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Critical conditions for wrinkling during the forming of anisotropic sheet metal
Journal of Materials Processing Technology, 35 (1992) 213-226
213
Elsevier
Critical conditions for wrinkling during the forming of anisotropic sheet metal A.M. Szacinski 1 and P.F. Thomson
Department of Materials Engineering, Monash University, Clayton, Victoria 3168, Australia (Received October 10, 1991; accepted March 9, 1992)
Industrial Summary In many applications, when sheet material is to be used near the limit of its capacity, it is important to be able to determine the critical failure strain, including that for wrinkling and to be able to correlate a number of properties of sheet materials, including the coefficient of normal plastic anisotropy, with critical wrinkling strain. The effect of preferred orientation on wrinkling behaviour under different loading and geometries was studied in numerical experiments employing the finite-element method, and confirmed in the Yoshida and other buckling tests, with the expectation that a material can be designed to increase the strain at which wrinkling occurs under a given state of stress or to permit appropriate orientation of the blank to maximise the allowable depth of draw. Critical elastic wrinkling strain can be represented on a diagram of surface principal-strains as a series of curves nested about the origin, expanding with elastic modulus, coefficient of normal elastic anisotropy (as defined in the present work) and coefficient of normal plastic anisotropy. It was found that the coefficients of elastic and normal plastic anisotropy correlate in copper and 70/30 brass but not in an annealed mild steel. The correlation can exist in materials in which the structure lacks symmetry or has symmetry of lower order than six-fold. They correlate in the same way with the critical strain for elastic wrinkling, and in such materials the former may then be used to predict the effect of orientation on the critical plastic wrinkling strain. In practice, such a relationship may be useful because the coefficient of normal elastic anisotropy is easier to calculate, given the principal orientations represented in the sheet, than is the normal plastic anisotropy, although the latter may be easier to measure.
1. Introduction The effect of preferred orientation on wrinkling behaviour has been studied in the present work, with the expectation that a material could be designed to increase the strain at which wrinkling occurs under a given state of stress: this
Correspondence to: P.F. Thomson, Department of Materials Engineering Monash University, Clayton, Victoria 3168, Australia. 1Present address: Gas and Fuel Corporation of Victoria, Scientific Services Department, Materials Section, Highett, Victoria 3190, Australia.
0924-0136/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
214
would permit a deeper draw before the onset of wrinkling. Another motive for investigating elastic wrinkling of anisotropic sheet metal was to determine whether a measure of elastic anisotropy obtained from a knowledge of the elastic constants of an anisotropic sheet could be correlated with normal plastic anisotropy and whether the tendency to plastic wrinkling could be correlated with it. Although plastic anisotropy is easier to measure, elastic anisotropy is easier to calculate, given the preferred crystallographic orientations represented in the sheet. Fracture is only one mode of failure in industrial pressings. Unsatisfactory pressings may result also from the presence of defects such as wrinkling, which arises from instability in compression due to non-uniform metal flow or nonuniform distribution of elastic recovery strain on unloading [ 1 ]. Such defects can lead to dimensional inaccuracies which cause difficulty during fit-up at the assembly stage or to a product which is unacceptable aesthetically. During the deformation of large thin sheet components, wrinkling may also be initiated elastically in the unsupported regions. Since the work of Geckeler [2], Baldwin and Howald [3] and Senior [4], extensive studies of wrinkling behaviour of sheet steel have been performed, including studies employing the Yoshida test [5,6] in which a square sheet is stretched diagonally until wrinkling occurs. The purpose of this latter test is to study the effect of material properties on resistance to wrinkling in large shallow, more or less unconfined, panels during forming. Although most sheet metals are anisotropic, relatively few studies of wrinkling behaviour have included the effect of anisotropy. Naziri and Pearce [7] concluded that the blank-holder pressure necessary to suppress wrinkling in a drawn cup increases with planar plastic anisotropy (Ar) and decreases with normal plastic anisotropy (r). Kleemola and Kumpulainen [8] found that when the flange of a drawn cup buckled into less than five waves, the number of waves depended on the anisotropy of the material. From their study of the critical conditions for elastic and plastic wrinkling of the flange of a circular blank during deep drawing, Yu and Johnson [9] concluded that a thin annular plate of large diameter may wrinkle elastically before the blank is drawn plastically, whereas a thick annular plate of small diameter wrinkles plastically because yield occurs before wrinkling. Stickels and Mould [10] found that a high average Young's modulus was associated with a high average coefficient of mean normal plastic anisotropy and correlated equally well with the limiting drawing ratio in the Swift cup test. Controlling elastic and plastic anisotropy can have a significant effect on forming performance [ 11,12 ]. However, Stickels and Mould [ 10 ] and Wilson [11,12] did not attempt to investigate whether the measures of elastic and plastic anisotropy could be related to initiation of wrinkling in ferrous or nonferrous sheet metals.
215
2. Computation and experiment The critical conditions for elastic wrinkling of anisotropic sheet metals in a number of different tests (Fig. 1 ) designed to produce different loading and geometries were predicted numerically using the finite-element method. These predictions were verified experimentally using the disc wrinkling test and the Yoshida test described in Fig. 1. The numerical and experimental results assisted in defining curves on the diagram of surface principal strains, which would describe the critical conditions for wrinkling in anisotropic elastic sheet metals.
2.1. Material properties The mechanical properties of the materials used in the experiments (Table 1 ) were obtained from tensile tests performed with an Instron testing machine according to the procedure recommended in A S T M Standard E8. The coefficient of normal plastic anisotropy of 3021 steel could not be determined, as the elongation at fracture was too low. 2.2. Modelling elastic anisotropy The effect of anisotropy on elastic wrinkling was modelled by assuming particular crystallographic orientations common in rolled sheets. The elastic compliances, sii, defined by sij = - aij
(1)
where ~j is the tensor strain component and aij is the tensor stress component, associated with those orientations, were calculated according to Lekhnitskii [13] and Hearmon [14].
r~ '
l
'
r2
(~)
(b)
(c)
Fig. 1. Schematic descriptions of tests: (a) disc-wrinklingtest; (b) Yoshida test; and (c) square plate wrinklingtest. ((i) TI=O (compression); (ii) TI=- T2 (shear); (iii) T~=-2T1; and (iv) T2= - 2T1)
216 TABLE 1
Properties of the sheet materials used Material description
Dominant Thickness Angle to RD orientation (ram) (°)
E (GPa)
Y (MPa)
n
cold-rolledsteel
{211}[0il]
0.60
0
231
682
0.10
{111}[112]
0.60
0
220
341
0.19 1.16
{100}[100] 200 annealed mild steel {111}[112]
0.60
0
223
255
0.21 1.32
0.68
0 45 90
70/30brass
{113}[211]
0.68
copper
{112}[111]
0.68
221 218 220 220 110 100 105 108 135 118 128 120
247 260 255 254 117 116 114 116 206 197 203 202
0.22 0.19 0.21 0.21 0.28 0.25 0.27 0.27 0.24 0.21 0.17 0.20
r
Ar
-
3021
Al-killedannealed steel A13 2%SisteelLycor
Mean value 0 45 90
Mean value 0 45 90
Mean value
1.27 1.43 0.15 1.89 1.50 0.95 1.05 - 0 . 2 2 0.71 0.94 0.72 1.28 - 0 . 6 6 0.52 0.95
Elastic wrinkling in tests (see Fig. 1 ) of sheets with the ideal orientations shown in Table 1 was predicted by the F E M using the ABAQUS programme. The eigenvalue prediction of elastic wrinkling adopted in this programme provides an estimate of the wrinkling load of a stiff plate or shell - i.e., one in which the pre-buckling response is essentially linear - when the collapse is not catastrophic (so that the plate is not excessively sensitive to imperfections). In ABAQUS, elastic wrinkling of an anisotropic material can be analysed by modifying the stiffness in the resulting problem Two measures of normal elastic anisotropy were adopted. The first was defined in a manner analogous to that of the coefficient of normal plastic anisotropy, as ~w re --
812 --
~t
(2)
Sl3
where ew and et are the logarithmic strains respectively in the width and thickness direction of a specimen deformed in uni-axial tension and s12 and Sl~ are the values of elastic compliance calculated from the dominant orientation using the method of Lekhnitskii [13 ]. This definition of elastic anisotropy is valid only when the stress is uni-axial, acting in the longitudinal direction.
217
Equation (2) can be interpreted as showing the ratio of the Poisson contractions in two directions (transverse and normal) due to the axial stress. The ratio of the elastic moduli En and Ew respectively in the directions normal to the rolling plane and transverse to the rolling direction was used to provide an alternative definition of normal elastic anisotropy (rE) ~w En rE _
_
Et
Ew
(3)
the ratio expressed in eqn. (3) being chosen as a measure of elastic anisotropy because the elastic moduli depend on the preferred crystallographic orientation within the sheet. It was found [15] that the value of the macroscopic elastic modulus of a homogeneous material was more significant than the value of the coefficient of normal plastic anisotropy in predicting the initiation of elastic wrinkling.
2.3. Modellingplastic anisotropy The value of the coefficient of normal plastic anisotropy was calculated from the crystallographic orientations representing ideal sheet textures, using the method of Lee and Oh [16]. Their formulation improves on the methods of Vieth and Whiteley [17] and Fukuda [18] by including all the operating slip systems in the calculation. Vieth and Whiteley used only the slip system with the largest Schmid factor, whereas Fukuda calculated an average r value from the five slip systems with Schmid factors of the greatest magnitude. Fukuda pointed out that the slip system with the largest Schmid factor never operates alone, but co-operates with other systems which have a Schmid factor of next to the largest magnitude. According to Lee and Oh [ 16 ], r-
ew F~[l(bp)(bd)]S] er •[[(tp)(td)lS]
(4)
where b, t, d and p are unit vectors along the width, thickness, slip directions and the direction normal to the slip plane, respectively, and Y~indicates summation over all operating slip systems. S is the corresponding Schmid factor calculated from
S= ](lp) (/d)]
(5)
where I is the unit vector along the tensile direction. It was assumed that the sheet had a texture (K1K2K3) [AIA2A~]. Unit vectors k and a respectively along the normal to the plane (K1K2K3) and in the direction (AIA2A3) were calculated as:
k=[k~,k2,k3]=[K~ K2 K3] i-~l,}K],[Kl
(6)
218 and
[ A 1 A 2 A3 1 a=[al,a2,a3]= i-AI']A]']A]
(7)
where
]gt=x/K~+g~+K~
]A]=x/A~+A~+A ~
and
The unit vectors l, t and b in eqns. (4) and (5) were obtained from the relationships
l= [ll,12,1,~] =
[al cos a +
(a2k3 -aak2)
sin a,
a2 cos a +
(a3kl -alk3)
sin a,
a~ cos a +
(aik2 -a2ki)
sin a ]
t= [ti,6,t3 ] = [ki,k2k.~ ] b= [bi,b2,b3] = [-al
(8) (9)
sin o~+
(a2k3-a3k2)
cos ~,
- a 2 sin ~ +
(a3kl -alk3)
cos ~,
- a 3 sin ~ +
(alk2 -a2k~)
cos ~]
(10)
where c~ is the angle between the rolling direction a and the tensile direction I. The unit vectors p and d (eqns. (4) and (5)) of a slip system (P1P2P3) [DID2D3] were calculated in the same way as the unit vectors k and a. The values of the coefficients of normal plastic anisotropy (eqn. (4)) calculated by the above method were compared with values obtained experimentally.
3. Experiment
3.1. Disc wrinkling test In this test (Fig. l ( a ) ) , wrinkling of a circular specimen was induced by indentation with opposed annular platens of smaller diameter. The dimensions of the circular plate and indenter used in the experiments were chosen as a result of preliminary tests to determine the diameters of plate and indenter for which wrinkling was easiest to produce and detect. A rig was designed to keep the indenter and plate concentric. The principal strains on the surface of the outer segment of the circular plate were measured to _+10 -~ using crossed foil electrical resistance strain gauges of 2 mm gauge length, placed with their centres at 5 mm from the indenter, with a d u m m y gauge for temperature compensation. Because the strains of interest were small - of elastic order - and rotations were small it was considered adequate to measure normal strains. The deflection of the plate during the test was measured with a digital trans-
219 ducer and three dial gauges positioned at 90 ° as close as possible to the outer edge of the plate: this permitted recording of displacement at the initiation of wrinkling. A film of PTFE, greased on one side, was used as a lubricant to minimise the friction between the indenter and the plate. 3.2. Yoshida test
Specimens 100 m m square were pulled between grips, 40 mm wide, in the Yoshida buckling test. The major and minor strains at the onset of wrinkling were measured with a crossed foil electrical resistance strain gauge positioned at the centre of the specimen. The deflection of the plate at wrinkling was recorded by a dial gauge attached to the mid-point of the surface on the face of the specimen opposite to that on which the strain gauge was placed. Wrinkling initiated in the above tests while the specimen was still elastic. The critical condition was confirmed by departure of the force-displacement curves from linearity. 4. Results and discussion
The loads required to initiate elastic wrinkling of square plates (Fig. 1 (c)) of pure iron with primitive orientation {100} [010], 2% Si steel Lycor 200 with dominant orientation {011} [100], Al-killed steel A13 with dominant orientation {111}[113] and cold-rolled steel 3021 with dominant orientation {211} [0il] stressed in compression parallel to the rolling direction were computed by the FEM, as implemented in the ABAQUS package (Table 2). The corresponding computed values of elastic modulus, coefficient of normal elastic anisotropy, re, as defined in the present work (Section 2.2 ), and coefficient of normal plastic anisotropy, are shown also. The coefficients of normal elastic and normal plastic anisotropy of Lycor 200, A13 steel and 3021 steel in directions 45 ° and 90 ° to that of rolling were not evaluated experimentally. The effect of the coefficients of normal elastic and plastic anisotropy on the elastic wrinkling load in Lycor 200, A13 steel and 3021 steel was only studied by computation. Elastic modulus was chosen as a measure of elastic anisotropy because it is orientation-dependent and it was expected that a close relationship between Young's modulus and the coefficient of normal plastic anisotropy would exist [10], although the manner of the relationship with orientation might be quite different. As expected, the wrinkling load increased with Young's modulus and a similar relationship was found in the results of other tests used [ 15 ]. The coefficients of normal elastic and normal plastic anisotropy related to wrinkling load in the same way as did the elastic modulus (Table 2). Wrinkling occurred earlier in material with orientations for which the coefficients of normal anisotropy, either elastic or plastic, were smaller. These findings confirmed the
22O TABLE 2 Computed critical wrinkling loads of steels and the corresponding coefficients of elastic and plastic anisotropy Material
Dominant orientation
Computed elastic modulus (GPa)
Computed coefficient of elastic anisotropy re
Computed coefficient of plastic anisotropy r
Computed critical load at onset of wrinkling (N/m)
pure iron Lycor 200 A13 steel 3021 steel
{100}[010] {011}[100] {111}[112] {211}[011 ]
176 210 221 236
1.00 1.00 1.10 2.02
0.50 0.69 1.16 1.59
620 998 1460 2320
conclusions of Naziri and Pearce [7] and of Kleemola and Kumpulainen [8], who studied only the effect of plastic anisotropy on wrinkling behaviour and established that the earliest wrinkles were observed in the steel sheets with the smallest coefficients of normal anisotropy. It appears that the critical elastic wrinkling load decreases with the elastic anisotropy, but the most deleterious form of anisotropy has not been determined; the possibility remains that some form of controlled elastic anisotropy, still undetermined, might be used to effect improved resistance to wrinkling. The computed wrinkling-limit curves for the different steels on the diagram of principal surface strains (Fig. 2 ), expand with the elastic modulus and the coefficients of normal elastic and plastic anisotropy. It seems reasonable to suppose that wrinkling would be resisted by material with a smaller coefficient s13 in the direction transverse to the axis of the wrinkles and also by a material with a larger coefficient of normal elastic anisotropy, s1Js13. The increase in critical wrinkling strain with elastic anisotropy as defined in this work may be advantageous in minimising the type of wrinkling that results from residual stress. As was also the case in a homogeneous material [ 15 ], the largest critical strain occurred in the test in which the ratio of surface strains corresponded to shear, so that elastic springback on unloading after forming may be minimised by promoting shear deformation. The relationship between elastic modulus and the calculated critical values of the strain ratio el/e2 at the onset of wrinkling of a square plate under edge compression for steels with different ideal orientations are shown in Fig. 3, with the corresponding values of elastic and plastic normal anisotropy parameters. It was tempting to correlate the ratio of eJe2 at the initiation of elastic wrinkling with the elastic modulus and the coefficients of normal elastic and plastic anisotropy in these anisotropic materials but it must be remembered that the test defined a particular strain path. The critical surface strains in the disc wrinkling test and in the Yoshida test on 3021 steel, A13 steel and Lycor
221
.7
1. Pure Iron 2. Lycor 200 3. A13 Steel 4. 3021 Steel
r,=l.00 r,--1.00 re=l.10 r,=2.02
r=0.50 r=0.69 r=l.16 r=1.59
E=176 E--210 E--221 E=236
GPa GPa GPa GPa
-6
-5 o x "4
-3
z
c~ ©
2
I
t
-6
-5
r
-4
-3
-2
-1
0
MINOR STRAIN x 104 Fig. 2. C o m p u t e d envelopes of surface strains at the onset of elastic wrinkling of square plate under edge compression, showing the effect of elastic and plastic anisotropy.
200 are shown in Table 3. These tests dictate different strain paths and the critical values of el/e2 now appear to vary independently of Young's modulus, re and f; and it is difficult to make a general conclusion concerning the relationship between anisotropy and the strain ratio at onset of wrinkling. The discrepancies between the computed and the observed critical strains shown in Table 3 are due to computation being based on the ideal orientation in each case, whereas the actual material would have a spread of orientations around the dominant one, which may have obscured also any direct relationship between the value of the strain ratio and the elastic modulus at the onset of wrinkling. It seems unlikely that the difference in absolute values of principal strains was caused by error in measurement, as strains were measured to + 10 -6, two orders of magnitude less than the discrepancy. The strain ratios el/e2 for an annealed mild steel, 70/30 brass and copper (described in Table 1 ) at the onset of elastic wrinkling in the Yoshida test, obtained by computation and experiment, are shown in Table 4, which shows also the corresponding values of Young's modulus, the coefficients of normal elastic anisotropy parameters re and rE and the coefficient of normal plastic
222
P R I N C I P A L ORIENTATION: 1.2.
z z
g
1-
{100}[0101
ro=l.00
r=0.50
2-
{011}[100]
r,=l.00
r=0.69
3-{m}[11~]
ro--MO
r=1.16
4 - {21~}[oi11 r~=2.oo ~=1.59
-1.0
/
-0.8
o cg
-0.6
.4
a~ oq
-0.4
-0.2 150
I 175
I 200
I 225
I 250
.
YOUNG'S MODULUS [GPa] Fig. 3. C o m p u t e d su rface s t r a i n r a t i o s at the o n s e t of elastic wrinkling in a s q u a r e p l a t e u n d e r edge c o m p r e s s i o n , as a f u n c t i o n of Y o u n g ' s m o d u l u s .
TABLE 3 Critical wrinklingstrains in the disc-wrinklingtest and the Yoshidatest: computedand observed Material
Elastic modulus (GPa)
L y c o r 200 {011}[100]
210
A13 steel {111}[115]
221
3021 steel {211}[0il]
236
el > 10 -4 e 2 X l 0 -4 el/e2 el X 10 -4 e 2 X 1 0 -4 eJe2 e~ X 10 -4 e 2 × 1 0 -4 e,/e2
Disc-wrinkling test
Yoshida test
Computed
Observed
Computed
Observed
9.15 4.22 2.17 6.25 - 3.32 - 1.88 56.80 -28.80 - 1.97
0.62 -0.14 -4.43 0.59 -0.22 -2.68 0.81 -0.46 -1.76
0.79 -0.48 -1.65 0.73 -0.32 -2.28 -
10.96 4.76 2.28 15.30 - 5.91 - 2.58 46.99 -24.99 - 1.88 -
-
anisotropy. In 70/30 brass and copper with the orientations approximating (113) [21i] and (112 ) [111 ], respectively, the critical strain ratio el/e2 at the onset of elastic wrinkling in the Yoshida test increased with the coefficients of
223 TABLE 4 Observed and stimulated results of the Yoshida test in annealed mild steel, copper a n d 70/30 brass Angle to RD (°)
Annealed mild steel (111)[li2]
70/30 brass {113)[21il
Copper (112}[i11]
Computed Observed Computed Observed Computed Observed E (GPa) re
rE
r
el/e2
0 45 90 0 45 90 0 45 90 0 45 90 0 45 90
220 220 220 2.29 1.79 2.29 1.29 1.29 1.29 1.22 1.36 1.51 --2.38 --2.38 --2.38
221 218 220
1.27 1.43 1.89 --1.95 --2.10 --2.45
111 75 89 0.60 0.95 0.40 0.88 1.04 0.70 0.83 1.01 0.62 --1.12 --1.65 --1.38
110 100 105
0.95 1.05 0.71 --2.11 --2.36 --1.86
191 94 131 1.0 2.15 0.37 1.0 1.39 0.68 0.60 1.62 0.50 --1.07 --2.80 --1.71
135 118 128
0.72 1.28 0.52 --2.31 --2.55 --1.80
:6 70/30 BRASS {113}[211] 1. 0° re=0.60 r=0.83 2. 45° r,=0.95 r=l.01 3. 900 re=0.40 r=0.62
rE=0.88 rE=l.04 rE=0.70
-5
-4
1
2
~
.1
-6
-4
-3
-z
-1
0
MINOR STRAIN x 104 Fig. 4. Computed envelopes of surface strains (wrinkling-limit curves) for 70/30 brass, tested at 0 °, 45 ° and 90 ° to the rolling direction.
224
COPPER {ll2}[il0] I. 0°
r~--l.00
r=0.60
2. 45° 3. 90o
re--2.15 rc=0.37
r--1.62 r=0.50
rE: 1.00 rE=l.39
] 87
rE=0.68 -6
"5
1
×
z
2
<
"4 u[~) -3
O ~-~ <
-2 -1 I
-7
-6
-5
-4
-3
-2
-1
0
MINOR STRAIN x 104 Fig. 5. Computed envelopes of surface strains 45 ° and 90 ° to the rolling direction.
(wrinkling-limit curves) for copper, tested at
0 °,
normal elastic and plastic anisotropy. It was found (Figs. 4 and 5) that the predicted components of strain at the initiation of elastic wrinkling varied directly with the normal elastic and plastic anisotropy parameters. The earliest wrinkling was predicted in each case for the orientation at which the coefficient of normal plastic anisotropy was smallest. The coefficients of normal elastic anisotropy, re and rE, for copper and 70/30 brass were related directly to the coefficients of normal plastic anisotropy (Table 4). It appears that it would be possible to predict the strains at which elastic wrinkling in copper and 70/30 brass occurs given the coefficients of normal elastic anisotropy as defined in the present work. However, the coefficients of normal elastic anisotropy in the annealed mild steel were not related to the coefficient of normal plastic anisotropy in that material (Table 4). This may be a result of the different symmetry in the texture of these materials [19], the dominant (111 ) [112] texture of the bcc mild steel sheet possessing six-fold symmetry, whereas the dominant textures of the fcc copper and 70/30 brass have two-fold symmetry. The six-fold symmetry of the annealed mild steel is manifested in its plastic properties (for example, in the different values
225 of the coefficient of normal plastic anisotropy and in the tendency to earing of deep-drawn cups), but not in its elastic properties, which can exhibit a maxim u m of four-fold symmetry, because elasticity is a linear fourth-order tensor [20]. In the annealed mild steel, the Young's modulus and the coefficients of normal elastic anisotropy, re and rE, were independent of the angle to the rolling direction as were the computed values of strain ratios at the onset of elastic wrinkling. It should be noted that the textures of the 3021 steel and Lycor 200 ({211 } [011 ] and {100} [ 100 ], respectively) had two-fold symmetry with respect to the loading direction used in the experiments, whereas the texture of the Al-killed A13 steel ( {111 } [ 112 ] ) had six-fold symmetry. For this reason, correlation between the coefficients of normal elastic and plastic anisotropy and the critical strain at initiation of wrinkling in the given orientation could be expected in 3021 steel and Lycor 200, but not in A13 steel. If the crystallographic structure of sheets is such that the plastic anisotropy lacks symmetry or has symmetry lower than six-fold, correlation with elastic anisotropy becomes more likely and it is possible to use the former to compare the critical wrinkling strains of sheet of different orientations. 5. Conclusions
(i) The conditions under which anisotropic sheet metals (steel, copper and 70/30 brass) wrinkled elastically, shown as elastic wrinkling-limit curves on the diagram of principal surface strains, were predicted by numerical experiments using the FEM to analyse the behaviour of sheet materials under different geometry and loading conditions. The curves expanded with the elastic modulus, the coefficient of normal elastic anisotropy (as defined in the present work) and the coefficient of normal plastic anisotropy. (ii) The coefficients of elastic and normal plastic anisotropy used herein can be correlated in materials which lack symmetry or have symmetry of order lower than six-fold. They correlated in the same way with the critical strain for elastic wrinkling and in such materials the former may then be used to predict the effect of orientation on the critical plastic wrinkling strain. High values of the coefficients of normal elastic anisotropy, as defined, coincided with high values of the coefficients of normal plastic anisotropy.
References
1 T. Shimomuraand M. Yoshida,Study of surface deflectionin door outer panel, Proc. 13th Biennial Congress oflDDRG, Melbourne,Australia, February 20-24, 1984,pp. 46-53. 2 J.W. Geckeler,Plastisches Knicken der Wandungvon Hohlzylinderund einigeandere Faltungsverscheinungenan Schalenund Blechen,Zeit. Angew. Math. Mech., 8 (5) (1928) 341352.
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9 10 11 12 13 14 15 16 17 18 19 20
W.M. Baldwin, Jr. and T.S. Howald, Folding in the cupping operation, Trans. A.S.M., 38 (1947) 757-788. B.W. Senior, Flange wrinkling in deep-drawing operations, J. Mech. Phys. Solids, 4 (1956) 235-246. K. Yoshida, H. Hayashi, M. Hirata, T. Hira and S. Ujihara, Yoshida buckling test, IDDRG Paper, DDR/WG III/81, Japan, (preprint), 1981. A.M. Szacinski and P.F. Thomson, The effect of mechanical properties on the wrinkling behaviour of sheet materials in the Yoshida test, J. Mech. Work. Technol., 10 (1984) 87-102. H. Naziri and R. Pearce, The effect of plastic anisotropy on flange-wrinkling behaviour during sheet metal forming, Int. J. Mech. Sci., 10 {1968) 681-694. H. Kleemola and J. Kumpulainen, Calculation method for determining the limit of flange wrinkling in the deep-drawing of cylindrical steel sheets without blankholding, Metall. Trans., l l A (1980) 1701-1710. T.X. Yu and W. Johnson, The buckling of annular plates in relation to the deep-drawing process, Int. J. Mech. Sci., 24(3) (1982) 175-188. C.A. Stickels and P.R. Mould, The use of Young's modulus for predicting the plastic-strain ratio of low-carbon steel sheets, Metall. Trans., 1 {1970) 1303-1312. D.V. Wilson, Plastic anisotropy in sheet metals, J. Inst. Metals, 94 (1966) 84-93. D.V. Wilson, Controlled directionality of mechanical properties in sheet meals, Metall. Rev., 139 (1971) 175-188. S.G. Lekhnitskii, Theory o/Elasticity o[ an Anisotropic Elastic Body, Holden-Day, San Francisco, 1963. R.F.S. Hearmon, An Introduction to Applied Anisotropic Elasticity. Oxford University Press, 1961. A.M. Szacinski, Wrinkling of sheet metal, Ph.D. Thesis, Monash University, Melbourne Australia, 1989. D.N. Lee and K.H. Oh, Calculation of plastic strain ratio from the texture of cubic metal sheet, J. Mater. Sci., 20 (1985) 3111-3118. R.W. Vieth and R.L. Whiteley, Influence of crystallographic orientation on plastic anisotropy in deep drawing sheet steel, Proc. IDDRG Congress, London, June 3 1964, pp. 1-10. M. Fukuda, Mathematical analysis on the relation between crystallographic texture and lankford r value in steel sheets, Trans. ISIJ, 8 (1968) 68-77. P. Bate, Private Communication, University of Birmingham, England, 1986. P.T. Welch, Private Communication, Technische Universitat-Clausthal, West Germany, 1986.
213
Elsevier
Critical conditions for wrinkling during the forming of anisotropic sheet metal A.M. Szacinski 1 and P.F. Thomson
Department of Materials Engineering, Monash University, Clayton, Victoria 3168, Australia (Received October 10, 1991; accepted March 9, 1992)
Industrial Summary In many applications, when sheet material is to be used near the limit of its capacity, it is important to be able to determine the critical failure strain, including that for wrinkling and to be able to correlate a number of properties of sheet materials, including the coefficient of normal plastic anisotropy, with critical wrinkling strain. The effect of preferred orientation on wrinkling behaviour under different loading and geometries was studied in numerical experiments employing the finite-element method, and confirmed in the Yoshida and other buckling tests, with the expectation that a material can be designed to increase the strain at which wrinkling occurs under a given state of stress or to permit appropriate orientation of the blank to maximise the allowable depth of draw. Critical elastic wrinkling strain can be represented on a diagram of surface principal-strains as a series of curves nested about the origin, expanding with elastic modulus, coefficient of normal elastic anisotropy (as defined in the present work) and coefficient of normal plastic anisotropy. It was found that the coefficients of elastic and normal plastic anisotropy correlate in copper and 70/30 brass but not in an annealed mild steel. The correlation can exist in materials in which the structure lacks symmetry or has symmetry of lower order than six-fold. They correlate in the same way with the critical strain for elastic wrinkling, and in such materials the former may then be used to predict the effect of orientation on the critical plastic wrinkling strain. In practice, such a relationship may be useful because the coefficient of normal elastic anisotropy is easier to calculate, given the principal orientations represented in the sheet, than is the normal plastic anisotropy, although the latter may be easier to measure.
1. Introduction The effect of preferred orientation on wrinkling behaviour has been studied in the present work, with the expectation that a material could be designed to increase the strain at which wrinkling occurs under a given state of stress: this
Correspondence to: P.F. Thomson, Department of Materials Engineering Monash University, Clayton, Victoria 3168, Australia. 1Present address: Gas and Fuel Corporation of Victoria, Scientific Services Department, Materials Section, Highett, Victoria 3190, Australia.
0924-0136/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
214
would permit a deeper draw before the onset of wrinkling. Another motive for investigating elastic wrinkling of anisotropic sheet metal was to determine whether a measure of elastic anisotropy obtained from a knowledge of the elastic constants of an anisotropic sheet could be correlated with normal plastic anisotropy and whether the tendency to plastic wrinkling could be correlated with it. Although plastic anisotropy is easier to measure, elastic anisotropy is easier to calculate, given the preferred crystallographic orientations represented in the sheet. Fracture is only one mode of failure in industrial pressings. Unsatisfactory pressings may result also from the presence of defects such as wrinkling, which arises from instability in compression due to non-uniform metal flow or nonuniform distribution of elastic recovery strain on unloading [ 1 ]. Such defects can lead to dimensional inaccuracies which cause difficulty during fit-up at the assembly stage or to a product which is unacceptable aesthetically. During the deformation of large thin sheet components, wrinkling may also be initiated elastically in the unsupported regions. Since the work of Geckeler [2], Baldwin and Howald [3] and Senior [4], extensive studies of wrinkling behaviour of sheet steel have been performed, including studies employing the Yoshida test [5,6] in which a square sheet is stretched diagonally until wrinkling occurs. The purpose of this latter test is to study the effect of material properties on resistance to wrinkling in large shallow, more or less unconfined, panels during forming. Although most sheet metals are anisotropic, relatively few studies of wrinkling behaviour have included the effect of anisotropy. Naziri and Pearce [7] concluded that the blank-holder pressure necessary to suppress wrinkling in a drawn cup increases with planar plastic anisotropy (Ar) and decreases with normal plastic anisotropy (r). Kleemola and Kumpulainen [8] found that when the flange of a drawn cup buckled into less than five waves, the number of waves depended on the anisotropy of the material. From their study of the critical conditions for elastic and plastic wrinkling of the flange of a circular blank during deep drawing, Yu and Johnson [9] concluded that a thin annular plate of large diameter may wrinkle elastically before the blank is drawn plastically, whereas a thick annular plate of small diameter wrinkles plastically because yield occurs before wrinkling. Stickels and Mould [10] found that a high average Young's modulus was associated with a high average coefficient of mean normal plastic anisotropy and correlated equally well with the limiting drawing ratio in the Swift cup test. Controlling elastic and plastic anisotropy can have a significant effect on forming performance [ 11,12 ]. However, Stickels and Mould [ 10 ] and Wilson [11,12] did not attempt to investigate whether the measures of elastic and plastic anisotropy could be related to initiation of wrinkling in ferrous or nonferrous sheet metals.
215
2. Computation and experiment The critical conditions for elastic wrinkling of anisotropic sheet metals in a number of different tests (Fig. 1 ) designed to produce different loading and geometries were predicted numerically using the finite-element method. These predictions were verified experimentally using the disc wrinkling test and the Yoshida test described in Fig. 1. The numerical and experimental results assisted in defining curves on the diagram of surface principal strains, which would describe the critical conditions for wrinkling in anisotropic elastic sheet metals.
2.1. Material properties The mechanical properties of the materials used in the experiments (Table 1 ) were obtained from tensile tests performed with an Instron testing machine according to the procedure recommended in A S T M Standard E8. The coefficient of normal plastic anisotropy of 3021 steel could not be determined, as the elongation at fracture was too low. 2.2. Modelling elastic anisotropy The effect of anisotropy on elastic wrinkling was modelled by assuming particular crystallographic orientations common in rolled sheets. The elastic compliances, sii, defined by sij = - aij
(1)
where ~j is the tensor strain component and aij is the tensor stress component, associated with those orientations, were calculated according to Lekhnitskii [13] and Hearmon [14].
r~ '
l
'
r2
(~)
(b)
(c)
Fig. 1. Schematic descriptions of tests: (a) disc-wrinklingtest; (b) Yoshida test; and (c) square plate wrinklingtest. ((i) TI=O (compression); (ii) TI=- T2 (shear); (iii) T~=-2T1; and (iv) T2= - 2T1)
216 TABLE 1
Properties of the sheet materials used Material description
Dominant Thickness Angle to RD orientation (ram) (°)
E (GPa)
Y (MPa)
n
cold-rolledsteel
{211}[0il]
0.60
0
231
682
0.10
{111}[112]
0.60
0
220
341
0.19 1.16
{100}[100] 200 annealed mild steel {111}[112]
0.60
0
223
255
0.21 1.32
0.68
0 45 90
70/30brass
{113}[211]
0.68
copper
{112}[111]
0.68
221 218 220 220 110 100 105 108 135 118 128 120
247 260 255 254 117 116 114 116 206 197 203 202
0.22 0.19 0.21 0.21 0.28 0.25 0.27 0.27 0.24 0.21 0.17 0.20
r
Ar
-
3021
Al-killedannealed steel A13 2%SisteelLycor
Mean value 0 45 90
Mean value 0 45 90
Mean value
1.27 1.43 0.15 1.89 1.50 0.95 1.05 - 0 . 2 2 0.71 0.94 0.72 1.28 - 0 . 6 6 0.52 0.95
Elastic wrinkling in tests (see Fig. 1 ) of sheets with the ideal orientations shown in Table 1 was predicted by the F E M using the ABAQUS programme. The eigenvalue prediction of elastic wrinkling adopted in this programme provides an estimate of the wrinkling load of a stiff plate or shell - i.e., one in which the pre-buckling response is essentially linear - when the collapse is not catastrophic (so that the plate is not excessively sensitive to imperfections). In ABAQUS, elastic wrinkling of an anisotropic material can be analysed by modifying the stiffness in the resulting problem Two measures of normal elastic anisotropy were adopted. The first was defined in a manner analogous to that of the coefficient of normal plastic anisotropy, as ~w re --
812 --
~t
(2)
Sl3
where ew and et are the logarithmic strains respectively in the width and thickness direction of a specimen deformed in uni-axial tension and s12 and Sl~ are the values of elastic compliance calculated from the dominant orientation using the method of Lekhnitskii [13 ]. This definition of elastic anisotropy is valid only when the stress is uni-axial, acting in the longitudinal direction.
217
Equation (2) can be interpreted as showing the ratio of the Poisson contractions in two directions (transverse and normal) due to the axial stress. The ratio of the elastic moduli En and Ew respectively in the directions normal to the rolling plane and transverse to the rolling direction was used to provide an alternative definition of normal elastic anisotropy (rE) ~w En rE _
_
Et
Ew
(3)
the ratio expressed in eqn. (3) being chosen as a measure of elastic anisotropy because the elastic moduli depend on the preferred crystallographic orientation within the sheet. It was found [15] that the value of the macroscopic elastic modulus of a homogeneous material was more significant than the value of the coefficient of normal plastic anisotropy in predicting the initiation of elastic wrinkling.
2.3. Modellingplastic anisotropy The value of the coefficient of normal plastic anisotropy was calculated from the crystallographic orientations representing ideal sheet textures, using the method of Lee and Oh [16]. Their formulation improves on the methods of Vieth and Whiteley [17] and Fukuda [18] by including all the operating slip systems in the calculation. Vieth and Whiteley used only the slip system with the largest Schmid factor, whereas Fukuda calculated an average r value from the five slip systems with Schmid factors of the greatest magnitude. Fukuda pointed out that the slip system with the largest Schmid factor never operates alone, but co-operates with other systems which have a Schmid factor of next to the largest magnitude. According to Lee and Oh [ 16 ], r-
ew F~[l(bp)(bd)]S] er •[[(tp)(td)lS]
(4)
where b, t, d and p are unit vectors along the width, thickness, slip directions and the direction normal to the slip plane, respectively, and Y~indicates summation over all operating slip systems. S is the corresponding Schmid factor calculated from
S= ](lp) (/d)]
(5)
where I is the unit vector along the tensile direction. It was assumed that the sheet had a texture (K1K2K3) [AIA2A~]. Unit vectors k and a respectively along the normal to the plane (K1K2K3) and in the direction (AIA2A3) were calculated as:
k=[k~,k2,k3]=[K~ K2 K3] i-~l,}K],[Kl
(6)
218 and
[ A 1 A 2 A3 1 a=[al,a2,a3]= i-AI']A]']A]
(7)
where
]gt=x/K~+g~+K~
]A]=x/A~+A~+A ~
and
The unit vectors l, t and b in eqns. (4) and (5) were obtained from the relationships
l= [ll,12,1,~] =
[al cos a +
(a2k3 -aak2)
sin a,
a2 cos a +
(a3kl -alk3)
sin a,
a~ cos a +
(aik2 -a2ki)
sin a ]
t= [ti,6,t3 ] = [ki,k2k.~ ] b= [bi,b2,b3] = [-al
(8) (9)
sin o~+
(a2k3-a3k2)
cos ~,
- a 2 sin ~ +
(a3kl -alk3)
cos ~,
- a 3 sin ~ +
(alk2 -a2k~)
cos ~]
(10)
where c~ is the angle between the rolling direction a and the tensile direction I. The unit vectors p and d (eqns. (4) and (5)) of a slip system (P1P2P3) [DID2D3] were calculated in the same way as the unit vectors k and a. The values of the coefficients of normal plastic anisotropy (eqn. (4)) calculated by the above method were compared with values obtained experimentally.
3. Experiment
3.1. Disc wrinkling test In this test (Fig. l ( a ) ) , wrinkling of a circular specimen was induced by indentation with opposed annular platens of smaller diameter. The dimensions of the circular plate and indenter used in the experiments were chosen as a result of preliminary tests to determine the diameters of plate and indenter for which wrinkling was easiest to produce and detect. A rig was designed to keep the indenter and plate concentric. The principal strains on the surface of the outer segment of the circular plate were measured to _+10 -~ using crossed foil electrical resistance strain gauges of 2 mm gauge length, placed with their centres at 5 mm from the indenter, with a d u m m y gauge for temperature compensation. Because the strains of interest were small - of elastic order - and rotations were small it was considered adequate to measure normal strains. The deflection of the plate during the test was measured with a digital trans-
219 ducer and three dial gauges positioned at 90 ° as close as possible to the outer edge of the plate: this permitted recording of displacement at the initiation of wrinkling. A film of PTFE, greased on one side, was used as a lubricant to minimise the friction between the indenter and the plate. 3.2. Yoshida test
Specimens 100 m m square were pulled between grips, 40 mm wide, in the Yoshida buckling test. The major and minor strains at the onset of wrinkling were measured with a crossed foil electrical resistance strain gauge positioned at the centre of the specimen. The deflection of the plate at wrinkling was recorded by a dial gauge attached to the mid-point of the surface on the face of the specimen opposite to that on which the strain gauge was placed. Wrinkling initiated in the above tests while the specimen was still elastic. The critical condition was confirmed by departure of the force-displacement curves from linearity. 4. Results and discussion
The loads required to initiate elastic wrinkling of square plates (Fig. 1 (c)) of pure iron with primitive orientation {100} [010], 2% Si steel Lycor 200 with dominant orientation {011} [100], Al-killed steel A13 with dominant orientation {111}[113] and cold-rolled steel 3021 with dominant orientation {211} [0il] stressed in compression parallel to the rolling direction were computed by the FEM, as implemented in the ABAQUS package (Table 2). The corresponding computed values of elastic modulus, coefficient of normal elastic anisotropy, re, as defined in the present work (Section 2.2 ), and coefficient of normal plastic anisotropy, are shown also. The coefficients of normal elastic and normal plastic anisotropy of Lycor 200, A13 steel and 3021 steel in directions 45 ° and 90 ° to that of rolling were not evaluated experimentally. The effect of the coefficients of normal elastic and plastic anisotropy on the elastic wrinkling load in Lycor 200, A13 steel and 3021 steel was only studied by computation. Elastic modulus was chosen as a measure of elastic anisotropy because it is orientation-dependent and it was expected that a close relationship between Young's modulus and the coefficient of normal plastic anisotropy would exist [10], although the manner of the relationship with orientation might be quite different. As expected, the wrinkling load increased with Young's modulus and a similar relationship was found in the results of other tests used [ 15 ]. The coefficients of normal elastic and normal plastic anisotropy related to wrinkling load in the same way as did the elastic modulus (Table 2). Wrinkling occurred earlier in material with orientations for which the coefficients of normal anisotropy, either elastic or plastic, were smaller. These findings confirmed the
22O TABLE 2 Computed critical wrinkling loads of steels and the corresponding coefficients of elastic and plastic anisotropy Material
Dominant orientation
Computed elastic modulus (GPa)
Computed coefficient of elastic anisotropy re
Computed coefficient of plastic anisotropy r
Computed critical load at onset of wrinkling (N/m)
pure iron Lycor 200 A13 steel 3021 steel
{100}[010] {011}[100] {111}[112] {211}[011 ]
176 210 221 236
1.00 1.00 1.10 2.02
0.50 0.69 1.16 1.59
620 998 1460 2320
conclusions of Naziri and Pearce [7] and of Kleemola and Kumpulainen [8], who studied only the effect of plastic anisotropy on wrinkling behaviour and established that the earliest wrinkles were observed in the steel sheets with the smallest coefficients of normal anisotropy. It appears that the critical elastic wrinkling load decreases with the elastic anisotropy, but the most deleterious form of anisotropy has not been determined; the possibility remains that some form of controlled elastic anisotropy, still undetermined, might be used to effect improved resistance to wrinkling. The computed wrinkling-limit curves for the different steels on the diagram of principal surface strains (Fig. 2 ), expand with the elastic modulus and the coefficients of normal elastic and plastic anisotropy. It seems reasonable to suppose that wrinkling would be resisted by material with a smaller coefficient s13 in the direction transverse to the axis of the wrinkles and also by a material with a larger coefficient of normal elastic anisotropy, s1Js13. The increase in critical wrinkling strain with elastic anisotropy as defined in this work may be advantageous in minimising the type of wrinkling that results from residual stress. As was also the case in a homogeneous material [ 15 ], the largest critical strain occurred in the test in which the ratio of surface strains corresponded to shear, so that elastic springback on unloading after forming may be minimised by promoting shear deformation. The relationship between elastic modulus and the calculated critical values of the strain ratio el/e2 at the onset of wrinkling of a square plate under edge compression for steels with different ideal orientations are shown in Fig. 3, with the corresponding values of elastic and plastic normal anisotropy parameters. It was tempting to correlate the ratio of eJe2 at the initiation of elastic wrinkling with the elastic modulus and the coefficients of normal elastic and plastic anisotropy in these anisotropic materials but it must be remembered that the test defined a particular strain path. The critical surface strains in the disc wrinkling test and in the Yoshida test on 3021 steel, A13 steel and Lycor
221
.7
1. Pure Iron 2. Lycor 200 3. A13 Steel 4. 3021 Steel
r,=l.00 r,--1.00 re=l.10 r,=2.02
r=0.50 r=0.69 r=l.16 r=1.59
E=176 E--210 E--221 E=236
GPa GPa GPa GPa
-6
-5 o x "4
-3
z
c~ ©
2
I
t
-6
-5
r
-4
-3
-2
-1
0
MINOR STRAIN x 104 Fig. 2. C o m p u t e d envelopes of surface strains at the onset of elastic wrinkling of square plate under edge compression, showing the effect of elastic and plastic anisotropy.
200 are shown in Table 3. These tests dictate different strain paths and the critical values of el/e2 now appear to vary independently of Young's modulus, re and f; and it is difficult to make a general conclusion concerning the relationship between anisotropy and the strain ratio at onset of wrinkling. The discrepancies between the computed and the observed critical strains shown in Table 3 are due to computation being based on the ideal orientation in each case, whereas the actual material would have a spread of orientations around the dominant one, which may have obscured also any direct relationship between the value of the strain ratio and the elastic modulus at the onset of wrinkling. It seems unlikely that the difference in absolute values of principal strains was caused by error in measurement, as strains were measured to + 10 -6, two orders of magnitude less than the discrepancy. The strain ratios el/e2 for an annealed mild steel, 70/30 brass and copper (described in Table 1 ) at the onset of elastic wrinkling in the Yoshida test, obtained by computation and experiment, are shown in Table 4, which shows also the corresponding values of Young's modulus, the coefficients of normal elastic anisotropy parameters re and rE and the coefficient of normal plastic
222
P R I N C I P A L ORIENTATION: 1.2.
z z
g
1-
{100}[0101
ro=l.00
r=0.50
2-
{011}[100]
r,=l.00
r=0.69
3-{m}[11~]
ro--MO
r=1.16
4 - {21~}[oi11 r~=2.oo ~=1.59
-1.0
/
-0.8
o cg
-0.6
.4
a~ oq
-0.4
-0.2 150
I 175
I 200
I 225
I 250
.
YOUNG'S MODULUS [GPa] Fig. 3. C o m p u t e d su rface s t r a i n r a t i o s at the o n s e t of elastic wrinkling in a s q u a r e p l a t e u n d e r edge c o m p r e s s i o n , as a f u n c t i o n of Y o u n g ' s m o d u l u s .
TABLE 3 Critical wrinklingstrains in the disc-wrinklingtest and the Yoshidatest: computedand observed Material
Elastic modulus (GPa)
L y c o r 200 {011}[100]
210
A13 steel {111}[115]
221
3021 steel {211}[0il]
236
el > 10 -4 e 2 X l 0 -4 el/e2 el X 10 -4 e 2 X 1 0 -4 eJe2 e~ X 10 -4 e 2 × 1 0 -4 e,/e2
Disc-wrinkling test
Yoshida test
Computed
Observed
Computed
Observed
9.15 4.22 2.17 6.25 - 3.32 - 1.88 56.80 -28.80 - 1.97
0.62 -0.14 -4.43 0.59 -0.22 -2.68 0.81 -0.46 -1.76
0.79 -0.48 -1.65 0.73 -0.32 -2.28 -
10.96 4.76 2.28 15.30 - 5.91 - 2.58 46.99 -24.99 - 1.88 -
-
anisotropy. In 70/30 brass and copper with the orientations approximating (113) [21i] and (112 ) [111 ], respectively, the critical strain ratio el/e2 at the onset of elastic wrinkling in the Yoshida test increased with the coefficients of
223 TABLE 4 Observed and stimulated results of the Yoshida test in annealed mild steel, copper a n d 70/30 brass Angle to RD (°)
Annealed mild steel (111)[li2]
70/30 brass {113)[21il
Copper (112}[i11]
Computed Observed Computed Observed Computed Observed E (GPa) re
rE
r
el/e2
0 45 90 0 45 90 0 45 90 0 45 90 0 45 90
220 220 220 2.29 1.79 2.29 1.29 1.29 1.29 1.22 1.36 1.51 --2.38 --2.38 --2.38
221 218 220
1.27 1.43 1.89 --1.95 --2.10 --2.45
111 75 89 0.60 0.95 0.40 0.88 1.04 0.70 0.83 1.01 0.62 --1.12 --1.65 --1.38
110 100 105
0.95 1.05 0.71 --2.11 --2.36 --1.86
191 94 131 1.0 2.15 0.37 1.0 1.39 0.68 0.60 1.62 0.50 --1.07 --2.80 --1.71
135 118 128
0.72 1.28 0.52 --2.31 --2.55 --1.80
:6 70/30 BRASS {113}[211] 1. 0° re=0.60 r=0.83 2. 45° r,=0.95 r=l.01 3. 900 re=0.40 r=0.62
rE=0.88 rE=l.04 rE=0.70
-5
-4
1
2
~
.1
-6
-4
-3
-z
-1
0
MINOR STRAIN x 104 Fig. 4. Computed envelopes of surface strains (wrinkling-limit curves) for 70/30 brass, tested at 0 °, 45 ° and 90 ° to the rolling direction.
224
COPPER {ll2}[il0] I. 0°
r~--l.00
r=0.60
2. 45° 3. 90o
re--2.15 rc=0.37
r--1.62 r=0.50
rE: 1.00 rE=l.39
] 87
rE=0.68 -6
"5
1
×
z
2
<
"4 u[~) -3
O ~-~ <
-2 -1 I
-7
-6
-5
-4
-3
-2
-1
0
MINOR STRAIN x 104 Fig. 5. Computed envelopes of surface strains 45 ° and 90 ° to the rolling direction.
(wrinkling-limit curves) for copper, tested at
0 °,
normal elastic and plastic anisotropy. It was found (Figs. 4 and 5) that the predicted components of strain at the initiation of elastic wrinkling varied directly with the normal elastic and plastic anisotropy parameters. The earliest wrinkling was predicted in each case for the orientation at which the coefficient of normal plastic anisotropy was smallest. The coefficients of normal elastic anisotropy, re and rE, for copper and 70/30 brass were related directly to the coefficients of normal plastic anisotropy (Table 4). It appears that it would be possible to predict the strains at which elastic wrinkling in copper and 70/30 brass occurs given the coefficients of normal elastic anisotropy as defined in the present work. However, the coefficients of normal elastic anisotropy in the annealed mild steel were not related to the coefficient of normal plastic anisotropy in that material (Table 4). This may be a result of the different symmetry in the texture of these materials [19], the dominant (111 ) [112] texture of the bcc mild steel sheet possessing six-fold symmetry, whereas the dominant textures of the fcc copper and 70/30 brass have two-fold symmetry. The six-fold symmetry of the annealed mild steel is manifested in its plastic properties (for example, in the different values
225 of the coefficient of normal plastic anisotropy and in the tendency to earing of deep-drawn cups), but not in its elastic properties, which can exhibit a maxim u m of four-fold symmetry, because elasticity is a linear fourth-order tensor [20]. In the annealed mild steel, the Young's modulus and the coefficients of normal elastic anisotropy, re and rE, were independent of the angle to the rolling direction as were the computed values of strain ratios at the onset of elastic wrinkling. It should be noted that the textures of the 3021 steel and Lycor 200 ({211 } [011 ] and {100} [ 100 ], respectively) had two-fold symmetry with respect to the loading direction used in the experiments, whereas the texture of the Al-killed A13 steel ( {111 } [ 112 ] ) had six-fold symmetry. For this reason, correlation between the coefficients of normal elastic and plastic anisotropy and the critical strain at initiation of wrinkling in the given orientation could be expected in 3021 steel and Lycor 200, but not in A13 steel. If the crystallographic structure of sheets is such that the plastic anisotropy lacks symmetry or has symmetry lower than six-fold, correlation with elastic anisotropy becomes more likely and it is possible to use the former to compare the critical wrinkling strains of sheet of different orientations. 5. Conclusions
(i) The conditions under which anisotropic sheet metals (steel, copper and 70/30 brass) wrinkled elastically, shown as elastic wrinkling-limit curves on the diagram of principal surface strains, were predicted by numerical experiments using the FEM to analyse the behaviour of sheet materials under different geometry and loading conditions. The curves expanded with the elastic modulus, the coefficient of normal elastic anisotropy (as defined in the present work) and the coefficient of normal plastic anisotropy. (ii) The coefficients of elastic and normal plastic anisotropy used herein can be correlated in materials which lack symmetry or have symmetry of order lower than six-fold. They correlated in the same way with the critical strain for elastic wrinkling and in such materials the former may then be used to predict the effect of orientation on the critical plastic wrinkling strain. High values of the coefficients of normal elastic anisotropy, as defined, coincided with high values of the coefficients of normal plastic anisotropy.
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