Crystallographical Calculation of Earing in Deep Drawing under Various Conditions

Earing in the cup drawing of sheet metals is calculated quantitatively using a crystallographical theory with measured texture data. In the analysis a polycrystalline sheet is simplified to be an aggregate of many single crystals with various orientations, and a crystallite orientation distribution function which is calculated from the measured texture is used as a volume fraction of a certain oriented crystal. The circumferential distribution ot radial strain in a flange of a blank being drawn is calculated by considering a restricted glide in a crystal, and then cup height at each peripheral position of a drawn cup is calculated from this distribution ot radial strain. For an aluminium, a copper and two steel sheets the calculated cup profiles are compared with experimental ones under various drawing conditions of dimension of a punch, a die and a blank. The result shows that earing in the drawn cup can be predicted satisfactorily by the present calculation for a wide range of drawing conditions and materials.


INTRODUCTION
There are many problems lying on an interdisciplinary research field and making them clear becomes very important nowadays. A problem lying on the boundary between the field of materials science and the field of sheet metal forming technology is treated in the present work. In the former field many researches are made energetically to clear quantitatively deformation behaviour of a polycrystalline material based on detailed deformation mode of a single crystal. However the macroscopic phenomena which can be made clear by them are only simple behaviours of isotropic materials under a uniaxial stress state at most. These results can't be used directly to solve technological problems. In the latter field, on the other hand, most of the problems are tried to be solved by use of a continuous theory, but an anisotropic behaviour of a sheet metal is very difficult to make clear by the theory. Although the behaviour is well known in the qualitative relation with behaviour of crystals, clearing the quantitative relation is a future subject. The authors have attempted to calculate quantitatively macroscopic phenomena in a deformation of sheet metals by use of a crystallographical theory (Kanetake, Tozawa and Kato, 1981, Kanetake, Tozawa and Otani, 1983and Kanetake, Tozawa and Yamamoto, 1985. In these works a polycrystal is assumed as an aggregate of many single crystals with a simple deformation mode, and measured texture data represented by a three-dimensional crystallite orientation distribution function is used with the crystallographical theory. The analyzing method was shown to be very useful for predicting quantitatively anisotropic behaviour of the sheet metals in practice use. In the present paper the analyzing method is applied to predict earing in a cup drawn under various drawing conditions for various textured sheets. On the earing, in the materials science field, the relationship between earing pattern and texture type has been found empirically, and the earing of a single crystal sheet has been predicted by use of a crystallographical theory (Tucker, 1961 andVieth andWhiteley, 1964). In the sheet metal forming field, on the other hand, the relationship between earing pattern and planar anisotropy of an r-value has been found empirically, and calculating the earing by use of a slip-line field theory or a finite element method has been carried out by some researchers. Through those many previous researches, the earing pattern of a drawn cup is predictable qualitatively from the texture or mechanical properties of a sheet blank, but is not quantitatively. The distribution of cup height along a circumference of the drawn cup is calculated and compared with experimental ones in the present work. From the results, it is confirmed that the analyzing method is useful for a wide range of drawing conditions and materials used in the practice.

THEORETICAL PREDICTIONS
Assumptions used in the analysis (1) At any stage during a deep drawing operation there is a tensile stress (tr) radially, a compressive stress (-tr) circumferentially and zero stress normal to the sheet surface at any point in a flange of the blank.
(2) As a deformation mode of a single crystal, only a restricted glide on following slip systems is assumed: (3) On the all slip systems a relationship between the shear stress, :, and the shear strain, ,, is expressed by a following equation, : =/tr k)," + :0 (1) where/ is the Schmid factor and k, n and :0 are constants.
(4) When more than one slip system operate simultaneously, the macroscopic strain is taken as a sum of its components calculated for each system separately.
(5) A textured polycrystalline sheet metal is simplified as an aggregate of many crystals with various orientations whose volume fractions are expressed by means of a crystallite orientation distribution function.
(6) The change in orientation of each grain, i.e. change of the texture, which occurs during the practical drawing, is ignored in resolved shear stress, but is taken into account in resolving strain.

Calculation method
Earing is attributed to an inhomogeneity of radial strain along the circumferential direction in a flange of a circular blank. Therefore, the circumferential distribution of radial strain during cup drawing is calculated, and then, using this result, the circumferential distribution of cup height is calculated. direction cosines of S, S2, S3, RD, TD and ND(Sa), with respect to the crystal axes, are defined by (11, ml, n), (12, m2, n2), (13, m3, n3), (14, m4, n4), (Is, ms, ns) and (13, m3, n3) respectively, they can be expressed in terms of the four angles % 0, q and tr as follows: l 3 -sin 0 cos , m 3 sin 0 sin q, follows" Then, the radial strain at a given te in the flange of the single crystal blank, e,(p, 0, ), can be calculated as follows.
N where K (orTM, /z0 0/cr and E expresses a sum over all the slip systems considered. N For the textured polycrystalline sheet, at the first step its texture is measured and represented by an orientation distribution function, w(p, 0, ). Then e,(ap, 0, ) calculated by Eq. (5) for each single crystallite orientation is weighted by w(p, 0, ) and averaged over all orientations, i.e. the radial strain at a given cr in the flange of the polycrystalline circular blank, ep, is calculated as follows: e(p, 0, )w(p, 0, q) sin 0 dap dO d (6) Using the ep, the cup height at each point along a circumference, h, is calculated as follows. h (h,/:p).ep + rp + to h (R,-R2)/(Rp + Ra) (7) R 2= 2(rp + to/2) 2 + (rp + to/2)(Rp rp)r + (Rp rp) 2 where Ra, Rp, rp, Rb and to are dimensions of a die, a punch and a The shear strain, YN, generated by the shear stress, N, in the slip system N is calculated as follows using equations (1), (2)  blank as shown in Figure 2, and p is an average value of the ep along a circumference. As described above, by measuring the texture of the sheet to be drawn and giving values of n and #o, a feature of the earing in the drawn cup, i.e. the distribution of the cup height along a circumference can be calculated under various drawing conditions. In the present work the calculation was made numerically at every 5 of p, 0, b and c respectively under n 1.0 and #0 0.05 Otani, 1983 andYamamoto, 1985).

Materials
As a FCC material, an annealed aluminium sheet and an annealed copper sheet, and as a BCC material, a low carbon steel sheet and a

Texture measurement
The average texture over the thickness was measured using the method proposed by Lopata and Kula (1962). A plane of a specimen to be measured and one quadrant of a pole figure to be determined are illustrated in Figure 3. A composite specimen for the texture measurement was prepared using the procedure of Elias and Heckler (1967). Three pole figures for each specimen, (111), material, were measured, and using those data the orientation distribution function was calculated according to the method of Bunge (1965) or Roe (1965).
Cup drawing For each material circular blanks were deep drawn with various dies, punches and blanks whose dimensions are shown in Table 2.
Of all the dimensions, a die diameter, dd, a punch diameter, dp, a punch profile-radius, rp, a blank diameter, Do, and a blank thickness, to, are used in the theoretical calculation. The height of the drawn cup was measured continuously along a circumference with a dial gauge which has a voltage signal output, and the data were sampled using a microcomputer. After the sampling was finished, two height data at diametrically opposite sites were averaged to exclude error due to eccentricity of the blank to the tool, then the cup profile was drawn on an X-Y plotter.    (Figure 11) have both typical four earing, 45-earing in the aluminium sheet cup and 0-90-earing in the high-strength steel sheet cup. A detailed observation in the latter can find that the ear at 90 is wider than that at 0. The cup of the copper sheet (Figure 9) has also four earing, but a position of the ears is a little out of 45 and the trough at 0 is deeper and narrower than that at 90. The cup of the low carbon steel sheet (Figure 10) has six earing. Although the ears are small on the whole, it's distinct that the ear at 0 is smaller than the others. In comparing the calculated cup profiles with the experimental ones it can be found, that there is a close agreement between them, in other words, for a principal feature of the experimental cups not only a number of ears but also a detailed position and height of each ear and trough can be predicted by the calculation method shown in the present work. This is a large advantage of the calculation method using the measured texture data for analyzing earing.

RESULTS AND DISCUSSION
In cup drawing with punches which have various profile-radii the calculated and experimental cup profiles agree similarly well for respective radii, though a little difference between both profiles is seen in the cups drawn with a spherical punch (See (a), (b) and (c) in each figure). Reducing of sheet thickness due to stretch deformation at the punch profile and increasing of it near a rim of a cup wall are both not taken into consideration in the present calculation. The effects of both these assumptions, consequently, compensate each other, and good agreement between calculated and experimental cup profiles can be seen under conditions of widely varied punch profile-radius.
The results of cups drawn with different diameter punches are seen in comparing (b) and (e) or (d) and (f) of each figure.
Clearances between a punch and a die are 1.25to and 1.9to for dp 38 and 37 mm respectively. Although the wall of the cup drawn under a condition of the smaller clearance is ironed slightly at a trough, the effect of ironing scarcely appears on the profile of the cup. In a case of drawing under a condition of the larger clearance, the experimental cup height is a little lower than the calculated one. This is due to the fact that the rim of the wall of the cup drawn under the condition is slightly curled because of remaining of bending deformation at a die shoulder. In spite of this fact, the calculated profile can predict very well the experimental one under the condition of a clearance range used in the practice.
The effect of a die profile-radius ra can be also seen in the figures, and ra 4 and 10 mm are equivalent to 5t0 and 12.5t0 respectively.
Although the radii are dose to a lower and an upper bound of that used in the practice, the effect of the radii on the cup profile is very small. The fact supports the present calculation method in which the die profile-radius is not taken into consideration.
For a BCC metal, as a deformation mode of a single crystal besides of the restricted glide considered in the present work, a pencil glide along { 111 } slip direction is observed in general. This problem as well as on the values of n and/0 was discussed in the previous works Otani, 1983 andYamamoto, 1985), and the assumptions used in the present work were decided to be suitable. Although many calculations of prastic properties have been made, up to now, using the Taylor theory or certain modification of the theory, another deformation model presented by Sachs (1928) was modified in the present calculation, which is the model for constant stress and violates the condition of minimum deformation work. Besides this, very bold assumptions are used for analyzing deformation behaviour of a textured polycrystalline sheet, that is, a change of the texture during cup drawing, an effect of a grain boundary or a next grain, an inhomogeneity of a stress state in a flange or of strain along a cup wall and so on are not considered at all in the present calculation. In spite of the fact, very close quantitative agreement between theory and practice was obtained in a wide range of drawing conditions and materials. This implies that considering only glide in each crystal grain separated from other grains is very useful for calculating deformation or strain, that a directional dependency of the deformation is affected significantly by the texture, and that a similar crystallographical calculation using measured texture data is very useful for analyzing not only earing of the drawn cup but also other macroscopic deformation behaviour of an anisotropic material.

CONCLUSION
Aluminium, copper and steel sheets were cup drawn under various conditions of geometry of a punch, a die and a blank, and their cup profiles were examined. On the other hand the cup profiles of those sheets were calculated using the crystallographical theory with the measured texture data for the same drawing conditions. As a result of comparing the experimental and calculated cup profiles, it was found that the principal feature of the cup, such as a number, a position and height of the ears can be predicted satisfactorily for a wide range of drawing conditions and materials by the present calculation method. Thus the calculation is very useful for technological purposes, for example a quantitative prediction of earing for various sheets or a computer simulation of a relationship between the texture and the earing. Furthermore, the similar calculation method using texture data is very useful for analyzing other problems lying between the materials science field and the sheet metal forming field.