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Wrinkling criterion for an anisotropic shell with compound curvatures in sheet forming
Pergamon
Int. J. Mech. $ci. Vol. 36, No. I0, pp. 945-960, 1994 Copyright © 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020-7403/94 $7.00 + 0.00
00~-740a(94)E000-0 WRINKLING CRITERION FOR AN ANISOTROPIC SHELL WITH C O M P O U N D CURVATURES IN SHEET FORMING CHUAN-TAO WANG, t GARY KINZEL ~ a n d TAYLAN ALTAN~ t General Motors Corporation, Technical Center, Box 9040, 30300 Mound Road, Warren, MI 48090-9040, U.S.A. and ~Engineering Research Center for Net Shape Manufacturing, The Ohio State University, Columbus, OH 43210, U.S.A. (Received 6 August 1993; and in revised form 15 February 1994)
Abstract--Wrinkling criteria are proposed for an elastic isotropie and plastic anisotropic shell with compound curvatures in the noncontact region of a sheet subjected to internal forming stresses. A quasi-shallow shell is modeled by DonnelI-Mushtari-Vlasov (DMV) shell theory. A bifurcation functional from Hill's general theory of uniqueness and bifurcation in elastic-plastic solids is used to model the local wrinkling phenomenon. Both strain hardening and transverse anisotropy are taken into consideration. This wrinkling criterion is especially useful as a failure criterion in three dimensional finite element modeling (FEM) of sheet forming. Given the principal stresses or strains and the geometry provided at each incremental deformation step, the criterion can be used to predict wrinkles in the elements in the unsupported region.
NOTATION A , B , C constants b~j curvature tensor of the middle surface in the prebuckling shell 'go prestrain effective strain d'go incremental strain tensor da~j incremental stress tensor 6o Kronecker index f(~,j) yield surface effective stress ~t loading index, flange angle incremental rotations CO compliance moduli incremental stretch strain tensor E elastic modulus Er instantaneous slope of the flow curve
H
(
scalar factor \ H = Erl _ _ I )
k strength coefficient incremental bending strains (or the changes of the curvatures) Lij instantaneous stiffness moduli M Hill index describing the shape of the yield locus dM o incremental stress resultants dN 0 incremental stress couples (per unit width) n strain hardening exponent normal anisotropy Ri radius of the principal curvature (RI or R2). 21,22 dimensionless wave numbers. 2]r critical wave number t thickness of the sheet. t3~ in-plane incremental displacements in the xl and x2 directions incremental buckling displacement normal to the shell mid-plane Xi coordinate (i = 1, 2, 3) 1. I N T R O D U C T I O N
In deep drawn sheet components, wrinkling and tearing are two major failure modes. Excessive compressive stresses cause wrinkling while excessive tensile stresses lead to 945
946
CHUAN-TAOWANGet
al.
fracturing. Careful control of the tensile restraining force generated by blankholders and drawbeads can avoid wrinkles and cracks. A number of approaches to control the blankholder force (BHF) and drawbead force (DBF) have been reported elsewhere I-1]. For prediction and control of fracture, a number of failure criteria, such as the forming limit diagram and localized necking instability may be used in conjunction with finite element modeling. However, there has been a lack of reliable wrinkling criteria for prediction and control of wrinkling, especially body wrinkling. There are three types of compressive instability problems that may occur in sheet forming operations. These instability problems are: flange wrinkling; wall wrinkling; and elastic buckling in the undeformed area due to residual elastic compressive stress. All of these can be observed in the forming of a common pie pan made from aluminium sheet. The flange wrinkling and wall wrinkling are basically the same and are caused by the compressive circumferential stress. However, wall wrinkling occurs far more easily than the wrinkling on the flat flange since the wall is relatively unsupported by the tool. The suppression of wall wrinkles by control of the blank holder force (BHF), which affects the radial tensile stress component, is more difficult than the suppression of flange wrinkles. It is known that wall wrinkling is more of a problem than flange wrinkling [2-7] in most press forming operations, and the blankholder force is far greater than that necessary to suppress flange wrinkling. The drawbead is then particularly designed to add an additional restraining force to prevent metal flowing into the die too freely. Therefore, wall wrinkling is the problem of most industrial importance and interest. However, very few studies I-8] have been conducted in understanding and controlling wall wrinkling. In modeling wall wrinkling, a curved shell model achieves better accuracy than a flat plate model. In the elastic buckling literature, DMV (Donnel-Mushtari-Vlasov) shell theory has been applied to study the wrinkling phenomenon of a three-dimensional shell with compound curvatures. The DMV model is based on shallow shell theory and is able to determine the critical condition (stress or strain) for a short wavelength wrinkling mode. Hutchinson [9] applied Hill's bifurcation criterion [10-11] to the DMV theory for a class of three-dimensional solids and proposed a general elastic-plastic bifurcation criterion for the DMV theory of plates and shells. Within the context of this model, a specific instability criterion was suggested to predict the conditions for the onset of wall wrinkling in doubly-curved and isotropic sheet metal undergoing a forming operation 1-12]. However, the importance of plastic anisotropy was not considered in this model and the enhancement of this model for anisotropic sheet materials is necessary. In this study, wrinkling criteria for an elastic isotropic and plastic anisotropic shell with compound curvature is proposed, based on DMV shell theory, bifurcation analysis and an incremental constitutive law incorporating Hill's nonquadratic yield criterion for transverse anisotropic materials [13]. The critical stresses and strains for the onset of wall wrinkling during sheet forming are related to the material properties (elastic modulus, Poisson's ratio, flow stress, strain hardening and plastic anisotropy), geometrical dimensions (wall inclination, die clearance, sheet thickness and principal curvatures) and loading conditions (ratios of radial tensile stress or strain to hoop compressive stress or strain). 2. INCREMENTAL STRESS-STRAIN CONSTITUTIVE EQUATIONS FOR ELASTIC ISOTROPIC AND PLASTIC ANISOTROPIC SOLIDS Most constitutive equations for the elasto-plastic deformation of a solid are derived based on the generalized Hook's law for elastic response and Von Mises' yield theory for isotropic solids undergoing plastic deformation [12, 14, 15]. A few studies deal with the anisotropic plastic response but none of these provide an explicit expression of the incremental moduli [16, 17]. In this section, a general expression for the constitutive equation will be given first for solids with the deformation behavior of elastic isotropy and plastic anisotropy. Then, an explicit formulation of the incremental stress-strain relation will be derived under plane stress conditions, which is the primary deformation mode in most sheet forming operations. Both strain hardening and plastic anisotropy are incorporated in this constitutive relation.
Wrinkling criterion for an anisotropic shell
947
2.1. Constitutive equation for elasto--plastic solids For a solid statically stressed beyond its elastic limit, a linear decomposition of the strain increments into elastic and plastic parts is valid because the recoverable elastic strains are small. The tensor expression for the strain increments is deij = de~j + de~ (i,j = 1, 2, 3),
(1)
where deo is the total increment strain, and the superscripts 'e' and 'p' stand for elastic and plastic, respectively. The generalized Hook's law relates the elastic strain increments to the stress increments (dtri~) as follows de~ = 1 [(1 + v)dtri~- vdtrkk6ij],
(2)
where E and v are Young's modulus and Poisson's ratio, dtTkk (k : 1, 2 and 3) is the sum of the three normal stresses according to the tensor summation convention and &j is the Kronecker delta defined by 6o =
if i ~ j.
The plastic strain increments may be defined in terms of a plastic potentialf(tro) or yield locus #(ao) if the flow (or incremental deformation) theory is applied. According to Hill [18-1, the plastic strain increments are related to the stress increments by
de p = ~H c3f(trij) d f = ~H Of (aii) ~f (au) dtrkt, ~aij O(Tij OUkl
(3)
where • is a parameter to identify plastic loading (~t = 1) and elastic loading and unloading (~ = 0), that is {1 =
if d f ~>0 if d f < 0
(4)
and H is a scalar factor related to strain history, current stress state and strain hardening and it is found from 1
1
H = Er
E'
(5)
where ET is the instantaneous slope of the flow curve, i.e. the effective stress-effective strain curve or d# ET = ~--~. (6) Substituting Eqns (2)-(6) into Eqn (1), the constitutive relationship between the strain increments and stress increments is given by
deo =
1
(1 1)af(ao)df((rk,)dau" [(1 + v)dao - vdakkbij] q- Ot ~T -Oaij Otrkl
(7)
2.2. Explicit formulae of a constitutive equation for elastic isotropic and plastic anisotropic solids The plastic potential, f(a~j), may take the form of the yield locus or effective stress, #(aij). To consider the plastic anisotropy, Hill's nonquadratic yield criterion for normal anisotropic materials [13] can be introduced. When the principal directions of the stresses are coaxial with the principal axes of anisotropy (which are the intersections of the mutually orthogonal planes of symmetry), Hill's 1979 nonquadratic yield function is given by # :
Cl{C2126ikbjlail•k
j - - 6 i j b k l t T i j a k l l M/2 +
16ijaijlM} x/M (i,j, k, l = 1, 2, 3),
(8)
with C~ =
2(1 + / ~ ) J
and C2 = (1 + 2R),
(9)
948
CHUAN-TAoWANGet al.
where 1~ is the normal anisotropy and M is a new index describing the shape of the yield locus. It was found that the effect of M on the shape of the yield locus is opposite to that of normal anisotropy, and the yield locus expands or the yield stress increases along the equal biaxial strain direction (45 ° line in firstquadrant of yield locus) as M decreases. Hill's 1979 nonquadratic yield function has bccn found more versatile for materials with different microstructures and anisotropic behavior [19, 20] and it covers the traditional isotropic yield theories proposed by Tresca (ifM = I and i~ = i), Von Mises (ifM = 2 and 1~ = I) and Hill's 1948 quadratic anisotropic yield criterion [21 ] (ifM = 2). This yield criterion is of particular interest for those materials having a normal anisotropy less than unity for which the quadratic yield criterion gives a significant underestimation of the flow stress in biaxial tension. For aluminum alloys, M is around 1.6-2.0 for a strain range of 0.02-0.18 [22]. M correlates well with the normal anisotropy for steel,brass, aluminum and copper using
the formula M = 1 + / ] for/~ <~ 1 [23] and M = 2 for/~ > 1 [24-1. Under plane stress conditions, Hill's nonquadratic yield function is simplified when the principal stresses are of interest, as in many sheet forming operations. The effective stress is related to the two in-plane principal stresses as follows: = Cl{C2lali - a221u + lall + a221u} TM.
(10)
An explicit expression for the constitutive equation (7) can then be derived under the two-dimensional principal stress state as follows:
=
[1 (1 1) ( 08 V,-
+
\o.1i/]
[__~E v+ ~ (1E-TT--1 ) 0 Oal, 8 06] ffa222 da22
1 d t722 - Vdall) + Gt( ~ T _ 1"~08 ~(
d~22
1
(1
(lla)
_ 88 _
1~ 06 ( 0 8
0~ da22 )
--p
=
E (dail + dail),
(llc)
or
dell = C l i d a l l + C12do'22 de22 = C12do'll + C22d0"22
(12)
-v
de33 = -~ (dalx + da22), where the compliance moduli C u are defined as '
=
' -v
(' (1
C12 = --~ + ~ ET
08 08 EJSa11 80"22
(13)
An inverse of the first two terms in Eqn (12) is found to be dall = Llldell + L12de22 do-22 = L1Edeil + L22de22
(14)
Wrinkling criterion for an anisotropic shell
949
and the instantaneous stiffness moduli L u are C22 L l l = C11C22 - C22 Ctt
(15)
L22 - C t t C 2 2 - C22 -- C12
L12
-- CllC22
-- C22"
With the yield criterion in Eqn (10), the derivatives are found to be c~"
~0-11 =
~C ~
~~
, L~2)a)')" --
0-22
iu - 1
÷
(0-11+ 0-221M-l] (16)
~0" : t~0-22
c I M l" - - C 2 ] 0 - 1 1 t~M- 1 k
--
0-22] M - 1 + I0-1x
+ 0-22)M-1]
•
If the strain hardening is described by Swift's equation, i.e. (17)
6 = k(eo + [)",
then the instantaneous slope Ev of the flow curve can be defined as 8t~
n
(18)
E-r = ~e = nk(eo + ~),,-1 = - - 8 ,
(eo + g)
where k is a strength coefficient, n is the strain hardening exponent, eo is the prestrain and is the effective strain. For a proportional loading path, the effective strain is given by M
= Dl[D21e11
-
M
M-!
e22[ h-z'i" "at- Jell+ e221~=T]-a-,
(19)
with 1 D1 = [2(1 +/~)]lm/2 = - 2C~ -1
D2 = [(1 + 2/~)] ~-~k-~,= C~~s.
(20)
Note that all of the instantaneous moduli C u (in Eqn (13)) or Lij (in Eqn (15)) are stress dependent in the elasto-plastic deformation, and they change as the current stress state changes for every incremental deformation step. 3. DONNELL-MUSHTARI-VLASOV (DMV) THEORY FOR QUASI-SHALLOW SHELLS OF GENERAL SHAPES A shell (Fig. l) is defined as a body in which the distance from any point within the body to some reference surface (usually the shell middle surface) is relatively small compared with any typical dimension of the reference surface (such as the radius of the curvature). Because of this smallness in the dimension normal to the surface, a three-dimensional deformation problem in a shell may be simplified to a two-dimensional one. For a shell stability analysis, the simplest possible form of the shell equations is the Donnell-Mushtari-Vlasov (DMV) equation for shells with general shape [25]. A quasi-shallow shell is the one that is relatively flat before deformation, and its displacement components in the deformed configuration are rapidly varying functions of the shell coordinates. For such quasi-shallow shells, the rotations about the normal are functions of tl~ displacement normal to the shell, but independent of the in-plane (tangential) displacements. 3.1. Assumptions (i) The shell is thin relative to its curvature, i.e. t / R is much less than unity (where R is the smallest principal radius of curvature of the undeformed middle surface). (ii) A plane stress state exists at each point through the thickness, i.e. the contributions of the transverse (normal and shear) stresses to the strain energy are ignored.
950
CHUAN-TAo WANG et al.
Q), N vx
)~
My ,~ ,
Nv
t
M, + \'~'~'x / dx
""
ry
N.+k~x/
+
(a)
,
- q
(b) FIG. 1. Shell element (a) force and moment equilibrium and (b) deformation and geometrical relations.
(iii) The strains are small compared with unity and hence the strains on the middle surface are linear functions of the middle surface displacements. (iv) The characteristic wavelength of the deformation is large compared with the shell thickness but small compared to the radii of curvature of its middle surface. 3.2. Incremental strains The geometry of a shell can be completely defined by the shape of its middle surface and its thickness at all points. Therefore, the displacement of any point inside the shell can be expressed in terms of the displacement of a corresponding point on the middle surface. Using a convective or comoving coordinate system x~(i = 1, 2, 3), the initial and deformed configurations of a shell are defined using the same coordinate system. That is, when the coordinate system is set in the middle surface of the undeformed (prebuckled) shell, the material points in the shell are identified by coordinates x~(i = 1, 2) lying in the middle
Wrinkling criterion for an anisotropic shell
951
surface of the undeformed body and coordinate xs normal to the undeformed middle surface. Let the shell coordinates x~ and x2 coincide with lines of principal curvatures (1/r~ and 1/ry) of the shell, dU~ or 0~ (i.= 1, 2) be the in-plane incremental displacements in the xl and x2 (or x and y) directions, W be the incremental buckling displacement normal to the middle surface of the shell and b 0 be the curvature tensor of the middle surface in the undeformed (prebuckled) shell. The incremental stretch strains dE~i (or /~i) and the incremental bending strains (or the changes of the curvatures) dK~j (or Ko) in the middle surface are created as
(i,j = 1,2)
Eij : ½(Ui,j '[- ~J]j,i) @ bij l~'r-)f- ½[~i[3j
(21a)
/~u = - l~,u,
(21b)
where the incremental rotations/~ are defined as /~, = - I42, =
(i = 1, 2).
(22)
In the above equations, the commas (,) denote the covariant differentiations with respect
to the general surface coordinates xl and x2, i.e.
0,.i
00,
= ~
O2I~i
and I~"m = -~x]" The non-
linear terms in Eqn (21a) come from the rotations about the normals (or the slopes along the xt directions). These nonlinear terms may be omitted in comparison with unity when the rotations are small. In such a case, the linear theory is recovered. The incremental Lagrangian strains for any point inside the shell with distance xs or z can then be defined as ~j = / ~ j + xs/¢~ /~,j = _ l~,i.
(23)
3.3. Incremental stress resultants and stress couples The incremental stress resultants and stress couples (per unit width) are defined as
:1 u =
f
tl2
:r~ dxs
d - t/2
~lr~j =
f t/2
(24) 6"/jxsdx3.
d - t/2
If we substitute the incremental strains in Eqn (23) into the constitutive equations in Eqn (14) and then substitute the results into Eqn (24), the incremental stress resultants and stress couples are found to be ]Vij =
[,/2¢Yijdx3 = f,/2 d - t/2
d - t/2
[,/2 ~/~ij =
d - t/2
Lokz~ktdxs = tLijkt(Ekt + ½ t~k~)
[,/2 dijx3dx3
=
d - t/2
?
.
(25)
Lijkl~klx3dx3 = - ~ LijklKkl •
When the rotations are small, the nonlinear terms can be neglected. Then Eqn (24) reduces to
1Vii = tLijktEkt t3 . I(/10 = -~ Lijkl Kkt ,
(26)
where the M 0 are bending moments when i = j and twisting moments when i ~ j. Note that the instantaneous moduli L 0 are rewritten as the fourth-order tensor (Luk~), for convenience of the general expression of the constitutive equations, and they will return to two index forms as L l a l l = L l l , L2222 = L22 and L~22 = Lx2, etc. in the explicit expression for the stress-strain relationship.
952
CHUAN-TAO WANG 4. W R I N K L I N G
et al.
CRITERIA
4.1. Bifurcation functional for a double curved shell Body wrinkling can be considered as the bifurcation of the in-plane deformation under the plane (membrane) stress condition to the out-of-plane bending type of deformation. Consider xl and x2 to coincide with the principal axes; let ~1 = al 1 and a2 = a22 be the principal in-plane (membrane) stresses along these axes; and let R t and R2 be the radii of the principal curvatures. A bifurcation functional was proposed by Hutchinson [9] based on Hill's general theory of uniqueness and bifurcation in elastic-plastic solids [10, 11, 21]. This bifurcation functional is given by
+ N' E'J + N,/3,1J } dS
#) =
t3 L'm Kk, . K,j . . + .tLoktEuE 0 + N,fl~ ,l~.i} dS, = fs{ -i2
(27)
where S denotes the region of the shell middle surface over which the wrinkles appear. This bifurcation functional is the total energy for wrinkling occurrence. In the right hand side of Eqn (27), the first term represents the bending energy (i = j) and twisting energy (i # j), the second term is the strain energy due to the membrane stresses, and the third term may be interpreted as the potential energy of the edge stress or the work done by the applied in-plane stresses in the middle surface. For all admissible incremental displacement fields 6"i and I,V, if F > 0, then the incremental solutions of the deformation are unique and bifurcation is not possible because to create wrinkles requires the energy supply, i.e. the total potential of the system, to increase. This is not a natural or spontaneous process from a thermodynamics point of view. Then F = 0 corresponds to the critical conditions for wrinkles to occur for some non-zero incremental displacement fields. That is, the wrinkling criterion may be written as: F ( 0 , I~) = 0
(not all 0i and I~ = 0).
(28)
4.2. Incremental displacements for wrinkling The incremental displacement fields given by Hutchinson and Neale [12] are employed for wrinkling in short-wavelength and shallow modes. In these fields, wrinkling is assumed to occur over a certain local region S of the sheet which spans many wavelengths of the wrinkling mode. Therefore, in a local wrinkling problem, the continuity conditions or boundary conditions along the boundaries of region S are relatively unimportant. The fields are given as follows: I/V = At
COS(~IXI/~)COS(~2X2/~)
(11 = e t sin(Xtxff~) cos(X2x2/~)
(29)
02 = Ct cos(21xl/~) sin(22x2/~), with = x/~,
(30)
where t = the thickness of the sheet; R = the radius of the principal curvature (RI or R=); A, B and C = constants representing the relative displacement amplitudes of the mode shape; and 21 and 22 = dimensionless wave numbers. 4.3. Wrinkling criterion for a double curved shell The membrane forces due to the in-plane stresses can be defined as =
-
Gli t
N22 =
-
t/o'22t
NIl
(31)
and ~/=
1 a22 is tensile stress -1 a22 is compressive stress.
(32)
Wrinklingcriterionfor an anisotropicshell
953
Finally, the bifurcation functional is obtained by substituting Eqn (31) and the strain increments [defined by the kinematic relationships in Eqns (21), (25) and (29)] into Eqn (27), i.e. F = (Jb + (J,. + Ou =
-~(t3s{(at)2 } - ~ ( L x ~ 2 ~ + Lz22"~ + (L~2 + L,,,,)(2122)z
-- ht3CS[0"11(~)2 -at-?]0"22(~)21,
(33)
or in the matrix form as 2
A
F = (St
t 2
l)
[ M ] { A B C } = (St ~
B?/M2
LCALM3a
MI2 MI3 1 M22 M2a M32 M33
{ABC}, (34)
where the matrix [M] is given by ?/t\
2
Mll = ]~ L~) [Ll12~ + L22224+ 2(L12 + 2L44)(2122)2] + I Ltl (~1) 2+L22 (~2) 2 + 2L12 (R--~x)(R--2)] - [O'l 1'~'2 + ?]0"22'~2]
M22 -- Lx12~ + L442~
(35)
M33 = L22,~-24+ L442~ M12 =M21 = Lll)-1 (~1) -I- L1221 (~-~2) and
M13 = M31 = L22~,2(~1)-I-L12~'2 (~-~2) M23 =
M32-- (L12 "1-L4,1.)J.I,~2,
where
(S = fs[cos(21xJ~)cos(22x2/~)]: dS = ~s [sin(21xl/~) sin(22x2/~)] 2 dS
(36)
= fs[Sin(21x1/~) cos(22x2/~)] 2 dS and ~'1/4 if both 21 and 2 2 ~ 0 (=/.1/2 if either 21 or 2 2 = 0 .
(37)
4.4. Criterion for wrinkling along two principal axes The wrinkling criterion F = 0 in Eqn (28) requires the determinant of the matrix [M] in Eqn (34) to vanish since the constants A, B, and C are not all zero; that is, M l l Mlz M13 [ det[M] = M21 M22 M23 }= 0 for wrinkling to occur.
M31 MS 36: IO-F
M32 M33
(38)
954
CHUAN-TAo WANG
et al.
The critical stress values of 0-11 and 022 can be obtained by minimizing this wrinkling equation with respect to the wave numbers hi and 22, i.e. OlMI - 0 O2i
(i = 1 , 2 ) .
(39)
4.5. Criterion for wrinklin9 alon9 one principal axis In most sheet forming operations, wrinkling takes place along one of the principal axes, that is, the wrinkles are aligned with one of the principal curvature. If wrinkles are assumed to be perpendicular to the xl direction, then the wave number 22 = 0, ¢ = ~ 2 t and M I 3 = M 3 1 = M 2 3 = M 3 2 = 0. Let the principal curvatures be
hi1 = 1/R1,
b22 = 1/R2
and other b u = 0,
(40)
where RI and R2 are the radii along the principal axes. Also let the instantaneous moduli be Lll = L l t x l ,
L22=L2222,
L12=Ll122
and
L44=L1212.
(41)
Then, the wrinkling criterion in Eqn (38) is simplified to det[M] = M 3 3 ( M 1 1 M 2 2 - M 2 2 ) = 0 , or ~/ = M l l M 2 2
(42)
-- M22 = 0
and '
M 2 2 ---- Lx12x4,
Mss = L442~,
(43)
M12=M21 = L,121(-~l)+Lx22,(~2). From Eqn (43), the wrinkling equation (42) depends explicitly on the stress component 0-11. The effect of the stress component 0-22 on wrinkling along the 0-11 direction appears implicitly in the incremental moduli L u, defined by Eqns (13), (15) and (16). With Eqns (42) and (43), a relationship between the wrinkling wave number and the stress can be established, i.e.
1(
0-1, = 1 2 \ 4 ]
L 2)
~
L-~/"
" , -'1/~0-11 )
Minimizing the stress with respect to the wave number [-5-7- = 0 , we obtain the critical wave number or wavelength 2y:
2clr = N//~(LllL~.~ L22)1/4=
(LxlL22L22)1/4. -~-~
1.8612 \
(45)
The critical wrinkling stress is then obtained by substituting 2y into Eqn (44), i.e.
0-], = ~.x/r~ ~22 t N/ /t
11 L 22
- L22 = 0.5774~22 x/L1 1L22 - L22"
(46)
Similarly, for wrinkling perpendicular to the x2 direction (¢ = . , f R : and hi = 0), the critical wavelength and stress are
(ZllZ~_2~ g22) 1/4 L22
(47)
0-~' = x /1-~-t ~ R l X/ /L 11L 2 2 - L 2 2 = 0.5774~-~11/L1 1 L 2 2 - L22 .
(48)
2~r = NIl(LIlting212) 1/4=
1.8612 \
and
Wrinkling criterion for an anisotropic shell
955
A procedure using those criteria is summarized as follows. The deformation geometry (the current thickness and principal radii) and the stress and strain components are assumed to be known or provided by the finite-element simulations. These components are used to calculate the effective stress by Eqn (10) and the effective strain by Eqn (19). With the effective stress and strain and the strain hardening equation (17), the moduli Cij and L~j are defined. Wrinkling conditions are predicted by substituting the moduli L o and the geometrical dimensions (the thickness and the principal radii) into Eqns (45)-(48). 4.6. Wrinklin9 criterion for shrink flanoin9 In shrink flanging operations (Fig. 2), the flat blank with an initial radius R1 is clamped at the location of Ro and forced to wrap around the die shoulder (with a radius of ra) to form a flange with a radius R2(Rz < R1) at the flange edge. Therefore, the stress state at the flange edge with a principal curvature of sin ~/R2 is uniaxial compression along the hoop direction. Wrinkles occur at the edge where the maximum compressive hoop stress or strain appears. If the hoop stress is assumed to be a l 1, then the effective stress 0 is identical to stress a~ a, i.e. = a11. Therefore, the moduli Cii in Eqn (13) are Cl1=~+
-
,
C22=~
and
C12=--~-
From Eqn (15), the incremental moduli Lij can be defined using Eqn (49), that is,
1
R1 ,ddr,_
L. RT I- r
FIG.2. Shrink flanging:the maximumcompressionappears at the flangeedge whichhas an initial radius of R~ and a final radius of Re.
CHUAN-TAo WANG et al.
956
I+Gt
-1
y
Elastic wrinkling. The elastic moduli are obtained by setting the loading index ~ to 0 in Eqn (50), i.e.
E Ltl - 1 -
E
vE
L22 - 1~ - v
1,2
L12 -
1~ - " v
(51)
Therefore, the elastic wrinkling criterion is given by substituting these moduli into the wrinkling criterion in Eqn (46), that is, 0"~r m.
1
t
E
---~ =--sin~ x/3 ~,2 x t/7------75.2' /l-v"
(52)
or, in terms of strain, 1
t
e]r = a]'/E = - - = - - sin0t x/3R2
1
lx/i--S~Sv 2'
(53)
where at is the flange angle.
Plastic wrinkling. For the plastic loading case (the loading index ~t -- 1), the incremental moduli are Lll
=
EET E
- - E T v2
L22 =
E
-
L12 =
vEET E -- ETV2"
E2 (54)
E T v2
Substituting these moduli into Eqn (46), the critical stress for wrinkling is cr], -
1
t sin~tE ~ 1 ET/E
X//3 R 2
-
(55)
v2ET " -g
For a flow curve described by Swift's hardening relation, Eqn (17), the slope of the flow curve is defined by Eqn (18), that is, dal E T --
det
-
nK(eo + ex)"-1 = nat/(e ° + el).
Finally, the critical strain for wrinkling to occur is found by substituting ET into Eqn (46), = ~ [ ~22sln~j
- eo.
(56)
5. E X A M P L E S
5.1. Shrink flanging experiments The validity of the wrinkling criteria proposed was investigated by comparing the predictions with axisymmetric shrink flanging experiments. The test material was the high
Wrinkling criterion for an anisotropic shell
957
strength (HS) steel sheets which are commonly used in automotive body panels. The material properties are listed in Table 1. The circular blanks were cut from HS sheets (thickness = 0.91 ram) to the initial diameters of 104.9, 114.5 and 126.7 ram. Two specimens for each diameter (a total of six specimens) were prepared. A multi-action press was used in the experiments. Figure 3 illustrates the geometry and dimensions of the tooling. The die corner radius and the punch corner radius are 1.52 mm (0.06") and the clearance between the punch and the die is 1.02 mm (0.04"). The circular blank is centered and clamped to the flat bottom of a cylindrical punch by a circular blankholder using three bolts. The punch moves down and flanges the sheet over the concave edge of the die. To detect the onset of the wrinkles (via visual observation), the incremental displacement of the punch was controlled by 0.25 mm (0.01"). The test stopped as soon as wrinkles appeared. For blanks with an initial diameter of 104.9 ram, no wrinkles were observed, while for blanks with larger initial diameters (114.5 mm), wrinkles appeared. For blanks with even larger initial diameters (126,7 mm), wrinkles occurred much earlier. Under the load, the diameters of the specimen were measured, and the pictures of the specimens and the wrinkled shapes were taken. Then, the punch was slowly lifted with the same incremental displacements in order to observe the elastic recovery of the wrinkles. No appreciable elastic recovery of the wrinkles was detected either by visual observation or by measuring the diameters after unloading. Therefore, there were no elastic wrinkles observed in all six specimens made from the HS steel sheets. The diameters and thickness at the edges of the initial blanks and the flanged parts, as well as the flange height (a vertical measure from the flat top to the flange edge) were measured and are presented in Table 2. Using the initial and final diameters, the maximum hoop strains at the flange edge can be calculated.
TABLE 1. MATERIAL PROPERTIES OF HS (HIGH STRENGTH) STEEL SHEETS
n
K (MPa)
eo
¢o (MPa)
,~
M
E (GPa)
v
Thickness (mm)
0.143
603.8
0.0003
289
1.4
2
206.8
0.3
0.91
~b104.9, 114.5, 126.7 mm mt
Punch ,99.3mm RI.52 mm
Sheet
I
I
Die
I
FIG. 3. Tooling set for axisymmetric shrink flanging (the circular sheet is clamped to the punch face by three bolts, and flanged around the punch shoulder when the punch moves down).
CHUAN-TAo WANG et al.
958
TABLE 2. MEASUREMENTS IN AXISYMMETRICSHRINK FLANGING OF HS STEEL SHEETS
to
D2
tf
Hoop strain
Sample
D1
number
(mm)
(mm) (mm) (mm) (mm)
A1 A2 B1 B2 C1 C2
104.90 104.90 114.50 114.50 126.7 126.64
0.91 0.91 0.91 0.91 0.91 0.91
100.89 100.89 106.68 106.96 124.90 124.86
H
0.94 0.94 0.94 0.94 0.92 0.92
%
3.00 3.42 8.00 7.90 4.65 4.50
-
Wrinkle
3.9 3.9 7.1 6.8 1.4 1.4
no no yes yes yes yes
Flange ratio D2/D 1 0.962 0.962 0.932 0.932 0.987 0.986
DI = initial blank diameter, D 2 = final diameter at the flange edge, to = initial thickness of blank, tf = final thickness at the flange edge, H = flange height.
0.08
,
,
,
i
]
,
,
,
,
I
'
'
'
'
t
'
'
'
'
.~ ~ g 0.06
o.o4 0.02 SI-IRINKFLANGE
,HSSTI ' . . . .
0 0.92
0.94 Flange
~U
, .... 0.96
, , , ,~.
0.98
1
Ratio ( R 2 / R 1 )
FIG. 4. The maximum hoop strains at flange edge vs flange ratio (R1 = initial blank radius, R2 = final radius at the flange edge).
5.2. Predictions and comparisons The axisymmetric shrink flanging of HS (high strength) sheet steels was simulated with the program FLANGE which was developed by these authors [26] and is able to predict the maximum compressive strains and wrinkles at the flange edge. In shrink flanges, the maximum hoop strains increase as the flange ratio decreases or as the initial blank radius (R1) increases. Figure 4 shows the comparison between the predicted and the measured maximum hoop strains at the flange edge. (Note that the strains are compressive and have negative values, but are plotted using the absolute values.) The experimental results match well with the proposed models for the maximum strains. The relationship between the flange height (the vertical measurement from the flat bottom to the flange edge, Fig. 2) and the flange ratio is illustrated in Fig. 5. Again, a good match is indicated. The predictions underestimate the flange height by 0.5 mm for the wrinkled specimens (about a half of the sheet thickness) and have a relative error of less than 15%. The proposed wrinkling criteria, Eqns (53) and (56), were investigated by the visual observation of the onset of the wrinkles during shrink flanging experiments. No elastic wrinkles were observed because the unloading does not remove the wrinkles. Both the measurements and the predictions via the proposed wrinkling criterion proved that the critical strains to onset the wrinkles are greater than the initial yield strain ( = yield stress/Young's modulus = 0.0014). Therefore, only plastic wrinkles occur in shrink flanges of the HS (high strength) steel sheets. The measured maximum strains in the wrinkled specimens (in these cases, the measured maximum strains may also be the critical strains due to the onset of the visible wrinkles at flange edge) were compared with the predicted wrinkling strains, as shown in Fig. 6. Wrinkles did not appear when a circular blank with
Wrinkling criterion for an anisotropic shell 10
959
' , , , , , ' , , •
Ptedkud
f Z. 6 4
_a c~ 2
SHRINK FLANGE HS Steel
0
. . . .
0.92
I
,
,
0.94
0.96
0.98
1
Flange Ratio (R2/R1)
FIG. 5. Flangeheight vs flange ratio (R1 = initial blank radius, R2
0.15
. . . .
.
,
. . . .
i
. . . .
I
. . . .
=
final radius at the flangeedge).
I
. . . .
lo-.l
"~ o.12
,
u ..~
SHRINK FLANGE HS Steel
0.09
I
/
0.06 °m
--2 0.03 o
0
0.005
0.01
0.015
0.02
0.025
(t/R2)sin(alpha)
FIG. 6. Critical strains for wrinkling in shrink flanging of high strength steel sheets (t -- sheet t h i c k n e s s , R2 = final radius at the flange edge, ct = flange angle).
a diameter of 104.9 m m was formed to a flange of diameter D 2 = 100.86 m m (the flange ratio is 0.96). The m a x i m u m strains measured and predicted are about 4%, which is less than the critical strain (about 6%) predicted by the proposed wrinkling model. Wrinkles did appear for flanges with diameters of D 2 = 106.68 and 124.98 mm, which are formed from the circular blanks with the initial diameters of Dt = 114.50 and 126.70 mm, respectively. The wrinkling limits decrease as the flange ratio decreases or the blank size increases. For a flange with initial blank diameter of 114.50 m m (flange ratio = 0.932), the predicted critical strain for wrinkling is 4.5%, which is less than the measured and predicted m a x i m u m strains (7%) at the flange edge. Therefore, wrinkles did appear in the experiment. For a flange with an even larger initial diameter (Dx = 126.70), the flange ratio has to be increased to 0.987 and the critical strain limit further reduces to 0.6%. At this flange ratio, the m a x i m u m strains measured and predicted are 1.4%, which is greater than the limit. Therefore, for those specimens, wrinkles did occur earlier during shrink flanging tests. 6. CONCLUSION Wrinkling criteria are developed for the local wrinkling phenomenon which occurs in the in die wall 'in the unsupported region of a sheet. These criteria are derived based on the application of the D M V shell theory and Hill's bifurcation model to an elastic isotropic and plastic anisotropic shell with c o m p o u n d curvatures. Experiments on the shrink flanging of
960
CHUAN-TAo WANG et al.
high strength steel sheets were conducted to verify the proposed wrinkling criteria. The predictions of the onset of the wrinkles at the flange edges agree well with the measurements. The wrinkling models are especially useful in conjunction with finite element modeling of sheet forming processes. A process control curve of restraining force (blankholder force or drawbead force) vs punch stroke could be obtained through the simulations. Acknowledgments--The authors would like to thank the following individuals in helping to conduct the shrink flanging experiments: Serdar Tufekci designed the tooling, prepared the specimens and conducted the tests, Mary Hartzler and S. Wm Lipson helped to machine the tooling and specimens.
REFERENCES I. C. T. WANG, G. KINZEL and T. ALTAN, Sheet Forming Technology: Operations and Process Control. Engineering Research Center for Net Shape Manufacturing, The Ohio State University, Report No. ERCfNSM-S-91-6 (1991). 2. B. W. SENIOR, Flange wrinkling in deep-drawing-operations. J. Mech. Phys. Solids 4, 235 (1956). 3. N. KAWAI, Critical conditions of wrinkling in deep drawing of sheet metals (1st report: fundamentals of analysis and results in case where a blank-holder is not used). Bull. JSME 4(13), 169 (1961). 4. N. KAWAI, Critical conditions of wrinkling in deep drawing of sheet metals (2nd report: analysis and considerations for conditions of blank-holding). Bull. JSME 4(13), 175 (1961). 5. N. KAWAI,Critical conditions of wrinkling in deep drawing of sheet metals (3rd report: predicting of critical blank-holding pressure). Bull. JSME, 4(13), 182 (1961). 6. J. HAVRANEK,Wrinkling limit of tapered pressing. J. Aust. Metals 20, 114 (1975). 7. T. X. Yu and W. JOHNSON,The buckling of annular plates in relation to the deep-drawing process. Int. J. Mech. Sci. 24(3), 175 (1982). 8. R. W. LOGAN, Sheet metal formability simulation and experiment. Ph.D Dissertation, The University of Michigan, MI, U.S.A. (1985). 9. J. W. HUTCHINSON,Plastic buckling, in Advances in Applied Mechanics 14, p. 67 (1974). 10. R. HILL, A general theory of uniqueness and stability in elastic/plastic solids. J. Mech. Phys. Solids 6, 236 (1958). 11 R. HILL, Bifurcation and uniqueness in Nonlinear Mechanics of Continua p. 236. Society of Industrial Applied Mathematics, (Muskhelishvili Vol.), Philadelphia, PA (1961). 12. J. W. HUTCmNSONand K. W. NEALE, Wrinkling of curved thin sheet metal, in Proc. Int. Syrup. on Plastic Instability, Considere Memorial (1841-1914), Paris, 9-13 September, p. 71 (1985). 13. R. HILL, Theoretical plasticity of textured aggregates, in Mathematical Proceedings of Cambridge Philosophical Society, Vol. 85, p. 179 (1979). 14. L. H. N. LEE, Inelastic buckling of initially imperfect cylindrical shells subject to axial compression. J. Aerospace Sci. 29, 87 (1962). 15. A. NEEDLEMANand J. R. RICE, Limit ductility set by plastic flow localization, in Mechanics of Sheet Metal Forming: Materials Behavior and Deformation Analysis p. 237 (edited by D. P. KOISTINENand N. M. WANG). Plenum Press, New York (1978). 16. N. TRIANTAFYLLIDISand A. NEEDLEMAN,An analysis of wrinkling in the Swift cup test. J. Engng Mater. Teehnol. 102, 241 (1980). 17. X. F. WANGand L. H. N. LEE, Wrinkling of an unevenly stretched sheet metal. J. Engn0 Mater. Technol. 111, 235 (1989). 18. R. HILL, The Mathematical Theory of Plasticity. Oxford (1950). 19. S. KOBAYASHI,R. M. CADDELLand W. F. HOSFORD,Int. J. Mech. Sci. 27(7/8), 509 (1985). 20. W. F. HOSFORD, Limitations of non-quadratic anisotropic yield criteria and their use in analyses of sheet forming. Proc. 15th Biennial 1DDRG Conor. Dearborn, MI, p. 163 (May 1988). 21. R. HILL, A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A193, 281 (1948). 22. R. H. WAGONER,Measurement and analysis of plane-strain work hardening. Metall. Trans. I1A, 165 (1980). 23. A. R. RAGABand A. T. AI~BAS,Assessment of work-hardening characteristics and limit strains of anisotropic aluminum sheets in biaxial stretching. J. Engng Mater. Technol. 108, 250 (1986). 24. J.D. BRESSANand J. A. WILLIAMS,The use of a shear instability criterion to predict local necking in sheet metal deformation. Int. J Mech. Sci. 25, 155 (1983). 25. D. BRUSHand B. O. ALMROTH,Buckling of Bars, Plates and Shells. McGraw-Hill (1975). 26. C.T. WANG,G. KINZELand T. ALTAN,Mechanics of Bending and Flanoino and A User's Manual for Computer Programs BEND and FLANGE. Engineering Research Center for Net Shape Manufacturing, The Ohio State University (in press).
Int. J. Mech. $ci. Vol. 36, No. I0, pp. 945-960, 1994 Copyright © 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020-7403/94 $7.00 + 0.00
00~-740a(94)E000-0 WRINKLING CRITERION FOR AN ANISOTROPIC SHELL WITH C O M P O U N D CURVATURES IN SHEET FORMING CHUAN-TAO WANG, t GARY KINZEL ~ a n d TAYLAN ALTAN~ t General Motors Corporation, Technical Center, Box 9040, 30300 Mound Road, Warren, MI 48090-9040, U.S.A. and ~Engineering Research Center for Net Shape Manufacturing, The Ohio State University, Columbus, OH 43210, U.S.A. (Received 6 August 1993; and in revised form 15 February 1994)
Abstract--Wrinkling criteria are proposed for an elastic isotropie and plastic anisotropic shell with compound curvatures in the noncontact region of a sheet subjected to internal forming stresses. A quasi-shallow shell is modeled by DonnelI-Mushtari-Vlasov (DMV) shell theory. A bifurcation functional from Hill's general theory of uniqueness and bifurcation in elastic-plastic solids is used to model the local wrinkling phenomenon. Both strain hardening and transverse anisotropy are taken into consideration. This wrinkling criterion is especially useful as a failure criterion in three dimensional finite element modeling (FEM) of sheet forming. Given the principal stresses or strains and the geometry provided at each incremental deformation step, the criterion can be used to predict wrinkles in the elements in the unsupported region.
NOTATION A , B , C constants b~j curvature tensor of the middle surface in the prebuckling shell 'go prestrain effective strain d'go incremental strain tensor da~j incremental stress tensor 6o Kronecker index f(~,j) yield surface effective stress ~t loading index, flange angle incremental rotations CO compliance moduli incremental stretch strain tensor E elastic modulus Er instantaneous slope of the flow curve
H
(
scalar factor \ H = Erl _ _ I )
k strength coefficient incremental bending strains (or the changes of the curvatures) Lij instantaneous stiffness moduli M Hill index describing the shape of the yield locus dM o incremental stress resultants dN 0 incremental stress couples (per unit width) n strain hardening exponent normal anisotropy Ri radius of the principal curvature (RI or R2). 21,22 dimensionless wave numbers. 2]r critical wave number t thickness of the sheet. t3~ in-plane incremental displacements in the xl and x2 directions incremental buckling displacement normal to the shell mid-plane Xi coordinate (i = 1, 2, 3) 1. I N T R O D U C T I O N
In deep drawn sheet components, wrinkling and tearing are two major failure modes. Excessive compressive stresses cause wrinkling while excessive tensile stresses lead to 945
946
CHUAN-TAOWANGet
al.
fracturing. Careful control of the tensile restraining force generated by blankholders and drawbeads can avoid wrinkles and cracks. A number of approaches to control the blankholder force (BHF) and drawbead force (DBF) have been reported elsewhere I-1]. For prediction and control of fracture, a number of failure criteria, such as the forming limit diagram and localized necking instability may be used in conjunction with finite element modeling. However, there has been a lack of reliable wrinkling criteria for prediction and control of wrinkling, especially body wrinkling. There are three types of compressive instability problems that may occur in sheet forming operations. These instability problems are: flange wrinkling; wall wrinkling; and elastic buckling in the undeformed area due to residual elastic compressive stress. All of these can be observed in the forming of a common pie pan made from aluminium sheet. The flange wrinkling and wall wrinkling are basically the same and are caused by the compressive circumferential stress. However, wall wrinkling occurs far more easily than the wrinkling on the flat flange since the wall is relatively unsupported by the tool. The suppression of wall wrinkles by control of the blank holder force (BHF), which affects the radial tensile stress component, is more difficult than the suppression of flange wrinkles. It is known that wall wrinkling is more of a problem than flange wrinkling [2-7] in most press forming operations, and the blankholder force is far greater than that necessary to suppress flange wrinkling. The drawbead is then particularly designed to add an additional restraining force to prevent metal flowing into the die too freely. Therefore, wall wrinkling is the problem of most industrial importance and interest. However, very few studies I-8] have been conducted in understanding and controlling wall wrinkling. In modeling wall wrinkling, a curved shell model achieves better accuracy than a flat plate model. In the elastic buckling literature, DMV (Donnel-Mushtari-Vlasov) shell theory has been applied to study the wrinkling phenomenon of a three-dimensional shell with compound curvatures. The DMV model is based on shallow shell theory and is able to determine the critical condition (stress or strain) for a short wavelength wrinkling mode. Hutchinson [9] applied Hill's bifurcation criterion [10-11] to the DMV theory for a class of three-dimensional solids and proposed a general elastic-plastic bifurcation criterion for the DMV theory of plates and shells. Within the context of this model, a specific instability criterion was suggested to predict the conditions for the onset of wall wrinkling in doubly-curved and isotropic sheet metal undergoing a forming operation 1-12]. However, the importance of plastic anisotropy was not considered in this model and the enhancement of this model for anisotropic sheet materials is necessary. In this study, wrinkling criteria for an elastic isotropic and plastic anisotropic shell with compound curvature is proposed, based on DMV shell theory, bifurcation analysis and an incremental constitutive law incorporating Hill's nonquadratic yield criterion for transverse anisotropic materials [13]. The critical stresses and strains for the onset of wall wrinkling during sheet forming are related to the material properties (elastic modulus, Poisson's ratio, flow stress, strain hardening and plastic anisotropy), geometrical dimensions (wall inclination, die clearance, sheet thickness and principal curvatures) and loading conditions (ratios of radial tensile stress or strain to hoop compressive stress or strain). 2. INCREMENTAL STRESS-STRAIN CONSTITUTIVE EQUATIONS FOR ELASTIC ISOTROPIC AND PLASTIC ANISOTROPIC SOLIDS Most constitutive equations for the elasto-plastic deformation of a solid are derived based on the generalized Hook's law for elastic response and Von Mises' yield theory for isotropic solids undergoing plastic deformation [12, 14, 15]. A few studies deal with the anisotropic plastic response but none of these provide an explicit expression of the incremental moduli [16, 17]. In this section, a general expression for the constitutive equation will be given first for solids with the deformation behavior of elastic isotropy and plastic anisotropy. Then, an explicit formulation of the incremental stress-strain relation will be derived under plane stress conditions, which is the primary deformation mode in most sheet forming operations. Both strain hardening and plastic anisotropy are incorporated in this constitutive relation.
Wrinkling criterion for an anisotropic shell
947
2.1. Constitutive equation for elasto--plastic solids For a solid statically stressed beyond its elastic limit, a linear decomposition of the strain increments into elastic and plastic parts is valid because the recoverable elastic strains are small. The tensor expression for the strain increments is deij = de~j + de~ (i,j = 1, 2, 3),
(1)
where deo is the total increment strain, and the superscripts 'e' and 'p' stand for elastic and plastic, respectively. The generalized Hook's law relates the elastic strain increments to the stress increments (dtri~) as follows de~ = 1 [(1 + v)dtri~- vdtrkk6ij],
(2)
where E and v are Young's modulus and Poisson's ratio, dtTkk (k : 1, 2 and 3) is the sum of the three normal stresses according to the tensor summation convention and &j is the Kronecker delta defined by 6o =
if i ~ j.
The plastic strain increments may be defined in terms of a plastic potentialf(tro) or yield locus #(ao) if the flow (or incremental deformation) theory is applied. According to Hill [18-1, the plastic strain increments are related to the stress increments by
de p = ~H c3f(trij) d f = ~H Of (aii) ~f (au) dtrkt, ~aij O(Tij OUkl
(3)
where • is a parameter to identify plastic loading (~t = 1) and elastic loading and unloading (~ = 0), that is {1 =
if d f ~>0 if d f < 0
(4)
and H is a scalar factor related to strain history, current stress state and strain hardening and it is found from 1
1
H = Er
E'
(5)
where ET is the instantaneous slope of the flow curve, i.e. the effective stress-effective strain curve or d# ET = ~--~. (6) Substituting Eqns (2)-(6) into Eqn (1), the constitutive relationship between the strain increments and stress increments is given by
deo =
1
(1 1)af(ao)df((rk,)dau" [(1 + v)dao - vdakkbij] q- Ot ~T -Oaij Otrkl
(7)
2.2. Explicit formulae of a constitutive equation for elastic isotropic and plastic anisotropic solids The plastic potential, f(a~j), may take the form of the yield locus or effective stress, #(aij). To consider the plastic anisotropy, Hill's nonquadratic yield criterion for normal anisotropic materials [13] can be introduced. When the principal directions of the stresses are coaxial with the principal axes of anisotropy (which are the intersections of the mutually orthogonal planes of symmetry), Hill's 1979 nonquadratic yield function is given by # :
Cl{C2126ikbjlail•k
j - - 6 i j b k l t T i j a k l l M/2 +
16ijaijlM} x/M (i,j, k, l = 1, 2, 3),
(8)
with C~ =
2(1 + / ~ ) J
and C2 = (1 + 2R),
(9)
948
CHUAN-TAoWANGet al.
where 1~ is the normal anisotropy and M is a new index describing the shape of the yield locus. It was found that the effect of M on the shape of the yield locus is opposite to that of normal anisotropy, and the yield locus expands or the yield stress increases along the equal biaxial strain direction (45 ° line in firstquadrant of yield locus) as M decreases. Hill's 1979 nonquadratic yield function has bccn found more versatile for materials with different microstructures and anisotropic behavior [19, 20] and it covers the traditional isotropic yield theories proposed by Tresca (ifM = I and i~ = i), Von Mises (ifM = 2 and 1~ = I) and Hill's 1948 quadratic anisotropic yield criterion [21 ] (ifM = 2). This yield criterion is of particular interest for those materials having a normal anisotropy less than unity for which the quadratic yield criterion gives a significant underestimation of the flow stress in biaxial tension. For aluminum alloys, M is around 1.6-2.0 for a strain range of 0.02-0.18 [22]. M correlates well with the normal anisotropy for steel,brass, aluminum and copper using
the formula M = 1 + / ] for/~ <~ 1 [23] and M = 2 for/~ > 1 [24-1. Under plane stress conditions, Hill's nonquadratic yield function is simplified when the principal stresses are of interest, as in many sheet forming operations. The effective stress is related to the two in-plane principal stresses as follows: = Cl{C2lali - a221u + lall + a221u} TM.
(10)
An explicit expression for the constitutive equation (7) can then be derived under the two-dimensional principal stress state as follows:
=
[1 (1 1) ( 08 V,-
+
\o.1i/]
[__~E v+ ~ (1E-TT--1 ) 0 Oal, 8 06] ffa222 da22
1 d t722 - Vdall) + Gt( ~ T _ 1"~08 ~(
d~22
1
(1
(lla)
_ 88 _
1~ 06 ( 0 8
0~ da22 )
--p
=
E (dail + dail),
(llc)
or
dell = C l i d a l l + C12do'22 de22 = C12do'll + C22d0"22
(12)
-v
de33 = -~ (dalx + da22), where the compliance moduli C u are defined as '
=
' -v
(' (1
C12 = --~ + ~ ET
08 08 EJSa11 80"22
(13)
An inverse of the first two terms in Eqn (12) is found to be dall = Llldell + L12de22 do-22 = L1Edeil + L22de22
(14)
Wrinkling criterion for an anisotropic shell
949
and the instantaneous stiffness moduli L u are C22 L l l = C11C22 - C22 Ctt
(15)
L22 - C t t C 2 2 - C22 -- C12
L12
-- CllC22
-- C22"
With the yield criterion in Eqn (10), the derivatives are found to be c~"
~0-11 =
~C ~
~~
, L~2)a)')" --
0-22
iu - 1
÷
(0-11+ 0-221M-l] (16)
~0" : t~0-22
c I M l" - - C 2 ] 0 - 1 1 t~M- 1 k
--
0-22] M - 1 + I0-1x
+ 0-22)M-1]
•
If the strain hardening is described by Swift's equation, i.e. (17)
6 = k(eo + [)",
then the instantaneous slope Ev of the flow curve can be defined as 8t~
n
(18)
E-r = ~e = nk(eo + ~),,-1 = - - 8 ,
(eo + g)
where k is a strength coefficient, n is the strain hardening exponent, eo is the prestrain and is the effective strain. For a proportional loading path, the effective strain is given by M
= Dl[D21e11
-
M
M-!
e22[ h-z'i" "at- Jell+ e221~=T]-a-,
(19)
with 1 D1 = [2(1 +/~)]lm/2 = - 2C~ -1
D2 = [(1 + 2/~)] ~-~k-~,= C~~s.
(20)
Note that all of the instantaneous moduli C u (in Eqn (13)) or Lij (in Eqn (15)) are stress dependent in the elasto-plastic deformation, and they change as the current stress state changes for every incremental deformation step. 3. DONNELL-MUSHTARI-VLASOV (DMV) THEORY FOR QUASI-SHALLOW SHELLS OF GENERAL SHAPES A shell (Fig. l) is defined as a body in which the distance from any point within the body to some reference surface (usually the shell middle surface) is relatively small compared with any typical dimension of the reference surface (such as the radius of the curvature). Because of this smallness in the dimension normal to the surface, a three-dimensional deformation problem in a shell may be simplified to a two-dimensional one. For a shell stability analysis, the simplest possible form of the shell equations is the Donnell-Mushtari-Vlasov (DMV) equation for shells with general shape [25]. A quasi-shallow shell is the one that is relatively flat before deformation, and its displacement components in the deformed configuration are rapidly varying functions of the shell coordinates. For such quasi-shallow shells, the rotations about the normal are functions of tl~ displacement normal to the shell, but independent of the in-plane (tangential) displacements. 3.1. Assumptions (i) The shell is thin relative to its curvature, i.e. t / R is much less than unity (where R is the smallest principal radius of curvature of the undeformed middle surface). (ii) A plane stress state exists at each point through the thickness, i.e. the contributions of the transverse (normal and shear) stresses to the strain energy are ignored.
950
CHUAN-TAo WANG et al.
Q), N vx
)~
My ,~ ,
Nv
t
M, + \'~'~'x / dx
""
ry
N.+k~x/
+
(a)
,
- q
(b) FIG. 1. Shell element (a) force and moment equilibrium and (b) deformation and geometrical relations.
(iii) The strains are small compared with unity and hence the strains on the middle surface are linear functions of the middle surface displacements. (iv) The characteristic wavelength of the deformation is large compared with the shell thickness but small compared to the radii of curvature of its middle surface. 3.2. Incremental strains The geometry of a shell can be completely defined by the shape of its middle surface and its thickness at all points. Therefore, the displacement of any point inside the shell can be expressed in terms of the displacement of a corresponding point on the middle surface. Using a convective or comoving coordinate system x~(i = 1, 2, 3), the initial and deformed configurations of a shell are defined using the same coordinate system. That is, when the coordinate system is set in the middle surface of the undeformed (prebuckled) shell, the material points in the shell are identified by coordinates x~(i = 1, 2) lying in the middle
Wrinkling criterion for an anisotropic shell
951
surface of the undeformed body and coordinate xs normal to the undeformed middle surface. Let the shell coordinates x~ and x2 coincide with lines of principal curvatures (1/r~ and 1/ry) of the shell, dU~ or 0~ (i.= 1, 2) be the in-plane incremental displacements in the xl and x2 (or x and y) directions, W be the incremental buckling displacement normal to the middle surface of the shell and b 0 be the curvature tensor of the middle surface in the undeformed (prebuckled) shell. The incremental stretch strains dE~i (or /~i) and the incremental bending strains (or the changes of the curvatures) dK~j (or Ko) in the middle surface are created as
(i,j = 1,2)
Eij : ½(Ui,j '[- ~J]j,i) @ bij l~'r-)f- ½[~i[3j
(21a)
/~u = - l~,u,
(21b)
where the incremental rotations/~ are defined as /~, = - I42, =
(i = 1, 2).
(22)
In the above equations, the commas (,) denote the covariant differentiations with respect
to the general surface coordinates xl and x2, i.e.
0,.i
00,
= ~
O2I~i
and I~"m = -~x]" The non-
linear terms in Eqn (21a) come from the rotations about the normals (or the slopes along the xt directions). These nonlinear terms may be omitted in comparison with unity when the rotations are small. In such a case, the linear theory is recovered. The incremental Lagrangian strains for any point inside the shell with distance xs or z can then be defined as ~j = / ~ j + xs/¢~ /~,j = _ l~,i.
(23)
3.3. Incremental stress resultants and stress couples The incremental stress resultants and stress couples (per unit width) are defined as
:1 u =
f
tl2
:r~ dxs
d - t/2
~lr~j =
f t/2
(24) 6"/jxsdx3.
d - t/2
If we substitute the incremental strains in Eqn (23) into the constitutive equations in Eqn (14) and then substitute the results into Eqn (24), the incremental stress resultants and stress couples are found to be ]Vij =
[,/2¢Yijdx3 = f,/2 d - t/2
d - t/2
[,/2 ~/~ij =
d - t/2
Lokz~ktdxs = tLijkt(Ekt + ½ t~k~)
[,/2 dijx3dx3
=
d - t/2
?
.
(25)
Lijkl~klx3dx3 = - ~ LijklKkl •
When the rotations are small, the nonlinear terms can be neglected. Then Eqn (24) reduces to
1Vii = tLijktEkt t3 . I(/10 = -~ Lijkl Kkt ,
(26)
where the M 0 are bending moments when i = j and twisting moments when i ~ j. Note that the instantaneous moduli L 0 are rewritten as the fourth-order tensor (Luk~), for convenience of the general expression of the constitutive equations, and they will return to two index forms as L l a l l = L l l , L2222 = L22 and L~22 = Lx2, etc. in the explicit expression for the stress-strain relationship.
952
CHUAN-TAO WANG 4. W R I N K L I N G
et al.
CRITERIA
4.1. Bifurcation functional for a double curved shell Body wrinkling can be considered as the bifurcation of the in-plane deformation under the plane (membrane) stress condition to the out-of-plane bending type of deformation. Consider xl and x2 to coincide with the principal axes; let ~1 = al 1 and a2 = a22 be the principal in-plane (membrane) stresses along these axes; and let R t and R2 be the radii of the principal curvatures. A bifurcation functional was proposed by Hutchinson [9] based on Hill's general theory of uniqueness and bifurcation in elastic-plastic solids [10, 11, 21]. This bifurcation functional is given by
+ N' E'J + N,/3,1J } dS
#) =
t3 L'm Kk, . K,j . . + .tLoktEuE 0 + N,fl~ ,l~.i} dS, = fs{ -i2
(27)
where S denotes the region of the shell middle surface over which the wrinkles appear. This bifurcation functional is the total energy for wrinkling occurrence. In the right hand side of Eqn (27), the first term represents the bending energy (i = j) and twisting energy (i # j), the second term is the strain energy due to the membrane stresses, and the third term may be interpreted as the potential energy of the edge stress or the work done by the applied in-plane stresses in the middle surface. For all admissible incremental displacement fields 6"i and I,V, if F > 0, then the incremental solutions of the deformation are unique and bifurcation is not possible because to create wrinkles requires the energy supply, i.e. the total potential of the system, to increase. This is not a natural or spontaneous process from a thermodynamics point of view. Then F = 0 corresponds to the critical conditions for wrinkles to occur for some non-zero incremental displacement fields. That is, the wrinkling criterion may be written as: F ( 0 , I~) = 0
(not all 0i and I~ = 0).
(28)
4.2. Incremental displacements for wrinkling The incremental displacement fields given by Hutchinson and Neale [12] are employed for wrinkling in short-wavelength and shallow modes. In these fields, wrinkling is assumed to occur over a certain local region S of the sheet which spans many wavelengths of the wrinkling mode. Therefore, in a local wrinkling problem, the continuity conditions or boundary conditions along the boundaries of region S are relatively unimportant. The fields are given as follows: I/V = At
COS(~IXI/~)COS(~2X2/~)
(11 = e t sin(Xtxff~) cos(X2x2/~)
(29)
02 = Ct cos(21xl/~) sin(22x2/~), with = x/~,
(30)
where t = the thickness of the sheet; R = the radius of the principal curvature (RI or R=); A, B and C = constants representing the relative displacement amplitudes of the mode shape; and 21 and 22 = dimensionless wave numbers. 4.3. Wrinkling criterion for a double curved shell The membrane forces due to the in-plane stresses can be defined as =
-
Gli t
N22 =
-
t/o'22t
NIl
(31)
and ~/=
1 a22 is tensile stress -1 a22 is compressive stress.
(32)
Wrinklingcriterionfor an anisotropicshell
953
Finally, the bifurcation functional is obtained by substituting Eqn (31) and the strain increments [defined by the kinematic relationships in Eqns (21), (25) and (29)] into Eqn (27), i.e. F = (Jb + (J,. + Ou =
-~(t3s{(at)2 } - ~ ( L x ~ 2 ~ + Lz22"~ + (L~2 + L,,,,)(2122)z
-- ht3CS[0"11(~)2 -at-?]0"22(~)21,
(33)
or in the matrix form as 2
A
F = (St
t 2
l)
[ M ] { A B C } = (St ~
B?/M2
LCALM3a
MI2 MI3 1 M22 M2a M32 M33
{ABC}, (34)
where the matrix [M] is given by ?/t\
2
Mll = ]~ L~) [Ll12~ + L22224+ 2(L12 + 2L44)(2122)2] + I Ltl (~1) 2+L22 (~2) 2 + 2L12 (R--~x)(R--2)] - [O'l 1'~'2 + ?]0"22'~2]
M22 -- Lx12~ + L442~
(35)
M33 = L22,~-24+ L442~ M12 =M21 = Lll)-1 (~1) -I- L1221 (~-~2) and
M13 = M31 = L22~,2(~1)-I-L12~'2 (~-~2) M23 =
M32-- (L12 "1-L4,1.)J.I,~2,
where
(S = fs[cos(21xJ~)cos(22x2/~)]: dS = ~s [sin(21xl/~) sin(22x2/~)] 2 dS
(36)
= fs[Sin(21x1/~) cos(22x2/~)] 2 dS and ~'1/4 if both 21 and 2 2 ~ 0 (=/.1/2 if either 21 or 2 2 = 0 .
(37)
4.4. Criterion for wrinkling along two principal axes The wrinkling criterion F = 0 in Eqn (28) requires the determinant of the matrix [M] in Eqn (34) to vanish since the constants A, B, and C are not all zero; that is, M l l Mlz M13 [ det[M] = M21 M22 M23 }= 0 for wrinkling to occur.
M31 MS 36: IO-F
M32 M33
(38)
954
CHUAN-TAo WANG
et al.
The critical stress values of 0-11 and 022 can be obtained by minimizing this wrinkling equation with respect to the wave numbers hi and 22, i.e. OlMI - 0 O2i
(i = 1 , 2 ) .
(39)
4.5. Criterion for wrinklin9 alon9 one principal axis In most sheet forming operations, wrinkling takes place along one of the principal axes, that is, the wrinkles are aligned with one of the principal curvature. If wrinkles are assumed to be perpendicular to the xl direction, then the wave number 22 = 0, ¢ = ~ 2 t and M I 3 = M 3 1 = M 2 3 = M 3 2 = 0. Let the principal curvatures be
hi1 = 1/R1,
b22 = 1/R2
and other b u = 0,
(40)
where RI and R2 are the radii along the principal axes. Also let the instantaneous moduli be Lll = L l t x l ,
L22=L2222,
L12=Ll122
and
L44=L1212.
(41)
Then, the wrinkling criterion in Eqn (38) is simplified to det[M] = M 3 3 ( M 1 1 M 2 2 - M 2 2 ) = 0 , or ~/ = M l l M 2 2
(42)
-- M22 = 0
and '
M 2 2 ---- Lx12x4,
Mss = L442~,
(43)
M12=M21 = L,121(-~l)+Lx22,(~2). From Eqn (43), the wrinkling equation (42) depends explicitly on the stress component 0-11. The effect of the stress component 0-22 on wrinkling along the 0-11 direction appears implicitly in the incremental moduli L u, defined by Eqns (13), (15) and (16). With Eqns (42) and (43), a relationship between the wrinkling wave number and the stress can be established, i.e.
1(
0-1, = 1 2 \ 4 ]
L 2)
~
L-~/"
" , -'1/~0-11 )
Minimizing the stress with respect to the wave number [-5-7- = 0 , we obtain the critical wave number or wavelength 2y:
2clr = N//~(LllL~.~ L22)1/4=
(LxlL22L22)1/4. -~-~
1.8612 \
(45)
The critical wrinkling stress is then obtained by substituting 2y into Eqn (44), i.e.
0-], = ~.x/r~ ~22 t N/ /t
11 L 22
- L22 = 0.5774~22 x/L1 1L22 - L22"
(46)
Similarly, for wrinkling perpendicular to the x2 direction (¢ = . , f R : and hi = 0), the critical wavelength and stress are
(ZllZ~_2~ g22) 1/4 L22
(47)
0-~' = x /1-~-t ~ R l X/ /L 11L 2 2 - L 2 2 = 0.5774~-~11/L1 1 L 2 2 - L22 .
(48)
2~r = NIl(LIlting212) 1/4=
1.8612 \
and
Wrinkling criterion for an anisotropic shell
955
A procedure using those criteria is summarized as follows. The deformation geometry (the current thickness and principal radii) and the stress and strain components are assumed to be known or provided by the finite-element simulations. These components are used to calculate the effective stress by Eqn (10) and the effective strain by Eqn (19). With the effective stress and strain and the strain hardening equation (17), the moduli Cij and L~j are defined. Wrinkling conditions are predicted by substituting the moduli L o and the geometrical dimensions (the thickness and the principal radii) into Eqns (45)-(48). 4.6. Wrinklin9 criterion for shrink flanoin9 In shrink flanging operations (Fig. 2), the flat blank with an initial radius R1 is clamped at the location of Ro and forced to wrap around the die shoulder (with a radius of ra) to form a flange with a radius R2(Rz < R1) at the flange edge. Therefore, the stress state at the flange edge with a principal curvature of sin ~/R2 is uniaxial compression along the hoop direction. Wrinkles occur at the edge where the maximum compressive hoop stress or strain appears. If the hoop stress is assumed to be a l 1, then the effective stress 0 is identical to stress a~ a, i.e. = a11. Therefore, the moduli Cii in Eqn (13) are Cl1=~+
-
,
C22=~
and
C12=--~-
From Eqn (15), the incremental moduli Lij can be defined using Eqn (49), that is,
1
R1 ,ddr,_
L. RT I- r
FIG.2. Shrink flanging:the maximumcompressionappears at the flangeedge whichhas an initial radius of R~ and a final radius of Re.
CHUAN-TAo WANG et al.
956
I+Gt
-1
y
Elastic wrinkling. The elastic moduli are obtained by setting the loading index ~ to 0 in Eqn (50), i.e.
E Ltl - 1 -
E
vE
L22 - 1~ - v
1,2
L12 -
1~ - " v
(51)
Therefore, the elastic wrinkling criterion is given by substituting these moduli into the wrinkling criterion in Eqn (46), that is, 0"~r m.
1
t
E
---~ =--sin~ x/3 ~,2 x t/7------75.2' /l-v"
(52)
or, in terms of strain, 1
t
e]r = a]'/E = - - = - - sin0t x/3R2
1
lx/i--S~Sv 2'
(53)
where at is the flange angle.
Plastic wrinkling. For the plastic loading case (the loading index ~t -- 1), the incremental moduli are Lll
=
EET E
- - E T v2
L22 =
E
-
L12 =
vEET E -- ETV2"
E2 (54)
E T v2
Substituting these moduli into Eqn (46), the critical stress for wrinkling is cr], -
1
t sin~tE ~ 1 ET/E
X//3 R 2
-
(55)
v2ET " -g
For a flow curve described by Swift's hardening relation, Eqn (17), the slope of the flow curve is defined by Eqn (18), that is, dal E T --
det
-
nK(eo + ex)"-1 = nat/(e ° + el).
Finally, the critical strain for wrinkling to occur is found by substituting ET into Eqn (46), = ~ [ ~22sln~j
- eo.
(56)
5. E X A M P L E S
5.1. Shrink flanging experiments The validity of the wrinkling criteria proposed was investigated by comparing the predictions with axisymmetric shrink flanging experiments. The test material was the high
Wrinkling criterion for an anisotropic shell
957
strength (HS) steel sheets which are commonly used in automotive body panels. The material properties are listed in Table 1. The circular blanks were cut from HS sheets (thickness = 0.91 ram) to the initial diameters of 104.9, 114.5 and 126.7 ram. Two specimens for each diameter (a total of six specimens) were prepared. A multi-action press was used in the experiments. Figure 3 illustrates the geometry and dimensions of the tooling. The die corner radius and the punch corner radius are 1.52 mm (0.06") and the clearance between the punch and the die is 1.02 mm (0.04"). The circular blank is centered and clamped to the flat bottom of a cylindrical punch by a circular blankholder using three bolts. The punch moves down and flanges the sheet over the concave edge of the die. To detect the onset of the wrinkles (via visual observation), the incremental displacement of the punch was controlled by 0.25 mm (0.01"). The test stopped as soon as wrinkles appeared. For blanks with an initial diameter of 104.9 ram, no wrinkles were observed, while for blanks with larger initial diameters (114.5 mm), wrinkles appeared. For blanks with even larger initial diameters (126,7 mm), wrinkles occurred much earlier. Under the load, the diameters of the specimen were measured, and the pictures of the specimens and the wrinkled shapes were taken. Then, the punch was slowly lifted with the same incremental displacements in order to observe the elastic recovery of the wrinkles. No appreciable elastic recovery of the wrinkles was detected either by visual observation or by measuring the diameters after unloading. Therefore, there were no elastic wrinkles observed in all six specimens made from the HS steel sheets. The diameters and thickness at the edges of the initial blanks and the flanged parts, as well as the flange height (a vertical measure from the flat top to the flange edge) were measured and are presented in Table 2. Using the initial and final diameters, the maximum hoop strains at the flange edge can be calculated.
TABLE 1. MATERIAL PROPERTIES OF HS (HIGH STRENGTH) STEEL SHEETS
n
K (MPa)
eo
¢o (MPa)
,~
M
E (GPa)
v
Thickness (mm)
0.143
603.8
0.0003
289
1.4
2
206.8
0.3
0.91
~b104.9, 114.5, 126.7 mm mt
Punch ,99.3mm RI.52 mm
Sheet
I
I
Die
I
FIG. 3. Tooling set for axisymmetric shrink flanging (the circular sheet is clamped to the punch face by three bolts, and flanged around the punch shoulder when the punch moves down).
CHUAN-TAo WANG et al.
958
TABLE 2. MEASUREMENTS IN AXISYMMETRICSHRINK FLANGING OF HS STEEL SHEETS
to
D2
tf
Hoop strain
Sample
D1
number
(mm)
(mm) (mm) (mm) (mm)
A1 A2 B1 B2 C1 C2
104.90 104.90 114.50 114.50 126.7 126.64
0.91 0.91 0.91 0.91 0.91 0.91
100.89 100.89 106.68 106.96 124.90 124.86
H
0.94 0.94 0.94 0.94 0.92 0.92
%
3.00 3.42 8.00 7.90 4.65 4.50
-
Wrinkle
3.9 3.9 7.1 6.8 1.4 1.4
no no yes yes yes yes
Flange ratio D2/D 1 0.962 0.962 0.932 0.932 0.987 0.986
DI = initial blank diameter, D 2 = final diameter at the flange edge, to = initial thickness of blank, tf = final thickness at the flange edge, H = flange height.
0.08
,
,
,
i
]
,
,
,
,
I
'
'
'
'
t
'
'
'
'
.~ ~ g 0.06
o.o4 0.02 SI-IRINKFLANGE
,HSSTI ' . . . .
0 0.92
0.94 Flange
~U
, .... 0.96
, , , ,~.
0.98
1
Ratio ( R 2 / R 1 )
FIG. 4. The maximum hoop strains at flange edge vs flange ratio (R1 = initial blank radius, R2 = final radius at the flange edge).
5.2. Predictions and comparisons The axisymmetric shrink flanging of HS (high strength) sheet steels was simulated with the program FLANGE which was developed by these authors [26] and is able to predict the maximum compressive strains and wrinkles at the flange edge. In shrink flanges, the maximum hoop strains increase as the flange ratio decreases or as the initial blank radius (R1) increases. Figure 4 shows the comparison between the predicted and the measured maximum hoop strains at the flange edge. (Note that the strains are compressive and have negative values, but are plotted using the absolute values.) The experimental results match well with the proposed models for the maximum strains. The relationship between the flange height (the vertical measurement from the flat bottom to the flange edge, Fig. 2) and the flange ratio is illustrated in Fig. 5. Again, a good match is indicated. The predictions underestimate the flange height by 0.5 mm for the wrinkled specimens (about a half of the sheet thickness) and have a relative error of less than 15%. The proposed wrinkling criteria, Eqns (53) and (56), were investigated by the visual observation of the onset of the wrinkles during shrink flanging experiments. No elastic wrinkles were observed because the unloading does not remove the wrinkles. Both the measurements and the predictions via the proposed wrinkling criterion proved that the critical strains to onset the wrinkles are greater than the initial yield strain ( = yield stress/Young's modulus = 0.0014). Therefore, only plastic wrinkles occur in shrink flanges of the HS (high strength) steel sheets. The measured maximum strains in the wrinkled specimens (in these cases, the measured maximum strains may also be the critical strains due to the onset of the visible wrinkles at flange edge) were compared with the predicted wrinkling strains, as shown in Fig. 6. Wrinkles did not appear when a circular blank with
Wrinkling criterion for an anisotropic shell 10
959
' , , , , , ' , , •
Ptedkud
f Z. 6 4
_a c~ 2
SHRINK FLANGE HS Steel
0
. . . .
0.92
I
,
,
0.94
0.96
0.98
1
Flange Ratio (R2/R1)
FIG. 5. Flangeheight vs flange ratio (R1 = initial blank radius, R2
0.15
. . . .
.
,
. . . .
i
. . . .
I
. . . .
=
final radius at the flangeedge).
I
. . . .
lo-.l
"~ o.12
,
u ..~
SHRINK FLANGE HS Steel
0.09
I
/
0.06 °m
--2 0.03 o
0
0.005
0.01
0.015
0.02
0.025
(t/R2)sin(alpha)
FIG. 6. Critical strains for wrinkling in shrink flanging of high strength steel sheets (t -- sheet t h i c k n e s s , R2 = final radius at the flange edge, ct = flange angle).
a diameter of 104.9 m m was formed to a flange of diameter D 2 = 100.86 m m (the flange ratio is 0.96). The m a x i m u m strains measured and predicted are about 4%, which is less than the critical strain (about 6%) predicted by the proposed wrinkling model. Wrinkles did appear for flanges with diameters of D 2 = 106.68 and 124.98 mm, which are formed from the circular blanks with the initial diameters of Dt = 114.50 and 126.70 mm, respectively. The wrinkling limits decrease as the flange ratio decreases or the blank size increases. For a flange with initial blank diameter of 114.50 m m (flange ratio = 0.932), the predicted critical strain for wrinkling is 4.5%, which is less than the measured and predicted m a x i m u m strains (7%) at the flange edge. Therefore, wrinkles did appear in the experiment. For a flange with an even larger initial diameter (Dx = 126.70), the flange ratio has to be increased to 0.987 and the critical strain limit further reduces to 0.6%. At this flange ratio, the m a x i m u m strains measured and predicted are 1.4%, which is greater than the limit. Therefore, for those specimens, wrinkles did occur earlier during shrink flanging tests. 6. CONCLUSION Wrinkling criteria are developed for the local wrinkling phenomenon which occurs in the in die wall 'in the unsupported region of a sheet. These criteria are derived based on the application of the D M V shell theory and Hill's bifurcation model to an elastic isotropic and plastic anisotropic shell with c o m p o u n d curvatures. Experiments on the shrink flanging of
960
CHUAN-TAo WANG et al.
high strength steel sheets were conducted to verify the proposed wrinkling criteria. The predictions of the onset of the wrinkles at the flange edges agree well with the measurements. The wrinkling models are especially useful in conjunction with finite element modeling of sheet forming processes. A process control curve of restraining force (blankholder force or drawbead force) vs punch stroke could be obtained through the simulations. Acknowledgments--The authors would like to thank the following individuals in helping to conduct the shrink flanging experiments: Serdar Tufekci designed the tooling, prepared the specimens and conducted the tests, Mary Hartzler and S. Wm Lipson helped to machine the tooling and specimens.
REFERENCES I. C. T. WANG, G. KINZEL and T. ALTAN, Sheet Forming Technology: Operations and Process Control. Engineering Research Center for Net Shape Manufacturing, The Ohio State University, Report No. ERCfNSM-S-91-6 (1991). 2. B. W. SENIOR, Flange wrinkling in deep-drawing-operations. J. Mech. Phys. Solids 4, 235 (1956). 3. N. KAWAI, Critical conditions of wrinkling in deep drawing of sheet metals (1st report: fundamentals of analysis and results in case where a blank-holder is not used). Bull. JSME 4(13), 169 (1961). 4. N. KAWAI, Critical conditions of wrinkling in deep drawing of sheet metals (2nd report: analysis and considerations for conditions of blank-holding). Bull. JSME 4(13), 175 (1961). 5. N. KAWAI,Critical conditions of wrinkling in deep drawing of sheet metals (3rd report: predicting of critical blank-holding pressure). Bull. JSME, 4(13), 182 (1961). 6. J. HAVRANEK,Wrinkling limit of tapered pressing. J. Aust. Metals 20, 114 (1975). 7. T. X. Yu and W. JOHNSON,The buckling of annular plates in relation to the deep-drawing process. Int. J. Mech. Sci. 24(3), 175 (1982). 8. R. W. LOGAN, Sheet metal formability simulation and experiment. Ph.D Dissertation, The University of Michigan, MI, U.S.A. (1985). 9. J. W. HUTCHINSON,Plastic buckling, in Advances in Applied Mechanics 14, p. 67 (1974). 10. R. HILL, A general theory of uniqueness and stability in elastic/plastic solids. J. Mech. Phys. Solids 6, 236 (1958). 11 R. HILL, Bifurcation and uniqueness in Nonlinear Mechanics of Continua p. 236. Society of Industrial Applied Mathematics, (Muskhelishvili Vol.), Philadelphia, PA (1961). 12. J. W. HUTCmNSONand K. W. NEALE, Wrinkling of curved thin sheet metal, in Proc. Int. Syrup. on Plastic Instability, Considere Memorial (1841-1914), Paris, 9-13 September, p. 71 (1985). 13. R. HILL, Theoretical plasticity of textured aggregates, in Mathematical Proceedings of Cambridge Philosophical Society, Vol. 85, p. 179 (1979). 14. L. H. N. LEE, Inelastic buckling of initially imperfect cylindrical shells subject to axial compression. J. Aerospace Sci. 29, 87 (1962). 15. A. NEEDLEMANand J. R. RICE, Limit ductility set by plastic flow localization, in Mechanics of Sheet Metal Forming: Materials Behavior and Deformation Analysis p. 237 (edited by D. P. KOISTINENand N. M. WANG). Plenum Press, New York (1978). 16. N. TRIANTAFYLLIDISand A. NEEDLEMAN,An analysis of wrinkling in the Swift cup test. J. Engng Mater. Teehnol. 102, 241 (1980). 17. X. F. WANGand L. H. N. LEE, Wrinkling of an unevenly stretched sheet metal. J. Engn0 Mater. Technol. 111, 235 (1989). 18. R. HILL, The Mathematical Theory of Plasticity. Oxford (1950). 19. S. KOBAYASHI,R. M. CADDELLand W. F. HOSFORD,Int. J. Mech. Sci. 27(7/8), 509 (1985). 20. W. F. HOSFORD, Limitations of non-quadratic anisotropic yield criteria and their use in analyses of sheet forming. Proc. 15th Biennial 1DDRG Conor. Dearborn, MI, p. 163 (May 1988). 21. R. HILL, A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A193, 281 (1948). 22. R. H. WAGONER,Measurement and analysis of plane-strain work hardening. Metall. Trans. I1A, 165 (1980). 23. A. R. RAGABand A. T. AI~BAS,Assessment of work-hardening characteristics and limit strains of anisotropic aluminum sheets in biaxial stretching. J. Engng Mater. Technol. 108, 250 (1986). 24. J.D. BRESSANand J. A. WILLIAMS,The use of a shear instability criterion to predict local necking in sheet metal deformation. Int. J Mech. Sci. 25, 155 (1983). 25. D. BRUSHand B. O. ALMROTH,Buckling of Bars, Plates and Shells. McGraw-Hill (1975). 26. C.T. WANG,G. KINZELand T. ALTAN,Mechanics of Bending and Flanoino and A User's Manual for Computer Programs BEND and FLANGE. Engineering Research Center for Net Shape Manufacturing, The Ohio State University (in press).