EXPERIMENTAL VERIFICATION OF THEORETICAL SOLUTIONS TO THE EXTRUSION PROBLEM

D. P. SNEGIREV, V. V. ISAEV

Presented by the scientific seminar of the Department of Applied Mechanics

Many technological operations are accompanied by steady-state plastic deformation. Examples of such operations are direct cold extrusion and drawing. To solve many problems associated with these processes, it is necessary to know the stress state of the metal being worked. Information on the stress state is necessary for assessing the deformability of the metal, i.e., the ability of the metal to undergo deformation without fracture, and for determining contact stresses and deforming forces.

Despite the considerable interest of researchers in axisymmetric problems, the stress state in the indicated technological operations has not been studied sufficiently. The analytical solution of axisymmetric problems is associated with certain mathematical difficulties even under serious assumptions. In the case of axisymmetric deformation, the differential equations for stresses and strain rates are not hyperbolic [1]. Therefore, most theoretical solutions have been obtained for an ideal-plastic body under the assumption of “complete plasticity,” which was first proposed by A. Haar and T. Karman. Regarding the applicability of the hypothesis of “complete plasticity,” R. Hill notes [1]: “...the Haar—Karman hypothesis for metals is physically unrealistic and introduces an error of unknown magnitude.”

In this connection, experimental verification of theoretical solutions becomes especially important. The number of such studies remains very limited. Among them, it is necessary to mention studies carried out by the experimental-computational method of visioplasticity developed by E. Thomsen and co-workers [2].

The investigations carried out were performed for billets extruded through rectangular dies: from M1 copper with various reductions—\((\psi = 64\%;\ \psi = 49\%;\ \psi = 36\%)\)—and billets extruded at one reduction—\(\psi = 64\%\), but from several materials—(brass L62; steel 10; duralumin D16).

\[ \psi = \frac{D^2 - d^2}{D^2}\cdot 100\% , \tag{1} \]

where \(D\) is the diameter of the container cross section;
\(d\) is the diameter of the calibrating section of the die.

The experimental procedure is described in [3].

The stress state was determined by an experimental-computational

method, consisting in a combination of methods for measuring the hardness of the deformed metal and dividing grids [4]. The stresses were determined on a BESM-4 computer at the nodes of the computational grid.

According to the assumption of A. Haar and T. Karman on “complete plasticity,” the circumferential stress $\sigma_\varphi$ is taken to be equal to one of the other two principal stresses. The principal stresses were determined by the formula

\[ \sigma_{1,3}=\frac{\sigma_r+\sigma_z}{2}\pm \frac{1}{2}\sqrt{(\sigma_r-\sigma_z)^2+4\tau_{rz}^2}. \tag{2} \]

In order to verify the indicated assumption, for each specimen we constructed plots of the distribution of the stresses $\sigma_\varphi$, $\sigma_1$, $\sigma_3$ along a radius located from the horizontal face of the die at a distance of three or four steps of the computational grid (Figs. 1, 2). The step of the computational grid was selected within the range $1.25\div 1.45$ mm. From the plots presented it is evident that neither of the principal stresses in any of the specimens studied is equal to the circumferential stress. However, in individual cases the stress $\sigma_r$ practically coincides with the principal stress $\sigma_3(\sigma_1>\sigma_3)$. For example, in specimens of copper M1 with reductions $\psi=49\%$ and $\psi=64\%$ such a coincidence is observed, but not over the entire length of the radius. The general regularity is a significant deviation of the circumferential stress from the principal $\sigma_1$.

Fig. 1. Verification of the assumption of A. Haar and T. Karman in the extrusion of one material with different reductions

Thus, the main premise of the theoretical solutions is not in full agreement with the experimental data.

For further comparison of the results obtained, the theoretical solution of T. I. Glebov [5] was used. By this method were cal-

the stresses in the plastic region were calculated for a specimen of M1 copper with a reduction \(\psi=64\%\). As was already indicated earlier, practically all theoretical solutions have been obtained for a non-hardening material. In individual cases an attempt is made to take hardening into account by introducing an increased yield strength into the calculation. In this connection, the calculation of the stresses was carried out for three conditions. In the first two cases the material was considered non-hardening, and the material properties were taken into account by different yield strengths: the initial and the increased ones. The increased yield strength was determined from the flow curve for M1 copper by the mean value of the strain intensity:

\[ e_{i\,\mathrm{av}}=\ln\frac{F}{f}, \tag{3} \]

where \(F\) is the area of the container;

\(f\) is the area of the through section of the die bearing land.

For the specimen under investigation \(e_{i\,\mathrm{av}}=1.0\), and the increased yield strength \(\sigma_s^{n}=39\ \text{kg/mm}^2\). In the third case the increased yield strength \(\sigma_i\), determined by measuring hardness for the calculated points, was taken into account. This calculation, of course, is also inaccurate, since in T. I. Glebov’s relations the derivatives of the yield strength with respect to the coordinates were taken to be equal to zero.

Fig. 2. Verification of the assumption of A. Haar and T. Kármán in the extrusion of different materials at the same reduction

According to the method described, axial stresses \(\sigma_z\) were calculated along a radius located at a distance of five steps of the calculation grid from the horizontal face of the die.

The experimental values of the stresses and the calculation results are presented in Fig. 3. It is apparent from this figure that the values of the axial stresses obtained with the initial yield strength proved to be too low, while with the increased yield strength they were too high in comparison with the results of the calculation using a variable yield strength. Fairly good agreement of the experimental results with the calculated ones is observed only in the case when the variable yield strength was used. However, this agreement is maintained only up to half the radius, starting from the axis of the billet. In the direction toward the periphery, the calculation results diverge sharply from the experimental data.

Fig. 3. Comparison of experimental results with the theoretical solution, taking into account:

◉ — initial yield limit;
□ — increased yield limit;
● — variable yield limit;
△ — experimental results.

Fig. 4. Comparison of the theoretical solution with experimental results:

● — theoretical solution.

Experimental results:
□ — for the series 5h
○ — the same 4h
△ — » 2h

— number of steps of the computational grid ($h = 1.25$ mm) from the horizontal land of the die.

In view of the fact that the theoretical solution with allowance for the variable yield limit gives the best agreement with the experimental results, additional calculations were carried out by this method for the stresses $\sigma_r$ and $\sigma_z$ along three radii located at different distances from the die funnel, and along two radii the tangential stress $\tau_{rz}$ was calculated.

From the graphs presented in Fig. 4 it is seen that there is fairly good agreement of the compared results from the axis of the billet to half of the meridional section. At the periphery, a significant discrepancy is observed between the experimental and calculated values. The compared results proved to be closest along radii located at a distance of four to five steps of the computational grid from the horizontal land of the die. This is explained by the fact that in this region the gradient of the yield limit is not significant. Along radii located closer to the entry into the die ring, a significant discrepancy of the compared results is observed near the axis of symmetry of the billet.

The good agreement between the experimental and theoretical results at the upper boundary of the plastic zone, near the axis of symmetry, makes it possible to use the results of the theoretical study as boundary conditions. Having determined, for example, one of the normal stresses on the axis of symmetry and taking the obtained stress as a boundary condition, further calculations can be carried out in the direction toward the periphery.

However, the theoretical solutions give a completely distorted picture in the region of large plastic deformations; therefore, the results of solutions by the methods of the theory of ideal plasticity are unsuitable for evaluating deformability in the processes under consideration.

REFERENCES

1. R. Hill. The Mathematical Theory of Plasticity. GITTL, 1956.

2. E. Thomsen, Jang, Sh. Kobayashi. Mechanics of Plastic Deformation in Metal Processing. Moscow, “Mashinostroenie,” 1969.

3. V. A. Mishunin, V. A. Ogorodnikov, D. P. Snegirev, P. A. Sokolov. In: Technology of Mechanical Engineering, issue 29. Investigations in the field of plasticity and metal forming. Tula, Publishing House of the Tula Polytechnic Institute, 1973, pp. 27–33.

4. G. D. Del, V. A. Ogorodnikov. Experimental-analytical investigation of the stressed state during axisymmetric extrusion of a hardening metal. Izvestiya VUZov, “Mashinostroenie,” 1969, No. 2.

5. T. I. Glebov. Determination of the stressed state in axisymmetric problems under static and dynamic extrusion processes. Metal Forming in Mechanical Engineering, issue 7, 1971.