Comparison of Some Strength Theories for Brittle Materials Under Plane Stress
Abstract Generated abstract
This paper compares several strength criteria for brittle materials under mixed plane stress, focusing on the effect of unequal tensile and compressive strength limits. Experimental data for 19 materials are approximated by a quadratic strength contour, with parameters fitted by least squares, and then compared with predictions from the maximum normal stress theory, Mohr, Balandin, Mirolyubov, Fischer, Lebedev, and a TPI criterion. Reliability is evaluated mainly by the maximum deviation between theoretical and approximated actual strength contours for equal principal stress ratios. The comparison indicates that the TPI and Lebedev criteria give the closest overall agreement, while the maximum normal stress and Mohr theories show the largest deviations, and Balandin’s criterion becomes less reliable for very brittle materials.
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COMPARISON OF SOME STRENGTH THEORIES FOR BRITTLE MATERIALS UNDER PLANE STRESS
V. N. Rakityanskii, L. M. Sedokov
(Presented by the scientific seminar of the Department of Strength of Materials)
Strength calculations for brittle materials under a complex stress state are of an approximate nature. Classical theories, which assess strength by the principal parameters of material resistance (normal stresses, linear strains, shear stresses, strain energy), give a large discrepancy with experimental results for brittle materials. The reason for this is the condition of equality of the tensile and compressive strength limits, whereas real brittle materials have substantially different values of the indicated strength characteristics. This circumstance led to the development of new strength theories, the main feature of which is allowance for the difference between the resistance of brittle materials to tension $\sigma_{\mathrm{p}}$ and to compression $\sigma_{\mathrm{c}}$. For such materials the brittleness coefficient is introduced
\[ \gamma = \frac{\sigma_{\mathrm{p}}}{\sigma_{\mathrm{c}}}, \tag{1} \]
which varies from 0 to 1.0. The new strength theories have been experimentally checked on a narrow set of investigated materials: cast irons [1], concretes [2], thermosetting plastics [3]. An assessment of reliability for a number of materials with different physicomechanical properties was made in [4].
Below a comparison is given of a number of strength theories for 19 materials (Table 1), the experimental results for which were obtained by different authors. In the region of biaxial tension, as shown in [4], the actual strength of brittle materials agrees satisfactorily with the first theory, as well as with the Fisher and Mohr criteria. The contours of these theories in this class of stress state coincide.
In the class of mixed plane stress state, the contours of limiting strength constructed according to different strength theories have a large discrepancy among themselves. Therefore, comparison of the theoretical calculation with actual values is of scientific interest.
The experimental results for the materials listed in Table 1 were approximated by equations of the form
\[ \sigma_1^2 + \gamma^2 \sigma_3^2 - C \gamma \sigma_1 \sigma_3 = \sigma_{\mathrm{p}}^2, \tag{2} \]
| No. | Material | $\gamma$ | $C$ | I theory | \multicolumn{6}{c}{ $\Delta_{\max}$ according to various strength theories, % } |
|---:|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| | | | | | Mohr | Mirolubov | Balandin | Fischer | Lebedev | TPI |
| 1 | Hardened steel U8A | 0,94 | 0,46 | 53 | 22 | 9,5 | 9,5 | 9,5 | 8,5 | 4,7 |
| 2 | Titanium alloy VT-14 | 1,0 | 0,70 | 64 | 17 | 5,1 | 5,1 | 5,1 | 5,1 | 9,6 |
| 3 | Titanium alloy VT-6S | 1,0 | 1,1 | 74 | 12 | 1,5 | 1,5 | 1,5 | 1,5 | 7,0 |
| 4 | Malleable cast iron | 0,65 | 0,17 | 45 | 27 | 16 | 14 | 15 | 9,7 | 1,8 |
| 5 | Malleable cast iron | 0,8 | 0,46 | 45 | 22 | 10 | 9,3 | 9,5 | 6,3 | 4,6 |
| 6 | Gray cast iron | 0,27 | 0,23 | 45 | 28 | 18 | 10 | 15 | 1,0 | 0,5 |
| 7 | Gray cast iron | 0,3 | 0,11 | 42 | 30 | 19 | 80 | 18 | 4,0 | 3,0 |
| 8 | Gray cast iron | 0,315 | 0,7 | 60 | 19 | 7,4 | 13 | 4,0 | 10 | 11 |
| 9 | Modified cast iron | 0,41 | 0,33 | 51 | 25 | 14 | 7,0 | 13 | 1,8 | 1,8 |
| 10 | Gray cast iron | 0,36 | 0,05 | 39 | 31 | 21 | 12 | 19 | 7,3 | 4,9 |
| 11 | Plastic K-18-2 | 0,24 | 0,39 | 50 | 25 | 14 | 15 | 12 | 3,6 | 3,3 |
| 12 | EZA-120 | 0,48 | 0,36 | 50 | 25 | 13 | 8,0 | 12 | 2,6 | 2,5 |
| 13 | Hardened steel U7 | 0,94 | 0,51 | 53 | 21 | 8,7 | 8,5 | 8,7 | 7,7 | 5,6 |
| 14 | High-silicon cast iron | 0,15 | 0,34 | 51 | 27 | 16 | 3,1 | 13 | 4,1 | 2,3 |
| 15 | Steel C-100 | 0,11 | 0,5 | 22 | 43 | 31 | 27 | 32 | 17 | 20 |
| 16 | Gypsum | 0,13 | 0,45 | 56 | 23 | 13 | 37 | 9,1 | 0,5 | 6,5 |
| 17 | Glass | −0,05 | −0,3 | 64 | 80 | 62 | 117 | 56 | 20 | 21,5 |
| 18 | Concrete | 0,11 | 0,4 | 56 | 23 | 13 | 41 | 9,6 | 0,2 | 5,6 |
| 19 | Plastic K-18-36 | 0,23 | 0,1 | 24 | 12 | 11 | 7 | 11 | 2,0 | 2,0 |
where \(\sigma_1, \sigma_3\) are coordinates along the axes of principal stresses. The value of the variable parameter was determined by the least-squares method. The values of the coefficient \(C\) entering into equation (2) for each material, at the minimum value of the root-mean-square deviation \(\Delta^{\mathrm{kv}}_{\mathrm{sr}}\), are given in the table.
The obtained curves approximating the experimental results were compared with the contours of the calculation equations according to the theory of greatest normal stresses, O. Mohr, P. P. Balandin, I. N. Mirolyubov, Fischer, A. A. Lebedev, and the criterion developed at the Department of Strength of Materials of TPI, based on the theory of maximum linear strains [5].
As the parameter for evaluating the criteria, the maximum deviation was adopted of the current radius of the actual strength contour \(\rho_\phi\) (of the approximation), which is the geometric sum of the principal stresses at the moment of fracture, and the radius according to the corresponding strength theory \(\rho_{\mathrm{t}}\), for the same ratio of principal stresses \(n=-\dfrac{\sigma_3}{\sigma_1}\)
\[ \Delta_{\max}=\frac{|\Delta \rho|_{\max}}{\rho_\phi} =\frac{(\rho_\phi-\rho_{\mathrm{t}})\,100\%}{\rho_\phi}, \tag{3} \]
\(\Delta_{\max}\) is the reliability of the theory in terms of maximum deviation, in percent. The results, summarized in Table 1, were obtained on a BESM-4 computer. The mean values of reliability for each strength theory for all materials are given in Table 2.
Table 2
| Theory | Calculation equation | \(\Delta^{\mathrm{sr}}_{\max}\), % |
|---|---|---|
| First theory | \(\sigma_1=\sigma_{\mathrm{p}};\quad \nu\sigma_3=-\sigma_{\mathrm{p}}\) | 50 |
| Mohr | \(\sigma_1-\nu\sigma_3=\sigma_{\mathrm{p}}\) | 27 |
| Balandin | \(\dfrac{1-\nu}{2}(\sigma_1-\sigma_3)+\dfrac{1}{2}\sqrt{(1-\nu)^2(\sigma_1+\sigma_3)^2+4\nu(\sigma_1^2+\sigma_3^2-\sigma_1\sigma_3)}=\sigma_{\mathrm{p}}\) | 20 |
| Mirolyubov | \(\dfrac{1-\nu}{2}(\sigma_1+\sigma_3)+\dfrac{1+\nu}{2}\sqrt{\sigma_1^2+\sigma_3^2-\sigma_1\sigma_3}=\sigma_{\mathrm{p}}\) | 16.7 |
| Fischer | \(\sigma_1^2+\nu^2\sigma_3^2-\nu\sigma_1\sigma_3=\sigma_{\mathrm{p}}^2\) | 14.4 |
| Lebedev | \(\sigma_1^2+\nu^2\sigma_3^2-\nu^2\sigma_1\sigma_3=\sigma_{\mathrm{p}}^2\) | 6.4 |
| TPI | \(\sigma_1^2+\nu^2\sigma_3^2-0.25\,\nu\sigma_1\sigma_3=\sigma_{\mathrm{p}}^2\) | 6.2 |
As a result of comparing the data in the tables, the following conclusions may be drawn:
- The reliability of the application of mechanical strength theories can be assessed in two ways:
a) by the root-mean-square deviation of the theoretical calculation from the approximated actual results;
b) by the maximum deviation from the actual results (3).
-
Reliability estimates of strength theories by these two parameters are fundamentally consistent [4].
-
The reliability estimate of strength theories by the maximum deviation is more sensitive to strength criteria.
-
The reliability estimate of the strength theories considered (except for P. P. Balandin’s) by the maximum deviation (3) proved to be relatively insensitive to the materials studied.
-
P. P. Balandin’s theory gives satisfactory agreement for materials with coefficient $\nu$ from 1 to 0.4. For very brittle materials ($\nu < 0.4$), the maximum deviation increases severalfold.
-
The best approximation to the experimental values for all the materials studied in the class of mixed plane stress state is given by two criteria—the TPI and A. A. Lebedev’s. The maximum deviation (3) for these strength criteria is 6.2% and 6.4%, respectively.
-
The greatest maximum deviation from the actual strength was given by two theories: the theory of greatest normal stresses (50%) and O. Mohr’s theory (27%).
REFERENCES
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J. Fischer, J. Holloman. Metals Technology. 1947, 14, No. 5.
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G. A. Geniev, K. N. Kissyuk. On the question of generalizing the strength theories of concrete. “Beton i zhelezobeton,” 1965, No. 2.
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V. M. Shinkov, K. N. Kan. Investigation of the statistical strength of rigid plastics under plane stress. “Plasticheskie massy,” 1964, No. 1.
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E. M. Mikhailovskii, L. M. Sedokov. Results of a study of the reliability of strength theories for brittle materials using a computer (in a forthcoming collection).
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L. M. Sedokov. Strength criteria for brittle materials under plane stress. “Vestnik mashinostroeniya,” 1972, No. 5.