Shear Resistance of Brittle and Hardening Materials
Abstract Generated abstract
This paper examines the choice of strength criteria for predicting shear resistance under complex stress states, especially where classical theories give divergent results. It presents previously derived relations for brittle and strain-hardening materials that incorporate additional mechanical characteristics, the tension to compression strength ratio for brittle materials and a hardening exponent from tensile curves for plastic materials. The author compares these formulas with experimental data for glass, gypsum, cast iron, hardened steels, metals, alloys, and cutting tests interpreted as shear resistance. The comparison suggests that the proposed dependences describe observed ratios of shear to tensile resistance over a broad range and help indicate the applicability limits of classical strength theories.
Full Text
Proceedings of the Tomsk Polytechnic Institute Named after S. M. Kirov, Holder of the Order of the October Revolution and the Order of the Red Banner of Labor
Vol. 188 1974
Shear Resistance of Brittle and Hardening Materials
L. M. Sedokov
(Presented by the scientific seminar of the Department of Strength of Materials)
Strength calculations under a complex state of stress are inevitably associated with the choice of a definite strength hypothesis. The available experiments do not give uniform results [1–4], which makes the choice of a strength criterion difficult.
It is usually considered that the first and second theories of strength should be applied to brittle materials, and the third and fourth to plastic materials. But such a recommendation follows from certain experiments and is not connected with the principles that underlie the derivation of the calculation formulas of the corresponding theories of strength. Under biaxial tension and under biaxial compression, for example, the first and third theories of strength give completely identical results. The division of materials into brittle and plastic is conditional; the degrees of “brittleness” and “plasticity” in different materials may differ substantially. All this leads to uncertainty in engineering calculations.
Using relations that follow from the conditions of equality of maximum elongations, equality of the specific works of plastic deformation per unit shear, and constancy of the largest component of the stress deviator, as well as additional characteristics of the mechanical properties of materials, new calculation dependences for determining shear resistance, \(\tau_{\mathrm{sd}}\), were obtained earlier [5, 6]. Experimental verification of these dependences is of considerable interest, since different theories of strength give the greatest discrepancies in shear.
For brittle materials it was obtained [6]
\[ \tau_{\mathrm{sd}}=\frac{2}{2+\nu^{2}}\,\sigma_{\mathrm{vr}}, \tag{1} \]
where
\[ \sigma_{\mathrm{vr}} \]
is the strength limit of the material under uniaxial tension,
\[ \nu=\sigma_{\mathrm{vr}}:\lvert\sigma_{\mathrm{zh}}\rvert, \tag{2} \]
\[ \sigma_{\mathrm{zh}} \]
is the strength limit of the material under uniaxial compression.
For plastic materials [5]
\[ \tau_{\mathrm{sh}}=\frac{2}{3(1+m)}\,\sigma_{\mathrm{br}}, \tag{3} \]
where
\(m\) is the hardening characteristic of the material, which is determined from the tensile curve when it is approximated by a power law,
\[ \sigma=\sigma_0\delta^m. \tag{4} \]
On the basis of (1) and (3), for an ideally plastic material \((v=1;\ m=0)\)
\[ \tau_{\mathrm{sh}}=\frac{2}{3}\,\sigma_{\mathrm{br}}. \tag{5} \]
Fig. 1. Calculated and actual shear resistance. Calculation according to strength theories: I — maximum normal stresses, II — maximum elongations \((=0.5)\), III — maximum tangential stresses, IV — octahedral stresses (energy theory), IIa — proposed, IIIa — Mohr, IVa — Balandin. Experimental results: □ — glass, ■ — gypsum, △ — gray cast iron, ▲ — hardened steels, ● — steels, copper, brass, aluminum (torsion of solid round rods), ○ — steels (cutting), and × — steels, copper, aluminum alloys (torsion of thin-walled tubes).
The proposed graph plots calculated values according to the four basic strength theories (horizontal dashed lines, since these theories do not contain additional characteristics of the mechanical properties of materials—\(v\) and \(m\)), according to the theories of O. Mohr—IIIa and II, and of Balandin—IVa.
The same graph also plots, as individual points, the experimental results of N. N. Davidenkov and A. N. Stavrogin (glass and gypsum), Z. M. Konyushko (hardened tool steels), Cornet and Grassi (gray cast iron), E. Davis (steel and copper), S. L. Zhukov, Ya. B. Fridman (cast aluminum alloys), and N. N. Davidenkov and N. I. Spiridonova (Armco iron) [2, 3, 7–10].
The ratio between the torsional resistance of solid round rods and the tensile resistance of steels, copper, brass, and aluminum was taken as an average—
...evaluated from the corresponding flow curves obtained by G. A. Smirnov-Alyaev [11]. Treating cutting resistance as shear resistance, one can compute the actual value of $\tau_{\mathrm{sh}}:\sigma_{\mathrm{vr}}$. It is shown by individual points in the graph according to the results of experiments by A. M. Rosenberg and A. N. Eremin [12].
As can be seen from the comparison presented, the actual values of $\tau_{\mathrm{sh}}:\sigma_{\mathrm{vr}}$ are satisfactorily described by equations (1) and (3), i.e., by curve IIa over the entire range of the actual variation of this ratio. Curve IIa intersects the straight lines of the classical strength theories, which makes it possible to establish the limits of applicability of each of them.
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