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Stability of an elastic circular tube of arbitrary wall thickness subjected to an external pressure
Int. .I. Non-Linear bfechmics, Vol. 4, pp. 143-158.
STABILITY ARBITRARY
Pergamon Press 1969.
Printed in Great Britain
OF AN ELASTIC CIRCULAR TUBE OF WALL THICKNESS SUBJECTED TO AN EXTERNAL PRESSURE* J. L. NOWINSKI and M. SHAHINPOOR
Department of Mecnanical and Aerospace Engineering, University of Delaware Abstrztct-Inthe last decade it became clear that the loss of stability may occur not only in slender but in thick and thick-walled bodies as well. In the present paper the stability of an infinitefy long circular cylinder of arbitrary wall thickness is investigated. The material of the cylinder is taken as incompressible and the cylinder is subjected to an external pressure. On a finite symmetric deformation of the cylinder is superposed a secondary infinitesimal deformation depending on the radial and hoop coordinates. A system of three coupled second order partial differential equations is obtained for three unknown functions, which is solved by the Frobenius series method. A nontrivial fulfillment of the boundary conditions leads to a characteristic equation for a deformation parameter. An approximate two term solution assuming Neo-Hookean material is analyzed. In the limit case of a thin shell the critical outer pressure is found to be in agreement with the known classical result. A numerical analysis for thick shells is also given.
DESPITE
two centuries of investigation of stability of equilibrium of elastic bodies only in the last decade has it become clear that the loss of stability need not necessarily be attributed to the slenderness of the bodies, but may occur in thick and thick-walled bodies as well. Moreover, it is argued that under sufficiently large strain even an unbounded medium (i.e. lacking a free surface) may become unstable [l], Elastic instability was also discovered in the presence of purely tensile loads [2]. Apparently the first explicit solution of a problem of the class considered belongs to Wilkes who in 1955, [3], discussed extensively the axially symmetric buckling of a thickwalled tube under axial thrust. Two years later Lubkin [IS] published a report in which instability of a column in plane strain as well as of a thick-walled cylinder and sphere under external pressure was investigated. The last two problems were reconsidered by Sensenig in 1964 [IS]. In both these investigations an energy technique was used based upon an extended form of Hooke’s law for principal directions as well as a particular form of the strain energy. In the opinion of theorists, see [14], such approach is considered as approximate and in a sense experimentally and theoretically special. Other problems of a similar class were investigated among others by Read [7] and Guo Zhong-Heng [8,91. Recently Wesolowski, apparently unaware of the work of Lubkin and Sensenig, analyzed spherical bodies (a solid and a hollow sphere) under uniform hydrostatic pressure. Wesolowski’s solution is based on a rigorous theory of finite deformations and leads to explicit numerical results (found with the help of the computer URAL II), [lo, 11-J. Lacking more irreproachable methods in most work the method of adjacent quilibfia * Research supported by a grant of the National Science Foundation. 143
144
.I. L. NOWINSK~ and M. SHAKINPOOK
or bifurcation of equilibrium is used despite the fact that some writers consider the method insecure and based on no more than Kirchhoffs uniqueness theorem, see [14], p. 255. From what is known as the method of adjacent equilibria, the critical load is defined as that for which uniqueness of equilibrium ceases to hold. More explicitly, using the theory of infinitesimal deformations superposed on finite deformations as developed by Green, Rivlin and Shield (see [12]) the body is subjected to a finite ordinarily homogeneous deformation and afterward exposed to a secondary small displacement from the finitely deformed state. The question is then raised as to whether or not equilibrium is possible in this slightly perturbed state under the unmodified load. The answer being in the positive the pertinent load is considered as critical and the finitely deformed state unstable. This technique is used in the present paper in which we reconsider the problem of instability of a thick-walled cylindrical tube subjected to an external hydrostatic pressure 4 using the rigorous theory of [12]. The tube is considered infinitely long and its material hyperelastic~ isotropic and for definiteness incompressible. However, there is no essential di~culty in extending the solution to the compressible case as well. The tube is first subjected to a finite symmetric deformation on which a secondary state of infmitesimal displacement is superposed consisting in a uniform distortion of the cross-sections. The equations of equilibrium in the final state of deformation in combination with the incompressibility condition provide a set of governing equations. These coupled homogeneous partial differential equations with variable coefficients are solved by the Frobenius method assuming for simplicity that the material of the tube is of the Neo-Hookean type. Upon disregarding all terms except the first two in the series obtained a nontrivial fulfillment of the boundary conditions leads to the characteristic equation of the fundamental problem. In the limit case ofa thin shell an order-of-magnitude analysis provides the value of the critical pressure in agreement with the classical result. For walis of arbitrary thickness a numerical analysis carried on with the aid of an electronic computer furnishes a relation between the relative wall thickness and the critical strain displayed on a diagram. 1. GENERAL
EQUATIONS
Let us consider an infinitely long tube of perfectly elastic isotropic material and circular cross-section, the internal and external radii in the undeformed state B, being respectively equal to A, and A,. The tube is subjected to a uniform hydrostatic pressure 4 on the external surface this being the pressure on the actual area directed normal to the surface at any stage of deformation. The pressure produces a finite axially symmetric deformation of the tube (state B) so that the inner and outer radii become respectively a1 = &A,* a2 = IL2& WI where pi, i = 1, 2, are unknown constant coefficients. We refer the tube to a cylindrical coordinate system r, 0, z in the deformed state B, so that assuming a plane strain deformation the point actually at r, 8, z in B was initially at R, 0, z in B,. It follows that the fundamental metric tensor in B is?
t Hereafter we use the notation probounded in Ref. [12] with some modifications proposed by Truesdell
Stability
of an elastic
wall thickness subjected fo an external pressure
circular tube of arbitrary
145
while before the defo~ation (1.3) where (1.4)
Q(r) = R/r and it was assumed that, for simplicity, the material of the tube is incompressible,
(1.5)
Q@ + &I) = 1.
The last ass~ption is not essential and may be removed at the expense of computational labor. An alternate fo~ulation of the incompressibility condition leads to a relation between the coefficients i”i, i = 1,2,
(1 - /m4/~2)2.
&=l-
(1.6)
The strain invariants presently become I = II =l+Q2+$
(1.7)
while the third invariant III = 1 by virtue of the incompressibility of the material. The stress components in the body B become
where
p
ril = Q26, + (1 + Q’)Jt + p,
7
p
1
=
223
=
*31
=I
0
9
may be treated as an (unknown) hydrostatic pressure and (1.9)
W = W(I, II) represents the stored energy density function. The equations of equilibrium of the medium in the state B are rfk (1.10) k = 1,2,3, .,i = 0 3 where the double semicolon denotes covariant differentiation with respect to gij and r, 8, z. We get finally rll = -L(r), 7
(
r2~22= Q2+
$
$3 = (1 - Q’)#
I
>
(1.11)
(q5f I& --L(r),
-
1 - -$ (
II, -L(r), >
i
146
J, L. NOWINSKI and M. SHAHWFQOR
with (1.12)
In deriving (1.11) the boundary condition r’l = 0
at
I = a,
(1.13)
was used. The second boundary condition ~~r=--q
r=a,
at
(1.14)
enables one to express the pressure 4 in terms of the unknown qu~tities one gets
a, i = I,2 and so (1.15)
4 = %r,). Upon using (1.5) it is easy to convert the integral (1.12) into the integrable form
(1.16) with Q1 = (AI/a,) = l/p1 and Qz = (&/a,) = 1,‘~~.Assuming for a moment that the sum (# + il/) is independent of Q one obtains finally (1.17) and from (1.15) the external pressure in terms of pI and ya, (1.18) We now consider a secondary infinitesimal deformation of the body B associated with the displacements wb i = 1,2,3, such that WI E w, = u(r, e),
w2 E wg = u(r, 0),
w,j = wz = 0.
(1.19)
This additional deformation carries the body into the state B’ and induces additional stresses @ and changes of the metric tensor represented by - 2vjr
U,@+ v,* - 2v/r
0
2(v,6 + ur>
0 , 0i
0
(1.19.1)
Stability
147
wall thickness subjected to an external pressure
of an elastic circular tube of arbitrary
The incremental invariants associated with the transition B --) B’ become
(1.20)
while in view of the incompressibility of the material the invariant III’ = 0; this condition is equivalent to 1 (1.21) @,r + f o,, + ; = 0. The incremental stresses are represented by where gij
= f*GU _ C;*@Q;,,
(1.23)
II/’= FI’ + MI’,
(6’ = AI’ + FII’, CT2W
A=zaIZ’
B=+,
iT2W
p’ is a hydrostatic p~udo-press~e function to be determined later. Longer calculations lead to the following expressions for the incremental stresses, $11 = Y11%r f P’, r 122
_ -
f
133
_ -
Y33b
z
(Y22%,
1 z rl2 _- - 7
+
Yl2
+
P’h
1
$9
7 i
all other stress components being identically equal to zero. The symbols employed in equations (1.24) have the following meaning, y11=2A(Q4-1)+2B
(1.25)
~33=2A~-~)+2~(Q4-~) + 2F Yl2
=
e
-t
P*
I
148
J. L.
NOWINSKI
and M.
SHAHINPOOR
To avoid too many technicalities let us assume at this stage that the material of the tube is of the so-called Neo-Hookean type. The stored energy function is then W(1) = C,(I - 3)
(1.26)
with C, as a material constant. If one requires that in the limit case of an infinitesimal deformation the Neo-Hookean constitutive equation reduces to that of Hooke one obtains the relation C, = E/6. With (1.26) in mind equations (1.9) and (1.23),_, now yield 4 = 2C,,
A=B=F=O,
* = 0,
(1.26.1)
so that the incremental stress components (1.24) become r ‘11 = -2pu,,
+ p’,
z r22= $(2pu,,
‘t
133
=
+ p’), $12
p’,
=
_
, (1.27) !j
u,e
+
v,,
-
2t’
r)
(
’
where (1.27.1) The incremental stress components must satisfy the following equilibrium equations, zt,i!
,.
r$ p + r; $j = 0,
i +
i,j = 1,2,3,
(1.28)
in which the Iyk symbols are constructed from the fundamental metric tensor in the state B and the components g’ij = wiiij + wjiii, see [12], equations (4.1.15) and (4.1.17). The field equation (1.28) corresponding to j = 3 is satisfied identically ; the two remaining equations are reduced to the following final form in the case of the Neo-Hookean material,
- %3
k +v,,.,4Q2 + pie= 0, Q2
(1.29)
(1.30) (~-P,~)+u,ee~-V,e~+P:r=O.
u,,.,Q24 + 2~3,
Upon adjoining the incompressibility
condition (1.21)
1 u,, + _z v,, + 11= 0, r
(1.31)
r
one obtains a system of three coupled homogeneous
partial differential equations with
~ra&il~ty of an elastic circular rube ~~arbitrary
wail thickne~~ su&ected to an e~ternalpres~re
149
variable coefficients, two of them of the second order, involving three unknown functions u, Dand p’ of two independent variables t and 8. To these one needs to apply the boundary conditions which follow from the demand that, simultaneously with the displacement field associated with the state B, there exists a nontrivial displacement field u, v corresponding to the same external load q on r = a 2 ; clearly, the surface r = a I has to remain free of load at all times. This requirement guarantees that for some pressure q (call it critical pressure qc,iJ a bifurcation of the equilibrium takes place in the sense that in concert with the state of equilibrium B there exists an adjacent nontrivial state of equilibrium B’ both states being associated with the same external load. In the actual case then? = (1.32) 0, for P = ffl,fa2, z flk _ Tllg’lk or in view of (1.11) and (1.27) in combination
with (X19.2)
25, Q21.GP + p’ = 0, for I = a1,a2. (1.33) 2V u,, -I- Ii,, - - = 0, r ~ This completes the formulation of the eigen-value problem which is thus reduced to a solution of the system of homogeneous governing equations (1.29)-(1.31) such that the homogeneous boundary conditions (1.33) be satisfied in a nontrivial way. Clearly this happens when the principal dete~inant of the linear system (1.33) vanishes providing the characteristic equation for the proper values of the problem. 2. SOLUTION
OF THE EIGEN-VALUF, PROBLEM
We first note that upon integrating (1.5) one gets Q = i (r2 -t-K)*, where K is a constant which by the incompressibility
(2.1) condition is represented by
K = Ai (1 - ~1:).
(2.2)
A substitution of (2.1) into (1.27.1) yields
(bK2
P‘r = r3(r2 t K)’
a derivative encountered in the governing equations (1.29) and (1.30). We now seek the solution of the eigen-value problem just formulated form,
t For a more explicit derivatiun see [12] or fl l],
(2.31
in a separable
1
J. L. NOWINSKIand M. SHAHINP~OR
150
The geometric interpretation of equations (2.4) is that the surfaces of the cylinder in the conceivable adjacent equilibrium state are assumed to display longitudes corrugations dependent on the angle 8 but independent on z. Hereafter we use only one general term of the series (2.4) and for brevity drop the subscript n. By appropriate transformation it is easy to reduce the system (1.29)-(1.31) to a single homogeneous fourth order differential equation for the variable u,
$!C-V+KU +&
1
-2-s+
F&]-F&+&]
+$[-6r-y+&[-++;-&-(r2~K)2] +g [-5+F+- 4K
d2U 1I( -._L+~_T2 +K1+a82 r2 + [ 1 1 _2_9K+ 4 U 1 (2.5 51 1 1 [ 8rK
(9 + K)’
8K
~-
+u
r2
which upon employing the representation
r4
r2 + K
(r2 + K)2
= O’
(2.4), yields
2n2
K(7 + n2)
K(n2 - 4)
~r2 + K - r2(r2 + K) + (9 + K)’
+ R;
-(5
+ n2) + K(7 + n2f + -r(4 - n2) + 2rK(4 - n2) (r2 + K)2 r3(r2 + K) (r2 + K)3 r(r2 + K)
+ RI
5 - n2
3K(3 + n2) + n4 - n2 - 4 +
+ r 42 (r r2(r2 + K)
+ K)
8K
(r2 -I- IQ3
(r2 + K)2
We now introduce non-dimensional variables r, = r/a2 and & = K/a; = l/,u: transform (2.6) into the following form, r,4R:V + riPl(r*)R;”
f r$P2(r*)R;
+ r*P3(r*)R;
+ P,(r,)R,
(2.6)
= O’
- 1 and
= 0
(2.7)
with
(5 - 2n2)ri - (7 + n2)K* + i (n2 - 4)r,2
P, = b
I
I [
P, = -!_ - (5 + n2)ri + (7 + n2)K* -t-f(4 G [
- n’)rz + 3(3 + n2)K, + i(n4 K, =
ri + K*.
,
1 1
- n2)r,4 + i2(4 - n2)rzK* , r - n2 -4)r$
+sr$K* 4
’ (2.8)
,
I
Stability
ofan
eiastic circular tube
ofarbitrary wali thickness subjected
to an external pressure
151
It is to be expected that under an external compressive load, as is the case in the present situation, we always have 0 < pl, y2 < 1. Hence K, > 0 and consequently K, > 0, so that the functions Pi(r*), i = 1, . . , , 4, in (2.7) are analytic in the neighborhood of r* = 0. The only singular point is then r* = 0 which, in fact, does not belong to the domain of definition of the function R,(r,). This stems from the fact that, by hypothesis, the inner surface of the tube is free of external load so that the domain of definition must be at least punctured at r* = 0. By known theorems [13], p. 396, the singular point r* = 0 of the equation of the form (2.7) is a regular singular point and it is possible to obtain, in the neighborhood of the singularity, explicit developments of the fund~ental solutions convergent for sufficiently small values of r,. We get such formal solutions of equation (2.7) by using the Frobenius method, i.e. representing R,(r,) in the form
where the parametric index p is obtained from the indicial equation P(P - I) (P - 2) (P - 3) + P(P - I) (P - 2) P,(O) f P(P - 1) P2(0) + P,(O)
+ P&$(O) = 0,
(2.10)
or explicitly from, P4 -4~3-(2+~z~~2~2(6+~2)~+3(3+~2)=0.
(2.11)
This equation has all four roots real and equal to 3, - 1, 1 + (4 + a”)*. It is easily seen that the case corresponding to n = 1 represents a rigid translation of the cross-sections of the tube, so that the first mode of instability of equilibria is associated with the value tl = 2. In what follows we confine the analysis to the mode n = 2 only. With this in mind the roots of the indicial equation in decreasing order become, p1 = 1 + 2y/2,p2
= 3,p, = -l,p,
= 1 - 242.
(2.12)
Since p2 and p3 differ by a positive integer they give rise to a formally identical solution and the solution corresponding to p3 must be replaced by an appropriate solution of the log type. As a result the function R,(r,) is expressed by the following equation, R, = A
c
m (- lPV(3) m (- l)“F,(3) v- 1 Jb(3 + v) yg+3 + BC1nr* - 16] cv=. jb(3 + v) r* v=o (2 13)
+C
*
where A, B, C and D are integration constants to be determined from the additional conditions, Jb(P + v) = UP + 1) f*(P f 2) &(P + 3) f - . _LAP+ 4,
(2.13.1)
f(r*, p + v) = (P + v)(p f v - l)(p + v - 2)(p + v - 3) + Pl@,)(P + v)cp f- v - l)(P + v - 2) +
P2h)co
+ +
P3@*)
$(P (P
+
v
+
4
-
1) +
P*@*k
(2.13.2)
152
J. L.No~r~mand
M.SHAHINPOOR
and
F,(P) =
f,(P f
v -
1)
.fAP +
v -
2) . . f fv-1@+1)
.MP
v -
1)
fAP
v -
2) . .' L-2(P
+
1)
.LIW
2). . . .A-&
+
1)
"fv-z.(P)
+
+
.A@)
0
OfXP
0
0
.I. .
0
0
. . . MP 3- 1) f,(P)
+
v -
-
(2,13.3)
Clearly f, (p + v) is equal to zero for N odd. For N even, say N = 2n, one obtains the
following relation, fin&J + v) = g
[20a - s/? - 4y + F(n)],
*
(2.13.4)
where a = p + v, fl = (p + v) (p + v - l), y = (p + v) (p + v - 1) (p + v - 2),
(2.13.5)
and F(n) is determined from
c
O”---rpF(n) (-1)
n=O
KS
20 = P.&T*) = 21 --r: K*
36 + Fr; *
60 - --& KS
Bearing in mind equations (2.4) and (2.13) it is straightforward equation (1.31) the function
(-1)“‘(2
92 + j-$$ *
- .._.
(2.13.6)
to find from the governing
+ 2 J2 -t- v)F,U + 2 J2)rz+zj2+” * 2Jb(l + 242 i-v) (2.14)
v=o We refrain from quoting explicitly the unwieldy expression obtained for R,(r ). Since the convergence of the series obtained seems to be fairly rapid, hereafter, in particular in numerical calculations, we confine ourselves to the first two terms of the series only. This procedure yields (2.15.1)
R, = A[r:]+ R, = A[-2r4,]
+ B [-+]
-t
Cl-(1
+ J2)r;+y
+
D[-(1
- t/2)r;-2Jz],
(2.15.2)
Stability
ofan elastic
+ D [-r;z-zJ2(
circular tube of arbitrary
waII thickness subjected to an external pressure
rz ?;;;c,)
- 12J2)
+ k;(l4
- r;zJz(14
153
(2.15.3)
- 8J2)l.
As remarked earlier the characteristic equation of the problem is obtained from the principal determinant of the homogeneous boundary conditions (1.33) upon substitution of the solution (2.4). For the mode of instability considered these conditions yield, 2Q24R; + R3 = 0, (2.16)
for I = ai, a2 -2R,
•l- Rf2-$R2
=O,
with Q from (2.1). It is convenient to introduce the non-dimensional variables r*i = al/u2 and r*2 = a2/a2 = 1. With these in mind the boundary conditions become,
A 14K* +
+~]~C[-10r~~2-~OK~+4r;j2+~~~2~~~)ri_2i2]
+I)
-
1orp2-
r,-1”- 2 J2
lOK* - 4K* J2 +
&.+ K*=f-4 2&
(2.17.1)
1 -lo-lOK,+4,/2K,-&
A[-
14r:,]
+ B
-fzrsl +g
*I
= 0,
(2.17.2)
-~C(-6-2,/2)r~~~J~ 1
1
+ D(- 6 + 2J2)r;;2J2 A[-- 141 + B[33] + C[-
6 - 2,,‘2] + D[- 6 t- 2421 = 0.
The vanishing of the principal determinant
(2.17.3) (2.17.4)
of the foregoing system gives
det A, = 0, where the following notation is used,
= 0,
(2.18)
J. L.
154
2K*G 1
A,, = 14K, +
A,,
t-2, + K,’
= -lOr*‘f’-
lOK* + 4K* J2 + [
A,, = - 10r;fJ2 A,,
= 14K,
and M. SHAHNPOOR
NOWINSKI
+ ~
-
lo& C
2rZ 1 ril
+
K,
2r’1
- 4K, J2 +
1 1 -2i2
r*l
d1 + K*
-2-2 r*l
J2
’ J2
y
2Ks
1 t-K,’ A,, = -66 - 68K, - 16 2K5 - 4 1 + K* ’ A,,=
-lo-lOK,
A,,=
-lo-
-442%
(2.19)
-&,
lOK, +442K,
-&, *
A,,
= -14rz,,
A,,
= -2--
A,,
= (-
+12_ r*l’ r*1 2J2)r::2J2, 6
A,,
= (-
6 + 2J2)riT2J2,
A,,
= -14,
A,,
=33,
A,,= A,,
ln r*1
-6-2J2, = -6
+ 2J2.
In what follows it is expedient to introduce the notation IC= (a2 - al)/a2. 3. THIN
SHELLS
If the thickness of the shell 6 = A, - A, is small as compared with the radii of the initial cross-section the shell may be treated as a moderately thin shell. For such shells it is legitimate to neglect the transverse deformations so that a, z a, + 6. In view of this the radial displacements of the inner and outer surfaces become indistinguishable from each other so that denoting the displacements by u we get a, x A, - u and a, z A, - u. Assuming that both 6/u, and u/al are negligible as compared with 1, we obtain K,
x 2uJu, and rc2 z d2/af.
(3.1)
For moderately thin shells their deflections, in particular at the beginning of buckling, are believed to be very small as compared with their thickness. It is reasonable, therefore, to consider K, and IC’as quantities of like order of smallness and, in the limit case of a thin shell, to neglect in the pertinent equations all terms involving products of K* and JC~or
Stability
of an elastic circular tube of arbdtrary wail thickness subjected to an external pressure
155
powers of K and K higher than the first and second, respectively. Similarly it is consistent to retain no more than three terms in the expansion
With these simplifications in mind the terms of the characteristic equation (2.18) become
4,
= 16&t,
A,, = -2 - 132K - 4tc - 61~~~ AI3 = - 12 - K,(8 + 4 ,/2) + (4 ,,/2 - 6)~ + (91 - 42) A,,
=
-
K’,
12 - & (8 - 4 ,,‘2) + (-4 42 + 6)~ -I- (91 + J2)rc2,
AzI = 16K,, A,,
= -2-132&,
A,,
= - 12 - K,(8 + 4 JZ),
A,,
= -12
- K*(8 - 4J2),
(3.3)
A,, = -14(1 - 3lc + 3rc7, A,,
= 3311 + K + rc2),
A,,
= (-6
A,, = (-6 A,,
= -14,
A,,
= 33
- 2 42) + 14(1 f J2)rc - 14(2 + +‘2&“,
:
+ 242) + 14(1 - ,/2)tc - 14(2 - 42)x2,
A,, = -6 - 2J2, A,,=
-6+2,/2,
J
so that equation (2.18) finally becomes rcZ(-K2 + 10-4 K + I(* + 10-4) = 0.
(3.4)
One solution of this equation, I? = 0 must be discarded as trivial ; to find the second solution we neglect the terms involving 10R4 as small? of higher order than rc2 or K*. This yields fkrally the critical condition
K* = K2.
(3.5)
To find the critical outer pressure CJfor a thin shell we insert the relation (3.5) into equation (1.18) and considering higher powers of K negligible as compared with the lowest powers of K appearing in the polynominals obtained we finally get 4 wit = Erc3/3.
7 Order of magnitude of K may be taken as
IO-‘.
(3.6)
156
J.L.
NOWINSKI
and M.
SHAHINPOOR
This result agrees with the classical result, see e.g. [14], if in the latter the material of the shell is assumed incompressible (i.e. Poisson’s ratio taken equal to one half) and the mean radius set equal to a, which is admissible for thin shells. 4. NUMERiCAL
ANALYSIS OF THICK SHELLS
For shells of arbitrary thickness no closed form solution of the type (3.6) could be found. Instead a numerical procedure of solving the characteristic determinant has been devised using the digital computer IBM 1620. Its final product is the diagram displayed in Fig. 1,
,” I4” II
x
o-5
06
0.7
0.8
0.9
I.0
Fro. 1. Deformation ratio vs. initial radii rate.
illustrating the dependence of the deformation ratio ,+ = a,/A, on the thickness ratio of the tube Al/A,. For compa~son the curves displaying the classical solution for thin shells and the result obtained in [ 1l] for thick-walled spherical shells under external hydrostatic pressure are given. It is seen that the graph for the thick~walled cylinder exhibits the same features as that for the thick-walled sphere, the stability of the mode investigated in [l l] for the thick-walled sphere (for the same wall-thickness) being smaller than the stability of the mode investigated in the present paper for the thick-walled cylinder. Figure 1 also displays the limit case when the deformed inner radius of the structure a, goes to zero. The pertinent relation can be found immediately from the incompressibility condition and is &Y’ Z Cl - @,,W2]”
(4.1)
&Ph = El - @WA3]“.
(4.2)
for the cylinder and
stability
of an elastic circular tube of arbitrary
wall thickness subjected to an external pressure
157
for the sphere. It is clear that in al1 cases J.$@> @‘I so that the mutual location of the diagrams in Fig. 1 found both for the spherical and cylindrical shell appears to be justified. It is of interest that both in [l l] and in the present case the computer program used was unable to supply data for the interval of thickness-ratio from about 0.8 to about 0.95 (or even to 1.0 as in [i 11). The singular character of this interval deserves certainly a separate investigation. The same is true of the upward trend of the curves associated with the decrease of thickness ratio Al/A, in the interval from 0 to approximately 0.8.
[l] M. A. BIOT,Internal buckling under initial stress in finite elasticity. Proc. R. Sot. A27, 306-328 (1963). [2] J. WESOLOWSKI, The axially symmetric problem of stability loss of an elastic bar subject to tension. Archwm Me&. stosow. 15, 383-395 (1963). [3] E. W. WILKES, On the stability of a circular tube under end thrust. Q. JI Me&. appi. Mafir. 8,88-100 (1955). [4] A. E. GREEN and A. J. M. SPENCER, The stability of a circular cylinder under finite extension and torsion. J. Math. Phys. 37,316338 (1959). is] S. LUBKIN,Dete~ination of buckling criteria by rnin~~a~on of total energy. Inst. Math. Sci. New York Univ., Rep. IMM-NYU241 (1957). [6] C. B. SENSENXG,Knstability of thick-elastic solids. Communspure appl. Math. 17,451-491 (1964). (71 H. E. ROAD,On the stability of equilibrium of thick- and thick-walled isotropic elastic solids. Ph.D. Dissert., University of Delaware (1964). [8] GLJOZHONG-HENG,The problem of stability and vibration of a circular plate subject to finite initial deformation. Archwm Mech. stosow. 14,239-252 (1962). [S] Guo ZHONG-HENG,Vibration and stability of a cylinder subject to finite deformation. Ar&wm Mech. stosow. 14, 757-768 (1962). [lo] Z. WESOLOWSKI, Stability of a fully elastic sphere uniformly loaded on the surface. Archwm Mech. stosow. 16, 1131-1150 (1964). [ll] Z. WESOLOWSKI, Stability of an elastic, thick-walled sphere uniformly loaded by an external pressure. Archwm Me& stosow. 19, 3-44 (1967). [12] A. E. GREENand W. ZERNA,Theoreticai Elasticity. Clarendon Press (1954). [I 9 E. L. INCE, ~rd~~a~_v ~~~~re~?ja~ ~~af~o~. Dover (1956). 1141C. TRUESDELL and W. NOLL, The non-linear field theories of mechanics. EncycI. Phys. IU/3. Springer (1965). I151 A. S. VOLMiR,Flexible Plates and Shells {in Russian). GITTL (1956). (Received 29 April 1968)
R&u&--Durant la derniere decermie il est apparu clairement que la perte de stab&t& peut se prod&e non seutement dams des corps minces mais aussi bien dam des corps Cpais ou & bords epais. On &die dam cet article la stabilite dun cylindre de revolution in~niment long, creux et d’epaisseur arbitraire. On consid& Ie cylindre fait en mattriau incompressible et soumis ii une pression exteme. Gn superpose B une deformation flnie et sym& trique du cylindre une deformation secondaire infmitesimale fonction des coordonn&s polaires On obtient un systeme de trois equations couplCes aux derivees partielles du denxieme ordre pour trois fonctions inconnues; ce systeme est resolu par la methode des series de Frohenius. Dans le but de satisfaire les conditions aux hmites dune fawn non triviale, on obtient une equation ~~~t~ris~que pour un parametre de d&formation. Fn supposant le mat&iau Neo-Hookeen on analyse une solution approximative contenant les deux premiers termes, Dans le eas limite d’un cylindre mince on trouve que la pression critique est en accord aver le resultat classique bien connu. On don& egabment une analyse numerique pour des cylindres Cpais.
Zusammenfassung-lm letzten Jahrzehnt wurde deutlich dass Stabilitiitsverluste nicht nur in schlanken sondern such in dicken und dickwandigen Kbrpem auftreten kbnnen. In der vorliegenden Arbeit wird die Stab&at eines unendlich langen Kreiszylinders mit beliebiger Wandstlrke untersucht. Das Zyhndermateriaf wird als inkompressibel ~geno~en, und der Zylinder ist liusserem Druck unterworfen. Einer endlichen, symmetrisch~ Verformung des Zylinders wird eine sekundZire in~nitesim~e Verformung tiberlagert, die von der Radial- und Tangentialkoordinate abh&tgig ist. Dabei erhglt man em System von drei gekoppelten, partiellen Differentiaigleichungen zweiter Grdnung Rlr drei unbekannte Fnnktionen. Zur L&sung wird die Reihenmethode uach Frobenius benutzt. Die nichttriviale Erfiillung der Randbedingungen ftibrt zu einer charakteristischen Gleichung
158
J. L. NOWWSKIand M.
SHAHINFOOR
fXir einen Ve~ormungsparameter. Mit der Annahme eines “Neo-Hookean” Materials wird eine durch zwei Glieder angengherte Lijsung untersucht. Im Grenzfall diinner Schalen ergibt sich ~bereinstimmung fiir den kritischen Aussendruck mit dem bekannten klassischen Ergebnis. Eine numerische Rechnung fiir dicke Schalen wird ebenfalls angegeben. ABCT~WI-B
IIOCjIeRHeM ~eCfITILm?T&iHCTUIO I13BWTHbIM, YTO IlOTepR ~CTOikWiBOCTki MOX(eT B03HllKIiJ'TbHOTOJIbHO BTOHKHX HO IiBTOJlCTLIX,3T3KHETO~CTOCTt?HHbIXTCJl~X. B HaCTOKIlJetipa6oTe UCC~O~~~TCR~CTOtYHBOCTb6OCKOHOYHO~JIIIHHO~T~~6bIK~~~~O~OCO9~HYL~'~OA3BOJlbHOttTOJI~IIH~ CTOHKM. ~PHHHMWTCFI, YTO MaTepllWI TPY6bl HOCH(HMaeMbIfi M 9TO Tpy6a ITOfiBOpIWTCR BbICLUeMy RaBJIeHHlo. Ha KOHBYHJ'IO CHMMOTpHWCKJ'IO ~~@OpMaL@iIO Tpybhl HaKJUlJ&IBaf!TCFl ~06aBOSHym, 6eCKOHeWO MaJIyIO @@OpMaIJEiIO, 33BMCHIIJyMl OT PaAEWIbHOZt kl YPJIOBOt KOOP~UH3TLd. nosyYaeTClr CNCTf?Mil TPOX COIZPR~OHHEdX WCTHMX J@@!pOH~H3JibHbiX J'paBHOHktt BTOpOl'O IIOpXJ(Ka Am Tpex Hell3necTnbrx ~~YHK~H#. 3Ta cEcTeMa pernaeTcn MeTogoia PRA~B Qpoaewayca. HeTpaswaXbHOe y~OB3ieTBOpeH~O KpaOBbIX YCJIOEHt IIpEfBOfiilT K X3p3KTep~~T~qOCKOMy ypaBHOHIlF0 Ha RapaMt?Tp ~O~pMa4~~. AH3~~3~pyOTCK ~p~~6~~~eHHOO PO~OH~O H3 ABYX YJEHOB ~P~HKMaK TOJIO P~n~~Ha-~pe~~oapa (Heo-ryKonc~oe Tezo). HaCQeHo, 9~0 B rpaH~~HoM cxyyae T~HKO& 060~0Y~~ KpUT~qeC~{OO BHelIfHOO ~ZIBJIOHHO COIVlaCyOTCR C ~3BeCTH~M K~aCC~iqeC~~~iM POSJVIbTaTOM. EpEBO,@fTCR:TPKH(e YMCJIt?HHb& [email protected] o6oJroueK.
STABILITY ARBITRARY
Pergamon Press 1969.
Printed in Great Britain
OF AN ELASTIC CIRCULAR TUBE OF WALL THICKNESS SUBJECTED TO AN EXTERNAL PRESSURE* J. L. NOWINSKI and M. SHAHINPOOR
Department of Mecnanical and Aerospace Engineering, University of Delaware Abstrztct-Inthe last decade it became clear that the loss of stability may occur not only in slender but in thick and thick-walled bodies as well. In the present paper the stability of an infinitefy long circular cylinder of arbitrary wall thickness is investigated. The material of the cylinder is taken as incompressible and the cylinder is subjected to an external pressure. On a finite symmetric deformation of the cylinder is superposed a secondary infinitesimal deformation depending on the radial and hoop coordinates. A system of three coupled second order partial differential equations is obtained for three unknown functions, which is solved by the Frobenius series method. A nontrivial fulfillment of the boundary conditions leads to a characteristic equation for a deformation parameter. An approximate two term solution assuming Neo-Hookean material is analyzed. In the limit case of a thin shell the critical outer pressure is found to be in agreement with the known classical result. A numerical analysis for thick shells is also given.
DESPITE
two centuries of investigation of stability of equilibrium of elastic bodies only in the last decade has it become clear that the loss of stability need not necessarily be attributed to the slenderness of the bodies, but may occur in thick and thick-walled bodies as well. Moreover, it is argued that under sufficiently large strain even an unbounded medium (i.e. lacking a free surface) may become unstable [l], Elastic instability was also discovered in the presence of purely tensile loads [2]. Apparently the first explicit solution of a problem of the class considered belongs to Wilkes who in 1955, [3], discussed extensively the axially symmetric buckling of a thickwalled tube under axial thrust. Two years later Lubkin [IS] published a report in which instability of a column in plane strain as well as of a thick-walled cylinder and sphere under external pressure was investigated. The last two problems were reconsidered by Sensenig in 1964 [IS]. In both these investigations an energy technique was used based upon an extended form of Hooke’s law for principal directions as well as a particular form of the strain energy. In the opinion of theorists, see [14], such approach is considered as approximate and in a sense experimentally and theoretically special. Other problems of a similar class were investigated among others by Read [7] and Guo Zhong-Heng [8,91. Recently Wesolowski, apparently unaware of the work of Lubkin and Sensenig, analyzed spherical bodies (a solid and a hollow sphere) under uniform hydrostatic pressure. Wesolowski’s solution is based on a rigorous theory of finite deformations and leads to explicit numerical results (found with the help of the computer URAL II), [lo, 11-J. Lacking more irreproachable methods in most work the method of adjacent quilibfia * Research supported by a grant of the National Science Foundation. 143
144
.I. L. NOWINSK~ and M. SHAKINPOOK
or bifurcation of equilibrium is used despite the fact that some writers consider the method insecure and based on no more than Kirchhoffs uniqueness theorem, see [14], p. 255. From what is known as the method of adjacent equilibria, the critical load is defined as that for which uniqueness of equilibrium ceases to hold. More explicitly, using the theory of infinitesimal deformations superposed on finite deformations as developed by Green, Rivlin and Shield (see [12]) the body is subjected to a finite ordinarily homogeneous deformation and afterward exposed to a secondary small displacement from the finitely deformed state. The question is then raised as to whether or not equilibrium is possible in this slightly perturbed state under the unmodified load. The answer being in the positive the pertinent load is considered as critical and the finitely deformed state unstable. This technique is used in the present paper in which we reconsider the problem of instability of a thick-walled cylindrical tube subjected to an external hydrostatic pressure 4 using the rigorous theory of [12]. The tube is considered infinitely long and its material hyperelastic~ isotropic and for definiteness incompressible. However, there is no essential di~culty in extending the solution to the compressible case as well. The tube is first subjected to a finite symmetric deformation on which a secondary state of infmitesimal displacement is superposed consisting in a uniform distortion of the cross-sections. The equations of equilibrium in the final state of deformation in combination with the incompressibility condition provide a set of governing equations. These coupled homogeneous partial differential equations with variable coefficients are solved by the Frobenius method assuming for simplicity that the material of the tube is of the Neo-Hookean type. Upon disregarding all terms except the first two in the series obtained a nontrivial fulfillment of the boundary conditions leads to the characteristic equation of the fundamental problem. In the limit case ofa thin shell an order-of-magnitude analysis provides the value of the critical pressure in agreement with the classical result. For walis of arbitrary thickness a numerical analysis carried on with the aid of an electronic computer furnishes a relation between the relative wall thickness and the critical strain displayed on a diagram. 1. GENERAL
EQUATIONS
Let us consider an infinitely long tube of perfectly elastic isotropic material and circular cross-section, the internal and external radii in the undeformed state B, being respectively equal to A, and A,. The tube is subjected to a uniform hydrostatic pressure 4 on the external surface this being the pressure on the actual area directed normal to the surface at any stage of deformation. The pressure produces a finite axially symmetric deformation of the tube (state B) so that the inner and outer radii become respectively a1 = &A,* a2 = IL2& WI where pi, i = 1, 2, are unknown constant coefficients. We refer the tube to a cylindrical coordinate system r, 0, z in the deformed state B, so that assuming a plane strain deformation the point actually at r, 8, z in B was initially at R, 0, z in B,. It follows that the fundamental metric tensor in B is?
t Hereafter we use the notation probounded in Ref. [12] with some modifications proposed by Truesdell
Stability
of an elastic
wall thickness subjected fo an external pressure
circular tube of arbitrary
145
while before the defo~ation (1.3) where (1.4)
Q(r) = R/r and it was assumed that, for simplicity, the material of the tube is incompressible,
(1.5)
Q@ + &I) = 1.
The last ass~ption is not essential and may be removed at the expense of computational labor. An alternate fo~ulation of the incompressibility condition leads to a relation between the coefficients i”i, i = 1,2,
(1 - /m4/~2)2.
&=l-
(1.6)
The strain invariants presently become I = II =l+Q2+$
(1.7)
while the third invariant III = 1 by virtue of the incompressibility of the material. The stress components in the body B become
where
p
ril = Q26, + (1 + Q’)Jt + p,
7
p
1
=
223
=
*31
=I
0
9
may be treated as an (unknown) hydrostatic pressure and (1.9)
W = W(I, II) represents the stored energy density function. The equations of equilibrium of the medium in the state B are rfk (1.10) k = 1,2,3, .,i = 0 3 where the double semicolon denotes covariant differentiation with respect to gij and r, 8, z. We get finally rll = -L(r), 7
(
r2~22= Q2+
$
$3 = (1 - Q’)#
I
>
(1.11)
(q5f I& --L(r),
-
1 - -$ (
II, -L(r), >
i
146
J, L. NOWINSKI and M. SHAHWFQOR
with (1.12)
In deriving (1.11) the boundary condition r’l = 0
at
I = a,
(1.13)
was used. The second boundary condition ~~r=--q
r=a,
at
(1.14)
enables one to express the pressure 4 in terms of the unknown qu~tities one gets
a, i = I,2 and so (1.15)
4 = %r,). Upon using (1.5) it is easy to convert the integral (1.12) into the integrable form
(1.16) with Q1 = (AI/a,) = l/p1 and Qz = (&/a,) = 1,‘~~.Assuming for a moment that the sum (# + il/) is independent of Q one obtains finally (1.17) and from (1.15) the external pressure in terms of pI and ya, (1.18) We now consider a secondary infinitesimal deformation of the body B associated with the displacements wb i = 1,2,3, such that WI E w, = u(r, e),
w2 E wg = u(r, 0),
w,j = wz = 0.
(1.19)
This additional deformation carries the body into the state B’ and induces additional stresses @ and changes of the metric tensor represented by - 2vjr
U,@+ v,* - 2v/r
0
2(v,6 + ur>
0 , 0i
0
(1.19.1)
Stability
147
wall thickness subjected to an external pressure
of an elastic circular tube of arbitrary
The incremental invariants associated with the transition B --) B’ become
(1.20)
while in view of the incompressibility of the material the invariant III’ = 0; this condition is equivalent to 1 (1.21) @,r + f o,, + ; = 0. The incremental stresses are represented by where gij
= f*GU _ C;*@Q;,,
(1.23)
II/’= FI’ + MI’,
(6’ = AI’ + FII’, CT2W
A=zaIZ’
B=+,
iT2W
p’ is a hydrostatic p~udo-press~e function to be determined later. Longer calculations lead to the following expressions for the incremental stresses, $11 = Y11%r f P’, r 122
_ -
f
133
_ -
Y33b
z
(Y22%,
1 z rl2 _- - 7
+
Yl2
+
P’h
1
$9
7 i
all other stress components being identically equal to zero. The symbols employed in equations (1.24) have the following meaning, y11=2A(Q4-1)+2B
(1.25)
~33=2A~-~)+2~(Q4-~) + 2F Yl2
=
e
-t
P*
I
148
J. L.
NOWINSKI
and M.
SHAHINPOOR
To avoid too many technicalities let us assume at this stage that the material of the tube is of the so-called Neo-Hookean type. The stored energy function is then W(1) = C,(I - 3)
(1.26)
with C, as a material constant. If one requires that in the limit case of an infinitesimal deformation the Neo-Hookean constitutive equation reduces to that of Hooke one obtains the relation C, = E/6. With (1.26) in mind equations (1.9) and (1.23),_, now yield 4 = 2C,,
A=B=F=O,
* = 0,
(1.26.1)
so that the incremental stress components (1.24) become r ‘11 = -2pu,,
+ p’,
z r22= $(2pu,,
‘t
133
=
+ p’), $12
p’,
=
_
, (1.27) !j
u,e
+
v,,
-
2t’
r)
(
’
where (1.27.1) The incremental stress components must satisfy the following equilibrium equations, zt,i!
,.
r$ p + r; $j = 0,
i +
i,j = 1,2,3,
(1.28)
in which the Iyk symbols are constructed from the fundamental metric tensor in the state B and the components g’ij = wiiij + wjiii, see [12], equations (4.1.15) and (4.1.17). The field equation (1.28) corresponding to j = 3 is satisfied identically ; the two remaining equations are reduced to the following final form in the case of the Neo-Hookean material,
- %3
k +v,,.,4Q2 + pie= 0, Q2
(1.29)
(1.30) (~-P,~)+u,ee~-V,e~+P:r=O.
u,,.,Q24 + 2~3,
Upon adjoining the incompressibility
condition (1.21)
1 u,, + _z v,, + 11= 0, r
(1.31)
r
one obtains a system of three coupled homogeneous
partial differential equations with
~ra&il~ty of an elastic circular rube ~~arbitrary
wail thickne~~ su&ected to an e~ternalpres~re
149
variable coefficients, two of them of the second order, involving three unknown functions u, Dand p’ of two independent variables t and 8. To these one needs to apply the boundary conditions which follow from the demand that, simultaneously with the displacement field associated with the state B, there exists a nontrivial displacement field u, v corresponding to the same external load q on r = a 2 ; clearly, the surface r = a I has to remain free of load at all times. This requirement guarantees that for some pressure q (call it critical pressure qc,iJ a bifurcation of the equilibrium takes place in the sense that in concert with the state of equilibrium B there exists an adjacent nontrivial state of equilibrium B’ both states being associated with the same external load. In the actual case then? = (1.32) 0, for P = ffl,fa2, z flk _ Tllg’lk or in view of (1.11) and (1.27) in combination
with (X19.2)
25, Q21.GP + p’ = 0, for I = a1,a2. (1.33) 2V u,, -I- Ii,, - - = 0, r ~ This completes the formulation of the eigen-value problem which is thus reduced to a solution of the system of homogeneous governing equations (1.29)-(1.31) such that the homogeneous boundary conditions (1.33) be satisfied in a nontrivial way. Clearly this happens when the principal dete~inant of the linear system (1.33) vanishes providing the characteristic equation for the proper values of the problem. 2. SOLUTION
OF THE EIGEN-VALUF, PROBLEM
We first note that upon integrating (1.5) one gets Q = i (r2 -t-K)*, where K is a constant which by the incompressibility
(2.1) condition is represented by
K = Ai (1 - ~1:).
(2.2)
A substitution of (2.1) into (1.27.1) yields
(bK2
P‘r = r3(r2 t K)’
a derivative encountered in the governing equations (1.29) and (1.30). We now seek the solution of the eigen-value problem just formulated form,
t For a more explicit derivatiun see [12] or fl l],
(2.31
in a separable
1
J. L. NOWINSKIand M. SHAHINP~OR
150
The geometric interpretation of equations (2.4) is that the surfaces of the cylinder in the conceivable adjacent equilibrium state are assumed to display longitudes corrugations dependent on the angle 8 but independent on z. Hereafter we use only one general term of the series (2.4) and for brevity drop the subscript n. By appropriate transformation it is easy to reduce the system (1.29)-(1.31) to a single homogeneous fourth order differential equation for the variable u,
$!C-V+KU +&
1
-2-s+
F&]-F&+&]
+$[-6r-y+&[-++;-&-(r2~K)2] +g [-5+F+- 4K
d2U 1I( -._L+~_T2 +K1+a82 r2 + [ 1 1 _2_9K+ 4 U 1 (2.5 51 1 1 [ 8rK
(9 + K)’
8K
~-
+u
r2
which upon employing the representation
r4
r2 + K
(r2 + K)2
= O’
(2.4), yields
2n2
K(7 + n2)
K(n2 - 4)
~r2 + K - r2(r2 + K) + (9 + K)’
+ R;
-(5
+ n2) + K(7 + n2f + -r(4 - n2) + 2rK(4 - n2) (r2 + K)2 r3(r2 + K) (r2 + K)3 r(r2 + K)
+ RI
5 - n2
3K(3 + n2) + n4 - n2 - 4 +
+ r 42 (r r2(r2 + K)
+ K)
8K
(r2 -I- IQ3
(r2 + K)2
We now introduce non-dimensional variables r, = r/a2 and & = K/a; = l/,u: transform (2.6) into the following form, r,4R:V + riPl(r*)R;”
f r$P2(r*)R;
+ r*P3(r*)R;
+ P,(r,)R,
(2.6)
= O’
- 1 and
= 0
(2.7)
with
(5 - 2n2)ri - (7 + n2)K* + i (n2 - 4)r,2
P, = b
I
I [
P, = -!_ - (5 + n2)ri + (7 + n2)K* -t-f(4 G [
- n’)rz + 3(3 + n2)K, + i(n4 K, =
ri + K*.
,
1 1
- n2)r,4 + i2(4 - n2)rzK* , r - n2 -4)r$
+sr$K* 4
’ (2.8)
,
I
Stability
ofan
eiastic circular tube
ofarbitrary wali thickness subjected
to an external pressure
151
It is to be expected that under an external compressive load, as is the case in the present situation, we always have 0 < pl, y2 < 1. Hence K, > 0 and consequently K, > 0, so that the functions Pi(r*), i = 1, . . , , 4, in (2.7) are analytic in the neighborhood of r* = 0. The only singular point is then r* = 0 which, in fact, does not belong to the domain of definition of the function R,(r,). This stems from the fact that, by hypothesis, the inner surface of the tube is free of external load so that the domain of definition must be at least punctured at r* = 0. By known theorems [13], p. 396, the singular point r* = 0 of the equation of the form (2.7) is a regular singular point and it is possible to obtain, in the neighborhood of the singularity, explicit developments of the fund~ental solutions convergent for sufficiently small values of r,. We get such formal solutions of equation (2.7) by using the Frobenius method, i.e. representing R,(r,) in the form
where the parametric index p is obtained from the indicial equation P(P - I) (P - 2) (P - 3) + P(P - I) (P - 2) P,(O) f P(P - 1) P2(0) + P,(O)
+ P&$(O) = 0,
(2.10)
or explicitly from, P4 -4~3-(2+~z~~2~2(6+~2)~+3(3+~2)=0.
(2.11)
This equation has all four roots real and equal to 3, - 1, 1 + (4 + a”)*. It is easily seen that the case corresponding to n = 1 represents a rigid translation of the cross-sections of the tube, so that the first mode of instability of equilibria is associated with the value tl = 2. In what follows we confine the analysis to the mode n = 2 only. With this in mind the roots of the indicial equation in decreasing order become, p1 = 1 + 2y/2,p2
= 3,p, = -l,p,
= 1 - 242.
(2.12)
Since p2 and p3 differ by a positive integer they give rise to a formally identical solution and the solution corresponding to p3 must be replaced by an appropriate solution of the log type. As a result the function R,(r,) is expressed by the following equation, R, = A
c
m (- lPV(3) m (- l)“F,(3) v- 1 Jb(3 + v) yg+3 + BC1nr* - 16] cv=. jb(3 + v) r* v=o (2 13)
+C
*
where A, B, C and D are integration constants to be determined from the additional conditions, Jb(P + v) = UP + 1) f*(P f 2) &(P + 3) f - . _LAP+ 4,
(2.13.1)
f(r*, p + v) = (P + v)(p f v - l)(p + v - 2)(p + v - 3) + Pl@,)(P + v)cp f- v - l)(P + v - 2) +
P2h)co
+ +
P3@*)
$(P (P
+
v
+
4
-
1) +
P*@*k
(2.13.2)
152
J. L.No~r~mand
M.SHAHINPOOR
and
F,(P) =
f,(P f
v -
1)
.fAP +
v -
2) . . f fv-1@+1)
.MP
v -
1)
fAP
v -
2) . .' L-2(P
+
1)
.LIW
2). . . .A-&
+
1)
"fv-z.(P)
+
+
.A@)
0
OfXP
0
0
.I. .
0
0
. . . MP 3- 1) f,(P)
+
v -
-
(2,13.3)
Clearly f, (p + v) is equal to zero for N odd. For N even, say N = 2n, one obtains the
following relation, fin&J + v) = g
[20a - s/? - 4y + F(n)],
*
(2.13.4)
where a = p + v, fl = (p + v) (p + v - l), y = (p + v) (p + v - 1) (p + v - 2),
(2.13.5)
and F(n) is determined from
c
O”---rpF(n) (-1)
n=O
KS
20 = P.&T*) = 21 --r: K*
36 + Fr; *
60 - --& KS
Bearing in mind equations (2.4) and (2.13) it is straightforward equation (1.31) the function
(-1)“‘(2
92 + j-$$ *
- .._.
(2.13.6)
to find from the governing
+ 2 J2 -t- v)F,U + 2 J2)rz+zj2+” * 2Jb(l + 242 i-v) (2.14)
v=o We refrain from quoting explicitly the unwieldy expression obtained for R,(r ). Since the convergence of the series obtained seems to be fairly rapid, hereafter, in particular in numerical calculations, we confine ourselves to the first two terms of the series only. This procedure yields (2.15.1)
R, = A[r:]+ R, = A[-2r4,]
+ B [-+]
-t
Cl-(1
+ J2)r;+y
+
D[-(1
- t/2)r;-2Jz],
(2.15.2)
Stability
ofan elastic
+ D [-r;z-zJ2(
circular tube of arbitrary
waII thickness subjected to an external pressure
rz ?;;;c,)
- 12J2)
+ k;(l4
- r;zJz(14
153
(2.15.3)
- 8J2)l.
As remarked earlier the characteristic equation of the problem is obtained from the principal determinant of the homogeneous boundary conditions (1.33) upon substitution of the solution (2.4). For the mode of instability considered these conditions yield, 2Q24R; + R3 = 0, (2.16)
for I = ai, a2 -2R,
•l- Rf2-$R2
=O,
with Q from (2.1). It is convenient to introduce the non-dimensional variables r*i = al/u2 and r*2 = a2/a2 = 1. With these in mind the boundary conditions become,
A 14K* +
+~]~C[-10r~~2-~OK~+4r;j2+~~~2~~~)ri_2i2]
+I)
-
1orp2-
r,-1”- 2 J2
lOK* - 4K* J2 +
&.+ K*=f-4 2&
(2.17.1)
1 -lo-lOK,+4,/2K,-&
A[-
14r:,]
+ B
-fzrsl +g
*I
= 0,
(2.17.2)
-~C(-6-2,/2)r~~~J~ 1
1
+ D(- 6 + 2J2)r;;2J2 A[-- 141 + B[33] + C[-
6 - 2,,‘2] + D[- 6 t- 2421 = 0.
The vanishing of the principal determinant
(2.17.3) (2.17.4)
of the foregoing system gives
det A, = 0, where the following notation is used,
= 0,
(2.18)
J. L.
154
2K*G 1
A,, = 14K, +
A,,
t-2, + K,’
= -lOr*‘f’-
lOK* + 4K* J2 + [
A,, = - 10r;fJ2 A,,
= 14K,
and M. SHAHNPOOR
NOWINSKI
+ ~
-
lo& C
2rZ 1 ril
+
K,
2r’1
- 4K, J2 +
1 1 -2i2
r*l
d1 + K*
-2-2 r*l
J2
’ J2
y
2Ks
1 t-K,’ A,, = -66 - 68K, - 16 2K5 - 4 1 + K* ’ A,,=
-lo-lOK,
A,,=
-lo-
-442%
(2.19)
-&,
lOK, +442K,
-&, *
A,,
= -14rz,,
A,,
= -2--
A,,
= (-
+12_ r*l’ r*1 2J2)r::2J2, 6
A,,
= (-
6 + 2J2)riT2J2,
A,,
= -14,
A,,
=33,
A,,= A,,
ln r*1
-6-2J2, = -6
+ 2J2.
In what follows it is expedient to introduce the notation IC= (a2 - al)/a2. 3. THIN
SHELLS
If the thickness of the shell 6 = A, - A, is small as compared with the radii of the initial cross-section the shell may be treated as a moderately thin shell. For such shells it is legitimate to neglect the transverse deformations so that a, z a, + 6. In view of this the radial displacements of the inner and outer surfaces become indistinguishable from each other so that denoting the displacements by u we get a, x A, - u and a, z A, - u. Assuming that both 6/u, and u/al are negligible as compared with 1, we obtain K,
x 2uJu, and rc2 z d2/af.
(3.1)
For moderately thin shells their deflections, in particular at the beginning of buckling, are believed to be very small as compared with their thickness. It is reasonable, therefore, to consider K, and IC’as quantities of like order of smallness and, in the limit case of a thin shell, to neglect in the pertinent equations all terms involving products of K* and JC~or
Stability
of an elastic circular tube of arbdtrary wail thickness subjected to an external pressure
155
powers of K and K higher than the first and second, respectively. Similarly it is consistent to retain no more than three terms in the expansion
With these simplifications in mind the terms of the characteristic equation (2.18) become
4,
= 16&t,
A,, = -2 - 132K - 4tc - 61~~~ AI3 = - 12 - K,(8 + 4 ,/2) + (4 ,,/2 - 6)~ + (91 - 42) A,,
=
-
K’,
12 - & (8 - 4 ,,‘2) + (-4 42 + 6)~ -I- (91 + J2)rc2,
AzI = 16K,, A,,
= -2-132&,
A,,
= - 12 - K,(8 + 4 JZ),
A,,
= -12
- K*(8 - 4J2),
(3.3)
A,, = -14(1 - 3lc + 3rc7, A,,
= 3311 + K + rc2),
A,,
= (-6
A,, = (-6 A,,
= -14,
A,,
= 33
- 2 42) + 14(1 f J2)rc - 14(2 + +‘2&“,
:
+ 242) + 14(1 - ,/2)tc - 14(2 - 42)x2,
A,, = -6 - 2J2, A,,=
-6+2,/2,
J
so that equation (2.18) finally becomes rcZ(-K2 + 10-4 K + I(* + 10-4) = 0.
(3.4)
One solution of this equation, I? = 0 must be discarded as trivial ; to find the second solution we neglect the terms involving 10R4 as small? of higher order than rc2 or K*. This yields fkrally the critical condition
K* = K2.
(3.5)
To find the critical outer pressure CJfor a thin shell we insert the relation (3.5) into equation (1.18) and considering higher powers of K negligible as compared with the lowest powers of K appearing in the polynominals obtained we finally get 4 wit = Erc3/3.
7 Order of magnitude of K may be taken as
IO-‘.
(3.6)
156
J.L.
NOWINSKI
and M.
SHAHINPOOR
This result agrees with the classical result, see e.g. [14], if in the latter the material of the shell is assumed incompressible (i.e. Poisson’s ratio taken equal to one half) and the mean radius set equal to a, which is admissible for thin shells. 4. NUMERiCAL
ANALYSIS OF THICK SHELLS
For shells of arbitrary thickness no closed form solution of the type (3.6) could be found. Instead a numerical procedure of solving the characteristic determinant has been devised using the digital computer IBM 1620. Its final product is the diagram displayed in Fig. 1,
,” I4” II
x
o-5
06
0.7
0.8
0.9
I.0
Fro. 1. Deformation ratio vs. initial radii rate.
illustrating the dependence of the deformation ratio ,+ = a,/A, on the thickness ratio of the tube Al/A,. For compa~son the curves displaying the classical solution for thin shells and the result obtained in [ 1l] for thick-walled spherical shells under external hydrostatic pressure are given. It is seen that the graph for the thick~walled cylinder exhibits the same features as that for the thick-walled sphere, the stability of the mode investigated in [l l] for the thick-walled sphere (for the same wall-thickness) being smaller than the stability of the mode investigated in the present paper for the thick-walled cylinder. Figure 1 also displays the limit case when the deformed inner radius of the structure a, goes to zero. The pertinent relation can be found immediately from the incompressibility condition and is &Y’ Z Cl - @,,W2]”
(4.1)
&Ph = El - @WA3]“.
(4.2)
for the cylinder and
stability
of an elastic circular tube of arbitrary
wall thickness subjected to an external pressure
157
for the sphere. It is clear that in al1 cases J.$@> @‘I so that the mutual location of the diagrams in Fig. 1 found both for the spherical and cylindrical shell appears to be justified. It is of interest that both in [l l] and in the present case the computer program used was unable to supply data for the interval of thickness-ratio from about 0.8 to about 0.95 (or even to 1.0 as in [i 11). The singular character of this interval deserves certainly a separate investigation. The same is true of the upward trend of the curves associated with the decrease of thickness ratio Al/A, in the interval from 0 to approximately 0.8.
[l] M. A. BIOT,Internal buckling under initial stress in finite elasticity. Proc. R. Sot. A27, 306-328 (1963). [2] J. WESOLOWSKI, The axially symmetric problem of stability loss of an elastic bar subject to tension. Archwm Me&. stosow. 15, 383-395 (1963). [3] E. W. WILKES, On the stability of a circular tube under end thrust. Q. JI Me&. appi. Mafir. 8,88-100 (1955). [4] A. E. GREEN and A. J. M. SPENCER, The stability of a circular cylinder under finite extension and torsion. J. Math. Phys. 37,316338 (1959). is] S. LUBKIN,Dete~ination of buckling criteria by rnin~~a~on of total energy. Inst. Math. Sci. New York Univ., Rep. IMM-NYU241 (1957). [6] C. B. SENSENXG,Knstability of thick-elastic solids. Communspure appl. Math. 17,451-491 (1964). (71 H. E. ROAD,On the stability of equilibrium of thick- and thick-walled isotropic elastic solids. Ph.D. Dissert., University of Delaware (1964). [8] GLJOZHONG-HENG,The problem of stability and vibration of a circular plate subject to finite initial deformation. Archwm Mech. stosow. 14,239-252 (1962). [S] Guo ZHONG-HENG,Vibration and stability of a cylinder subject to finite deformation. Ar&wm Mech. stosow. 14, 757-768 (1962). [lo] Z. WESOLOWSKI, Stability of a fully elastic sphere uniformly loaded on the surface. Archwm Mech. stosow. 16, 1131-1150 (1964). [ll] Z. WESOLOWSKI, Stability of an elastic, thick-walled sphere uniformly loaded by an external pressure. Archwm Me& stosow. 19, 3-44 (1967). [12] A. E. GREENand W. ZERNA,Theoreticai Elasticity. Clarendon Press (1954). [I 9 E. L. INCE, ~rd~~a~_v ~~~~re~?ja~ ~~af~o~. Dover (1956). 1141C. TRUESDELL and W. NOLL, The non-linear field theories of mechanics. EncycI. Phys. IU/3. Springer (1965). I151 A. S. VOLMiR,Flexible Plates and Shells {in Russian). GITTL (1956). (Received 29 April 1968)
R&u&--Durant la derniere decermie il est apparu clairement que la perte de stab&t& peut se prod&e non seutement dams des corps minces mais aussi bien dam des corps Cpais ou & bords epais. On &die dam cet article la stabilite dun cylindre de revolution in~niment long, creux et d’epaisseur arbitraire. On consid& Ie cylindre fait en mattriau incompressible et soumis ii une pression exteme. Gn superpose B une deformation flnie et sym& trique du cylindre une deformation secondaire infmitesimale fonction des coordonn&s polaires On obtient un systeme de trois equations couplCes aux derivees partielles du denxieme ordre pour trois fonctions inconnues; ce systeme est resolu par la methode des series de Frohenius. Dans le but de satisfaire les conditions aux hmites dune fawn non triviale, on obtient une equation ~~~t~ris~que pour un parametre de d&formation. Fn supposant le mat&iau Neo-Hookeen on analyse une solution approximative contenant les deux premiers termes, Dans le eas limite d’un cylindre mince on trouve que la pression critique est en accord aver le resultat classique bien connu. On don& egabment une analyse numerique pour des cylindres Cpais.
Zusammenfassung-lm letzten Jahrzehnt wurde deutlich dass Stabilitiitsverluste nicht nur in schlanken sondern such in dicken und dickwandigen Kbrpem auftreten kbnnen. In der vorliegenden Arbeit wird die Stab&at eines unendlich langen Kreiszylinders mit beliebiger Wandstlrke untersucht. Das Zyhndermateriaf wird als inkompressibel ~geno~en, und der Zylinder ist liusserem Druck unterworfen. Einer endlichen, symmetrisch~ Verformung des Zylinders wird eine sekundZire in~nitesim~e Verformung tiberlagert, die von der Radial- und Tangentialkoordinate abh&tgig ist. Dabei erhglt man em System von drei gekoppelten, partiellen Differentiaigleichungen zweiter Grdnung Rlr drei unbekannte Fnnktionen. Zur L&sung wird die Reihenmethode uach Frobenius benutzt. Die nichttriviale Erfiillung der Randbedingungen ftibrt zu einer charakteristischen Gleichung
158
J. L. NOWWSKIand M.
SHAHINFOOR
fXir einen Ve~ormungsparameter. Mit der Annahme eines “Neo-Hookean” Materials wird eine durch zwei Glieder angengherte Lijsung untersucht. Im Grenzfall diinner Schalen ergibt sich ~bereinstimmung fiir den kritischen Aussendruck mit dem bekannten klassischen Ergebnis. Eine numerische Rechnung fiir dicke Schalen wird ebenfalls angegeben. ABCT~WI-B
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