ACCURACY OF APPROXIMATION OF FLOW CURVES OF PLASTIC MATERIALS

D. E. Vyazun, K. K. Pakhotin, L. M. Sedokov

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

The relationship between stress and strain in the plastic region reflects the material’s capacity for strain hardening. In a number of cases it becomes necessary to express this relationship analytically, which leads to additional errors.

To estimate these errors, studies were carried out on the flow curves of a number of plastic materials that differ substantially in their mechanical properties, including ultimate strength, relative elongation at rupture, and the material’s capacity for strain hardening.

Table 1 gives the mechanical characteristics of the materials investigated. The true ultimate tensile strength, \(\bar{\sigma}_k\), and the limiting relative logarithmic elongation, \(e_k\), were determined

Table 1

taking into account the change in the cross-sectional area after statistical processing of the test results. Here \(\xi\) is the boundary of the confidence interval of the mean flow curve,

\[ \xi = \frac{t\delta}{\sqrt{N}} = \frac{ t \sqrt{\displaystyle\sum_{1}^{N}(\bar{\sigma}_k-\bar{\sigma}_{ki})^2} }{ \sqrt{N(N-1)} }, \]

where

\(t\) — parameter of Student’s distribution;
\(N\) — number of tests;
\(\sigma_{ki}\) — value of the true ultimate strength in an individual test;
\(T\%\) — level of accuracy in determining the tensile strength;

\[ T=\frac{\xi}{\sigma_k}\cdot 100. \]

For each material, from 3 to 15 specimens were tested. Figure 1 shows the tensile curves of R6M3 steel obtained as a result of tests on five specimens.

The mean value of the limiting elongation was found to be \(\bar e_h=0.357\). The mean ultimate stress at the indicated limiting strain was found to be \(\sigma_k=97.9\ \text{kg/mm}^2\).

For obtaining the statistical flow curve of St. 45 steel, 40 specimens were tested in tension. The accuracy of the obtained values of the ultimate strengths and flow curves is close to the accuracy of measuring loads and strains during mechanical tests. A large number of specimens of different diameters made of St. 45 made it possible to identify two modifications of this steel, which were subsequently also confirmed by torsion tests.

Fig. 1. Tensile curves of five specimens of R6M3 steel

It turned out that the statistical ultimate strength of St. 45 in bars of diameter 12, 14, 16 mm is equal to \(99.2\ \text{kg/mm}^2\), while in the same steel in bars of diameter 18, 20, 10 mm it proved to be lower—\(92.6\ \text{kg/mm}^2\).

The tensile curve can be approximated by a power function

\[ \sigma=Ae^m \tag{1} \]

or by a linear function

\[ \sigma=\sigma_0(1+ce) \tag{2} \]

In these expressions, \(\sigma\) is the stress at the given instant of deformation; \(A\), \(m\), \(\sigma_0\), \(c\) are material constants; \(e\) is the strain.

An assessment of the accuracy of approximation of tensile curves can be made using dimensionless parameters of the resistance of materials to plastic deformation [1].

If the tensile curve corresponds exactly to relation (1), then all experimental values of the materials investigated coincide with the straight line

\[ \frac{\sigma_\phi}{A}=e^m. \]

For the power approximation (1), the dimensionless stress is expressed by the quantity \(\dfrac{\sigma_\phi}{A}\), where \(\sigma_\phi\) is the actual stress taken from the actual tensile curve; \(A\) is the stress corresponding to unit elongation on the approximated curve; \(e^m\) may be called the reduced strain.

For the linear approximation (2), the dimensionless stress is equal to \(\dfrac{\sigma_\phi}{\sigma_0}\), where \(\sigma_0\) is the conventional yield strength of the material, the intercept cut off on the stress axis by the straight line (2).

Figures 2 and 3 show tensile curves for some of the materials studied in dimensionless coordinates corresponding

Fig. 2. Tensile curves in dimensionless coordinates for the linear approximation: ●—St.3; ○—2Kh13; ×—copper; ▲—LS59; △—1Kh18N9T; □—45a; ▲—R6M3; ■—R12; △—R18; +—45v.

to the power-law (Fig. 2) and linear (Fig. 3) approximations of the tensile curves.

A characteristic of the accuracy of the approximation may also be the ratio of the area \(a\) enclosed between the actual and approximated tensile curves to the interval of reduced strain \((e_n^m - e_0^m)\) and to the mean dimensionless stress over the interval; that is, for the power-law approximation the error is

\[ \Delta = \frac{a}{(e_n^m - e_0^m)\dfrac{\sigma_\phi}{A}}, \]

and for the linear approximation

\[ \Delta = \frac{a}{(c e_n - c e_0)\dfrac{\sigma_\phi}{\sigma_0}}. \]

The results of determining the approximation errors of the tensile curves for all the materials indicated above are given in Table 2.

It is evident from the table that the errors associated with the mathematical notation of the portion of the tensile curve at large plastic strains do not exceed 4% for all the materials investigated.

Fig. 3. Tensile curves in dimensionless coordinates for the power approximation. The material designations are the same as in Fig. 2.

Approximation (2) is convenient in that the relationship between stress and strain is expressed very simply and, at the same time, as the data presented in Fig. 2 and in Table 2 show, is satisfied with sufficient accuracy.

Table 2

For an additional check of the accuracy of the linear approximation, hardening diagrams of metals and alloys presented in [2] were used; these were replotted in the coordinates “true stress—logarithmic elongation.”

To compare the adopted linearized dependence with the actual one, statistical methods of data processing were used, and an estimate of the accuracy of such a solution was given.

1. A dependence was found for the relative hardening of the material \(\frac{\sigma_k}{\sigma_0}\) on \(e e_k\), where \(\sigma_k\) and \(e_k\) are the true stress and strain at the final deformation (at the moment of rupture).

All points, under such a comparison, fall on one straight line; there is almost no scatter, with small deviations for the hardening curves of silicon iron, sheet copper, and annealed Armco iron.

1. For each material, the actual work \(a_\phi\) up to fracture was calculated by planimeter. The calculated work on the basis of (2) is

\[ a_p=\frac{\sigma_0+\sigma_k^p}{2} e_k, \]

where \(\sigma_k^p\) is the stress at the final deformation, calculated by formula (2).

The discrepancy between \(a_\phi\) and \(a_p\), as a rule, is no more than 5%. The largest deviations (8%) were obtained on hardening curves in tension for steel containing 0.23% C and 24.6% Ni, cobalt iron containing 3.16% Co, annealed carbon steel containing 0.17% C, and annealed alloy steel containing 0.81% C, 0.28% Si, 0.68% Mn, and 0.90 Cr.

1. The root-mean-square value of the maximum deviations for all 166 materials was \(1.88\ \text{kg}/\text{cm}^2\).

2. The difference between the actual curve and the linearized curve occurs at the initial stage of deformation. Let us denote by \(e_B\) the deformation starting from which the difference between the linear and actual diagrams is greater than the maximum deviation.

Of the 166 approximated curves, 106 diagrams have the ratio

\[ \frac{e_B}{e_k}<0.1, \]

43 diagrams have this ratio within the limits \(0.1\text{--}0.2\), and 10 diagrams have the indicated ratio within the limits \(0.2\text{--}0.4\).

1. The correlation coefficient [3] was calculated for dependence (2): \(r=0.994\). Such a value of the correlation coefficient indicates the high reliability of the linear expression of the flow curve.

Conclusions

1. Power-law and linear approximation of the hardening curves of all 166 materials considered, which differ very substantially in their mechanical properties, quite satisfactorily describe the relationship between stress and strain up to fracture.

2. In linear approximation (2), its deviation from the actual flow curve, as a rule, does not exceed 3%, and only in 14 cases out of 166 is it from 3 to 7%.

3. In power-law approximation (1), the calculated flow curve gives a deviation from the actual one that, as a rule, does not exceed 2%, and only in two cases out of 10 was about 4%.

4. The high accuracy of the approximation was obtained according to the first variant without subsequent refinement.

5. The greatest deviation of the analytical dependences from the actual flow curves occurs in the initial period of plastic de-

formation. 106 materials out of 166 are quite satisfactorily approximated by a linear dependence, starting from a strain equal to 10% of the residual strain at fracture.

REFERENCES

1. L. M. Sedokov. Dimensionless parameters of the resistance of a material to plastic deformation. Zavodskaya laboratoriya, No. 10, 1964.

2. V. L. Krokha. Strain-hardening curves of metals under cold deformation. Moscow, Mashinostroenie, 1968.

3. R. S. Guter, B. V. Ovchinskii. Elements of numerical analysis and mathematical processing of experimental results. Moscow, Nauka, 1970.