STUDY OF THE RELATIONSHIP BETWEEN HARDNESS, STRESSES, AND DEFORMATION UNDER BIAXIAL STRESS STATE

G. D. Del, K. N. Tsukublina, I. N. Temnik

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

The method for studying plastic deformations by measuring hardness [1, 2] is based on the assumption that, for different types of stress states and loading paths, there exists a single relationship between the stress intensity and the hardness of a plastically deformed metal. Work [3] is devoted to the experimental substantiation of the possibility of determining the intensity of stresses and deformations in the general case of plastic deformation from hardness, using a calibration curve constructed from the results of testing the material in tension and compression. However, in that work the results were obtained at comparatively small plastic deformations.

The present article sets forth the results of a study of the relationship between stress intensity, hardness, and strain intensity in the region of large plastic deformations. Testing of specimens was carried out according to the procedure proposed in [4], [5]. The essence of the procedure is as follows. Two solid cylindrical specimens of different diameters are loaded by an axial force and a torque. It is assumed that the difference in diameters is sufficiently small, and that the force and moment for both specimens vary according to the same law. From the test results, graphs are constructed of the dependence of torque and axial force on shear at the surface or on elongation.

Then, by subtracting from the torque and axial force applied to the specimen of larger diameter the corresponding loads that acted on the smaller specimen at the moment when the deformations on the surfaces of the rods were identical, one determines \(M'\) and \(P'\) for a fictitious tube supplementing the specimen of smaller diameter to the larger specimen. The stresses on the surfaces of the rods are then regarded as identical and are determined from \(M'\) and \(P'\) in the same way as in testing tubes:

\[ \sigma_z=\frac{P'}{\pi\left(R_b^2-R_m^2\right)},\quad \tau=\frac{M'}{2\pi\left(R_b-R_m\right)R_0^2}, \]

where \(R_b\) and \(R_m\) are, respectively, the radii of the larger and smaller specimens, \(R_0=\dfrac{R_b+R_m}{2}\) is the mean radius of the tube. The longitudinal strain,

the logarithmic shear and the shear angle on the surface are determined, respectively, by

\[ e_z=\ln \frac{l}{l_0}, \]

\[ q=\ln \left[1+\frac{\gamma}{2}\left(\gamma+\sqrt{4+\gamma^2}\right)\right], \]

\[ \gamma=\frac{4\pi}{180l}\cdot R_0\sqrt{\frac{l_0}{l}}, \]

where \(l_0\) is the initial length, \(L\) is the length of the working part of the specimen. The stress intensity was determined by the formula

\[ \sigma_i=\sqrt{\sigma^2+3\tau^2}. \]

The strain intensity

\[ e_i=\sqrt{e_z^2+\frac{q^2}{3}}. \]

The procedure is justified for the case of simple loading; here it was assumed that the transverse sections of the deformed specimens remain plane and the radii remain straight.

Six pairs of specimens from four materials were tested (Kh18N9T steel, St. 3 steel, copper, L62 brass). Blanks for specimens of the same material were cut from a single bar and, after machining, were subjected to recrystallization annealing. Subsequent sorting of the specimens was carried out by hardness.

Before testing, a scribed line was made along a generatrix, and two annular notches were made to fix the initial length. The hardness \(H_B\) in the initial state, as well as after each loading stage, was measured at two points of the cylindrical surface with a ball 5 mm in diameter, under a load of 500 kg in testing steels and 250 kg in testing copper and brass. The diameters of the impressions were measured on a toolmaker’s microscope.

The test was carried out on a ZDMU-30 machine by loading with a tensile force and a torque. At each deformation stage, after unloading, the current dimensions \(l\), \(d\), and the hardness were measured.

A plane stress state was realized with the ratio of the principal stresses

\[ n=\frac{\sigma_2}{\sigma_1}, \]

equal to \(n=0\) (tension), \(n=-0.2\); \(n=-0.4\); \(n=-0.6\); \(n=-0.8\), and \(n=-1\) (torsion).

The experimental results are presented in Figs. 1, 2, 3, 4. The deviations of the points corresponding to the biaxial stress state from the calibration curve do not exceed \(\pm 15\%\) for graphs relating hardness to stress intensity, and \(\pm 20\%\) for curves relating hardness to strain intensity, constructed from the results of testing steels. For nonferrous metals the deviations of the experimental points corresponding to the biaxial stress state from the calibration curve are somewhat larger. However, the scatter is random in character and is evidently explained to a considerable extent by the inaccuracy in determining the stresses.

Thus, the results obtained show that the relationship between indentation hardness, stress intensity, and strain is practically independent of the type of stress state.

Fig. 1. Relationship between hardness, stresses, and deformation under conditions of a biaxial stress state for stainless steel X18H9T.

Fig. 2. Relationship between hardness, stresses, and deformation under conditions of a biaxial stress state for steel 3.

Fig. 3. Relationship between hardness, stresses, and deformation under conditions of a biaxial stress state for copper

Fig. 4. Relationship between hardness, stresses, and deformation under conditions of a biaxial stress state for brass

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. G. D. Del’. Hardness of a deformed metal. Bulletin of the USSR Academy of Sciences, “Metals,” No. 4, 1967.

4. V. I. Maksak. A method for studying large plastic deformations. Abstracts of papers of the IV All-Union Conference on Strength and Plasticity, Nauka Publishing House, Moscow, 1967.

5. V. I. Maksak, G. A. Doshinskii. Methodology and study of large plastic deformations under simple loading. Bulletin of the Tomsk Polytechnic Institute, vol. 173, 1970.