Application of the Theory of Strength Based on Maximum Elongations for Brittle Materials

L. M. Sedokov

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

An engineering strength calculation must provide a quantitative assessment of the danger of failure or of the onset of another type of limiting state. The initial data for such an assessment are the known components of the stress state and the mechanical properties of the material. This assessment is carried out on the basis of the chosen strength criterion. However, no strength criterion can reflect the entire variety of factors that in one way or another affect the state of a material [1].

At the same time, the selection of an appropriate theory of strength and the development of methods for applying them to various materials under different conditions continue to be important questions in engineering calculations, the significance of which is increased by the desire to reduce actual strength margins and by the creation of modern high-strength materials.

It is usually considered that, for brittle materials, the first and second theories of strength should be used. Meanwhile, the calculation formulas according to the second theory of strength (the theory of maximum elongations) do not reflect the principal feature of brittle materials: their different resistance to tension and compression.

As is known, this feature of brittle materials is taken into account by the theories of O. Mohr and Balandin. The latter may be regarded as generalizations, respectively, of the theories of maximum shear stresses and of the energy theory, recommended for strength calculations of plastic materials [1].

In this connection there arises the need to develop such calculation formulas, based on the theory of maximum elongations, that would take into account the different strength of brittle materials in tension and compression.

In Fig. 1, in dimensionless coordinates, the strength contour is shown according to the first (square) and according to the second (rhombus) theories of strength for materials that resist tension and compression equally [2, 3].

The strength contour according to the second theory of strength (the rhombus) is constructed without taking into account the fact that, when two principal stresses are approximately equal numerically and identical in sign, the greatest elongation in absolute value will occur in the direction of the third principal stress, even if the latter is equal to zero.

The strength contour according to the second theory of strength for a plane stress state is completely determined by six equations (the coefficient of transverse deformation is taken to be 0.5):

\[ 2\sigma_1-\sigma_2=\pm 2\sigma_s, \]
\[ 2\sigma_2-\sigma_1=\pm 2\sigma_s, \]
\[ \sigma_1+\sigma_2=\pm 2\sigma_s, \]

which corresponds to the hexagon \(B_1-B_6\). The two contours constructed in Fig. 1 have 6 common points: 4 on the coordinate axes and 2 at equal principal stresses [4].

Fig. 1. Strength contours of materials equally resistant to tension and compression according to the theory of maximum normal stresses (square) and according to the theory of maximum elongations

In Fig. 2 a strength contour is constructed for materials having different resistance to tension and compression, according to the theory of maximum normal stresses (dashed line) [2] and according to the theory of maximum elongations with restrictions in the region of like-signed principal stresses [4] and transition lines \(A_2B_3\) and \(A_6B_6\).

The lines \(A_2A_3\) and \(A_5A_6\) correspond to O. Mohr’s theory.

The ray \(OB_3\) separates the stress states leading to failure from compressive stresses according to the theory of maximum normal stresses, and the stress states leading to failure from tensile stresses according to the same theory. In constructing the strength contour \(B_1-B_6\), it was assumed that the straight lines \(B_3B_4\) and \(B_5B_6\) continue only up to their intersection with the rays \(OB_3\) and \(OB_6\). The lines \(A_2B_3\) and \(A_6B_6\) are transitions from tension to compression and vice versa.

The equation of the straight line \(B_3B_4\)

\[ \nu \frac{\sigma_2}{\sigma_{\mathrm{br}}}-\mu \nu \frac{\sigma_1}{\sigma_{\mathrm{br}}}=-1. \tag{1} \]

The equation of the straight line

\[ \frac{\sigma_1}{\sigma_{\mathrm{br}}}-\mu \nu^2 \frac{\sigma_2}{\sigma_{\mathrm{br}}}=1; \qquad \nu=\frac{\sigma_{\mathrm{br}}}{|\sigma_{bd}|}. \tag{2} \]

Equations (1) and (2) make it possible to obtain calculation formulas for determining the failure stresses. If \(\sigma_1 : |\sigma_2| > \nu\), then the tensile failure stress is determined by the line

\[ \sigma_1 = \frac{\sigma_{\mathrm{B}}}{1-\mu \frac{\sigma_2}{\sigma_1}\nu^2}. \tag{3} \]

If \(\sigma_1 : |\sigma_2| < \nu\), then the compressive failure stress will be determined by the line

\[ \sigma_2 = -\frac{\sigma_{bd}}{1-\mu \frac{\sigma_1}{\sigma_2}}. \tag{4} \]

The coefficient of transverse strain changes in the course of deformation, approaching its limiting value \(\mu = 0.5\). As the experiments of I. M. Tyuneeva [5] show, for Steel 45 and copper under plastic

Fig. 2. Strength contours for brittle materials according to the theory of maximum normal stresses (dashed line) and according to the theory of maximum elongations with restrictions in the region of like-named stresses and transition lines \(A_2A_3\) and \(A_6B_6\). The lines \(A_2A_3\) and \(A_5A_6\) correspond to O. Mohr’s theory.

deformation of 1%, the coefficient of transverse strain increases from 0.30 to 0.45. Before failure, brittle materials exhibit a certain residual deformation. This deformation under compression of gray cast-iron specimens \((HB = 160\text{—}180)\) amounted to 8% [6]. Although the residual deformation preceding the failure of brittle materials is very small, it is evidently sufficient for the coefficient \(\mu\) to approach its limiting value substantially.

Z. M. Konyushko [7] experimentally determined the strength of heat-treated tool steels in tension, in compres-

compression and in shear. The shear strength limit was determined by testing thin-walled short tubular specimens.

Knowing the actual tensile and compressive strength of materials, according to the adopted strength criterion one can calculate the shear strength. The table gives three variants of the calculation:

Table

Calculated and actual values of the shear strength of hardened tool steels according to the experiments of Z. M. Konyushko

a) according to O. Mohr’s theory of strength

\[ \tau_{\mathrm{sh}}=\frac{1}{1+\nu}\,\sigma_{\mathrm{br}}; \tag{5} \]

b) according to P. Balandin’s theory of strength

\[ \tau_{\mathrm{sh}}=\frac{1}{\sqrt{3\nu}}\,\sigma_{\mathrm{br}}; \tag{6} \]

c) according to the proposed method (3)

\[ \tau_{\mathrm{sh}}=\frac{1}{1+0.5\nu^2}\,\sigma_{\mathrm{br}}. \tag{7} \]

The data given in the table show poor agreement of O. Mohr’s theory with experiment, as had also been noted earlier, and approximately equally satisfactory confirmation of the calculated formulas (6) and (7). Formula (6), at small values of the coefficient $\nu$, gives a shear strength limit considerably exceeding the tensile strength limit, which is unlikely.

The foregoing makes it possible to recommend formulas (3) and (4), obtained on the basis of the theory of maximum elongations, as possible expressions for the strength criterion of brittle materials.

References

1. S. D. Ponomarev et al. Strength Calculations in Mechanical Engineering. Vol. 1, Mashgiz, Moscow, 1956.

2. M. M. Filonenko-Borodich et al. Course in Strength of Materials. State Publishing House of Technical-Theoretical Literature, Moscow, 1956.

3. J. Marin. “Plasticity and creep of engineering materials.” In: Mechanics, No. 4 (80), 1963.

4. L. M. Sedokov. Resistance of Materials to Large Plastic Deformations. Tomsk, Tomsk Polytechnic Institute Publishing House, 1966.

5. I. M. Tyuneeva. “On the change of volume during plastic deformation.” Scientific Reports of Higher Education Institutions, Physics and Mathematics, No. 3, 111–115, 1959.

6. L. M. Sedokov. “Investigation of the plasticity and strength of gray cast iron.” Proceedings of Higher Educational Institutions. Physics, No. 5, 1959.

7. Z. M. Konyushko. “Investigation of the strength of heat-treated tool steels.” Anniversary collection of the Bauman Moscow Higher Technical School. Strength Calculations in Mechanical Engineering, No. 46, Mashgiz, 1955.