EFFECT OF COMPLEX LOADING OF CONSTANT INTENSITY ON THE YIELD SURFACE

A. M. KORENEVA, G. A. DOSHCHINSKII

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

Analytical description of the behavior of a plastically deforming, strain-hardening material presupposes the establishment of the shape of the yield surface, which separates, in stress space, the regions of elastic and plastic states, and of its transformation in the process of plastic deformation.

This article presents the results of experimental studies of the influence of preliminary complex loading of constant intensity on the shape of subsequent yield surfaces, and also establishes several additional regularities of plastic deformation for this type of loading.

The postulates of Ilyushin [1] and Drucker [2], while establishing certain general properties of yield surfaces (for example, convexity), do not give a complete idea of the specific shape of subsequent yield surfaces or of the influence of the deformation history on their geometry. The available experimental works devoted to the systematic study of the shape of subsequent yield surfaces and the influence on them of the deformation history are insufficient for definite conclusions, and at times they are contradictory (3÷5), etc.

The experiments described were carried out under conditions of biaxial tension, produced by stretching tubular specimens with additional internal pressure.

Specimens of 40Kh steel were made from solid rods with a diameter of 40 mm. The outer diameter of the specimens was 30 mm, \(t = 1\) mm. The specimens of U8 steel were analogous. The duralumin specimens were sections of cold-drawn tubes with an outer diameter of 30 mm, a wall thickness of 1 mm, and a gauge length of 120 mm. All specimens were subjected to recrystallization annealing. Longitudinal and transverse strains were determined with the aid of mechanical strain gauges. At each loading stage of the specimen, instrument readings were taken after a hold of 5÷7 min, in order to eliminate the influence of creep. Special experiments were carried out to check the degree of isotropy. These experiments consisted in testing specimens in tension in the axial and tangential directions.

The tests were carried out on a ZMDU-30 machine and were programmed in Ilyushin’s plane [1]:

\[ S_1 = \sqrt{\frac{2}{3}}\left(\sigma_z - \frac{1}{2}\sigma_\theta\right) \]

\[ S_2=\frac{\sqrt{2}}{2}\sigma_\theta \]

\[ S=S_1 i+S_2 j \]

\(S_1, S_2\) are the components of the stress vector;
\(\sigma_z, \sigma_\theta\) are the axial and circumferential stresses.

Initially, specimens of 40Kh steel were stretched by an axial force until plastic deformations appeared, and then the internal pressure and axial force were varied so that loading was carried out on a surface of constant intensity.

After unloading, the specimens were reloaded along rays \(\dfrac{S_2}{S_1}=0;\ 0.3;\ 0.5;\ 0.8\). From the measured deformation values the strain intensity \(e_i\) was calculated, and from the stresses \(\sigma_z\) and \(\sigma_\theta\), the stress intensity. From the \(\sigma_i-e_i\) diagram, the value of \(\sigma_{is}\) was determined at the conventional yield limit with an allowance for residual strain \(\approx 0.173\%\).

The determination of the stresses \(S_1\) and \(S_2\) corresponding to the value \(\sigma_{is}\) was carried out from the graphs \(\sigma_z-\sigma_i,\ \sigma_\theta-\sigma_i\).

Fig. 1

For 40Kh steel, experiments were also carried out to study the influence of preliminary plastic deformation under simple loading on the yield surface. It turned out that the experimental results coincided with those obtained earlier in [6].

Specimens of D1T duralumin were initially loaded along a surface of constant intensity, along an arc length up to \(90^\circ\), and after unloading the loading of each specimen was carried out along its own ray in the coordinates \(S_1-S_2\). The results obtained are similar to those for 40Kh steel; in the segments of motion along the arc, \(\sigma_i=\mathrm{const}\).

The experimental results show that motion along the arc \(\sigma_i=\mathrm{const}\) leads, in the vicinity of the sector traversed, to a local

transformation of the yield surface to a form corresponding to isotropic hardening (Fig. 1).

Experiments show that an increase in \(\sigma_i\) of the preliminary loading also leads to an expansion of the yield surface; however, this expansion is limited.

Thus, in experiments carried out on specimens of U8 steel, it was established that an increment of plastic strain is observed only up to the value \(\sigma_z=\sigma_\theta\) (Fig. 2). This makes it possible to suppose that plastic strain is affected by the mean stress

\[ \sigma_{\mathrm{cp}}=\frac{\sigma_1+\sigma_2+\sigma_3}{3}. \]

The realization of loading \(\sigma_i=\mathrm{const}\) on tubular specimens under biaxial tension is associated with an inevitable change in the spherical part of the stress tensor; the latter may also be regarded as the cause of the observed changes in the plastic state under conditions \(\sigma_i=\mathrm{const}\).

Fig. 2

The influence of hydrostatic pressure on plastic deformation was noted in the works of Stüssi [7], Yagn and Chaplygin [8], and others.

REFERENCES

1. A. A. Ilyushin. On the foundations of the general mathematical theory of plasticity. Publishing House of the Academy of Sciences of the USSR, 1961.

2. D. C. Drucker. Stress-strain relations in the plastic range of metals. Experiments and basic concepts. Rheology, v. 1, 1956.

3. A. Philips. Conical points on the yield surface. In: Mechanics, No. 4, 1961.

4. A. M. Zhukov. Some features of the curve of neutral loading. Izv. AN SSSR, No. 8, 1958.

5. G. B. Talypov. Influence of preliminary plastic deformation on the yield and fracture limits of copper. In: Studies on Elasticity and Plasticity, No. 4, 1965.

6. A. M. Koreneva, G. A. Doshchinsky. Experimental study of the curve of neutral loading. Izvestiya TPI, 1967.

7. Fr. Stussi. Grundlagen der mathematischen Plastizitatstheorie und der versush. ZAMP, 1950, Bd. 1, Fact. 4.

8. Yu. I. Yagn, I. A. Chaplygin. On the resistance of metals to plastic deformation. Doklady AN SSSR, vol. XI, No. 6, 1953.