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Exact solution to the problem of forced large amplitude radial oscillations of a thin hyperelastic tube
Inl. J. Non-Linear
Mechanics.
VoL 6. pp. 193-207. Pergamon
Press 1971. Printed in Great Britain
EXACT SOLUTION T.0 THE PROBLEM OF FORCED LARGE AMPLITUDE RADIAL OSCILLATIONS OF A THIN HYPERELASTIC TUBE? M. SHAHINPOCIR$ AND
J. L. NOWINSKI
Department of Mechanical and Aerospace Engineering, University of Delaware, Newark, Delaware, U.S.A.
Abstract-The title problem is analysed on the basis of a rigorous finite deformation theory of elasticity. The material of the tube is considered incompressible and of Mooney-Rivlin type. The governink equation is that of Knowles simplified to thin-walled tubes. An exact solution is found for a general forcing function, and specific cases are investigated involving: free oscillations, Heaviside type step load, blast load, harmonic load and periodic step pulse load. The method permits an explicit derivation of displacement and stress fields undetermined in the earlier work. A numerical example is solved and graphs are given for harmonic forcing function. It appears that the actual oscillations display two separate components: a high speed fluctuation component (almost negligible) related to free oscillatory motions, and a low speed fluctuation component (dominant, with large amplitude). 1. INTRODUCTION
the theory of finite elastic deformations has a rather long history, the associated dynamic phenomena have mostly been analysed with relation to the propagation of discontinuities in infinite media. The work, originated with the classical investigations of Hadamard, was extended and enriched in recent years by important contributions of Ericksen, Truesdell, Nell, Toupin, Hill, John and others& The first dynamic problem in finite elasticity of bounded media found its explicit solution in two remarkable papers by Knowles published in 1960 [4] and 1962 [S] . I/ and analysing large amplitude radial oscillations of a thick-walled circular cylinder. Due to the assumed incompressibility of the material, the problem could be reduced to that of an autonomous motion of a system with a single degree of freedom. Two cases were discussed: free oscillations and a motion resulting from suddenly applied and subsequently maintained uniform surface pressures. Knowles dynamic problem was later placed as a particular item upon a broad list of quasi-equilibrated motions of incompressible bodies established by Truesdell tt Guided by a remark of Heng and Sole& [7] and independently Knowles and Jakub [8] investigated finite radial oscillations of spherical bodies. Wang [9] treated a similar problem for a thin-walled shell. Finite longitudinal shear (telescopic) oscillations of thick-walled tubes were first studied by Nowinski and Schultz [ 10,111. A similar problem was recently recognised by Wang [12]. An inspection of Knowles’s two papers concerning hollow circular cylinders makes it 7 Research supported partially by the National Science Foundation and by the Themis project. ALTHOUGH
$ Now at the Department of Mechanical Engineering, Pahlavi University, Shiraz, Iran. 5 For an extensive bibliography compare [l], [2] and [3]. 11Some of these papers have been recognized as representative of the resurgence of continuum mechanics in the past decade [6]. tt Compare [Z, p. 208, sqq.]. 193
M. SHAHINPOOR and J. L. NOWIYSKI
194
evident that the author’s interest was primarily centered on the period of motions, for which he derived both the general equation and the explicit equation for the thin-walled shell of Mooney-Rivlin material. The actual displacements (and consequently the state of stress) were not determined. It is, however, apparent that in a number of instances information on the intensity of the displacement and stress fields may be desired. This incompleteness was partially removed in a paper by Nowinski and Wang [ 131, in which, by means of Galerkin’s procedure, an approximate equation for the displacements of a tube of arbitrary wall thickness, undergoing free large amplitude radial oscillations, was derived. It is the purpose of the present note to furnish an explicit exact solution for the displacement and stress fields in a thin-walled tube, subjected to forced finite radial oscillations generated by arbitrary time varying pressures. The material of the tube is considered incompressible and of Mooney-Rivlin type. The governing equation is the one derived originally by Knowles for a thick-walled tube if simplified to a thin wall. A solution to this equation is found for a general periodic forcing function. Specific solutions are investigated for free oscillatory motions (Knowles, and Nowinski and Wang case), for a Heaviside step loading, for a blast loading, a harmonically varying load and a periodic step pulse loading As already mentioned, the method permits an explicit derivation of the displacement field (and, consequently of the stress field) undetermined in the earlier work. A numerical example is solved and graphs are given for the displacement and pressure fluctuation versus time, for various values of the parameter characterizing the ratio of pressure to material characteristics and geometry of the tube. It appears that the actual oscillations display two distinct components : one, called high speed fluctuation component of small amplitude, and the other--low speed fluctuation component of large amplitude. The first bears relation to the free oscillatory (may be called “natural”) motions of the tube, and represents some kind of (almost negligible) ripples on the second dominant component linked to the forcing load. 2. GOVERNING
EQUATION
Consider a cylindrical coordinate system fixed in space, whose z-axis coincides with the longitudinal axis of an infinite circular tube of arbitrary thickness and of inner and outer radii R, and R,, respectively, both in the unstressed state. The material of the tube is hyperelastic, isotropic and incompressible and a generic particle of the medium which at time c has the coordinates r, C#J,z, is assumed to be, in the unstressed state, at the point R, 4, z. It follows that the motion of the tube is axially symmetric and entirely described by the function r = r (R, t). We assume that the (purely radial) motions of the tube are periodic, of arbitrary magnitude and induced by time dependent hydrostatic surface pressures P,(t) and P2(t) on the inner and outer curved surface of the tube, respectively. For an incompressible material of mass density p, the associated governing equation of motion was originally derived by Knowles [4]. If the hyperelastic material of the tube is specialized to a rubber-like material, whose strain-energy function is of the Mooney-Rivlin type, the corresponding equation takes the form x($)ln(l
+$)+[ln(l +K
+-$)-*I($) (2.1)
Exact solution to theproblem
offorced large amplituk
radial oscillations of a thbr hyperelastic
tube
195
where x
-
rl(c) RI
K=
a+B
(2.2)
PR:’
’
with r,(O) = R, and a and /I the Rivlin-Mooney material coeff1cients.t Clearly, the coefficient ,Uis a measure of the wall thickness of the tube, and the coefficient K is characteristic of its material. In the limiting case of a thin-walled shell, of interest in the present note, we may assume that p2 Q p 4 1, so that the governing equation reduces to PI(l) -
1x-
P2@)
4PR:.
Kx-~
= 0
(2.3)
and was used in [ 131 with a vanishing forcing term to analyse free oscillatory motions. The last equation is of the form d’x(t) dt2 + P*(t) x(t) - Kx(t)- 3 = 0, and can be supplemented by the initial conditions x(0) = x0,
1(O) = xb,
(2.5)
with a dot as time derivative, and x0 and xb as the initial displacement and velocity at the inner surface of the tube. According to a paper by Pinney [14], the solution of the initial value problem (2.4), (2.5) has the form x(t) = [u’(t) ; KW2u2(t)]f,
(2.6)
where u and u are the fundamental set of solutions of the linear equation i;(t) + P’(t)&)
= 0,
(2.7)
with the initial conditions u(0) = x0, o(0) = 0,
$0) = xb, $0) # 0;
(2.8)
the symbol W denotes the Wronskian of the problem (ut; - tie) which, in the present case, is equal to a non-zero constant. It follows, that whatever the function P*(t), that is whatever the forcing function, the problem can be reduced to a linear problem represented by a linear differential equation with a variable coefficient. In particular, for periodic functions P*(t), associated with forced periodic motions, equation (2.7) becomes that of a general Hill’s type [15, p. 21 and can be solved explicitly. 3. SOLUTION A. Let
us
FOR VARIOUS
FORCING
FUNCTIONS
first investigate the case of free oscillations, when PI(c) - P2(t) z 0. Then & u = x0 cos Kft + -sin K+
t The strain energy density is taken in the form W(I,, I,) = ia(l first two strain invariants.
Kit, - 3) + i/?(lI
(3.1) - 3), where I and II are the
M. SHAHINP~~R
1%
and J.
L.
NOWWSKI
and B # 0,
v = Bsin K’t,
(3.2)
W = x,BK+,
so that finally
1 2
x(t)
x0
=
cos
Kft
& -sin
+
K*t
K+
+ !- sin2Kft 4
i
,
(3.3)
or in another form x(t) =
x2 + 1 2x2
Xf +
2K
+
1
0
(3.4)
[
For xb = 0, the last equation reduces to the one derived directly for free oscillatory motions in [13, equation (2.2)]. It is apparent that in the latter case the period of oscillations is (3.5) as derived in [4, equation (6.7)] in the limiting case of a thin-walled shell. The usefulness of the present solution, however, consists in that it furnishes, as already mentioned, the explicit solution for the displacement field (3.4) and eventually for the stress field. B. The next case to be investigated consists in forced oscillations induced by a Heaviside type step loading starting at time t, PI(f)
-
P2(0
SM:
=
i
0 for t < 0, PO fort > 0,
(3.6)
where PO is a constant. Let us assume that, before the application of the load, the tube remained at rest, so that x(0) = 1,
(3.7)
a(0) = 0. As earlier, in the present case u = cos (K - PO)+ t, u = B* sin (K - PO)* t,
B* # 0
(3.8)
W = B*(K - PO)+,
and finally from (2.6) x(t) =
2K - PO PO cos [4(K - P,)]‘t 2(K - PO) - 2(K - PO)
t ,
(3.9)
under the condition that (K - PO) > 0 in order for the motions to be periodic. The last condition can also be written in the form P,-P,<
r+B c 2
> p.
(3.10)
Exact solution to ihe problem offorced large amplitude radial oscillation
of a thbt hyperelastic
tube
197
Note that the condition for large periodic motions of thick-walled shells obtained by Knowle in [3, equation (25)] has the form
(- >
P, - P, < a+B 2
1P(1 + I.49
(3.11)
and clearly as cc + 0 this reduces to (3.10). Also note that the period of oscillations as obtained from (3.9) is 4
Aa + B) da + B) -
The result (3.12) coincides with the approximation (33)], for a thin walled shell, that is with Tz
2(P,
-
P2)
’
obtained by Knowles in [S, equation
K’ ’ 1p-0 na
[
where
a=[ (1 + p)exp[-(PIY
(3.12)
(3.13)
t 1
P,)/(a + S)] - 1 p+o’
(3.14)
or in the limit
f
Aa + B)
a z [ p(a + j3) - 2(P, - Pz) 1 ’
(3.15)
C Let us now investigate the response of a tube to a blast load that decays linearly as follows, P,(r) - PM &WR:
=
A(1 - t/T*) for 0 G t < T*, fort > T*v. I0
(3.16)
Here A is the initial (maximum) intensity of the blast, and T* is its duration considered very short (T * 4 1 set). In the present case equation (2.7) takes the form
d2y
dt2+
Upon the transformation
K-A+&
1
y=O.
(3.17)
of the independent variable from t to q, where ,=K-A++,
(3.18)
the last equation reduces to 0,
(3.19)
M. SHMNPOOR and J. L.
198
NOWINSKI
with two linearly independent solutions in the form [lS], (3.20) and
where J, and Y+are Bessel functions of the order 4 and of the first and second kind, respectively. Clearly, du, drl du,_ ---= dt d’t dt
,
and (3.21)
Again it may be proved [lS] that ;;q*
2
[J,Y_,
- J_+Y+] = ;,
(3.22)
=
(3.23)
so that the Wronskian of the problem becomes Wu,,
u2)
2.
With the initial conditions (2.8) in mind, we finally get the general solution in the form,
(3.24)
where x&c - A) Y-4
-&(K-A)'Y, >
A, =
(
g-(K-A)’
>
3A
9
T*7t
x,(K - A)* J, B, =
;+A)%
>
-x,(K-A)J_,
3A T*Z
(3.25) >
,
Exact solutitm to tbeprobiem offorced large amplitude rad~o~c~llatio~ ofa thin hyperelastic tube
199
and -(K
-
A)3 Y+
;y(K - A)3
)
( 3A T%
AZ =
, (3.26)
2 T* ?--$K
(
(K - A)+J, Bz =
- A)” ).
3A
Let us note that in order for the constants to be real, the inequality X > A should hold, and one obtains for the Wronskian the simple result W(u, u) = &
(A,&
- &A,)
= x0.
(3.27)
In view of the equation (2.6) the final form of the displacement field becomes
x(t) = [(A: + $A+;(;:+)
+ (B: + $B:)qY@‘)
Since by virtue of (3.27) A,&
- B,A,
= xb
T*E ( -SC’ )
(3.29)
then (3.30)
A;B$ + B:A$ = The a~umption relation,
that T* + 1 set puts in place of the foregoing equation the approximate A;B; + A$:
cz 2A,A,B1Bz,
(3.31)
which, if employed, enables one to cast the equation (3.28) into a relatively simpler form,
M. SHAHINPOOR and J. L. NOWINSKI
200
As in .subsection B one obtains for t > T* 2 x(t)
=
x0*cos K+(t - T*) + $
sin K*(t - T*) I 3 +
sin2 K+(t -
--$
t > T*
(3.33)
0
where
and
play the role of the initial conditions at the time t = T* of the cessation of the external loading. D. Let us now analyse a more general problem of the response of the tube to an arbitrary periodic loading with the period 27~. This case can be symbolically represented by pi(f)
-
p2(Q
=
(3.35)
A$(@,
&PR:.
where Il/(ut) = l&Ot + 24, and I ci/bQ Imax= 1, provided an appropriate min
normalization
(3.36) was made.
Equation (2.7) now reduces to the general Hill’s equation
d2y -@-+ [K -
AJl(ot)] y = 0.
(3.37)
We can rewrite the equation (3.37) as (3.38) so that the substitution y = exp [y i w d(t)], with y2 = K, yields 0
(3.39)
Exact solution to the problem offorced large amplitude radial oscillations of a thin hyperelasric tube
201
In view of the last two equations and upon using the notation <‘=[l equation (3.37) can be transformed
-&Or)],
into the form,
Idw+w*+(*=Q Y
(3.40)
dt
This equation is, of course, of the Riccati type and its solution can be taken in the form 1
w=w,+;w,+Tw,+o
1 Y
A, 0Y
as suggested in [15, p. 931. In the last equation the term O(l/y3) can be neglected, due to the fact that K is of the order of E/p, which is generally a large number in the system of in.lb.sec. units adopted in the present paper. Clearly wo, wi and w1 are functions of time to be determined later. At this junction it is instrumental to model the solution on the procedure given by McLachlan in [15]. With (3.41) in mind we first reduce equation (3.40) to w; + t[* + d(Go + 2w,w,) + +:
+ w: + 2w,w*) + 0 ; 0
= 0.
(3.42)
Upon making the coefficients of y”, y- ‘, and y-* vanish one obtains w; + 52 = 0, tie + 2w,w, = 0,
(3.43)
GJt,+ wt + 2w,w, = 0, o(l/Y3). . . = 0. Thus the fundamental set of solutions of (3.28) is found as Yl Y2
zz const t[-+ coS[+##)dr], sin
(3.44)
where (3.45) It is convenient to rewrite equation (3.45) in a more explicit form, namely Yl
const = [A - Kl&Ot)]+ Y2
.
$(wt) +
24 4K [l
3/*w - (A/IO&t)]
dt 1 [I - (A/K) $@r)l*
(3.46)
M. SHAHINP~~Rand J. L. NOWIN~KI
202 In
view of (2.6H2.8) we now obtain
r
A’(“)j+cos[K+j$(L)d~] u=xoKK -- AIj(ot) I-
+ {[K _ :s(w~),*sin[Ki~mc~)d’ll,
0
(3.47)
0
where (3.48) and (3.49) with B, # 0 as an arbitrary constant. Moreover WW V) = &x,[K
(3.50)
- @(0)-j*,
so that finally,
+ 5 [K - Ati(O)
[K - A+(ot)]-”
sin I[XiSm(L)dt]j+.
(3.51)
0
This result completes the solution of the general problem discussed in this subsection. Note that since at time t = 0 the tube is under a pressure equal to A$(O), the x0 represents the displacement associated with the static case, in which a uniform pressure equal to A+(O) is applied. It is easily found from equation (2.1) that in the case of a thin shell (3.52)
In view of the last equation, equation (3.51) reduces to A;K-*
___2
+ A,K-‘sin[?Ki$m(~)dt]
(3.53)
-~co+K+/&)d$ 0
Exact solution to theproblem offorced large amplitude radial oscillations ofa thin hyperelastic
tube
203
with the notation a = A/K. It is evident that the periodic solutions are stable if, and only if, K > A. From the last equation there follows that the oscillations can be decomposed into two components: one-a high speed fluctuation component, seemingly related to the natural oscillations of the tube, and another-a low speed fluctuation component related to the forcing function .t The frequency of the high speed fluctuation component is defined by _f = f
4(t) =
?{(I - EJl(ot)f + &&t)
[l - rz$(ot)]-f
+
-_;$‘(cot) [l - al&t)]-*
.
(3.54)
I
Again, by the high speed periodicity of x(t) is meant here the reciprocal of (3.54). In the case in which the forcing function vanishes (A = 0), equation (3.54) reduces to the one derived directly by Knowles for a freely oscillating tube [4, equation (1.7)]. Note that the low speed fluctuation component of x(t) is periodic and has the same period pxct, as the applied pressure, so that 2x Pxw = w
(3.55)
.
To f= the ideas, let us assume that +(ot) = sin cot,
(3.56)
and (3.57) so that the motions of the tube start from its rest, unstretched, position and the tube executes harmonic oscillations. In this case equation (3.48) furnishes the value of A, where A, =
-UK-+ 4[1 + &(tL202/K)]’
13.58)
and equation (3.53) reduces to x(t) = [l - cl sin ot]‘-*
a”’ {I - cos[2K+S#‘)d~]} 32[i + & E2/1212 0
where /I2 = 02/K is a very small number for all practical purposes. t A somewhat similar phenomenon moving-coil loud-speakers.
was discovered by McLachlan [IS, p. 2721 in an electrical problem involving
204
M. SHAHINP~~R
and J. L. NOWINSKI
As an illustration let us consider a blood vessel in the vascular system, into which the blood is being pumped, periodically by the contractions of the heart with a period equal, say (roughly), to 1 sec. Then o = 2n set- ’ and for a vessel of large inner radius equal to 0.1 in. K+ - 14,400 see- I. This gives /? z 0900435 and, therefore, for all practical purposes the value of x(t) is dominated by the factor (1 - a sin ot)-* associated with the low speed fluctuation component of oscillations.
Ressure
------
0.3L
-0.71 0
I T/4
I */P
I 3714
I ‘I
I %lrB
I 3*/Z
I 7*/4
I 2r
I
(WI)
FIG. 1. Square of non-dimensional
inner radius x (t), and non-dimensional various values of a’.
pressure Z sin cot versus time for
Variations of the square of the inner radius of the tube x(t), and of the reduced pressure for various values of o! are shown in a representative diagram in Fig. 1. It is seen that in the scale used, the high speed fluctuation component of the oscillations is not discernible, and the graphs seem to represent perfectly smooth curves. With an increasing value of ol the amplitude of both the reduced pressure and the displacement increase. The increase and its rate are different for the expansion and the contraction periods of the tube, in the sense that it is harder for the tube to contract than to expand. Such a behavior is rather understandable from the physical point of view. The amplitude of the expansion mode is always larger than the one of the contraction mode, but this difference decreases with a decreasing pressure.
P*(t)/K
E. Consider now a periodic step pulse loading defined by
0,
t=g
(3.60)
Exact solution to theproblem
offorced large amplitude radial oscillation
I
ofa thin hyperelastk
tube
205
< t < T*, t = T*
0
this pattern being repeated ad infinitum. Equation (2.1) can now be brought into the form
izxz ln
( >
l.5 + K(1
1 +
x2
-
x2)
lnc:++ “,*‘I=+P*(t)(X2
-
i),
(3.61)
provided that the initial conditions (3.57) are satisfied. According to a known argument (cf. [4]) equation (3.61) describes a periodic motion if there exists a value of x(t) different from unity (either, say, x1 > 1 or x2 < 1, both positive in the present case) such that -the associated value of i(t) = o = 0. Posing u = 0 in (3.61) we get
t1 f 1
1
‘I=
(l+p)exp;(-PO/IC]-l
(3.62)
,
and
x2= [ (1 +fl)exp[+P,/K]
(3.63)
- 1 ’
The fact that x1 and x2 are real implies that P,/K < In (1 + p). As shown in the subsection B [equations (3.13) and (3.14)] the period of oscillations of a thin shell, in the case of a Heaviside loading, was found by Knowles in the form
T= 5
r
(1 + p)exp [-I;P*/@
+ I
(3.64)
+ /.I)] - 1 .
This result, confirmed above, reveals an interesting fact that a sudden application of load sets the tube into a periodic motion with the period of oscillations of the order of nK-*, which is very small for practical purposes. The last quantity is equal to the period of natural oscillations of a thin-walled tube. A similar result was found in the subsection D, in the case of a sinusoidal loading, where the effect of high speed fluctuations was shown to be negligible, as compared with the low speed component of motion. In the sinusoidal loading case, the period of high speed vibrations is [cf. equation (3.54)], T = KK-*
- o2 sin ot) + ozol 8K (1 - a sin cot)*
(1 - Esinot)f
5a02 cos2 of
+
4[1 - ~Zsinot]*
11’ -I
(3.65)
It is evident that the above period is itself, a periodic function of time. In the presently analysed periodic pulse loading, for a tube of arbitrary wall thickness one obtains T=
x1-= dx s
X2
V
X1 s
X2
(x2
-
x2 In (1 + p/x2) dx. 1) [P*(t) - K In (1 + p) + K In (1 + p/x2)
(3.66)
M.
206
SHAHINPOOR
and
J. L. NOWIKSKI
In order to investigate the low speed periodicity of x(t), let us return to the equation (2.3). In view of the equation (2.7) and the boundary conditions (3.57) we arrive at a solution for the displacement field in the form, x(t) =
2/S - P*(t) p*(t) ]coS[4(K-~)lff~. i 2[pK - P*(t)] - [ 2[pK - P*(t)]
(367)
Note that P*(t) satisfies the Dirichlet conditions and can be expanded in a Fourier series of the well known form m P*(t) =
c
#l=i*3.5;~.
4po sin 2nnt T*’ nx
(3.68)
Thus
[I - (; x(t) =
2
+cos[4(K - g+t}]
$E${l
n=1,3.5:~,
(3.691 (I -6
2
Asin?)
n=l.3.5*.”
where E = P,IyK. From the last equation it is clear that the actual motion represents a coupled motion which, as stated earlier, consists of two periodic components. Its high speed component has the period [see (3.69)], (3.70) which itself is a periodic function of time, with the period T* equal to period of applied loading. The low speed component has a constant period equal to T*. It is easy to show that the equation (3.66) reduces to equation (3.70), when the thickness of the wall of the shell p + 0. It seems, therefore, conceivable to obtain an approximate solution for a tube of arbitrary wall thickness subjected to a periodic pulse loading, by inserting the exact expression for the period of oscillations of such a tube, equation (3.66), into the equation (3.69). In this way we obtain an approximation to the complete solution for a tube of arbitrary thickness, in an explicit form x(t) =
x;ln (1 + p/x2) )+dx] - 1) [P*(r) - K In (1 + q) + K In (1 + p/x2)
’
+ r)eIp(E)
-1
f 1 ’
-‘t
H’.
Exact solution to the problem offorced large amplitude radial oscillations of a thin hyperelastic tube
207
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111C. TRUFSDIU and R TOUPIN,The Classical Field Theories, Encycl. Phys., Vol. III/I, Springer (1960).
[21 C. TRUESDJXLand W. NOLL, The Non-Linear Field Theories of Mechanics, Encycl. Phys., Vol. III/3, Springa (1965). [31 A. C. ERINGEN, Nonlinear Theory of Conrinuow Media. McGraw-Hill (1962). [41 J. K. KNOWL~, Large amplitude oscillations of a tube of incompressible elastic material. Q. appl. Math. 18, 71 (1960). 151J. K. KNOWLES,On a class of oscillations in the finite-deformation theory of elasticity. J. appl. Mech. 29, 283 (1962). ed., Continuum Mechunice IK Problems of Non-Lineur Elasticity. Gordon & Breach (1965). [61 C. TRUE~D~LL [71 G. Z. HENG and R. SOLECKI, Free and forced finite amplitude oscillations of an elastic thick-walled hollow sphere of incompressible material. Arch. Me&. Sros. 15, 424 (1964). PI J. K. KNOWLESand M. T. JAKIJB,Finite dynamic deformation of an incompressible elastic medium containing a spherical cavity. Arch. Rat. Mech. An. 14, 367 (1965). [91 C. C. WANG, On the radial oscillations of a spherical thin shell in the finite elasticity theory. Arch. Rat. Mech. Anal., 23, 270 (1965). [lOI J. L. NO~INSK~and A. B. SCHULTZ,Note on a class of finite longitudinal oscillations of thick-walled cylinders. Proc. Ind. Nat. Congr. Theor. appl. Mech., 31 (1964). J. L. NOWINSKI,On a dynamic problem in finite elastic shear. Inf. J. Eng. Sci. 4, 501 (1966). ;::I A. S. D. WANG, On free oscillations of elastic incompressible bodies in finite shear. (h publ.) (1969). (131 J. L. NO~INSIUand S. D. WANG, Gale&in’s solution to a severely non-linear problem of finite elastodynamics. Int. J. Non-Lineor Mech. 1, 219 (1966). [141 E. PINNEY,The nonlinear differential equation y” + p(x) y + cy- ’ = 0. Proc. Am. Math. Sot. 1, 68 (1950). [15] N. W. MCLACHLAN,neory and Application of Mathieu Functions. Dover (1964). (Received 27 February 1970) Resmd--On
analyse. le problbme du titre eh partant de la thCorie de l’tlasticit6 rigoureuse pour les dfformations finies. On consid&e le mat&au du tube incompressible et du type Mooney-Rivlin. L’&uation fondamentale est celle de Knowles simplifiQ pour des tubes minces. On trouve la solution exacte pour une vibration for& g&&rale e.t on Ctudie des cas particuliers faisant intervenir : les oscillations libres, une charge en escalier du type Heaviside, une explosion, une charge harmonique et une charge avec des impulsions p&iodiques. La m&hode permet un calcul explicite des champs de ddplacement et de contraintes rest& indCterminb dans le travail prbc&-lent. On r&out un exemple numtrique et on donne des graphes pour des vibrations for&es harmoniques. 11apparalt que les oscillations rklles pr&entent deux composantes sCparCes: une composante a fluctuation rapide (presque negligeable) relib aux mouvements P oscillations libres et une composante & fluctuation lente (dominante et d’amplitude BevQ). Zos;lmwnfassmg-Das in der uberschrift erwtinte Problem tird unter Zugrundelegung einer rigorosen Elastizit&stheorie endlicher Verformung untersucht Dabei wird angenommen, dass das Material des Rohres inkompressibel und vom Mooney-Rivlin Typ ist Die Hauptgleichung ist diejenige von Knowles, vereinfacht fur dilnnwandige Rohre. Fur eine allgemeine Druckfunktion wird eine exakte Lbsung gefunden, und einige spezielle Fiille werden untersucht, unter anderen : freie Schwingungen, eine Stufenbelastung vom Heaviside Typ, eine Explosionsbelastung, eine harmonische Belastung- und eine periodische Stufenimpuls Belastung Die Methode erlaubt eine explizite Herleitung der Verschiebungs- und Spannungsfelder, die in der friiheren Arbeit unbestimmt waren, Ein numerisches &spiel wird gel&t, und Diagramme fur harmonische Druckfunktionen werden dargestellt Es stellt sich heraus, dass die tatsiichlichen Schwingungen zwei getrennte Komponenten aufweisen: eine schnelle Fluktuationskomponente (kann beinahe vernachltisigt werden), die mit den freien oszillierenden Bewegungen in Verbindung steht, und eine langsame Fluktuationskomponente (vorherrschend, mit grosser Amplitude) .&EEOTaqRsI-&yYeHne npo6nern B~Hy?U~eliHbIX panHaJlbHhIX Hone6aHu# 6OnbmO# aMnnUTyAM TOHKOfil%nepynpyrOfi Tpy6bl OCHOBaHOHa CTpOl’OfiTeOpllM ynpyrOCTU RJlfl KOHe’lHblX&IOpMal@. MaTepHan ~py6b1 nonaraeTcx HecHu?MaeMblMa T&ina @He-P%iB.znHa (Mooney-Rivlin) lIpo6nema onpe~enJ%eTca YpaBHeHHeM Hoynca (Knowles) C ynpomeaaen! aJIB ToHKOCTeHHbIXTpy6. TosHoe pemenHe HaiReHo HJIH o6mefi BbxHymxammeti I$~HKUHN. npvl ATOM mcnenymTcs cneqw@mecKKe cnyqaa, a EIMeHHO: cBo6o~nbIe KOJIe6aHUH, ncTynHbsaTyI0 Harpy3KyD, T&ma XeBucattJ&a HarpyaKy RrnMIUKoft.rap~oHKseCKy~
C
Harpy3Ky
A IIepAO~UYeCKyIO
*CTyBeHb~aTylOlpAMIlyJIbCHyH)
Harpy3Ky.
Mechanics.
VoL 6. pp. 193-207. Pergamon
Press 1971. Printed in Great Britain
EXACT SOLUTION T.0 THE PROBLEM OF FORCED LARGE AMPLITUDE RADIAL OSCILLATIONS OF A THIN HYPERELASTIC TUBE? M. SHAHINPOCIR$ AND
J. L. NOWINSKI
Department of Mechanical and Aerospace Engineering, University of Delaware, Newark, Delaware, U.S.A.
Abstract-The title problem is analysed on the basis of a rigorous finite deformation theory of elasticity. The material of the tube is considered incompressible and of Mooney-Rivlin type. The governink equation is that of Knowles simplified to thin-walled tubes. An exact solution is found for a general forcing function, and specific cases are investigated involving: free oscillations, Heaviside type step load, blast load, harmonic load and periodic step pulse load. The method permits an explicit derivation of displacement and stress fields undetermined in the earlier work. A numerical example is solved and graphs are given for harmonic forcing function. It appears that the actual oscillations display two separate components: a high speed fluctuation component (almost negligible) related to free oscillatory motions, and a low speed fluctuation component (dominant, with large amplitude). 1. INTRODUCTION
the theory of finite elastic deformations has a rather long history, the associated dynamic phenomena have mostly been analysed with relation to the propagation of discontinuities in infinite media. The work, originated with the classical investigations of Hadamard, was extended and enriched in recent years by important contributions of Ericksen, Truesdell, Nell, Toupin, Hill, John and others& The first dynamic problem in finite elasticity of bounded media found its explicit solution in two remarkable papers by Knowles published in 1960 [4] and 1962 [S] . I/ and analysing large amplitude radial oscillations of a thick-walled circular cylinder. Due to the assumed incompressibility of the material, the problem could be reduced to that of an autonomous motion of a system with a single degree of freedom. Two cases were discussed: free oscillations and a motion resulting from suddenly applied and subsequently maintained uniform surface pressures. Knowles dynamic problem was later placed as a particular item upon a broad list of quasi-equilibrated motions of incompressible bodies established by Truesdell tt Guided by a remark of Heng and Sole& [7] and independently Knowles and Jakub [8] investigated finite radial oscillations of spherical bodies. Wang [9] treated a similar problem for a thin-walled shell. Finite longitudinal shear (telescopic) oscillations of thick-walled tubes were first studied by Nowinski and Schultz [ 10,111. A similar problem was recently recognised by Wang [12]. An inspection of Knowles’s two papers concerning hollow circular cylinders makes it 7 Research supported partially by the National Science Foundation and by the Themis project. ALTHOUGH
$ Now at the Department of Mechanical Engineering, Pahlavi University, Shiraz, Iran. 5 For an extensive bibliography compare [l], [2] and [3]. 11Some of these papers have been recognized as representative of the resurgence of continuum mechanics in the past decade [6]. tt Compare [Z, p. 208, sqq.]. 193
M. SHAHINPOOR and J. L. NOWIYSKI
194
evident that the author’s interest was primarily centered on the period of motions, for which he derived both the general equation and the explicit equation for the thin-walled shell of Mooney-Rivlin material. The actual displacements (and consequently the state of stress) were not determined. It is, however, apparent that in a number of instances information on the intensity of the displacement and stress fields may be desired. This incompleteness was partially removed in a paper by Nowinski and Wang [ 131, in which, by means of Galerkin’s procedure, an approximate equation for the displacements of a tube of arbitrary wall thickness, undergoing free large amplitude radial oscillations, was derived. It is the purpose of the present note to furnish an explicit exact solution for the displacement and stress fields in a thin-walled tube, subjected to forced finite radial oscillations generated by arbitrary time varying pressures. The material of the tube is considered incompressible and of Mooney-Rivlin type. The governing equation is the one derived originally by Knowles for a thick-walled tube if simplified to a thin wall. A solution to this equation is found for a general periodic forcing function. Specific solutions are investigated for free oscillatory motions (Knowles, and Nowinski and Wang case), for a Heaviside step loading, for a blast loading, a harmonically varying load and a periodic step pulse loading As already mentioned, the method permits an explicit derivation of the displacement field (and, consequently of the stress field) undetermined in the earlier work. A numerical example is solved and graphs are given for the displacement and pressure fluctuation versus time, for various values of the parameter characterizing the ratio of pressure to material characteristics and geometry of the tube. It appears that the actual oscillations display two distinct components : one, called high speed fluctuation component of small amplitude, and the other--low speed fluctuation component of large amplitude. The first bears relation to the free oscillatory (may be called “natural”) motions of the tube, and represents some kind of (almost negligible) ripples on the second dominant component linked to the forcing load. 2. GOVERNING
EQUATION
Consider a cylindrical coordinate system fixed in space, whose z-axis coincides with the longitudinal axis of an infinite circular tube of arbitrary thickness and of inner and outer radii R, and R,, respectively, both in the unstressed state. The material of the tube is hyperelastic, isotropic and incompressible and a generic particle of the medium which at time c has the coordinates r, C#J,z, is assumed to be, in the unstressed state, at the point R, 4, z. It follows that the motion of the tube is axially symmetric and entirely described by the function r = r (R, t). We assume that the (purely radial) motions of the tube are periodic, of arbitrary magnitude and induced by time dependent hydrostatic surface pressures P,(t) and P2(t) on the inner and outer curved surface of the tube, respectively. For an incompressible material of mass density p, the associated governing equation of motion was originally derived by Knowles [4]. If the hyperelastic material of the tube is specialized to a rubber-like material, whose strain-energy function is of the Mooney-Rivlin type, the corresponding equation takes the form x($)ln(l
+$)+[ln(l +K
+-$)-*I($) (2.1)
Exact solution to theproblem
offorced large amplituk
radial oscillations of a thbr hyperelastic
tube
195
where x
-
rl(c) RI
K=
a+B
(2.2)
PR:’
’
with r,(O) = R, and a and /I the Rivlin-Mooney material coeff1cients.t Clearly, the coefficient ,Uis a measure of the wall thickness of the tube, and the coefficient K is characteristic of its material. In the limiting case of a thin-walled shell, of interest in the present note, we may assume that p2 Q p 4 1, so that the governing equation reduces to PI(l) -
1x-
P2@)
4PR:.
Kx-~
= 0
(2.3)
and was used in [ 131 with a vanishing forcing term to analyse free oscillatory motions. The last equation is of the form d’x(t) dt2 + P*(t) x(t) - Kx(t)- 3 = 0, and can be supplemented by the initial conditions x(0) = x0,
1(O) = xb,
(2.5)
with a dot as time derivative, and x0 and xb as the initial displacement and velocity at the inner surface of the tube. According to a paper by Pinney [14], the solution of the initial value problem (2.4), (2.5) has the form x(t) = [u’(t) ; KW2u2(t)]f,
(2.6)
where u and u are the fundamental set of solutions of the linear equation i;(t) + P’(t)&)
= 0,
(2.7)
with the initial conditions u(0) = x0, o(0) = 0,
$0) = xb, $0) # 0;
(2.8)
the symbol W denotes the Wronskian of the problem (ut; - tie) which, in the present case, is equal to a non-zero constant. It follows, that whatever the function P*(t), that is whatever the forcing function, the problem can be reduced to a linear problem represented by a linear differential equation with a variable coefficient. In particular, for periodic functions P*(t), associated with forced periodic motions, equation (2.7) becomes that of a general Hill’s type [15, p. 21 and can be solved explicitly. 3. SOLUTION A. Let
us
FOR VARIOUS
FORCING
FUNCTIONS
first investigate the case of free oscillations, when PI(c) - P2(t) z 0. Then & u = x0 cos Kft + -sin K+
t The strain energy density is taken in the form W(I,, I,) = ia(l first two strain invariants.
Kit, - 3) + i/?(lI
(3.1) - 3), where I and II are the
M. SHAHINP~~R
1%
and J.
L.
NOWWSKI
and B # 0,
v = Bsin K’t,
(3.2)
W = x,BK+,
so that finally
1 2
x(t)
x0
=
cos
Kft
& -sin
+
K*t
K+
+ !- sin2Kft 4
i
,
(3.3)
or in another form x(t) =
x2 + 1 2x2
Xf +
2K
+
1
0
(3.4)
[
For xb = 0, the last equation reduces to the one derived directly for free oscillatory motions in [13, equation (2.2)]. It is apparent that in the latter case the period of oscillations is (3.5) as derived in [4, equation (6.7)] in the limiting case of a thin-walled shell. The usefulness of the present solution, however, consists in that it furnishes, as already mentioned, the explicit solution for the displacement field (3.4) and eventually for the stress field. B. The next case to be investigated consists in forced oscillations induced by a Heaviside type step loading starting at time t, PI(f)
-
P2(0
SM:
=
i
0 for t < 0, PO fort > 0,
(3.6)
where PO is a constant. Let us assume that, before the application of the load, the tube remained at rest, so that x(0) = 1,
(3.7)
a(0) = 0. As earlier, in the present case u = cos (K - PO)+ t, u = B* sin (K - PO)* t,
B* # 0
(3.8)
W = B*(K - PO)+,
and finally from (2.6) x(t) =
2K - PO PO cos [4(K - P,)]‘t 2(K - PO) - 2(K - PO)
t ,
(3.9)
under the condition that (K - PO) > 0 in order for the motions to be periodic. The last condition can also be written in the form P,-P,<
r+B c 2
> p.
(3.10)
Exact solution to ihe problem offorced large amplitude radial oscillation
of a thbt hyperelastic
tube
197
Note that the condition for large periodic motions of thick-walled shells obtained by Knowle in [3, equation (25)] has the form
(- >
P, - P, < a+B 2
1P(1 + I.49
(3.11)
and clearly as cc + 0 this reduces to (3.10). Also note that the period of oscillations as obtained from (3.9) is 4
Aa + B) da + B) -
The result (3.12) coincides with the approximation (33)], for a thin walled shell, that is with Tz
2(P,
-
P2)
’
obtained by Knowles in [S, equation
K’ ’ 1p-0 na
[
where
a=[ (1 + p)exp[-(PIY
(3.12)
(3.13)
t 1
P,)/(a + S)] - 1 p+o’
(3.14)
or in the limit
f
Aa + B)
a z [ p(a + j3) - 2(P, - Pz) 1 ’
(3.15)
C Let us now investigate the response of a tube to a blast load that decays linearly as follows, P,(r) - PM &WR:
=
A(1 - t/T*) for 0 G t < T*, fort > T*v. I0
(3.16)
Here A is the initial (maximum) intensity of the blast, and T* is its duration considered very short (T * 4 1 set). In the present case equation (2.7) takes the form
d2y
dt2+
Upon the transformation
K-A+&
1
y=O.
(3.17)
of the independent variable from t to q, where ,=K-A++,
(3.18)
the last equation reduces to 0,
(3.19)
M. SHMNPOOR and J. L.
198
NOWINSKI
with two linearly independent solutions in the form [lS], (3.20) and
where J, and Y+are Bessel functions of the order 4 and of the first and second kind, respectively. Clearly, du, drl du,_ ---= dt d’t dt
,
and (3.21)
Again it may be proved [lS] that ;;q*
2
[J,Y_,
- J_+Y+] = ;,
(3.22)
=
(3.23)
so that the Wronskian of the problem becomes Wu,,
u2)
2.
With the initial conditions (2.8) in mind, we finally get the general solution in the form,
(3.24)
where x&c - A) Y-4
-&(K-A)'Y, >
A, =
(
g-(K-A)’
>
3A
9
T*7t
x,(K - A)* J, B, =
;+A)%
>
-x,(K-A)J_,
3A T*Z
(3.25) >
,
Exact solutitm to tbeprobiem offorced large amplitude rad~o~c~llatio~ ofa thin hyperelastic tube
199
and -(K
-
A)3 Y+
;y(K - A)3
)
( 3A T%
AZ =
, (3.26)
2 T* ?--$K
(
(K - A)+J, Bz =
- A)” ).
3A
Let us note that in order for the constants to be real, the inequality X > A should hold, and one obtains for the Wronskian the simple result W(u, u) = &
(A,&
- &A,)
= x0.
(3.27)
In view of the equation (2.6) the final form of the displacement field becomes
x(t) = [(A: + $A+;(;:+)
+ (B: + $B:)qY@‘)
Since by virtue of (3.27) A,&
- B,A,
= xb
T*E ( -SC’ )
(3.29)
then (3.30)
A;B$ + B:A$ = The a~umption relation,
that T* + 1 set puts in place of the foregoing equation the approximate A;B; + A$:
cz 2A,A,B1Bz,
(3.31)
which, if employed, enables one to cast the equation (3.28) into a relatively simpler form,
M. SHAHINPOOR and J. L. NOWINSKI
200
As in .subsection B one obtains for t > T* 2 x(t)
=
x0*cos K+(t - T*) + $
sin K*(t - T*) I 3 +
sin2 K+(t -
--$
t > T*
(3.33)
0
where
and
play the role of the initial conditions at the time t = T* of the cessation of the external loading. D. Let us now analyse a more general problem of the response of the tube to an arbitrary periodic loading with the period 27~. This case can be symbolically represented by pi(f)
-
p2(Q
=
(3.35)
A$(@,
&PR:.
where Il/(ut) = l&Ot + 24, and I ci/bQ Imax= 1, provided an appropriate min
normalization
(3.36) was made.
Equation (2.7) now reduces to the general Hill’s equation
d2y -@-+ [K -
AJl(ot)] y = 0.
(3.37)
We can rewrite the equation (3.37) as (3.38) so that the substitution y = exp [y i w d(t)], with y2 = K, yields 0
(3.39)
Exact solution to the problem offorced large amplitude radial oscillations of a thin hyperelasric tube
201
In view of the last two equations and upon using the notation <‘=[l equation (3.37) can be transformed
-&Or)],
into the form,
Idw+w*+(*=Q Y
(3.40)
dt
This equation is, of course, of the Riccati type and its solution can be taken in the form 1
w=w,+;w,+Tw,+o
1 Y
A, 0Y
as suggested in [15, p. 931. In the last equation the term O(l/y3) can be neglected, due to the fact that K is of the order of E/p, which is generally a large number in the system of in.lb.sec. units adopted in the present paper. Clearly wo, wi and w1 are functions of time to be determined later. At this junction it is instrumental to model the solution on the procedure given by McLachlan in [15]. With (3.41) in mind we first reduce equation (3.40) to w; + t[* + d(Go + 2w,w,) + +:
+ w: + 2w,w*) + 0 ; 0
= 0.
(3.42)
Upon making the coefficients of y”, y- ‘, and y-* vanish one obtains w; + 52 = 0, tie + 2w,w, = 0,
(3.43)
GJt,+ wt + 2w,w, = 0, o(l/Y3). . . = 0. Thus the fundamental set of solutions of (3.28) is found as Yl Y2
zz const t[-+ coS[+##)dr], sin
(3.44)
where (3.45) It is convenient to rewrite equation (3.45) in a more explicit form, namely Yl
const = [A - Kl&Ot)]+ Y2
.
$(wt) +
24 4K [l
3/*w - (A/IO&t)]
dt 1 [I - (A/K) $@r)l*
(3.46)
M. SHAHINP~~Rand J. L. NOWIN~KI
202 In
view of (2.6H2.8) we now obtain
r
A’(“)j+cos[K+j$(L)d~] u=xoKK -- AIj(ot) I-
+ {[K _ :s(w~),*sin[Ki~mc~)d’ll,
0
(3.47)
0
where (3.48) and (3.49) with B, # 0 as an arbitrary constant. Moreover WW V) = &x,[K
(3.50)
- @(0)-j*,
so that finally,
+ 5 [K - Ati(O)
[K - A+(ot)]-”
sin I[XiSm(L)dt]j+.
(3.51)
0
This result completes the solution of the general problem discussed in this subsection. Note that since at time t = 0 the tube is under a pressure equal to A$(O), the x0 represents the displacement associated with the static case, in which a uniform pressure equal to A+(O) is applied. It is easily found from equation (2.1) that in the case of a thin shell (3.52)
In view of the last equation, equation (3.51) reduces to A;K-*
___2
+ A,K-‘sin[?Ki$m(~)dt]
(3.53)
-~co+K+/&)d$ 0
Exact solution to theproblem offorced large amplitude radial oscillations ofa thin hyperelastic
tube
203
with the notation a = A/K. It is evident that the periodic solutions are stable if, and only if, K > A. From the last equation there follows that the oscillations can be decomposed into two components: one-a high speed fluctuation component, seemingly related to the natural oscillations of the tube, and another-a low speed fluctuation component related to the forcing function .t The frequency of the high speed fluctuation component is defined by _f = f
4(t) =
?{(I - EJl(ot)f + &&t)
[l - rz$(ot)]-f
+
-_;$‘(cot) [l - al&t)]-*
.
(3.54)
I
Again, by the high speed periodicity of x(t) is meant here the reciprocal of (3.54). In the case in which the forcing function vanishes (A = 0), equation (3.54) reduces to the one derived directly by Knowles for a freely oscillating tube [4, equation (1.7)]. Note that the low speed fluctuation component of x(t) is periodic and has the same period pxct, as the applied pressure, so that 2x Pxw = w
(3.55)
.
To f= the ideas, let us assume that +(ot) = sin cot,
(3.56)
and (3.57) so that the motions of the tube start from its rest, unstretched, position and the tube executes harmonic oscillations. In this case equation (3.48) furnishes the value of A, where A, =
-UK-+ 4[1 + &(tL202/K)]’
13.58)
and equation (3.53) reduces to x(t) = [l - cl sin ot]‘-*
a”’ {I - cos[2K+S#‘)d~]} 32[i + & E2/1212 0
where /I2 = 02/K is a very small number for all practical purposes. t A somewhat similar phenomenon moving-coil loud-speakers.
was discovered by McLachlan [IS, p. 2721 in an electrical problem involving
204
M. SHAHINP~~R
and J. L. NOWINSKI
As an illustration let us consider a blood vessel in the vascular system, into which the blood is being pumped, periodically by the contractions of the heart with a period equal, say (roughly), to 1 sec. Then o = 2n set- ’ and for a vessel of large inner radius equal to 0.1 in. K+ - 14,400 see- I. This gives /? z 0900435 and, therefore, for all practical purposes the value of x(t) is dominated by the factor (1 - a sin ot)-* associated with the low speed fluctuation component of oscillations.
Ressure
------
0.3L
-0.71 0
I T/4
I */P
I 3714
I ‘I
I %lrB
I 3*/Z
I 7*/4
I 2r
I
(WI)
FIG. 1. Square of non-dimensional
inner radius x (t), and non-dimensional various values of a’.
pressure Z sin cot versus time for
Variations of the square of the inner radius of the tube x(t), and of the reduced pressure for various values of o! are shown in a representative diagram in Fig. 1. It is seen that in the scale used, the high speed fluctuation component of the oscillations is not discernible, and the graphs seem to represent perfectly smooth curves. With an increasing value of ol the amplitude of both the reduced pressure and the displacement increase. The increase and its rate are different for the expansion and the contraction periods of the tube, in the sense that it is harder for the tube to contract than to expand. Such a behavior is rather understandable from the physical point of view. The amplitude of the expansion mode is always larger than the one of the contraction mode, but this difference decreases with a decreasing pressure.
P*(t)/K
E. Consider now a periodic step pulse loading defined by
0,
t=g
(3.60)
Exact solution to theproblem
offorced large amplitude radial oscillation
I
ofa thin hyperelastk
tube
205
< t < T*, t = T*
0
this pattern being repeated ad infinitum. Equation (2.1) can now be brought into the form
izxz ln
( >
l.5 + K(1
1 +
x2
-
x2)
lnc:++ “,*‘I=+P*(t)(X2
-
i),
(3.61)
provided that the initial conditions (3.57) are satisfied. According to a known argument (cf. [4]) equation (3.61) describes a periodic motion if there exists a value of x(t) different from unity (either, say, x1 > 1 or x2 < 1, both positive in the present case) such that -the associated value of i(t) = o = 0. Posing u = 0 in (3.61) we get
t1 f 1
1
‘I=
(l+p)exp;(-PO/IC]-l
(3.62)
,
and
x2= [ (1 +fl)exp[+P,/K]
(3.63)
- 1 ’
The fact that x1 and x2 are real implies that P,/K < In (1 + p). As shown in the subsection B [equations (3.13) and (3.14)] the period of oscillations of a thin shell, in the case of a Heaviside loading, was found by Knowles in the form
T= 5
r
(1 + p)exp [-I;P*/@
+ I
(3.64)
+ /.I)] - 1 .
This result, confirmed above, reveals an interesting fact that a sudden application of load sets the tube into a periodic motion with the period of oscillations of the order of nK-*, which is very small for practical purposes. The last quantity is equal to the period of natural oscillations of a thin-walled tube. A similar result was found in the subsection D, in the case of a sinusoidal loading, where the effect of high speed fluctuations was shown to be negligible, as compared with the low speed component of motion. In the sinusoidal loading case, the period of high speed vibrations is [cf. equation (3.54)], T = KK-*
- o2 sin ot) + ozol 8K (1 - a sin cot)*
(1 - Esinot)f
5a02 cos2 of
+
4[1 - ~Zsinot]*
11’ -I
(3.65)
It is evident that the above period is itself, a periodic function of time. In the presently analysed periodic pulse loading, for a tube of arbitrary wall thickness one obtains T=
x1-= dx s
X2
V
X1 s
X2
(x2
-
x2 In (1 + p/x2) dx. 1) [P*(t) - K In (1 + p) + K In (1 + p/x2)
(3.66)
M.
206
SHAHINPOOR
and
J. L. NOWIKSKI
In order to investigate the low speed periodicity of x(t), let us return to the equation (2.3). In view of the equation (2.7) and the boundary conditions (3.57) we arrive at a solution for the displacement field in the form, x(t) =
2/S - P*(t) p*(t) ]coS[4(K-~)lff~. i 2[pK - P*(t)] - [ 2[pK - P*(t)]
(367)
Note that P*(t) satisfies the Dirichlet conditions and can be expanded in a Fourier series of the well known form m P*(t) =
c
#l=i*3.5;~.
4po sin 2nnt T*’ nx
(3.68)
Thus
[I - (; x(t) =
2
+cos[4(K - g+t}]
$E${l
n=1,3.5:~,
(3.691 (I -6
2
Asin?)
n=l.3.5*.”
where E = P,IyK. From the last equation it is clear that the actual motion represents a coupled motion which, as stated earlier, consists of two periodic components. Its high speed component has the period [see (3.69)], (3.70) which itself is a periodic function of time, with the period T* equal to period of applied loading. The low speed component has a constant period equal to T*. It is easy to show that the equation (3.66) reduces to equation (3.70), when the thickness of the wall of the shell p + 0. It seems, therefore, conceivable to obtain an approximate solution for a tube of arbitrary wall thickness subjected to a periodic pulse loading, by inserting the exact expression for the period of oscillations of such a tube, equation (3.66), into the equation (3.69). In this way we obtain an approximation to the complete solution for a tube of arbitrary thickness, in an explicit form x(t) =
x;ln (1 + p/x2) )+dx] - 1) [P*(r) - K In (1 + q) + K In (1 + p/x2)
’
+ r)eIp(E)
-1
f 1 ’
-‘t
H’.
Exact solution to the problem offorced large amplitude radial oscillations of a thin hyperelastic tube
207
REFERENCES
111C. TRUFSDIU and R TOUPIN,The Classical Field Theories, Encycl. Phys., Vol. III/I, Springer (1960).
[21 C. TRUESDJXLand W. NOLL, The Non-Linear Field Theories of Mechanics, Encycl. Phys., Vol. III/3, Springa (1965). [31 A. C. ERINGEN, Nonlinear Theory of Conrinuow Media. McGraw-Hill (1962). [41 J. K. KNOWL~, Large amplitude oscillations of a tube of incompressible elastic material. Q. appl. Math. 18, 71 (1960). 151J. K. KNOWLES,On a class of oscillations in the finite-deformation theory of elasticity. J. appl. Mech. 29, 283 (1962). ed., Continuum Mechunice IK Problems of Non-Lineur Elasticity. Gordon & Breach (1965). [61 C. TRUE~D~LL [71 G. Z. HENG and R. SOLECKI, Free and forced finite amplitude oscillations of an elastic thick-walled hollow sphere of incompressible material. Arch. Me&. Sros. 15, 424 (1964). PI J. K. KNOWLESand M. T. JAKIJB,Finite dynamic deformation of an incompressible elastic medium containing a spherical cavity. Arch. Rat. Mech. An. 14, 367 (1965). [91 C. C. WANG, On the radial oscillations of a spherical thin shell in the finite elasticity theory. Arch. Rat. Mech. Anal., 23, 270 (1965). [lOI J. L. NO~INSK~and A. B. SCHULTZ,Note on a class of finite longitudinal oscillations of thick-walled cylinders. Proc. Ind. Nat. Congr. Theor. appl. Mech., 31 (1964). J. L. NOWINSKI,On a dynamic problem in finite elastic shear. Inf. J. Eng. Sci. 4, 501 (1966). ;::I A. S. D. WANG, On free oscillations of elastic incompressible bodies in finite shear. (h publ.) (1969). (131 J. L. NO~INSIUand S. D. WANG, Gale&in’s solution to a severely non-linear problem of finite elastodynamics. Int. J. Non-Lineor Mech. 1, 219 (1966). [141 E. PINNEY,The nonlinear differential equation y” + p(x) y + cy- ’ = 0. Proc. Am. Math. Sot. 1, 68 (1950). [15] N. W. MCLACHLAN,neory and Application of Mathieu Functions. Dover (1964). (Received 27 February 1970) Resmd--On
analyse. le problbme du titre eh partant de la thCorie de l’tlasticit6 rigoureuse pour les dfformations finies. On consid&e le mat&au du tube incompressible et du type Mooney-Rivlin. L’&uation fondamentale est celle de Knowles simplifiQ pour des tubes minces. On trouve la solution exacte pour une vibration for& g&&rale e.t on Ctudie des cas particuliers faisant intervenir : les oscillations libres, une charge en escalier du type Heaviside, une explosion, une charge harmonique et une charge avec des impulsions p&iodiques. La m&hode permet un calcul explicite des champs de ddplacement et de contraintes rest& indCterminb dans le travail prbc&-lent. On r&out un exemple numtrique et on donne des graphes pour des vibrations for&es harmoniques. 11apparalt que les oscillations rklles pr&entent deux composantes sCparCes: une composante a fluctuation rapide (presque negligeable) relib aux mouvements P oscillations libres et une composante & fluctuation lente (dominante et d’amplitude BevQ). Zos;lmwnfassmg-Das in der uberschrift erwtinte Problem tird unter Zugrundelegung einer rigorosen Elastizit&stheorie endlicher Verformung untersucht Dabei wird angenommen, dass das Material des Rohres inkompressibel und vom Mooney-Rivlin Typ ist Die Hauptgleichung ist diejenige von Knowles, vereinfacht fur dilnnwandige Rohre. Fur eine allgemeine Druckfunktion wird eine exakte Lbsung gefunden, und einige spezielle Fiille werden untersucht, unter anderen : freie Schwingungen, eine Stufenbelastung vom Heaviside Typ, eine Explosionsbelastung, eine harmonische Belastung- und eine periodische Stufenimpuls Belastung Die Methode erlaubt eine explizite Herleitung der Verschiebungs- und Spannungsfelder, die in der friiheren Arbeit unbestimmt waren, Ein numerisches &spiel wird gel&t, und Diagramme fur harmonische Druckfunktionen werden dargestellt Es stellt sich heraus, dass die tatsiichlichen Schwingungen zwei getrennte Komponenten aufweisen: eine schnelle Fluktuationskomponente (kann beinahe vernachltisigt werden), die mit den freien oszillierenden Bewegungen in Verbindung steht, und eine langsame Fluktuationskomponente (vorherrschend, mit grosser Amplitude) .&EEOTaqRsI-&yYeHne npo6nern B~Hy?U~eliHbIX panHaJlbHhIX Hone6aHu# 6OnbmO# aMnnUTyAM TOHKOfil%nepynpyrOfi Tpy6bl OCHOBaHOHa CTpOl’OfiTeOpllM ynpyrOCTU RJlfl KOHe’lHblX&IOpMal@. MaTepHan ~py6b1 nonaraeTcx HecHu?MaeMblMa T&ina @He-P%iB.znHa (Mooney-Rivlin) lIpo6nema onpe~enJ%eTca YpaBHeHHeM Hoynca (Knowles) C ynpomeaaen! aJIB ToHKOCTeHHbIXTpy6. TosHoe pemenHe HaiReHo HJIH o6mefi BbxHymxammeti I$~HKUHN. npvl ATOM mcnenymTcs cneqw@mecKKe cnyqaa, a EIMeHHO: cBo6o~nbIe KOJIe6aHUH, ncTynHbsaTyI0 Harpy3KyD, T&ma XeBucattJ&a HarpyaKy RrnMIUKoft.rap~oHKseCKy~
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