RESULTS OF A STUDY OF THE RELIABILITY OF STRENGTH THEORIES FOR BRITTLE MATERIALS USING A COMPUTER

E. M. MIKHAILOVSKII, L. M. SEDOKOV

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

The considerable number of existing strength theories for brittle materials, which differ both in their initial assumptions and in their experimental confirmation, greatly complicates the application of these theories in engineering practice.

Attempts to assess the reliability of strength theories by means of a diagram of limiting stress states [1], or by calculating the deviation of theoretical calculations from actual data for individual types of stress state and for only one material [2], cannot be considered convincing.

Below, a method is set forth for quantitatively evaluating criteria of brittle strength for entire regions of a plane stress state using the “Minsk-1” digital computer. Both results of studies of the strength of various brittle materials known from the literature and our own results were used. Almost all tests were carried out on thin-walled tubular specimens. A list of the materials whose test results under a plane stress state are used in the present work is given in the table.

The calculation scheme is simple. In each experiment the limiting values of the principal stresses $\sigma_1$ and $\sigma_2$ (or $\sigma_3$) were recorded, whose ratio is $n = \dfrac{\sigma_2}{\sigma_1}$ or $n = \dfrac{\sigma_3}{\sigma_1}$. The calculation formula for a given strength criterion can be transformed to the form

$$ \sigma_1 = f(\sigma_p, \gamma, n), \tag{1} $$

where $\sigma_p$ is the tensile strength;
$\gamma$ is the brittleness coefficient of the material;

$$ \gamma = \frac{\sigma_p}{|\sigma_c|}; $$

$\sigma_c$ is the compressive strength;
$\sigma_1$ is the theoretical value of the limiting magnitude of one of the principal stresses for a given ratio of the principal stresses, $n$.

By comparing the actual and theoretical values of the limiting principal normal stress, one can, for an entire region of stress—

stressed states, to compute the relative root-mean-square deviation of the theoretical calculations from the experimental data by the formula

\[ \Delta_{\mathrm{av}}^{\mathrm{rms}}= \sqrt{\frac{1}{K}\sum_{i=1}^{K} \left(\frac{\sigma'_1-\sigma_1}{\sigma_1}\right)^2}, \tag{2} \]

where \(K\) is the number of tests in the region of plane stress state under investigation;

\(\sigma'_1\) is the theoretical value of one of the principal stresses, calculated by formula (1);

\(\sigma_1\) is the actual value of the principal stress.

Table 1

To analyze the reliability of the strength theories, the analytical expression of a number of them was approximated by the equation

\[ \sigma_{\mathrm{eqv}}^{2}=\sigma_{1}^{2}+\nu^{2}\sigma_{2}^{2} -C\nu\sigma_{1}\sigma_{2}=\sigma_{p}^{2}. \tag{3} \]

The coefficient \(C\) in equation (3), for ideally plastic materials that have the same resistance to tension and compression \((\nu=1)\), depends on the resistance to pure shear:

\[ C=\left(\frac{\sigma_p}{\tau_{\mathrm{sd}}}\right)^2-2. \tag{4} \]

Using the relations of the classical strength theories for pure shear, one can find the value of the coefficient \(C\). Thus, according to the theory of maximum normal stresses \(C=-1\), according to the theory of maximum shear stresses \(C=2\), according to the theory of maximum elongations \(C=0.25\), and according to the energy theory \(C=1\).

Substituting the indicated values of the coefficient \(C\) into relation (3), we obtain the calculation equations according to the classical theories, general-

evaluated for brittle materials. Transforming equation (3) to the form (1), we obtain

\[ \sigma'_1=\frac{\sigma_p}{\sqrt{1+\nu^2 n^2-C\nu n}} . \tag{5} \]

For the “tension—compression” region, using relations (2) and (5), the relative root-mean-square deviation was calculated on the Minsk-1 computer. The value of \(C\) was specified in increments of 0.5. The results of these calculations are shown in Fig. 1. As can be seen from the figure, for all the materials studied the minimum discrepancy between the theoretical and experimental results lies near the value of \(C\) corresponding to the maximum-elongation theory.

Fig. 1. Variation of the relative root-mean-square error of the calculations as a function of the coefficient \(C\) for several brittle materials: 1—gypsum (\(\nu=0.13\)); 2—hardened steel U7 (\(\nu=0.94\)); 3—plastic EZI-120 (\(\nu=0.48\)); 4—plastic K-18-2 (\(\nu=0.24\)); 5—gray cast iron (\(\nu=0.36\)); 6—gray cast iron (\(\nu=0.31\)); 7—glass (\(\nu=0.11\)).

For strength theories not described by equation (3), the root-mean-square deviations (2) were also calculated by direct computations.

Summing, for each strength theory, the root-mean-square deviations for all materials in the “tension—tension” and “tension—compression” regions (separately for each region),

\[ \Delta=\frac{\displaystyle\sum_{i=1}^{m}\Delta^{\mathrm{sq}}_{\mathrm{cp}\,i}}{m}, \tag{6} \]

where \(m\) is the number of materials studied, we determine the accuracy of the calculations for each strength criterion in the region under study:

\[ P=(1-\Delta)\cdot 100\%. \tag{7} \]

It follows from Fig. 2 that, for the tested materials in the analyzed regions of a plane stress state, the theory of A. A. Lebedev and the criterion of maximum elongations for brittle materials, written by the equations,

\[ \sigma_1^2+\nu^2\sigma_2^2-\nu^2\sigma_1\sigma_2=\sigma_p^2, \tag{8} \]

\[ \sigma_1^2+\nu^2\sigma_2^2-0.25\nu\sigma_1\sigma_2=\sigma_p^2. \tag{9} \]

Fig. 2. Accuracy of various strength theories for brittle materials according to the calculation results: horizontal hatching — for the “tension—tension” zone; diagonal hatching — for the “tension—compression” zone.

Thus, the use of an electronic digital computer made it possible, for the first time, to provide a quantitative analysis of the reliability of strength theories for brittle materials under a plane stress state and to substantiate the reliability of choosing one or another strength criterion in engineering practice.

REFERENCES

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

2. V. M. Shannikov, K. N. Kan. Study of the static strength of rigid plastics under a plane stress state, Plasticheskie massy. “Khimiya,” 1964, No. 1.