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Bifurcation of rotating thick-walled elastic tubes
0022-5096!80’0201-0059 $02.00’0
J. Mech. Phyx Solids Vol. 28, pp. 59-74 tc PergamonPressLtd. 1980. Printedin GreatBritain
BIFURCATION
OF ROTATING THICK-WALLED ELASTIC TUBES
D. M. HAUGHT~N
and R. W. OGDEN
School of Mathematics, University of Bath, Claverton Down. Bath BA2 7AY. England
(Received
8 October
1979)
ABSTRACT THE DEFORMATION of a circular cylindrical elastic tube of finite wall thickness rotating about its axis is examined. A circular cylindrical deformed configuration is considered first, and the angular speed analysed as a function of an azimuthal deformation parameter at fixed axial extension for an arbitrary form of incompressible, isotropic elastic strain-energy function. This extends the analysis given previously (HAUGHTON and OGDEN, 1980) for membrane tubes. Bifurcation from a circular cylindrical configuration is then investigated. Prismatic, axisymmetric and asymmetric bifurcation modes are discussed separately. Their relative importance is assessed in relation to the wall thickness and length of the tube, the magnitude of the axial extension, and the angular speed turning-points. Numerical results are given for a specific form of strain-energy function. Amongst other results it is found that (i) for long tubes, asymmetric modes of bifurcation can occur at low values of the angular speed and before any possible axisymmetric or prismatic modes and (ii) for short tubes, there is a range of values of the axial extension (including zero) for which no bifurcation can occur during rotation.
1.
INTRODUCTION
IN A RECENT paper (HAUGHTON and OGDEN, 1980) the possibility of bifurcation from a circular cylindrical configuration of a circular cylindrical elastic membrane tube rotating about its axis was investigated. The corresponding investigation for tubes of finite wall thickness is the subject of the present paper. To our knowledge this problem has not been tackled previously within the context of finite elasticity theory, although PATTERSON and HILL (1977) have examined briefly the corresponding problem for a solid cylinder. Under the assumption that the circular cylindrical shape is maintained during rotation, CHADWICK, CREASY and HART (1977) have studied, in some detail, the rotation of tubes and cylinders under a variety of boundary conditions. Their aim was to find conditions on the elastic constants occurring in the elastic strain-energy function that ensured the existence and, in some cases, uniqueness of the type of deformation considered for all values of the angular speed. Our starting point, in Section 2, is the analysis of the circular cylindrical configuration. However, we adopt boundary conditions different from those used by Chadwick. Creasy and Hart and our objectives are also different. Moreover, our
59
D. M. HAUGHTW and R. W. OGDEN
60
analysis in Section 2 is carried through for a completely general form of inconrpressihlr, isotropic elastic strain-energy function. An appropriate choice of variables enables us to obtain a concise expression (in the form of an integral) for the angular speed as a function of a single deformation parameter. This leads to a precise criterion, in terms of material properties and not involving an integral, for the existence of turning-points of the angular speed with respect to the chosen deformation parameter. Discussion of this in relation to previous work by the writers shows that turning-points can be expected, and hence the deformation is not in general ~rni+el~ determined by the angular speed. In Section 3 a general incremental deformation from an arbitrary circular cylindrical current configuration is considered, and the appropriate equations and boundary conditions specialized to cylindrical polar coordinates. Our purpose is to investigate the possible existence of circular cylindrical configurations at which the deformation can lose its circular symmetry during rotation. As in the case of a membrane (HAUGHTON and OGDEN. 1980), prismatic, axisymmetric and asymmetric modes of bifurcation are examined separately and their relative importance discussed. A limited number of analytical results are obtained but, for the most part. the equations have to be solved numerically, and this is done in respect of a particular class of elastic strain-energy functions. We find that a prismatic mode of bifurcation is possible for thin-walled tubes approximately at the turning-points of the angular speed. In the membrane limit, the result is exact. as shown by HAUGHTON and OGDEN (1980). For thick-walled tubes, bifurcation into a prismatic mode can occur before the maximum of the angular speed is reached. For tubes shorter than a certain length, axisymmetric bifurcations cannot occur when the axial extension lies within some critical range of values (dependent on the length and thickness of the tube). The same applies (with different critical values of the axial extension in general) to asymmetric bifurcations. Thus, there can exist a range of values of the axial extension for which no bifurcation can occur during rotation (the end-conditions being chosen to exclude prismatic bifurcations). Moreover, outside this range of values either an axisymmetric or asymmetric bifurcation may have priority, depending on the tube length and thickness and the axial extension. For long tubes, or large values of the axial extension, an asymmetric (or wobbling) mode of bifurcation has priority, as might be expected.
3
a.
THE CIRCULAR CYLINDRICAL CONFIGURATION
We suppose that the geometry of the tube is specified in terms of cylindrical coordinates R. 0, 2, according to 24 < R < 8,
0 < 0 < 27t,
polar
OGZdL
in the undeformed and stress-free configuration. The tube is constrained to rotate about its axis with uniform angular speed U. Let & be the (uniform) axial stretch in the material. This may be associated entirely with (I)
(01) WV os JaPu@
‘0 =
(q)EELl = (D)“O
aql Jo sa3eJlns Iwale[ aql uo uo!ix.~~ ou s! alaql leql asoddns i3M ~le!~aie!ruay3 JO &suap (mogun) aql s! d alaqM
s! A[p+y payspes IOU uogenba t.un!lqq!nba Quo aqL .sa+my_tap pglEd alouap ‘y pue y sldymqns aql pue
(5) ‘(E) UIOIJ urwqo aM “7 c y pue ‘y saq3lam luapuadapu! ayl 01 Zu!zyyads
‘(9) pue uo
‘lUp?JlSUO3
‘(E ‘z ‘1 = !)
(s)
8uye
Ai!yq!ssarduIo+
aql troy
amssald
cye~so~pLy he.wq-r~
aql s! d pue
aiaym d-!Q
= !!Q
~eyl OS ‘(E ‘z ‘1 = !)!!D Aq losual ssalls iCqme3 aql JO siuauochuo~ ledptyd aql alouap ah ‘fy bZy‘ty uo iC~~t?yatutu.k spuadap ‘au.mloA i!un .~ad‘(Ey ‘z-f *Iv)M s M uogmy Ghaua-ugs aql pqos ysela ydollos! a[q!ssaldtuoxy ue .IO,I ‘( I) ~UyxJ
‘YP -_= “p
/
c_
b’.,
.
=
H
zy
_
‘-=
.
.I
I./ .
aAey aM lu!e.wuo3 hiq!q!ssaldtuoxy aql 01 %u!p~omv .LlaA!lDadsal suo!wal!p -(A‘z ‘0) ayl 01 ihpuodsalro3 saq3la.w Ied!x+d aql fy ‘zy ‘I’! icq alouap am .auy s! 3 a.xaqm ‘Z”/ = Z
(z)
‘Jrn = o-6)
‘(zv-zzr)II.J
(I) Lq paq!map
s! uoymmojap
= z~-z*~
aql leql OS alq!ssa~dwoxy
JO
suoymba paumsse s! le!lawu au
aql
‘q 3.1 3 v
‘7”( Z 1 $G2 > () suxlal II! paypads s! [e!.Iair?uraql
leql OS ‘= ‘6,‘1 sawupJoo3 JeIod p+~puqlCs JO uogemE&xo~ (pawoJap) lua.t.uw aqi asoddns ‘(samoJ le!xo ayldoldde j! JO (pasodtu! axoJ ~e!xe ou s! aIaql uaqm)
JO uo!iexIdd~ aql Lq) pax_~Jpplayaq hu 19
D. M. HAUGHTONand R. W.
62
OGDEN
Integration of (9) and use of (10) leads, after change of variable from r to I by means of (l), to (11) In deriving (11) use has been made of (7), and we have adopted the notation A0= a/A, & = b/B. From (1) we obtain the connexions
/I,‘&-- 1 =$(;lpi-l)=4(1.:i;--l).
(12)
where q = B2jA2 2 1.
(13)
i, > /I 3 E.,
(14)
It follows that
with equality holding if and only if CI)= 0 (non-trivially loading, in which case the deformation is homogeneous). It follows from (ll), with the help of (12), that (B2-A’)(~,-i._‘;L,‘)pw
g
= eL(i,,, L)/i,a
in the presence of axial
@*(i,, 1,)/l.,.
(15)
This reduces to the corresponding formula given by HAUGHTON and OGDEN(1980) for the membrane in the limit B -+ A. The existence of a turning-point of rr) at fixed & is ensured if there exists a value of L, > 1 at which the right-hand side of (1.5) vanishes, recalling the connection (12). Precisely the same criterion is found for the existence of pressure turning-points for an internally pressurized thick-walled tube (HAUGHTON and OGDEN, 1979b), and the reader is referred to that paper for further discussion in relation to material properties. A necessary condition for turning-points to exist at fixed i, is that ;lC&-- @A= 0
(16)
for some 2 in the interval (&, i,i. The equation (16) is necessary and sufficient for the existence of turning-points of o in respect of a membrane. This has been discussed by HAUGHTON and OGDEN(1980) and its implications anaiysed for specific forms of strain-energy function. No further discussion is required here. Finally, we note that the total axial force N on the ends of the tube required to maintain the axial stretch 2; is given by N = 2n
‘3 f122r dr. sb
(17)
The right-hand side of (17) can be rewritten in a number of different ways [see, for example, TRUESDELL and NOLL (1965, Section 56)], none of which we shall require here.
63
Bifurcation of rotating thick-walled elastic tubes
3.
THE
AND BIFURCATION
INCREMENTALEQUATIONS
ANALYSIS
We write the equation of motion as div s = px,,
(18)
in general, where s is the nominal stress tensor, div is the divergence operator relative to the undeformed configuration, and p is the density of the material. Also, x denotes the position vector of a material particle in the current (deformed) configuration and the subscript t refers to the material time-derivative. We assume that the point x undergoes an incremental displacement, ir say, with corresponding increment S in s. Takjng the increment in (18) and then evaluating (18) in the current configuration leads to the incremental equation of motion div S,, = &,,
(19)
where the subscript zero denotes evaluation in the current configuration and div is the divergence operator there. Let e,, e,, e3 be unit basis vectors corresponding to the cylindrical polar coordinates 8. z, r respectively, and write x = uer +we,+ue,.
(20)
Then the incremental acceleration due to the angular speed o and its increment &I is given by it, = - (w’u + Zo.Gr)e, - cozcel.
(21)
Note that there is no time-dependence in (2 1) and the problem may be treated as a (quasi-)static one with a non-zero body force. Assuming no traction is imposed on the lateral surfaces of the cylinder we find that the incremental boundary conditions on these surfaces may be written S& = 0
on
r = a, b,
(22)
where n denotes the (outward) normal and T denotes the transpose. The incremental constitutive relation is written s, = &l+m-$,
(23)
where n = &/dx, 6 denotes the identity and i, the increment in p, and W is the (fourth-order) tensor of instantaneous moduli. Full details of W and, in particular, its components on the principal axes e,, e,, e3 can be found in HAUGHTON and OGDEN (1978. 1979a). Because of the incompressibility constraint, we have tr (n) = 0.
(24)
Relative to the chosen basis. n =[:;zjY;
z
s],
(25)
D.
64
M. HAUGHTON and R. W. OGDEN
and (24) reduces to 11,+ (II + rR)/‘r+ 1(‘: = 0.
(26)
where the subscripts denote partial derivatives. Use of (21), (23) and (25) in (19) leads to the equations
r&/r = ./A3131v,,+(r.d;13,
+.~3131)Cn/r-(r.~jt;131
-I-.3,,,,)c/r’
+(,~~I12++.~~22r)~B,/r+(r.~~lj1+.~~313++.~3111,+pr2~2)~~g/r2 +(./A1331+.~~133)uIH/‘r+.~~11Lt’eer~+.Az121c,,,
(28)
ji = (r.~;3;,,,+.~~232)M’,lr+.~3232\Cr, +(r-~3;Z,z+.~~23Z+-~‘112-^~+pr2Lf~2)ur/r +
(3,332
+.d22,,)u,r+./A,,,2~,,/r2
+-d2222w_z+(.d,221
+A,,z2)ceZ/r,
(29)
and rearrangements of the terms on the right-hand sides of (27)-(29) may be made by means of (26). The prime denotes differentiation with respect to r, since the underlying deformation, and hence .diju, depends only on r. Expressions for the components .~ijal occurring above can be found in HAUGHTON and OGDEN(1978), for example. We merely note here the connexions Bijji = J,,-
fli = .ajiij
(i #
j),
On use of (10) and (293 and the assumption .dijij # 0 (i the boundary conditions (22) become
#
j) it is easily shown that
rv, + u, - 1’ = 0, \L’,+uz = 0, (J~~33+p)u,+.~1133(11+o,)/r+-~2233~._-_
on r = a, b.
(30)
= 0, I
From (5) and (9) we have rp’-pr2c02
= ra;+o,--a,,
(31)
and this may be used to eliminate p’ or to2 from (27)-(29). The equations are solved by means of, for example, the substitutions u = f(r) cos (mtl)
sin (az),
c = g(r) sin (me) sin (c(z), \L’= h(r) cos (mU) cos (CC), jr = k(r) cos (me) sin (~2). I
(32)
Bifurcation of rotating thick-walled elastic tubes
65
By taking appropriate combinations of such solutions the boundary conditions on the ends of the tube can be met, as discussed by HAUGHTON and OGDEN ~1979b~. In particular, and in order to illustrate the results, we may take r=tm/l
(n=O,1.2
,... ).
(33)
The values (33) are consistent with the end conditions if = r = Sat2 = 0
on
2 = 0, I,
(34)
for example. The latter of (34) corresponds to holding the current axial loading fixed po~~f~~jse. When h(r) is eliminated by use of (26), equations (27)-(29) with (32) become k’= (~.6;,33-r.~;233-.~,1*1+-~11*2 4-8
2332 -m*.~,3,3-n21-2.~,323fpr202)f;lf2
+()‘~~333-r.~2233+rp’+.~33333S..?jll122-2~)2133-.~2332)f)/r +- @,,,, +(r.@;,,,-
-.@,,,YJ -d2331)f” r.~~233+~~,122f~~2332-~-iA11,1-~~13r3)mgl~2
(35)
+(.~,,33+.~,331-.~2233-.~2332)mCI’l~r
mk/r= (~.~~,,,+.JA,,31+~1,11-.~31122-.~12tl+pr202)mf/r” +(@ir33+.@133i-. 1122-.~1221)mf’/r +(‘.4?p;i3i +63r3i +: 2 r 2,.@2121 -m2(J?I122+.@3,22t
-Jfl,II)~d~2
(361
-(Y.~S;,,l+.~3131)g’lr--P31318~.
a2k = .n,,,,~‘“+(r-~~,,,+2.~,,,,~f“~r +~r.;P;,,2-.~~~~2-n12.~~212-~~32(.~~222-~2~~~-.~Z233f)~/P2 -ilr%232-.~3232+m2~~,212 -r2r2(r.~2332+rp’+.~2332+.d,,22-.~,222)}f/r3 +~~3232m~“l~+~~~~~232--~3232)m~‘/r2
(37)
-(~~~~232-.~3232fm2.~~2,2-r2r2(.~,,22-.W2222+.~,22,~)mg/~3,
&.Ihaving been set equal to zero since this corresponds to a cylindrically symmetric motion. Correspondingly, the boundary conditions (30) become 0,
rg’-,4-mf==
r2y + rf’ + (a’? - 1 )j”+ m2.f= 0, (.~3333-.lA2233+63)r~SI+(,~lr33-.R2233)(f~mg)-rk
=
on r = a, b.
(3X)
0,
3.1 Fr~smat~c bj~~rcat~ons First we examine solutions of the governing equations (27tf29) that are independent of z. Equation (29) reduces to an equation for W, decoupled from (27) and (28). with boundary conditions W, = 0 on r = a, b. The solution satisfying the
D. M. HALJGHTON and R. W.
66
OGDEN
end-conditions u’ = 0 on z = 0,l is easily shown to be the trivial one and we therefore set w 3 0. The problem now becomes a two-dimensional one and we replace (32) by U = f(r) cos (me), c = g(r) sin (m8),
(39)
J! = k(r) cos (me), I and we note that end-conditions on the cylinder are no longer relevant [in particular, (34) can only be satisfied trivially]. The governing equations are now (35) and (36) with u = 0, (37) being satisfied identically, and f(r) and g(r) are connected by rf’+f+mg
= 0
(40)
from (26) with (39). Elimination of k(r) and g(r) leads to the equation [s,,,,r3f”‘f(r.8;131
+3.?83,3,)rZf”
+(r.?$;r3r -.a,,,,
+m2(2d
+(m2-
+r.O;,,,
I){r2.8;131
1133+2~1331-.~~11,-.~3333):rf’1’ +(m2-
I~~,,,,
+m2(o,
-a,)}j/r
= 0,
(41)
while the boundary conditions (38) reduce to r2f”+rf’+(m2-l)f=O
on
r=a,b
(42)
r = a, b.
(43)
and 1 3131r3fnr+(r.ST3,31
+4.h?,,,,)r2f”
+(r!~;131+,~3L31+~(2~1133-.~~111-.~3333+.~~331-~3))rf +{(m2-l)(r.B’3,3,
+.W3,,,)+pr202m2)f=
0
on
The equation (41) and the boundary condition (42) are precisely the same as those obtained for the corresponding problem of a pressurized tube (HAUGHTON and OGDEN, 1979b), but (43) differs by the inclusion of the term in CC). For m = 1 the solution forf’(r) from (41) may be written formally as
where cr. c2, c3 are constants, and the boundary conditions reduce to rf”+f’ prZw2f+c1
= 0, = 0. I
on
r = a, b.
(45)
However, the term involving o makes it impossible to obtain a concise and explicit bifurcation criterion like that given by HAIJGHTON and OGDEN (1979b) for a pressurized tube (for which cr = 0 is required to satisfy the boundary conditions). For a rotating membrane tube, HAUGHTON and OGDEN (1980) have shown that the mode m = 1 prismatic bifurcation is not possible under the (physically justified) constitutive assumptions they adopted. Since the constitutive inequalities necessary
Bifurcation of rotating thick-walled elastic tubes
67
and sufficient for the existence of bifurcations of the membrane are necessary for the existence of the corresponding bifurcations of a thick-walled tube, the system (44) and (45) does not have a non-trivial solution in general. We recall that the m = 1 prismatic mode is not possible in respect of a pressurized thick-walled tube when a certain constitutive inequality is imposed (HAUGHTON and OGDEN, 1979b). HAUGHTON and OGDEN (1980) also showed that bifurcation into a prismatic mode first becomes possible for a membrane ~*/ren the angular speed reaches a maximum. This corresponds to mode number m = 2. Moreover, higher-order modes do not occur for the forms of strain-energy function used by Haughton and Ogden. For m L 2, the equation (41) cannot be integrated even once analytically (even for very simple forms of strain-energy function), so that the coincidence of the m = 2 bifurcation and maximum angular speed points cannot be established for thickwalled tubes (and cannot in general be expected). For m = 2, we have obtained numerical solutions of (41)-(43). The resulting bifurcation curves are plotted in the (i.,, &)-plane and compared with the curves of the angular speed turning-points for A/B = O-99, 0.85, 0.5 in Fig. 1. For A/B = 0.99, the bifurcation curve cannot be distinguished from the curve of turning-points of w on the scale used in Fig. 1. This is to be expected, since a tube of thickness ratio 0.99 is effectively a membrane. For A/B = O-85, the same applies except when i., and A, are both greater than about 3. Then the difference between the curves is only small. Even for A/B = O-5there is little difference in the curves for R, less than about 4. but in this case we can see that bifurcation occurs before the maximum of (I) is reached. The divergence of the curves increases with increasing B/A.
/g
___________________
1. Plot of the m = 2 prismatic mode bifurcation curves in the (j.,, i,Fplane for A.‘B = 0.99. 0.85, 03 (continuous curves) compared with the (broken) curves of turning-points of o. Also shown is the curve of w = 0 (independent of .4.!B).
FIG.
D. M. HALJGHTON and R. W. OGDEN
68
The calculations have been carried out in respect of the strain-energy function described in detail by HAUGHTON and OGDEN (1979a). Briefly, we record that
where c(, = 1.3, p: = 1.491,
x2 = 5.0,
x3 = -2.0,
/J: = 0003,
(47)
/J: = - 0.023
and &! = p,/,u (r = 1,2,3), p being the ground-state
shear modulus defined by
As in the case of a membrane the values (47) rule out bifurcations corresponding
to
m 3 3.
3.2 Axisymmetric
bifurcations
Next, we examine solutions of (27)-(29) that are independent of 8. Equation (28) leads to an equation for v and the boundary condition (30) gives rv,- v = 0 on r = a, b. It is easily shown that the resulting Sturm-Liouville system has only the trivial solution at any stage of the deformation when the boundary conditions (34) are imposed and the inequalities .#ijij > 0, i # j, adopted. We now take m = 0 and eliminate k(r) from (35) and (37) to obtain rS{r-1[r-‘.~3232(r2f,,+rfl-f)]‘)’ -~2r2(r[r(.~2222
+.I,,,,
- 2282233 - 253223m’ +(r2a’;,-rZ~)‘;2j2+r.~;222+r.~;133-r:~;122-r~~;233-r.~;223 +2d,,22+~,22,-.~,,,,-.~~,,2+pr2~2)fJ + r4r4.W,,,,
The corresponding
f = 0.
(49)
boundary conditions are r*f”+rf’+(a*r*-l)f=
0
on
r = a, b
(50)
and r*[r-‘.~,,,,(r*f”+rf,-f)]’ -u2r2[(.d,222
+.aJjj3
- 2.A,,,, -2e@‘3223+d,,,,)rf
+(ra~,-r.JA~2,2+.JA2,,2+.W,,,,-.~,,2,-.~22,,-.~A3223)fl
and a is given by (33). For s( = 0, equation (49) leads to the solution
= 0 on r = a, b, (51)
f(r) = Cr+Dr-‘,
where C and D are arbitrary constants, and the boundary conditions (50) and (51) are automatically satisfied. When the tube has finite length this solution corresponds to maintenance of the circular cylindrical shape, but for L + rx an axisymmetric solution of the type discussed by HAUGHTON and OGDEN (1979b) is possible.
Bifurcation of rotating thick-walled elastic tubes
69
For x # 0, we choose n = 1 in (33) in view of the dual character of the quantities n and L (recalling I = &L) that was exploited previously by the writers (HAUGHTON and OGDEN, 1979a,b, 1980). Thus, al = x and the parameters L/B and A/B are (independently) at our disposal. We have carried out numerical calculations for a number of values of L/B, A/B and & in respect of the strain-energy function (46) with the material constants (47). In Fig. 2(a, b), for A/B = 0235 and 0.5 respectively, the bifurcation curves calculated from (49)-(51) are plotted in the (A,, &)-plane for L/B = 10,20,40 in order to illustrate the dependence of the curves on the length of the tube. The value A/B = 0.85 is chosen as representative of a thin-walled tube so that comparison can be made with the corresponding results for a membrane (HAUGHTON and OGDEN, 1980). The general features shown in Fig. 2(a) are also those obtained for a membrane with some minor differences of detail. Thick-walled tubes are represented by the value A/B = 05, and it can be seen from Fig. 2(a, b) that the character of the results for a thick-walled tube is broadly similar to that for a thin-walled tube. The picture changes for very thick-walled tubes as Fig. 3 illustrates. The curves of(i) w = 0 and (ii) turning-points of w at fixed A, are included in Fig. 2(a. b) for reference purposes. We remark that bifurcation into an axisymmetric mode may occur before or after the maximum of o depending on the values of L/B, A/B and AZ.Moreover, for values of & less than some critical value no turning-points of 0 occur. In Fig. 3 the critical values of i., are plotted against A/B for iZ = 2,3,4,5 to illustrate the dependence of bifurcation on the tube thickness for the representative value L/B = 10. For very thick tubes bifurcation into an axisymmetric mode of deformation cannot occur. For a pressurized tube the asymmetric (or bending) mode of bifurcation is generally more important than the axisymmetric mode when the tube is in axial compression. Roughly speaking, the converse is true when the tube is in axial tension (HAUGHTON and OGDEN, 1979b). For a rotating tube, on the other hand, the priorities are not so clear-cut, as we show in Section 3.3. Indeed, the asymmetric (or wobbling) mode will dominate when the tube is in Jxial tension except for short tubes and certain ranges of values of &. This behaviour depends markedly on the value of A/B. In axial compression, the asymmetric mode also dominates in general. Note that the axisymmetric bifurcation points on o = 0 in Fig. 2(a, b) correspond to bifurcation of a tube under pure axial compression, coinciding with those calculated by HAUGHTONand OGDEN (1979b). The curves shown in Fig. 2(a, b) are similar to those obtained by HAUGHTONand OGDEN (1979b) for the axisymmetric bifurcation of a pressurized tube. Further discussion of such curves is contained in that paper. We now examine asymmetric modes of bifurcation. 3.3 Asymmetric
bifurcations
With dependence on both 0 and z retained no progress can be made analytically. We have therefore carried out numerical calculations on the basis of equations (35)(37) with boundary conditions (38). For m 2 2, our results show that there are no
D. M. HAUGHTON
and R. W. OGDEN
UI=O
I i I 1
1
,
3
2
J
I
4
5
AZ (Iif
I
t 2
t 3
I
f
4
5
A,
FIG. 2. Plot of the axisymmetric mode bifurcation curves in the (A_,A&plane for L/B = 10.20.40 together with the curve of OJ = 0: (a) A!B = 085 and (b) A 6 = 0.5. The curve of turning-points of cu is also shown (broken curve).
bifurcations (r # 0): but for m = 1 and the values of u given by (33), bifurcations are possible. In fact, we choose xl= x and a selection of values of Ljl3 as in Section 3.2, The numerical method is described briefty in Section 3.4. In Fig. 4(a, b, c) for A/B = @99,0%5,0.5 respectively, we plot i, against i_, for the
Bifurcation
of rotating
1
elastic tubes
I
I
08
1.0
thick-walled
06
04
71
I
,
02
C
f-JIB
FIG. 3. Plot of the axisymmetric
mode critical
values of i, against
A/B for L/B = 10 and i,, = 2.3.4.5.
values L/B = 5,6,7, 10. For longer tubes the initial (or n = 1) bifurcation curves lie in the narrow range between the curve for L/B = 10 and that for o = 0. This shows that long tubes “wobble” sooner than shorter tubes as o is increased from zero, which is consistent with physical intuition.
I
I I
I
2
I
3
A* (a! FIG. 4. Plot
of the asymmetric
mode
bifurcation
curves
in the (i.,, i,)_plane
AIB = 0.99. (b) A:B = 0.85and (c) A,B = 0.5.
for LIB = 5.6.7.
10: (a)
?2
D. M. HAUGHTON and R. W. OGDEN
The curves, in Fig. 4(a) particularly. shoutd be compared with those obtained by for a rotating membrane tube. It should be pointed out, however, that for a given value of LIB, more curves appear in the ~~~~ran~ case than are shown in Fig. 4(a). This is because integration through the tube wail thickness with respect to r is implicit for a membrane. ~~n~~ue~tiy, the boundary conditions applied to the g~~~r~~solution at the ends of the tube art: not poj~t~ise co~dit~~~s such as we have used to obtain (33). The “average” e~d-co~d~t~o~s yield HAUGHTON and OGDEN (2980)
Bifurcation of rotating thick-walled elastic tubes
13
values of r other than (33) and, therefore, bifurcation curves additional to those shown in Fig. 4(a). Application of “average” end-conditions to tubes of finite wall thickness would also yield additional values of z, possibly dependent in a complicated way on iL, and I.,. However, the (i, &)-dependent values of x obtained (implicitly) by HAUGHTON and OGDEN (1980) for a membrane do not yield bifurcation phenomena
whose features are essentially different from those corresponding to (33). The same applies to tubes of finite wall thickness. Figure 4 shows that under axial tension, tubes “wobble” at quite low values of the angular speed (well below the maximum in general). Exceptions to this are provided by short tubes. For a short tube there is a range of values of i., (in axial tension) within which either (i) an axisymmetric mode of bifurcation has priority over any poss;ble asymmetric mode, or (ii) only an axisymmetric mode is possible, or (iii) no axisymmetric or asymmetric bifurcation can occur (refer to Figs 2 and 4). In particular, for a short tube there is a critical value of i.: (> l), AZ, say, depending on L/B such that asymmetric mod_es cannot occur if iZ < AZCin axial tension. This is consistent with physical intuition since increase in length (either through L. or A,) can be expected to increase the tendency to “wobble”. The competition between asymmetric and axisymmetric bifurcations in axial tension described here contrasts markedly with that for a pressurized tube (HAUGHTON and OGDEN, 1979b).
In axial compression, comparison of Figs 2 and 4 shows that an asymmetric mode of bifurcation always has priority over any possible axisymmetric mode, but for short tubes there is again a range of values of AZ(G 1) for which no bifurcation occurs. The dependence of bifurcation on L/B for short tubes and on A/B for thick-walled tubes can be particularly marked in the case of a pressurized tube. The same is true for a rotating tube, as Fig. 3 illustrates. However, rather than elaborate on the details for short and/or thick tubes we have chosen here merely to highlight the mainfeatures of the bifurcation phenomena. Reference can be made to the paper by HAUGHTON and OGDEN (1979b) for the detailed changes which can be expected for short or thick tubes. 3.4 The numerical method Equations form
(35)-(37)
after suitable non-dimensionalization, u:=F,(u,r),
i=l,...,
6,
are written in the (52)
where u(r) z (f,f’,f”, g, g’, k), and the prime denotes differentiation with respect to r. For each value k = 1, . . ., 6, the initial-value problem defined by (52) with ui(a) = fiik, i = 1, . . ., 6,
is solved using a fourth-order Runge-Kutta method. We write uk (k = 1, . .. 6) for the six solutions and h u = c CkUk k=l
(53)
D. M. HAUGHT~Nand R. W. OGDEN
74
for the solution of (52) subject to the boundary conditions (38), recalling the connection (12) between u and b, where the ct are arbitrary constants. Substitution of (53) into (38) leads to a non-trivial solution for u provided a certain 6 x 6 determinant vanishes. This bifurcation criterion can be satisfied as accurately as required by appropriate values of a provided such values exist in the domain under consideration. ACKNOWLEDGEMENT
The work of D.M.H. was supported by a U.K. Science Research Council Research Grant.
REFERENCES CHADWICK, P., CREASY,
1977
J. Aust. math. Sot. B20, 62.
C. F. M. and HART, V. G. HAUGHTON, D. M. and OGDEN,, R. W.
1978
J. Mech. Phys. Solids 26, I 11
1979a b 1980
Ibid. 27, 179. Ibid. 27, 489. Math. Proc. Cambridge Phil. Sot. 87, 357. Mech. Res. Comm. 4, 69. Handbuch der Physik (edited by S. Fliigge), Vol. 11113. The Non-linear Field Theories of Mechanics. Springer-Verlag, Berlin.
PATTERSON,J. C. and HILL, J. M. TRUESDELL,C. and NOLL, W.
1977 1965
J. Mech. Phyx Solids Vol. 28, pp. 59-74 tc PergamonPressLtd. 1980. Printedin GreatBritain
BIFURCATION
OF ROTATING THICK-WALLED ELASTIC TUBES
D. M. HAUGHT~N
and R. W. OGDEN
School of Mathematics, University of Bath, Claverton Down. Bath BA2 7AY. England
(Received
8 October
1979)
ABSTRACT THE DEFORMATION of a circular cylindrical elastic tube of finite wall thickness rotating about its axis is examined. A circular cylindrical deformed configuration is considered first, and the angular speed analysed as a function of an azimuthal deformation parameter at fixed axial extension for an arbitrary form of incompressible, isotropic elastic strain-energy function. This extends the analysis given previously (HAUGHTON and OGDEN, 1980) for membrane tubes. Bifurcation from a circular cylindrical configuration is then investigated. Prismatic, axisymmetric and asymmetric bifurcation modes are discussed separately. Their relative importance is assessed in relation to the wall thickness and length of the tube, the magnitude of the axial extension, and the angular speed turning-points. Numerical results are given for a specific form of strain-energy function. Amongst other results it is found that (i) for long tubes, asymmetric modes of bifurcation can occur at low values of the angular speed and before any possible axisymmetric or prismatic modes and (ii) for short tubes, there is a range of values of the axial extension (including zero) for which no bifurcation can occur during rotation.
1.
INTRODUCTION
IN A RECENT paper (HAUGHTON and OGDEN, 1980) the possibility of bifurcation from a circular cylindrical configuration of a circular cylindrical elastic membrane tube rotating about its axis was investigated. The corresponding investigation for tubes of finite wall thickness is the subject of the present paper. To our knowledge this problem has not been tackled previously within the context of finite elasticity theory, although PATTERSON and HILL (1977) have examined briefly the corresponding problem for a solid cylinder. Under the assumption that the circular cylindrical shape is maintained during rotation, CHADWICK, CREASY and HART (1977) have studied, in some detail, the rotation of tubes and cylinders under a variety of boundary conditions. Their aim was to find conditions on the elastic constants occurring in the elastic strain-energy function that ensured the existence and, in some cases, uniqueness of the type of deformation considered for all values of the angular speed. Our starting point, in Section 2, is the analysis of the circular cylindrical configuration. However, we adopt boundary conditions different from those used by Chadwick. Creasy and Hart and our objectives are also different. Moreover, our
59
D. M. HAUGHTW and R. W. OGDEN
60
analysis in Section 2 is carried through for a completely general form of inconrpressihlr, isotropic elastic strain-energy function. An appropriate choice of variables enables us to obtain a concise expression (in the form of an integral) for the angular speed as a function of a single deformation parameter. This leads to a precise criterion, in terms of material properties and not involving an integral, for the existence of turning-points of the angular speed with respect to the chosen deformation parameter. Discussion of this in relation to previous work by the writers shows that turning-points can be expected, and hence the deformation is not in general ~rni+el~ determined by the angular speed. In Section 3 a general incremental deformation from an arbitrary circular cylindrical current configuration is considered, and the appropriate equations and boundary conditions specialized to cylindrical polar coordinates. Our purpose is to investigate the possible existence of circular cylindrical configurations at which the deformation can lose its circular symmetry during rotation. As in the case of a membrane (HAUGHTON and OGDEN. 1980), prismatic, axisymmetric and asymmetric modes of bifurcation are examined separately and their relative importance discussed. A limited number of analytical results are obtained but, for the most part. the equations have to be solved numerically, and this is done in respect of a particular class of elastic strain-energy functions. We find that a prismatic mode of bifurcation is possible for thin-walled tubes approximately at the turning-points of the angular speed. In the membrane limit, the result is exact. as shown by HAUGHTON and OGDEN (1980). For thick-walled tubes, bifurcation into a prismatic mode can occur before the maximum of the angular speed is reached. For tubes shorter than a certain length, axisymmetric bifurcations cannot occur when the axial extension lies within some critical range of values (dependent on the length and thickness of the tube). The same applies (with different critical values of the axial extension in general) to asymmetric bifurcations. Thus, there can exist a range of values of the axial extension for which no bifurcation can occur during rotation (the end-conditions being chosen to exclude prismatic bifurcations). Moreover, outside this range of values either an axisymmetric or asymmetric bifurcation may have priority, depending on the tube length and thickness and the axial extension. For long tubes, or large values of the axial extension, an asymmetric (or wobbling) mode of bifurcation has priority, as might be expected.
3
a.
THE CIRCULAR CYLINDRICAL CONFIGURATION
We suppose that the geometry of the tube is specified in terms of cylindrical coordinates R. 0, 2, according to 24 < R < 8,
0 < 0 < 27t,
polar
OGZdL
in the undeformed and stress-free configuration. The tube is constrained to rotate about its axis with uniform angular speed U. Let & be the (uniform) axial stretch in the material. This may be associated entirely with (I)
(01) WV os JaPu@
‘0 =
(q)EELl = (D)“O
aql Jo sa3eJlns Iwale[ aql uo uo!ix.~~ ou s! alaql leql asoddns i3M ~le!~aie!ruay3 JO &suap (mogun) aql s! d alaqM
s! A[p+y payspes IOU uogenba t.un!lqq!nba Quo aqL .sa+my_tap pglEd alouap ‘y pue y sldymqns aql pue
(5) ‘(E) UIOIJ urwqo aM “7 c y pue ‘y saq3lam luapuadapu! ayl 01 Zu!zyyads
‘(9) pue uo
‘lUp?JlSUO3
‘(E ‘z ‘1 = !)
(s)
8uye
Ai!yq!ssarduIo+
aql troy
amssald
cye~so~pLy he.wq-r~
aql s! d pue
aiaym d-!Q
= !!Q
~eyl OS ‘(E ‘z ‘1 = !)!!D Aq losual ssalls iCqme3 aql JO siuauochuo~ ledptyd aql alouap ah ‘fy bZy‘ty uo iC~~t?yatutu.k spuadap ‘au.mloA i!un .~ad‘(Ey ‘z-f *Iv)M s M uogmy Ghaua-ugs aql pqos ysela ydollos! a[q!ssaldtuoxy ue .IO,I ‘( I) ~UyxJ
‘YP -_= “p
/
c_
b’.,
.
=
H
zy
_
‘-=
.
.I
I./ .
aAey aM lu!e.wuo3 hiq!q!ssaldtuoxy aql 01 %u!p~omv .LlaA!lDadsal suo!wal!p -(A‘z ‘0) ayl 01 ihpuodsalro3 saq3la.w Ied!x+d aql fy ‘zy ‘I’! icq alouap am .auy s! 3 a.xaqm ‘Z”/ = Z
(z)
‘Jrn = o-6)
‘(zv-zzr)II.J
(I) Lq paq!map
s! uoymmojap
= z~-z*~
aql leql OS alq!ssa~dwoxy
JO
suoymba paumsse s! le!lawu au
aql
‘q 3.1 3 v
‘7”( Z 1 $G2 > () suxlal II! paypads s! [e!.Iair?uraql
leql OS ‘= ‘6,‘1 sawupJoo3 JeIod p+~puqlCs JO uogemE&xo~ (pawoJap) lua.t.uw aqi asoddns ‘(samoJ le!xo ayldoldde j! JO (pasodtu! axoJ ~e!xe ou s! aIaql uaqm)
JO uo!iexIdd~ aql Lq) pax_~Jpplayaq hu 19
D. M. HAUGHTONand R. W.
62
OGDEN
Integration of (9) and use of (10) leads, after change of variable from r to I by means of (l), to (11) In deriving (11) use has been made of (7), and we have adopted the notation A0= a/A, & = b/B. From (1) we obtain the connexions
/I,‘&-- 1 =$(;lpi-l)=4(1.:i;--l).
(12)
where q = B2jA2 2 1.
(13)
i, > /I 3 E.,
(14)
It follows that
with equality holding if and only if CI)= 0 (non-trivially loading, in which case the deformation is homogeneous). It follows from (ll), with the help of (12), that (B2-A’)(~,-i._‘;L,‘)pw
g
= eL(i,,, L)/i,a
in the presence of axial
@*(i,, 1,)/l.,.
(15)
This reduces to the corresponding formula given by HAUGHTON and OGDEN(1980) for the membrane in the limit B -+ A. The existence of a turning-point of rr) at fixed & is ensured if there exists a value of L, > 1 at which the right-hand side of (1.5) vanishes, recalling the connection (12). Precisely the same criterion is found for the existence of pressure turning-points for an internally pressurized thick-walled tube (HAUGHTON and OGDEN, 1979b), and the reader is referred to that paper for further discussion in relation to material properties. A necessary condition for turning-points to exist at fixed i, is that ;lC&-- @A= 0
(16)
for some 2 in the interval (&, i,i. The equation (16) is necessary and sufficient for the existence of turning-points of o in respect of a membrane. This has been discussed by HAUGHTON and OGDEN(1980) and its implications anaiysed for specific forms of strain-energy function. No further discussion is required here. Finally, we note that the total axial force N on the ends of the tube required to maintain the axial stretch 2; is given by N = 2n
‘3 f122r dr. sb
(17)
The right-hand side of (17) can be rewritten in a number of different ways [see, for example, TRUESDELL and NOLL (1965, Section 56)], none of which we shall require here.
63
Bifurcation of rotating thick-walled elastic tubes
3.
THE
AND BIFURCATION
INCREMENTALEQUATIONS
ANALYSIS
We write the equation of motion as div s = px,,
(18)
in general, where s is the nominal stress tensor, div is the divergence operator relative to the undeformed configuration, and p is the density of the material. Also, x denotes the position vector of a material particle in the current (deformed) configuration and the subscript t refers to the material time-derivative. We assume that the point x undergoes an incremental displacement, ir say, with corresponding increment S in s. Takjng the increment in (18) and then evaluating (18) in the current configuration leads to the incremental equation of motion div S,, = &,,
(19)
where the subscript zero denotes evaluation in the current configuration and div is the divergence operator there. Let e,, e,, e3 be unit basis vectors corresponding to the cylindrical polar coordinates 8. z, r respectively, and write x = uer +we,+ue,.
(20)
Then the incremental acceleration due to the angular speed o and its increment &I is given by it, = - (w’u + Zo.Gr)e, - cozcel.
(21)
Note that there is no time-dependence in (2 1) and the problem may be treated as a (quasi-)static one with a non-zero body force. Assuming no traction is imposed on the lateral surfaces of the cylinder we find that the incremental boundary conditions on these surfaces may be written S& = 0
on
r = a, b,
(22)
where n denotes the (outward) normal and T denotes the transpose. The incremental constitutive relation is written s, = &l+m-$,
(23)
where n = &/dx, 6 denotes the identity and i, the increment in p, and W is the (fourth-order) tensor of instantaneous moduli. Full details of W and, in particular, its components on the principal axes e,, e,, e3 can be found in HAUGHTON and OGDEN (1978. 1979a). Because of the incompressibility constraint, we have tr (n) = 0.
(24)
Relative to the chosen basis. n =[:;zjY;
z
s],
(25)
D.
64
M. HAUGHTON and R. W. OGDEN
and (24) reduces to 11,+ (II + rR)/‘r+ 1(‘: = 0.
(26)
where the subscripts denote partial derivatives. Use of (21), (23) and (25) in (19) leads to the equations
r&/r = ./A3131v,,+(r.d;13,
+.~3131)Cn/r-(r.~jt;131
-I-.3,,,,)c/r’
+(,~~I12++.~~22r)~B,/r+(r.~~lj1+.~~313++.~3111,+pr2~2)~~g/r2 +(./A1331+.~~133)uIH/‘r+.~~11Lt’eer~+.Az121c,,,
(28)
ji = (r.~;3;,,,+.~~232)M’,lr+.~3232\Cr, +(r-~3;Z,z+.~~23Z+-~‘112-^~+pr2Lf~2)ur/r +
(3,332
+.d22,,)u,r+./A,,,2~,,/r2
+-d2222w_z+(.d,221
+A,,z2)ceZ/r,
(29)
and rearrangements of the terms on the right-hand sides of (27)-(29) may be made by means of (26). The prime denotes differentiation with respect to r, since the underlying deformation, and hence .diju, depends only on r. Expressions for the components .~ijal occurring above can be found in HAUGHTON and OGDEN(1978), for example. We merely note here the connexions Bijji = J,,-
fli = .ajiij
(i #
j),
On use of (10) and (293 and the assumption .dijij # 0 (i the boundary conditions (22) become
#
j) it is easily shown that
rv, + u, - 1’ = 0, \L’,+uz = 0, (J~~33+p)u,+.~1133(11+o,)/r+-~2233~._-_
on r = a, b.
(30)
= 0, I
From (5) and (9) we have rp’-pr2c02
= ra;+o,--a,,
(31)
and this may be used to eliminate p’ or to2 from (27)-(29). The equations are solved by means of, for example, the substitutions u = f(r) cos (mtl)
sin (az),
c = g(r) sin (me) sin (c(z), \L’= h(r) cos (mU) cos (CC), jr = k(r) cos (me) sin (~2). I
(32)
Bifurcation of rotating thick-walled elastic tubes
65
By taking appropriate combinations of such solutions the boundary conditions on the ends of the tube can be met, as discussed by HAUGHTON and OGDEN ~1979b~. In particular, and in order to illustrate the results, we may take r=tm/l
(n=O,1.2
,... ).
(33)
The values (33) are consistent with the end conditions if = r = Sat2 = 0
on
2 = 0, I,
(34)
for example. The latter of (34) corresponds to holding the current axial loading fixed po~~f~~jse. When h(r) is eliminated by use of (26), equations (27)-(29) with (32) become k’= (~.6;,33-r.~;233-.~,1*1+-~11*2 4-8
2332 -m*.~,3,3-n21-2.~,323fpr202)f;lf2
+()‘~~333-r.~2233+rp’+.~33333S..?jll122-2~)2133-.~2332)f)/r +- @,,,, +(r.@;,,,-
-.@,,,YJ -d2331)f” r.~~233+~~,122f~~2332-~-iA11,1-~~13r3)mgl~2
(35)
+(.~,,33+.~,331-.~2233-.~2332)mCI’l~r
mk/r= (~.~~,,,+.JA,,31+~1,11-.~31122-.~12tl+pr202)mf/r” +(@ir33+.@133i-. 1122-.~1221)mf’/r +(‘.4?p;i3i +63r3i +: 2 r 2,.@2121 -m2(J?I122+.@3,22t
-Jfl,II)~d~2
(361
-(Y.~S;,,l+.~3131)g’lr--P31318~.
a2k = .n,,,,~‘“+(r-~~,,,+2.~,,,,~f“~r +~r.;P;,,2-.~~~~2-n12.~~212-~~32(.~~222-~2~~~-.~Z233f)~/P2 -ilr%232-.~3232+m2~~,212 -r2r2(r.~2332+rp’+.~2332+.d,,22-.~,222)}f/r3 +~~3232m~“l~+~~~~~232--~3232)m~‘/r2
(37)
-(~~~~232-.~3232fm2.~~2,2-r2r2(.~,,22-.W2222+.~,22,~)mg/~3,
&.Ihaving been set equal to zero since this corresponds to a cylindrically symmetric motion. Correspondingly, the boundary conditions (30) become 0,
rg’-,4-mf==
r2y + rf’ + (a’? - 1 )j”+ m2.f= 0, (.~3333-.lA2233+63)r~SI+(,~lr33-.R2233)(f~mg)-rk
=
on r = a, b.
(3X)
0,
3.1 Fr~smat~c bj~~rcat~ons First we examine solutions of the governing equations (27tf29) that are independent of z. Equation (29) reduces to an equation for W, decoupled from (27) and (28). with boundary conditions W, = 0 on r = a, b. The solution satisfying the
D. M. HALJGHTON and R. W.
66
OGDEN
end-conditions u’ = 0 on z = 0,l is easily shown to be the trivial one and we therefore set w 3 0. The problem now becomes a two-dimensional one and we replace (32) by U = f(r) cos (me), c = g(r) sin (m8),
(39)
J! = k(r) cos (me), I and we note that end-conditions on the cylinder are no longer relevant [in particular, (34) can only be satisfied trivially]. The governing equations are now (35) and (36) with u = 0, (37) being satisfied identically, and f(r) and g(r) are connected by rf’+f+mg
= 0
(40)
from (26) with (39). Elimination of k(r) and g(r) leads to the equation [s,,,,r3f”‘f(r.8;131
+3.?83,3,)rZf”
+(r.?$;r3r -.a,,,,
+m2(2d
+(m2-
+r.O;,,,
I){r2.8;131
1133+2~1331-.~~11,-.~3333):rf’1’ +(m2-
I~~,,,,
+m2(o,
-a,)}j/r
= 0,
(41)
while the boundary conditions (38) reduce to r2f”+rf’+(m2-l)f=O
on
r=a,b
(42)
r = a, b.
(43)
and 1 3131r3fnr+(r.ST3,31
+4.h?,,,,)r2f”
+(r!~;131+,~3L31+~(2~1133-.~~111-.~3333+.~~331-~3))rf +{(m2-l)(r.B’3,3,
+.W3,,,)+pr202m2)f=
0
on
The equation (41) and the boundary condition (42) are precisely the same as those obtained for the corresponding problem of a pressurized tube (HAUGHTON and OGDEN, 1979b), but (43) differs by the inclusion of the term in CC). For m = 1 the solution forf’(r) from (41) may be written formally as
where cr. c2, c3 are constants, and the boundary conditions reduce to rf”+f’ prZw2f+c1
= 0, = 0. I
on
r = a, b.
(45)
However, the term involving o makes it impossible to obtain a concise and explicit bifurcation criterion like that given by HAIJGHTON and OGDEN (1979b) for a pressurized tube (for which cr = 0 is required to satisfy the boundary conditions). For a rotating membrane tube, HAUGHTON and OGDEN (1980) have shown that the mode m = 1 prismatic bifurcation is not possible under the (physically justified) constitutive assumptions they adopted. Since the constitutive inequalities necessary
Bifurcation of rotating thick-walled elastic tubes
67
and sufficient for the existence of bifurcations of the membrane are necessary for the existence of the corresponding bifurcations of a thick-walled tube, the system (44) and (45) does not have a non-trivial solution in general. We recall that the m = 1 prismatic mode is not possible in respect of a pressurized thick-walled tube when a certain constitutive inequality is imposed (HAUGHTON and OGDEN, 1979b). HAUGHTON and OGDEN (1980) also showed that bifurcation into a prismatic mode first becomes possible for a membrane ~*/ren the angular speed reaches a maximum. This corresponds to mode number m = 2. Moreover, higher-order modes do not occur for the forms of strain-energy function used by Haughton and Ogden. For m L 2, the equation (41) cannot be integrated even once analytically (even for very simple forms of strain-energy function), so that the coincidence of the m = 2 bifurcation and maximum angular speed points cannot be established for thickwalled tubes (and cannot in general be expected). For m = 2, we have obtained numerical solutions of (41)-(43). The resulting bifurcation curves are plotted in the (i.,, &)-plane and compared with the curves of the angular speed turning-points for A/B = O-99, 0.85, 0.5 in Fig. 1. For A/B = 0.99, the bifurcation curve cannot be distinguished from the curve of turning-points of w on the scale used in Fig. 1. This is to be expected, since a tube of thickness ratio 0.99 is effectively a membrane. For A/B = O-85, the same applies except when i., and A, are both greater than about 3. Then the difference between the curves is only small. Even for A/B = O-5there is little difference in the curves for R, less than about 4. but in this case we can see that bifurcation occurs before the maximum of (I) is reached. The divergence of the curves increases with increasing B/A.
/g
___________________
1. Plot of the m = 2 prismatic mode bifurcation curves in the (j.,, i,Fplane for A.‘B = 0.99. 0.85, 03 (continuous curves) compared with the (broken) curves of turning-points of o. Also shown is the curve of w = 0 (independent of .4.!B).
FIG.
D. M. HALJGHTON and R. W. OGDEN
68
The calculations have been carried out in respect of the strain-energy function described in detail by HAUGHTON and OGDEN (1979a). Briefly, we record that
where c(, = 1.3, p: = 1.491,
x2 = 5.0,
x3 = -2.0,
/J: = 0003,
(47)
/J: = - 0.023
and &! = p,/,u (r = 1,2,3), p being the ground-state
shear modulus defined by
As in the case of a membrane the values (47) rule out bifurcations corresponding
to
m 3 3.
3.2 Axisymmetric
bifurcations
Next, we examine solutions of (27)-(29) that are independent of 8. Equation (28) leads to an equation for v and the boundary condition (30) gives rv,- v = 0 on r = a, b. It is easily shown that the resulting Sturm-Liouville system has only the trivial solution at any stage of the deformation when the boundary conditions (34) are imposed and the inequalities .#ijij > 0, i # j, adopted. We now take m = 0 and eliminate k(r) from (35) and (37) to obtain rS{r-1[r-‘.~3232(r2f,,+rfl-f)]‘)’ -~2r2(r[r(.~2222
+.I,,,,
- 2282233 - 253223m’ +(r2a’;,-rZ~)‘;2j2+r.~;222+r.~;133-r:~;122-r~~;233-r.~;223 +2d,,22+~,22,-.~,,,,-.~~,,2+pr2~2)fJ + r4r4.W,,,,
The corresponding
f = 0.
(49)
boundary conditions are r*f”+rf’+(a*r*-l)f=
0
on
r = a, b
(50)
and r*[r-‘.~,,,,(r*f”+rf,-f)]’ -u2r2[(.d,222
+.aJjj3
- 2.A,,,, -2e@‘3223+d,,,,)rf
+(ra~,-r.JA~2,2+.JA2,,2+.W,,,,-.~,,2,-.~22,,-.~A3223)fl
and a is given by (33). For s( = 0, equation (49) leads to the solution
= 0 on r = a, b, (51)
f(r) = Cr+Dr-‘,
where C and D are arbitrary constants, and the boundary conditions (50) and (51) are automatically satisfied. When the tube has finite length this solution corresponds to maintenance of the circular cylindrical shape, but for L + rx an axisymmetric solution of the type discussed by HAUGHTON and OGDEN (1979b) is possible.
Bifurcation of rotating thick-walled elastic tubes
69
For x # 0, we choose n = 1 in (33) in view of the dual character of the quantities n and L (recalling I = &L) that was exploited previously by the writers (HAUGHTON and OGDEN, 1979a,b, 1980). Thus, al = x and the parameters L/B and A/B are (independently) at our disposal. We have carried out numerical calculations for a number of values of L/B, A/B and & in respect of the strain-energy function (46) with the material constants (47). In Fig. 2(a, b), for A/B = 0235 and 0.5 respectively, the bifurcation curves calculated from (49)-(51) are plotted in the (A,, &)-plane for L/B = 10,20,40 in order to illustrate the dependence of the curves on the length of the tube. The value A/B = 0.85 is chosen as representative of a thin-walled tube so that comparison can be made with the corresponding results for a membrane (HAUGHTON and OGDEN, 1980). The general features shown in Fig. 2(a) are also those obtained for a membrane with some minor differences of detail. Thick-walled tubes are represented by the value A/B = 05, and it can be seen from Fig. 2(a, b) that the character of the results for a thick-walled tube is broadly similar to that for a thin-walled tube. The picture changes for very thick-walled tubes as Fig. 3 illustrates. The curves of(i) w = 0 and (ii) turning-points of w at fixed A, are included in Fig. 2(a. b) for reference purposes. We remark that bifurcation into an axisymmetric mode may occur before or after the maximum of o depending on the values of L/B, A/B and AZ.Moreover, for values of & less than some critical value no turning-points of 0 occur. In Fig. 3 the critical values of i., are plotted against A/B for iZ = 2,3,4,5 to illustrate the dependence of bifurcation on the tube thickness for the representative value L/B = 10. For very thick tubes bifurcation into an axisymmetric mode of deformation cannot occur. For a pressurized tube the asymmetric (or bending) mode of bifurcation is generally more important than the axisymmetric mode when the tube is in axial compression. Roughly speaking, the converse is true when the tube is in axial tension (HAUGHTON and OGDEN, 1979b). For a rotating tube, on the other hand, the priorities are not so clear-cut, as we show in Section 3.3. Indeed, the asymmetric (or wobbling) mode will dominate when the tube is in Jxial tension except for short tubes and certain ranges of values of &. This behaviour depends markedly on the value of A/B. In axial compression, the asymmetric mode also dominates in general. Note that the axisymmetric bifurcation points on o = 0 in Fig. 2(a, b) correspond to bifurcation of a tube under pure axial compression, coinciding with those calculated by HAUGHTONand OGDEN (1979b). The curves shown in Fig. 2(a, b) are similar to those obtained by HAUGHTONand OGDEN (1979b) for the axisymmetric bifurcation of a pressurized tube. Further discussion of such curves is contained in that paper. We now examine asymmetric modes of bifurcation. 3.3 Asymmetric
bifurcations
With dependence on both 0 and z retained no progress can be made analytically. We have therefore carried out numerical calculations on the basis of equations (35)(37) with boundary conditions (38). For m 2 2, our results show that there are no
D. M. HAUGHTON
and R. W. OGDEN
UI=O
I i I 1
1
,
3
2
J
I
4
5
AZ (Iif
I
t 2
t 3
I
f
4
5
A,
FIG. 2. Plot of the axisymmetric mode bifurcation curves in the (A_,A&plane for L/B = 10.20.40 together with the curve of OJ = 0: (a) A!B = 085 and (b) A 6 = 0.5. The curve of turning-points of cu is also shown (broken curve).
bifurcations (r # 0): but for m = 1 and the values of u given by (33), bifurcations are possible. In fact, we choose xl= x and a selection of values of Ljl3 as in Section 3.2, The numerical method is described briefty in Section 3.4. In Fig. 4(a, b, c) for A/B = @99,0%5,0.5 respectively, we plot i, against i_, for the
Bifurcation
of rotating
1
elastic tubes
I
I
08
1.0
thick-walled
06
04
71
I
,
02
C
f-JIB
FIG. 3. Plot of the axisymmetric
mode critical
values of i, against
A/B for L/B = 10 and i,, = 2.3.4.5.
values L/B = 5,6,7, 10. For longer tubes the initial (or n = 1) bifurcation curves lie in the narrow range between the curve for L/B = 10 and that for o = 0. This shows that long tubes “wobble” sooner than shorter tubes as o is increased from zero, which is consistent with physical intuition.
I
I I
I
2
I
3
A* (a! FIG. 4. Plot
of the asymmetric
mode
bifurcation
curves
in the (i.,, i,)_plane
AIB = 0.99. (b) A:B = 0.85and (c) A,B = 0.5.
for LIB = 5.6.7.
10: (a)
?2
D. M. HAUGHTON and R. W. OGDEN
The curves, in Fig. 4(a) particularly. shoutd be compared with those obtained by for a rotating membrane tube. It should be pointed out, however, that for a given value of LIB, more curves appear in the ~~~~ran~ case than are shown in Fig. 4(a). This is because integration through the tube wail thickness with respect to r is implicit for a membrane. ~~n~~ue~tiy, the boundary conditions applied to the g~~~r~~solution at the ends of the tube art: not poj~t~ise co~dit~~~s such as we have used to obtain (33). The “average” e~d-co~d~t~o~s yield HAUGHTON and OGDEN (2980)
Bifurcation of rotating thick-walled elastic tubes
13
values of r other than (33) and, therefore, bifurcation curves additional to those shown in Fig. 4(a). Application of “average” end-conditions to tubes of finite wall thickness would also yield additional values of z, possibly dependent in a complicated way on iL, and I.,. However, the (i, &)-dependent values of x obtained (implicitly) by HAUGHTON and OGDEN (1980) for a membrane do not yield bifurcation phenomena
whose features are essentially different from those corresponding to (33). The same applies to tubes of finite wall thickness. Figure 4 shows that under axial tension, tubes “wobble” at quite low values of the angular speed (well below the maximum in general). Exceptions to this are provided by short tubes. For a short tube there is a range of values of i., (in axial tension) within which either (i) an axisymmetric mode of bifurcation has priority over any poss;ble asymmetric mode, or (ii) only an axisymmetric mode is possible, or (iii) no axisymmetric or asymmetric bifurcation can occur (refer to Figs 2 and 4). In particular, for a short tube there is a critical value of i.: (> l), AZ, say, depending on L/B such that asymmetric mod_es cannot occur if iZ < AZCin axial tension. This is consistent with physical intuition since increase in length (either through L. or A,) can be expected to increase the tendency to “wobble”. The competition between asymmetric and axisymmetric bifurcations in axial tension described here contrasts markedly with that for a pressurized tube (HAUGHTON and OGDEN, 1979b).
In axial compression, comparison of Figs 2 and 4 shows that an asymmetric mode of bifurcation always has priority over any possible axisymmetric mode, but for short tubes there is again a range of values of AZ(G 1) for which no bifurcation occurs. The dependence of bifurcation on L/B for short tubes and on A/B for thick-walled tubes can be particularly marked in the case of a pressurized tube. The same is true for a rotating tube, as Fig. 3 illustrates. However, rather than elaborate on the details for short and/or thick tubes we have chosen here merely to highlight the mainfeatures of the bifurcation phenomena. Reference can be made to the paper by HAUGHTON and OGDEN (1979b) for the detailed changes which can be expected for short or thick tubes. 3.4 The numerical method Equations form
(35)-(37)
after suitable non-dimensionalization, u:=F,(u,r),
i=l,...,
6,
are written in the (52)
where u(r) z (f,f’,f”, g, g’, k), and the prime denotes differentiation with respect to r. For each value k = 1, . . ., 6, the initial-value problem defined by (52) with ui(a) = fiik, i = 1, . . ., 6,
is solved using a fourth-order Runge-Kutta method. We write uk (k = 1, . .. 6) for the six solutions and h u = c CkUk k=l
(53)
D. M. HAUGHT~Nand R. W. OGDEN
74
for the solution of (52) subject to the boundary conditions (38), recalling the connection (12) between u and b, where the ct are arbitrary constants. Substitution of (53) into (38) leads to a non-trivial solution for u provided a certain 6 x 6 determinant vanishes. This bifurcation criterion can be satisfied as accurately as required by appropriate values of a provided such values exist in the domain under consideration. ACKNOWLEDGEMENT
The work of D.M.H. was supported by a U.K. Science Research Council Research Grant.
REFERENCES CHADWICK, P., CREASY,
1977
J. Aust. math. Sot. B20, 62.
C. F. M. and HART, V. G. HAUGHTON, D. M. and OGDEN,, R. W.
1978
J. Mech. Phys. Solids 26, I 11
1979a b 1980
Ibid. 27, 179. Ibid. 27, 489. Math. Proc. Cambridge Phil. Sot. 87, 357. Mech. Res. Comm. 4, 69. Handbuch der Physik (edited by S. Fliigge), Vol. 11113. The Non-linear Field Theories of Mechanics. Springer-Verlag, Berlin.
PATTERSON,J. C. and HILL, J. M. TRUESDELL,C. and NOLL, W.
1977 1965