Strength of Glass Under a Plane Stress State
Abstract Generated abstract
This paper reports experimental measurements of the strength of S90-1 glass under plane stress, addressing the limited prior data for mixed stress states near uniaxial compression. Thin-walled tubular specimens were loaded by combined internal hydrostatic pressure and axial tension or compression, with preliminary uniaxial tests used to establish tensile isotropy, tensile to compressive resistance differences, and elastic constants. The resulting strength data for biaxial tension and mixed stress states are compared with several brittle-material strength criteria. The authors find that maximum normal stress and simplified criteria agree well in parts of the stress plane, but the generalized maximum elongation theory gives the best overall correspondence with the experiments.
Full Text
STRENGTH OF GLASS UNDER PLANE STRESS
D. E. Vyazun, E. M. Mikhailovskii
(Presented by the scientific seminar of the Department of Strength of Materials)
Glass is the classical brittle material on which physical as well as phenomenological theories of strength have been created, substantiated, and verified experimentally. Whereas by now there exists a sufficient number of experiments revealing the influence of various factors on the strength of glass under a linear stress state, only one work [1] has been devoted to its strength under a plane stress state. However, the experiment described in that work was not carried through to completion in the region of mixed stress states, where, because of the insufficient power of the setup, a large number of ratios of the principal stresses close to uniaxial compression were not realized. In connection with this, it became necessary to carry out a detailed and complete experiment that would fill the existing gap and make it possible to analyze the agreement between experiment and the theories proposed for brittle materials.
For the tests, tubular specimens made of S90-1 glass were used, 120 mm long and with an outside diameter of 19.2 mm, with a wall thickness of 1.2 mm.
The plane stress state was produced by loading the tubes simultaneously with internal hydrostatic pressure and an axial tensile or compressive load on a universal TsDMU-30 machine.
In tests for uniaxial and biaxial tension, a special separable fixture was used, which made it possible to avoid destruction of the specimen when it was installed in the grips and which assisted in centering the axial load. Only careful observance of the procedure for preparing specimens for testing [1] and attentive work by two operators at the machine made it possible to obtain sufficiently reliable results. First, results were obtained for the strength of glass under uniaxial stress states: axial tension, axial compression, and circumferential tension. The results obtained were processed statistically.
The tests carried out under a linear stress state made it possible:
1) to establish the isotropy of the glass tested under circumferential and axial tension;
2) to determine the coefficient taking into account the difference in the resistance of glass to uniaxial tension and compression;
Table 1
Results of testing C90-1 glass under plane stress
| $\sigma_1$, kg/mm² | 0 | 5.05 | 0 | 4.92 | 3.55 | 2.63 | 1.23 | 5.06 | 5.09 | 5.02 | 5.18 | 5.00 | 4.61 | 3.85 | 2.76 |
| $\sigma_2$, kg/mm² | 5.17 | 0 | −44.2 | 4.76 | 5.29 | 5.29 | 5.03 | −5.07 | −9.97 | −15.63 | −20.02 | −29.72 | −37.12 | −38.43 | 40.70 |
| $n$ | 32 | 10 | 16 | 10 | 11 | 10 | 10 | 7 | 8 | 7 | 9 | 8 | 9 | 9 | 8 |
| $\xi$ | 2.5 | 4.3 | 6 | 3 | 8 | 9 | 6 | 5 | 4 | 5 | 5 | 3.5 | 3 | 4 | 3 |
3) to compile a program of stress states covering the entire region of the mixed stress state.
In addition, using wire resistance strain gauges, diagrams of tension and compression of glass were obtained, as well as the moduli of normal elasticity and Poisson’s ratios under the indicated tests. After this, a plane stress state was realized at various ratios of the principal stresses.
Table I gives the results of testing glass, where
$n$ is the number of specimens tested at each ratio of the principal stresses;
$\xi$ is the accuracy coefficient for determining the arithmetic mean value of strength [3];
$\sigma_1$ and $\sigma_2$ are the arithmetic mean values of the principal stresses, calculated from the known formulas for thin-walled tubes [4].
Fig. 1. Strength contours for glass according to various theories: 1—maximum normal stresses; 2—simplified Mohr criterion; 3—generalized maximum strains; 4—according to Mirolubov; 5—according to Balandin; o—experiment.
Figure 1 shows theoretical contours constructed according to various strength theories: 1—according to the theory of maximum normal stresses; 2—according to the simplified Mohr criterion; 3—according to the theory of maximum strains generalized to brittle materials [2]; 4—according to Mirolubov’s theory; and 5—according to Balandin’s criterion. The points indicate the arithmetic mean values of the principal stresses.
Analyzing the constructed contours, it is necessary to note:
1) in the region of biaxial tension, the best agreement with experiment is given by the first and second theories and by Mohr’s criterion; Mirolubov’s and Balandin’s theories underestimate the calculated values by 40–50%;
2) in the region of the mixed stress state, the first and second theories also give good agreement; Mohr’s and Mirolubov’s criteria underestimate, while Balandin’s criterion substantially overestimates, the calculated values;
3) for ratios of the principal stresses close to uniaxial compression, a discrepancy is observed for both the first and the second strength theories, with the first strength theory giving the larger error.
On the basis of the analysis carried out for the indicated quadrants of the plane stress state of glass, it may be considered that the best agreement with experiment is given by the theory of maximum elongations generalized to brittle materials.
REFERENCES
-
N. N. Davidenkov, A. N. Stavrogin. On a strength criterion under brittle fracture and a plane stress state. Izv. AN SSSR, OTN, No. 8, 1954.
-
L. M. Sedokov. Limits of applicability of basic strength theories under the plane stress state of brittle materials. Vestnik mashinostroeniya, No. 10, 1969.
-
A. M. Dlin. Mathematical Statistics in Engineering. Sovetskaya nauka, Moscow, 1958.
-
S. D. Ponomarev et al. Strength Calculations in Mechanical Engineering. Mashgiz, vol. 1, Moscow, 1956.