Investigation of the Stress State During Upsetting of Prisms by Hardness Measurement

E. S. Vaksman, G. D. Del, L. M. Sedokov

(Submitted for publication by the scientific seminar of the Department of Strength of Materials)

Upsetting of prisms between parallel plates is a common operation in metal forming by pressure. As a result of the action of friction forces on the contact surfaces during upsetting, a nonuniform field of stresses and strains is created. Knowledge of the nature of the distribution and the magnitude of the stresses is of great interest, since it makes it possible to study a whole series of questions associated with this technological operation (clarification and elimination of the causes of crack formation in blanks, evaluation of residual stresses, calculation of the tool for strength, etc.).

L. Prandtl [1], using the methods of the theory of ideal-plastic bodies, obtained an analytical solution of the problem under consideration, which was further developed by V. V. Sokolovskii [2], L. D. Tomlenov, and others. As a result of these studies, formulas were obtained for calculating the forces in upsetting, and diagrams of contact stresses were constructed. E. P. Unksov [3] experimentally determined the normal contact stress, and also established, by the photoelastic method, the stress distribution over the transverse section of a strip. In the latter case it was assumed that, qualitatively, the pattern of stress distribution in elastic and plastic upsetting is the same.

The present work gives the results of an investigation of the stress state during upsetting of a prism made of a strain-hardening material, carried out by hardness measurement [5]. In this, the stress-strain state over the transverse section of the prism was determined, diagrams of contact stresses were constructed, and the influence of material hardening on the stress state was investigated. It is known that, for large ratios of the length of the prism to its width, in its middle part during upsetting the deformation is plane. To specify the region of plane strain, four specimens with cross-sectional dimensions \(15 \times 15\) mm and lengths of 15, 30, 75, and 150 mm were tested.

For this purpose, on the contact surface of the prisms along the axis of symmetry, points were prick-punched at intervals of 4 mm, after which the specimens were upset by 10%. Measurement of the distances between the prick-punched points after upsetting made it possible to determine the axial strain \(e_z\). It was established that at a distance equal to one height from the ends, the axial strain is practically absent.

Taking this into account, for the main series prisms of square cross section with side 24 mm and length 77 mm were adopted, made of

stainless steel Kh18N9T ($\sigma_t = 26\ \mathrm{kg/mm^2}$, $\sigma_{\mathrm{uts}} = 62\ \mathrm{kg/mm^2}$). The choice of material is explained by the high work-hardening capacity of this steel. Two specimens were investigated, upset by 10% and 30%. The upset specimens were cut through the middle along the length of the section. In the resulting section, after polishing, hardness was measured with a diamond pyramid under a load of 20 kg.

Hardness was measured at 80–100 points. The results of the hardness measurements were averaged and isoscalars were drawn, shown in the right half of Fig. 1. Owing to the symmetry of the prism, the calculation was performed for one quarter of the section.

Fig. 1. Epures of contact stresses, network of slip lines, and isoscalars during 10% upsetting: ○ — for steel Kh18N9T, □ — for an ideally plastic material.

A calibration graph relating hardness to the intensities of shear stresses and strains was obtained beforehand. It was constructed from the results of testing the steel under investigation in axial compression. For this purpose, nine cylindrical specimens with a height-to-diameter ratio equal to 1.5 were upset, under conditions of careful lubrication, to different degrees of deformation. Hardness was measured in the meridional section after polishing.

Using the calibration graph, from the hardness distribution in the upset prisms the function $k(xy)$ was determined for each of the specimens.

The results of the hardness measurements were interpreted by the method described in [5]. In the interpretation, in the region $ABL$ (Fig. 1) the Cauchy problem was solved; in the region $ACD$ an initial characteristic problem was solved; and in the remaining regions mixed problems were solved. The boundary conditions adopted in the interpretation were the following: the lateral surface is free from load; along the axes of symmetry the angle of inclination of the slip lines is 45°; the friction at point $A$ (Fig. 1) on the contact surfa-

...was neglected. As a result of the solution, the network of slip lines shown in the left half of Fig. 1 and the stress hodographs were obtained. For the larger upsetting, it was not possible to construct the stress hodographs for the vertical axis of symmetry because of insufficient boundary conditions. For the same reason, the stresses $\sigma_x$ and $\sigma_y$ for the horizontal axis of symmetry were not determined in the region close to the center of the section.

Fig. 2. Hodographs of the principal stresses along the horizontal axis of symmetry: ○ — upsetting 10%, □ — upsetting 30%, △ — for an ideally plastic material.

This part of the hodograph was constructed by analogy with the smaller upsetting and is shown on the graph by a dashed line.

The hodographs of $\sigma_x$ and $\sigma_y$ along the horizontal axis of symmetry are shown in Fig. 2. They are not monotonic and have two local extrema each. With smaller upsetting, the maxima are not pronounced, especially for the hodograph of $\sigma_y$, but with an increase in the degree of upsetting they become very sharp. At the same time, the extrema also shift along the horizontal axis. With smaller upsetting they are located

at a distance of \(1/3\,b\) from the edge of the section (where \(b\) is the width of the prism), and for greater upsetting—at a distance of \(1/4\,b\). At 10% upsetting the stress \(\sigma_x\) is tensile at all points. With an increase in the degree of upsetting it changes sign at the center of the specimen, and there a triaxial compressive stress state is produced. The plots of \(\sigma_x\) and \(\sigma_y\) differ from the plots obtained in studying the stress state by the polarization-optical method [3]. The difference consists in the presence of a sharply expressed maximum at a large degree of upsetting and of local extrema.

The plots of \(\sigma_x\) and \(\sigma_y\) along the vertical axis of symmetry at the smaller upsetting are shown in Fig. 3. The stress plots are not monotonic; the stress \(\sigma_x\) changes sign twice. It is positive at the contact surface and at the center of the section. In character, the plots do not coincide with the plots obtained by E. P. Unksov [3]. In the study by the polarization-optical method, the stress \(\sigma_x\) decreases monotonically toward the center, remaining negative along the entire axis, while \(\sigma_y\) increases slightly toward the center. Along the vertical axis the stress state is everywhere homogeneous.

In interpreting the results of hardness measurement for a specimen upset by 10%, it was possible to obtain plots of the contact stresses, presented in Fig. 1. The curve of normal stresses \(\sigma_y\) has a concave character with a minimum on the axis of symmetry, whereas according to the study [3] it has a maximum value at this point. However, analysis of the results of [3] shows that, as the ratio of the specimen width to its height decreases, the plot of \(\sigma_y\) becomes increasingly leveled.

It was suggested that the concave character obtained for the plot of \(\sigma_y\) is determined by the strain-hardening capacity of the material. To verify this assumption, a calculation was carried out for an ideally plastic material with the same boundary shape. The distribution of \(\sigma_y\) for this case is shown by the dashed line in Fig. 1 on the scale \(\sigma_y/k\), where \(k\) is the intensity of shear stresses, the same at every point. It turned out that the plot of \(\sigma_y\) is considerably straightened. Some concavity is evidently explained by the curvilinearity of the side contour.

Earlier, A. D. Tomlenov [4], using the slip-line method, constructed a plot of the normal contact stress for specimens with a small ratio of width to height. The plot obtained by us for an ideally plastic material agrees very well with the latter.

The contact shear stress \(\tau_{xy}\) increases from 0 at a certain distance from the edge of the section to a maximum, then falls to 0 at a distance of \(1/6\,b\) from the center. The plot of \(\tau_{xy}\) coincides with the plot obtained by E. P. Unksov [3] in plane upsetting of a lead specimen.

A calculation of the deforming force, performed from the plots of \(\sigma_y\) on the horizontal axis of symmetry for both upsettings and from the plot of \(\sigma_y\) on the contact surface for 10% upsetting, showed that in all cases the calculated deforming force is overestimated in comparison with the actual one by 7%, 5%, and 10%, respectively.

Such a comparatively small error indicates the sufficiently high accuracy of the calculation performed.

In the case of plane deformation, the deformed state along the axes of symmetry can be determined from hardness [6]. Figure 3 presents the plots of the strains \(e_x=-e_y\). The strains increase monotonically toward the center of the specimen. For verification, the average strain for the vertical and horizontal axes of symmetry at both degrees of upsetting was compared with the strain found from the change in

Fig. 3. Diagrams of the principal stresses along the vertical axis of symmetry at an upsetting reduction of 10%, and diagrams of the principal strains along the axes of symmetry at upsetting reductions of 10% and 30%. ○ — upsetting reduction 10%, □ — upsetting reduction 30% (for the strain diagrams), △ — for the stress diagrams.

of the dimensions of the cross section. For the smaller upsetting, the average deformation proved to be overestimated by 30%, whereas for the larger upsetting it coincides with the actual value. This can be explained by the fact that the accuracy of the method for investigating the deformed state from the hardness distribution is greater in the region of medium reductions (20%–30%) and comparatively low in the region of small deformations [6].

From the results of the investigation carried out, the following principal conclusions may be drawn:

1. The stress–strain state during plane upsetting of prisms is nonuniform. At small degrees of upsetting, in the central part of the specimen there is a tensile stress state of like sign. As the degree of upsetting increases, the stress \(\sigma_x\) changes sign in the center to negative, and triaxial compression arises there.

2. The stress state depends on the strain-hardening capacity of the material. A stress state of the same name in the more strongly hardening material occurs at a smaller upsetting than in the less strongly hardening material.

REFERENCES

1. L. Prandtl. Examples of the application of Hencky’s theorem to the equilibrium of plastic bodies. In: Theory of Plasticity. Publishing House of the Academy of Sciences of the USSR, 1946.

2. V. V. Sokolovskii. Theory of Plasticity. Publishing House of the Academy of Sciences of the USSR, 1946.

3. E. P. Unksov. Engineering Theory of Plasticity. Mashgiz, 1951.

4. A. D. Tomlenov. Mechanics of Metal-Forming Processes. Mashgiz, 1963.

5. G. D. Del’. Investigation of plastic deformation by measuring hardness. Proceedings of TPI, vol. 138, 1965.

6. G. D. Del’. Determination of the deformed state in the plastic region from the hardness distribution, in the present collection.