DETERMINATION OF STRESSES IN RAPID PROCESSES OF PLASTIC DEFORMATION FROM HARDNESS DISTRIBUTION

G. D. Del, V. I. Livshits

(Presented by the scientific seminar of the Laboratory of Plastic Deformations of the Department of Strength of Materials, TPI)

In [1] a method is set forth for determining the stress state in the plastic region from the hardness distribution. By testing the material in tension, compression, or torsion and by parallel measurement of the hardness of the tested specimens, a calibration graph “hardness \(H\)—intensity of shear stresses \(K\)” is constructed. It has been experimentally confirmed that the indicated graph may, with sufficient accuracy, be regarded as unique for different stress states and loading paths. From the hardness distribution in a deforming body, using the calibration graph, the function \(K(x,y,z,q)\) is determined, where \(q\) characterizes the stage of deformation of the body as a whole. By integrating the differential equilibrium equations for a given stage of deformation \((q=\mathrm{const})\), the stress field is found in statically determinate problems.

In the present article the indicated method is extended to the case of rapid processes of plastic deformation. As has been shown experimentally, in the region of large plastic deformations the function \(K(H)\) is only slightly sensitive to the strain rate. This makes it possible, in studying such processes, to use a calibration graph constructed from the results of static tests. In contrast to static processes, determination of the stress state in the case under consideration is complicated by the necessity of taking into account inertial mass forces.

The method has been developed as applied to plane strain. The equations of motion

\[ \frac{\partial \sigma_x}{\partial x}+\frac{\partial \tau_{xy}}{\partial y}=X;\qquad \frac{\partial \tau_{xy}}{\partial x}+\frac{\partial \sigma_y}{\partial y}=Y, \tag{1} \]

by the substitution

\[ \begin{aligned} \sigma_x &= \sigma - K\cdot \sin 2\Theta,\\ \sigma_y &= \sigma + K\cdot \sin 2\Theta,\\ \tau_{xy} &= K\cdot \cos 2\Theta, \end{aligned} \tag{2} \]

where \(\sigma\) is the hydrostatic pressure, and \(\Theta\) is the angle of inclination of the tangent to the slip line, are transformed into a system of hyperbolic type

\[ \frac{\partial \sigma}{\partial x} -2K\cos 2\Theta\,\frac{\partial \Theta}{\partial x} -2K\sin 2\Theta\,\frac{\partial \Theta}{\partial y} = \sin 2\Theta\,\frac{\partial K}{\partial x} -\cos 2\Theta\,\frac{\partial K}{\partial y} +X. \tag{3} \]

\[ \frac{\partial \sigma}{\partial y} - 2K \sin 2\Theta \frac{\partial \Theta}{\partial x} + 2K \cos 2\Theta \frac{\partial \Theta}{\partial y} = - \cos 2\Theta \frac{\partial K}{\partial x} - \sin 2\Theta \frac{\partial K}{\partial y} + Y. \]

The characteristics of this system coincide with the slip lines; the relations on these lines are

\[ \frac{\sigma}{2K} - \Theta = -A_\alpha;\qquad \frac{\sigma}{2K} + \Theta = -A_\beta. \tag{4} \]

Here it is assumed that

\[ A_\alpha = \frac{1}{2K} \int_{\alpha} \left( \frac{\partial K}{\partial \beta} - X \cos \Theta - Y \sin \Theta \right)\, d\alpha + \psi_1(\beta) \]

\[ A_\beta = \frac{1}{2K} \int_{\beta} \left( \frac{\partial K}{\partial \alpha} + X \sin \Theta - Y \cos \Theta \right)\, d\beta + \psi_2(\alpha), \tag{5} \]

where \(\alpha\) and \(\beta\) are measured along the slip lines of the \(\alpha\)- and \(\beta\)-families, whose directions are fixed in such a way that they form a right-handed coordinate system, where the shear stress has a positive direction. The quantities \(\psi_1\) and \(\psi_2\) are determined from the boundary conditions.

Fig. 1

The boundary-value problems arising in the interpretation are solved by the methods described in [1]. The integrals \(A_\alpha\) and \(A_\beta\) are determined from the approximate relations (Fig. 1):

\[ (A_\alpha)_3 = (A_\alpha)_2 - \frac{l_{2-3}}{2K_3} \left[ \frac{K_A - K_C}{AC} - X_B \cos \Theta_2 - Y_B \sin \Theta_2 \right], \]

\[ (A_\beta)_3 = (A_\beta)_1 - \frac{l_{1-3}}{2K_3} \left[ \frac{K_D - K_F}{DF} + X_E \sin \Theta_1 - Y_E \cos \Theta_1 \right]. \tag{6} \]

Here the subscripts correspond to the designations of points in the figure. By the developed method, the stress state in the chip-formation zone during free cutting of steel 45G17Yu3Kh with a hard-alloy tool was investigated under the following conditions: cutting speed \(17.5\ \text{m/min}\); feed \(0.9 \div 1.1\ \text{mm}\); depth \(3.5\ \text{mm}\). Under these conditions, a shear chip was formed.

A division grid with a base of 0.25 mm was applied to the side surface of the specimen by scratching a system of mutually perpendicular grooves of triangular cross section. The width of the grooves did not exceed 0.1 of the grid base.

With the aid of a special device, an instantaneous stop of the cutting process was carried out, with withdrawal of the cutting tool. The zone of chip formation fixed in this way (the “root” of the chip) was cut out of the specimen. The chip “roots” were arranged successively according to

Fig. 2. Accelerations of particles in the chip-formation zone

the stages of formation of a chip element—from the shear of the preceding element to the complete formation of the next one.

Knowing the grid base, one can determine the coordinates of various grid nodes before the beginning of formation of an element. One of the nodes lying outside the plastic region, with initial coordinates identical for all stages of element formation, was taken as the origin of coordinates. From the change in the \(X\) coordinate of the tool-tip apex and the cutting speed, the time corresponding to each stage of formation of a chip element was established.

Fig. 3. Stresses in the cutting zone

Then the distances of various nodes from the origin were measured at different stages of formation of the element, and the graphs \(x(t)\), \(y(t)\) were plotted for nodes with identical initial coordinates. By double graphical differentiation of these graphs, the distribution of accelerations \(a_x(x,y)\) and \(a_y(x,y)\) was established. Figure 2 shows the acceleration field at one of the last stages of formation of the chip element. The magnitude of the inertial terms in the equations of motion (1) was found by multiplying \(a_x\) and \(a_y\) by the density of the material.

Strictly speaking, the described procedure for determining inertial mass forces is applicable only to steady-state processes of plastic deformation (for example, in continuous chip formation). In the case considered here, however, the distribution of accelerations may be influenced to some extent by inertial loads arising as a result of vibrations of the material being machined, which were not taken into account in the calculation.

After photographing the deformed grid, the “root” of the chip was ground down to the middle of its thickness, and the microhardness on the prepared polished section was measured using a PMT-3 instrument. The deformation in this section may be regarded as plane; it was also assumed that the acceleration field in it is the same as on the side surface of the chip.

By the method described above, the stress components were calculated with allowance for the inertial mass forces (Fig. 3). The stresses were then determined by the procedure used for static processes. Comparison of the results showed that their discrepancy does not exceed fractions of a percent. It follows from this that, when studying stresses in the process of formation of a shear chip at cutting speeds of an industrial level, inertial mass forces may be neglected. The question of the influence of these forces under other cutting conditions, especially at speeds up to \(10{,}000 \div 20{,}000\) m/min, requires further experimental verification, which can be carried out by the method described.

In order to verify the results obtained, the mean shear stresses along the shear line were compared with data obtained from dynamometric measurements of the cutting forces. The discrepancy did not exceed 3%, which indicates the high accuracy of the method for determining stresses from hardness.

REFERENCES

1. G. D. Del. Investigation of plastic deformation by hardness measurement. Izv. TPI, vol. 138, 1965.