Relationship Between Hardness, Strain, and Stresses in the Region of Large Plastic Deformations

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

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

The method for investigating plastic deformations by measuring hardness \[1, 2\] is based on the assumption that, for different stress states and loading paths, there exists a single relationship between the stress intensity and the hardness of a deformed body. Previously this assumption was verified in the region of comparatively small plastic deformations by testing tubular specimens \[3\]. The purpose of the present investigation is to verify the assumption in the region of large plastic deformations.

The investigation was carried out according to the method developed by V. I. Maksak \[4\]. The essence of the method is as follows. Two cylindrical specimens of different diameters are loaded by an axial force \(P\) and a torque \(M\) in such a way that the ratio \(\frac{P}{M}\) is the same for the specimens. It is assumed that the difference between the specimen diameters is sufficiently small. From the test results, graphs are constructed of the change in the deforming force and torque as functions of the logarithmic shear at the surface \(q\). From these graphs, for different \(q\), the differences are determined:

\[ M = M_2 - M_1,\quad P = P_2 - P_1, \]

where

\(P_{1,2}, M_{1,2}\) are, respectively, the force and moment in the deformation of the smaller and larger specimens. From the results obtained, graphs \(M(q)\), \(P(q)\) are constructed for a fictitious tube that supplements the specimen of smaller diameter to the larger one. The stresses in the walls of this tube are determined by the formulas

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

where

\(R_b, R_m\) are, respectively, the radii of the larger and smaller specimens,

\[ R_0 = \frac{R_b + R_m}{2} \]

is the mean radius of the “tube.” From the change in length \(l\) and shear \(\gamma\), the longitudinal strain and logarithmic shear at the surface were determined.

\[ e_z=\ln \frac{l}{l_0}, \qquad q=\ln \left[1+\frac{\gamma}{2}\left(\gamma+\sqrt{4+\gamma^2}\right)\right]. \]

Stress and strain intensities

\[ \sigma_i=\sqrt{\sigma^2+3\tau^2}, \qquad e_i=\sqrt{e_z^2+\frac{q^2}{3}}. \]

The procedure is justified for the case of simple loading.

Six pairs of specimens made of two materials, St. 3 and steel X18H9T, were tested. A plane stress state was realized with a ratio of principal stresses \(n\) equal to 0 (tension), \(-0.8\), \(-0.6\), \(-0.4\), \(-0.2\), and \(-1\) (torsion). To maintain constancy of \(n\), the ratio \(\dfrac{P}{M}\) must

Fig. 1

Fig. 2

be determined at each loading stage of the specimens. The deformations at one loading stage for each pair were kept identical.

Hardness was measured after unloading with a ball of \(d = 5\ \mathrm{mm}\) under a load of \(500\ \mathrm{kg}\) at two points on the cylindrical surface. The diameters of the impressions were measured on a measuring microscope. In Figs. 1 and 2, the arithmetic mean of four measurements on each pair of specimens is compared with the intensity of stresses and strains. The deviations of individual points from the calibration curve constructed from the results of tensile and torsion tests lie within the limits of the experimental accuracy. Taking into account the previously obtained results of a study of the relationship between hardness and stresses under nonmonotonic loading [5], it may be considered, with accuracy sufficient for engineering investigations, that the relationship between stress intensity and hardness is the same for different stress states and in the region of large plastic deformations.

REFERENCES

1. G. A. Smirnov-Alyaev. Resistance of Materials to Plastic Deformation. Mashgiz, Moscow, 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 (in press).

4. V. I. Maksak. A method for studying large plastic deformations. Abstracts of Reports of the IV All-Union Conference on Strength and Plasticity.

5. G. D. Del, V. A. Ogorodnikov. Variation of the hardness of metals under nonmonotonic loading. Izv. TPI (in press).