Generalization of the Average Strain Condition to Brittle Materials

G. A. Doshchinsky

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

For the theoretical prediction of the danger of fracture of brittle materials under various states of stress, a whole series of proposals is known, beginning with the “classical theories of strength” and ending with theories proposed in recent times [1, 2, 3, 4]. A number of detailed works [1, 2, 3, 10] have been devoted to the review and criticism of various theories, and there is hardly any real need for their additional analysis here. It will be sufficient to note only the principal shortcomings of the theories.

The decisive circumstance that casts doubt on the reliability of a number of theories is the considerable discrepancy between the theoretical strength of brittle materials and the data of experimental verification [1]. The significance of some theories is substantially diminished by the narrow ranges of stress states that are reliably characterized by the theories. The possibility of applying some theories (Mohr’s limiting envelope theory [9], Yagn’s theory [2]) is limited by the necessity of carrying out a series of preliminary tests that realize a whole range of stress states. Taking into account also a number of shortcomings from the standpoint of physical meaning and convenience of practical application, it must be recognized that the problem of establishing equally dangerous conditions under various stress states is not fully, or even sufficiently, solved by any theory. Hence follows the necessity of further investigations in the theory of strength of brittle materials.

Comparison of various theories and assessment of the logic of their mathematical formulation are usually carried out on the basis of the geometrical interpretation of the theories. In an orthogonal coordinate system, along whose axes the principal stresses $\sigma_1$, $\sigma_2$, $\sigma_3$ are laid off, each theory is represented by a certain surface form corresponding to it. The coordinates of points located on this surface determine the stress ratios in equally dangerous states.

The principal observations on the behavior of brittle materials and considerations of a logical nature make it possible to outline the approximate form of the limiting surface. The equality of the axes of the principal stresses for isotropic materials determines the symmetry of the strength surface with respect to the axis equally inclined to the axes $\sigma_1$, $\sigma_2$, $\sigma_3$ and passing through the origin of coordinates. The assumed possibility of fracture under triaxial uniform tension requires the closedness

of the limiting surface from the side of the positive octant and, conversely, the surface may be open from the side of the negative octant, since brittle materials are capable of withstanding rather high hydrostatic pressures. The high resistance of brittle materials to compression and their low resistance to tension predetermine a broadening of the surface toward the negative octant. The latter circumstance usually finds analytical expression in theories through the dependence of the function determining the limiting state on the linear invariant of the stress tensor. The continuity of the change in the stresses $\sigma_1$, $\sigma_2$, $\sigma_3$ permits one to expect that the strength condition will be represented in the coordinate system $\sigma_1$, $\sigma_2$, $\sigma_3$ by a smooth curvilinear surface. There is even the opinion [3, 2] that the representation of certain limiting surfaces by polyhedra indicates the crude approximate character of the corresponding theories. To all this it should be added that the strength surface must be a surface of the simplest form and, accordingly, the strength equation should not be a cumbersome equation of high degree; otherwise the practical application of the theory will be difficult.

Comparison of different stressed states of plastic materials on the basis of using, as a common measure of strain, $e_{cp}$,

\[ e_{cp}=\sqrt{\frac{e_1^2+e_2^2+e_3^2}{3}} \tag{1} \]

shows good agreement of this criterion with experimental data for testing plasticity conditions [7, 8]. It appears expedient to analyze the possibility of extending this criterion also to materials that behave differently in tension and compression, which include materials with predominant properties of brittleness.

The simplest tensile and compressive tests of brittle materials show that, for them, with a certain degree of approximation, it is possible to adopt a linear relation between stress and strain up to states close to fracture. In this case it may also be considered that the elastic characteristics are quantities independent of the type of stressed state. For a brittle material that fractures at stresses $\sigma_p$ and $\sigma_c$ (in tension and compression), the values of the average strains (in the form (1)) are determined as:

\[ e_{cp}^{\text{tens}}= \sqrt{\frac{e_p^2+\mu^2 e_p^2+\mu^2 e_p^2}{3}} = e_p\sqrt{\frac{1+2\mu^2}{3}}, \]

\[ e_{cp}^{\text{comp}}= \sqrt{\frac{e_c^2+\mu^2 e_c^2+\mu^2 e_c^2}{3}} = e_c\sqrt{\frac{1+2\mu^2}{3}}, \tag{2} \]

where $\mu$ is an elastic constant of the material—the coefficient of transverse deformation;

$e_p$ and $e_c$ are the relative longitudinal strains at the instant of fracture in tension and compression.

Since $\sigma_c>\sigma_p$, it follows that $e_{cp}^{\text{comp}}>e_{cp}^{\text{tens}}$.

Consequently, the magnitude of the average strain for a brittle material in equivalent states (at fracture) is not constant, but, as already follows from these two tests, increases in passing from

from tension to compression. The regularity of this increase for each material can be found only if data from additional tests under other states of stress are used. However, the convenience of applying a strength theory is determined by the minimum of data taken from the simplest tests, even if this is partly to the detriment of the accuracy of the theory. Therefore, taking as a basis only the initial tensile and compressive tests, i.e., having only two experimental points for the limiting value \(e_{cp}^{k}\), in the first approximation one may adopt a linear law of variation. As the variable parameter of this linear dependence one may take the value of the relative volumetric strain \(\Theta = e_1 + e_2 + e_3\), which perhaps characterizes the principal difference in the deformed states of a material under tension and compression. It is practically inadvisable to adopt a dependence of a more complicated form, since determining the shape of the curve would require a larger number of preliminary tests (the shape of a curve of the second order, for example, is already determined by five experimental points).

We write the initial expression for the limiting value of the average strain in the form

\[ e_{cp}^{k} = A \cdot \Theta + B, \tag{3} \]

where \(A\) and \(B\) are experimental constants of the material. The admissibility of the initial form, in the absence of additional assumptions in the subsequent derivations, can be substantiated by comparing the final results of the theory with experimental data. Let us also note that in most strength theories for brittle materials an analogous kind of linear dependence of the limiting-state criterion on the mean normal stress is assumed

\[ \sigma_0 = \frac{\sigma_1 + \sigma_2 + \sigma_3}{3}. \tag{4} \]

The values of the constants \(A\) and \(B\) are determined on the basis of testing the material in tension and compression, for which equation (3) gives:

\[ e_p \sqrt{\frac{1 + 2\mu^2}{3}} = A \cdot (1 - 2\mu)e_p + B, \]

\[ e_c \sqrt{\frac{1 + 2\mu^2}{3}} = -A \cdot (1 - 2\mu)e_c + B. \]

Solving this system of equations simultaneously, we find:

\[ A = \frac{(e_p - e_c)\cdot \sqrt{1 + 2\mu^2}}{(e_p + e_c)\cdot (1 - 2\mu)\cdot \sqrt{3}}, \qquad B = \frac{2e_p \cdot e_c}{e_p + e_c}\sqrt{\frac{1 + 2\mu^2}{3}}. \tag{5} \]

The strength condition for any state of stress in its initial form will be

\[ e_{cp} < e_{cp}^{k}. \tag{6} \]

In expanded form we obtain:

\[ \sqrt{\frac{e_1^2+e_2^2+e_3^2}{3}} < \frac{(e_p-e_c)\sqrt{1+2\mu^2}}{(e_p+e_c)(1-2\mu)\sqrt{3}} (e_1+e_2+e_3) + \frac{2e_p e_c}{e_p+e_c}\sqrt{\frac{1+2\mu^2}{3}} . \]

Substituting the values of the principal strains and the limiting tensile and compressive strains, expressed through stresses:

\[ e_1=\frac{1}{E}(\sigma_1-\mu\sigma_2-\mu\sigma_3); \qquad e_2=\frac{1}{E}(\sigma_2-\mu\sigma_3-\mu\sigma_1); \]

\[ e_3=\frac{1}{E}(\sigma_3-\mu\sigma_1-\mu\sigma_2); \]

\[ e_p=\frac{\sigma_p}{E}; \qquad e_c=\frac{\sigma_c}{E}, \]

we pass to the form of the strength condition in stresses

\[ \frac{1}{E} \sqrt{(1+2\mu^2)(\sigma_1^2+\sigma_2^2+\sigma_3^2) -(4\mu-2\mu^2)(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1)} < \]

\[ < \frac{1}{E} \frac{(\sigma_p-\sigma_c)\sqrt{1+2\mu^2}}{\sigma_p+\sigma_c} (\sigma_1+\sigma_2+\sigma_3) + \]

\[ + \frac{2}{E} \frac{\sigma_p\cdot\sigma_c\sqrt{1+2\mu^2}}{\sigma_p+\sigma_c}. \]

Carrying out further transformations and reducing, we arrive at the expression

\[ \sqrt{ \sigma_1^2+\sigma_2^2+\sigma_3^2 -\frac{4\mu-2\mu^2}{1+2\mu^2} (\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1) } + \]

\[ + \frac{\sigma_c-\sigma_p}{\sigma_p+\sigma_c} (\sigma_1+\sigma_2+\sigma_3) < \frac{2\sigma_p\sigma_c}{\sigma_p+\sigma_c}. \tag{7} \]

Passing to the allowable stresses \(|\bar{\sigma}|_p\) and \(|\bar{\sigma}|_c\), we can finally represent the strength condition in a form more convenient for practical calculations

\[ \frac{1+\nu}{2} \sqrt{ 9\sigma_0^2 -k_\mu(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1) } + \frac{3(1-\nu)}{2}\sigma_0 \le |\bar{\sigma}|_p, \tag{8} \]

where

\[ \nu=\frac{|\bar{\sigma}|_p}{|\bar{\sigma}|_c}, \qquad k_\mu=\frac{2(1+\mu)^2}{1+2\mu^2}. \]

Unlike the theory of I. N. Mirolyubov [4], this condition assumes an influence of the elastic characteristics of the material on strength, which is not excluded owing to the fact that the elastic behavior of brittle materials is observed almost up to failure.

For various materials \(\mu = 0 \div 0.5\), \(k_\mu = 2 \div 3\).

The resistance to failure in shear for brittle materials, according to condition (8), is determined by the relation

\[ \tau_k=\frac{2\sigma_p}{(1+\nu)\sqrt{k_\mu}} =(1.41\div 1.15)\frac{\sigma_p}{1+\nu}. \]

Let us establish the form of the limiting surface corresponding to condition (8). For this purpose, let us write the equation

\[ \frac{1+\nu}{2} \sqrt{9\sigma_0^2-k_\mu(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1)} +\frac{3(1-\nu)}{2}\sigma_0=\sigma_p \]

in a system of orthogonal coordinates \(x, y, z\), in which the \(x\)-axis is equally inclined to the axes \(\sigma_1,\sigma_2,\sigma_3\). Using, in the transformation, the obvious relations

\[ x=\frac{\sigma_1+\sigma_2+\sigma_3}{\sqrt{3}}=\sqrt{3}\,\sigma_0;\qquad x^2+y^2+z^2=\sigma_1^2+\sigma_2^2+\sigma_3^2, \]

\[ \sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1 = \frac{1}{2}\left[(\sigma_1+\sigma_2+\sigma_3)^2 -(\sigma_1^2+\sigma_2^2+\sigma_3^2)\right] = \]

\[ =\frac{1}{2}(2x^2-y^2-z^2), \]

we have

\[ \frac{1+\nu}{2} \sqrt{3x^2-k_\mu x^2+\frac{k_\mu}{2}(y^2+z^2)} + \frac{\sqrt{3}(1-\nu)x}{2} =\sigma_p. \]

Freeing ourselves from the radical and reducing like terms, we obtain the equation of the limiting surface in the form:

\[ Ax^2+Cy^2+Cz^2+2Dx+F=0, \tag{9} \]

where

\[ A=\left[3-k_\mu-\frac{3(1-\nu)^2}{(1+\nu)^2}\right]; \qquad C=\frac{k_\mu}{2}; \]

\[ D=\frac{2\sqrt{3}\,\sigma_p(1-\nu)}{(1+\nu)^2}; \qquad F=-\frac{4\sigma_p^2}{(1+\nu)^2}. \]

The section of the limiting surface by the planes \(x=\mathrm{const}\) gives circles

\[ y^2+z^2=C_1^2. \]

Consequently, the limiting surfaces determined by equation (9) are surfaces of revolution about the \(x\)-axis.

Let us determine the possible forms of the surface as a function of the ratio of the material characteristics. For a surface of revolution, the form

surface can be established from the equation of the generator. We find the equation of the generator by intersecting the limiting surface with the plane \(z=0\). Substituting \(z=0\) into equation (9), we see that the equation of the generator is the equation of a second-order curve

\[ Ax^{2}+Cy^{2}+2Dx+F=0. \tag{10} \]

The invariants of this curve are respectively determined as:

\[ J_{1}=(A+C)=\frac{12\nu}{(1+\nu)^{2}}-\frac{k_{\mu}}{2}; \]

\[ J_{2}=AC=\frac{k_{\mu}}{2}\left[\frac{12\nu}{(1+\nu)^{2}}-k_{\mu}\right]; \]

\[ J_{3}= \begin{vmatrix} A & O & D\\ O & C & O\\ D & O & F \end{vmatrix} =C(AF-D^{2})=-\frac{2\sigma_{p}^{2}\cdot k_{\mu}}{(1+\nu)^{2}}\,(3-k_{\mu}). \]

Since \(\nu \le 1;\ k_{\mu}=2\div 3\) \((\mu=0\div 0.5)\), the conditions

\[ CF \ge 0 \quad \text{when } J_{2}=J_{3}=0;\qquad J_{3}=0 \quad \text{when } J_{2}>0; \]

\[ J_{1}J_{3}>0 \quad \text{when } J_{2}>0 \quad J_{3}\ne 0 \]

are incompatible.

Consequently, only real values correspond to the points of curve (10), and in the geometric representation of condition (8) imaginary surfaces have no place.

Limiting surfaces with a linear generator are determined by the condition

\[ J_{3}=-\frac{2\sigma_{p}^{2}\cdot k_{\mu}}{(1+\nu)^{2}}(3-k_{\mu}), \]

whence

\[ k_{\mu}=\frac{2(1+\mu)^{2}}{1+2\mu^{2}}=3, \]

i.e., they occur for materials with \(\mu=0.5\).

In this case, if

\[ J_{2}=\frac{3}{2}\left[-\frac{3(1-\nu)^{2}}{(1+\nu)^{2}}\right]=0,\quad (\nu=1), \]

we obtain the Huber—Mises—Hencky cylinder; if

\[ J_{2}=\frac{3}{2}\left[-\frac{3(1-\nu)^{2}}{(1+\nu)^{2}}\right]<0 \]

\((\nu<1)\)—the Mirolubov cone.

For materials with \(\mu\ne 0.5\) the generator is curvilinear \((J_{3}\ne 0)\).

Depending on the relation between the material characteristics \(\mu\) and \(\nu\), the possible forms of the limiting surface will be ellipsoid, paraboloid, hyperboloid surfaces. The type of surface is determined by the conditions:

\[ \text{when }\quad \frac{12\nu}{(1+\nu)^{2}}>\frac{2(1+\mu)^{2}}{1+2\mu^{2}} \quad \text{— ellipsoid,} \]

\[ \text{when }\quad \frac{12\nu}{(1+\nu)^{2}}=\frac{2(1+\mu)^{2}}{1+2\mu^{2}} \quad \text{— paraboloid,} \]

\[ \text{when }\quad \frac{12\nu}{(1+\nu)^{2}}<\frac{2(1+\mu)^{2}}{1+2\mu^{2}} \quad \text{— hyperboloid.} \]

The possibility of representing a strength condition of this type by surfaces as a function of the properties of the material was noted by Ya. I. Yagn [2] and W. Burzynski [10].

Condition (8) may be applied not only to brittle fracture. Under simple loading conditions it may also be used to determine equivalent states in the presence of plastic deformations, if the characteristics of the elastic-plastic properties of the material are introduced into it [8]. A change in the characteristics of the material during elastic-plastic deformation will also lead to a change in the shape of the limiting surface.

The limits of applicability of condition [8] are determined by the interval

\[ \Theta_{\text{comp}}=-\frac{(1-2\mu)\sigma_c}{E}< \]

\[ <\frac{(1-2\mu)(\sigma_1+\sigma_2+\sigma_3)}{E}< \]

\[ <\Theta_p=\frac{(1-2\mu)\sigma_p}{E}. \]

Fig. 1. Approximate form of the dependence
\[ \varepsilon_{\text{av}}^{k}=\Phi(\Theta) \]
(according to experimental facts)

Fig. 2

1) theory of average strain;
2) Mohr theory;
3) theory of normal stresses;
4) theory of strains (maximum).

The extension of the linear dependence of the limiting value of the average strain beyond these limits may lead to significant discrepancies with experimental data (Fig. 1).

Comparison of condition (8) with experimental data for brittle materials known in the technical literature [1, 5, 6] is given in Fig. 2. The results corresponding to several of the best-known and most widely used theories are also shown there. On the basis of the experimental data presented, condition (8) may be recommended for strength calculations of brittle materials under mixed (tension–compression) states of stress. For biaxial tension and biaxial compression, the strength of brittle materials, according to experiments [5, 6], can be determined respectively by the theory of maximum normal stresses and the theory of shear stresses.

REFERENCES

1. Ponomarev S. D. et al. Foundations of Modern Methods of Strength Calculation in Mechanical Engineering, Mashgiz, 1950.

2. Yagn Yu. I. New methods of strength calculation. Bulletin of Engineers and Technicians, No. 6, 1931.

3. Balandin P. P. On the question of strength hypotheses. Bulletin of Engineers and Technicians, No. 1, 1937.

4. Mirolyubov I. N. On the question of generalizing the theory of strength of octahedral shear stresses to brittle materials. Proceedings of the Leningrad Technological Institute named after Lensovet, issue 25, 1953.

5. Davidenkov N. N. and Stavrogin A. N. On a strength criterion for brittle fracture and a plane state of stress. Izv. AN SSSR, OTN, 1954.

6. Davidenkov N. N. and Yarov V. A. Brittle fracture under biaxial compression. Journal of Technical Physics, vol. 25, issue 12, 1955.

7. Doshinskii G. A. Theory of the limiting elastic state. Proceedings of the Tomsk Polytechnic Institute, vol. 85, 1957.

8. Doshinskii G. A. On the theory of elastic-plastic deformation. Proceedings of the Tomsk Polytechnic Institute, vol. 85, 1957.

9. O. Mohr. “Technische Mechanik,” Berlin, 1906.

10. W. Burzynski. “Ueber die Anstrengungshypothesen.” Schweizerische Bauzeitung, Band 94, No. 21, 1929.