of X, the kinematical constraints (5.10) can be shown to hold. The radial stretch, R,, has the form: R s

_r(R,)-

(7.6)

Ro

In Fig. 5, we plot (7.6) for the allowable range of X together with the radial stretch for the standard type I material given in (5.9). We note that the two plots have the same qualitative features: both have stretch monotonically increasing for increasing angular velocity with a large rate of increase of stretch as the appropriate angular velocity limits are approached. The most notable difference between the two materials is that the range of allowable X for the standard type I material is three times that for the standard material of type III. Acknow/edgemen~s-The encouragement.

authors would like to thank Professors M. M. Carroll and J. Casey for their advice and

REFERENCES [I] [2] :3] [4] [5] [6]

A. E. GREEN and R. T. SHIELD, Proc. R. Sec. A202,407-419 (1950). A. E. GREEN and W. ZERNA, Theoretical Elasticity, 1st edn. Oxford University Press. P. CHADWICK, C. F. M. CREASY and V. G. HART, J. Ausrral. Math. Sot. U)B, 62-96 (1977). D. M. HAUGHTON and R. W. OGDEN, Q 1. Mech. Appl. Math. 33,251-265 (1980). J. C. PATTERSON and J. M. HILL, Mech. Rex Commun. 4,69-74 (1977). M. M. CARROLL, J. Elasticity 20, 65-92 (1988). 171 C. B. SENSENIG, Commun. Pure Appl. Math. 17,451-491 (1964). _8] C. B. SENSENIG, Commun. Pure Appl. Math. 18, 147-156 (1965). (Received

25 April

1991; accepted

18 June 1991)