Localized axisymmetric plastic buckling deformation in cylindrical cosserat shells under axial compression

International Journal of Plasticity, Vol. 3, pp. 193-210, 1987

0749-6419/87 $3.00 + .00 © 1987 Pergamon Journals Ltd.

Printed in the U.S.A.

LOCALIZED AXISYMMETRIC PLASTIC BUCKLING DEFORMATION IN CYLINDRICAL COSSERAT SHELLS UNDER AXIAL COMPRESSION

H.

RAMSEY

The University of British Columbia Abstract-The theory of a Cosserat surface is applied in the analysis of localized axisymmetric buckling deformation in cylindrical shells under axial compression. The material is treated as rigid-plastic with isotropic work hardening. It is found that neither deformation due to transverse shear nor work hardening of the material has significant effect on the shape of the buckling mode. Localization of the buckling deformation is governed by a characteristic length formed from the shell radius and thickness.

I. INTRODUCTION

A striking feature of plastic buckling of axially compressed cylindrical shells of moderate thickness, shells with a radius/thickness ratio of less than 40, is that buckling occurs in an axisymmetric bellows, or concertina mode, in which buckles form and collapse one at a time. Extensive experimental studies of this phenomenon have been presented by ALLAN [1968], M ~ I s • JOHNSON [1983], and ANDREWSet al. [1983]. An elementary analysis of the folding mechanism in the finite deflection range was given by ALEXASDER [1960], and more recently, WI~RZmCKI ~, BrtAX [1986] have presented a more refined analysis. In uniform cylinders, the first buckle usually forms at one end and appears to be initiated by friction between the loading plate and the end of the specimen that restrains the outward radial expansion that accompanies compression. To eliminate the uncertainty associated with the boundary conditions in an experiment, R~SEY [1980,1981 ] tested cylindrical shells with an artificial axisymmetric imperfection well away from the ends. These latter papers included linearized analyses, based on elementary shell theory, of the initial stages of buckling, which attempted to establish the mechanism for the localization of the buckling deformation. Since the ratio of thickness/wavelength in the buckles is relatively large, deformation due to transverse shear could be important. In the present paper, the theory of a Cosserat surface, which includes transverse shear deformation, is applied to study the localization of plastic buckling deformation in axially compressed cylindrical shells. I1. KINEMATIC DESCRIPTION OF A COSSERAT SURFACE

The current deformed configuration of a Cosserat surface is specified by a position vector r of a material point on the surface and by a director displacement vector d. Both r a n d d are functions of convected coordinates 0 ~ (a = 1 , 2 ) o n the surface and of time t. Referred to the convected coordinate system 0 ~ in the current configuration, surface base vectors are denoted by a~, a n, the surface metric tensors by a~a, a ~B, and the coef193

194

H. RAMSEY

ficients in the second fundamental form by b,~a. To complete the set of base vectors, a unit vector a3 = a 3 normal to the surface is introduced such that [ai} =

[a,,a3},

[a i} = { a ~ ' , a 3 }

form right-handed systems. The initial values, in a reference configuration, of r, d, ai, a i, a,a, a ~ , and b ~ are denoted by R, D, Ai, A ~, A ~ , A ~ , and B ~ , respectively. The set o f thirteen kinematic variables,

where Greek indices have the values 1, 2, and Roman indices the values 1, 2, 3, are the strains. The extension of the surface is described by the components e~a, where 2e,~ = a ~ - A,,a.

(1)

The strain components x~, which relate to bending and transverse shear deformation, are defined in terms of additional kinematic variables X~, where

A ~ = d~l ~ - b ~ d 3 ,

A3~ = d3.~ + b~d~

(2)

and di = d.ai

b~ = a~Vb~.,. The vertical stroke [ denotes covariant differentiation with respect to a~a, and the comma denotes partial differentiation with respect to 0 ~. In the reference configuration, Ai,,, di, b~ have the values Ai~, Di, B~. Then, Kic~ ~- /~iol -- A i ~ ,

(3)

The strain components ~i contribute to transverse shear deformation and thickness change and are the increments to the components of the director displacement vector, ~i = d i - D i .

(4)

IlL STRESS C O M P O N E N T S , LOADING FUNCTION. AND CONSTITUTIVE RELATIONS

In the general theory o f an elastic-plastic Cosserat surface (GREEN et al. [1968]) and in the theory of superposed small deformations on a large deformation of an elastic Cosserat surface (G~a~Er~ • NAGrmX [1971]), stress components are referred to current arc length s, current area a, and current base vectors a;. However, there is considerable computational convenience in buckling problems, in which buckling is treated as a small deformation superposed on a large deformation, to refer stress components to arc length S and area E in a reference state, and to current base vectors ai ( R ~ s E Y [1987]). Accordingly, the contact force N and contact director couple M are referred to arc length

Plastic bucklingof compressedcylindricalshells

195

S in a reference state, and the intrinsic director couple m is referred to area Z' in a reference state. The stress components {N "i, M=i, m i] are now defined by the relations N = Naiv,~ai,

M -- Mc~iv~ai,

m = miai

(5)

where

and r is the outward unit normal vector, in the reference state, associated with a smooth closed curve comprising material points on the surface along which S is the arc length. The mechanical power 6) per unit area in the reference state is then given by 6) = ~ ' o ~ , a + M~'iiq,~ + mini,

(6)

~,~a = ~ , ~ = N~,~ _ m~,d ~ _ M w , A .a

(7)

where

and d '~ = d . a ~',

,A?a = a ' % ~ .

(8)

The superposed dot denotes differentiation with respect to t, the coordinates 0 ~ being held constant. In the general theory of an elastic-plastic Cosserat surface, plastic strains are introduced, in addition to the strain components defined by the kinematic relations, eqns 0)-(4). Here the idealization of rigid-plastic material response is introduced at the outset, so no distinction between the two sets of strains need be made. Constitutive relations for the strain rates in a rigid-plastic Cosserat plate were developed by RAMSEY [1986]. These equations, which model the Von Mises yield condition and Lgvy-Mises flow rule in three-dimensional plasticity and which include isotropic work hardening, have the general form

~.a= A af

c~,"--"~ ,

.

af

xi~, = A OM~------7,

~;=A af

Orni

(9)

where

(lO) H, the hardening modulus, is the slope of the tangent to the uniaxial stress-strain curve, h is the plate thickness in the reference state, and f, the loading function, is a quadratic form in the stresses {P7ai, M ~i, m i }. The form o f f for axisymmetric deformations in a cylindrical Cosserat shell is now specified for a particular choice of reference state, namely, a rigid stress-free state in which the shell thickness is uniform. The shell may have undergone previous finite

196

H. RAMSE¥

homogeneous deformation. For a shell of uniform thickness in the reference state, it is appropriate to put O~, = 0 ,

D 3 = 1.

(11)

Then

A,~ = -D3B~,~

(12)

is symmetric. The loading function defined by 2 f = (~IA,~aA.~ + 2~'2A~,~Aa~)N"~/V ~ + ~3A,~m~'m ~ + ~'4(m3)2 + (~sA,~A.~ + ~6A~,.~A~ + ~TA~A&~)M~';3M "~

(13)

+ ~sAo,~M~'3M ~3 + 2~9A~,2N°';Jm3dj -l + 2~IoA,~aA.~N"3M "y~ is invariant under transformation o f the surface coordinates 0 ~ and is also invariant under reflection o f the director, D --, - D and d --, - d . When B ~ = 0, f as defined by eqn (13) reduces to a form appropriate for a fiat plate. The values for the coefficients g'~. . . . , ~'9 determined by RAMSEV [1986] tO describe the buckling of a rectangular plate under uniform uniaxial compression are appropriate for the present problem as well, in which small buckling deformation is superposed on uniform uniaxial compression. These values are:* 1

1

7

2

(14)

= - 1 6 / h 2,

= ~ = 24/h 2.

A value for ~'8 was not determined. However, this coefficient governs nonuniform thickness change, an unimportant effect in buckling deformations when bending predominates. The presence o f the current value d3 of the normal component of the director displacement vector, in the term with coefficient ~'9, incorporates the dependence of f on plastic strain. The dependence o f f on d3 is required, along with the term in ~'10, to account for the change in radius in a cylindrical shell produced by axial load. The coefficient ~'~ois determined in Section IV. The condition ~'6 = ~'7, together with the symmetry condition A,,a = A~,~, implies that f depends just on the symmetric part of M ~a. As a result, the flow rule, eqn (9), yields k~,a = kay,. Hence, while f, as defined by eqn (13), is suitable for the axisymmetric deformations presently considered, it may not be suitable when there is asymmetric bending. IV. U N I F O R M FINITE D E F O R M A T I O N UNDER AXIAL LOAD

In the reference state, the 0 ~ coordinate curves are taken as straight lines parallel to the axis o f the cylinder, and the 0 2 c o o r d i n a t e curves, as an o r t h o g o n a l s y s t e m o f *The result ~'3 = 5/3 reported by RAMSEY [1986] is in error.

Plastic buckling of compressed Cy[indricalshells

197

closed circles. The surface metric tensors, in the reference state at time t = 0, are chosen as

:__ [.o 0].

.1,.

Positive directions along the 01, 02 coordinate curves are such that the unit normal vector Aa = A 3 is positive outward. With • denoting the curvature o f the 02 coordinate curves, the coefficients in the second fundamental form are given by

In view of eqn (15), the formulas of Gauss reduce to A~,,~ = B ~ A 3 , which yield At,~ = AI,2 -- Az, t = 0,

A2,2 = - K A 3 .

(17)

The formulas of Weingarten A3,~ = -B~A.~

yield A3,1 = 0,

A3,2 -- KA2.

(18)

Axial loading is described by putting

N~ =

0

'

where N = N ( t ) is a monotonically increasing function. The resulting uniform deformation is described by the extension ratios k~ = k~(t), k2 = k2(t), d3 = d3(t) in the axial, circumferential, and radial directions. These extension ratios have the initial values kl(0) -- k2(0) -- d3(0) = I.

(20)

Hence, for deformation of the cylindrical shell by uniform finite extensions, al = k i A l ,

a2 = k2A2,

a3 = a 3 = A3,

d = d3a 3

(21)

and

[.2 oj a~a=

k

'

a~a=

0

k2_2 .

(22)

198

H. RAMSEY

The coefficients in the second fundamental form are written as 0 o] 0 -K "

b.~ =

(23)

The circumference in the deformed shell is determined by the extension ratio k2, and hence the curvature of the 0 z coordinate curves. Calculation of a2.z using eqns (17) and (21), and also using the formulas of Gauss in the current state, leads to the result (24)

K = kzR.

Other kinematic results, which follow from eqns (22), (23), and the definitions, eqn (2), are noted: b~ = - ~ k £ 2

b~ = b~ = b? = O,

All

=

A,2

=

A21 = A3,

=

A32 =

(25)

A n = Kd3

O,

l!, =l!2 = l . 2, =A 3, =A32 = 0 ,

A22

:

Kk£2d3

.

The strains associated with the deformation described by eqns (21)-(24) are:

1

e.=~(kl

1

2-1),

r22=rd3-K,

(5, =(52 = 0 ,

e22=5(k22-1),

e,2=0,

Ku = ~ 1 2 = r 2 , =K3, = K 3 2 = 0 ,

(53 = d 3 -

(26)

1.

The function N ( t ) is chosen such that plastic flow begins at time t = 0. When the loading function, eqn (13), is evaluated using eqns (11), (12), (15), (16), and (19), it is found that Of

. . . o)V 22 =

N

. 3'

of . Om 3 =

N d_ ,

~- 3 ,

Of = ~'loKN, aM":

(27)

where, in view of eqns (7) and (19), .N" = N. Then, the flow rule, eqn (9), along with the kinematic relations, eqn (26), leads to the relations, for t >_ 0, 022=kzkz=A

_

,

b3=d3=A

-~dF

1 .

(28)

Hence k2/¢2 = d3 d3

(29)

and, in view of the initial conditions, eqn (20), k2(t) = d3(t).

(30)

Plastic buckling of compressed cylindrical shells

199

From eqns (24) and (26), it follows that (31)

/~22 = (k2d3 + k2d3)l~ Substitution from eqn (27) in the flow rule yields

(32)

/~22 "- A ( ~ ' 1 o ~ N ) .

The coefficient ~'i0 can now be found using eqns (28), (30), (31), and (32). Its value is 2

(33)

g'lO = -- ~ "

Thus the loading function, eqn (13), and associated flow rule, eqn (9), admit a solution for uniform finite deformation of a cylindrical shell in which N I~ is the only nonzero stress component, and the strains are given by eqns (24) and (26). Equilibrium is satisfied trivially. V. FIELD EQUATIONS FOR AXISYMMETRIC BUCKLING DEFORMATION SUPERPOSED ON CONTINUING UNIFORM FLOW UNDER AXIAL COMPRESSION

Kinematic relations and equilibrium conditions describing axisymmetric buckling of a cylindrical Cosserat shell under axial compression were derived previously by RArctSE¥ [1987]. The results are summarized here for convenience. The buckling deformation is described by writing increments to the position vector r and director displacement vector d as ~u, ~b, where ~ is a small positive constant. All quantities and all equations are subsequently linearized with respect to e. For axisymmetric buckling deformation, u = u l a l + w a 3,

b = ' b l a l + b 3 a 3,

(34)

where ul=ut(OI,t),

w=w(OI,t),

bl=bl(Ol,t),

b3 =b3(Ol,t).

Referred to the coordinate system 0 ~ in the buckled configuration, the surface base vectors are denoted by a~ + ea~, a ~ + ea ''~, the surface metric tensors by a,~ + ~a'a, a ~a + ~a '~a, and the coefficients in the second fundamental form by b~a + eb~,~. The unit normal vector becomes a3 + ea] = a 3 + ea 'a. When substitutions are made, using eqns (22), (23), and (24), in the kinematic relations for superposed small deformations on a large deformation of a Cosserat surface (GP~E~r ~ N^Grn3i [1971]), the following results are obtained: a~ = k l - 2 U l , l a l + w, l a 3 a-~ ---- ~ k ~ l wa2

a~ = kl-2w,! a! a~l = 2u U, a l "l -'- W, l l ,

a~2 = 2Kk2 w, b"22 ~ -I72w,

(35) a[2 = O b~2 = O.

200

H. RAMSEY

In the buckled configuration, the strains are written [ e,~ + ~e~,~,

Ki, + eK;,~, 6, + ~6"~I.

Then, from eqns (1)-(14) and (35), the strain increments [e~,~,KL,6; } are given by ell = ul.l,

e;.2 = g k 2 w ,

e;2 = 0,

' ~--"b l , t , KII

' ~---K k 2 b 3 + K 2 d 3 w, K22

" =K;.i = 0 , K12

(36) r,j2 : O,

r~l = b3,1,

tS~ = bl + da w.l,

5; = O,

o~ = b3.

Also in the buckled state, the stresses defined by eqn (5) have the representatio n {N ~ + eN'~'i,M ~'i + e M ' % m i + em'i}. For the axisymmetric deformation presently considered, N '12 = N '21 = N '23 = M ' l z = M '21 = M '23 =

m '2 = 0,

(37)

and it is recalled that N n = .~n = N is the only nonzero stress component in the neighboring state of uniform finite extension. The nonzero components of eqn (7) are ~,u

= N,n,

/~,22 =

(38)

N,22 _ ~kf|d3M,22.

The constitutive relations, eqns (9) and (10), are next evaluated in terms of the reference values, eqns (l 1), (15), and (16), and the state of stress described by eqns (19) and (37). Also, the values for the coefficients given by eqns (14) and (33) are introduced, except for ~'3 and ~'s. The coefficients ~'3 and ~'s are retained in order to trace the effect of transverse shear deformation. The nontrivial components of eqn (9) become On + ~0[l = A

., = e22 + ~e22

N + ~ (2N 'n

A [ - ~N + ~e

33 + e6; = A

-

( 2 / ~ '22 -- d ~ l m '3

d~-lN + ~ (2m '3

-N'n)

-

-~

KM

'22 ],

- d3-l/~ '22) ,

kn + ek~l = A e [ 1 6 h - Z ( 2 M 'll - M'22)],

t~EEq-et~2----A[ 2- ~ K N + e . 1 6 h

_2(2M,22

(40)

(41) (42)

M , II) _

2~ _~_~(~,tl +,~,22) ] ,

(43)

/~13 -I" 6/~3 = A e ~ s M '13,

(44)

~l + e6~ = Ae~3m 'l.

(45)

Plastic buckling of compressed cylindrical shells

201

The loading rate/V enters into the evaluation of A, as specified by eqn (10). Since the constitutive relations, eqns (9) and (10), are homogeneous equations of degree one in the time derivatives, they contain no characteristic time. The loading rate can be specified arbitrarily provided A > 0, this condition ensuring that continuing plastic flow prevails. It is convenient to choose N o t e t. Then = N

(46)

and, as a consequence, the coefficients in the constitutive relations for the superposed buckling deformation are somewhat simpler. To the first order in e, eqn (10) yields A =

3 -

3

[[2/~,11 _ / ~ , 2 2

[22V'zI _ N ,22 _ d3-1/,r/,3 _

_ d;Ith,3

_ 2/~j~,22]

(47)

2KM'22]}.

Substitution for A from eqn (47) in eqns (39)-(45) and partitioning into e-independent and e-dependent sets yields H h ~ n = -2Hhe22 = - 2 H h d 3 6 3 = N,

nhr22 = - K N

(48)

and

Hh~

1 ~,22

-- ~I d 3-I m• ,3 - KAY/'22,

1 = /~,11 -- 2

2 H h ~ 2 = - l ~ TM + ~

+~3 ./~t22 2Hh~ = df I

+

(49)

d ; J r h '3

~3 0'3""1m ,3 - 3 K M '22,

(50)

l --1 m" '3 + ~/~/,22 ] ~ ' l l jr. 21 ~ , 2 2 + ~d3

+ ( 2 - - ~1d 3 -2) m "3 - ~3 d3_1~,22 - d ~ - l K m '22, Hh3~fi = 24(2M TM - M'22),

(52)

Hh3r22 = 2 4 ( 2 M ' 2 2 - M ' I I ) - K h 2

3 .~,22

_ gM,22 + 2

(51)

l

I~,ll- ~l ~,22- ~d3 I --I rn",3

+ 2 d~m,3

]

(53)

+/.7M,22 ,

2Hhi¢]i = 3~'sM '13,

(54)

2Hh~

(55)

= 3 ~3 m " I.

202

H. RAMSEY

In addition to the constitutive relations, eqns (48)-(55), expressions for the strain rates appearing on the left side of eqns (49)-(55) are needed in terms of the displacement components us, w, bl, b3. Differentiation of the kinematic relations, eqn (36), with respect to time yields ell -~- /~l,l,

e22 = / ~ ( / ~ 2 W -1"-k 2 w )

~, = bLl,

t~2 = K ( k 2 b 3 + kzb3) + .Kz(d3 w + d3 ~'~')

(56)

~1 = b3,1,

61 = b, + d3w, l + d3 fv,,,

~; = b3.

Differential equations of equilibrium complete the set of field equations for the superposed small buckling deformation. They can be obtained by applying Green's theorem on a surface to the global equilibrium conditions:

fa ( N + e N ' ) d S = O ,

fa ( M + e M ' ) d S = r

fz (m+em')d£',

fa (r + eu) x (N + eN')dS + fa (d + eb) x (M + ¢M')dS =O,

(57)

(58)

(59)

where the surface integral on the right side of eqn (58) extends over an arbitrary simply connected region $" with a smooth boundary aS, and the line integrals are taken around the boundary a S in the positive sense with respect to a3, and where N + eN'

=

( N c~i + ~N'c~i)(ai + ea:)t,~,

(60)

M + ~M' = ( M ai + ~M'~'i)(ai + ea:)v~,

(61)

m + era' = (m i + em'i)(ai + ca'i).

(62)

The condition of force equilibrium, eqn (57), yields two nontrivial components: N'f

I11 + ki-2Nua,11 = 0,

N'113 - k21~N "22 + Nw.ll = 0,

(63) (64)

when eqns (15)-(19), (21), (24), (34), (35), and (37) are noted. Similarly, the condition of director couple equilibrium, eqn (58), yields two equations,

M'~ 11 -- m ' x,

(65)

A l l 13 - k z K M "z2 = m '3.

(66)

Plastic buckling of compressedcylindrical shells

203

The condition of moment equilibrium, eqn (59), reduces to just one equation when eqns (57) and (58) are used: N ' 13 _ d3 m' I = 0.

(67)

There is no constitutive equation for the stress component N't3 appearing in eqns (64) and (67). However, N ' 1 3 can be eliminated using these two equations, with the result d3m', i -

R2a~N '22 + N W , l I = O.

(68)

YI. SOLUTION FOR SINUSOIDAL BUCKLING

The extension ratios k~, k2, d3 appear in the coefficients of the field equations, along with the extension rates k2, d3. Even in moderately thick cylindrical shells, of the common ductile metals, the uniform axial compressive strain would be, at most, on the order of a few percent at the maximum value of the axial compressive load. Accordingly, the extension ratios would differ from unity by only a few percent, and the approximation k~(t)

=k2(t) =d3(t) = I

(69)

is now made in the coefficients of the field equations. The extension rates k2, d3 can be found using eqns (26), (30), and (48). When k2, da in these expressions are approximated by unity, these expressions reduce to ]~2 = d3 =

N

2tfh "

(70)

The growth of the buckling deformation is followed for a small time interval t - tt 0, where t~ _> 0. A new time variable T is introduced, defined by r = (P - P~)/H,

(71)

where -Ph

= N

and PI = P ( t l ) . P can be identified as the uniform compressive stress in the cylindrical shell viewed as a three-dimensional body. The hardening modulus H is treated as a constant in eqn (71), implying that the actual stress-strain curve of the material is approximated by the tangent to the curve at the point where P = P~. The time variable r is the increment in the uniform axial compressive strain over the small time interval that the growth of the buckling deformation is followed. Since N m e t, P ~ e t, and hence 4" = P / H .

(72)

Since the extension ratios have been approximated by the constant value of unity, it is appropriate to treat N and P as constants in eqns (70) and (72), respectively. All of the field equations now have constant coefficients.

204

H. RAMSEY

A solution for sinusoidal buckling in an infinitely long cylindrical shell is now constructed by putting tt 1 = Uer~sinsO ~,

w = WerTcos s0 l, b~ = B2 e ~ sin s0 l, b3 = B3er~cosc~Ol, ~,11 = P h N l e r , cos sO l, ~ , 2 : = PhNze,-~ cos s0 t,

(73)

m,3 = P h N 3 e r ~ c o s s O l, M'll = ph2Mle~cos

a0 t,

M'22 = p h E M z e r r cos sO I, M,~3 = P h 2 M 3 e r ~ s i n s O 1, rn '1 = PhVe~" sin sO 1,

where U, W, . . . . V and r are functions of a. Since .~,11 = N , n , by eqn (38), substitution from eqn (73) in eqn (63), recalling that N = - P h , yields Ni = sU.

(74)

Next, substitution in the constitutive relations, eqns (49)-(51), and in the kinematic relations, eqn (56), from eqn (73), and noting eqns (69), (70), (72), and (74), leads to the results -2KhM2

1

= r ( s U + B3) + ~ (1 + 2r)KW,

/ 3Nz = 1 - 3

\

H

+ r+ 3)sU+

(75)

(1 + 2r)/~W,

(76)

P

H +2r+3 3N 3 = ( ~3 \ P

) aU

t

+ 3rB3 + ~ (1 + 2 r ) K W .

(77)

Substitution of eqn (73) in eqn (52) and using eqn (56) gives directly 24 (2Mj - M z ) = cthrB2.

(78)

Plastic buckling of compressed cylindrical shells

205

Eqn (53) for K~2 yields 48 (2M2 - MI) = /'~h [ ( - 4 -H- + 4 + p

~8 r) c ~ U + ( l + 2 r ) B j + ~ ( l5+ 2 r ) K

W]

(79)

when eqns (75)-(77) are used. The two remaining constitutive relations, eqns (54) and (55), become 3~'sh2M3 = -2ahrB3, 3~3V= 2rBl -- (1 + 2r)c~W.

(80)

(81)

Three of the equilibrium equations have not yet been used. Substitution of eqn (73) in eqn (65) yields -c~hMl = V

(82)

a h V - . K h ( N 2 + KhM2) + a z h W = 0

(83)

and substitution in eqn (68) yields

when eqn (38) is noted. The remaining equilibrium condition, eqn (66), becomes ,~hM3 - g h M 2 = N3.

(84)

Eqns (75)-(84) constitute a system of ten homogeneous equations in ten unknowns, the six stress components Mi, M2, M3, N2, Na, V, and four displacement components U, W, B~, B3. For a nontrivial solution to exist, r is determined as a function of a. When K = 0, this system of equations reduces to those for sinusoidal buckling in a flat rectangular plate under uniaxial compression, treated previously by ~ E ~ " [1986]. Eqns (75), (77), (80), and (84) become an independent set for M3, N3, U, B3, which govern bulging instability. For a nontrivial solution, r is determined as 3(I - H / P ) r = 1 + 2o~2~'8"-t "

(85)

Eqn (85) differs from the earlier result (compare RA~sr~x, [1986], eqn (6.12)), due to the difference in the definitions of stress. The hardening modulus H in the present paper is the slope of the tangent to a stress-strain curve in which N n is referred to arc length in the reference state, while H in the earlier paper is the slope of the tangent to a stressstrain curve in which N ~t is referred to arc length in the current state. While the distinction between the two tangent moduli is insignificant when P / H << 1, it becomes significant when P / H is of order one. In the present case, r > 0, the condition for instability, occurs only when P / H > 1. Also when/7 = 0, eqns (78), (79), (81), (82), and (83) become an independent set for M~, M2, V, B~, W, which describe the bending instability included in elementary plate theory. For a nontrivial solution for bending instability, 1

r = ~ (3~'3 -- 1) + 36~-2h -2.

(86)

206

H. RAMSEY

It is to be noted that eqn (86) is independent o f P/H, and r > 0 for all values of ~, with r ~ oo as h --* 0. Also, recalling that ~'3 = 7/3, r remains positive as h ~ oo. In the case of the flat plate, growth o f the bending instability commences as soon as the plate is loaded into the plastic state, where P / H << 1 for materials with a stress-strain curve with a continuously turning tangent. Hence a compressed plate always buckles in a bending mode, without appearance of the bulging mode. In the case o f a cylindrical shell, K :~ 0, solving eqns (75)-(84) can be simplified by suppressing the bulging instability, which apparently is never observed in experiments. By setting ~'8 = 0, the loading function f, eqn (13), is independent of M ~3, and the flow rule, eqn (9), then requires that k~3 = 0. Hence, by eqn (56), b3.~ = 0, and for sinusoidal buckling, B 3 = 0.

(87)

Thus, putting ~'8 = 0 introduces a kinematic constraint and M3 becomes a reaction determined by equilibrium, eqn (84). Eqn (80) is evanescent. It can be noted from eqn (85), when ~'s = 0, that r is zero, and bulging instability does not arise. When B 3 = 0, eqns (75) to (79) and (81) to (83) constitute a system of seven homogeneous equations for the unknowns Mr, M2, N2, V, U, W, BI. An approximate eigencondition for r is now obtained, in which terms in (/7h)2 are neglected in comparison with terms o f order one. First, eqns (78) and (79) are used to eliminate MI. An equation for M2 in terms of U, W, Bl results. This expression for M2 is then substituted in the left side o f eqn (75), with the result

(88) + 1Kh.uh.rBl

= O,

where

3 = H / P - 1 > 0.

(89)

When K ~ 0, r is given by eqn (86). Hence r is of order one or greater./~h would typically be o f order l0 -~ or less. When terms o f order ( . ~ h ) 2 a r e neglected compared to terms o f order one or greater, eqn (88) simplifies to

r ~ U + ~ (1 + 2 r ) K W +

Khc~h.rBl = 0.

(9O)

The term Kh(N2 + KhM2), which appears in eqn (83), can be expressed in terms o f U, W using eqns (75) and (76), and subsequently in terms of W, B~ when eqn (90) is used. Then, eqn (83) can be written 2

[

3---~3~h.rB1 + [ 1

1 +2r

- 3~'3 -

]

ct2hw-

1( r)]

+ ~rr 13+ ~

(1 + 2 r ) K 2 h W = 0

(91)

Plastic buckling of compressed cylindrical shells

207

when eqn (81) is used for V, and terms in (Kh) 2 are neglected in the coefficient of (txh.rBi). A second independent equation for W, Bl can be found by solving eqns (78) and (79) for MI and using eqn (90) to express the result in terms of W, BI. Another expression for MI in terms of W, BI can be found from eqns (81) and (82). The result _

4

+

~2h2 rBI + ~

(1 + 2 r ) ¢ W = O

(92)

then follows from equating the two expressions for M I and dropping terms in ( z ~ h ) 2 in the coefficient of (~ W). Eqn (91) contains the term of leading order in K, and eqns (91) and (92) revert to the case of a flat plate when K = 0. It should be noted that the procedure in obtaining eqns (91) and (92) has not involved division by K or cancellation o f / ~ as a common factor in any intermediate equations. For eqns (91) and (92) to have a nontrivial solution, r must satisfy the quadratic equation (o~4h 4 -t-

12/~2h2)r 2 -

[1

]

~ (3['3 - 1)'v4h 4 + 36ot2h 2 - 6(I + 6/3)~2h 2 r (93)

+ 1 8 ~ 2 h 2 = 0.

In obtaining eqn (93), terms in f~2h2 have been dropped in the coefficient of ot2h 2. It is convenient now to scale tx by putting =

31/4(I~/h)I/2~.

(94)

Then the roots rl, 1"2of eqn (93) can be written

l

]

rl = (~4 + 4)-1 12.31/2(~h)-1~2 + 2 (3['3 - 1 ) ~ 4 - 2(1 + 6/3) + O(/Th),

(95)

r2 = 0 (Kh).

(96)

Since terms in ,~2h2 have been dropped in comparison with terms of order one in obtaining eqns (95) and (96), the terms of order Kh must now be dropped in the expressions for ri, r2, since the leading term is of order (/Th) -l. The root r2 is then zero and does not contribute to instability. For a thin shell, ~ h ~ 0 and rl is given essentially by rl = 12"31/2(~h)-1~2(~ 4 + 4) -1.

(97)

In this case, instability is unaffected by the shear coefficient ['3 and the hardening modulus H. For a thicker shell, since (3['3 - 1) > 0, shear deformation has a destabilizing effect, while work hardening of the material is stabilizing. Vll. SOLUTION FOR LOCALIZED BUCKLING

Of the eleven dependent variables listed in eqn (73), the radial displacement w is most readily observed. For nonsinusoidal buckling, when w is an even function of 0 i, w can be expressed as a Fourier cosine integral:

208

H. RAMSEY

W(X,r) = (2/~') 1/2

er~W(~)cos (~x) d~,

(98)

where x = 31/4 (K/h)1/201

(99)

and W(~) is the Fourier cosine transform of w at time tl. Eqn (98) governs the evolution o f the mode shape as buckling progresses. A typical range of values o f r would be 0
10 -3,

and, when eqn (95) is used, e r, can be approximated by erT = ek~(l + zg),

(100)

where 1

k = ~ ( 3 ~ 3 - l) > 0

OOI)

and g = (~4 + 4)-1[12.31/2(~h)-1~2 _ 2(3~'3 + 6B)].

(102)

The function g ( ~ ) is a well-defined Fourier cosine transform, inasmuch as g = Oil ~]-2) as [~] --, ~ and g remains bounded for 0 _< ~ < ~ . The inverse Fourier cosine transform of g(~) is denoted by G ( x ) , that is, G ( x ) = (2~') -1/2

g(~)e~X~d~.

The inversion integral is easily evaluated by means o f the residue theorem, with the result, for x ~- 0, G ( x ) = -Trl/2Ce-Xsin ( x + 0 ) ,

(103)

where i

C2

=

tant~ =

108(Kh)-2 + 4 (3~'3 + 63) 2, -12-31/2(Kh) -l + (3~'3 + 6~) 12,31~(/~h)_ l + (3~3 + 6~)

Then, using the convolution theorem for the Fourier cosine transform, eqn (98) can be written as

Plastic buckling of compressed cylindrical shells

w(x,r) = ek'w(x,O) + rek'(2r) -I/2

fO°°

w(~,O) [ G ( x + ~) + G ( l x - ~ 1)] d~

209

(lO4)

when the approximation indicated in eqn (100) is used. In the experiments reported by RAMSEY[1980], localized plastic buckling deformation in axially compressed cylindrical shells was initiated away from the ends by means of a local artificial imperfection. The localized buckling deformation observed can be characterized by putting w(x,0) = e -bx:,

(105)

where b is large. When e -~2 is substituted for w(x,0) in the convolution integral in eqn (I04), Laplace's method can be applied to evaluate the integral. Retention of just the leading term in the asymptotic expansion, as b --* ~ , of this integral leads to the result w(x,r) = e*~[w(x,O) + r(2b)-l/2G(x)].

(106)

VIII. DISCUSSION

The results, eqns (103) and (106), indicate that neither transverse shear deformation nor work hardening of the material (which enters through the parameter ~ defined in eqn (89)) has significant effect on the buckling deformation. Localization of the buckling deformation is governed by the factor e -x, which depends just on the characteristic length (h/i~) ~/2. The rate of decay, in terms of distance from the site of the imperfection, obtained here differs by less than 10 percent from the earlier result, RAMSEY [1980,1981], which was based on elementary shell theory. Experimental evidence that the shape and extent of plastic buckling deformation in shells does not depend on the work-hardening characteristics of the material has been presented by RAMSEY[1977]. Geometrically similar steep truncated conical shells of aluminum alloy and stainless steel were loaded by axial compression. Even though the tangent moduli, in the plastic region, of the two materials differ by an order of magnitude, the buckling deformations observed were virtually identical.

REFERENCES 1960 1968 1968 1971 1977 1980 1981

A~EX.'~DER,J.M., "An Approximate Analysis of the Collapse of Thin Cylindrical Shells under Axial Loading," Quart. J. Mech. Appl. Math., 13, 10. AI.L~N,T., "Experimental and Analytical Investigation of the Behaviour of Cylindrical Tubes Subject to Axial Compressive Forces," J. Mech. Eng. Sci., 10, 182. GtzEE~,A.E., NAGHDI,P.M., and OSBORN,R.B., "Theory of an Elastic-Plastic Cosserat Surface," Int. J. Solids Struct., 4, 907. GREEN,A.E., and NAGHDI,P.M., "On Superposed Small Deformations on a Large Deformation of an Elastic Cosserat Surface," J. Elast., 1, 1. RAMSl~Y,H., "Plastic Buckling of Conical Shells under Axial Compression," Int. J. Mech. Sci., 19, 257. RAMSI~Y,H., "Localized Plastic Buckling Deformation in Axially Compressed Cylindrical Shells," J. Eng. Math., 14, 283. RAMS~,H., "Buckling Deformation in Axially Compressed Elastic-PlasticCylindricalShells Initiated by Local Axisymmetric Imperfections," J. Eng. Math., IS, 171.

210

1983 1983 1986 1986 1987

H. RAMSEY

ANDREWS,K.R.F., ENGLAND,G.L., and GRAN1, E., "Classification of the Axial Collapse of Cylindrical Tubes under Quasi-Static Loading," Int. J. Mech. Sci. 25, 687. MAMAtaS,A.G., and JOrt'NSON,W., "The Quasi-Static Crumpling of Thin-Walled Circular Cylinders and Frusta under Axial Compression," Int. J. Mech. Sci., 25, 713. RAMSEY,H. "Comparison of Buckling Deformations in Compressed Rigid-Plastic Cosserat Plates with Three-Dimensional Plates," Int. J. Solids Struct., 22,239. WlERZmCKI,T., and BRAT, S.U., "A Moving Hinge Solution for Axisymmetric Crushing of Tubes," Int. J. Mech. Sci., 28, 135. RAMSEY,H., "Axisymmetric Buckling of a Cylindrical Elastic Cosserat Shell under Axial Compression," Quart. J. Mech. Appl. Math. (in press).

Department of Mechanical Engineering The University of British Columbia Vancouver, B.C. V6T IW5 Canada

(Received 15 October 1986; In final revised form 9 January 1987)