Pentagon
International Journal of Plasticity, Vol. 13, Nos. 6-7, pp. 551-570, 1997
© 1997Elsevier ScienceLtd
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0749-6419/97$17.00+0.00
PH: S0749-6419(97)00025-9
INTERACTION CURVES CORRESPONDING TO THE
DECOHESIVE CARRYING CAPACITY OF A CYLINDRICAL
SHELL U N D E R COMBINED LOADING
M. 7.yczkowski and Tran-le Binh*
Politechnika Krakowska (Cracow University of Technology)
(Received in final revisedform 2 April 1997)
Akaract--An elastic-perfectly plastic cylindrical shell under combined ring of radial forces and
axial tension at infinity is studied in detail. The Huber-Mises-Hencky yield condition is assumed.
The classical limit load-carrying capacity is evaluated, but, in general, an elastic-plastic deformation process cannot reach the limit state. Earlier an infinite increase of axial strains results in the
formation of inadmissible discontinuities and it determines the decohesive carrying capacity (DCC)
of the shell. Complete interaction curves corresponding to the elastic carrying capacity, to DCC
and to the limit carrying capacity are constructed and compared. The domain between the concave
curve corresponding to DCC, and the convex curve describing the limit carrying capacity is
unobtainable for an elastic-perfectly plastic shell without violation of continuity requirements.
© 1997 Elsevier Science Ltd
I. INTRODUCTORY REMARKS
The limit load-carrying capacity o f a structure is usually defined either as the loading parameter corresponding to the onset o f deformations o f a rigid-plastic hardening structure, or
as that resulting in uneontained plastic flow o f an elastic-perfecty plastic structure, it means
in f o r m a t i o n o f a mechanism o f plastic collapse. These two definitions often coincide, but
in some cases they m a y lead to various results. Namely, a mechanism o f plastic collapse
m a y be preceded by infinitely large strains, leading to inadmissible discontinuities, and
then such a m e c h a n i s m c a n n o t be realized in a continuous body: it will be preceded by a
decohesion. I n real materials various forms o f decohesion m a y take place, either as a
result o f void initiation and growth, or as a result o f crack initiation and growth, or w i t h o u t
any o f these phenomena. A c o m p a r i s o n o f various physical criteria o f decohesion is given,
e.g. by Clift et al. (1990); the authors conclude that a limited value o f unit plastic w o r k is
in the best agreement with experimental results. Plastic strains are then limited as well.
But even if we assume the idealized elastic-perfectly plastic material, decohesion m a y
occur as a result o f f o r m a t i o n o f inadmissible discontinuities due to infinite increase o f
n o r m a l strains e. The c o r r e s p o n d i n g loading parameter was called by Szuwalski and
7.yczkowski (1973) the decohesive carrying capacity (DCC). This ' m a t h e m a t i c a l ' criterion
*Present address: Hal Phong Maritime University, Hal Phong, Vietnam.
551
552
M. 7.yczkowskiand L. B. Tran
of decohesion, connected with formation of inadmissible discontinuities, may be regarded
as an upper bound to all physical criteria of decohesion, since none of them admits infinitely large strains.
In the engineering approach to limit analysis the strains or strain rates are often
assumed in the form of a distribution (generalized function, e.g. of the Dirac type); then
the displacements or velocities are discontinuous as the Heaviside functions. The first
attempt to form a theoretical background for such an approach was made by Hill (1952),
who proposed a 'relaxed' treatment of discontinuities, regarding jumps in normal velocities in plane stress problems as admissible, leading to necking. Further attempts to admit
the solutions showing some kinds of discontinuities are connected mainly with variational
approach; we mention here the pagers by Temam and Strang (1980), Anzellotti and Giaquinta (1980, 1982), Seregin (1985), and Repin (1991, 1994). In fact, such jumps may be
treated as admissible just at the end of process, but if the process is to be continued, then
they lead to inadmissible discontinuities in displacements, at least within the framework of
the small-strain theory. However, since the strains increase infinitely, a more rigorous
treatment requires here the application of a finite-strain theory. Such an analysis was
carried out in several papers listed below. Then in most cases it turns out that the jumps
due to positive normal strains remain inadmissible and result in a certain DCC whereas
the jumps due to negative strains disappear in view of the increasing cross-sectional area,
and in the latter case the process can be continued. In such cases decohesion may also take
place, but it should be analyzed in a quite different approach, e.g. proposed by Needleman
(1990).
In the present paper we do not admit normal discontinuities of displacements and
analyze the decohesive carrying capacity defined by infinite increase of normal strains.
Similar problems were analyzed for bar systems by Szuwalski (1980), for beams by Tran
and Zyczkowski (1976), for disks by Szuwalski and 7,yczkowski (1973), Zyczkowski and
Szuwalski (1975), Szuwalski (1986a,b), Skoczefi and Szuwalski (1988), for toroidal shells
by Skrzypek (1982), Skrzypek and 7,yczkowski (1983), Skrzypek and Muc (1988), Bielski
and Skrzypek (1989). These papers were based mostly on small-strain theory; finite strains
were analyzed by Zyczkowski and Szuwalski (1982), Skrzypek and Zyczkowski (1983),
Szuwalski and Zyczkowski (1984), Zyczkowski e t al. (1992). Such an analysis, much more
complicated, changes the criterion of decohesion from Ou/Ox - , oo to the type & r / O x - * c~,
but for positive stresses and strains numerical values of DCC remain almost without
change. A somewhat different approach to DCC is due to Mrrz and Kowalczyk (1989):
they introduced an additional constitutive relation between the rate of displacement
discontinuity and the respective traction rate along the material discontinuity line. Then
the 'brittle' decohesion is replaced by a continuous process with softening behaviour. A
certain kind of DCC may also take place if shearing strains increase infinitely, and a tangential discontinuity appears (7,yczkowski, 1956; Seregin, 1985; Repin, 1994). Extensive
reviews of the results obtained are given by 7,yczkowski (1981), Szuwalski (1990), and
Skrzypek (1993).
The purpose of the present paper is to construct all interaction curves for an elasticperfectly plastic sandwich cylindrical shell under a ring of radial forces 2Q combined with
axial tension at infinity p~ (Fig. 1). This problem was initiated by Tran and 7,yczkowski
(1984), where just one-side plastification was studied in detail. Here we consider two-side
plastification and derive equations of interaction curves for the whole range of positive
loading parameters. The assumptions are as follows:
Deeohesivecarrying capacityof a cylindrical shell
553
Fig. 1. Loadingsof the shell.
1. The material is elastic-perfectly plastic with Young's modulus E and yield-point
stress ~ro, incompressible and subject to the Huber-Mises-Hencky (HMH) yield
condition;
2. The analysis is restricted to small strains and small deflections;
3. The shell is of the sandwich type, with the thickness of carrying layers h, the distance
between middle surfaces of the layers H, and the mean radius R;
4. The Love-Kirchhoff hypothesis of straight normals is adopted and shear deformations are neglected; for sandwich shells this assumption may be regarded as questionable, and hence a relevant discussion will be given at the end of the present
paper;
5. The Hencky-Ilyushin deformation theory is employed. The Prandtl-Reuss theory
would essentially complicate the calculations, but--in view of a comparison performed by Szuwalski and Zyczkowski (1973) for disks--only minor changes in
results may be expected.
Similar problems for cylindrical shells were discussed by Klement (1962), Sayir (1968),
Ubaydillayev (1976), and Brooks (1988). However, none of the above papers studied the
problem of termination of the process due to an infinite increase of strains, and this will be
the main goal of the present paper.
H. GOVERNINGEQUATIONS
First we recall briefly the derivation of governing equations for a sandwich cylindrical
shell under consideration, given by Tran and Zyczkowski (1984). The Love-Kirchhoff
hypothesis determines the strains as follows
e~ = ~.i:i:Kill,
i = x, O,
(I)
where the superscripts refer to the outer layer and inner layer of a sandwich shell,
respectively. Expressing the elongations X; and changes of curvatures Ki in terms of displacements in linearized form, we rewrite (1) as follows
e +x = (d+w")a°
-E'
Ex
(u' w")'T°
~-,
~
E~
,~o
E
(2)
554
M. 7.yczkowskiand L. B. Tran
We introduced here dimensionless variables
x*
x= R~,
Eu*
u=tr0~,
Ew*
w=troR,
(3)
where the physical variables are starred, deflections w are positive inwards, and primes
denote differentiation with respect to dimensionless axial coordinate x.
The equilibrium equations for a cylindrical shell without external distributed loading
and with neglected beam-column effect may be written in the form
?
nx = O ,
nx=px=const.,
I!
(4)
mx+no=O,
where the following dimensionless normal forces ni, bending moments mi, and axial
loading Px are introduced:
n~
mi-
n, - 2troh'
m~
p*
2tr0h------H' Px = 2tro---'~-
(5)
Expressing generalized stresses m; and ni in terms of layer stresses ~ , i = x, O, we obtain
1
1
mi:~ao (ai---¢Ti+), ni.-~~ao (~7Z -..~o'+),
(6)
cry + a+ = 2croPx, (#~-- a+)" + (cry- + a+) = O.
(7)
hence
Further equations depend on the range of work of the cross-section under consideration.
If just the outer layer is plastic, we parametrize the H M H yield condition in this layer by
the Nadai-Sokolovsky formulae
~+=~0sin
(w + +
,
o+=~a0sinto+,
(8)
thus replacing two unknowns, ~+ and a+, by one unknown to+ only. Eliminating ~- from
the first eqn (7), making use of the Hencky-Ilyushin equations of shape change (similarity
of stress and strain deviators) and of geometric eqns (2) we obtain
w
u' - 2
2 w tan w+ - w".
(9)
To the inner layer we apply Hooke's law for an incompressible body
,
4
4
(lO)
Deeohesivecarryingcapacity of a cylindricalshell
555
The equilibrium equations (7) with substituted (10) and u' eliminated from (9) result in the
following system of equations for w and to+:
~/3
~
w" + - ~ - w tan co+ = y
3
sin to+ - ~ Px
(11)
w, Vc3
3
+ - ~ - w tanco+ + ~ w = (co~ sinco+ -co+ cosco+)~/'3 + -~dr~ sin co+ + ~3 cos to+.
Subtracting the first eqn (11) from the second one we can eliminate the deflection w,
w= ~
4
t!
(co~ sin to+ - to+ cos co+) + pO + cos co+,
(12)
and substitution of (12) into any of eqn (11) yields the governing equation for co+:
-~-co~ tan2co+ +
4co+co++ 3o)+ - - f c o + - c o ~
tanco+
(13)
3px
(tanco++d'~)"
+ oco+co+ - -3,- co+ + 16cosco+
." ,2 ,,
"¢c3 ,2
Similarly, for one-side plastification of the inner layer, we parametrize % and trx by the
formulae of the type (8) with to_ as a parameter, and subsequently obtain
w
u' . . . .
w tanco_ + w",
2
2
w-
_ --
4 (co~_ sin co_ - co'_' cosco_) + cosco_ + P x ,
~/~
~ co_ tanEco_+
(14)
(15)
4to' co"_'+3to"2+-~-co_-o)4 tanco_
(16)
~,2
3px
(tanco_ + ~/~)"
+6co'2co " +--~--co_ -t 16cosco_
Two-side plastification introduces some qualitative differences and will be considered in
detail here. Now the stresses in both layers are parametrized by formulae of the type (8).
Equilibrium equations (7) with substituted both formulae (8) contain co+ and co_ and do
not contain w, hence the system is partly uncoupled. Eliminating from the first equation
co--,
sin co_ = p x ~ f 3 - sin co+,
(17)
cos co_ = =Effcos2 co+ + 2 d ' 3 p x sin to+ - 3p2x,
(18)
556
M. Zyczkowski and L. B. Tran
we should choose sign ' + ' for to_ < ~, and sign ' - ' for to_ > ~; in what follows, just the
sign ' + ' will be considered. Generalized stresses, expressed in terms o f to+, are equal is
this case
2
mx = Px ~/~ sin to+,
nx = Px,
mo = ~
p x - C O S to+
1(
no = -~ Px + cos to+ +
)
,.
(19)
~¢~smto+ + x/cos2 to+ + 2~,/"3px sin to+ - 3p2x ,
)
cos 2 to+ + 2~/3px sin to+ - 3p 2 .
The second equilibrium eqn (7) yields now the following non-linear second-order equation
f o r to+
to+ = to~ tan to+ + to--------+
4 cos
Px + cos to+ +
os 2 to+ + 2d"3px sin to+ - 3p 2 .
(20)
Finally, making use o f the Hencky-Ilyushin equations for both layers we obtain (9) and
(14); eliminating u' and to_ we arrive at the following linear second-order equation for w:
(21)
= -~- w
~/cos 2 to+ + 2"d'3px sin to+ - 3p 2
HI. A SHELL UNDER A RING OF FORCES AND AXIAL LOADING
Consider a circular cylindrical shell under a ring o f radial forces 2Q at x = 0 and axial
loading p~, at infinity (Fig. 1). In the elastic range we have the general solution
w = e-aX(Al cos/~x + A2 sin/~x) + e#X(A3 cosflx + A4 sin fix) + Vpx
(22)
where ~4 = (1 - v2)/4 and, for an incompressible body, f f = 3/16. Conditions at infinity
yield A3 = A4 = 0, and making use o f the conditions at x = 0 we obtain
w = - q f l e -#x (cos fix + sin/~x) + 2 '
(23)
where q denotes dimensionless ring force
q = 2- oh
Q"
(24)
Deeohesive carrying capacity of a cylindrical shell
557
The corresponding elastic stress distribution is given by
Crx = + ~
e -ax (cos/~x - sin ~lx) + pxcro,
z.l,,,
(25)
~r0 = ~--~-ax[2/~2(cos #x + sin/3x) 4- v(cos #x - sin #x)]
where the upper and the lower signs refer to the outer and the inner layer, respectively.
The stress intensity Crr~areaches its upper bound at x = 0, either in outer or in inner layer,
and equating; this upper bound to the yield-point stress ~r0 we determine the elastic interaction curve as follows
(
px4-
~)2
,2q2
(
q) q
- ~ = 1.
+(2,8 2 + v) 4 - ~ - (2/~2 + v) px'4-~-~
(26)
These two ellipses intersect each other at q = 0, Px = + l , and for an incompressible
material at Px = 0, q = +V/2/v/3 = +1.0746, and describe a curvilinear convex tetragon.
In what follows, we confne ourselves to the quadrant q > O, px > O. Then the maximal
stress intensity in the elastic range is reached in the outer layer; exceeding the elastic
carrying capacity we arrive at a certain zone 0 < x < Xl in which the outer layer is plastic
(except for p~ = 0, when the elastic range is followed immediately by plastification of both
layers, and for q = 0, when the shell becomes plastic as a whole). For x > xl the shell is
elastic. This case was considered in detail by Tran and Zyczkowski (1984) and will not be
repeated here. We are going to describe limitations of one-side plastification and to consider the case with a zone of two-side plastification, However, summary of results will
include the results of the above-mentioned paper as well.
For large ratio q/Px the range of one-side plastification is followed by two-side plastification starting from x = 0. To describe onset of this range we require the stresses in the
inner, elastic: layer of one-side plastification (10) to satisfy the H M H yield condition at
x = 0. Substituting (9), (11) and (12) into (10) we express the stresses in the inner layer in
terms of the parameter CO+for the outer layer:
(
2
)
o x = ~ro 2px - - ~ sin CO+ ,
(27)
a s i n c o + + ~4 co+
. cos CO+) .
Oo = or0 ( - ~ 1 sin CO+- cos co+ - ~4 co+
Denote CO+l~ar x = 0 by COo+,and similarly the derivatives at x = 0. Then, substituting (27)
into the H M H yield condition we derive the condition of onset of two-side plastification
in the vicinity of x = 0,
6p2x+ (3 coscoo +-3,¢c3 sin coo+ + 4~/3w~+ sin co0+ -4~/-3co~+ coscoo+)Px
+ 4~/'3 (co~+ sin COo+-CO~+ cos COo+) cos COo+
+ 8 (CO'o2+sin COo+-co~+ cos COo+)2= 0.
(28)
558
M. Zyczkowskiand L. B. Tran
The second loading parameter, q, does not appear in (28) explicitly, but is hidden in too+.
Now, if the loadings Px and q exceed (28) and the ratio q/Px is large (precise limitation
will be given later), we have to consider at least three zones: two-side plastification
(denoted by an additional subscript 2), one-side plastification of the outer layer (subscript
1) and elastic zone (subscript e). In the zone of two-side plastification we have the eqn (20)
and eqn (21) for o~+ and w2. Boundary conditions at the centre are as follows:
w6(0) = 0,
s2(0) = q
(29)
where
(30)
s = m'x
denotes dimensionless transverse shearing force. Making use of (19) and (30) we rewrite
the second condition (29) in the form
2
t
qt~ Wo+ COStoo+ = q.
(31)
The zone of two-side plastification ends if
3e~2
1
~°2(Xl) -- 2 t ~ -- ax2 -- 2G
3
2E'
(32)
where ~0 denotes modulus in the Hencky-Ilyushin equations. Substituting into (32) the
relevant formulae for strains and stresses we obtain the condition determining the
boundary coordinate xl:
w2(x~) =
- cos[~_
-
(33)
(x0],
or, making use of (18),
W2(XI) = --~/COS 20)2+(XI) -Jr-2~¢r3pxsin O)2+(XI) -- 3/02 .
(34)
At this point continuity of deflections and of generalized stresses is required,
wl(xl)
= w2(xl),
w~(xl) =
w~(xl)
(35)
m~l(Xl) =
mx2(Xl)
m" 1( x l ) = m ' 2 ( x l ) .
The last two equations yield directly
a,l+(x2) = o~+(xO,
o~;+(xl) = o~+(xl).
(36)
In acertain neighbouring zone Xl < X < X2 we have one-side plastification, with outer
layer plastic, and inner-elastic. This case is described by eqns (12) and (13). It ends at a
certain point x2, evaluated from the condition
Decohesive carrying capacity of a cylindrical shell
oi (x2)-
1
:-
3
+
559
(37)
It may be reduced to the form
wl(x2) = - cos[col+(xg],
(38)
p~Vr3 + [2vr3 - 4co'(+(x2)] cos [COl+(x2)] + 4co'12+(x2) sin [COl+(x2)] = 0.
(39)
or, making use of (12),
At this point we have the continuity conditions
We(X2) = WI(X2),
W~(X2)= W;(X2)
mxe(X2) = mxl(X2)
mtxe(X2) = mtxl(X2).
(40)
The last two conditions may be reduced to the form
w (x2) = w'((x2),
(41)
14{,~'(x2) = w I (x2) + T
{W/l(X2) -- CJl+(X2) sin [col+(x:)]} t a n [col+(X2)l,
where the general integral for we is given by (22).
So, in the case of two-side plastification at the centre of the shell, we have to integrate
governing equations in three zones: two-side plastification, one-side plastification, and
elastic. Twelve boundary conditions are needed: we have two boundary conditions (29)
and (31) at x = 0, four continuity conditions (35) and (36) at x = Xl evaluated from (33)
or (34), four continuity conditions (40) and (41) at x = x2 evaluated from (38) or (39), two
boundary conditions at infinity. The last boundary conditions yield A3 = A4 = 0 in (22).
So, integrating numerically along the axis x, we have a 2 + 2 boundary value problem: we
have to assume unknown values w(0) and COo+and adjust them in such a way as to satisfy
four boundary conditions (40) and (41) at x = x2, having just two constants, A1 and A2, at
our disposal
It should also be mentioned, that for very large values of the ratio q/Px the plastic zone
in the inner layer is longer than that in the outer layer. Then the condition (32) should be
replaced by ~o-~(xl) = 3/2E, and in the zone Xl < x < x2 eqns (12) and (13) should be
replaced by (15) and (16). The condition for x2 takes then the form ~o{(x2) = 3/2E. So,
effective forms of the conditions (34) and (39) are changed, but the central idea of calculations remains.
IV. DECOHESIVE CARRYING CAPACITY OF THE SHELL
The solution of the system of eqns (13), (20) and (21) terminates with infinitely
increasing strains e+ at x = 0; this condition determines DCC of the shell. Indeed, strains
Ex+ in the plastic zone are larger than other strain components and they reach their upper
bound at x = 0.
560
M. Zyczkowskiand L. B. Tran
Making use of eqns (2), (8), and similarity of deviators, we may write
e~_
E+
u'+w"_sin(to+-~)
W
(42)
COS tO+
and hence the condition ex+ ~ oo is equivalent to ~o+(0) = 7r/2 at w(0) # 0 (in order to
eliminate e+ (0) = 0). This additional equation joins Px and q and describes the interaction
curve of DCC of the shell. The case of one-side plastification, determining a part of this
interaction curve, was considered by Tran and Zyczkowski (1984); here we are going to
consider the case of two-side plastification at the centre and to construct the whole interaction curve.
Equations (20) and (21), describing two-side plastification, become singular at the
starting point x = 0 and a numerical solution of these equations must be completed by
appropriate generalized power series valid in the vicinity of that point. It turns out that
the following generalized power series hold (for x > 0):
fO+(X) ~- E CjXj/2 = CO "~- ClXI/2 "l- C2x
j=O
-~- . . . .
(43)
W(X) = E BjxJ/2 = B 0 4- Bl xl/2 -~ B2x --}-. . . .
j=O
(44)
The condition w'(0) = 0, (29), yields BI = B2 = 0, whereas the condition of decohesion
results in Co = 7r/2, and (31) gives
-~ = q.
(45)
Substituting (43) into (20) and performing suitable operations on generalized power series,
7.yczkowski (1965), Feldmar and K61big (1986), we find subsequently
C2 = O,
c3-2
v5
(74_---15
and substituting (44) into (21) we obtain
(46)
Deeohesivecarryingcapacityof a cylindricalshell
B3-
B4 =
561
B0
G#3'
dr3(1
- p,`'v~)Bo
8~2 ff3p,` - 3p2x '
B5 = [ - Cl4---6--+40-~l(Px+~2q~px-3p2)]Bo,
4~
(47)
So
B6 =
48(2V/3P ,` _ 3p2x)3/2,
The constant Bo = w(0) remains free, to be found from the boundary condition at x = x2,
(40). According to the system of coordinates adopted, for q > 0 we have w ( 0 ) < 0
(deflection goes outwards in the range under consideration), and hence Bo < 0. Further,
absolute value of deflection decreases with x, hence B3 should be positive, and this argument results iLnnegative CI."
Cl = _~q~/r~.
(48)
It should be recalled that in the range of one-side plastification at the centre, considered
by Tran and Zyczkowski (1984), the constant Cl was either positive (for smaller values
q/Px), or negative (for larger values q/p,` inside that range). Namely, Cl was positive
inside the interval 0.3120 < q < 0.5562, 0.9720 >p,` > 0,7321, it means 0.321 < q/p,, < 0.760,
and C1 was negative inside the interval 0.5562 < q < 1.2879, 0.7321 > p > 0.2695, it means
0.760 < q/p,` < 4.779. The value q/Px = 4.779 corresponds to the onset of two-side plastification at the centre. Here we consider just this case and assume q/p,` > 4.779, and hence
the sign of C~ remains without change.
Starting from the expansions (43) and (44) we may integrate governing equations in
subsequent zones. For given q we have to assume B0 and Px and adjust them so as to
satisfy four continuity conditions (40) having just two constants, AI and A2, at our disposal. Combiining p,` obtained in such a way with given q we obtain individual points of
the interaction curve of DCC of the shell, F(~, /~,`)= 0 (parameters corresponding to DDC
are denoted by a hat).
Making use of (9), (2), and performing operations on generalized power series, we
may in turn analyze the behaviour of displacements u and strains e+ and e; in the
vicinity of x := 0 at the moment of decobesion (strains e0 are simply proportional to w).
Substitution of (43) + (47) into (9) and integration with the initial condition u(0) = 0
yields
562
M. Zyczkowski and L. B. Tran
u=Bo-~-C-~lx
+
+4
~/;~px_--3p2x J
Cl
16 [~/r3 --~1l (px + x/2~,/3px -3p2x)] x3/2
1
+ lO-C,
(49)
16 (2.,/~pf--3p~)3/2 x2 + . . . .
and from (2) we obtain
e +
. /~
-,/2
l
2
/
16
~
(50)
= So
/2-
x-3pBj
-4(2~px-3P2x)3/2x+...}.
(51)
It is seen that for x ~ 0 the strains ef increase infinitely, according to the definition of
DCC adopted.
Accuracy of the series derived is rather poor, in particular the series for ex, (50) and
(51), are sufficiently accurate just inside the interval 0 < x < 0.05 (though they include all
the constants evaluated above, up to (?4 and B6). Nevertheless, these series are very useful
to start numerical integration from the singular point x = 0.
V. NUMERICAL EXAMPLE
An example of DCC with two-side yielding at the centre may be obtained for
q > 1.2879. We assume ~ = 1.558, then, from (48), C1 = -1.643. Integrating all governing
equations and satisfying necessary continuity conditions we find Bo = - 1.400. ~x = O.111,
hence ~/~,` = 14.02. Figure 2 shows distribution of ~o+, w and w' in the case under consideration, whereas Fig. 3---strain distribution, and Fig. 4--stress distribution.
The boundary coordinates are xl = 0.802, x2 = 0.815, hence the zone of one-side plastification of the outer layer is very short. For larger values of q/p,,, roughly q/Px > 15, a
zone of one-side plastitication of the inner layer is formed and then followed by an elastic
zone for x > x2. Equation (14) + eqn (16) should then be employed.
If p,` ~ 0, then B4 and B6 in (47) increase infinitely, and the series (47) becomes divergent. It turns out, that the corresponding value of ~ tends to the limit load-carrying
capacity ~ determined below.
Decohesive carrying capacity of a cylindrical shell
563
f
1.0
0.5
plostification
~O.a02
/ of the outer layer
0.815
1.0
1~5
•
0.5
two-side
plaslificalion
2.0
'
elastic zone
0.5
~ 2~ ,' ~5 2j I1 5
~
3.0 " x
w
ii
1,0
Fig. 2. Distribution of deflections, slopes, and of the parameter o~+in the shell at decohesion.
+1
plastification
~
nn~of the outer layer
).815
.
II
Sx
/
elastic zone 2
Fig. 3. Distribution of strains at decohesion.
564
M. Zyczkowski and L. B. Tran
UO
/
plastiticollon
of the outer" Io'
0.80:
0.5
I
two-side
plostificotion
.S
2.0
2.~"'-----..~
3.0
x
elastic zone
+/
(YX
Fig. 4. Distribution of stresses at decohesion.
VI. L I M I T LOAD---CARRYING CAPACITY OF THE S H E L L
Usually the limit load-carrying capacity of a plastic structure determines both the end
of a deformation process in an elastic-perfectly plastic structure (change into a mechanism
of plastic collapse) and the onset of deformation of a rigid-plastic hardening structure. In
the shell under consideration this is not the case and we derive the limit interaction curve
just to compare it with the curve o f DCC derived above.
First assume any yield condition in generalized stresses, written in parametrical form
mx = mx(toi), no = no(toi), nx = n x ( t o i ) , with toi being parameters. In our case nx = const.,
so this value may be assumed as a second parameter, and just one parameter to is sufficient. We rewrite the first equilibrium equation in the form
ds
r
-d--~ww + no = 0,
(52)
where the shearing force s is given by
s=
dmx o9',
do;
and co' = dto/dx. Eliminating to' from (52) and (53) we may write
(53)
Decohesive carrying capacity of a cylindrical shell
565
dmx
s-~+no--d---~-=O,
(54)
and after integration
~-+ C=-
i no-~l-~todto,
mx
(55)
too
with to for x = 0 denoted by too. Boundary condition at the centre (29) gives C = -q2/2.
Moreover, at a certain point x = xl, to = tol, the plastic zone meets the rigid zone, and
then s = 0. Making use of this condition we may determine the limit load-carrying capacity ~ by
~ = 2 no-d-~ d
I..
.
(56)
tOo
N o w we return to the H M H yield condition. In the rigid-plastic case just two-side plastifieation is possible; making use of the expressions for this case (19) and regarding to+ as
the paramete:r to we obtain
7ooso(+
q 1/2
costo + ~/cos~ to + 2~,x sin to- 3~) dto]
.
(57)
In the static approach employed we should assume too and tol in such a way as to maximize ~ for given Px. The parameter to decreases along the axis of shell, hence the sign
'minus' in (57). The value too will be determined from the condition of vanishing integrand
in (57), hence too = ~r/2. Indeed, for to > 7r/2 the integrand would be negative, and hence
the result smaller contrary to the assumption. On the other hand, the value to1 will be
determined fi:om the condition of vanishing expression under the radical, since complex
values of the integrand are not admitted:
toi = arcsin(pxVr3 - 1).
(58)
Finally, integrating (57) within these limits we obtain
~ = - ~ 3 ~ 3 [ r r - 2 a r c s i n ( p ~ C c j - 1 ) -4- 2p~(2 - px~C3)-2(pxVr3 - 1)~/2~¢C3px-3/~] v2. (59)
For Px = 0 one obtains
~=~-
1.9045,
(60)
566
M. ~:yczkowskiand L. B. Tran
and this is the well-known solution for pure radial ring forces. On the other hand, for
Px = 1 one obtains, rather surprisingly,
1 [Tr--2arcsin(~f3-1)+2(2-V~)-2(~/3-1)~]l/2=0.7739.
(61)
Of course, the value Px = 1 results in the limit state of the shell under consideration even
without any q and it cannot be exceeded. The result (61) means that the limit interaction
curve contains the straight segment px = 1, 0 <_ q < 0.7739, whereas (59) determines just
the remaining part of the curve.
The above solution is statically admissible. However, it may be shown that a kinematically admissible velocity field, w = cs, found by Sawczuk and Hodge (1960) for pure
radial ring load, holds here as well. This field gives as additional argument for the
boundary condition s = 0 for x = Xl, used above. Hence we obtained a complete solution
from the point of view of rigid-plastic analysis.
VII. SUMMARY OF RESULTS
Interaction curves for an infinite sandwich cylindrical shell under ring of radial forces
and tension are gathered in Fig. 5. Elastic interaction curve is given by (26), and limit
interaction curve obtained via rigid-plastic analysis--by (59) and (61). Exceeding elastic
interaction curve we enter the elastic-plastic range with one-side plastification of the
external layer in a certain interval 0 < x < xl. Just for pure ring of forces, Px = 0, the
elastic range is followed immediately by two-side plastification, and for large ratio q/px
one-side plastification turns soon into two-side plastification. In general, the limit interaction curve cannot be attained by an elastic-perfectly plastic shell, since the elastic-plastic
range terminates with infinitely increasing longitudinal strains e + at x = 0, followed by
decohesion.
Part of the interaction curve, corresponding to decohesive carrying capacity reached at
one-side plastification, was determined by Tran and Zyczkowski (1984). It corresponds to
the interval 0.3120 < ~ < 1.2879, 0.9720 >/~x > 0.2695. We observe tangency of the concave interaction curve corresponding to DCC to the convex elastic interaction curve. It
takes place for/~x = ~ 1 = 0.7321, ~ = ( x / ~ - 1)/¢/3 = 0.5562 and is due to the fact
that in the condition of decohesion (42) evaluating e + we have a symbol 0/0 (e0 ~ 0,
cos~o+ -~ 0).
In the present paper we discussed DCC reached at two-side plastification. It is determined by the series (43) and (44) with the coefficients (46), (47) and (48), subsequent
numerical integration of (20) and (21), further of (13), and evaluation of B0 and Px from
continuity conditions (40). It holds for 1.2879 < ~ < 1.9045, 0.2695 > ,3x > 0, it means for
q/Px > 4.779. The remaining interval 0.9720 </~x < 1 is the most dificult for integration.
One-side plastification of the outer layer is followed by an elastic zone, then by a zone of
one-side plastification of the inner layer, finally by another elastic zone up to infinity.
Even larger number of plastic zones is possible. Effective calculations were not carried
out, but this small portion of the interaction curve was extrapolated.
Assuming the Love-Kirchhoff hypothesis we mentioned that for sandwich shells it is
often regarded as questionable. Indeed, for small shearing stiffness of the core one should
take shear effects into account. Then (2) is replaced by
Decohesive carrying capacity of a cylindrical shell
567
(62)
where Cs denotes shearing stiffness. However, if we introduce these formulae into (42)
defining DCC of the shell, then just the middle term is subject to alteration. The strain ex+
increases infinitely if to+ = zr/2 as before, so qualitatively the problem remains without
change, just effective calculations would be much more complicated and some quantitative differences may be expected.
VIII. EFFECT OF ELASTIC COMPRESSIBILITY
The present analysis was restricted to incompressible materials. However, as it was
shown previously in many other cases, elastic compressibility usually affects the decohesive carrying capacity. For example, considering a disk with rigid inclusion Szuwalski and
7.yczkowski 111973, 1984) proved that both under surface and thermal loadings the compressibility results in a decrease of DCC.
In the case of the cylindrical shell under consideration introduction of elastic compressibility starting from eqns (10) presents no particular problems, but the resulting equations
and power series expansions are much more complicated. Such an analysis will not given
here, but some conclusions may be drawn basing just on equations for the elastic range.
The equation of the elastic carrying capacity (26) is given for an arbitrarily compressible
material. For pure axial tension, substituting q = 0, we obtain obviously Px = Px = 1
irrespective of Poisson's ratio v. For pure radial loading, substituting Px = 0, we obtain
q -----~ =
(63)
~/(2 - v) :F (1 - 2 u ) ~ i - v2
p~
1.0
0.9
0.8
0.7
. . . . . . . .
I
I
I
I
~ . . . .
L
d~o~tin~itTes
0.6
0.5
0.4
0.3
0.2
0.1
0
=
I
I
I
I
~ge"
'1carryin,g 1 ~
~
~
. . . .
"
"~
carry'rag capacHy
........
\
I
t
I
I
I
I
~ ' ,
' #'_,__ - . !/. , .
,
,
,V
,
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
~ah'on
\
/
It w o - -
side.,,._
/"
[
IDlaslificoflon'~
I
-9-~7 ~. . . . .
~
[I " 9045
.
~.
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
q
Fig. 5. Interaction curves for the shell under consideration.
568
M. Zyczkowski and L. B. Tran
and this value changes from 1.0746 for u = 1/2 (for both layers) to 1.4142 for v = 0 for
outer layer, and to 0.8165 for inner layer. Since always the smaller value gives the correct
result, we realize that for a compressible material the elastic range terminates here with
plastification of the inner layer. So, for a compressible material the curve of the elastic
carrying capacity consists of arcs of two ellipses: of a large part of ellipse constructed for
the outer layer, with upper signs in (26), and of a small part of ellipse for the inner layer,
with lower signs. The elipse for the outer layer is longer in the horizontal direction than
for an incompressible material and in this range compressibility results in an increase of
the elastic carrying capacity. The same conclusion may be expected for the whole curve of
decohesive carrying capacity. We can easily find the point of tangency of both interaction
curves since the condition of DCC, namely oJ+(0) = zr/2 remains without change. Substituting a + (0) = 2tr+ (0) into (25) and (26) we find the corresponding p and q. For example,
for v = 0 we obtain p =/~ = 1/v~ = 0.5774, and ~ = ~ = ~
= 0.8165. This point lies
slightly outside the DCC curve shown in Fig. 5, and hence an increase of DCC is seen. On
the other hand, if the shell is subject to pure loadings, p or q, the DCC does not depend on
elastic compressibility: we obtain/3 = 1 and ~ = 1.9045 for any value of v.
IX. CONNECTION WITH PLASTIC LOCALIZATION PHENOMENA
Related problems of discontinuous bifurcation connected with the loss of ellipticity of
stress regimes and strain localization were discussed by many authors; we mention here
the review by Rice (1976), and more recent results for arbitrary yield conditions with
plastic hardening and associated or non-associated flow rules by Ottosen and Runesson
(1991), Runesson et al. (1991) (plane states) and Larsson and Runesson (1993) (numerical
implementation). The DCC is also connected with the loss of ellipticity, but is defined by
infinitely large normal strain whereas discontinuous bifurcations may occur also without
this feature. In many cases a post-critical analysis is possible, like that shown by Mr6z and
Kowalczyk (1989), and from this point of view the interaction curves of DCC may also be
termed 'localization onset curves'. On the other hand, post-critical response in perfectly
plastic structures corresponds up to decohesion to a softening behaviour and may be
analyzed just under displacement--controlled loadings. Under force-controlled loadings
the decohesion takes place immediately, and hence the term 'decohesive carrying capacity'
seems better justified.
X. CONCLUSIONS
1. The whole interaction curve, corresponding to decohesive carrying capacity of an
incompressible shell under consideration has been determined, and the effects of
shear and elastic compressibility have been estimated.
2. The DCC curve, which may also be termed 'localization onset curve' is concave.
This concavity is typical for interaction curves and was revealed, e.g. by Szuwalski
and 7.yczkowski (1984), Zyczkowski et al. (1992).
3. The domain outside the DCC curve is inadmissible for an elastic-perfectly plastic
material since it is preceded by formation of inadmissible discontinuity of displacements. This fact should be taken into account when employing purely numerical
methods of solution, like finite element method. It may work just in weak formulation if we admit discontinuous displacement approximation (Johnson and Scott,
1981; Stephan and Temam, 1987; Larsson et al., 1993).
Decohesive carrying capacity of a cylindrical shell
569
Acknowledgement~rant PB 238/T07/95/08 is gratefully acknowledged.
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