CHANGE IN METAL HARDNESS UNDER NONMONOTONIC LOADING

G. D. DEL, V. A. OGORODNIKOV

(Presented by the Department of Strength of Materials)

In studying the stress–strain state in the plastic region by the method of hardness measurement [1, 2, 3], it is assumed that the relationship “stress intensity \(\sigma_i\)—hardness \(H\)—plastic strain intensity \(e_i\)” is the same for different stress states. This is expressed in the extension of the calibration graph “\(\sigma_i\)—\(H\)—\(e_i\),” constructed by testing materials in tension, compression, or torsion and measuring the hardness of deformed specimens, to various processes of plastic deformation. Already from the divergence of the flow curves \(\sigma_i\)—\(e_i\) for certain loading paths it follows that the indicated relationship cannot be universal.

Fig. 1.

The greatest divergence from the calibration curve should be expected under nonmonotonic loading with intermediate unloading. Figure 1 gives a calibration graph characteristic of plastic metals. Hardness is measured after unloading; therefore elastic deformation does not lead to a change in it. A consequence of the latter is the horizontal segment on the right branch of the graph, corresponding to a change in stresses from zero to the yield point. Subsequently, plastic deformation is accompanied by a monotonic increase in hardness.

If, after tension to point A, the specimen is unloaded and compressed in the direction of the preceding tension, then the plastic state will occur at a stress \(\sigma_{iB}\), smaller than \(\sigma_{iA}\). The hardness, however, remains equal to \(H_A\), since in passing from point A to point B the deformation remained elastic. Thus, point B, corresponding to the plastic state, does not lie on the curve \(OA\), and the graph \(\sigma_i\)—\(H\) is not the same for the entire plastic region.

The aim of the present work is to determine the regularity of the change in hardness during subsequent plastic deformation in the second direction and to investigate the possibility of applying the hardness-measurement method

...hardness in the study of technological processes accompanied by nonmonotonic loading of a metal.

In works devoted to the study of the Bauschinger effect [4], the greatest decrease in the yield point was observed when a twisted specimen was twisted in the reverse direction. It is also indicated that the ratio of the shear stresses in torsion in the forward and reverse directions initially decreases with increasing preliminary deformation and, when the latter reaches about 2%, is approximately 2.5. With a further increase in this deformation, the indicated ratio remains approximately constant.

In view of the above, tests were carried out on thin-walled tubular specimens in torsion. Tubes with an outside diameter of 30 mm and a wall thickness of 1 mm were made of steel 20. One of them was twisted until loss of stability; the other was deformed to \(e_i = 0.035\) and then twisted in the reverse direction. The conditional yield point \(\sigma_{0.2}\) in this case amounted to 0.65 of the stress preceding unloading. Torsion was carried out with a number of intermediate unloadings for the purpose of measuring hardness. Hardness was measured with a diamond

Fig. 2. Torsion of a tube in the forward and reverse directions.
1 — torsion in the forward direction; 2 — torsion in the reverse direction; 3 — calibration curve

pyramid under a load of 30 kg at ten points on the outer surface. The mean of these measurements is compared in Fig. 2 with the intensity of stresses and strains, determined from the relations:

\[ \sigma_i = \frac{\sqrt{3}M}{2\pi R_{\mathrm{cp}}^{2}t}, \qquad e_i = \frac{\varphi R_{\mathrm{cp}}}{l\sqrt{3}}, \tag{1} \]

where \(M\) is the twisting moment; \(R_{\mathrm{cp}}\) and \(t\) are the mean radius and the wall thickness of the tube; \(\varphi\) is the angle of twist, measured by a mirror strain gauge over a gauge length \(l = 100\) mm.

The results obtained show that plastic torsion in the reverse direction is accompanied by an insignificantly small decrease in hardness. After the stresses preceding unloading are reached, the \(\sigma_i - H\) curve coincides with the curve obtained under simple loading. The \(e_i - H\) curve, however, is somewhat shifted relative to the calibration graph.

To verify these conclusions at large deformations and along other loading paths, experiments were carried out on torsion of stretched specimens, compression after tension, and compression of specimens in a direction perpendicular to the preliminary compression. Annealed specimens of steel 3 were tested.

Torsion tests were carried out according to the method proposed by V. I. Maksakov [5]. According to this method, two cylindrical specimens of somewhat different diameters are twisted. The difference in torques at one and the same angle of twist is attributed to a hypothetical tube that supplements the smaller specimen to the specimen of larger diameter. The stress intensity is determined from the formula:

\[ \sigma_i=\frac{\sqrt{3}(M_1-M_2)} {2\pi\left(\frac{R_1+R_2}{2}\right)^2(R_1-R_2)}, \tag{2} \]

where \(M\) is the torque and \(R\) is the specimen radius. The subscript 1 refers to the specimen of larger diameter, and 2 to the smaller specimen.

Before twisting, specimens with diameters of 17 and 20 mm were stretched to true strains of 0.02 (pair of specimens No. 4), 0.06 (No. 5), 0.1 (No. 6), and 0.17 (No. 7). After unloading, at three points on the cylindrical surface the hardness HB was measured with a ball 5 mm in diameter under a load of 250 kg. The diameters of the impressions were measured on a toolmaker’s microscope with an accuracy of \(\pm 0.01\) mm. The hardness was determined as the mean of six tests on each pair of specimens.

Fig. 3. Torsion of stretched specimens. 1, 2—calibration curves (1—torsion, 2—tension before torsion; 4, 5, 6, 7—torsion after tension)

The twisting was carried out in steps. After each stage of deformation, hardness measurements were made. The results obtained are shown in Fig. 5. The strain intensity \(e_i\) was determined as the sum of the strain in tension and all the preceding stages of twisting. The same figure also gives the calibration curves constructed from the results of the torsion test (specimens No. 1) and tension test (No. 2).

The results of these experiments confirm the conclusions drawn earlier: the \(\sigma_i—H\) curves practically coincided, while the \(e_i—H\) curves under complex loading shifted somewhat to the left relative to the calibration graphs.

For compression testing after tension, four cylindrical specimens 20 mm in diameter were made and stretched. Their strains were 0.023 (specimen No. 2), 0.064 (No. 3), 0.12 (No. 4), and 0.14 (No. 5). From the stretched specimens, 5–6 cylinders were cut from each.

with a height equal to the diameter. The hardness HB was measured on their end faces, and from this, taking into account the stresses and strains in tension, a calibration graph was constructed (Fig. 4). Cylinders cut from one specimen were upset, with lubrication, to different strains. After this they were cut along a meridional section, and at 5–6 points on the resulting surface, sufficiently far from the end faces, the hardness was measured.

Fig. 4. Compression after tension. 1 — tension; 2, 3, 4, 5 — compression after tension

The corresponding stress and strain intensities were determined from the relations:

\[ \sigma_i = \frac{4P}{\pi D^2}, \qquad e_i = e_{ip} + \ln \frac{h_0}{h}; \tag{3} \]

where \(P\) is the compressive force; \(D\) is the diameter of the upset specimen; \(e_{ip}\) is the strain intensity in tension; \(h_0\) and \(h\) are the specimen heights before and after upsetting. The experimental results are given in Fig. 4.

To carry out the experiments of the last series, 20 specimens 24 mm in diameter and 36 mm high were made. They were sorted by hardness and divided into 4 groups. Specimens belonging to one group were compressed, with lubrication, to the same strain. The latter was, for the different groups, 0.02 (group No. 2), 0.09 (No. 3), 0.17 (No. 4), and 0.3 (No. 5). Then, in a direction perpendicular to the axis, cylinders with height and diameter equal to 10 mm were turned out. Specimens of one group were compressed to different strains. After upsetting, hardness was measured in two sections parallel to and sufficiently close to the meridional one. The strain intensity was determined as the sum of the absolute values of the strains in the first and second upsetting operations. The calibration graph shown in Fig. 5 was constructed from the results of compression, with lubrication, of 8 specimens with a height and diameter of 24 mm.

The results of the investigation performed indicate the possibility of studying the stress state under nonmonotonic loading of metal by the method of hardness measurement. However, if earlier it was assumed that the calibration graph could be extended to the entire plastic region, then in the case considered it is necessary to confine oneself to the region in which the strain intensity in the last stage of loading is greater than 0.02. This is not associated with a complication of the experiment. As in the study of monotonic processes, here it is necessary

it is necessary to distinguish the elastic and plastic regions. As a criterion for such a distinction, the condition of increasing hardness in the deformation process was proposed [2]. As follows from the results obtained, this same condition, applied to the last loading stage, makes it possible to exclude from consideration the region in which the increment of \(e_i\) in the last stage is less than 0.02.

Fig. 5. Compression of specimens cut in the transverse direction from upset cylinders. 1—calibration curve; 2, 3, 4, 5—compression in the transverse direction

Earlier it was assumed [2] that plastic deformation could be studied by measuring hardness only in those cases when unloading is elastic. This substantially limited the range of application of the method, since even in such a comparatively simple process as torsion of solid specimens in the region of large strains, unloading is partially plastic. The indicated assumption is connected with the fact that hardness during unloading should not change—this makes it possible, from the hardness of the unloaded body, to judge the hardness, and consequently also the stresses, during plastic deformation. The results obtained by us show that such a restriction is not necessary: the deformation during unloading may be plastic, but it must not exceed 0.02.

As for strain, only the “accumulated” intensity of strain can be determined from hardness. Moreover, the accuracy of such a determination under nonmonotonic loading is comparatively low.

REFERENCES

1. G. A. Smirnov-Alyaev. Resistance of Materials to Plastic Deformation. Mashgiz, 1961.

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

3. N. N. Zorew, G. D. Del, G. L. Kufarew, M. G. Goldschmidt. Spannungszustand in Der Schnittzone, Girp, Liege, Belgium, 1965.

4. G. B. Talyshov. Investigation of the Bauschinger effect. Izvestiya AN SSSR, Mekhanika i mashinostroenie, No. 6, 1954.

5. V. I. Maksak. Method for constructing a shear diagram at large strains. Abstracts of reports of the scientific-technical conference of young scientists and specialists. Mechanics Section, Novosibirsk, 1966.

6. P. Bridgman. Studies in Large Plastic Flow and Fracture. IIL, 1955.

7. G. D. Del. Relation between hardness and stresses in the plastic region. Izv. vuzov—Fizika, issue 3, 1966.