DEFORMED STATE UNDER PLANE AND AXISYMMETRIC UPSETTING

E. S. VAKSMAN, V. A. OGORODNIKOV, G. D. DEL

(Presented by the scientific seminar of the Department of Strength of Materials)

Upsetting is one of the common operations in metal forming. For the rational execution of this operation it is necessary to know the deformed state of the billets being upset.

The aim of the present work is to investigate the deformed state during upsetting of billets made of materials with different strain-hardening capacity. The deformations of specimens made of steels Kh18N9T, St. 3, and copper were determined from the hardness distribution [1, 2, 3]; the deformed state of lead was studied from the distortion of a grid, applied in advance to one of the halves of the composite specimen.

In determining deformation by measuring hardness, a calibration curve “hardness—strain intensity” is constructed. For this purpose eight cylindrical specimens of each material, 30 mm in diameter and 45 mm high, were compressed in the axial direction to different degrees of deformation. Compression was carried out with lubrication of the end faces by a paste of colloidal graphite and glycerin. The strain intensity corresponding to each upsetting was determined from the relation

\[ e_i = \ln \frac{H_0}{H}, \tag{1} \]

where \(H_0\) and \(H\) are the height of the specimen before and after upsetting. The deformed specimens were cut along a meridional section. The resulting surface was carefully ground, and at 4–5 points its Brinell hardness was measured with a ball 5 mm in diameter under a load of 500 kg (when testing copper the load was 250 kg). The indicated conditions for determining hardness were chosen in order to obtain the largest number of impressions during the subsequent measurement of the hardness of specimens whose deformed state was being studied. Deformations were determined under plane and axisymmetric deformation.

Plane deformation. From the hardness distribution, the deformed state of cylindrical and prismatic specimens made of steel Kh18N9T was determined. Three cylindrical specimens 45 mm in diameter and 135 mm long were upset in the radial direction between parallel press plates to three reductions (10, 20, and 30%). Rectangular prisms with dimensions \(45 \times 30 \times 90\) mm were upset in the direction of the largest transverse dimension to the same reductions. Subsequently, the deformed specimens were cut along the section midway along their length. The resulting surface was ground, and at various points of its axes of symmetry (along

and transverse to the direction of compression) hardness was measured. From the hardness, using a calibration graph, the corresponding value of the strain intensity \(e_i\) was determined. For the selected ratio of specimen dimensions, the deformed state in the section at the mid-length of the specimen is plane on average. Under plane strain [2]

\[ e_1=-e_3=\frac{\sqrt{3}}{2}e_i . \tag{2} \]

The strain \(e_3\) is directed along, and \(e_1\) transverse to, the direction of upsetting.

The diagrams of the principal strains along the axes of symmetry are shown in Figs. 1 and 2. The results obtained were checked by comparing the mean strains along the coordinate axes:

\[ e_{x\,\mathrm{av}}=\ln\frac{b}{b_0},\qquad e_{y\,\mathrm{av}}=\ln\frac{H}{H_0}, \tag{3} \]

where \(b, H\) are the width and height of the cross section after upsetting, with the strains found from the obtained diagrams:

\[ e'_{x\,\mathrm{av}}=\int_{-b/2}^{+b/2} e_x\,dx,\qquad e'_{y\,\mathrm{av}}=\frac{1}{H}\int_{-H/2}^{+H/2} e_y\,dy . \tag{4} \]

The error in determining the strains did not exceed 9%, except for upsetting by 10%, for which it was 20%.

Lead specimens were made with the same dimensions as the steel ones. The specimens were cut through the middle along the length of the cross section, and on one of the cut planes a coordinate grid with a base of 1 mm was rolled. In the cells of the rolled grid, impressions in the form of circles 1.75 mm in diameter were made with a special punch. The dimensions of the impressions obtained were measured on a toolmaker’s microscope. Then both halves of the specimen were soldered with Wood’s alloy, and the resulting composite specimen was compressed to the required degree of upsetting (as in the preceding case, the upsetting was 10, 20, and 30%). Subsequently the specimen was unsoldered, and the deformed state was determined from the distortion of the original circles. The principal strains were calculated from the relations:

Fig. 1. Strains during upsetting of prisms.

\[ e_1=-\ln\frac{d_0}{a},\qquad e_3=\ln\frac{b}{d_0} \tag{5} \]

where \(d_0\) is the initial diameter of the impression;
\(a, b\) are the minor and major axes of the ellipse.

The diagrams of the principal strains along the axes of symmetry during upsetting of lead prisms and during radial compression of a cylinder are shown, respectively, in Figs. 1 and 2. As is seen from the graphs, the deformed state during upsetting is nonuniform; with increasing reduction, the degree of nonuniformity increases.

From a comparison of the deformation diagrams of specimens made of different materials (Kh18N9T steel, lead), it follows that the deformed state is practically independent of the hardenability of the material.

Axisymmetric upsetting. The deformed state was determined from the hardness distribution. Nine cylindrical specimens 36 mm in diameter and 54 mm high, made of Kh18N9T steel, St. 3 steel, and copper, were upset in the axial direction between parallel press plates to the same reduction (10, 20, and 30%). After measuring the hardness at the center of the end face of the deformed specimens, the latter were cut along the meridional section. The resulting surface was ground, and hardness was measured at various points along its axes of symmetry. From the hardness, using calibration curves, the corresponding value of the strain intensity \(e_i\) was determined.

Fig. 2. Deformations under radial compression of a cylinder

Under axisymmetric deformation, along the axes of symmetry of the meridional section, three strains are determined: circumferential, \(e_\varphi\); radial, \(e_r\); and axial, \(e_z\). The strain intensity is

\[ e_i=\frac{\sqrt{2}}{3}\sqrt{(e_r-e_\varphi)^2+(e_r-e_z)^2+(e_\varphi-e_z)^2+\frac{3}{2}\gamma_{rz}^{2}} . \tag{6} \]

The incompressibility condition is

\[ e_r+e_\varphi+e_z=0 . \tag{7} \]

Along the axis \((z)\) of symmetry, which coincides with the direction of upsetting, \(\gamma_{rz}=0\), \(e_r=e_\varphi\), and therefore

\[ -e_z=2e_r=2e_\varphi=e_i . \tag{8} \]

Along the second axis of symmetry the equation holds

\[ \frac{d e_\varphi}{dr}+\frac{1}{r}(e_r-e_\varphi). \tag{9} \]

The method for the numerical solution of the system of equations (6), (7), (9) is presented in [2]. The calculation performed was checked by comparing the circumferential strain obtained in the calculation on the contour with the actual

\[ e_\varphi=\ln\frac{R}{R_0}, \tag{10} \]

where \(R_0\) and \(R\) are the radii before and after deformation, averaged over the height of the specimen section. The greatest discrepancy did not exceed 8%. The results of checking the plots of strain intensity \(e_i\), constructed along the vertical axis \(z\), also indicate a high accuracy in determining strains from the hardness distribution. The discrepancy between the mean value of the strain \(e_i\) found from these plots and \(\ln \dfrac{H_0}{H}\) did not exceed 3%.

Fig. 3. Strains during axisymmetric upsetting of a cylinder

Figure 3 presents plots of the strains \(e_r, e_\varphi\) during axisymmetric upsetting. The results obtained indicate a significant nonuniformity of the strain distribution along the vertical axis \(z\). Along the horizontal axis of symmetry the strains are distributed more uniformly. As in the case of plane upsetting, the deformed state is practically independent of the strain-hardening capacity of the material.

The investigation carried out shows that the deformed state during upsetting in the hot state can be determined on models deformed cold. The strains of the models can be determined from the hardness distribution.

REFERENCES

1. G. D. Del. Investigation of plastic deformation by hardness measurement. Izvestiya TPI, vol. 138, 1961.

2. G. D. Del. Determination of the deformed state from the hardness distribution. Izvestiya TPI, vol. 157, 1967.

3. Ya. B. Fridman, T. K. Zilova, N. N. Demina. Study of plastic deformation and fracture by the method of rolled grids, Oborongiz, 1962.