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Circumferential waves for a cylindrical shell in smooth contact with a continuum
Journal
of Sound and Vibration (1979) H(2),
CIRCUMFERENTIAL
209-22 1
WAVES FOR A CYLINDRICAL SHELL
IN SMOOTH CONTACT
WITH A CONTINUUM
H. I. EPSTEIN Department of Civil Engineering,
University of Connecticut,
Storrs, Connecticut
(Received 29 August 1978, and in revisedform 20 November
06268, U.S.A.
1978)
Dispersion relations are determined for circumferential waves propagating in a layered, circular cylinder by using shell equations to approximate the behavior of the outer layer. These equations include the effects of transverse shear deformation and rotatory inertia. The cylinder consists of an elastic core in smooth contact with a hollow, circular cylinder of distinctly different elastic properties. Two distinct modes exist as the shell thickness reduces to zero. One mode is recognized to be surface waves on the convex cylindrical surface of the core; the second mode is associated with long longitudinal waves in the shell. The approximate dispersion curves for these modes are compared with curves obtained by employing elasticity equations for the layer. As the curvature increases, the agreement of the two theories becomes progressively poorer whether or not any disagreement exists for the case of no curvature. The agreement of the two theories is better when the layer is relatively stiff than when the layer is relatively soft. The shell equations simplify the calculations necessary to produce the dispersion curves.
1. INTRODUCTION
Dispersion relations have been presented for circumferential waves in a composite, circular cylinder [l]. The cylinder consists of an elastic circular core encased in a hollow, circular cylinder (layer) of distinctly different elastic properties. The effects of curvature and layer thickness on the phase velocity of the lowest modes for both an acoustically softer and an acoustically stiffer layer in smooth and bonded contact were considered. Various limiting cases for these waves are of interest in that they produce previously derived results corresponding to other geometries. For the case in which the outer radius of the cylinder is infinite, while the radius of the core remains finite, the motion reduces to interface waves and the effect of curvature on Stoneley waves is reproduced [2]. In the limit as the ratio of layer thickness to wavelength vanishes, the motion of the core reduces to that of Rayleigh waves on the curved surface [3,4]. In the limit as the curvature vanishes, the motion reduces to the waves which occur for a layer joined to a half-space [5]. Extreme values for the layer thickness for this limiting case result in Rayleigh surface waves [6] or Stoneley waves [7] if the material properties satisfy certain restrictive conditions [5, 81. For the composite, circular cylinder, the thickness of the covering layer is usually not small, as compared to the wavelength, for applications in geophysics. Conversely, for many applications in engineering it is sufficient to consider only waves of length larger than or equal to the thickness ofthe layer. A prime example of such a system is a pavement supported by a continuous subgrade. Dynamic disturbances in the subgrade may result from earthquakes, explosions, pile driving or underground traffic [9]. Either smooth or bonded contact may be considered at the pavement-subgrade interface. Curvature in the layer occurs as a result of the crown in the road. The extent to which an approximate shell theory can be used in lieu of elasticity equations 209 0022-460X/79/ 100209t 13$02.00/O
01979 Academic Press Inc. (London) Limited
210
H. I. EPSTEIN
to represent the dynamic behavior of the curved layer which encases and is bonded to the circular core has been investigated [lo]. The shell equations included the effects of transverse shear deformation and rotatory inertia. It was shown that some simplifications result when the shell theory is used. The approximation gave better results for an acoustically stiffer shell than for a softer shell. The shell approximation became better with decreasing curvature. The present paper is concerned with the case in which the outer circular layer is in smooth contact with the circular core. At the interface, normal displacements and stresses are continuous while the shear stresses vanish for both media. Dispersion relations are derived for propagating free waves in this composite cylinder by employing cylindrical shell equations for the layer and elasticity equations for the core. The purpose of this paper is to determine whether these dispersion relations are valid approximations to the dispersion relations found by employing elasticity theory for the layer (shell) [l]. The shell equations which are used are those which have been derived previously for the case of bonded contact [lo].
3. GOVERNING
2.1. CYLINDRICAL
EQUATIONS
SHELL EQUATIONS
A cylindrical shell is shown in Figure the inner face of the shell, the thickness
1. In this figure a, h and 1are defined as the radius to of the shell, and the distance from the inner face of
0
’
v /
/
I
e
Centerof curvcture
Figure 1. The cylindrical
shell.
the shell to the neutral axis, respectively (a list of notation is given in the Appendix). The co-ordinate system referenced to the neutral axis of the shell is indicated by x and y. The subscript 1 will be used to denote the material properties of the shell. It is also useful to introduce R, the radius measured to the neutral axis, which is given by R=a+l. The boundary
conditions
for the shell are in terms of the normal
(1) and shear stresses:
i.e.,
WAVES FOR A SMOOTHLY
aty = h - 1:
SUPPORTED
211
SHELL
cJy = oxy = 0,
(2)
and at y = - [: oY = p(x, t), ox. = z(x, tk
(3)
where p(x, t) and z(x, t) are the normal and tangential tractions on the inner face of the shell. The equations of motion for the cylindrical shell were obtained previously [lo] in a manner analogous to Morley’s treatment of a curved rod [ll]. The basic assumptions made in the derivation of the shell equations are as follows: x and y shell displacements are assumed to be of the form U(.%y, t) = W.% r) - yw,
t), v(x, y, t) = w,
t)
(435)
where U(x, t) and V(x, t) are the displacements of the curved surface y = 0, and Y(x, t) is the rotation of the cross-section; a Timoshenko shear coefficient K is used, found from
s h-l
axrdy=Khpl
-I
av ax
--x
u
(6)
+ Y
where ,u~is the shear modulus; and the stress 0,(x, y, t) is assumed to be small. These assumptions lead to the following equations of motion [lo]:
(8) (9)
where I =
0, if rotatory inertia is ignored 1, if rotatory inertia is included EP = E,/(l
1’
- v;);
(10) (11)
in which E, is Young‘s modulus, v1 is Poisson’s ratio, and p1 is the mass density of the shell material. Also, (12,13) M, = f - rj + q= + (1 - 4~1+ 6~/= - 4q3)/4(1/5 + q),
(14)
in which, q = l/h, [ = h/a
(1% 16)
and q, which defines the location of the neutral axis, can be found from v = [l - (l/C)ln(l
+ i)l/ln(l
+ 0
(17)
2.2. EQUATIONS FOR THE CORE
For the core, when harmonic wave forms are substituted into the elastic, plane strain equations of motion in polar co-ordinates (r, O), displacements, uVand uO,and stresses, dr
212
H. I. EPSTEIN
in terms of the following quantities: II. the number of complete and flrH can be found circumferential waves: (I, the radius of the core; k, the wave number which is equal to rliu [ 1] ; to. the radian frequency which is equal to kc, c being the phase velocity; J, J; and Jy. the 11th order Bessel function of the first kind and its first and second derivatives with respect to r’; 1’.,LL and p, the properties of the core material; t. time: two constants. .4 and B for this presentation: and x and fl which are dimen\ionleas phase velocities CIC~and c’cr., respectively. where c,. and cr. are the phase velocities of dilatation and shear waves given by c,..: = 2p( 1 - ?)//I( 1 - 21’).c..; = ,Ll,‘/I. The stresses and displacements
are given by
U* = [AJi(crkr) + (n/r)BJ$kr)] u0 = - [(nlr)AJ,,(akr)
0
rtl
I’
= -2p{A(:)[Jk(nkr)
eiw’cos rz9,
(20)
+ BJ&G’kr)] eiot sin ~0,
(Ctk)2 J”(akr)
J;(ctkr) - &
(18, 19)
+ B :
1
(21)
e'"" cosn9, (22)
Ji(fikr) - $- J,(Bkr)
oi
- i JH(okr)] + B[[(zr
11
-$(bk)2]J#?kr)-
:J~(/Ikr)~~e””
sin&. (23)
3. THE COMPOSITE 3.1.
CYLINDER
DISPERSION RELATIONS
Consider the cylindrical shell, of material 1, to be in smooth contact with the circular core, of radius a, of distinctly different material properties. The interface of the two materials is the curved plane y = --I or, equivalently, r = a (see Figure 1). Along this interface, the boundary conditions for smooth contact are a#,
0, t) = p(x, t), u,(a, 8, t) = V(x, f),
(24,25)
frTlO(u, 8, f) = 0, z(x, t) = 0.
(2627)
Substitution of equation (20) into equation (25), and using Bowman [12] for derivatives of Bessel functions, results in V = jkA[aJn_ Similarly,
when equation
l(tm) -
J,,(m)] + kBJ,(fin)j
(22) is substituted
p = 2pk2{A[(1/2n)(2n
into equation
cos
relations
given
no.
of the cross-section
and further
+ lirl)J,,(fin)]j
eIot cos no.
(29)
be of the form (30)
let U = D eiot sin ntl,
where D is also a constant. using the relation
(28)
(24) the result is
Vr(X.t) = C eiot sin nH, where C is a constant,
by
+ 2 - nP2)Jn(an) - (cqn)Jn_ l(an)] + B[BJ,_,(Bn) -(1
Let the rotation
eiot
some
Substituting
equations
(27), (28), (30) and (31) into equation
(31) (7),
WAVES
FOR A SMOOTHLY a/ax
=
SUPPORTED
213
SHELL
(32)
(i/R)a/ae,
and solving for D results in D =
+ K) + ky[L’ B(H + ~)J,(j?n)
(l/@‘){kA[crJn_ r(cln) - J,(an)]y#(H
- hC[XB2M,Z
+ KYiL/t]:,
(33)
where D’ = 6fi2M - yHL2 - Kyc2i?/<2.
(34)
In these equations, a dimensionless wave number, 5, and material parameters, y and 6, are introduced where (35-37)
5 = kk Y = P~/P>6 = P,/P. The other new symbols used in these equations are 2/(1 - v,), L = l/(1 + vi), M = 1 + M,t2L2.
H =
(38-40)
Also, it will be useful to introduce JA = aJ,- ,(an)/Jn(an), JB = BJ,_ ,(Bn)/J,(LW.
(41,42)
Substitution of equations (27) to (33) into equations (8) and (9) results in two homogeneous equations in the constants A, B, and hC. A third homogeneous equation is obtained from equations (23) and (26). For these three equations, non-trivial solutions exist when the determinant of the resulting 3 x 3 matrix is equal to zero. The matrix [S] is given by
[S]
=
D’
Sll
x
s21
[ S31
s12
s13
~2~
s23
‘32
s33
where sll = [Ky{L + (1/2
s13
=
=
-KY@
-2y12HL2Ml
-
+
K)(Ky&
(1/2t;D’)yiL2(H
+ 2~y - 26fi2t2M3Z
+
1
(43)
,
6B2t2M2)][1
- JAI,
(49
+ K)(Ky[L + SP’t’M,), + (1/t2D’)(~y[L
(44b)
+ SP2[‘M2)
x (KytL + W25’M2Z), szl = 2 - p’ + [JA
-
l]{KytL2
+(1/25D’)[~i(H s22
=
2JB - 2 - (25/t) + lcytL2 + (yi2HL2/<) ‘23
=
sgI = JA -
-2KyL
-
[yc(H
+ (yi2HL2/5)
(44c)
- SP2tM Wd)
+ K)L212 - (2@$
- Sb25M
+ K)L~/D’][~~~~M,Z
+ (1/25D’)(y[(H + Ky[L/t2].
1 - [it, s32 = 1 + (c/()(1 - JB) - b2/2, s33 = 0.
+ K)L’)~,
(44e) (440
(4% h, i)
The approximation made by using a shell theory for the layer is seen to reduce the 6 x 6 determinant obtained from elasticity theory [l] to the determinant of the 3 x 3 matrix [S] given in equation (43). The 3 x 3 determinant can be expanded into a polynomial equation in < with coefficients in terms of the material parameters, the phase velocity, and the curvature.
214 3.2.
H. I. EPSTEIN LIMITING
CASES
For preScri&d curvature and prescribed material properties, setting the determinant of [S] equal to zero gives a dispersion relation between the dimensionless phase velocity, B, and the dimensionless wave number, t( = kh). For a given geometry. phase velocities can be computed by specifying the material parameters v, v,, y and 6. One way of presenting the results is to plot p us. l/ka for prescribed material properties and for various values of kh. Hence, for constant wave number, k, the plot would show phase velocity vs. curvature, with curvature increasing as 1 !ka increases. For this way of presenting the results, < -+ 0 represents the limit as the shell thickness vanishes. In the limit 2 -+ 0, equation (43) yields
i? x lim (D’) = 0,
(45)
t-0
where R = ‘,(/I” - 2)2 - 4 - (4/32/n) + JB[4 + (2/P/n) - (2/@] + JA[~ + (2jP/n) - (2/r@ - JB(~ - (2/rij2)] jJ,,k4J,,UW,
(46)
R = 0 being the equation for the phase velocity of surface waves on a convex cylindrical surface [2]. This is the same limit as predicted by elasticity theory [ 11,and is one of the modes that arises for the limit r + 0. The second mode, as seen from equation (49, must satisfy the relation lim (D’) = 0.
(47)
C-0
From equation (34), the limiting phase velocity for this mode is given by c2= (Eplp,)(l + ic/Hn2).
(48)
A second way of presenting the results is to plot fi vs. t( =!h) for various values of the integer n( = ka). Such curves can be interpreted by considering the wave number, k, to be constant. From this point of view, the plot would show phase velocity vs. shell thickness for various values of n. Since n = ka, and k is considered to be constant, varying II corresponds to varying curvature, where the larger the value of n the smaller the curvature. For this way of presenting the results, n -+ x, represents the plate and half-space problem. In the limit as the curvature vanishes, the shell matrix [S] in equation (43) reduces to the plate matrix [P] given by
[P] =
- 4r&
- rcyS/s
2~y
2 - p’ + q
2 + (K/s
-2K:’
s+l/s
0
i 29
+
Nt2i6 ,
(49)
J
in which K = KY - Sp’, N = Hy - 6[j21, q = (1 - a2)lj2, s = (1 - p”)“‘.
(D-53)
When the determinant of [P] is set equal to zero, the result is (qP2NK/12rcy)t3
- (NRf/12~y)t2
- qSg4t - R’ = 0,
(54)
R’ = lim R = (2 - p2)2 - 4(1 - a2)1’2(l - b2)‘/2, “-a
(55)
where
R’ = 0 being the equation
for the phase velocity of Rayleigh’s surface waves [6]. The
WAVES FOR A SMOOTHLY
SUPPORTED
215
SHELL
polynomial expression given in equation (54) was presented by Achenbach and Keshava [13] for a plate in smooth contact with a half-space. 3.3.
THE SHELL MODE
As was seen from equation (45) in the previous section, two distinct modes exist as the shell thickness reduces to zero (5 + 0). One mode is recognized to be surface waves on a convex cylindrical surface which reduce to Rayleigh surface waves in the core as the curvature vanishes. This will be termed the surface wave mode. The second mode has a phase velocity given by equation (48). As the curvature vanishes, the phase velocity for this mode reduces to c = (EJpJ? (56) This is the velocity of long longitudinal waves in the plate. For a given Poisson’s ratio in the layer, H is given by equation (38). Hence, for prescribed Poisson’s ratio, vl, the nondimensional phase velocity, c~/(EJP~), is a function of n. For a given value of n, and for the limit 5 + 0, the phase velocity is larger than (EJp,) I” . As n increases, the phase velocity decreases to its limiting value. The appearance of this shell mode indicates that for small 5, the longitudinal motion of the curved plate is only weakly coupled to the motion of the core, and this coupling vanishes altogether in the limit c + 0. This behavior is due to the smooth contact at the interface.
4. DISPERSION
CURVES
Dispersion curves and comparisons between the elasticity and shell theories are presented in this section. The results depend somewhat on the value chosen for the Timoshenko shear coefficient, K. For the curves which follow, K is chosen in the manner suggested by Mindlin [14], which was shown to give (57)
I.0
P 0-a
0.6
Figure 2. Phase velocity versus wave number for the surface wave mode of a composite Y = Y, = 0.25, 6 = 0.75, y = 0.1, R = 0365.
soft shell:
cylinder
with
a relatively
216 .ii r
I.1
I.0
P
? 0.9
il
0.8
5.7
I
1
I 2.0
1
I.0
0.5
0
I.5
E
Figure 3. Phase velocity versus wave number for the lowest mode of a composite shell: Y = y1 = 0.25, 6 = 144, y = 8.0, C2 = 2.36.
cylinder
with a relatively
stiff
---
_--0.9
_---
P 0.8
-
-
-
-
-
0.02
I
1
0.04
0.06
1
0.08
0.13
I /ka Figure
Elasticity
Comparison theory; -------,
4.
of elastlclt) shell theory.
and shell theories
for the relatively
soft shell-surface
____--
Rave mode.
___-
B \
r
5 = 1.0
5 = 0.5 h
7
I
h
c
,
n
elast.
shell
elast.
shell
1 2 3 4 10
2.02 0.975 1.00 0.992 0.931
2.45 1.01 1.04 1.03 0.953
1.14 O-742 0.753 0,751 0.738
1.37 0.877 0.907 0.902 0.859
~,
WAVES FOR A SMOOTHLY
SUPPORTED
217
SHELL
for this problem, where cp, and cT, represent the velocities of Rayleigh and shear waves in the shell material, respecttvely. Plots of the dimensionless phase velocity b us. l/ka or us. 5 are presented herein for the surface wave mode and for the shell mode. A useful parameter in specifying the materials in the problem is the ratio of shear wave velocities 52 = CT,&
=
(y/s)‘?
(58)
Curves for the two modes are presented for the acoustically softer shell, Q = O-365, and for the acoustically stiffer shell, Q = 2.36, which have been investigated previously by using elasticity theory for the composite cylinder [l], for the case of no curvature [S], for bonded contact [lo], and for comparisons of the no curvature case with plate theory [13].
4.1.
THE SURFACE
WAVE MODE
For the approximate shell theory, by using equation (43), the plot of phase velocity us. shell thickness, /I us. 5, is shown in Figure 2 for the soft shell, Q = 0.365. The same plot for the stiff shell, Sz = 2.36, is shown in Figure 3. As II -+ o~i,no solution is possible for /I > 1 [l]
OS8 310 I
I
0
0.02
I
I
I
0.04
0.06
0.08
I 0.10
I /ko
Figure 5. Comparison of elasticity Elasticity theory; ------, shell theory.
and shell theories
for the relatively
stiff shell-surface
r
1 2 3 4 10
-----,
,
5 = 0.5 n
wave mode.
5 = 1.0
n
I 2.38 0.836 0.916 0.938 0.916
243 0.860 0.936 0.953 0.923
2.23 0.728 0.866 0.918 0.970
2.31 0.769 0.892 0.937 0.973
and hence that portion of the curve is shown as a dotted line in Figure 3. The comparison of elasticity theory and shell theory for the soft plate is shown by plotting j? us. l/ka in Figure 4. The comparison for the stiff plate is shown in Figure 5. Values for fi for n < 10 are shown in tabular form. The curves for the elasticity theory used in Figures 4 and 5 were shown previously in reference [ 11.
218
H. I. EPSTEIN
0.6
Figure 6. Phase velocity versus wave number for the shell mode of a composite shell: v = Y, = 0.25, 6 = 0.75, y = 0.1, 0 = 0365.
0.501 “*co
I 0
I n=50
I
n=30
I 0.02
1
I
n=ZO
n=15
1 0.04
/ 0.06
cylinder
with a relatively
soft
n=lC I 0.08
I 0.10
I/k0
Figure 7. Comparison theory:------,shelltheory.
of elasticity
r
and shell theories
;=o *
for the relatively
soft shell-shell
l = 0.5
n
shell
b elast.
1 2 3 4 10
0.684 0.618 0606 0601 0.597
0.503 0.491 0.524 0.541 0.572
f=l,O
mode. -,
’
b shell
elast.
shell
0.524 O-520 0.545 a558 0.581
0.425 0401 O-452 0.480 0.532
0443 0.441 0.500 0.523 0.566
Elasticity
WAVES
4.2.
FOR A SMOOTHLY
SUPPORTED
SHELL
219
THE SHELL MODE
For the stiff layer, 52 = 2.36, the shell mode is not investigated because, for this case, the velocity of long longitudinal waves in the layer has a value for the non-dimensional phase velocity, /I, that is greater than unity for the case of no curvature. The plot of /I us. 5 for the shell mode of the soft plate, Sz = 0.365, is shown in Figure 6. The case of no curvature, ii -+ c;o, is seen to be the line /I = /I,. The phase velocity for this value of /I is equal to &,/P,) w .For a constant wave number, k, and for small values of c, /I > B,. As < increases, the phase velocity decreases and /3 < b,. For large values of 5, as the curvature increases, the phase velocity decreases. The plot of /I us. l/ka for the shell mode of the soft plate, Q = 0.365, is shown in Figure 7 for both the elasticity theory and the shell theory. The curves for the elasticity theory were shown previously in reference [I].
5. DISCUSSION The accuracy of the approximate theory is checked by comparing the phase velocities of the lowest modes obtained by using the approximate shell theory and the exact elasticity theory. Good agreement is obtained if the ratio of the layer thickness to the wavelength is small. For a constant layer thickness, this case represents small wave numbers. Achenbach and Keshava [13] showedlthat the plate theory, which includes the effects of rotatory inertia and transverse shear deformation, gave perfect agreement with the elasticity theory for phase velocities of the lowest mode of the stiff plate over the entire permissible range of t( = kh). For the soft plate, good agreement was found for small values of 5 and there was little advantage in including transverse shear and rotatory inertia. As 5 increased, the agreement became poorer for the shell mode. For the surface wave mode, the two theories agreed again at an intermediate 5, but diverged again with increasing 5. As seen in Figures 4,5 and 7, as the curvature increases, the agreement of the two theories becomes progressively poorer whether or not any disagreement exists for the case of no curvature. The agreement of the two theories is better for the acoustically stiffer plate than for the acoustically softer plate. With the exception of the shell mode, the limit r + 0 gives phase velocities that are independent of the theory used in the layer. The simplification made by introducing a shell theory for the curved layer is in the reduction of the dispersion relations from the roots of a 6 x 6 to a 3 x 3 determinant. Achenbach and Keshava [13] showed that the use of the plate theory reduces the calculations significantly. However, the approximation made by using the shell theory, when there is curvature, does not simplify the calculations as much as the plate theory when there is no curvature. While the computational savings associated with the approximate equation are not great, there are some advantages associated with using a shell theory. First, the dispersion equation can be written in polynomial form, and thus the effect of the variation of individual material parameters may be more quickly identified. Also, the stresses and displacements are more readily computed when shell equations are used.
ACKNOWLEDGMENT The author’s thanks are due to Professor Jan D. Achenbach of the Technological tute, Northwestern University, for his guidance and advice.
Insti-
220
H. I.
EPSTEIK
REFERENCES 1. H. I. EPSTEIN 1976 Journal oj‘Sound and Vibration 48, 57-71. Circumferential waves in a composite circular cylinder. 2. H. 1. EPSTEIN 1976 Journal of Sound and Vibration 46. 59-66. The effect of curvature on Stoneley waves. 3. I. A. VIKTOROV1958 Soviet Physics-Acoustics: American Institute of Physics Translation 4, 13 l-l 36. Rayleigh-type waves on a cylindrical surface. 4. K. SEZAWA 1927 Bulletin-Earthquake Research Institute (Tokyo) 3, I-18. Dispersion of elastic waves propagated on the surface of stratified bodies and on curved surfaces. 5. J. D. ACHENBACHand H. I. EPSTEIN 1967 Journalof the Engineering Mechanics Dioision. American Society oj Civil Engineers 93, 27-42. Dynamic lnlcractlon of a layer and a half-space. 6. J. W. STRUTT,LORD RAYLEIGH1885 Proceedings ofthe London Mathematicul Society 17, 4411. On waves propagated along the plane surface of an elastic solid. 7. R. STONELEY1924 Proceedings of the Rqval Society (London) 106.416428. Elastic waves at the surface of separation of two solids. 8. J. G. SCHOLTE1942 Proceetlings Nederlandse Akadernic [Ian Weterschappen 45. 159-164. On the Stoneley wave equation. 1966 Journal oj’the Engineering Mechanics 9. J. D. ACHENBACH,S. P. KESHXJ~ and G. HEKRMANN Dicision, American Societ_voj’CiG/ Engineers 92. 113-129. Waves in a smoothly joined plate and half-space. 10. H. 1. EPSTEIN 1978 Journal of’Sound and Vibration 58, 155-166. Circumferential waves for a cylindrical shell supported by a continuum. 11. L. S. D. MORLEY 1961 Quarterly Journal ofMechanics and Applied Mathematics XIV, 155-172. Elastic waves in a naturally curved rod. 12. F. BOWMAN1938 Introduction to Bessel Functions. London: Longmans, Green and Co. 13. J. D. ACHENBACHand S. P. KESHAVA1967 Journal of Applied Mechanics, Transaitions of the American Society of Mechanical Engineers 34,397404. Free waves in a plate supported by a semiinfinite continuum. 14. R. D. MINDLIN 1951 Journal of‘Applied Mechanics, Transactions oj the Americun Society oj‘ Mechanical Engineers 18, 31-38. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates.
APPENDIX a A, B. C, D C CL CT
E EP h H, L, M 1
J” JA, JB k K, N
1
radius of the core constants phase velocity phase velocity of dilatational waves phase velocity of shear waves Young’s modulus material parameter given by equation (11) shell thickness dimensionless quantities given by equations (35) to (37) 1 if rotatory inertia included, 0 if not nth order Bessel function of the first kind dimensionless quantities given by equations (41) and (42) wave number dimensionless quantities given by equations (50) and (51) distance from inner face of shell to the neutral axis
: NOTATION M,, M,, M, n
r. 0 R
it, R’
PI II, I’
LT.v
-%Y 1
dimensionless quantities given by equations (12) to (14) number of complete circumferential modes normal traction 3 x 3 plate matrix, equation (49) dimensionless quantities given by equations (52) and (53) polar co-ordinates radius to the neutral axis of the shell dimensionless quantities given by equations (46) and (55) 3 x 3 shell matrix, equation (43) time displacements displacements on the curved surface y = 0 shell co-ordinates dimensionless phase velocity (= (,/CL) tic,
221
WAVES FOR A SMOOTHLY SUPPORTED SHELL
PI/P
P
P,IP hIa Vh Timoshenko shear coefficient shear modulus Poisson’s ratio dimensionless wave number (=kh)
0 5 Y
mass density stress tangential traction rotation of the shell cross-section radian frequency
: c&T Note: subscript 1 pertains
to the shell.
of Sound and Vibration (1979) H(2),
CIRCUMFERENTIAL
209-22 1
WAVES FOR A CYLINDRICAL SHELL
IN SMOOTH CONTACT
WITH A CONTINUUM
H. I. EPSTEIN Department of Civil Engineering,
University of Connecticut,
Storrs, Connecticut
(Received 29 August 1978, and in revisedform 20 November
06268, U.S.A.
1978)
Dispersion relations are determined for circumferential waves propagating in a layered, circular cylinder by using shell equations to approximate the behavior of the outer layer. These equations include the effects of transverse shear deformation and rotatory inertia. The cylinder consists of an elastic core in smooth contact with a hollow, circular cylinder of distinctly different elastic properties. Two distinct modes exist as the shell thickness reduces to zero. One mode is recognized to be surface waves on the convex cylindrical surface of the core; the second mode is associated with long longitudinal waves in the shell. The approximate dispersion curves for these modes are compared with curves obtained by employing elasticity equations for the layer. As the curvature increases, the agreement of the two theories becomes progressively poorer whether or not any disagreement exists for the case of no curvature. The agreement of the two theories is better when the layer is relatively stiff than when the layer is relatively soft. The shell equations simplify the calculations necessary to produce the dispersion curves.
1. INTRODUCTION
Dispersion relations have been presented for circumferential waves in a composite, circular cylinder [l]. The cylinder consists of an elastic circular core encased in a hollow, circular cylinder (layer) of distinctly different elastic properties. The effects of curvature and layer thickness on the phase velocity of the lowest modes for both an acoustically softer and an acoustically stiffer layer in smooth and bonded contact were considered. Various limiting cases for these waves are of interest in that they produce previously derived results corresponding to other geometries. For the case in which the outer radius of the cylinder is infinite, while the radius of the core remains finite, the motion reduces to interface waves and the effect of curvature on Stoneley waves is reproduced [2]. In the limit as the ratio of layer thickness to wavelength vanishes, the motion of the core reduces to that of Rayleigh waves on the curved surface [3,4]. In the limit as the curvature vanishes, the motion reduces to the waves which occur for a layer joined to a half-space [5]. Extreme values for the layer thickness for this limiting case result in Rayleigh surface waves [6] or Stoneley waves [7] if the material properties satisfy certain restrictive conditions [5, 81. For the composite, circular cylinder, the thickness of the covering layer is usually not small, as compared to the wavelength, for applications in geophysics. Conversely, for many applications in engineering it is sufficient to consider only waves of length larger than or equal to the thickness ofthe layer. A prime example of such a system is a pavement supported by a continuous subgrade. Dynamic disturbances in the subgrade may result from earthquakes, explosions, pile driving or underground traffic [9]. Either smooth or bonded contact may be considered at the pavement-subgrade interface. Curvature in the layer occurs as a result of the crown in the road. The extent to which an approximate shell theory can be used in lieu of elasticity equations 209 0022-460X/79/ 100209t 13$02.00/O
01979 Academic Press Inc. (London) Limited
210
H. I. EPSTEIN
to represent the dynamic behavior of the curved layer which encases and is bonded to the circular core has been investigated [lo]. The shell equations included the effects of transverse shear deformation and rotatory inertia. It was shown that some simplifications result when the shell theory is used. The approximation gave better results for an acoustically stiffer shell than for a softer shell. The shell approximation became better with decreasing curvature. The present paper is concerned with the case in which the outer circular layer is in smooth contact with the circular core. At the interface, normal displacements and stresses are continuous while the shear stresses vanish for both media. Dispersion relations are derived for propagating free waves in this composite cylinder by employing cylindrical shell equations for the layer and elasticity equations for the core. The purpose of this paper is to determine whether these dispersion relations are valid approximations to the dispersion relations found by employing elasticity theory for the layer (shell) [l]. The shell equations which are used are those which have been derived previously for the case of bonded contact [lo].
3. GOVERNING
2.1. CYLINDRICAL
EQUATIONS
SHELL EQUATIONS
A cylindrical shell is shown in Figure the inner face of the shell, the thickness
1. In this figure a, h and 1are defined as the radius to of the shell, and the distance from the inner face of
0
’
v /
/
I
e
Centerof curvcture
Figure 1. The cylindrical
shell.
the shell to the neutral axis, respectively (a list of notation is given in the Appendix). The co-ordinate system referenced to the neutral axis of the shell is indicated by x and y. The subscript 1 will be used to denote the material properties of the shell. It is also useful to introduce R, the radius measured to the neutral axis, which is given by R=a+l. The boundary
conditions
for the shell are in terms of the normal
(1) and shear stresses:
i.e.,
WAVES FOR A SMOOTHLY
aty = h - 1:
SUPPORTED
211
SHELL
cJy = oxy = 0,
(2)
and at y = - [: oY = p(x, t), ox. = z(x, tk
(3)
where p(x, t) and z(x, t) are the normal and tangential tractions on the inner face of the shell. The equations of motion for the cylindrical shell were obtained previously [lo] in a manner analogous to Morley’s treatment of a curved rod [ll]. The basic assumptions made in the derivation of the shell equations are as follows: x and y shell displacements are assumed to be of the form U(.%y, t) = W.% r) - yw,
t), v(x, y, t) = w,
t)
(435)
where U(x, t) and V(x, t) are the displacements of the curved surface y = 0, and Y(x, t) is the rotation of the cross-section; a Timoshenko shear coefficient K is used, found from
s h-l
axrdy=Khpl
-I
av ax
--x
u
(6)
+ Y
where ,u~is the shear modulus; and the stress 0,(x, y, t) is assumed to be small. These assumptions lead to the following equations of motion [lo]:
(8) (9)
where I =
0, if rotatory inertia is ignored 1, if rotatory inertia is included EP = E,/(l
1’
- v;);
(10) (11)
in which E, is Young‘s modulus, v1 is Poisson’s ratio, and p1 is the mass density of the shell material. Also, (12,13) M, = f - rj + q= + (1 - 4~1+ 6~/= - 4q3)/4(1/5 + q),
(14)
in which, q = l/h, [ = h/a
(1% 16)
and q, which defines the location of the neutral axis, can be found from v = [l - (l/C)ln(l
+ i)l/ln(l
+ 0
(17)
2.2. EQUATIONS FOR THE CORE
For the core, when harmonic wave forms are substituted into the elastic, plane strain equations of motion in polar co-ordinates (r, O), displacements, uVand uO,and stresses, dr
212
H. I. EPSTEIN
in terms of the following quantities: II. the number of complete and flrH can be found circumferential waves: (I, the radius of the core; k, the wave number which is equal to rliu [ 1] ; to. the radian frequency which is equal to kc, c being the phase velocity; J, J; and Jy. the 11th order Bessel function of the first kind and its first and second derivatives with respect to r’; 1’.,LL and p, the properties of the core material; t. time: two constants. .4 and B for this presentation: and x and fl which are dimen\ionleas phase velocities CIC~and c’cr., respectively. where c,. and cr. are the phase velocities of dilatation and shear waves given by c,..: = 2p( 1 - ?)//I( 1 - 21’).c..; = ,Ll,‘/I. The stresses and displacements
are given by
U* = [AJi(crkr) + (n/r)BJ$kr)] u0 = - [(nlr)AJ,,(akr)
0
rtl
I’
= -2p{A(:)[Jk(nkr)
eiw’cos rz9,
(20)
+ BJ&G’kr)] eiot sin ~0,
(Ctk)2 J”(akr)
J;(ctkr) - &
(18, 19)
+ B :
1
(21)
e'"" cosn9, (22)
Ji(fikr) - $- J,(Bkr)
oi
- i JH(okr)] + B[[(zr
11
-$(bk)2]J#?kr)-
:J~(/Ikr)~~e””
sin&. (23)
3. THE COMPOSITE 3.1.
CYLINDER
DISPERSION RELATIONS
Consider the cylindrical shell, of material 1, to be in smooth contact with the circular core, of radius a, of distinctly different material properties. The interface of the two materials is the curved plane y = --I or, equivalently, r = a (see Figure 1). Along this interface, the boundary conditions for smooth contact are a#,
0, t) = p(x, t), u,(a, 8, t) = V(x, f),
(24,25)
frTlO(u, 8, f) = 0, z(x, t) = 0.
(2627)
Substitution of equation (20) into equation (25), and using Bowman [12] for derivatives of Bessel functions, results in V = jkA[aJn_ Similarly,
when equation
l(tm) -
J,,(m)] + kBJ,(fin)j
(22) is substituted
p = 2pk2{A[(1/2n)(2n
into equation
cos
relations
given
no.
of the cross-section
and further
+ lirl)J,,(fin)]j
eIot cos no.
(29)
be of the form (30)
let U = D eiot sin ntl,
where D is also a constant. using the relation
(28)
(24) the result is
Vr(X.t) = C eiot sin nH, where C is a constant,
by
+ 2 - nP2)Jn(an) - (cqn)Jn_ l(an)] + B[BJ,_,(Bn) -(1
Let the rotation
eiot
some
Substituting
equations
(27), (28), (30) and (31) into equation
(31) (7),
WAVES
FOR A SMOOTHLY a/ax
=
SUPPORTED
213
SHELL
(32)
(i/R)a/ae,
and solving for D results in D =
+ K) + ky[L’ B(H + ~)J,(j?n)
(l/@‘){kA[crJn_ r(cln) - J,(an)]y#(H
- hC[XB2M,Z
+ KYiL/t]:,
(33)
where D’ = 6fi2M - yHL2 - Kyc2i?/<2.
(34)
In these equations, a dimensionless wave number, 5, and material parameters, y and 6, are introduced where (35-37)
5 = kk Y = P~/P>6 = P,/P. The other new symbols used in these equations are 2/(1 - v,), L = l/(1 + vi), M = 1 + M,t2L2.
H =
(38-40)
Also, it will be useful to introduce JA = aJ,- ,(an)/Jn(an), JB = BJ,_ ,(Bn)/J,(LW.
(41,42)
Substitution of equations (27) to (33) into equations (8) and (9) results in two homogeneous equations in the constants A, B, and hC. A third homogeneous equation is obtained from equations (23) and (26). For these three equations, non-trivial solutions exist when the determinant of the resulting 3 x 3 matrix is equal to zero. The matrix [S] is given by
[S]
=
D’
Sll
x
s21
[ S31
s12
s13
~2~
s23
‘32
s33
where sll = [Ky{L + (1/2
s13
=
=
-KY@
-2y12HL2Ml
-
+
K)(Ky&
(1/2t;D’)yiL2(H
+ 2~y - 26fi2t2M3Z
+
1
(43)
,
6B2t2M2)][1
- JAI,
(49
+ K)(Ky[L + SP’t’M,), + (1/t2D’)(~y[L
(44b)
+ SP2[‘M2)
x (KytL + W25’M2Z), szl = 2 - p’ + [JA
-
l]{KytL2
+(1/25D’)[~i(H s22
=
2JB - 2 - (25/t) + lcytL2 + (yi2HL2/<) ‘23
=
sgI = JA -
-2KyL
-
[yc(H
+ (yi2HL2/5)
(44c)
- SP2tM Wd)
+ K)L212 - (2@$
- Sb25M
+ K)L~/D’][~~~~M,Z
+ (1/25D’)(y[(H + Ky[L/t2].
1 - [it, s32 = 1 + (c/()(1 - JB) - b2/2, s33 = 0.
+ K)L’)~,
(44e) (440
(4% h, i)
The approximation made by using a shell theory for the layer is seen to reduce the 6 x 6 determinant obtained from elasticity theory [l] to the determinant of the 3 x 3 matrix [S] given in equation (43). The 3 x 3 determinant can be expanded into a polynomial equation in < with coefficients in terms of the material parameters, the phase velocity, and the curvature.
214 3.2.
H. I. EPSTEIN LIMITING
CASES
For preScri&d curvature and prescribed material properties, setting the determinant of [S] equal to zero gives a dispersion relation between the dimensionless phase velocity, B, and the dimensionless wave number, t( = kh). For a given geometry. phase velocities can be computed by specifying the material parameters v, v,, y and 6. One way of presenting the results is to plot p us. l/ka for prescribed material properties and for various values of kh. Hence, for constant wave number, k, the plot would show phase velocity vs. curvature, with curvature increasing as 1 !ka increases. For this way of presenting the results, < -+ 0 represents the limit as the shell thickness vanishes. In the limit 2 -+ 0, equation (43) yields
i? x lim (D’) = 0,
(45)
t-0
where R = ‘,(/I” - 2)2 - 4 - (4/32/n) + JB[4 + (2/P/n) - (2/@] + JA[~ + (2jP/n) - (2/r@ - JB(~ - (2/rij2)] jJ,,k4J,,UW,
(46)
R = 0 being the equation for the phase velocity of surface waves on a convex cylindrical surface [2]. This is the same limit as predicted by elasticity theory [ 11,and is one of the modes that arises for the limit r + 0. The second mode, as seen from equation (49, must satisfy the relation lim (D’) = 0.
(47)
C-0
From equation (34), the limiting phase velocity for this mode is given by c2= (Eplp,)(l + ic/Hn2).
(48)
A second way of presenting the results is to plot fi vs. t( =!h) for various values of the integer n( = ka). Such curves can be interpreted by considering the wave number, k, to be constant. From this point of view, the plot would show phase velocity vs. shell thickness for various values of n. Since n = ka, and k is considered to be constant, varying II corresponds to varying curvature, where the larger the value of n the smaller the curvature. For this way of presenting the results, n -+ x, represents the plate and half-space problem. In the limit as the curvature vanishes, the shell matrix [S] in equation (43) reduces to the plate matrix [P] given by
[P] =
- 4r&
- rcyS/s
2~y
2 - p’ + q
2 + (K/s
-2K:’
s+l/s
0
i 29
+
Nt2i6 ,
(49)
J
in which K = KY - Sp’, N = Hy - 6[j21, q = (1 - a2)lj2, s = (1 - p”)“‘.
(D-53)
When the determinant of [P] is set equal to zero, the result is (qP2NK/12rcy)t3
- (NRf/12~y)t2
- qSg4t - R’ = 0,
(54)
R’ = lim R = (2 - p2)2 - 4(1 - a2)1’2(l - b2)‘/2, “-a
(55)
where
R’ = 0 being the equation
for the phase velocity of Rayleigh’s surface waves [6]. The
WAVES FOR A SMOOTHLY
SUPPORTED
215
SHELL
polynomial expression given in equation (54) was presented by Achenbach and Keshava [13] for a plate in smooth contact with a half-space. 3.3.
THE SHELL MODE
As was seen from equation (45) in the previous section, two distinct modes exist as the shell thickness reduces to zero (5 + 0). One mode is recognized to be surface waves on a convex cylindrical surface which reduce to Rayleigh surface waves in the core as the curvature vanishes. This will be termed the surface wave mode. The second mode has a phase velocity given by equation (48). As the curvature vanishes, the phase velocity for this mode reduces to c = (EJpJ? (56) This is the velocity of long longitudinal waves in the plate. For a given Poisson’s ratio in the layer, H is given by equation (38). Hence, for prescribed Poisson’s ratio, vl, the nondimensional phase velocity, c~/(EJP~), is a function of n. For a given value of n, and for the limit 5 + 0, the phase velocity is larger than (EJp,) I” . As n increases, the phase velocity decreases to its limiting value. The appearance of this shell mode indicates that for small 5, the longitudinal motion of the curved plate is only weakly coupled to the motion of the core, and this coupling vanishes altogether in the limit c + 0. This behavior is due to the smooth contact at the interface.
4. DISPERSION
CURVES
Dispersion curves and comparisons between the elasticity and shell theories are presented in this section. The results depend somewhat on the value chosen for the Timoshenko shear coefficient, K. For the curves which follow, K is chosen in the manner suggested by Mindlin [14], which was shown to give (57)
I.0
P 0-a
0.6
Figure 2. Phase velocity versus wave number for the surface wave mode of a composite Y = Y, = 0.25, 6 = 0.75, y = 0.1, R = 0365.
soft shell:
cylinder
with
a relatively
216 .ii r
I.1
I.0
P
? 0.9
il
0.8
5.7
I
1
I 2.0
1
I.0
0.5
0
I.5
E
Figure 3. Phase velocity versus wave number for the lowest mode of a composite shell: Y = y1 = 0.25, 6 = 144, y = 8.0, C2 = 2.36.
cylinder
with a relatively
stiff
---
_--0.9
_---
P 0.8
-
-
-
-
-
0.02
I
1
0.04
0.06
1
0.08
0.13
I /ka Figure
Elasticity
Comparison theory; -------,
4.
of elastlclt) shell theory.
and shell theories
for the relatively
soft shell-surface
____--
Rave mode.
___-
B \
r
5 = 1.0
5 = 0.5 h
7
I
h
c
,
n
elast.
shell
elast.
shell
1 2 3 4 10
2.02 0.975 1.00 0.992 0.931
2.45 1.01 1.04 1.03 0.953
1.14 O-742 0.753 0,751 0.738
1.37 0.877 0.907 0.902 0.859
~,
WAVES FOR A SMOOTHLY
SUPPORTED
217
SHELL
for this problem, where cp, and cT, represent the velocities of Rayleigh and shear waves in the shell material, respecttvely. Plots of the dimensionless phase velocity b us. l/ka or us. 5 are presented herein for the surface wave mode and for the shell mode. A useful parameter in specifying the materials in the problem is the ratio of shear wave velocities 52 = CT,&
=
(y/s)‘?
(58)
Curves for the two modes are presented for the acoustically softer shell, Q = O-365, and for the acoustically stiffer shell, Q = 2.36, which have been investigated previously by using elasticity theory for the composite cylinder [l], for the case of no curvature [S], for bonded contact [lo], and for comparisons of the no curvature case with plate theory [13].
4.1.
THE SURFACE
WAVE MODE
For the approximate shell theory, by using equation (43), the plot of phase velocity us. shell thickness, /I us. 5, is shown in Figure 2 for the soft shell, Q = 0.365. The same plot for the stiff shell, Sz = 2.36, is shown in Figure 3. As II -+ o~i,no solution is possible for /I > 1 [l]
OS8 310 I
I
0
0.02
I
I
I
0.04
0.06
0.08
I 0.10
I /ko
Figure 5. Comparison of elasticity Elasticity theory; ------, shell theory.
and shell theories
for the relatively
stiff shell-surface
r
1 2 3 4 10
-----,
,
5 = 0.5 n
wave mode.
5 = 1.0
n
I 2.38 0.836 0.916 0.938 0.916
243 0.860 0.936 0.953 0.923
2.23 0.728 0.866 0.918 0.970
2.31 0.769 0.892 0.937 0.973
and hence that portion of the curve is shown as a dotted line in Figure 3. The comparison of elasticity theory and shell theory for the soft plate is shown by plotting j? us. l/ka in Figure 4. The comparison for the stiff plate is shown in Figure 5. Values for fi for n < 10 are shown in tabular form. The curves for the elasticity theory used in Figures 4 and 5 were shown previously in reference [ 11.
218
H. I. EPSTEIN
0.6
Figure 6. Phase velocity versus wave number for the shell mode of a composite shell: v = Y, = 0.25, 6 = 0.75, y = 0.1, 0 = 0365.
0.501 “*co
I 0
I n=50
I
n=30
I 0.02
1
I
n=ZO
n=15
1 0.04
/ 0.06
cylinder
with a relatively
soft
n=lC I 0.08
I 0.10
I/k0
Figure 7. Comparison theory:------,shelltheory.
of elasticity
r
and shell theories
;=o *
for the relatively
soft shell-shell
l = 0.5
n
shell
b elast.
1 2 3 4 10
0.684 0.618 0606 0601 0.597
0.503 0.491 0.524 0.541 0.572
f=l,O
mode. -,
’
b shell
elast.
shell
0.524 O-520 0.545 a558 0.581
0.425 0401 O-452 0.480 0.532
0443 0.441 0.500 0.523 0.566
Elasticity
WAVES
4.2.
FOR A SMOOTHLY
SUPPORTED
SHELL
219
THE SHELL MODE
For the stiff layer, 52 = 2.36, the shell mode is not investigated because, for this case, the velocity of long longitudinal waves in the layer has a value for the non-dimensional phase velocity, /I, that is greater than unity for the case of no curvature. The plot of /I us. 5 for the shell mode of the soft plate, Sz = 0.365, is shown in Figure 6. The case of no curvature, ii -+ c;o, is seen to be the line /I = /I,. The phase velocity for this value of /I is equal to &,/P,) w .For a constant wave number, k, and for small values of c, /I > B,. As < increases, the phase velocity decreases and /3 < b,. For large values of 5, as the curvature increases, the phase velocity decreases. The plot of /I us. l/ka for the shell mode of the soft plate, Q = 0.365, is shown in Figure 7 for both the elasticity theory and the shell theory. The curves for the elasticity theory were shown previously in reference [I].
5. DISCUSSION The accuracy of the approximate theory is checked by comparing the phase velocities of the lowest modes obtained by using the approximate shell theory and the exact elasticity theory. Good agreement is obtained if the ratio of the layer thickness to the wavelength is small. For a constant layer thickness, this case represents small wave numbers. Achenbach and Keshava [13] showedlthat the plate theory, which includes the effects of rotatory inertia and transverse shear deformation, gave perfect agreement with the elasticity theory for phase velocities of the lowest mode of the stiff plate over the entire permissible range of t( = kh). For the soft plate, good agreement was found for small values of 5 and there was little advantage in including transverse shear and rotatory inertia. As 5 increased, the agreement became poorer for the shell mode. For the surface wave mode, the two theories agreed again at an intermediate 5, but diverged again with increasing 5. As seen in Figures 4,5 and 7, as the curvature increases, the agreement of the two theories becomes progressively poorer whether or not any disagreement exists for the case of no curvature. The agreement of the two theories is better for the acoustically stiffer plate than for the acoustically softer plate. With the exception of the shell mode, the limit r + 0 gives phase velocities that are independent of the theory used in the layer. The simplification made by introducing a shell theory for the curved layer is in the reduction of the dispersion relations from the roots of a 6 x 6 to a 3 x 3 determinant. Achenbach and Keshava [13] showed that the use of the plate theory reduces the calculations significantly. However, the approximation made by using the shell theory, when there is curvature, does not simplify the calculations as much as the plate theory when there is no curvature. While the computational savings associated with the approximate equation are not great, there are some advantages associated with using a shell theory. First, the dispersion equation can be written in polynomial form, and thus the effect of the variation of individual material parameters may be more quickly identified. Also, the stresses and displacements are more readily computed when shell equations are used.
ACKNOWLEDGMENT The author’s thanks are due to Professor Jan D. Achenbach of the Technological tute, Northwestern University, for his guidance and advice.
Insti-
220
H. I.
EPSTEIK
REFERENCES 1. H. I. EPSTEIN 1976 Journal oj‘Sound and Vibration 48, 57-71. Circumferential waves in a composite circular cylinder. 2. H. 1. EPSTEIN 1976 Journal of Sound and Vibration 46. 59-66. The effect of curvature on Stoneley waves. 3. I. A. VIKTOROV1958 Soviet Physics-Acoustics: American Institute of Physics Translation 4, 13 l-l 36. Rayleigh-type waves on a cylindrical surface. 4. K. SEZAWA 1927 Bulletin-Earthquake Research Institute (Tokyo) 3, I-18. Dispersion of elastic waves propagated on the surface of stratified bodies and on curved surfaces. 5. J. D. ACHENBACHand H. I. EPSTEIN 1967 Journalof the Engineering Mechanics Dioision. American Society oj Civil Engineers 93, 27-42. Dynamic lnlcractlon of a layer and a half-space. 6. J. W. STRUTT,LORD RAYLEIGH1885 Proceedings ofthe London Mathematicul Society 17, 4411. On waves propagated along the plane surface of an elastic solid. 7. R. STONELEY1924 Proceedings of the Rqval Society (London) 106.416428. Elastic waves at the surface of separation of two solids. 8. J. G. SCHOLTE1942 Proceetlings Nederlandse Akadernic [Ian Weterschappen 45. 159-164. On the Stoneley wave equation. 1966 Journal oj’the Engineering Mechanics 9. J. D. ACHENBACH,S. P. KESHXJ~ and G. HEKRMANN Dicision, American Societ_voj’CiG/ Engineers 92. 113-129. Waves in a smoothly joined plate and half-space. 10. H. 1. EPSTEIN 1978 Journal of’Sound and Vibration 58, 155-166. Circumferential waves for a cylindrical shell supported by a continuum. 11. L. S. D. MORLEY 1961 Quarterly Journal ofMechanics and Applied Mathematics XIV, 155-172. Elastic waves in a naturally curved rod. 12. F. BOWMAN1938 Introduction to Bessel Functions. London: Longmans, Green and Co. 13. J. D. ACHENBACHand S. P. KESHAVA1967 Journal of Applied Mechanics, Transaitions of the American Society of Mechanical Engineers 34,397404. Free waves in a plate supported by a semiinfinite continuum. 14. R. D. MINDLIN 1951 Journal of‘Applied Mechanics, Transactions oj the Americun Society oj‘ Mechanical Engineers 18, 31-38. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates.
APPENDIX a A, B. C, D C CL CT
E EP h H, L, M 1
J” JA, JB k K, N
1
radius of the core constants phase velocity phase velocity of dilatational waves phase velocity of shear waves Young’s modulus material parameter given by equation (11) shell thickness dimensionless quantities given by equations (35) to (37) 1 if rotatory inertia included, 0 if not nth order Bessel function of the first kind dimensionless quantities given by equations (41) and (42) wave number dimensionless quantities given by equations (50) and (51) distance from inner face of shell to the neutral axis
: NOTATION M,, M,, M, n
r. 0 R
it, R’
PI II, I’
LT.v
-%Y 1
dimensionless quantities given by equations (12) to (14) number of complete circumferential modes normal traction 3 x 3 plate matrix, equation (49) dimensionless quantities given by equations (52) and (53) polar co-ordinates radius to the neutral axis of the shell dimensionless quantities given by equations (46) and (55) 3 x 3 shell matrix, equation (43) time displacements displacements on the curved surface y = 0 shell co-ordinates dimensionless phase velocity (= (,/CL) tic,
221
WAVES FOR A SMOOTHLY SUPPORTED SHELL
PI/P
P
P,IP hIa Vh Timoshenko shear coefficient shear modulus Poisson’s ratio dimensionless wave number (=kh)
0 5 Y
mass density stress tangential traction rotation of the shell cross-section radian frequency
: c&T Note: subscript 1 pertains
to the shell.