ON THE RELATIONSHIP BETWEEN RESIDUAL STRAIN TOLERANCES FOR DETERMINING YIELD STRENGTHS IN TENSION AND SHEAR (TORSION)

G. A. Doshchinskii

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

One of the most commonly used material characteristics employed in strength calculations is the yield strength, i.e., the stress at which the system of developing irreversible shears encompasses a significant volume of the deforming polycrystalline body. An external indication of this state of the material may be a considerable increase in strain with only a small increase, or even a constant value, of the load. In a graphical representation of the deformation process, the latter is characterized by the appearance of the so-called yield plateau. The stress corresponding to this state is sometimes called the “physical yield strength.”

However, such a phenomenon is observed in a limited group of materials, among which annealed low-carbon steel belongs. For most materials, the characteristic development of plastic deformation is such that the behavior of the material is graphically represented by a monotonic curve $\sigma—e$. The value of the yield strength of such a polycrystalline body is entirely devoid of physical definiteness.

The practical need for knowing a strength characteristic by means of which it would be possible to distinguish states with predominantly elastic deformation from those with plastic deformation, and conversely, leads to the necessity of using conventional yield strengths, established according to some tolerance on the magnitude of residual strain. At present, the most commonly used convention in tensile tests is to take as the conventional yield strength the stress at which the residual elongation is $0.2\%$.

\[ \Delta \lambda = 0.002 = 0.2\%. \]

For plane and three-dimensional stress states and, in particular, for such a type of deformation as shear (torsion), the corresponding state of the material is established on the basis of theories of equivalence of strains under different types of loading.

Proceeding from concepts of equivalence of stress states at equal values of the maximum shear stresses, Ludwik in 1909 gave a solution asserting that, for the tolerance

for tension of 0.2%, the equally effective tolerance in torsion will be 0.4% relative shear, i.e.

\[ \Delta\gamma_{\text{res}}=2\cdot\Delta\delta . \tag{1} \]

However, the wide use of this tolerance was associated with a misunderstanding, for, being based on the theory of equality of maximum shear stresses at equal maximum relative shears, relation (1) is in no way consistent with this theory. In attempting to find, for torsion, the shear of equal magnitude in tension, Ludwik erroneously expressed it again from the torsional deformation scheme. This error was noticed only in 1944 by N. N. Davidenkov, who pointed out the need to change the ratio of the tolerances to the form [3]

\[ \Delta\gamma_{\text{res}}=1.5\cdot\Delta\delta=0.3\%. \tag{2} \]

Accordingly, for a tolerance of 0.2% in tension, the equivalent tolerance in torsion will be 0.3% relative residual shear. If the hypothesis of equality of octahedral shear stresses at equality of octahedral shears is taken as the basis, then the equivalent tolerance in torsion will be

\[ \Delta\gamma_{\text{res}}=0.35\%. \tag{3} \]

It should be noted that relations (2) and (3) are established under the assumption of incompressibility of materials during plastic deformation.

“Further, more detailed analysis shows that even this tolerance requires refinement. It turns out that, in order to establish correctly the tolerance for residual deformation in torsion and in complex stress states, proceeding from a specified tolerance in tension, it is also necessary to be guided by the magnitude of Poisson’s ratio” [4].

The values of the transverse-deformation coefficient depend on the volumetric deformability of materials and on the degree of deformation. The assumption of incompressibility of the material corresponds to adopting, for the coefficient of transverse plastic deformation, determined from the ratio of residual transverse and longitudinal deformation, the value 0.5.

\[ \nu_0 \approx 0.5. \]

The nonideal continuity of real structural materials (especially foam materials) is associated with the possibility of residual volumetric deformations and the corresponding deviation of the values of \(\nu_0\) from 0.5. This circumstance cannot fail to be taken into account when establishing the equivalence of tolerances.

In the general case, when deciding the question of the ratio of tolerances, the different resistance of materials to tension and compression must also be taken into consideration. A number of stress states, including shear, may represent combinations of tension–compression. The universality of an equivalent tolerance established by the hypotheses of maximum or octahedral stresses, while having a one-sided advantage in generality, at the same time may also leave doubts with respect to reliability when applied to the broad range of modern structural materials. This may lead to the strength capabilities of individual materials being underutilized in a number of cases. On the contrary, any justified increase in the tolerances for residual deformation may be a source of additional strength reserves, especially for high-quality (high-strength) materials with a steep characteristic curve.

Another circumstance may also be regarded as an imperfection: expressions (2) and (3) do not take into account the intensity of the development of plasti-

…plastic deformations as a function of the degree of deformation and the type of stress state. Plastic deformation increases progressively as the degree of total deformation grows; moreover, stress states of the shear type are more favorable for its development than stress states close to hydrostatic. In this respect tension occupies an intermediate position.

The correspondence of residual-strain tolerances is determined mainly by the reliability of the theory on the basis of which the equivalence of material states under different types of deformation is established. The shortcomings of these theories and the assumptions adopted in them correspondingly reduce the reliability of the equivalent tolerances. It may be supposed that the frequently observed scatter in the value of the ratio of the yield strengths in shear and in tension, noted, for example, in [5], is to a considerable extent connected with the above-mentioned imperfections and with the inaccuracy of relations of type (2) or (3).

In [1], [2] a generalized condition was considered that establishes a relation between the stressed and the deformed state, refined by the inclusion of additional information about the material, allowing one also to take into account residual volumetric deformability and, above all, agreeing well with numerous experimental data. According to this theory, every deformation is elastic-plastic; moreover, the principal stresses at any stage of deformation are related to the principal strains by the dependence:

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

\[ =E(\lambda)\cdot \sqrt{\frac{e_1^2+e_2^2+e_3^2}{1+2\mu(\lambda)^2}} \qquad \text{or} \qquad \sigma_{\mathrm{eq}}=E(\lambda)\cdot e_{\mathrm{eq}}, \]

—where \(E(\lambda)\) and \(\mu(\lambda)\) are the current values of the secant modulus of the tensile diagram and of the coefficient of transverse strain.

Under elastic deformation these material characteristics correspond to the values of the elastic constants—the modulus of normal elasticity \(E\) and Poisson’s ratio \(\mu\). According to this theory it follows that the condition establishing the equivalence of stress states in the Mises–Hencky form is a condition inherent in such stages of already developed deformation when the elastic part of the deformation and its influence on the general regularity of the interrelation between the stressed and the deformed state may be neglected.

On the basis of this theory it appears possible to consider also the question of the equivalence of tolerances in the experimental determination of yield strengths under different deformations.

The principal components of elastic-plastic (total) deformation and their elastic and plastic parts, respectively, can be determined by the following relations:

\[ \text{Total deformation}\qquad e_j=\frac{1}{E(\lambda)} \left[\sigma_j-\mu(\lambda)(\sigma_i+\sigma_k)\right]. \]

\[ \text{Elastic part}\qquad \varepsilon_j=\frac{1}{E} \left[\sigma_j-\mu(\sigma_i+\sigma_k)\right]. \]

\[ \text{Plastic}\qquad \delta_j=\frac{1}{D} \left[\sigma_j-\mu_0(\sigma_i+\sigma_k)\right], \qquad (j,i,k=1,2,3). \]

Here

\[ D=\frac{E(\lambda)\cdot E}{E-E(\lambda)}, \qquad \mu_0=\frac{E\cdot\mu(\lambda)-E(\lambda)\cdot\mu}{E-E(\lambda)}. \]

These relations hold under loadings with an increase of all principal stresses (the total parts, and not only the deviatoric parts) proportional to one parameter \(\lambda\).

The components of the total and elastic strains under tensile stress corresponding to the yield point are determined by the values

\[ e_1^p=\frac{1}{E(\lambda)}\sigma_{\mathrm{т}}, \qquad e_2^p=e_3^p=-\frac{\mu(\lambda)}{E(\lambda)}\sigma_{\mathrm{т}}, \]

\[ e_1^e=\frac{1}{E}\sigma_{\mathrm{т}}, \qquad e_2^e=e_3^e=-\frac{\mu}{E}\sigma_{\mathrm{т}} . \]

The residual elongation in tension is

\[ \delta=e_1^p-e_1^e =\delta_{\mathrm{т}}\left(\frac{1}{E(\lambda)}-\frac{1}{E}\right) =\frac{\sigma_{\mathrm{т}}}{\dfrac{E(\lambda)\cdot E}{E-E(\lambda)}}=\frac{\sigma_{\mathrm{т}}}{D}, \]

whence

\[ \sigma_{\mathrm{т}}=\delta\,\frac{E(\lambda)\cdot E}{E-E(\lambda)} . \]

In the state corresponding to the yield point in shear (torsion), the total, elastic, and plastic relative shears are expressed by the relations

\[ \gamma_{\mathrm{tot}}^c=e_1^c-e_3^c=\frac{2\tau_{\mathrm{т}}}{E(\lambda)}[1+\mu(\lambda)], \qquad \gamma_{\mathrm{el}}^c=\varepsilon_1^c-\varepsilon_3^c=\frac{2\tau_{\mathrm{т}}}{E}(1+\mu), \]

\[ \gamma_{\mathrm{res}}^c =2\tau_{\mathrm{т}}\left[\frac{1+\mu(\lambda)}{E(\lambda)}-\frac{1+\mu}{E}\right]. \]

Using the concept of the equivalence index \(\eta\), which is the ratio of the yield points in shear and tension,

\[ \eta=\frac{\tau_{\mathrm{т}}}{\sigma_{\mathrm{т}}}, \]

we express the residual shear in torsion in the form

\[ \gamma_{\mathrm{res}}^c =2\eta\cdot\delta\,\frac{E(\lambda)\cdot E}{E-E(\lambda)} \left[\frac{1+\mu(\lambda)}{E(\lambda)}-\frac{1+\mu}{E}\right] =2\eta\cdot(1+\mu_0)\cdot\delta . \]

According to the theory of maximum shear stresses, \(\eta=0.5\); according to the theory of octahedral stresses, \(\eta=0.577\). Therefore, the equivalent tolerances, taking account of volumetric compressibility, according to these theories will respectively be equal to

\[ \Delta\gamma=(1+\mu_0)\cdot\Delta\delta \quad \text{and} \quad \Delta\gamma=1.154(1+\mu_0)\cdot\Delta\delta . \]

In our case, in the state corresponding to yielding,

\[ \sqrt{\tau_{\mathrm{т}}^2+\tau^2+\frac{4\mu(\lambda)-2\mu(\lambda)}{1+2\mu(\lambda)^2}\cdot\tau_{\mathrm{т}}^2} =\sigma_{\mathrm{т}} \quad \text{and} \quad \eta=\frac{\sqrt{1+2\mu(\lambda)^2}}{\sqrt{2[1+\mu(\lambda)]}}, \]

the equivalent tolerance for shear is

\[ \Delta\gamma =\frac{\sqrt{2\cdot[1+2\mu(\lambda)^2]}}{1+\mu(\lambda)}(1+\mu_0)\cdot\Delta\delta, \]

i.e., according to the given theory, the magnitude of the equally effective tolerance is found as a function of the degree of deformation and the type of material, which makes it possible to take more fully into account the characteristics of different materials. For a tolerance of \(0.2\%\) in tension, for different materials at

\[ \varphi(\lambda)=0 \div 0.5 \]

the equivalent will be the relative shear

\[ \Delta \gamma = (1.41 \div 1.154)\cdot(1+\nu_0)\cdot \Delta\delta . \]

For a volumetrically plastically incompressible material, \(\nu_0=0.5\),

\[ \Delta \gamma = (0.42 \div 0.35)\%. \]

In [2] a generalization of the theory under consideration is given for materials that behave differently in tension and compression. If the ratio between the yield limits in tension and compression (corresponding to identical values of the secant modulus) is used,

\[ \nu=\frac{\varepsilon_T^p}{\varepsilon_T^{\mathrm{comp}}}, \quad \text{then} \quad \eta_i= \frac{2}{(1+\nu)\cdot\sqrt{\dfrac{2[1+\varphi(\lambda)]^2}{1+2\varphi(\lambda)^2}}} = \frac{\sqrt{2\cdot[1+2\varphi(\lambda)^2]}}{(1+\nu)[1+\varphi(\lambda)]} \]

and the equivalent tolerance for different materials will, in the general case, be determined by the expression

\[ \Delta\gamma= \frac{\sqrt{2[1+2\varphi(\lambda)^2]}}{(1+\nu)[1+\varphi(\lambda)]} \cdot(1+\nu_0)\cdot\Delta\delta . \]

For torsion of a solid shaft, as well as for bending and other nonuniform stress states, the magnitude of the tolerance can be refined by taking into account the influence of residual stresses.

Analogously, the ratio of residual-strain tolerances can also be found for other stress states.

REFERENCES

1. Doshchinskii G. A. On the theory of elastoplastic deformation. Izvestiya TPI, vol. 85, 1957.

2. Doshchinskii G. A. Generalization of the condition of mean deformation to brittle materials. Izvestiya TPI, vol. 96, 1959.

3. Davidenkov N. N. On Ludwik’s flow curve. Izvestiya AN SSSR, OTN, No. 9, 1950.

4. Odinq I. A. The Problem of Structural Strength in Mechanical Engineering. Publication of the Society for the Dissemination of Political and Scientific Knowledge, 1949.

5. Kishkin S. T., Ratner S. I. Experimental verification of the fundamental law of the theory of plasticity. Zhurnal tekhnicheskoi fiziki, vol. 19, issue 3, 1949.