Method for Determining Stresses in Sheet Rolling Using a Digital Computer

V. D. Del, N. F. Zima

(Presented by the scientific seminar of the Laboratory of Plastic Deformations)

Despite the prevalence and comparative simplicity of the sheet-rolling process, information on the stressed state of the strip being rolled remains very limited because of the lack of boundary conditions for an analytical calculation of stresses and the great complexity of their experimental determination.

In connection with this, an experimental-analytical method has been developed for determining stresses in sheet rolling from the distribution of hardness and the distortion of a rectangular grid, similar to that previously used in the study of continuous chip-formation processes [1, 2]. The coordinate grid is applied to a plate made of the material under investigation, which is inserted into a slit made in the plane of symmetry of the strip being rolled, perpendicular to the roll axes.

Rolling of a wide strip is a steady-state plane-strain process; therefore, grid lines that before rolling were parallel to the direction of strip motion are transformed during deformation into the trajectories of particle motion. The process is stopped, and in the region under investigation, from the distortion of the coordinate grid, the fields of the angles $\alpha$ between the tangents to the trajectories of particle motion and the $x$ axis, and of the distances $a$ between the grid nodes in the direction of the trajectory, are determined (Fig. 1).

The surface is then machined; its hardness is measured at various points, and from it and from a previously constructed calibration curve, the intensity of shear stresses $\kappa$ is determined. The results are averaged by drawing isolines.

To calculate the stresses at the nodes of the coordinate grid shown in Fig. 1, the values of $a$, $\alpha$, and $\kappa$ are determined.

The angles of inclination of the tangents to the trajectories of slip lines at the grid nodes are calculated from the formula [1]

$$ \theta = \frac{\pi}{2} + \frac{1}{2}\arctan \left[ \frac{ \tan \alpha \left( \frac{\partial a}{\partial y} + a \frac{\partial \alpha}{\partial x} \right) - \frac{\partial a}{\partial x} + a \frac{\partial \alpha}{\partial y} }{ \tan \alpha \left( \frac{\partial a}{\partial x} - a \frac{\partial \alpha}{\partial y} \right) + \frac{\partial a}{\partial y} + a \frac{\partial \alpha}{\partial x} } \right]. \tag{1} $$

Substitution of the known relations

$$ \sigma_x = \sigma - \kappa \cdot \sin 2\theta; \qquad \sigma_y = \sigma + \kappa \cdot \sin 2\theta; \qquad \tau_{xy} = \kappa \cdot \cos 2\theta \tag{2} $$

into the differential equilibrium equations

\[ \frac{\partial \sigma_x}{\partial x}+\frac{\partial \tau_{xy}}{\partial y}=0;\qquad \frac{\partial \tau_{xy}}{\partial x}+\frac{\partial \sigma_y}{\partial y}=0 \tag{3} \]

makes it possible, after simple transformations, to obtain the following relations for the derivatives of the hydrostatic pressure \(\sigma\) with respect to the coordinates:

\[ \frac{\partial \sigma}{\partial x} = \left(\frac{\partial \kappa}{\partial x}+2\kappa\frac{\partial \theta}{\partial y}\right)\sin 2\theta + \left(2\kappa\frac{\partial \theta}{\partial x}-\frac{\partial \kappa}{\partial y}\right)\cos 2\theta, \tag{4} \]

\[ \frac{\partial \sigma}{\partial y} = \left(2\kappa\frac{\partial \theta}{\partial x}-\frac{\partial \kappa}{\partial y}\right)\sin 2\theta - \left(2\kappa\frac{\partial \theta}{\partial y}+\frac{\partial \kappa}{\partial x}\right)\cos 2\theta. \tag{5} \]

Numerical integration of the expressions obtained is carried out by the finite-difference method; for calculating \(\sigma\), a difference scheme of the form

\[ \sigma_1=\sigma_0+f_{01}\cdot \Delta, \]

was used, where \(f_{01}\) is the right-hand side of expressions (4) and (5) for a point lying between the nodes with indices 0 and 1, and \(\Delta\) is the step of the coordinate grid (in what follows we take \(\Delta=1\)).

Fig. 1. Scheme for calculating stresses during rolling

The boundary conditions for calculating the hydrostatic pressure are the conditions at point \(D\) of the free contour in the immediate vicinity of the contact, where \(\sigma=-\kappa\).

From point \(D\), the values of \(\sigma\) are calculated along the ray \(DC\), the direction of which, in order to avoid entering the elastic region, is chosen at an angle \(-\pi/4\) to the \(x\)-axis; expression (4) in finite differences is then written as follows:

\[ \begin{aligned} \sigma_{\kappa,l} &= \sigma_{\kappa+1,l-1} + \bigl[ \kappa_{\kappa,l}-\kappa_{\kappa+1,l-1} + (\kappa_{\kappa,l}+\kappa_{\kappa+1,l-1})(\theta_{\kappa+1,l} -\theta_{\kappa,l-1}) \bigr] \sin(\theta_{\kappa,l}+\theta_{\kappa+1,l-1}) \\ &\quad + \bigl[ (\kappa_{\kappa,l}+\kappa_{\kappa+1,l-1})(\theta_{\kappa,l}-\theta_{\kappa+1,l-1}) -\kappa_{\kappa+1,l}+\kappa_{\kappa,l-1} \bigr] \cos(\theta_{\kappa,l}+\theta_{\kappa+1,l-1}). \end{aligned} \]

Fig. 2. Flowchart for stress calculation

• Initial data: \(m, n, \alpha^{0}, a, k\)
• Conversion to the binary system and arrangement

Subprogram for calculating \(\theta\)

• Initial data: \(a, \alpha\)
• Calculation of \(\theta\) at the nodes of a row
• Check for completion of the row
• Check for completion of the computations

Subprogram for calculating \(\sigma\)

• Initial data: \(\theta, k\)
• Calculation of \(\sigma\) on the boundary ray
• Calculation of \(\sigma\) on the axis of symmetry
• Check for the presence of closed data
• Linear extrapolation
• Calculation of \(\sigma\) at the nodes of a row
• Calculation of the components of the stress tensor
• Check for completion of the row
• Print \(k, \theta^{0}, \sigma, \sigma_{xy}, \sigma_{x}, \sigma_{y}\) at the nodes of the row

• Check for completion of the computations

• Stop

The subsequent calculation of \(\sigma\) can be performed in the following ways:

1. The calculation for all nodes except those belonging to the contact is carried out by numerical integration of expression (4); the values of \(\sigma\) at the nodes on the contact are calculated from expression (5).

2. From point \(C\), \(\sigma\) is calculated along the axis of symmetry, and then, using the values obtained on the ray \(DC\) and the axis of symmetry, the calculation is carried out over the entire region by integrating expression (4) with rotation of the \(x\)-axis through an angle \(\pi/4\), or expression (5) with rotation of the \(y\)-axis through an angle \(-\pi/4\).

3. Using the values of \(\sigma\) on the ray \(DC\) and on the axis of symmetry, the calculation is carried out over the entire region by integrating expression (5).

On the axis of symmetry, owing to the constancy of \(\Theta\) \((\Theta = 3\pi/4)\) and the symmetry of \(\kappa\) and \(\Theta\) with respect to it, the computational expression for the hydrostatic pressure is considerably simplified:

\[ \sigma_{k,l}=\sigma_{k,l-1}-\kappa_{k,l}+\kappa_{k,l-1}. \tag{4} \]

After the hydrostatic pressure has been calculated at the nodes of the coordinate grid, all stress components are calculated from relations (2).

The flow chart for calculating the stresses by the method described is presented in Fig. 2. The calculation was performed on an M-20 digital computer. The stressed state of copper, steel, duralumin, and lead during rolling at microspeeds and at different thicknesses and degrees of strip deformation was investigated. Figure 3 presents stress diagrams for rolling a copper strip 6 mm thick to a deformation degree of 17% on a laboratory mill with rolls 50 mm in diameter. From the nature of the diagrams it may be concluded that the process of shape change must occur mainly in the region where the strip is gripped, which is confirmed by the form of the deformed grid.

The strip material near the contact is under conditions close to triaxial compression. This explains why the values of the acting stresses exceed the yield strength of the strip material several times. It has been established that strain hardening of the material increases unproductive energy losses and leads to an unfavorable stress distribution. A similar effect accompanies a reduction in the thickness of the rolled strip. Increasing its degree of deformation, however, reduces the fraction of the spherical tensor; consequently, from the standpoint of energy expenditure, it is more advantageous to reduce the number of rolling passes by increasing the degree of deformation in each of them.

REFERENCES

1. G. D. Del, M. G. Goldshmidt. Determination of stresses in the cutting zone. Izv. TPI, vol. 162, Tomsk, 1967.

2. N. N. Zorev, G. D. Del, M. G. Goldshmidt, V. D. Del. The stressed state in continuous chip formation. Vestnik mashinostroeniya, No. 7, 1969 (in press).

Fig. 3. Stress diagrams during rolling of a copper strip

Visible labels in the figure: \(\sigma_y\), \(\sigma_x\), \(\tau_{xy}\), \(x\), \(y\), \(I\!-\!I\), \(II\!-\!II\), “deformation zone length,” \(b/b_0\).