CALCULATION OF STRESSES IN COLD FORWARD EXTRUSION USING THE “BESM-4” COMPUTER

G. L. KALINICHENKO; D. P. SNEGIREV

(Presented by the scientific seminar of the Department of Engineering and Computational Mathematics)

A wide variety of circumstances require machine building to create reliable, productive machines with the lowest economic costs. Cold extrusion of metals, when applied rationally, makes it possible to significantly reduce the labor intensity of manufacturing and the cost of products.

In the process of cold extrusion, as a result of strain hardening of the metal, especially of its surface layers, and the presence of a texture that follows the contours of the products, an increase is observed in the wear resistance and fatigue strength of parts. Preliminary studies have shown an increase in strength under alternating-sign loads by 20–50% [1].

For the rational construction of technological operations of cold extrusion of strain-hardening metals, it is important to know the distribution of stresses, which depend on many process parameters: the properties of the material, the die semi-angle, the deformation rate, lubrication, the degree of reduction \(\psi\), etc.

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

where \(D\) is the diameter of the container section,
\(d\) is the diameter of the sizing section of the die.

The stresses were determined by an experimental–computational method, consisting in a combination of methods for measuring the hardness of the deformed metal and grid-line methods [2]. The calculation was performed on the “BESM-4” computer.

Specimens for extrusion were made with a height \(H \approx 1.6D\), from two symmetrical halves separated along the meridional plane. The meridional surfaces of these specimens were finished to class 9 or 10 surface cleanliness. On the machined surface of one of the halves of the composite specimen, a normal grid with a spacing of 1–2 mm was applied with a diamond tip. The grid spacing was selected with allowance for the magnitude of reduction and the die semi-angle.

The composite specimens were extruded with lubrication in a special container on a 500-ton hydraulic press. In the established stage of extrusion, under constant deformation force, the process ...

was stopped, the specimen was removed from the container and divided into two halves.

The meridional surface of one half of the extruded specimen was prepared for measuring Vickers hardness; from the other half, the deformed grid, magnified 10 times, was copied on a projector. Then a normal grid with a constant spacing was applied to the enlarged deformed region. The spacing was chosen so that an integral number of spacings fit into the container radius, and the number of nodes of the normal grid was no less than 10. Subsequently, the increments of strains and stresses were calculated at the nodes of the superimposed grid.

If, at the established stage of plastic deformation, some family of lines in the undeformed metal is oriented along the extrusion axis, then these lines, as they deform, are transformed into the trajectories of the particles located on them relative to the walls of the container and die.

The magnitude of the velocity of these particles can be determined from marks applied beforehand, while the particle-velocity vector is directed along the tangent to their trajectory.

Fig. 1. Scheme of axisymmetric extrusion for calculating strain increments.

If marks are applied at equal intervals \(a_0\) on a line oriented along the extrusion axis, then, in its motion relative to the tool, point 1 occupies the position of point 2, point 2 the position of point 3, etc. (Fig. 1), during one and the same time interval.

\[ \Delta t=\frac{a_0}{v_0}. \tag{1} \]

The relative velocity of particle displacement is

\[ v=\frac{a}{\Delta t}=v_0\frac{a}{a_0}, \tag{2} \]

where \(a\) is the distance between marks after deformation.

The components of these velocities in a cylindrical coordinate system are

\[ v_r=v\cdot \cos\alpha,\qquad v_z=v\cdot \sin\alpha, \tag{3} \]

where \(\alpha\) is the angle of inclination of the tangent to the particle trajectory, measured from the coordinate \(r\) in the positive direction.

The values of the strain increments were determined from the distortion of the normal grid, arranged so that one family of lines was oriented along the extrusion axis.

Under axisymmetric deformation, the components of the strain tensor are equal to

\[ \left. \begin{aligned} \dot{\varepsilon}_r&=\frac{\partial v_r}{\partial r} =\frac{v_0}{a_0}\left(\frac{\partial a}{\partial r}\cos\alpha -a\cdot\sin\alpha\,\frac{\partial \alpha}{\partial r}\right),\\ \dot{\varepsilon}_z&=\frac{\partial v_z}{\partial z} =\frac{v_0}{a_0}\left(\frac{\partial a}{\partial z}\sin\alpha +a\cdot\cos\alpha\,\frac{\partial \alpha}{\partial z}\right),\\ \dot{\gamma}_{rz}&=\frac{\partial v_r}{\partial z}+\frac{\partial v_z}{\partial r} =\frac{v_0}{a_0}\left[\cos\alpha\left(\frac{\partial a}{\partial z} +a\frac{\partial \alpha}{\partial r}\right) +\sin\alpha\left(\frac{\partial a}{\partial r} -a\frac{\partial \alpha}{\partial z}\right)\right],\\ \dot{\varepsilon}_\varphi&=\frac{v}{r}\cos\alpha =\frac{v_0}{a_0}\cdot\frac{a}{r}\cos\alpha . \end{aligned} \right\} \tag{4} \]

Taking into account that the constant multiplier in relations (4) cancels in what follows, the strain increments were then determined as:

\[ \Delta\varepsilon_{ij}=\dot{\varepsilon}_{ij}\cdot\Delta t . \tag{5} \]

The derivatives were determined from the relations [3]:

a) derivatives on the axis

\[ \left(\frac{\partial a}{\partial r}\right)_0 =\frac{-3a_0+4a_1-a_2}{2\cdot h},\quad \left(\frac{\partial \alpha}{\partial r}\right)_0 =\frac{-3\alpha_0+4\alpha_1-\alpha_2}{2\cdot h}; \]

b) derivatives at the boundary

\[ \left(\frac{\partial a}{\partial r}\right)_0 =\frac{a_{n-2}-4a_{n-1}+3a_n}{2\cdot h},\quad \left(\frac{\partial \alpha}{\partial r}\right)_n =\frac{\alpha_{n-2}-4\alpha_{n-1}+3\alpha_n}{2\cdot h}; \]

c) intermediate derivatives

\[ \left(\frac{\partial a}{\partial r}\right)_3 =\frac{a_1-a_4}{2\cdot h},\quad \left(\frac{\partial \alpha}{\partial r}\right)_3 =\frac{\alpha_1-\alpha_4}{2\cdot h}, \tag{6} \]

where \(h\) is the base of the computational grid.

Using similar relations, the derivatives in the direction of the coordinate \(z\) were determined.

The increment of the strain intensity is

\[ \Delta\varepsilon_i=\frac{2}{\sqrt{3}} \sqrt{\Delta\varepsilon_r^2+\Delta\varepsilon_z^2+\Delta\varepsilon_r\cdot\Delta\varepsilon_z +\frac{\Delta\gamma_{rz}^2}{4}} . \tag{7} \]

The components of the stress deviator were calculated from the relations:

\[ S_r=\sigma_r-\sigma=\frac{2}{3}\frac{\sigma_i}{\Delta\varepsilon_i}\Delta\varepsilon_r,\quad S_z=\sigma_z-\sigma=\frac{2}{3}\frac{\sigma_i}{\Delta\varepsilon_i}\Delta\varepsilon_z, \]

\[ S_\varphi=\sigma_\varphi-\sigma=\frac{2}{3}\frac{\sigma_i}{\Delta\varepsilon_i}\Delta\varepsilon_\varphi,\quad \tau_{rz}=\frac{1}{3}\frac{\sigma_i}{\Delta\varepsilon_i}\Delta\gamma_{rz}, \]

here \(\sigma\) is the hydrostatic pressure.

To determine the characteristic \(a\) at the nodes of the computational grid, isolines of \(\alpha\) were constructed. The values of \(a\) were determined by direct measurement: without

construction of isolines. The stress intensity $\sigma_i$ in the plastic region was determined from the hardness distribution using a calibration curve constructed from the results of compression tests of the material with lubrication.

The initial data $a$, $\alpha$, and $\sigma_i$ are specified row by row by three arrays. Rows are separated by an empty cell, and arrays by two empty cells. This method of entering the initial data makes it possible, with a single program block, to establish the boundaries, i.e., to determine the number of nodes of the computational mesh in the direction of the coordinate $r$ and the coordinate $z$.

The program includes a standard subroutine for computing the derivatives of functions specified in tabular form. It is sufficient to transfer the values of any row or column into the working field of the standard subroutine, and, by calling the latter, the derivatives with respect to the coordinate $r$ or $z$ are computed. Thus the following derivatives are determined in the array:

\[ \frac{\partial a}{\partial r};\quad \frac{\partial a}{\partial z};\quad \frac{\partial \alpha}{\partial r};\quad \frac{\partial \alpha}{\partial z}. \]

Next, two arrays, $\sin \alpha$ and $\cos \alpha$, are computed, and the strain increments are determined: $\Delta \varepsilon_r$; $\Delta \varepsilon_z$; $\Delta \varepsilon_\varphi$; $\Delta \gamma_{rz}$ and $\Delta \varepsilon_i$; then the components of the stress deviator are found: $s_r$; $s_z$; $s_\varphi$; $\tau_{rz}$.

In the second part of the program the stresses are computed. Having determined, from the equilibrium condition [2], the boundary stress at the contact of the billet with the die, the machine computes all stresses at the nodes of the computational mesh on this radius. The stresses obtained on the radius are used as boundary conditions. This makes it possible to compute the stresses throughout the entire array.

The machine prints out the following information:

1) all initial data entered for the calculation: extrusion force $P$ in kg; mesh spacing $h$; $a$; $\alpha$; $\sigma_i \varepsilon_i$;

2) strain increments: $\Delta \varepsilon_r$; $\Delta \varepsilon_z$; $\Delta \varepsilon_\varphi$; $\Delta \gamma_{rz}$; $S$; $\Delta \varepsilon_i$;

3) components of the stress deviator: $s_r$; $s_z$; $s_\varphi$;

4) stresses: $\sigma_1$; $\sigma_3$; $\sigma$; $\sigma_r$; $\sigma_z$; $\tau_{rz}$.

Having all the input information, the intermediate results, and the final results, one can check the correctness of the computations performed by the machine.

REFERENCES

1. V. A. Mishunin, L. M. Reshetnikova, et al. Quality and accuracy of parts in cold extrusion. Kuznechno-shtampovochnoe proizvodstvo, 1971, No. 9.

2. G. D. Del’, V. A. Ogorodnikov. Experimental-analytical study of the stress state in axisymmetric pressing of a strain-hardening metal. Izv. vuzov, Mashinostroenie, 1969, No. 2.

3. B. P. Demidovich, I. A. Maron. Fundamentals of Computational Mathematics. Moscow, 1963.