Relationship Between Hardness and Stress of a Deformable Body

Yu. S. Bogomolov, G. D. Del, L. M. Sedokov

(Presented by the scientific seminar of the Department of Strength of Materials)

In the practice of studying plastic deformation, the method of measuring hardness is widely used [1, 2, 3]. Knowing the hardness at various points of a deformed body and having a graph relating hardness to stresses and strains, one can obtain an idea of the stress–strain state of the body under investigation.

To obtain calibration graphs of “hardness—stress” and “hardness—strain,” it is necessary to measure hardness at different stages of deformation of a body whose stress–strain state is known. The stress–strain state can be characterized sufficiently reliably in the simplest cases: 1) tension, 2) compression, and 3) torsion.

The procedure for constructing diagrams of the dependence of the hardness of a deformable body on the magnitudes of stresses and strains in each of the cases listed has its own special features and difficulties.

When obtaining calibration graphs from the results of tensile testing, a number of specimens of the material under investigation are stretched to different degrees of deformation (some are brought to fracture). Flats are made on the deformed specimens, on which hardness is measured at various points along the specimen. The magnitude of the strain intensity corresponding to this hardness is determined from the diameter \(d\) of the transverse section passing through the point at which the hardness was measured,

\[ e_i = 2 \ln \frac{d_0}{d}, \tag{1} \]

where \(d_0\) is the initial diameter of the specimen.

The stress intensity is determined from the deforming force \(P\) for the given transverse section

\[ \sigma_i = \frac{4P}{\pi d^2}. \tag{2} \]

In parallel, hardness may be measured in the transverse sections of the specimen.

The dimensions of the specimens and the surface finish are determined by the method of hardness measurement: micro- or macro-hardness.

The described procedure gives sufficiently satisfactory results for small deformations. With the appearance of a neck, the stressed-hasp?

the state is no longer linear, which must be taken into account when determining the stress intensity [4], and it substantially complicates the method of constructing a calibration graph from tensile-test results.

However, when measuring hardness in the neck of a specimen, it is impossible to obtain, from tensile-test results, calibration graphs for large strains, which is the principal drawback of this method.

When obtaining the calibration dependence “hardness—stress” and “hardness—strain” from the results of axial-compression tests, a number of difficulties arise that are associated with the presence of friction at the ends. Friction between the ends of the specimen and the press platens leads to an increase in the deforming force and to nonuniformity of the deformed state over the volume of the specimen, the external manifestation of which is barreling.

There are various methods for reducing the influence of friction on the test results. The most effective way is to reduce friction by lubricating the ends of the compressed specimen. A paste of glycerin and colloidal graphite, lead gaskets, etc., are used as lubricants.

The results of a compression test are significantly affected by the ratio of the specimen height to its diameter. On the basis of an analysis of compression curves obtained by M. A. Bol’shanina [5], this ratio is recommended to be taken as about 2. The use of specimens with an even larger height-to-diameter ratio leads to loss of stability of the specimen during its plastic deformation.

In the course of plastic deformation the indicated ratio decreases. Therefore, specimens are subjected to periodic turning of the outside diameter. In this way it is possible to construct a calibration graph in the region of very large strains (\(e_i\) amounts to several units). This is the main advantage of the method described.

In the case of compression without lubrication, the stress state within the volume of the specimens being tested will be nonuniform. This considerably complicates the testing procedure. A series of specimens is made from the material under investigation, each of which is compressed to a specified value of strain and then cut perpendicular to the axis at the midsection. Hardness is measured at various points of the section. The stress intensity is determined from the formula

\[ \sigma_i = \frac{P}{F_0}(1-\varepsilon_{\text{comp}}), \tag{3} \]

where
\(P\) is the load, in kg;
\(F_0\) is the area of the initial section, in mm\(^2\);

\[ \varepsilon_{\text{comp}}=\frac{\Delta h}{h} \]

is the relative strain in compression.

The strain intensity is

\[ e_i=\ln\frac{h_0}{h}. \tag{4} \]

Here \(h_0\) is the initial height of the specimen before deformation, in mm;
\(h\) is the current height of the specimen, in mm.

A calibration graph can also be constructed from torsion-test results. A cylindrical specimen of radius \(R\) is twisted to a specified value of the unit angle of twist \(\Theta\).

Therefore the shear in the outer layer of the specimen is determined by \(\gamma = \theta R\), and from the latter the logarithmic shear [6]

\[ g=\ln\left[1+\frac{\gamma}{2}\left(\gamma+\sqrt{4+\gamma^2}\right)\right]. \tag{5} \]

The strain intensity is \(e_i=\dfrac{1}{\sqrt{3}}\,g\).

The tangential stress is determined from the torque \(M\) from the relation [5]

\[ \tau=\frac{3}{2\pi R^3}\left(M+\frac{1}{3}\gamma\frac{dM}{d\gamma}\right). \tag{6} \]

The stress intensity is \(\sigma_i=\sqrt{3}\,\tau\).

At several points (10–15) along the outer diameter, the hardness is measured.

Figure 1 gives calibration curves for a number of materials, constructed from the results of torsion tests. As is seen from the graphs, the scatter of the experimental points is quite small.

The greatest strain in this case was limited by fracture: for steel Kh13 and ShKh15 it is equal to 0.6, and for steel Kh18N9T it is about 1.0.

Figure 1 gives the paired graphs [2]. If the condition is adopted that the relation between stress intensity and strain intensity is independent of the stress state of the deformed material, then the calibration graphs obtained will be common to all cases of deformation.

Fig. 1. Paired calibration graphs:
1 — Kh18N9T, 2 — ShKh15, 3 — Kh13

For comparison of different methods of obtaining the “hardness—stress” relationship, Fig. 2 is presented, where the lower curve was constructed from the results of torsion tests, and the upper one from the results of axial compression tests without lubrication, with hardness measured on the end faces of the specimen.

As is seen from the graphs (Fig. 2), the initial portions of the curves (up to a hardness of \(HRD=40\)) differ only slightly from one another. The values of stress intensity from the calibration graph obtained in compression are greater than the corresponding values of stress intensity from the calibration graph constructed in torsion by 15%. At a larger value of the hardness this difference increases substantially and at \(HRD=54\) amounts to about 35%.

It is also necessary to note the considerable scatter of the results in constructing the calibration graph “hardness—stress intensity” under compression (Fig. 2), which lies within the range of \(\pm 15\text{--}20\%\).

Fig. 2. Comparison of calibration graphs based on the results of axial compression and torsion tests

REFERENCES

1. P. Bridgman. Studies in Large Plastic Deformations and Fracture. Foreign Literature Publishing House, 1955.

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

3. A. M. Rozenberg and L. A. Khvorostukhin. Hardness and stress in a plastically deformed body. Journal of Technical Physics, vol. XXV, 1955, no. 2.

4. N. N. Davidenkov, N. I. Spiridonova. Analysis of the stressed state in the neck of a tensile specimen. Zavodskaya Laboratoriya, no. 6, 1945.

5. V. D. Kuznetsov. Physics of the Solid State. Vol. 2, Tomsk, 1941.

6. A. Nadai. Plasticity and Fracture of Solids. Foreign Literature Publishing House, 1954.