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The vibrations of a thin-walled elastic cylinder under axial stress
THE VIBRATIONS OF A THIN-WALLED ELASTIC CYLINDER UNDER AXIAL STRESS J. R. THOMPSON and A. I. WLLSON Department of Mathematics, University of Leicester, England Abstract-The dispersion equation governing the a%isymmet~cvibrations of an elastic thin-walled cylinder under axial stress is derived and its solutions, together with their implications for stability, are discussed. Numerical results are given for a material with a strain-energy function of Ko form.
1. INTRODUCTION THE FR~PAGA~~N of
waves of small amplitude in various bounded and unbounded hyperelastic media, upon which a large primary stress is imposed, has been analysed by many workers. Usmani and Beatty [l] and Willson[21, for example, have considered the propagation of waves across the free surface of a semi-infinite medium for which the strain-energy function is of Hadam~d form. In [3], (and there references are given to other work in this field), Wiison has studied wave-propagation in thin pre-stressed plates and has obtained important genera1 results for isotropic materials with any str~n-energy function. Free harmonic axisymmetric vibrations in a hoiiow circular cylinder have been investigated within the context of linear elasticity by McFadden[rl], Ghosh[S] and by Herrmann and Mirsky[6], and these results have been extended by Gazis[7,8] to include the non-axisymmet~c case. When a huge primary axial load, compressive or tensile, is imposed upon the cylinder all of these modes of vibration undergo some modification. These changes are especially signiticant for modes of low frequency and the in~oduetion of the primary stress may indeed cause them to become unstabIe. In this paper the results are reported of a study of these effects for the important case in which the thickness of the cylinder wall is much smaller than the mean radius of the cylinder itseif, with particular emphasis being placed upon considerations of stability. The plan of the paper is as follows: in Section 2 the basic equations governing the axisymmetric vibrations of smal! amplitude in the presence of a large axial stress are given in terms of the strain-energy function for the material. The dispersion equation for a thin-walled cylinder is derived in Section 3 and the nature of its solutions in various limiting cases is explored. Section 4 is devoted to an analysis of the implications for the stability of these disturbances. Fin~iy in Section 5 nurne~~~ results are given for the special case in which the strain-energy function is of Ko form. 2. THE BASXC EQUATIONS
The theory of non-linear elasticity is now well established and a very clear account has been given by Eringen and Suhubi[9]. It is suthcient here, therefore, to summarise the main results, In the first instance it is convenient to work with a rectangular Cartesian system fixed in space. The typical particle of the elastic medium in its natural stress-free state occupies a position whose coord~ates are denoted .by Xk, (k = 1, 2, 31, and in the 8eneraf state of the medium the position of the same particle has coordinates xi, (i = 1,2,3), where Xidepends upon the X, and the time f. These relations may be inverted to give X, in terms of Xi,t. The elements of a matrix c are defined by the relations
where suffixes appearing after the comma indicate partial differentiation with respect to the coordinates Xi, and where the double suffix convention has been used. The inverse of c is denoted by c-l, and Is, I,, I3 are the usual three invariants of cl’. The s~a~~nergy function for the medium, assumed to be hyperelastic, isotropic and homogeneous, is denoted by &‘(I,, It, &). Then the stresses pg are given by the equation Pij = 21f”2 WlC,’ + 21S”2(ZzW2 + IJWJ)&[- 21$2W2Cii, 725
(2)
J. R. THOMPSON and A. J. WILLSON
126
where 8, is the Kronecker delta and W,, (a = 1,2, 3), denotes the partial derivative of W with respect to I,. The axis of a hollow cylinder of this elastic material is taken to be along 0~~. The cross-section of the cylinder is a circular annulus and for the moment the length of the cylinder is taken to be infinite. The cylinder is now placed under an axial load so that its new state, which will be called its primary state, is specified by the equations Xl =
@XI,
x2 =
/3x2,
x3 =
(3)
rx3,
where 13,y are positive constants; in this state the axial principal stress will be denoted by p”. It follows that I, = 2j.P + y2, 12= /34+ 2p2y2,
I3 =
f34y2.
(4)
Following Eringen and Suhubi[9], apart from changes in notation, define 0 = 21:‘2w,, B, = 21;‘” WII, c,
21;“2 w23,
=
Q,= 21;“2W 1,
VI= 21;“2 w 2,
B2 = 21;‘12W22,
B3 = 21;‘” W339
c, = 21;“2Ws,,
c3
=
(5)
21;“2 w 12,
with the partial derivatives of W all calculated at the point in (I,, I,, I,) space given by (4). Then from (2) it is readily shown that p” = ( y2-
p2&.D + /32P\ar)
(6)
and, since this is the only non-vanishing stress component in the primary state, 8-20 + @+
(p2+ y2)V= 0.
(7)
Equations (6) and (7) determine B and y in terms of the primary axial stress p”. The densities po and p of the medium, in its neutral and primary states, respectively, satisfy p=
/3-2y-‘po.
(8)
In order to study the vibrations of the cylinder it is now convenient to use cylindrical coordinates r, 4, z, where r = (xi + x:)“~, z = x3 and 4 is the usual polar angle. The vibration is regarded as a perturbation of the primary state. Consideration will be restricted here to axisymmetric vibrations in the planes of constant 4. The displacement components due to the perturbation are denoted by u(r, z, t) radially and w(r, z, t) axially. The spatial derivatives of u, w are regarded as small, and their squares and products will be neglected. In addition to the primary axial stress p”, small additional stresses will be created by the perturbation and these will be denoted in the usual way by pm p++, pzz and pa. Their values have been calculated by Eringen and Suhubi[9] and we shall follow their notation closely. Accordingly define al = 2/34[B1 t (b2 t y2J2B2+ P4y4B3 t 2/32y2(p2 t y2)C, t 2BZr2C2t 2(p2 t r2)c3] - W2[@ t (P2 t a2
=
W2y2[B1
t
r2w1,
W2U32
t
y2)B2 + B6y2Bs
t /3”(/3’+ 3y2)C, t B2(B2t r2)c2 t (313~t r2)~3]
- 2&@ t pP), a3
=
a4 =
a2 + (P2-
y2)(@
(9) t
/32q),
2y4(B, t 4fi4B2 t j3’B3 t 4B6C, t 2p4C2t 4B2C3)t (jS2- ~‘)(a t p2q),
a5 = p+D t pP),
‘+, = f12(@t r2p),
a7 =
r2@
+ Pn
The vibrationsof a thin-walledelastic cylinderunderaxial stress
721
and further R=a,-28,
T=ad+2p0.
S= a,+2/?4,
(10)
Then pH= Su,fRF+a2w,,
pn = Ru, + SF+ azw,, PZZ= a3ur + y
+ Tw,,
pR = w(u,
+ w,) + pour.
(11)
It is important to note the Baker-EricksenPO] inequality R > S, and that a7=aS+po,
a3=
a2 - p”.
(12)
Then the equations of motion in the absence of body forces require + a7u,, + (a2 + as) w,, = pi,
R(u,,+T-;)
(13)
where dots indicate successive partial differentiation with respect to time t; and for the curved surfaces of the cylinder to be free from traction Ru,+$+a,w,
~0,
&+w,=Oonr=a+h,
(14)
where 2h is the wall thickness and a the mean radius of the cylinder in the primary state. Of course, the quantities a, h themselves depend upon p”, and their values are PA, /3H, respectively, where A, H refer to the neutral state. Finally, when p” vanishes R=T=h+2p,
S=aZ=h,
a5 = a7 = p,
(15)
where A, CLare the usual Lame constants. 3. THE DISPERSION
EQUATION
In the absence of primary stress the modes of vibration of a hollow cylinder fall into two groups, according to whether their frequencies do or do not remain bounded as the wall thickness tends to zero. The modes that are of principal interest in the present context are the vibrations of low frequency, and for these we may proceed as follows. Solutions are sought in series form u = [a0 + ad&x) + a2(kx)2 + - - -1exp [i(wt - kz)], w = i[bo+ b,(&x) + b2(kr)*+ - . -1exp [i(wt - kz)],
(16)
where x = r-a, and o and k are the angular frequency and axial wavenumber of the perturbation. For the intended order of working it is necessary to take the series developments in (16) up to the terms a&)’ and b&~)~. Substitution of (16) into (13) yields six equations for the coefficients a2, as, ad, b2, b3, b4 in terms of ao, al, bo, b,. The boundary conditions (14) then give a system of four homogeneous linear equations for ao, al, bo, bl correct to second order in both kh and h/a, and the vanishing of the determinant of this system is the required dispersion relation. It should be noted that there is no independent restriction upon the value of &a. The reduction of the determinant, to this order of working, involves lengthy but straight-
J. R. THOMPSON and A. J. WILLSON
728
forward calculation, The result is S+R ka
1
R-jtkh)’
t
@-PO+
R-i-4S rkaP
a:
]
0
22
0,
(17)
9-T
where 8 = pw21k2,
l=e-a7+1Y2+ag=B-p”$a2.
(18)
Equation (17) is one of the main results of this investigation: it governs all the low-frequency modes of vibration (that is, all the modes whose frequencies remain bounded as h tends to zero) provided that h is much smaller than both a and k-‘. It is convenient now to study its form in various special cases. Suppose first that there is no primary stress (PO= 0). Then in terms of Young’s modulus E and Poisson’s ratio a; (17) becomes
_ (1 f 3o)(kh)’ -8 -e 3(ka)2 E EE
1+4a 1+4a+
3~~1
=o.
(19)
When the cylinder radius is so large that (ka)-’ may be neglected entirely, the roots of this quadratic equation in t? are E(kh); + O(k4h4), e1 = 3(1 -(r )
e2 = $
+ O(k2h2).
(20)
These are the plate waves whose modifications under primary stress have been studied by Willson[3]. Another interesting limiting case arises when the wall thickness is so small by comp~son with the wavelength that in (19) the terms in kh may be disregarded. Then
As (ka)-’ increases from zero to infinity, both roots &, e2 increase monotonically, 8, from zero to E, and & from E/(1 - c?) to infinity. In particular when (ka) is unity 8, = E/( 1 + u) = 2p,
02= E/( 1 - u) = 2~( 1 f (T)/(1 - u).
This case is of some historical interest as the vibration corresponding to &, when ka = 1, is the limiting form for thin-wiled cylinders of the equivolumin~ waves discovered by Lame in about 1852. Suppose now that the primary stress p” is taken into account. For a cylinder with very thin walls, so that the terms in kh may again be neglected, the quadratic eqn (17) for 0 becomes R~2+~[a~-R(T-tpO)+(SZ-R2)(ka)-2]+[po(RT-a~+(S-R)(2a~-RT-ST)(ka)-2]=0. (21)
Its solutions for 0 again vary monotonically with (ku), the limiting cases being when ka % 1, 0 = p”
or (RT - a:)/R,
when ka Q 1, 8 = [(R + S)T - 2a$]/(R + S) or (R2 - S2)~R(ka)2.
(22)
The vibrations of a thin-walled elastic cylinder under axial stress
129
For fairly small primary stresses and when (&a)-’ and (kh) are both small, approximate forms for the roots may be obtained by assigning to the coefficients of #a)-’ and (kh) in (17) the values appropriate to the neutral state. This gives E
E(kh)2
E e2+(1_02).
@1=P”+&$+3(1_cr2y
Attention turns now to the implications for stability of the root 8, exhibited in (23).
4. STABILITY
It is clearly of great practical importance to determine the value of the primary stress sufficient to cause instability, the onset of which is taken to be indicated by the vanishing of one of the solutions for 8. For a medium whose strain-energy is known the critical values of p” can be determined numerically from (17) and in the next section one such case is considered in detail. But when these critical values are small, and these cases are especially important in practice, some progress can be made by approximate means. Under the conditions in which (23) is valid it is apparent from a consideration of the root 8, that instability may be produced by a small compressive primary stress. Since
e,, regarded as a function of k, has a minimum at k = + kc, where k: = [3(1 - a2)lu2/(ah),
(24)
2Eh Pc = [3(1 - ,2)]“% *
(25)
and then 8, % (p” + pJ, where
Thus if - p” < pC all of these waves are stable, but if - p” just exceeds this critical value pe then waves whose wavenumbers are in the vicinity of 5 kc will be unstable, and for the wave with the greatest rate of growth the wavenumber, 2 k. say, is given by k: = - 3(1- (r2)po/(2Eh2) = k%-PO/p,).
(26)
The quantities (koh)‘, (koa)-2, (kch)2, (kca)-2 and pJE are all of order h/a. For a cylinder of finite length, account must be taken of the boundary conditions on the terminal planes. The determinant in (17) is even in k and a solution may be constructed by superposition so that both w and pn depend upon z through a factor (sin kz). So for a cylinder confined between two smooth rigid plates at z = 0 and z = 1 in the primary state, it is appropriate to take k=ndl,
(n=lJJ,...).
(27)
Accordingly, for -p” > pc, the axisymmetric instability will be present in the lowest (n = 1) mode for cylinders with length of the order of (ah)“‘, and appears first, as the compressive primary stress is slowly increased, in a cylinder of length I,, where
(28) and when -p” > pc instability will be present in some higher mode for all sufficiently long cylinders. The instability disappears for cylinder lengths significantly less than lC,at least in the context of the present analysis based upon (23), since the relevant value of kh is no longer
I. R. THOMPSON and A. J. WILLSON
730
small. But the physical concept of a thin-walled cylinder would then give way to that of a thick annular plate. Finally, the form of (23) and a consideration of (17) suggest that for small values of ka instability will not appear in axisymmetric modes except under very high primary stresses. In practice, instability will appear in modes which are not axially symmetric before these high stress values are attained. 5. AN EXAMPLE:
THE Ko MODEL
In [ 111,Ko proposed for polyurethane foam rubber the strain-energy function W = p(&+
I:“),
(p constant),
(29)
and because of its simple form this has been used for purposes of general illustration by many writers, a precedent followed here. From (6), (7) and (29), 84 = y-1,
PO= /.L (1 - y-s’*),
(30)
and the other quantities of interest are found to be R=~/.L,
s=(r*=p,
T=3(p-p@).
(31)
The quantity p appearing in (29) is, for the theory of linear elasticity in this model, identical with the Lam& constant Jo, and the Poisson condition A = 1~is satisfied, so that E = 5~12,
u = l/4.
(32)
This model may be used to give a simple instance of the results obtained in Sections 3 and 4. For a cylinder composed of material obeying (29), and with a wall so thin that terms in h may be neglected, (17) and (21) take the form 3p + F[6P - 8 - 8(ku)-*] - [9P* + P{24(ka)-* - 8) - 2O(ka)-*I= 0,
(33)
where F = elp = po*lpk*,
P = tfO/jL.
(34)
Investigation of this equation reveals that its solutions for F are real for all P and (ku). These solutions are shown in Fig. 1. Moreover, for any given value of (ku), there will be a negative root for F, implying instability, if P lies outside the range (PI, P2) determined by the roots of 9P2 + P{24(ku)-‘- 8) - 2O(ku)-*= 0,
(35)
where PI and P2 denote respectively the negative and the positive root. The stability results of Section 4 may be illustrated in the same way. For cylinders of Ko material, (25) and (28) yield for the critical stress pC and the critical length f, PC=
80 “*ph
0
i-y
at
c =
16 “’
z
0
dab)
t/z ,
(36)
and by virtue of (23) and (27) the stress required to produce instability in the lowest (n = 1) mode is given by (37)
The vibrationsof a thin-walledelastic cylinderunderaxial stress
731
Fig. 1. Variationof po*/pk*(F) againstp’/p (P) for Ko material,for three values of ka, when termsin the cylinderthickness are neglected;negative values for F indicateinstability.The curves have one common asymptote,and a commondirectionfor the second asymptote.The curve for ka = 10 is typical of all large ka; for small ka the second asymptoterecedes into the second quadrant.
Figure 2 shows the contours of loglO(-pa/p) against logio(hll) and logio(2a/I). These values have been obtained by direct solution of (17) with B set equal to zero, with particular values of J, R, S, T given by (18) and (31), but the approximation (37) yields an almost identical picture. One further consideration must be borne in mind. In the last section brief reference was made to the modes of vibration which are not axisymmetric, and it was asserted there that for small values of (kc) instability in these modes would appear for values of the primary compressive stress smaller than those required for instability in the axisymmetric modes. For the Ko model it is possible to test the validity of this assertion.
0.5
I
-2 5
-20
1
log (h/l)
-1.5
I
-1.0
Fig, 2. Contoursof log (-#/a), for a finite cylindricalshell of Ko material,againstlog (/I//) and log&/I). Here -pO is the compressivestress requiredto produceinstabilityin the lowest mode.
132
J. R. THOMPSONand A. J. WILLSON
For the general non-axisymmetric vibration, the components of displacement are denoted by u (radially), u (transverse) and w (axially), where u, u, w depend upon the cylindrical coordinates r, 4, z and the time f. The corresponding stresses are readily calculated from (l), (2) and (29), and then the equations of motion for these vibrations in Ko material may be cast into the form piil/.L=3 *,+p-;)+~+Uz*++-4!$+(2-P)w~, ( piil/.& = ++
4u 3
+vW+;-++*+,+(2-P)+
p@/p=(2-P)(u,+T+
(38)
~~+(~-p)~w~+~+~~3w~).
and the boundary conditions are 3u,+E+%+ r r
w, = ,,-~+~= r
r
u,+ w,=o ,
(39)
on the curved surfaces. For considerations of stability, the most important of these modes is found to be the flexural mode in which, for example, w = n(r) sin fj sin kz cos ot. This mode may be analysed by methods identical with those used above for the axisymmetric case. For the cylinder with very thin walls, so that again the terms in (M) may be neglected completely, one finds in analogy with (35), 16P(P - 1) + (3P3 + 17P2 + 8P -20)(h)‘+
P(9P -8)(h)‘=
0.
W)
This equation has, for all (ka), precisely one root in the interval (-0.6,O) and this root will be
Fig. 3. Variation-of-PI and -PI againstka for Ko material.Thesexre, apartfrom the factor p. values of the compressivestress needed to produceinstabilitywhen the cylinderthickaess may be neglectedentirely, respectivelyin the axisymmetricand non-axisymmetricmodes.
The vibrations of a thin-walled elastic cylinder under axial stress
133
denoted by Pi. One readily finds from (35) and (40) that
forkoel:
Pi--
8 qG$’
Fi--7.
S(ku)’
Figure 3 shows how Pi and F, depend upon (ka) generally. Since P, becomes small with (ka)‘, it is clear that for long cylinders one must investigate these non-axisymmetric modes, as well as the high order axisymmetric modes arising from large values of n in (27) and whose instability was discussed in the last section. It must be emphasised that Pi and Pi in (35) and (40) are the limiting values of pa/p for marginal stability when the terms in (kh) are neglected entirely. Their values, therefore, should not be used to predict the primary stresses needed for instability in those waves for which (ka) is so large that it is comparable with (k/t)-‘. The role played by the non-axisymmetric modes of vibration in materials other than those described by Ko’s form of the strain-energy function, and the dependence of the corresponding critical stresses upon the ratio of wall-thickness to cylinder radius, fall outside the scope of the present investigation, but work on these matters is in progress. REFERENCES
111S. A. USMANI and M. F. BEATTY, J. EIasticity 4, 249 (1974). [21 A. J. WILLSON, J. Elasticity 7, 103(1977). [31 A. J. WILLSON, Int. 1. Engng Sci. 15,245 (1977). [41 J. A. MCFADDEN,J. Acoust. Sot. Am. 26,714 (1954). IS1 I. GHOSH. Bull. Calcutta Math. Sot. 14.31 11923). i6j G. HERRMANN and I. MIRSKY, Trans. Ah So;. Mech. Engrs 78,563 (1956). 171D. C. GAZIS, 1. Acoust. Sot. Am. 31,568 (1959). [8] D. C. GAZIS, 1. Acoust. Sot. Am. 31,573 (1959). [91 A. C. ERINGEN and E. S. SUHUBI, Elastodynamics, Vol. I. Academic Press, New York (1974). IO] M. BAKER and J. L. ERICKSEN, J. Wash. Acad. Sci. 44.33 (1954). 111W. L. KO, Ph.D. Thesis. California Institute of Technology, Pasadena (1%3). (Received 17 July 1978)
1. INTRODUCTION THE FR~PAGA~~N of
waves of small amplitude in various bounded and unbounded hyperelastic media, upon which a large primary stress is imposed, has been analysed by many workers. Usmani and Beatty [l] and Willson[21, for example, have considered the propagation of waves across the free surface of a semi-infinite medium for which the strain-energy function is of Hadam~d form. In [3], (and there references are given to other work in this field), Wiison has studied wave-propagation in thin pre-stressed plates and has obtained important genera1 results for isotropic materials with any str~n-energy function. Free harmonic axisymmetric vibrations in a hoiiow circular cylinder have been investigated within the context of linear elasticity by McFadden[rl], Ghosh[S] and by Herrmann and Mirsky[6], and these results have been extended by Gazis[7,8] to include the non-axisymmet~c case. When a huge primary axial load, compressive or tensile, is imposed upon the cylinder all of these modes of vibration undergo some modification. These changes are especially signiticant for modes of low frequency and the in~oduetion of the primary stress may indeed cause them to become unstabIe. In this paper the results are reported of a study of these effects for the important case in which the thickness of the cylinder wall is much smaller than the mean radius of the cylinder itseif, with particular emphasis being placed upon considerations of stability. The plan of the paper is as follows: in Section 2 the basic equations governing the axisymmetric vibrations of smal! amplitude in the presence of a large axial stress are given in terms of the strain-energy function for the material. The dispersion equation for a thin-walled cylinder is derived in Section 3 and the nature of its solutions in various limiting cases is explored. Section 4 is devoted to an analysis of the implications for the stability of these disturbances. Fin~iy in Section 5 nurne~~~ results are given for the special case in which the strain-energy function is of Ko form. 2. THE BASXC EQUATIONS
The theory of non-linear elasticity is now well established and a very clear account has been given by Eringen and Suhubi[9]. It is suthcient here, therefore, to summarise the main results, In the first instance it is convenient to work with a rectangular Cartesian system fixed in space. The typical particle of the elastic medium in its natural stress-free state occupies a position whose coord~ates are denoted .by Xk, (k = 1, 2, 31, and in the 8eneraf state of the medium the position of the same particle has coordinates xi, (i = 1,2,3), where Xidepends upon the X, and the time f. These relations may be inverted to give X, in terms of Xi,t. The elements of a matrix c are defined by the relations
where suffixes appearing after the comma indicate partial differentiation with respect to the coordinates Xi, and where the double suffix convention has been used. The inverse of c is denoted by c-l, and Is, I,, I3 are the usual three invariants of cl’. The s~a~~nergy function for the medium, assumed to be hyperelastic, isotropic and homogeneous, is denoted by &‘(I,, It, &). Then the stresses pg are given by the equation Pij = 21f”2 WlC,’ + 21S”2(ZzW2 + IJWJ)&[- 21$2W2Cii, 725
(2)
J. R. THOMPSON and A. J. WILLSON
126
where 8, is the Kronecker delta and W,, (a = 1,2, 3), denotes the partial derivative of W with respect to I,. The axis of a hollow cylinder of this elastic material is taken to be along 0~~. The cross-section of the cylinder is a circular annulus and for the moment the length of the cylinder is taken to be infinite. The cylinder is now placed under an axial load so that its new state, which will be called its primary state, is specified by the equations Xl =
@XI,
x2 =
/3x2,
x3 =
(3)
rx3,
where 13,y are positive constants; in this state the axial principal stress will be denoted by p”. It follows that I, = 2j.P + y2, 12= /34+ 2p2y2,
I3 =
f34y2.
(4)
Following Eringen and Suhubi[9], apart from changes in notation, define 0 = 21:‘2w,, B, = 21;‘” WII, c,
21;“2 w23,
=
Q,= 21;“2W 1,
VI= 21;“2 w 2,
B2 = 21;‘12W22,
B3 = 21;‘” W339
c, = 21;“2Ws,,
c3
=
(5)
21;“2 w 12,
with the partial derivatives of W all calculated at the point in (I,, I,, I,) space given by (4). Then from (2) it is readily shown that p” = ( y2-
p2&.D + /32P\ar)
(6)
and, since this is the only non-vanishing stress component in the primary state, 8-20 + @+
(p2+ y2)V= 0.
(7)
Equations (6) and (7) determine B and y in terms of the primary axial stress p”. The densities po and p of the medium, in its neutral and primary states, respectively, satisfy p=
/3-2y-‘po.
(8)
In order to study the vibrations of the cylinder it is now convenient to use cylindrical coordinates r, 4, z, where r = (xi + x:)“~, z = x3 and 4 is the usual polar angle. The vibration is regarded as a perturbation of the primary state. Consideration will be restricted here to axisymmetric vibrations in the planes of constant 4. The displacement components due to the perturbation are denoted by u(r, z, t) radially and w(r, z, t) axially. The spatial derivatives of u, w are regarded as small, and their squares and products will be neglected. In addition to the primary axial stress p”, small additional stresses will be created by the perturbation and these will be denoted in the usual way by pm p++, pzz and pa. Their values have been calculated by Eringen and Suhubi[9] and we shall follow their notation closely. Accordingly define al = 2/34[B1 t (b2 t y2J2B2+ P4y4B3 t 2/32y2(p2 t y2)C, t 2BZr2C2t 2(p2 t r2)c3] - W2[@ t (P2 t a2
=
W2y2[B1
t
r2w1,
W2U32
t
y2)B2 + B6y2Bs
t /3”(/3’+ 3y2)C, t B2(B2t r2)c2 t (313~t r2)~3]
- 2&@ t pP), a3
=
a4 =
a2 + (P2-
y2)(@
(9) t
/32q),
2y4(B, t 4fi4B2 t j3’B3 t 4B6C, t 2p4C2t 4B2C3)t (jS2- ~‘)(a t p2q),
a5 = p+D t pP),
‘+, = f12(@t r2p),
a7 =
r2@
+ Pn
The vibrationsof a thin-walledelastic cylinderunderaxial stress
721
and further R=a,-28,
T=ad+2p0.
S= a,+2/?4,
(10)
Then pH= Su,fRF+a2w,,
pn = Ru, + SF+ azw,, PZZ= a3ur + y
+ Tw,,
pR = w(u,
+ w,) + pour.
(11)
It is important to note the Baker-EricksenPO] inequality R > S, and that a7=aS+po,
a3=
a2 - p”.
(12)
Then the equations of motion in the absence of body forces require + a7u,, + (a2 + as) w,, = pi,
R(u,,+T-;)
(13)
where dots indicate successive partial differentiation with respect to time t; and for the curved surfaces of the cylinder to be free from traction Ru,+$+a,w,
~0,
&+w,=Oonr=a+h,
(14)
where 2h is the wall thickness and a the mean radius of the cylinder in the primary state. Of course, the quantities a, h themselves depend upon p”, and their values are PA, /3H, respectively, where A, H refer to the neutral state. Finally, when p” vanishes R=T=h+2p,
S=aZ=h,
a5 = a7 = p,
(15)
where A, CLare the usual Lame constants. 3. THE DISPERSION
EQUATION
In the absence of primary stress the modes of vibration of a hollow cylinder fall into two groups, according to whether their frequencies do or do not remain bounded as the wall thickness tends to zero. The modes that are of principal interest in the present context are the vibrations of low frequency, and for these we may proceed as follows. Solutions are sought in series form u = [a0 + ad&x) + a2(kx)2 + - - -1exp [i(wt - kz)], w = i[bo+ b,(&x) + b2(kr)*+ - . -1exp [i(wt - kz)],
(16)
where x = r-a, and o and k are the angular frequency and axial wavenumber of the perturbation. For the intended order of working it is necessary to take the series developments in (16) up to the terms a&)’ and b&~)~. Substitution of (16) into (13) yields six equations for the coefficients a2, as, ad, b2, b3, b4 in terms of ao, al, bo, b,. The boundary conditions (14) then give a system of four homogeneous linear equations for ao, al, bo, bl correct to second order in both kh and h/a, and the vanishing of the determinant of this system is the required dispersion relation. It should be noted that there is no independent restriction upon the value of &a. The reduction of the determinant, to this order of working, involves lengthy but straight-
J. R. THOMPSON and A. J. WILLSON
728
forward calculation, The result is S+R ka
1
R-jtkh)’
t
@-PO+
R-i-4S rkaP
a:
]
0
22
0,
(17)
9-T
where 8 = pw21k2,
l=e-a7+1Y2+ag=B-p”$a2.
(18)
Equation (17) is one of the main results of this investigation: it governs all the low-frequency modes of vibration (that is, all the modes whose frequencies remain bounded as h tends to zero) provided that h is much smaller than both a and k-‘. It is convenient now to study its form in various special cases. Suppose first that there is no primary stress (PO= 0). Then in terms of Young’s modulus E and Poisson’s ratio a; (17) becomes
_ (1 f 3o)(kh)’ -8 -e 3(ka)2 E EE
1+4a 1+4a+
3~~1
=o.
(19)
When the cylinder radius is so large that (ka)-’ may be neglected entirely, the roots of this quadratic equation in t? are E(kh); + O(k4h4), e1 = 3(1 -(r )
e2 = $
+ O(k2h2).
(20)
These are the plate waves whose modifications under primary stress have been studied by Willson[3]. Another interesting limiting case arises when the wall thickness is so small by comp~son with the wavelength that in (19) the terms in kh may be disregarded. Then
As (ka)-’ increases from zero to infinity, both roots &, e2 increase monotonically, 8, from zero to E, and & from E/(1 - c?) to infinity. In particular when (ka) is unity 8, = E/( 1 + u) = 2p,
02= E/( 1 - u) = 2~( 1 f (T)/(1 - u).
This case is of some historical interest as the vibration corresponding to &, when ka = 1, is the limiting form for thin-wiled cylinders of the equivolumin~ waves discovered by Lame in about 1852. Suppose now that the primary stress p” is taken into account. For a cylinder with very thin walls, so that the terms in kh may again be neglected, the quadratic eqn (17) for 0 becomes R~2+~[a~-R(T-tpO)+(SZ-R2)(ka)-2]+[po(RT-a~+(S-R)(2a~-RT-ST)(ka)-2]=0. (21)
Its solutions for 0 again vary monotonically with (ku), the limiting cases being when ka % 1, 0 = p”
or (RT - a:)/R,
when ka Q 1, 8 = [(R + S)T - 2a$]/(R + S) or (R2 - S2)~R(ka)2.
(22)
The vibrations of a thin-walled elastic cylinder under axial stress
129
For fairly small primary stresses and when (&a)-’ and (kh) are both small, approximate forms for the roots may be obtained by assigning to the coefficients of #a)-’ and (kh) in (17) the values appropriate to the neutral state. This gives E
E(kh)2
E e2+(1_02).
@1=P”+&$+3(1_cr2y
Attention turns now to the implications for stability of the root 8, exhibited in (23).
4. STABILITY
It is clearly of great practical importance to determine the value of the primary stress sufficient to cause instability, the onset of which is taken to be indicated by the vanishing of one of the solutions for 8. For a medium whose strain-energy is known the critical values of p” can be determined numerically from (17) and in the next section one such case is considered in detail. But when these critical values are small, and these cases are especially important in practice, some progress can be made by approximate means. Under the conditions in which (23) is valid it is apparent from a consideration of the root 8, that instability may be produced by a small compressive primary stress. Since
e,, regarded as a function of k, has a minimum at k = + kc, where k: = [3(1 - a2)lu2/(ah),
(24)
2Eh Pc = [3(1 - ,2)]“% *
(25)
and then 8, % (p” + pJ, where
Thus if - p” < pC all of these waves are stable, but if - p” just exceeds this critical value pe then waves whose wavenumbers are in the vicinity of 5 kc will be unstable, and for the wave with the greatest rate of growth the wavenumber, 2 k. say, is given by k: = - 3(1- (r2)po/(2Eh2) = k%-PO/p,).
(26)
The quantities (koh)‘, (koa)-2, (kch)2, (kca)-2 and pJE are all of order h/a. For a cylinder of finite length, account must be taken of the boundary conditions on the terminal planes. The determinant in (17) is even in k and a solution may be constructed by superposition so that both w and pn depend upon z through a factor (sin kz). So for a cylinder confined between two smooth rigid plates at z = 0 and z = 1 in the primary state, it is appropriate to take k=ndl,
(n=lJJ,...).
(27)
Accordingly, for -p” > pc, the axisymmetric instability will be present in the lowest (n = 1) mode for cylinders with length of the order of (ah)“‘, and appears first, as the compressive primary stress is slowly increased, in a cylinder of length I,, where
(28) and when -p” > pc instability will be present in some higher mode for all sufficiently long cylinders. The instability disappears for cylinder lengths significantly less than lC,at least in the context of the present analysis based upon (23), since the relevant value of kh is no longer
I. R. THOMPSON and A. J. WILLSON
730
small. But the physical concept of a thin-walled cylinder would then give way to that of a thick annular plate. Finally, the form of (23) and a consideration of (17) suggest that for small values of ka instability will not appear in axisymmetric modes except under very high primary stresses. In practice, instability will appear in modes which are not axially symmetric before these high stress values are attained. 5. AN EXAMPLE:
THE Ko MODEL
In [ 111,Ko proposed for polyurethane foam rubber the strain-energy function W = p(&+
I:“),
(p constant),
(29)
and because of its simple form this has been used for purposes of general illustration by many writers, a precedent followed here. From (6), (7) and (29), 84 = y-1,
PO= /.L (1 - y-s’*),
(30)
and the other quantities of interest are found to be R=~/.L,
s=(r*=p,
T=3(p-p@).
(31)
The quantity p appearing in (29) is, for the theory of linear elasticity in this model, identical with the Lam& constant Jo, and the Poisson condition A = 1~is satisfied, so that E = 5~12,
u = l/4.
(32)
This model may be used to give a simple instance of the results obtained in Sections 3 and 4. For a cylinder composed of material obeying (29), and with a wall so thin that terms in h may be neglected, (17) and (21) take the form 3p + F[6P - 8 - 8(ku)-*] - [9P* + P{24(ka)-* - 8) - 2O(ka)-*I= 0,
(33)
where F = elp = po*lpk*,
P = tfO/jL.
(34)
Investigation of this equation reveals that its solutions for F are real for all P and (ku). These solutions are shown in Fig. 1. Moreover, for any given value of (ku), there will be a negative root for F, implying instability, if P lies outside the range (PI, P2) determined by the roots of 9P2 + P{24(ku)-‘- 8) - 2O(ku)-*= 0,
(35)
where PI and P2 denote respectively the negative and the positive root. The stability results of Section 4 may be illustrated in the same way. For cylinders of Ko material, (25) and (28) yield for the critical stress pC and the critical length f, PC=
80 “*ph
0
i-y
at
c =
16 “’
z
0
dab)
t/z ,
(36)
and by virtue of (23) and (27) the stress required to produce instability in the lowest (n = 1) mode is given by (37)
The vibrationsof a thin-walledelastic cylinderunderaxial stress
731
Fig. 1. Variationof po*/pk*(F) againstp’/p (P) for Ko material,for three values of ka, when termsin the cylinderthickness are neglected;negative values for F indicateinstability.The curves have one common asymptote,and a commondirectionfor the second asymptote.The curve for ka = 10 is typical of all large ka; for small ka the second asymptoterecedes into the second quadrant.
Figure 2 shows the contours of loglO(-pa/p) against logio(hll) and logio(2a/I). These values have been obtained by direct solution of (17) with B set equal to zero, with particular values of J, R, S, T given by (18) and (31), but the approximation (37) yields an almost identical picture. One further consideration must be borne in mind. In the last section brief reference was made to the modes of vibration which are not axisymmetric, and it was asserted there that for small values of (kc) instability in these modes would appear for values of the primary compressive stress smaller than those required for instability in the axisymmetric modes. For the Ko model it is possible to test the validity of this assertion.
0.5
I
-2 5
-20
1
log (h/l)
-1.5
I
-1.0
Fig, 2. Contoursof log (-#/a), for a finite cylindricalshell of Ko material,againstlog (/I//) and log&/I). Here -pO is the compressivestress requiredto produceinstabilityin the lowest mode.
132
J. R. THOMPSONand A. J. WILLSON
For the general non-axisymmetric vibration, the components of displacement are denoted by u (radially), u (transverse) and w (axially), where u, u, w depend upon the cylindrical coordinates r, 4, z and the time f. The corresponding stresses are readily calculated from (l), (2) and (29), and then the equations of motion for these vibrations in Ko material may be cast into the form piil/.L=3 *,+p-;)+~+Uz*++-4!$+(2-P)w~, ( piil/.& = ++
4u 3
+vW+;-++*+,+(2-P)+
p@/p=(2-P)(u,+T+
(38)
~~+(~-p)~w~+~+~~3w~).
and the boundary conditions are 3u,+E+%+ r r
w, = ,,-~+~= r
r
u,+ w,=o ,
(39)
on the curved surfaces. For considerations of stability, the most important of these modes is found to be the flexural mode in which, for example, w = n(r) sin fj sin kz cos ot. This mode may be analysed by methods identical with those used above for the axisymmetric case. For the cylinder with very thin walls, so that again the terms in (M) may be neglected completely, one finds in analogy with (35), 16P(P - 1) + (3P3 + 17P2 + 8P -20)(h)‘+
P(9P -8)(h)‘=
0.
W)
This equation has, for all (ka), precisely one root in the interval (-0.6,O) and this root will be
Fig. 3. Variation-of-PI and -PI againstka for Ko material.Thesexre, apartfrom the factor p. values of the compressivestress needed to produceinstabilitywhen the cylinderthickaess may be neglectedentirely, respectivelyin the axisymmetricand non-axisymmetricmodes.
The vibrations of a thin-walled elastic cylinder under axial stress
133
denoted by Pi. One readily finds from (35) and (40) that
forkoel:
Pi--
8 qG$’
Fi--7.
S(ku)’
Figure 3 shows how Pi and F, depend upon (ka) generally. Since P, becomes small with (ka)‘, it is clear that for long cylinders one must investigate these non-axisymmetric modes, as well as the high order axisymmetric modes arising from large values of n in (27) and whose instability was discussed in the last section. It must be emphasised that Pi and Pi in (35) and (40) are the limiting values of pa/p for marginal stability when the terms in (kh) are neglected entirely. Their values, therefore, should not be used to predict the primary stresses needed for instability in those waves for which (ka) is so large that it is comparable with (k/t)-‘. The role played by the non-axisymmetric modes of vibration in materials other than those described by Ko’s form of the strain-energy function, and the dependence of the corresponding critical stresses upon the ratio of wall-thickness to cylinder radius, fall outside the scope of the present investigation, but work on these matters is in progress. REFERENCES
111S. A. USMANI and M. F. BEATTY, J. EIasticity 4, 249 (1974). [21 A. J. WILLSON, J. Elasticity 7, 103(1977). [31 A. J. WILLSON, Int. 1. Engng Sci. 15,245 (1977). [41 J. A. MCFADDEN,J. Acoust. Sot. Am. 26,714 (1954). IS1 I. GHOSH. Bull. Calcutta Math. Sot. 14.31 11923). i6j G. HERRMANN and I. MIRSKY, Trans. Ah So;. Mech. Engrs 78,563 (1956). 171D. C. GAZIS, 1. Acoust. Sot. Am. 31,568 (1959). [8] D. C. GAZIS, 1. Acoust. Sot. Am. 31,573 (1959). [91 A. C. ERINGEN and E. S. SUHUBI, Elastodynamics, Vol. I. Academic Press, New York (1974). IO] M. BAKER and J. L. ERICKSEN, J. Wash. Acad. Sci. 44.33 (1954). 111W. L. KO, Ph.D. Thesis. California Institute of Technology, Pasadena (1%3). (Received 17 July 1978)