POISSON’S RATIO DURING THE PROCESS OF DEFORMATION AND FRACTURE

E. M. Mikhailovskii, K. K. Pakhotin, L. M. Sedokov

(Presented by the City Scientific Conference of Young Scientists and Specialists)

Within the elastic range, the ratio of transverse to longitudinal deformation under simple tension of a specimen remains approximately constant and expresses one of the constants of the mechanical properties of the material, called Poisson’s ratio—\(\mu\).

Beyond the elastic range, the coefficient of transverse deformation changes substantially and may be characterized by three quantities, found either from the elastic, or from the plastic, or from the total (elastic-plastic) deformation.

The practical use in engineering calculations of the equation of the theory of maximum elongations [1, 2]

\[ \sigma_1-\mu(\sigma_2+\sigma_3)\leq \sigma_p \tag{1} \]

requires precise knowledge of the magnitude of the coefficient of transverse deformation beyond the elastic range, since real brittle materials before fracture give a residual deformation exceeding \(1\text{–}2\%\).

Extending condition (1) also to the region of uniaxial compression [2] leads to the result that the compressive strength of brittle materials is always equal to

\[ |\sigma_c|=\frac{\sigma_p}{\mu}, \tag{2} \]

where

\[ \sigma_c \text{—the strength limit in axial compression,} \]

\[ \sigma_p \text{—the strength limit in axial tension.} \]

Let us call the ratio of the strength limits in tension and in compression the “brittleness coefficient”

\[ \nu=\frac{\sigma_p}{|\sigma_c|}. \tag{3} \]

From equations (2) and (3) it follows that the brittleness coefficient is always equal to the coefficient of transverse deformation \(\mu\). However, this equality has neither theoretical nor experimental justification. The actual value of these quantities is given in Table 1.

It follows from Table 1 that the numerical values of the coefficients \(\mu\) and \(\nu\) for different materials do not coincide: for glass, \(\mu\) is 5 times greater than \(\nu\), while for hardened steel U7, on the contrary, \(\nu\) is three times greater than \(\mu\). With respect to cast iron, these quantities are fairly close, but nevertheless the discrepancy exceeds \(10\%\).

For small elastic-plastic deformation, Zhukov [7] showed that the experimental data agree well with those calculated by the formula

\[ \mu_0=\frac{3K-E_z}{6K}, \tag{4} \]

where \(K\) is the bulk-deformation modulus,
\(E_z\) is the secant modulus.

At deformations of the order of 1%, the calculated value of \(\mu_0\) from relation (4) differs from \(\mu=0.5\) by no more than 5%. Taking into account that for brittle materials the residual deformation usually exceeds 1%, one may assume that at the moment of fracture, for brittle materials \(\mu\) is always close to 0.5 [8].

Table 1

Davidenkov and Vasil’ev [10] proposed determining the coefficient of transverse plastic deformation \(\mu^*\) from the equation

\[ \mu^*=\frac{1}{\varepsilon^*}\left(1-\sqrt{\frac{1+\theta}{1+\varepsilon^*}}\right), \tag{5} \]

where \(\theta\) is the residual volume deformation,
\(\varepsilon^*\) is the relative conventional residual elongation.

Meyer [10], Andrew and Lee [11], and Davidenkov and Vasil’ev established that, when metals are stretched to \(\varepsilon^*=10\%\), the residual volume deformation lies within \((10\div15)\times10^{-4}\); then, according to relation (5), the coefficient \(\mu^*\) is equal to 0.47.

It was suggested [10] that at large deformations \(\mu^*\) should approach 0.5.

If the residual volume deformation is neglected, owing to its obvious smallness, we obtain

\[ \mu^*=\frac{1}{\varepsilon^*}\left(1-\sqrt{\frac{1}{1+\varepsilon^*}}\right). \tag{6} \]

The graph \(\mu^*=f(\varepsilon^*)\) according to (6) is shown in Fig. 1. The results for tensile tests of solid cylindrical specimens of copper and brass in the as-received condition are also plotted there.

Before testing, the diameters of the specimens were measured with a micrometer at the ends and in the middle of the gauge length in two mutually perpendicular planes. Specimens were selected for the experiments for which the discrepancy between diameter values along the gauge length did not exceed 0.01 mm.

The gauge length was marked by two scratches perpendicular to the axis of the specimen, made on a dividing apparatus, and was equal to 20 mm. Tension was carried out on a TsDMU-30 machine up to \(\varepsilon^*=60\%\).

During the experiment, the diameters over the gauge length were measured with a micrometer, and the distance between marks with a special microscope with a scale division of 0.02 mm. The data obtained are given in Table 2.

Fig. 1. Graph of the dependence of the coefficient of plastic deformation on the conventional residual relative strain: ● — copper (experiment); ○ — brass (experiment).

in Table 2. From Table 2 and Fig. 1 it follows that \(\mu^*\) is considerably less than 0.5 for deformations exceeding 10%.

Table 2

A decrease in \(\mu^*\) was to be expected, since in experiments the coefficient of plastic transverse deformation is usually determined as the result of dividing the readings of instruments recording the absolute residual longitudinal and transverse deformation.

The actual coefficient of plastic deformation, calculated from the ratio of the true (logarithmic) deformations, will always be equal to 0.5, as shown in Fig. 2 by the solid line.

At small degrees of deformation, the elastic component of the total deformation must also be taken into account. In this case the coefficient of transverse deformation increases, asymptotically approaching the value 0.5; moreover, the intensity of the increase in its value is the greater, the smaller the strain hardening of the material, and vice versa. This follows from the following formula [9]:

\[ \mu_0=\frac{D}{E}\left(\mu-\mu_1^*\right)+\mu_1^*, \tag{7} \]

where

\(E\) is the modulus of initial normal elasticity,

\(D\) is the tangent modulus—the tangent of the angle of inclination of the tangent to the flow curve.

In Fig. 2, dependence (7) for copper and brass is shown by dashed lines.

Fig. 2. 1 — graph of the coefficient of plastic deformation found from the relation of logarithmic strains; 2 — graph of the coefficient \(\mu_0\) according to dependence (7) for copper. ● — copper (experiment); ○ — brass (experiment).

Conclusions

1. The coefficient of transverse plastic deformation \(\mu^*\), calculated as the ratio of strain-gauge readings, decreases with increasing deformation and approaches the value of Poisson’s ratio in the elastic region.

2. The actual coefficient of transverse plastic deformation will be equal to 0.5 if it is calculated from the relation of true (logarithmic) strains.

3. With accuracy sufficient for engineering calculations, when determining the reduced stress according to the theory of maximum elongations as applied to brittle materials (with the introduction of the brittleness coefficient), the coefficient of transverse deformation should be taken equal to 0.5, since this value of the coefficient is the most probable at the moment of fracture.

References

1. V. A. Gastev, A Short Course in Strength of Materials. Fizmatgiz, Moscow, 1959.

2. Ya. B. Fridman, Mechanical Properties of Metals. Oborongiz, Moscow, 1952.

3. N. N. Davidenkov, I. N. Stavrogin. Izv. AN SSSR, OTN, No. 8, 1954.

4. J. Cornet and R. C. Grassi, J. of Appl. Mech., vol. 22, No. 2, 1955.

5. F. S. Savitskii, O. P. Burmakina. FMM, vol. 7, issue 3, 1959.

6. Z. M. Konyushko. “Strength Calculations in Mechanical Engineering,” collection 46, MVTU, Mashgiz, 1955.

7. A. M. Zhukov. Izv. AN SSSR, OTN, No. 12, 1954.

8. L. M. Sedokov. Izv. TPI, vol. 157, Tomsk, 1968.

9. N. N. Davidenkov, D. M. Vasil’ev. Zavodskaya laboratoriya, vol. 18, No. 5, Moscow, 1952.

10. Mayer, Metals Technology, vol. 111, issue 3, 1936.

11. Andrew, Lee, Iron and Steel Institute 165, page 2, 1950.

12. A. V. Gur’ev. FMM, vol. 2, issue 3, 1956.