INVESTIGATION OF THE INHOMOGENEITY OF THE STRESS STATE DURING PLANE UPSETTING

E. S. VAKSMAN, L. M. SEDOKOV

(Recommended for publication by the scientific conference of the Faculty of Mechanics)

To solve technological problems in metal forming by pressure, it is necessary to know the distribution of stresses in the workpiece undergoing deformation.

As a result of the action of contact-friction forces during the upsetting process, a nonuniform distribution of stresses and strains occurs. Analytical methods for calculating the stress state [1], [2], based on the relations of the mathematical theory of plasticity, do not reflect the ability of a real metal to harden during plastic deformation. Meanwhile, because of the localization of deformation in certain volumes of the workpiece being deformed, this characteristic of real metals must influence the stress state.

In the present work the stress state was determined for a number of materials during upsetting under conditions of plane strain. Specimens of stainless steel Kh18N9T, steel St. 3, steel ShKh15, and aluminum were upset. The choice of material was determined by their different capacities for hardening during plastic deformation.

In [3] a method is presented for determining stresses in the plastic region from the distribution of hardness. For this purpose, by testing in axial compression with lubrication and subsequently measuring the hardness of specimens compressed to different degrees of deformation, a calibration graph is constructed that relates the intensity of shear stresses \(K\) to hardness. Then, by measuring the hardness at various points of the deformed body, the function \(K(x, y)\) is determined with the aid of the calibration graph.

The stresses are calculated by numerical integration of the differential equilibrium equations, taking into account the established distribution of \(K\). The length of the specimens was three times greater than their width; consequently, deformation in the middle part along the length was considered plane. The hardness of the deformed specimens was measured in the plane of deformation; the surface tested was carefully polished. Measurements were made with a diamond pyramid under a load of \(30\ \text{kg}\) (for testing aluminum specimens, \(10\ \text{kg}\)). For each specimen, 100–150 measurements were made. The results of the hardness measurements were averaged by drawing isoscalar lines of equal hardnesses.

To construct calibration graphs, 8–10 specimens of each of the materials under study were made, with a diameter of \(15\ \text{mm}\) and a height of \(23\ \text{mm}\).

The hardness of the compressed specimens was measured at points of meridional sections sufficiently far from the end faces.

The interpretation was carried out by the following procedure. In the region \(ABC\) (Fig. 1), the Cauchy problem was solved. The free boundary was divided into a number of segments by the points \(1, 2\ldots\). The integrals \(A_\alpha, A_\beta\) [3] at these points were determined from the relations:

\[ (A_\alpha)_1 = 0.5 + \Theta,\qquad (A_\beta)_1 = 0.5 - \Theta, \]

where \(\Theta\) is the angle between the \(x\)-axis and the tangent to the slip line drawn at an angle of \(45^\circ\) to the free boundary. Here and below, the subscript indicates the number of the point for which the corresponding quantity is being determined.

Fig. 1. On the method for determining the stress state

From the points \(1, 2\ldots\), rays were drawn at angles \(\gamma_{1-3}, \gamma_{2-3}\ldots\) to the coordinate axes until they intersected at points of the second layer. The angles were calculated from the relations:

\[ \gamma_{1-3}=\frac{1}{4}\left[2\Theta_1+(A_\alpha)_1-(A_\beta)_2\right], \]

\[ \gamma_{2-3}=\frac{1}{4}\left[2\Theta_2+(A_\alpha)_1-(A_\beta)_2\right]. \]

At the points of the second layer

\[ (A_\alpha)_3=(A_\alpha)_1+\frac{\kappa_M-\kappa_N}{2\kappa_3 l_{MN}}\,l_{1-3}, \]

\[ (A_\beta)_3=(A_\beta)_2+\frac{\kappa_{M'}-\kappa_{N'}}{2\kappa_3 l_{M'N'}}\,l_{2-3}. \]

The hydrostatic pressure \(\sigma\) and the angle \(\Theta\) are determined from the formulas:

\[ \sigma=\kappa(-A_\alpha-A_\beta),\qquad \Theta=\frac{1}{2}(A_\alpha-A_\beta), \]

the stresses:

\[ \sigma_x=\sigma-\kappa\sin 2\Theta,\qquad \sigma_y=\sigma+\kappa\sin 2\Theta. \]

The points of the third layer were calculated in the same way.

Along the axis of symmetry the angle \(\Theta = 0.786\) is known. Therefore, the stresses in the remaining region were determined by solving the mixed problem. Such a decoding procedure is possible only when point \(D\) is located sufficiently close to the center of the section. If this condition is not satisfied, in the region BDL (Fig. 1, b) we solve the initial characteristic problem. In doing so it was assumed that there is no friction at point \(B\). This calculation path is possible only in those cases when point \(L\) is located sufficiently close to the center of the section. Therefore, the proposed procedure is applicable only for determining the stress state during upsetting of sufficiently narrow strips.

Fig. 2. Stress-component diagrams for different degrees of upsetting

Figure 2 shows stress diagrams along the horizontal axis of symmetry for various upsetting reductions of prisms made of stainless steel X18H9T with initial height \(h = 24\) mm and cross-sectional width \(b = 16\) mm. A distinctive feature of the diagrams is the increase in the nonuniformity of the distribution of the stress components \(\sigma_x, \sigma_y\) with increasing upsetting. The intense increase in the compressive stress \(\sigma_y\) at the center of the specimen is accompanied by a much smaller increase in it in the regions near the free boundary. With increasing upsetting, the component \(\sigma_x\) changes sign, and in the central part of the specimen a state of triaxial compression arises.

An increase in the nonuniformity is also caused by an increase in the strain-hardening capacity of the material (Fig. 3). In the highly hardening steels Kh18N9T and ShKh15 the nonuniformity is greater than in the weakly hardening steel—St. 3.

Fig. 3. Diagrams of the components of the stress state during upsetting of different materials

Although the strain-hardening capacity of the metal decreases with increasing plastic deformation, the nonuniformity of the stress state increases with increasing upsetting. This can be explained by the predominant influence of the friction that increases during deformation. The latter is confirmed by curve 4 (Fig. 2), which was obtained under conditions in which the end faces of the upset specimen were lubricated with a mixture of colloidal graphite and glycerin, with the simultaneous use of lead-foil gaskets 0.05 mm thick. All the other diagrams were obtained by compression between ground plates without lubrication; in all cases the contact surface of the specimens was machined by planing.

Figure 4 presents diagrams of the components of the stress state during 30% upsetting of specimens made of stainless steel Kh18N9T with different ratios of the dimensions of the transverse section. In the low specimen (curve 2) the role of the component $\sigma_x$ is more significant; the state of triaxial compression in low specimens occurs at a smaller degree of upsetting.

The obtained diagrams were checked by comparing the calculated deforming forces, determined as the product of the area of the diagram by the length of the specimen, with the actual value of these forces. As can be seen from the table, the discrepancy between the calculated forces and the actual ones did not exceed 15%.

On the basis of the results of the investigation carried out, the following principal conclusions can be drawn:

1. The stress state during plane upsetting of prisms is inhomogeneous. At small degrees of upsetting, in the central part of the specimen a stress state of unlike signs occurs. With an increase in the degree of upsetting, the stress \(\sigma_x\) changes sign in the center to negative, and triaxial compression arises there.

Fig. 4. Diagrams of the components of the stress state during upsetting of specimens with different dimensional ratios

1. The stress state depends on the character of the material hardening curve. The steeper the hardening curve, the more nonuniform the stress distribution. A like-signed stress state in a more strongly hardening material occurs at a smaller upsetting than in a less strongly hardening one.

2. The state of triaxial compression in low specimens occurs at a smaller degree of upsetting than in high ones.

REFERENCES

1. S. I. Gubkin. Plastic Deformation of Metals, vol. 1. Metallurgizdat, 1961.
2. A. A. Pozdeev, V. I. Ternovskii. Investigation of the stress state during upsetting. Izv. vuzov. Ferrous Metallurgy, No. 11, 1962.
3. G. D. Del’. Investigation of plastic deformation by hardness measurement. Izv. TPI, vol. 138, 1965.