Conditions for the Application of Classical Strength Theories

P. M. Sedokov

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

The division of strength theories into classical and new ones [1—3] is based not only on the history of their development. All classical strength theories assume a very substantial, indeed maximal, idealization of the mechanical properties of materials. Nevertheless, in a number of cases, calculation according to the classical theories of strength, including the theory of maximum normal stresses, which ignores the existence of the two other principal stresses and completely removes the problem of developing strength theories, proves to be very close to the actual values of strength [1]. Therefore it is necessary to establish those conditions under which the application of the classical theories of strength does not lead to significant errors.

In Fig. 1 the diagrams of linear tension and linear compression of a real material are presented in general form. There, the idealization of the mechanical properties of this material that underlies calculations according to the classical theories of strength is also shown by dashed lines. If the material, in the course of plastic deformation preceding fracture, undergoes hardening, then the strength criterion splits into two: a) the condition for the onset of plastic deformation and b) the fracture criterion. In the first case the material constant is the yield limit $\sigma_{t p}$, and in the second—the strength limit $\sigma_p$. In the latter case it is assumed that Hooke’s law is valid up to the breaking stress $\sigma_p$.

Fig. 1. Physical and idealized diagrams of tension and compression of real brittle and plastic materials

All classical theories of strength are also based on the assumption that the resistance of materials to tension and compression is equal:

\[ \sigma_c=\sigma_p;\qquad \sigma_{tc}=\sigma_{tp}, \]

The discrepancy between theoretical and actual strength is explained primarily by the fact that the behavior of real materials under load differs substantially from the indicated idealization. But the range of applicability of the classical strength theories is considerably broader than might be supposed by comparing the ideal and actual properties of materials and from the brief recommendations in textbooks and monographs.

Figure 2 shows a comparison of four classical strength theories for a biaxial stress state in dimensionless coordinates. To preserve the indices of the coordinate axes, it was necessary to abandon the widely used condition

\[ \sigma_1 > \sigma_2 > \sigma_3 . \]

The analytical expression of the strength boundaries according to the four classical strength theories can be written in the following form:

\[ \sigma_{\mathrm{eqv}}^{\mathrm{I}} = |\sigma_1;\ \sigma_2;\ \sigma_3|_{\max} = \sigma_p, \]

\[ \sigma_{\mathrm{eqv}}^{\mathrm{II}} = |\sigma_1-\sigma_0;\ \sigma_2-\sigma_0;\ \sigma_3-\sigma_0|_{\max} = \frac{2}{3}\sigma_p, \]

\[ \sigma_{\mathrm{eqv}}^{\mathrm{III}} = |\sigma_1-\sigma_2;\ \sigma_2-\sigma_3;\ \sigma_3-\sigma_1|_{\max} = \sigma_p, \]

\[ \sigma_{\mathrm{eqv}}^{\mathrm{IV}} = (\sigma_1^2+\sigma_2^2+\sigma_3^2-\sigma_1\sigma_2-\sigma_2\sigma_3-\sigma_3\sigma_1)^{1/2} = \sigma_p . \]

The strength boundaries according to the first three theories are singular. The strength criterion according to the second strength theory is written in terms of the constancy of the largest component of the stress deviator [2]. This condition coincides completely with the theory of maximum elongations if the coefficient of transverse deformation has the limiting value \((\mu = 0.5)\) [1], and also with the condition of equality of those parts of the greatest elongations which go into change of shape [3]:

\[ 2\sigma_{\mathrm{eqv}}^{\mathrm{II}} = |2\sigma_k-\sigma_l-\sigma_m|_{\max} = 2\sigma_p . \]

Here \(k, l, m = 1, 2, 3\) and \(k \ne l;\ l \ne m\).

The four strength contours according to the classical strength theories, shown in Fig. 2, have six common points \(a, b, c, d, e\), and \(f\).

Fig. 2. Contours of limiting strength according to the four classical strength theories for a plane stress state in dimensionless coordinates.

Therefore, in a biaxial stress state, when one of the principal stresses is numerically much smaller than the other (points \(a, c, d, f\)), or when they are numerically close to each other but have different signs (points \(b\) and \(e\)), calculations by all the classical theories of strength, including the first, give approximately the same value of the equivalent stress, and the choice of any particular theory of strength for the indicated classes of stress states loses practical meaning.

The classical theories of strength give the greatest discrepancy when the principal stresses having different signs are numerically equal (shear), and when the signs are the same and the numerical value of one stress is half that of the other (see the dashed lines in Fig. 2).

It is convenient to represent graphically the relationship between the equivalent stresses calculated according to the classical theories of strength, as is done in Fig. 3, a, where stress circles for triaxial tension, their upper halves, are constructed. An equilateral triangle \(ABC\) is constructed on the diameter of the large stress circle. From point \(D\), corresponding to the mean principal stress, a straight line is drawn to vertex \(C\). From point \(D\) a perpendicular \(DE\) is also dropped to side \(AC\). Then

\[ \sigma_{\text{eqv}}^{\mathrm{I}}=\overline{OB}, \]

\[ \sigma_{\text{eqv}}^{\mathrm{II}}=\overline{EC}, \]

\[ \sigma_{\text{eqv}}^{\mathrm{III}}=\overline{AC}, \]

\[ \sigma_{\text{eqv}}^{\mathrm{IV}}=\overline{DC}. \]

The segments \(\overline{OB}\) and \(\overline{AC}\) correspond to the indicated theories of strength by construction. The proof that \(\overline{DC}=\sigma_{\text{eqv}}^{\mathrm{IV}}\) was given by A. M. Rozenberg and G. S. Smirnov-Alyaev [4]. It remains to prove that \(\overline{EC}=\sigma_{\text{eqv}}^{\mathrm{II}}\). From point \(H\) (Fig. 3, b) we drop a perpendicular to \(AC\). Then

\[ CE = CK + EK. \]

But \(CK = CH \cdot \cos 30^\circ\), and

\[ CH=\frac{\sqrt{3}}{2}(\sigma_1-\sigma_3). \]

Consequently,

\[ CK=\frac{3}{4}(\sigma_1-\sigma_3). \]

From the drawing in Fig. 3, b it is seen that

\[ EK = HD\sin 30^\circ = \frac{-2\sigma_2+\sigma_1+\sigma_3}{2}\cdot\frac{1}{2}. \]

Then

\[ CE=\frac{3}{4}\sigma_1-\frac{3}{4}\sigma_3-\frac{\sigma_2}{2} +\frac{\sigma_1}{4}+\frac{\sigma_3}{4}, \]

\[ CE=\sigma_1-\frac{1}{2}(\sigma_2+\sigma_3)=\sigma_{\text{eqv}}^{\mathrm{II}}, \]

which was to be proved.

For the case when point \(D\), corresponding to the mean principal stress, is closer to point \(B\) (Fig. 3, c), it is similarly easy to prove

\[ CF=\left|\sigma_3-\frac{\sigma_1+\sigma_2}{2}\right| =\sigma_{\text{eqv}}^{\mathrm{II}}. \]

The graphical determination of equivalent stresses, shown in Fig. 3, makes it possible to compare the equivalent stresses visually and to draw three conclusions:

Fig. 3. Graphical determination of reduced stresses according to the classical theories of strength

1. If the intermediate principal stress \(\sigma_2\) is, in sign and magnitude, close to \(\sigma_1\) or to \(\sigma_3\), then the values of the equivalent stresses calculated according to the second, third, and fourth theories of strength will be close to one another: \(AC \approx DC \approx EC\).

2. If the intermediate principal stress is close to the half-difference of the other two principal stresses,

\[ \sigma_2 \approx \frac{\sigma_1-\sigma_3}{2}, \]

then the values of the equivalent stresses calculated according to the second, third, and fourth theories of strength will have the greatest divergence:

\[ AC : HC : CE = 1 : \frac{1}{\sqrt{3}} : \frac{2}{3}. \]

1. Under triaxial tension and triaxial compression, the calculated value of the equivalent stresses according to the theory of greatest normal stresses differs substantially from the corresponding values according to the other classical theories of strength and contradicts experimental results.

The behavior of real materials under load differs substantially from the idealization that is assumed when using

of the classical strength theories (Fig. 1). It is precisely this difference that leads, in experiments, to the confirmation of one or another of the three theories.

Figure 4a schematically shows several actual strength limits for a number of brittle materials under a plane stress state, when the principal stresses have different signs. If the ratio between the tensile strength \(\sigma_p\) and the compressive strength \(\sigma_c\)

\[ \nu=\frac{\sigma_p}{|\sigma_c|} \]

decreases from 1 to 0, then, when comparing the actual and calculated value of the shear resistance, both the theory of maximum elongations \((\nu \to 1)\) and the theory of maximum normal stresses \((\nu \to 0)\) can be confirmed.

Fig. 4. Strength boundary of brittle (a) and plastic (b) materials under a plane stress state, when one of the principal stresses is compressive and the other tensile

Figure 4b schematically shows the strengths for hardening materials on the basis of an analysis of experimental results [1]. Hardening of a material can be characterized by the ratio of the strength and yield limits, the relative uniform narrowing, or by the exponent in the power-law approximation of the tensile curve \(m\).

Comparison of the actual and calculated value of the shear resistance of hardening materials can confirm both the theory of maximum elongations \((m \to 0)\), and the theory of octahedral stresses \((m \approx 0.15)\), and the theory of maximum shear stresses \((m \approx 0.3)\).

Fig. 5. Comparison of the actual and calculated, according to the classical strength theories, ratios between shear strength and tensile strength

Considering Fig. 5, where the actual values of the ratios \(\tau_{\mathrm{cd}}/\sigma_p\) are plotted as functions of the coefficient \(\nu\) and the exponent \(m\), one can

the following recommendations may be made on the application of the classical theories of strength for stress states close to shear:

1. The theory of maximum normal stresses should be applied only to very brittle materials ($\nu < 0.3$).

2. The theory of maximum elongations, for $\mu = 0.5$, may also be applied to brittle materials if $\nu \to 1$, and to plastic materials if $m \to 0$.

3. Calculation according to the theory of octahedral stresses gives satisfactory agreement with experimental data for weakly hardening materials ($m \approx 0.15$).

4. The theory of maximum shear stresses is applicable to materials that, in the course of the plastic deformation preceding failure, harden substantially ($m \approx 0.3$).

Returning to Fig. 2, additional recommendations may be made for the plane state.

All the classical theories may also be applied approximately to brittle materials if

a) the ratio of the principal stresses is close to points $a$, $b$, $c$, and the strength characteristic of the material is taken to be its resistance to uniaxial tension.

b) the ratio of the principal stresses is close to points $d$, $e$, $f$, and the strength characteristic of the material is taken to be its strength limit in uniaxial compression.

REFERENCES

1. L. M. Sedokov. Limits of applicability of the basic theories of strength for brittle materials under a plane stress state. Vestnik Mashinostroeniya, 1969, No. 10.

2. L. M. Sedokov. Resistance of Materials to Large Plastic Deformations. Tomsk, Tomsk State University Press, 1966.

3. A. Yu. Ishlinskii. Hypothesis of form-change strength. Uchenye Zapiski MGU, “Mechanics,” 46, Moscow, 1940.

4. G. A. Smirnov-Alyaev, V. M. Rozenberg. Theory of Plastic Deformations of Metals. Mashgiz, 1956.