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Bifurcation of elastic-plastic spherical shells subject to internal pressure
J.Mech. Phys. Solids, 1975, Vol. 23, pp. 357 to 367.
Pergamon Press.
Printed in Great Britain.
BIFURCATION OF ELASTIC-PLASTIC SPHERICAL SHELLS SUBJECT TO INTERNAL PRESSURE By A. NEEDLEMAN? Department
of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, U.S.A. (Received 2nd January 1975)
THE BIFURCATION of elastic-plastic
spherical shells subject to internal pressure loading is investigated. Two cases are considered; in one case, the pressure is prescribed, while in the other, the change in volume enclosed by the shell is prescribed. Attention is restricted to axisymmetric bifurcations. Separation of variables is employed to introduce a complete set of bifurcation modes. For one of these modes a simple approximate bifurcation criterion is obtained. A numerical method is then employed to investigate the bifurcation of elastic-plastic spherical shells in a variety of modes. A comparison is made between the numerical results and the predictions of the approximate criterion. Of particular interest is whether bifurcation can occur prior to the maximum pressure point. The numerical results obtained here show that this does not occur for the axisymmetric modes considered.
1. INTRODUCTION GENERALLY, failure of spherical shells subject to internal pressure is taken to occur when the pressure, given by an analysis assuming spherical symmetry, reaches a maximum. Such a failure criterion excludes the possibility that some bifurcation mode could be activated before this maximum is attained. For spherically-symmetric loading conditions, MILES (1969) found the most general bifurcation mode available for a rigid-plastic material with a non-singular yield surface. He also found for a materially homogeneous rigid-plastic spherical shell with a non-singular yield surface that uniqueness of deformation is guaranteed until the maximum pressure is attained. More recently, STRIFORS and STOR~;KERS (1973) showed that Miles’ sufficient condition for uniqueness is also necessary, in the sense that bifurcation is possible at the instant when Miles’ sufficient condition ceases to hold. However, the possibility remains that an elastic-plastic spherical shell could bifurcate into a mode not available in the rigid-plastic case. For example, there is no three-dimensional rigid-plastic bifurcation mode that can adequately describe the necking of tensile bars, although there is such a mode available for elastic-plastic materials. Here, the bifurcation of elastic-plastic spherical shells subject to internal pressure is investigated. The boundary-value problem is posed in two ways. In one case, the pressure is taken to be prescribed, while in the other, the change in volume
t Now at Division of Engineering, Brown University, Providence, Rhode Island 02912, U.S.A. 357
A. NEEDLEMAN
358
enclosed by the shell is prescribed. In both, the eigenvalue problem governing bifurcation is posed in terms of a variational principle due to HILL (1961, 1962), and in this formulation full account is taken of both geometrical and material nonlinearities. Consideration is restricted to axisymmetric bifurcations and separation of variables is employed to introduce a complete set of bifurcation modes. Although the functional governing bifurcation is identical for both types of loading, the admissible functions differ. When the change in enclosed volume is prescribed, there is a constraint condition that must be satisfied by the bifurcation mode. For one mode that satisfies this constraint condition, a simple approximate bifurcation criterion is obtained. This approximate bifurcation criterion for elastic-plastic materials is formally identical to an exact criterion for rigid-plastic materials obtained by MILES (1969). A numerical method, based on the finite element method, is then employed to investigate the bifurcation of elastic-plastic spherical shells for a variety of modes. A comparison is made between the numerical results and the predictions of the approximate criterion. 2.
PROBLEMFORMULATION
The formulation of the field equations employed in this study has recently been reviewed by HUTCHINSON (1973a), where further details and more complete references are given. For completeness, some of the basic equations are presented here. This formulation is Lagrangian in character and employs the undeformed configuration of the body as reference. This choice is a convenient one for implementing the numerical method that will be described subsequently. However, many of the basic equations do take on a simpler form if the current configuration of the body is taken to be the reference configuration. The volume and surface of a body in the undeformed configuration are denoted by V and S respectively, and each particle is labelled by a set of convected coordinates xi (i = 1,2,3) which serve as independent variables. In the undeformed body, the covariant components of the metric tensor are denoted by gij and its determinant by g ; while in the deformed body, these quantities are denoted by G, and G respectively. The Langrangian strain increments are given by ?jij = +(tii, i Urn, where uk are the covariant components of the displacement vector referred to the undeformed base vectors, and ( ), i denotes covariant differentiation with respect to the undeformed metric tensor. Here, the operation (.) denotes differentiation with respect to some monotonically increasing parameter that characterizes the load history, and corresponding derivatives are termed ‘increments’ (and not ‘rates’). This parameter is taken to be the radial displacement at the spherical shell inner radius W. The contravariant components of the Kirchhoff stress tensor on the embedded deformed-coordinates are denoted by yik. These are related to the contravariant components of the Cauchy stress tensor aik by qik = (G/g)‘/20’k_ (2) j
+
ti j,
i +
glm(li’,
j
+
U’,
iti”,
j)),
(1)
Bifurcation of elastioplastic
For the problems considered takes the form
spherical shells
here the incremental
359
principle of virtual work
J (4iJ’81jij+qiiglmti1,i6ri”,j) dV = -fi J a”n,&& dS_p SP
$ c?“n,&& dS, SP
(3)
where the strain increment variations S#j are related to the displacement increment variations by (1) and S, is that surface on which the pressure or the change in enclosed volume is prescribed. The second surface-integral on the right-hand side of (3) arises from the configuration dependence of the loading, and c$’ and d” are given by (see, for example, SEWELL(1965)) ui’ = ~sjk’s’“‘(gj~+ UJ,I)(gkm+ uk ,), 6” = girGl,I _ gQr, I + &lmrUj, I’&, ~,1
(4)
where siki is the alternating tensor. The stress increments 4” are related to the strain increments 311by a constitutive relation (to be stated subsequently) that is of the form 4” = LiMg,. (5) The instantaneous moduli have the symmetries L!jkl = Lkzij= tikij = Lk’ji and have two branches, one corresponding to plastic loading and the other to elastic unloading. For a spherical shell subject to a prescribed internal pressure or to a prescribed change in enclosed volume, one equilibrium state, corresponding to a sphericallysymmetrical deformation, is available for all values of W. This state is termed the fundamental solution and is unique for sufficiently small values of W. The bifurcation criterion employed is an application of HILL’S (1958, 1961, 1962) general theory of bifurcation and uniqueness in elastic-plastic solids to this particular problem. Bifurcation from the fundamental solution first becomes possible at some critical displacement WC,corresponding to a pressure pc. Then, with the body in a state characterized by a displacement field u and a stress field q, a non-trivial solution exists to the set of homogeneous equations I = S(~*i’i~+qiig,mti*z,~ti*m,j) dV+p, 6I=‘;.
J c?*“n,$ SP
dS = 0,
(6) >
The superscript * denotes the difference between field quantities associated with the fundamental solution and those associated with the second solution, and q:j and d**’ are related to the displacement increments ti: by (1) and (42), respectively. The identity (6,) is derived by considering the difference between two fields satisfying (3). If the increment of pressure is prescribed, then p* vanishes, while if the change in enclosed volume is prescribed, then the change in volume due to ti: is zero, so that I ai'nr ti: dS = 0. (7) SP A sufficient condition for uniqueness of deformation is that the functional I in (6,) be positive for all admissible displacement fields ii:, and bifurcation first becomes possible when this functional vanishes. Although the bifurcation functional I in (6,) is employed for both types of loading, the admissible functions differ. When the change in volume enclosed by the shell is prescribed, the displacements ti: must satisfy the constraint (7).
360
A. NEEDLEMAN
If the increment in the fundamental solution at WChas the property that loading occurs everywhere in the current plastic zone, then the stress-rates in (4,) are related to the strain-rates by * *ii = Ejk$,;. (8) 4 Furthermore, the branch of the tensor of moduli for plastic loading is to be used in the current plastic zone in (8). Thus, the system (4) and (8) is independent of the loading-unloading criterion; and in HILL’S (1961, 1962) terminology a linear comparison solid has been introduced. Although no attempt is made here to study the post-bifurcation behavior of the spherical shell, we note that the amplitude of the bifurcation mode is uniquely determined by the requirement that loading occurs everywhere in the current plastic zone, except at one point or along one curve where neutral loading takes place (HUTCHINSON,1973b). The particular stress-strain law used in this study is a large-strain generalization of J,-flow theory with isotropic hardening, due to Professor B. Budiansky (unpublished work, 1968). This stress-strain law is meant to apply to situations in which the elastic strains remain small and the total strain increment is written as the sum of the elastic strain increment s~j and the plastic strain increment Qfj. The elastic strain-increment/stress-increment relation is taken to be Sj
=
~
((1 + V)G,G, - VGijGk,)G“t
(9)
where E is Young’s modulus, v is Poisson’s ratio, and ek’ are the contravariant components of the symmetric Jaumann derivative of the Cauchy stress, which are related to the contravariant components of the convected derivative 6k’ by bk’ = ekl + Gkm&j,, + GL”ckmdmn.
(10) The stress-strain relation (9) is not actually elastic in that it does not admit a potential in the large strain region, but when linearized it does give Hooke’s law. The yield function for J,-flow theory with isotropic hardening is F = oe-- Y, where Y, the flow stress, is either the maximum value of rre over the stress history or the initial yield stress g,,, whichever is the greater, and u, is given by Ue = (3GikGjrSiiSkz)“‘.
(11)
where Sij = oii _ gGiiGkle”.
(12)
The generalized flow rule takes the form q$ =
%1/G- 1IEPij &Joe if Q, = Y and b, 2 0,
1 0
if (i) (TV< Y or (ii) (T, = Y and 5’, I 0.
(13)
Here, Et is a function of the stress invariant 6, and is the slope of the uniaxial truestress/natural-strain curve. The particular model of uniaxial stress-strain behavior that will be employed is a piecewise continuous power-law with a continuous tangent modulus, i.e.
Bifurcation of elastic-plastic spherical shells
361
where E is the natural strain, 0 is the true stress, a,, is the uniaxial yield stress, E,,is the yield strain, given by sy = a,,/E, and m is the strain-hardening exponent. The stress-strain relation (14) can be inverted to obtain the tensor of moduli C relating the contravariant components of Jaumann derivative of the Cauchy stress 8’j to the strain increments rjkrwhere
and h =
0 if (i) ce < Y or (ii) ce = Y and S”Qij I 0. (16) > 1 1 if g’e= Y and S”~, 2 0. The tensor of moduli L in (5), relating the convected derivative of Kirchhoff stress 4” to the strain increments ljklr is found from (IO), (15) and 4 rj = ( Glg)3$j + 4 ijGk$,, (17) to be Vkl= (GIg)+Cijkr_ +(Gik@ + Gjk# + Gil&k+ Gjrqrk)+ qijGkl. L” (18) The moduli L!jkl do not possess the symmetry L!jkz= Lk”’ necessary for the variational principle (42) to hold unless the material is elasticully incompressible, in which case (G/g)* = 1 and 8 = 4”. Therefore, in order to obtain the necessary symmetry we make the approximation that oii zz q’j, and 8 will be replaced by qij in the constitutive equations (7) to (11). This approximation will involve little error, since in (2) the factor relating & and q” is equal to the ratio of the volume in the deformed configuration to that in the undeformed configuration. The volume change is due entirely to the elastic strains which, by assumption, remain small. With this approximation, rfjkl
=
cijkl
_J(Gikqji
+
Gjkqi’
+
Gilqjk
+
Gjlqik). (19)
HUTCHINS~N(1973a)
has proposed a slightly modified version of this constitutive equation in which the Cauchy stress components and derivatives appearing in (10) to (12) are replaced by the corresponding Kirchhoff stress components and derivatives. In this alternative formulation, (19) is obtained without approximation, but the tangent modulus Et differs slightly from the slope of the true-stress/natural-strain curve. The tensor of moduli (19) obtained here is identical to that employed in previous studies (see, for example, NEEDLEMAN(1972)). However, the present development has followed that given by HUTCHINSON(1973a) except that, jn line with Professor B. Budiansky’s original proposal (as mentioned- earlier in this section), the Cauchy stress tensor has been employed as the basic stress measure in the formulation.
3.
BIFURCATIONANALYSIS
The initial mean radius of the spherical shell is R, and the initial thickness is t,. Spherical coordinates are employed with x1 = r, x2 = 8, x3 = 4. In the fundamental solution the non-vanishing displacement and stress components are ul = M), rr:(r) = G, 1d ‘,
a;(r) = G22~22 = G33g33.
(20)
362
A. NEEDLEMAN
As stated previously, consideration is restricted to axisymmetric bifurcations and we seek bifurcation modes of the form tiy = W,(r)P,(cos
e),
tiz* = rU,(r)
dP,(cos de
e) ’
where P, is the Legendre polynomial of degree n (= 0, 1,2, . . .). The set of bifurcation modes (21) gives an exact separation of variables and is complete in that any sufficiently smooth axisymmetric admissible functions ai, tiz can be expressed as a linear combination of modes of the form (21). Here, we note that an essential condition on the function tiz is that it vanish at the poles, 8 = (0, n). Each of the functions W, and U, depends on the material parameters: oJE, the yield strain, v, Poisson’s ratio, and m, the strain-hardening exponent, although this dependence is not explicitly noted in (21). For the case of prescribed pressure loading, any mode of the form (21) is admissible, whereas when the change in volume enclosed by the shell is prescribed, the constraint (7) excludes the mode with n = 0. If the elastic strains are neglected so that the total strain increment satisfies the flow rule (13), then the available modes are severely restricted. MILES (1969) has shown that in this case, for pressure loading, the only admissible modes are those with n = Oandn = 1. When the mode assumption (21) is substituted into the bifurcation functional (6,), we obtain Ro+tt. I, = j {K,W,2+(2K3+2K4+n(n+l)K,)W,Z+[n2(n+1)2K3+n(n+1)(K,-KK,) R,- *to +n(n+l)K,]U,2+n(n+1)K6U~2+4K2W,‘W,-2n(n+1)(K,+K,+K,)W,U, -2n(n+l)K,
W,lU,+2K,n(n+l)U~W,-2K,n(n+1)U~U,}r2
dr
-P[R,-~t,+u,(R,-~t,)][2W,‘-22n(n+l)W,U,+n(n+l)Ui,Z]/.~-_t,~
= 0,
(22)
where ( )’ denotes d( )/dr and K3 = GZZ(Gz2 L2222 -I-&,
K2 = (G11G22)*L!122, K _ G22G G L2233
K5 = G22(G11G22L1212+q;),
K4,:
K1 = G”(G:lL1”l+q;),
K, = (G”G22)+(G,lG2ZL1212), G” = (1 +u;)-‘,
(23)
G11(b:I;232L121’+q;), *I G33 = G22 sin2 0
G22 = (r + u,)-2,
(24)
A sufficient condition for uniqueness is that 1, > 0 for all admissible n and a numerical method to be described subsequently will be employed to implement a numerical solution to (22) for arbitrary n. Here, the special case II = 1 will be analyzed in greater detail. The bifurcation functional (22) reduces to ;_f*yk&
[w:,F+4v
+c2
(?7-
Av
4wI,i
wl,~+~(wl-u,) -
(y9]
) 2+
&
[ul,‘+yq2
(
--4: W&+2q:U:,i
(g/G)*?
di-2p(R-~t)(WI-Ul)Z
= 0, R-ft
(25)
Bifurcation of elastic-plastic
363
spherical shells
where ? - r+u,,
(),P = (l/(1+4))
(26)
dz,
while R and t denote the current radius and thickness, respectively. The change of variable by (26) is equivalent to changing the reference configuration from the undeformed configuration to the current one. The constants C, and C, are given by c, = l/3(1 +v)(1 -$(l-2V)E,/E),
(27) > C, = S{Cl -(v/(1 +v))(l -WW1/[1-4(~ -NWlh The quantity in (25) multiplied by E/(1 -2~) is the volume change G’jrjjfj. It is anticipated that bifurcation will occur far into the plastic range and that the effect of elastic compressibility will be negligible. Since the remainder of the analysis is considerably simplified if elastic incompressibility is assumed, we let v + l/2 and G’jrj$ --f 0 such that G’j$/(l -2v) remains finite, and obtain Et - $0; -
i+ 1W:,
(4Y
(+E + 2~:)
(5E+24
x Ul,F - Wl,, +
&
1
WlJ}r2 dr -
= 0, (28)
$p(R -+t)3 Wf, i R-at
where the incompressibility condition G’+$.= Wl,F+~(Wl-U,)=O
(29)
has been employed in simplifying (25). We now make the approximation that the shear strain vanishes, viz. U1,F+Y1,
F = 0,
(30)
whose validity will be tested by the numerical results presented in Section 4. The requirement that the shear strain vanish, equation (30), together with the incompressibility condition (29), determines WI and U1 uniquely (to within a multiplicative constant), viz. U,(r) = l/i = l/(r+u,). W,(r) = 2/F = 2/(r + u,), (31) This is precisely the bifurcation mode available for rigid-plastic spherical shells (MILES, 1969). For this mode, the bifurcation mode deflections correspond to thinning at one pole with a corresponding thickening at the other. Anticipating that bifurcation will occur after the sphere has become fully-plastic, the equation of equilibrium in the fundamental state gives 0:,i =2(&-r&/i:
= 2Y/?,
(32) The assumption that bifurcation where Y is the current value of the flow stress. occurs after the shell has become fully-plastic is verified by the numerical results given in Section 4. When (31) and (32) are employed in (28), the approximate bifurcation condition can be expressed as R++~E -y di: = 0. (33) I,= s +R-at
A. NEEDLEMAN
364
For pressure loading, but not for the case of prescribed change in volume, the mode n = 0 is also admissible. An analysis along similar lines gives the bifurcation condition
ICI=
R+f’2E,-3Y
s ~ F4
dr = 0.
R-&i
(34)
Unlike (33), this condition is exact for an incompressible material since the incompressibility condition alone determines W, to within a multiplicative constant. Furthermore, for either type of loading it can be shown that (34) is also the condition for the pressure in the fundamental state to reach a maximum. MILES(1969) has shown that if, considered as functions of ?, Et is monotonically increasing and Y is monotonically decreasing, then I, > I, regardless of the shell thickness. For materially homogeneous shells, but not necessarily composite ones, this requirement will be met. For a rigid-plastic shell subject to prescribed change in enclosed volume, I, > 0 is an exact sufficient condition for uniqueness (MILES, 1969). More recently, STRIFORS and STORAKERS (1973) have shown that it is also a necessary condition, in the sense that bifurcation is possible when I, = 0. Furthermore, in the rigid-plastic case, it = 1 is the only mode available for this type of loading. Thus, for a rigid-plastic spherical shell subject to prescribed change in enclosed volume, bifurcation cannot take place until after the maximum pressure has been attained. When pressure loading is prescribed, the mode n = 0 is also admissible in the bifurcation functional, and, thus, the uniqueness condition fails to hold at the maximum pressure point. At this point the shell becomes unstable in the sense of HILL (1958). For an elastic-plastic spherical shell the criterion (33) is only approximate. Furthermore, bifurcation could first occur into a mode not available for a rigidplastic material. Thus, the possibility remains that an elastic-plastic shell could undergo bifurcation prior to the maximum pressure point for either prescribed pressure loading or prescribed change in enclosed volume. This possibility is investigated by a numerical method described in Section 4.
4.
NUMERICALMETHODAND RESULTS
The method employed to implement a numerical solution to (22) for arbitrary mode number n is based on the finite element method. The interval CR,-+t,, R,+&,J is divided into sub-intervals of length h = t&i - 1) where j is the number of nodal points. An increment of the fundamental solution is calculated by a finite element method similar to that employed in previous finite-strain plasticity studies (see, for example, NEEDLEMAN (1972)). The displacement increment ti, is taken to be linear in r within each sub-interval. In the numerical calculation of the fundamental solution, increments of W are prescribed and the corresponding pressure p is determined by extrapolating 0: to the inner radius r = R,-+t,. The numerical results show that no unloading occurs in the fundamental solution, even after the maximum pressure has been attained, thus justifying the use of (8), the linear comparison solid. After each increment a check is made for bifurcation in several modes, usually those with mode numbers n = 1,2, 3.
Bifurcationof elastic-plasticsphericalshells
365
For a given pressure p and given mode number n the variational equation (6J for the functional (22) becomes A(p; n)x = 0,
(35) where the symmetric matrix A is the assembled stiffness matrix determined from (16) and x is the vector of bifurcation mode displacement. In formulating (35), the functions u,(r) and W,(r) are taken to be linear in each sub-interval. Cholesky decomposition is due to factor A into A = M*M,
where M is a lower triangular matrix and MT its transpose.
(36) The determinant of A is
det A = (det M)‘,
(37) and det M is the product of its diagonal entries. If one of these diagonal entries is zero, bifurcation has occurred in the nth mode at this value of the pressurep and the The next increment of the fundamental bifurcation mode is then determined. solution is calculated and the complete fundamental solution is determined in a linear incremental fashion. For the mode n = 1, it is necessary to constrain the spherical shell against rigidbody translation. This is accomplished by prescribing W&J = 0. (38) Several checks were made on the accuracy of this numerical procedure. For a thin shell, the fundamental solution obtained by this method was in excellent agreeSecondly, the bifurcation functional (22) ment with the thin-shell approximation. can readily be changed to apply to shells subject to external pressure. In this case, for sufficiently thin shells, agreement was obtained between the results of this numerical method and a thin-shell theory analysis, both for the critical pressures and the critical mode numbers. The results to be presented here were obtained with values of j between 11 (10 sub-intervals) and 41 (40 sub-intervals). For a thick shell, to/R, = 0.4, calculations with j = 11 and j = 21, using the same number of increments, gave agreement within 1%. Increasing the number of increments usually resulted in a greater improvement in accuracy, particularly in locating the precise value of W at the maximum pressure point. Generally, 40 to 60 increments were employed in a calculation. This program was run on an IBM 370/165 computer and a complete calculation took between 6 and 25 s, depending on the number of nodal points, the number of bifurcation modes checked, and the number of increments. Numerical results were obtained for various values of the material parameters and for shells varying in thickness between to/R,, values of 0.002 and 1.0. Some of these results are summarized in Table 1, wherein the superscript “c” denotes quantities evaluated at bifurcation, the superscript “max” denotes quantities evaluated at the maximum pressure, and Y,, is the average value of the flow stress through the thickness. In every case it was found that bifurcation first occurred in the mode n = 1, which satisfies the constraint (7). Furthermore, bifurcation always occurred after the shell had become fully-plastic and after the maximum pressure had been attained. As can be seen in Table 1, the bifurcation pressure is only slightly less than the maximum pressure. However, the value of the normal displacement at the inner
366
A. NEEDLEMAN TABLE1. Bifurcation
m
%I
V
&I&
Critical mode number,
LI$
results
P”l%
P”lP”“”
W O/R0 WC/ W-”
0.0148 0.00650 0.0787 1.29 3.03 0.0526 0.0592
0.974 0.990 0.987 0.983 0.918 0.996 0.990
0.18 0.065 0.065 0.092 0.20 0.024 0.071
nc
3 8 8 8 8 20 8
0.005 0*005 0.001 0*005 0.005 O*OOl 0.01
0.3 0.3 0.3 o-3 0.3 0.3 0.3
0*002 0.002 0.02 0.4 1.0 0.02 0.02
1 1 1 1 1 1 1
5.848 1a938 2.374 1.939 I.976 1.412 1.775
1.5 1.48 1.52 1.55 2.6 1.5 1.42
radius at bifurcation WC is significantly larger than its value at the maximum pressure IV”““. Even for very thin shells, the difference between the maximum pressure point and the bifurcation point is significant. This can be seen directly from (33) and (34). For a thin shell for which the flow stress Y is nearly constant through the thickness, equation (33) reduces to Y” = E; (3% and (34) gives the well-known result (see, for example, JOHNSONand MELLOR(1962)) Ymax= W,“““. (40) Thus, bifurcation inherently occurs after the maximum pressure is attained. This is in contrast to the necking bifurcation of elastic-plastic tensile bars. For thin tensile bars the bifurcation load approaches the maximum load (HUTCHINSONand MILES, 1974). However, for spherical shells subject to prescribed change in enclosed volume, bifurcation occurs significantly after the maximum pressure even for very thin shells. Here, at each stage of the computation, the approximate criteria (33) and (34) were evaluated using the computed stress distributions. For thin shells (t,/R, values of OGO2to O-02), excellent agreement was obtained between the finite element results shown in Table 1, the approximate criteria (33) and (34), and the simple formulae (39) and (40), even though Poisson’s ratio was taken to be 0.3 in the numerical computations. As stated previously, equation (33) is approximate even for an incompressible elastic-plastic material, since the bifurcation mode shear-strain was neglected in the derivation. Evidently, for thin shells the geometry constrains the shear strain to be negligible. Of course, the precise specification of “thin” depends on the material properties. For a thin spherical shell, the fundamental solution corresponds to an approximate state of equal biaxial tension and, when the change in enclosed volume is prescribed, bifurcation first takes place when (39) holds. In contrast to this result for a spherical shell, a thin flat plate subject to equal biaxial tension, with displacement boundary conditions, does not bifurcate until the flow stress is of the order of an elastic modulus (DUBEYand ARIARATNAM, 1969). For a thick shell, t,/R, = O-4, there was a significant discrepancy between the predictions of (33) and (34) for IV”“” and w” and the numerical results. This calculation was repeated with Poisson’s ratio v equal to 0.48 and all other parameters fixed. In this case there was good agreement between the value of IV”“” obtained
Bifurcation of elastic-plastic spherical shells
367
by evaluating (34) and the numerical value. However, the value of WC/R,, obtained from (33) was O-072while that obtained from the finite element computation was O-097. Since bifurcation occurs after the limit point of the fundamental solution, the critical pressure obtained by the numerical method is less than that obtained from the approximate criterion (33). The present results show for a materially homogeneous elastic-plastic spherical shell subject to internal pressure loading that uniqueness of deformation is maintained until the maximum pressure is attained, at least when consideration is restricted to axisymmetric bifurcation modes. If the pressure is prescribed, then the shell becomes unstable at the maximum pressure point; on the other hand, if the change in enclosed volume is prescribed, then the limit point can be passed and bifurcation occurs in the mode n = 1. However, the possibility remains that an imperfection could activate this bifurcation mode before the maximum pressure is attained. ACKNOWLEDGMENT The writer is indebted to Professor J. W. Hutchison (Division of Engineering and Applied Physics, Harvard University) for valuable comments. This work was partially supported by U.S. NSF. Grant GP 22796.
REFERENCES DUBEY,R. N. and ARIARATN~, S. T. HILL, R.
HUTCHINSON,J. W.
HUTCHINSON,J. W. and MILES, J. P. JOHNSON,W. and MELLOR,P. B. MILES, J. P. NEEDLEMAN, A. SEWELL,M. J. S-IRIFORS,H. and STO&ERS, B.
1969
Quart. appl. Math. 27, 381.
1958 1961
J. Mech. Phys. Solids 6, 236. Problems of Continuum Mechanics (Contributions in Honor of the Seventieth Birthday of Academician N. I. Muskhelishvili, 16th February 1961; edited by M. A. LAVRENT’EV et al.) (English edition; edited by RADOK, J. R. M.), p. 155. Society for Industrial and Applied Mathematics, Philadelphia. J. Mech. Phys. Solids 10,185. Numerical Solution of Nonlinear Structural Problems (edited by HARTNUNG, R. F.), p. 17. American Society of Mechanical Engineers, New York. J. Mech. Phys. Solids 21, 163. Ibid. 22, 61.
1962 1973a
1973b 1974 1962 1969 1972 1965 1973
Plasticity for Mechanical Engineers, p. 184. Wiley, New York. J. Mech. Phys. Solids 17, 303. Ibid. 20, 111. Proc. Roy. Sot. London A286,402. J. Mech. Phys. Solidr 21, 125.
Pergamon Press.
Printed in Great Britain.
BIFURCATION OF ELASTIC-PLASTIC SPHERICAL SHELLS SUBJECT TO INTERNAL PRESSURE By A. NEEDLEMAN? Department
of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, U.S.A. (Received 2nd January 1975)
THE BIFURCATION of elastic-plastic
spherical shells subject to internal pressure loading is investigated. Two cases are considered; in one case, the pressure is prescribed, while in the other, the change in volume enclosed by the shell is prescribed. Attention is restricted to axisymmetric bifurcations. Separation of variables is employed to introduce a complete set of bifurcation modes. For one of these modes a simple approximate bifurcation criterion is obtained. A numerical method is then employed to investigate the bifurcation of elastic-plastic spherical shells in a variety of modes. A comparison is made between the numerical results and the predictions of the approximate criterion. Of particular interest is whether bifurcation can occur prior to the maximum pressure point. The numerical results obtained here show that this does not occur for the axisymmetric modes considered.
1. INTRODUCTION GENERALLY, failure of spherical shells subject to internal pressure is taken to occur when the pressure, given by an analysis assuming spherical symmetry, reaches a maximum. Such a failure criterion excludes the possibility that some bifurcation mode could be activated before this maximum is attained. For spherically-symmetric loading conditions, MILES (1969) found the most general bifurcation mode available for a rigid-plastic material with a non-singular yield surface. He also found for a materially homogeneous rigid-plastic spherical shell with a non-singular yield surface that uniqueness of deformation is guaranteed until the maximum pressure is attained. More recently, STRIFORS and STOR~;KERS (1973) showed that Miles’ sufficient condition for uniqueness is also necessary, in the sense that bifurcation is possible at the instant when Miles’ sufficient condition ceases to hold. However, the possibility remains that an elastic-plastic spherical shell could bifurcate into a mode not available in the rigid-plastic case. For example, there is no three-dimensional rigid-plastic bifurcation mode that can adequately describe the necking of tensile bars, although there is such a mode available for elastic-plastic materials. Here, the bifurcation of elastic-plastic spherical shells subject to internal pressure is investigated. The boundary-value problem is posed in two ways. In one case, the pressure is taken to be prescribed, while in the other, the change in volume
t Now at Division of Engineering, Brown University, Providence, Rhode Island 02912, U.S.A. 357
A. NEEDLEMAN
358
enclosed by the shell is prescribed. In both, the eigenvalue problem governing bifurcation is posed in terms of a variational principle due to HILL (1961, 1962), and in this formulation full account is taken of both geometrical and material nonlinearities. Consideration is restricted to axisymmetric bifurcations and separation of variables is employed to introduce a complete set of bifurcation modes. Although the functional governing bifurcation is identical for both types of loading, the admissible functions differ. When the change in enclosed volume is prescribed, there is a constraint condition that must be satisfied by the bifurcation mode. For one mode that satisfies this constraint condition, a simple approximate bifurcation criterion is obtained. This approximate bifurcation criterion for elastic-plastic materials is formally identical to an exact criterion for rigid-plastic materials obtained by MILES (1969). A numerical method, based on the finite element method, is then employed to investigate the bifurcation of elastic-plastic spherical shells for a variety of modes. A comparison is made between the numerical results and the predictions of the approximate criterion. 2.
PROBLEMFORMULATION
The formulation of the field equations employed in this study has recently been reviewed by HUTCHINSON (1973a), where further details and more complete references are given. For completeness, some of the basic equations are presented here. This formulation is Lagrangian in character and employs the undeformed configuration of the body as reference. This choice is a convenient one for implementing the numerical method that will be described subsequently. However, many of the basic equations do take on a simpler form if the current configuration of the body is taken to be the reference configuration. The volume and surface of a body in the undeformed configuration are denoted by V and S respectively, and each particle is labelled by a set of convected coordinates xi (i = 1,2,3) which serve as independent variables. In the undeformed body, the covariant components of the metric tensor are denoted by gij and its determinant by g ; while in the deformed body, these quantities are denoted by G, and G respectively. The Langrangian strain increments are given by ?jij = +(tii, i Urn, where uk are the covariant components of the displacement vector referred to the undeformed base vectors, and ( ), i denotes covariant differentiation with respect to the undeformed metric tensor. Here, the operation (.) denotes differentiation with respect to some monotonically increasing parameter that characterizes the load history, and corresponding derivatives are termed ‘increments’ (and not ‘rates’). This parameter is taken to be the radial displacement at the spherical shell inner radius W. The contravariant components of the Kirchhoff stress tensor on the embedded deformed-coordinates are denoted by yik. These are related to the contravariant components of the Cauchy stress tensor aik by qik = (G/g)‘/20’k_ (2) j
+
ti j,
i +
glm(li’,
j
+
U’,
iti”,
j)),
(1)
Bifurcation of elastioplastic
For the problems considered takes the form
spherical shells
here the incremental
359
principle of virtual work
J (4iJ’81jij+qiiglmti1,i6ri”,j) dV = -fi J a”n,&& dS_p SP
$ c?“n,&& dS, SP
(3)
where the strain increment variations S#j are related to the displacement increment variations by (1) and S, is that surface on which the pressure or the change in enclosed volume is prescribed. The second surface-integral on the right-hand side of (3) arises from the configuration dependence of the loading, and c$’ and d” are given by (see, for example, SEWELL(1965)) ui’ = ~sjk’s’“‘(gj~+ UJ,I)(gkm+ uk ,), 6” = girGl,I _ gQr, I + &lmrUj, I’&, ~,1
(4)
where siki is the alternating tensor. The stress increments 4” are related to the strain increments 311by a constitutive relation (to be stated subsequently) that is of the form 4” = LiMg,. (5) The instantaneous moduli have the symmetries L!jkl = Lkzij= tikij = Lk’ji and have two branches, one corresponding to plastic loading and the other to elastic unloading. For a spherical shell subject to a prescribed internal pressure or to a prescribed change in enclosed volume, one equilibrium state, corresponding to a sphericallysymmetrical deformation, is available for all values of W. This state is termed the fundamental solution and is unique for sufficiently small values of W. The bifurcation criterion employed is an application of HILL’S (1958, 1961, 1962) general theory of bifurcation and uniqueness in elastic-plastic solids to this particular problem. Bifurcation from the fundamental solution first becomes possible at some critical displacement WC,corresponding to a pressure pc. Then, with the body in a state characterized by a displacement field u and a stress field q, a non-trivial solution exists to the set of homogeneous equations I = S(~*i’i~+qiig,mti*z,~ti*m,j) dV+p, 6I=‘;.
J c?*“n,$ SP
dS = 0,
(6) >
The superscript * denotes the difference between field quantities associated with the fundamental solution and those associated with the second solution, and q:j and d**’ are related to the displacement increments ti: by (1) and (42), respectively. The identity (6,) is derived by considering the difference between two fields satisfying (3). If the increment of pressure is prescribed, then p* vanishes, while if the change in enclosed volume is prescribed, then the change in volume due to ti: is zero, so that I ai'nr ti: dS = 0. (7) SP A sufficient condition for uniqueness of deformation is that the functional I in (6,) be positive for all admissible displacement fields ii:, and bifurcation first becomes possible when this functional vanishes. Although the bifurcation functional I in (6,) is employed for both types of loading, the admissible functions differ. When the change in volume enclosed by the shell is prescribed, the displacements ti: must satisfy the constraint (7).
360
A. NEEDLEMAN
If the increment in the fundamental solution at WChas the property that loading occurs everywhere in the current plastic zone, then the stress-rates in (4,) are related to the strain-rates by * *ii = Ejk$,;. (8) 4 Furthermore, the branch of the tensor of moduli for plastic loading is to be used in the current plastic zone in (8). Thus, the system (4) and (8) is independent of the loading-unloading criterion; and in HILL’S (1961, 1962) terminology a linear comparison solid has been introduced. Although no attempt is made here to study the post-bifurcation behavior of the spherical shell, we note that the amplitude of the bifurcation mode is uniquely determined by the requirement that loading occurs everywhere in the current plastic zone, except at one point or along one curve where neutral loading takes place (HUTCHINSON,1973b). The particular stress-strain law used in this study is a large-strain generalization of J,-flow theory with isotropic hardening, due to Professor B. Budiansky (unpublished work, 1968). This stress-strain law is meant to apply to situations in which the elastic strains remain small and the total strain increment is written as the sum of the elastic strain increment s~j and the plastic strain increment Qfj. The elastic strain-increment/stress-increment relation is taken to be Sj
=
~
((1 + V)G,G, - VGijGk,)G“t
(9)
where E is Young’s modulus, v is Poisson’s ratio, and ek’ are the contravariant components of the symmetric Jaumann derivative of the Cauchy stress, which are related to the contravariant components of the convected derivative 6k’ by bk’ = ekl + Gkm&j,, + GL”ckmdmn.
(10) The stress-strain relation (9) is not actually elastic in that it does not admit a potential in the large strain region, but when linearized it does give Hooke’s law. The yield function for J,-flow theory with isotropic hardening is F = oe-- Y, where Y, the flow stress, is either the maximum value of rre over the stress history or the initial yield stress g,,, whichever is the greater, and u, is given by Ue = (3GikGjrSiiSkz)“‘.
(11)
where Sij = oii _ gGiiGkle”.
(12)
The generalized flow rule takes the form q$ =
%1/G- 1IEPij &Joe if Q, = Y and b, 2 0,
1 0
if (i) (TV< Y or (ii) (T, = Y and 5’, I 0.
(13)
Here, Et is a function of the stress invariant 6, and is the slope of the uniaxial truestress/natural-strain curve. The particular model of uniaxial stress-strain behavior that will be employed is a piecewise continuous power-law with a continuous tangent modulus, i.e.
Bifurcation of elastic-plastic spherical shells
361
where E is the natural strain, 0 is the true stress, a,, is the uniaxial yield stress, E,,is the yield strain, given by sy = a,,/E, and m is the strain-hardening exponent. The stress-strain relation (14) can be inverted to obtain the tensor of moduli C relating the contravariant components of Jaumann derivative of the Cauchy stress 8’j to the strain increments rjkrwhere
and h =
0 if (i) ce < Y or (ii) ce = Y and S”Qij I 0. (16) > 1 1 if g’e= Y and S”~, 2 0. The tensor of moduli L in (5), relating the convected derivative of Kirchhoff stress 4” to the strain increments ljklr is found from (IO), (15) and 4 rj = ( Glg)3$j + 4 ijGk$,, (17) to be Vkl= (GIg)+Cijkr_ +(Gik@ + Gjk# + Gil&k+ Gjrqrk)+ qijGkl. L” (18) The moduli L!jkl do not possess the symmetry L!jkz= Lk”’ necessary for the variational principle (42) to hold unless the material is elasticully incompressible, in which case (G/g)* = 1 and 8 = 4”. Therefore, in order to obtain the necessary symmetry we make the approximation that oii zz q’j, and 8 will be replaced by qij in the constitutive equations (7) to (11). This approximation will involve little error, since in (2) the factor relating & and q” is equal to the ratio of the volume in the deformed configuration to that in the undeformed configuration. The volume change is due entirely to the elastic strains which, by assumption, remain small. With this approximation, rfjkl
=
cijkl
_J(Gikqji
+
Gjkqi’
+
Gilqjk
+
Gjlqik). (19)
HUTCHINS~N(1973a)
has proposed a slightly modified version of this constitutive equation in which the Cauchy stress components and derivatives appearing in (10) to (12) are replaced by the corresponding Kirchhoff stress components and derivatives. In this alternative formulation, (19) is obtained without approximation, but the tangent modulus Et differs slightly from the slope of the true-stress/natural-strain curve. The tensor of moduli (19) obtained here is identical to that employed in previous studies (see, for example, NEEDLEMAN(1972)). However, the present development has followed that given by HUTCHINSON(1973a) except that, jn line with Professor B. Budiansky’s original proposal (as mentioned- earlier in this section), the Cauchy stress tensor has been employed as the basic stress measure in the formulation.
3.
BIFURCATIONANALYSIS
The initial mean radius of the spherical shell is R, and the initial thickness is t,. Spherical coordinates are employed with x1 = r, x2 = 8, x3 = 4. In the fundamental solution the non-vanishing displacement and stress components are ul = M), rr:(r) = G, 1d ‘,
a;(r) = G22~22 = G33g33.
(20)
362
A. NEEDLEMAN
As stated previously, consideration is restricted to axisymmetric bifurcations and we seek bifurcation modes of the form tiy = W,(r)P,(cos
e),
tiz* = rU,(r)
dP,(cos de
e) ’
where P, is the Legendre polynomial of degree n (= 0, 1,2, . . .). The set of bifurcation modes (21) gives an exact separation of variables and is complete in that any sufficiently smooth axisymmetric admissible functions ai, tiz can be expressed as a linear combination of modes of the form (21). Here, we note that an essential condition on the function tiz is that it vanish at the poles, 8 = (0, n). Each of the functions W, and U, depends on the material parameters: oJE, the yield strain, v, Poisson’s ratio, and m, the strain-hardening exponent, although this dependence is not explicitly noted in (21). For the case of prescribed pressure loading, any mode of the form (21) is admissible, whereas when the change in volume enclosed by the shell is prescribed, the constraint (7) excludes the mode with n = 0. If the elastic strains are neglected so that the total strain increment satisfies the flow rule (13), then the available modes are severely restricted. MILES (1969) has shown that in this case, for pressure loading, the only admissible modes are those with n = Oandn = 1. When the mode assumption (21) is substituted into the bifurcation functional (6,), we obtain Ro+tt. I, = j {K,W,2+(2K3+2K4+n(n+l)K,)W,Z+[n2(n+1)2K3+n(n+1)(K,-KK,) R,- *to +n(n+l)K,]U,2+n(n+1)K6U~2+4K2W,‘W,-2n(n+1)(K,+K,+K,)W,U, -2n(n+l)K,
W,lU,+2K,n(n+l)U~W,-2K,n(n+1)U~U,}r2
dr
-P[R,-~t,+u,(R,-~t,)][2W,‘-22n(n+l)W,U,+n(n+l)Ui,Z]/.~-_t,~
= 0,
(22)
where ( )’ denotes d( )/dr and K3 = GZZ(Gz2 L2222 -I-&,
K2 = (G11G22)*L!122, K _ G22G G L2233
K5 = G22(G11G22L1212+q;),
K4,:
K1 = G”(G:lL1”l+q;),
K, = (G”G22)+(G,lG2ZL1212), G” = (1 +u;)-‘,
(23)
G11(b:I;232L121’+q;), *I G33 = G22 sin2 0
G22 = (r + u,)-2,
(24)
A sufficient condition for uniqueness is that 1, > 0 for all admissible n and a numerical method to be described subsequently will be employed to implement a numerical solution to (22) for arbitrary n. Here, the special case II = 1 will be analyzed in greater detail. The bifurcation functional (22) reduces to ;_f*yk&
[w:,F+4v
+c2
(?7-
Av
4wI,i
wl,~+~(wl-u,) -
(y9]
) 2+
&
[ul,‘+yq2
(
--4: W&+2q:U:,i
(g/G)*?
di-2p(R-~t)(WI-Ul)Z
= 0, R-ft
(25)
Bifurcation of elastic-plastic
363
spherical shells
where ? - r+u,,
(),P = (l/(1+4))
(26)
dz,
while R and t denote the current radius and thickness, respectively. The change of variable by (26) is equivalent to changing the reference configuration from the undeformed configuration to the current one. The constants C, and C, are given by c, = l/3(1 +v)(1 -$(l-2V)E,/E),
(27) > C, = S{Cl -(v/(1 +v))(l -WW1/[1-4(~ -NWlh The quantity in (25) multiplied by E/(1 -2~) is the volume change G’jrjjfj. It is anticipated that bifurcation will occur far into the plastic range and that the effect of elastic compressibility will be negligible. Since the remainder of the analysis is considerably simplified if elastic incompressibility is assumed, we let v + l/2 and G’jrj$ --f 0 such that G’j$/(l -2v) remains finite, and obtain Et - $0; -
i+ 1W:,
(4Y
(+E + 2~:)
(5E+24
x Ul,F - Wl,, +
&
1
WlJ}r2 dr -
= 0, (28)
$p(R -+t)3 Wf, i R-at
where the incompressibility condition G’+$.= Wl,F+~(Wl-U,)=O
(29)
has been employed in simplifying (25). We now make the approximation that the shear strain vanishes, viz. U1,F+Y1,
F = 0,
(30)
whose validity will be tested by the numerical results presented in Section 4. The requirement that the shear strain vanish, equation (30), together with the incompressibility condition (29), determines WI and U1 uniquely (to within a multiplicative constant), viz. U,(r) = l/i = l/(r+u,). W,(r) = 2/F = 2/(r + u,), (31) This is precisely the bifurcation mode available for rigid-plastic spherical shells (MILES, 1969). For this mode, the bifurcation mode deflections correspond to thinning at one pole with a corresponding thickening at the other. Anticipating that bifurcation will occur after the sphere has become fully-plastic, the equation of equilibrium in the fundamental state gives 0:,i =2(&-r&/i:
= 2Y/?,
(32) The assumption that bifurcation where Y is the current value of the flow stress. occurs after the shell has become fully-plastic is verified by the numerical results given in Section 4. When (31) and (32) are employed in (28), the approximate bifurcation condition can be expressed as R++~E -y di: = 0. (33) I,= s +R-at
A. NEEDLEMAN
364
For pressure loading, but not for the case of prescribed change in volume, the mode n = 0 is also admissible. An analysis along similar lines gives the bifurcation condition
ICI=
R+f’2E,-3Y
s ~ F4
dr = 0.
R-&i
(34)
Unlike (33), this condition is exact for an incompressible material since the incompressibility condition alone determines W, to within a multiplicative constant. Furthermore, for either type of loading it can be shown that (34) is also the condition for the pressure in the fundamental state to reach a maximum. MILES(1969) has shown that if, considered as functions of ?, Et is monotonically increasing and Y is monotonically decreasing, then I, > I, regardless of the shell thickness. For materially homogeneous shells, but not necessarily composite ones, this requirement will be met. For a rigid-plastic shell subject to prescribed change in enclosed volume, I, > 0 is an exact sufficient condition for uniqueness (MILES, 1969). More recently, STRIFORS and STORAKERS (1973) have shown that it is also a necessary condition, in the sense that bifurcation is possible when I, = 0. Furthermore, in the rigid-plastic case, it = 1 is the only mode available for this type of loading. Thus, for a rigid-plastic spherical shell subject to prescribed change in enclosed volume, bifurcation cannot take place until after the maximum pressure has been attained. When pressure loading is prescribed, the mode n = 0 is also admissible in the bifurcation functional, and, thus, the uniqueness condition fails to hold at the maximum pressure point. At this point the shell becomes unstable in the sense of HILL (1958). For an elastic-plastic spherical shell the criterion (33) is only approximate. Furthermore, bifurcation could first occur into a mode not available for a rigidplastic material. Thus, the possibility remains that an elastic-plastic shell could undergo bifurcation prior to the maximum pressure point for either prescribed pressure loading or prescribed change in enclosed volume. This possibility is investigated by a numerical method described in Section 4.
4.
NUMERICALMETHODAND RESULTS
The method employed to implement a numerical solution to (22) for arbitrary mode number n is based on the finite element method. The interval CR,-+t,, R,+&,J is divided into sub-intervals of length h = t&i - 1) where j is the number of nodal points. An increment of the fundamental solution is calculated by a finite element method similar to that employed in previous finite-strain plasticity studies (see, for example, NEEDLEMAN (1972)). The displacement increment ti, is taken to be linear in r within each sub-interval. In the numerical calculation of the fundamental solution, increments of W are prescribed and the corresponding pressure p is determined by extrapolating 0: to the inner radius r = R,-+t,. The numerical results show that no unloading occurs in the fundamental solution, even after the maximum pressure has been attained, thus justifying the use of (8), the linear comparison solid. After each increment a check is made for bifurcation in several modes, usually those with mode numbers n = 1,2, 3.
Bifurcationof elastic-plasticsphericalshells
365
For a given pressure p and given mode number n the variational equation (6J for the functional (22) becomes A(p; n)x = 0,
(35) where the symmetric matrix A is the assembled stiffness matrix determined from (16) and x is the vector of bifurcation mode displacement. In formulating (35), the functions u,(r) and W,(r) are taken to be linear in each sub-interval. Cholesky decomposition is due to factor A into A = M*M,
where M is a lower triangular matrix and MT its transpose.
(36) The determinant of A is
det A = (det M)‘,
(37) and det M is the product of its diagonal entries. If one of these diagonal entries is zero, bifurcation has occurred in the nth mode at this value of the pressurep and the The next increment of the fundamental bifurcation mode is then determined. solution is calculated and the complete fundamental solution is determined in a linear incremental fashion. For the mode n = 1, it is necessary to constrain the spherical shell against rigidbody translation. This is accomplished by prescribing W&J = 0. (38) Several checks were made on the accuracy of this numerical procedure. For a thin shell, the fundamental solution obtained by this method was in excellent agreeSecondly, the bifurcation functional (22) ment with the thin-shell approximation. can readily be changed to apply to shells subject to external pressure. In this case, for sufficiently thin shells, agreement was obtained between the results of this numerical method and a thin-shell theory analysis, both for the critical pressures and the critical mode numbers. The results to be presented here were obtained with values of j between 11 (10 sub-intervals) and 41 (40 sub-intervals). For a thick shell, to/R, = 0.4, calculations with j = 11 and j = 21, using the same number of increments, gave agreement within 1%. Increasing the number of increments usually resulted in a greater improvement in accuracy, particularly in locating the precise value of W at the maximum pressure point. Generally, 40 to 60 increments were employed in a calculation. This program was run on an IBM 370/165 computer and a complete calculation took between 6 and 25 s, depending on the number of nodal points, the number of bifurcation modes checked, and the number of increments. Numerical results were obtained for various values of the material parameters and for shells varying in thickness between to/R,, values of 0.002 and 1.0. Some of these results are summarized in Table 1, wherein the superscript “c” denotes quantities evaluated at bifurcation, the superscript “max” denotes quantities evaluated at the maximum pressure, and Y,, is the average value of the flow stress through the thickness. In every case it was found that bifurcation first occurred in the mode n = 1, which satisfies the constraint (7). Furthermore, bifurcation always occurred after the shell had become fully-plastic and after the maximum pressure had been attained. As can be seen in Table 1, the bifurcation pressure is only slightly less than the maximum pressure. However, the value of the normal displacement at the inner
366
A. NEEDLEMAN TABLE1. Bifurcation
m
%I
V
&I&
Critical mode number,
LI$
results
P”l%
P”lP”“”
W O/R0 WC/ W-”
0.0148 0.00650 0.0787 1.29 3.03 0.0526 0.0592
0.974 0.990 0.987 0.983 0.918 0.996 0.990
0.18 0.065 0.065 0.092 0.20 0.024 0.071
nc
3 8 8 8 8 20 8
0.005 0*005 0.001 0*005 0.005 O*OOl 0.01
0.3 0.3 0.3 o-3 0.3 0.3 0.3
0*002 0.002 0.02 0.4 1.0 0.02 0.02
1 1 1 1 1 1 1
5.848 1a938 2.374 1.939 I.976 1.412 1.775
1.5 1.48 1.52 1.55 2.6 1.5 1.42
radius at bifurcation WC is significantly larger than its value at the maximum pressure IV”““. Even for very thin shells, the difference between the maximum pressure point and the bifurcation point is significant. This can be seen directly from (33) and (34). For a thin shell for which the flow stress Y is nearly constant through the thickness, equation (33) reduces to Y” = E; (3% and (34) gives the well-known result (see, for example, JOHNSONand MELLOR(1962)) Ymax= W,“““. (40) Thus, bifurcation inherently occurs after the maximum pressure is attained. This is in contrast to the necking bifurcation of elastic-plastic tensile bars. For thin tensile bars the bifurcation load approaches the maximum load (HUTCHINSONand MILES, 1974). However, for spherical shells subject to prescribed change in enclosed volume, bifurcation occurs significantly after the maximum pressure even for very thin shells. Here, at each stage of the computation, the approximate criteria (33) and (34) were evaluated using the computed stress distributions. For thin shells (t,/R, values of OGO2to O-02), excellent agreement was obtained between the finite element results shown in Table 1, the approximate criteria (33) and (34), and the simple formulae (39) and (40), even though Poisson’s ratio was taken to be 0.3 in the numerical computations. As stated previously, equation (33) is approximate even for an incompressible elastic-plastic material, since the bifurcation mode shear-strain was neglected in the derivation. Evidently, for thin shells the geometry constrains the shear strain to be negligible. Of course, the precise specification of “thin” depends on the material properties. For a thin spherical shell, the fundamental solution corresponds to an approximate state of equal biaxial tension and, when the change in enclosed volume is prescribed, bifurcation first takes place when (39) holds. In contrast to this result for a spherical shell, a thin flat plate subject to equal biaxial tension, with displacement boundary conditions, does not bifurcate until the flow stress is of the order of an elastic modulus (DUBEYand ARIARATNAM, 1969). For a thick shell, t,/R, = O-4, there was a significant discrepancy between the predictions of (33) and (34) for IV”“” and w” and the numerical results. This calculation was repeated with Poisson’s ratio v equal to 0.48 and all other parameters fixed. In this case there was good agreement between the value of IV”“” obtained
Bifurcation of elastic-plastic spherical shells
367
by evaluating (34) and the numerical value. However, the value of WC/R,, obtained from (33) was O-072while that obtained from the finite element computation was O-097. Since bifurcation occurs after the limit point of the fundamental solution, the critical pressure obtained by the numerical method is less than that obtained from the approximate criterion (33). The present results show for a materially homogeneous elastic-plastic spherical shell subject to internal pressure loading that uniqueness of deformation is maintained until the maximum pressure is attained, at least when consideration is restricted to axisymmetric bifurcation modes. If the pressure is prescribed, then the shell becomes unstable at the maximum pressure point; on the other hand, if the change in enclosed volume is prescribed, then the limit point can be passed and bifurcation occurs in the mode n = 1. However, the possibility remains that an imperfection could activate this bifurcation mode before the maximum pressure is attained. ACKNOWLEDGMENT The writer is indebted to Professor J. W. Hutchison (Division of Engineering and Applied Physics, Harvard University) for valuable comments. This work was partially supported by U.S. NSF. Grant GP 22796.
REFERENCES DUBEY,R. N. and ARIARATN~, S. T. HILL, R.
HUTCHINSON,J. W.
HUTCHINSON,J. W. and MILES, J. P. JOHNSON,W. and MELLOR,P. B. MILES, J. P. NEEDLEMAN, A. SEWELL,M. J. S-IRIFORS,H. and STO&ERS, B.
1969
Quart. appl. Math. 27, 381.
1958 1961
J. Mech. Phys. Solids 6, 236. Problems of Continuum Mechanics (Contributions in Honor of the Seventieth Birthday of Academician N. I. Muskhelishvili, 16th February 1961; edited by M. A. LAVRENT’EV et al.) (English edition; edited by RADOK, J. R. M.), p. 155. Society for Industrial and Applied Mathematics, Philadelphia. J. Mech. Phys. Solids 10,185. Numerical Solution of Nonlinear Structural Problems (edited by HARTNUNG, R. F.), p. 17. American Society of Mechanical Engineers, New York. J. Mech. Phys. Solids 21, 163. Ibid. 22, 61.
1962 1973a
1973b 1974 1962 1969 1972 1965 1973
Plasticity for Mechanical Engineers, p. 184. Wiley, New York. J. Mech. Phys. Solids 17, 303. Ibid. 20, 111. Proc. Roy. Sot. London A286,402. J. Mech. Phys. Solidr 21, 125.