ON THE REGULARITIES OF PLASTIC STRAIN INCREMENT UNDER COMPLEX LOADING OF CONSTANT INTENSITY WITH A FIXED DIRECTION OF THE PRINCIPAL STRESSES

G. A. Doshchinskii, N. V. Pogozheva

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

The results of a number of studies carried out in recent years in the field of complex loading [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11] note that the motion of the loading point on the Mises surface is accompanied by a change in the plastic state of the material and by an increment of plastic strains. Attempts to determine the true form of the neutral surface led to curves coinciding neither with the Mises ellipse nor with the Saint-Venant hexagon. However, despite the obvious conclusion that the Mises loading surface is not a surface of neutral loading, and that motion along it may be accompanied by a substantial increment of plastic strain, the regularities of this strain remained unclear. Some clarity on this question was introduced by works [12], [13], [14], [15], [16]. The experiments in these works were carried out on thin-walled tubular specimens of various metals according to the following procedure. By simple tension the specimens were brought into the plastic region up to a certain value of \(\sigma_i\), and then complex loading was carried out by simultaneous variation of the longitudinal force \(P\) and the internal pressure \(q\) so that \(\sigma_i\) remained constant in magnitude throughout the continuation of the experiment. If, for graphical representation, one adopts the five-dimensional orthonormalized coordinate space of A. A. Ilyushin, then such loadings will be illustrated in the \(S_1—S_2\) plane by deviation of the vector \(\bar S\) from the initial ray \(S_1\). The end of the vector \(\bar S\) describes, under loading \(\sigma_i=\mathrm{const}\), an arc of a circle. The loading parameter is the length of the loading trajectory \(L(S)\), or the corresponding angle of deviation of the vector \(\bar S\) from \(S_1\)—\(\varphi_S\). The experiments in works [12], [13], [14], [15], [16] were carried out mainly in the range \(\varphi_S = 0 \div 60^\circ\). As a result, the regularity was obtained

\[ \Delta Э = k \cdot L(S), \tag{1} \]

which indicates that the increment of plastic strains under loading conditions \(\sigma_i=\mathrm{const}\) with a fixed direction of the principal stresses is directly proportional to the length of the loading path.

In connection with the results obtained in works [12], [13], [14], [15], [16], it became necessary to trace the regularities of the increment of plastic strains over a broader range of the indicated type of complex loading. In the experiments described in the present work, by introducing into the loading program a compressive longitudinal load with \(\sigma_i=\mathrm{const}\), it was possible to bring the angle of deviation of the stress vector \(S\) to \(180^\circ\). Such a range of investigations makes it possible, during loading, to pass from the region of pure tension along the arc of the circle \(R-S=\dfrac{\sqrt{2}}{\sqrt{3}}\sigma_i\) into the opposite region—compression (Fig. 1), and practically exhausts the possibilities of such loading. The tests were carried out on the universal testing machine ZDMU-30, which makes it possible to load the specimen simultaneously and independently by an axial force, internal pressure, and torque.

Fig. 1

A series of thin-walled tubular specimens of steels 35 and 45 was tested according to programs providing for deviation of the stress vector \(S\) from \(S_1\) by \(\varphi_s=59^\circ\) and \(\varphi_s=180^\circ\), and a series of copper specimens for which \(\varphi_s=90^\circ\). The isotropy of each specimen studied was checked during the experiment. The absence, at the end of the ray of simple loading, of strains represented in the plane \(Э_1-Э_2\) by the component \(Э_2\) made it possible to consider the isotropy of the material satisfactory, while the presence of some value of \(Э_2\) made it possible to reject the corresponding tests from the total number.

Measurement of strains was carried out with the aid of an instrument, a detailed description of which is given in [17]. Taking into account the vector character of the investigated increment of plastic strain, the regularities of the latter should be considered both in absolute value and in direction. As a scalar measure of the additional plastic strain, the modulus of the plastic-strain-increment vector was taken, or the proportional quantity, the intensity of the plastic-strain increment,

\[ \Delta Э_i=\frac{\sqrt{2}}{\sqrt{3}}\left|\Delta \overline{Э}_i\right| =\frac{\sqrt{2}}{\sqrt{3}}\Delta Э. \tag{2} \]

The components of plastic strain were determined by excluding the elastic part from the total strain. The increment of the components of plastic strain over a segment of complex loading with \(\sigma_i=\mathrm{const}\) was determined from the difference between the current strains and the values at the beginning of the process of complex loading,

\[ \Delta Э_j=Э_j-Э_{j0}\qquad (j=1,2). \]

The intensity of the strain increment is determined (under the condition of incompressibility) by the expression

\[ \Delta Э_i=\frac{2}{\sqrt{3}}\sqrt{\Delta Э_z^2+\Delta Э_\theta^2+\Delta Э_z\Delta Э_\theta}. \tag{3} \]

On the basis of tests of copper and steel specimens at various values of \(\sigma_i=\mathrm{Const}\), graphs were constructed for the variation of the additional \(\Delta Э_i\) and total \(Э_i\) strains as functions of the loading path length \(L(S)\) (or of the angle of deviation \(\varphi_S\)). The experiments showed that proportionality of relation (1) is observed only up to \(\varphi_S=60^\circ\). Under further loading, a deviation of this relation from a straight line appears (Fig. 2). The intensity of the total strain also increases with an increase in the angle of rotation \(\varphi_S\). But the limiting increase in the intensity of the total strain \(Э_i\) is determined by the angle \(\varphi_S=90^\circ\). With a further increase in the angle \(\varphi_S\), the strain intensity decreases substantially. This latter fact indicates that, under conditions of complex loading with \(\sigma_i=\mathrm{Const}\), the relation between the intensities of stresses and plastic strains is not single-valued. At a fixed value of the intensity of the acting stresses, the strain intensity under conditions of complex loading undergoes certain changes associated with a change in the type of stress state. Establishing the vector character of the increment of plastic strain became possible on the basis of an analysis of strain rates under loading conditions \(\sigma_i=\mathrm{Const}\), which, owing to the nonessential role of time for the concept of plastic strain, was replaced by the study of strain increments at individual stages of loading, which are the differentials of the plastic-strain vector \(d\overline{Э}\).

Fig. 2

The determination of the plastic-strain increment at individual stages of loading \(\sigma^i=\mathrm{Const}\) was carried out by graphical differentiation of the experimentally obtained dependences: \(Э_1=f_1[L(S)]\), \(Э_2=f_2[L(S)]\), with a finite-difference representation of the differentials \((dЭ)_j\approx \Delta Э_j,\ j=1,2)\). The differentiation step was taken equal to \(5^\circ\).

The modulus of the plastic-strain increment vector was determined by the formula

\[ dЭ=|d\overline{Э}|=\sqrt{dЭ_1^2+dЭ_2^2}. \tag{4} \]

In the plane of the combined coordinates \(S_1(\Э_1)—S_2(\Э_2)\), by constructing the plastic-strain-increment vectors at the ends of the corresponding stress vectors, patterns of the process under study were constructed for several specimens; one of these is shown in Fig. 3. In analyzing the increments of plastic strains and in constructing patterns of the process, an approximate constancy was observed in the modulus of the plastic-strain-increment vector \(|d\bar{\Э}|^*\) and in the angular divergence \(\chi^*\) of this vector from the corresponding vector \(S\). This observation, together with results obtained earlier ([12], [13], [14], [15], [16]), leads to the conclusion that there exist characteristic regularities of the given process, which, for a constant differentiation step, have the form

\[ |d\bar{\Э}|^*=\mathrm{const};\qquad \chi^*=\mathrm{const}. \tag{5} \]

The second of these regularities denies the collinearity of the stress vectors and the plastic-strain-increment vectors, which is a necessary consequence of the principle of gradientality under conditions of isotropic hardening of materials. This makes it possible to regard the anisotropy of the hardening process as the cause of the established angular divergence.

Fig. 3

As a consequence of the experimentally observed constancy of \(\chi^*\) and \(|d\bar{\Э}|^*\), the constancy of the curvature of the plastic-strain trajectory follows. This makes it possible to note that the circular loading process is also accompanied by a circular process of plastic deformation. However, because of the presence of a preliminary plastic deformation \(\bar{\Э}_0\) and the angular divergence of the vectors \(d\bar{\Э}^*\) and \(S\), the circles of the loading trajectory and of the corresponding deformation trajectory are not concentric in the space of combined coordinates \(S_1(\Э_1)—S_2(\Э_2)\). Fig. 4 confirms that the actual arrangement of the experimental points corresponds to a circular one; the deviations do not exceed the experimental error. This fact leads to the conclusion that there is proportionality between the length of the plastic-strain circle and the length of the loading-trajectory circle \(\sigma_i=\mathrm{Const}\), which may be written in the form

\[ \Delta L(\Э)=n\cdot L(S). \tag{6} \]

This dependence is confirmed experimentally throughout the entire process \(\varphi_S=0\div180^\circ\), whereas regularity (1), obtained in [12], [13], [14], [15], [16], is valid only on the interval \(\varphi_S=0\div60^\circ\). The regularities of the plastic-strain increment (1), (6) include the loading parameter \(\varphi_S\), which, under the conditions of the given loading, is proportional to the angle of the type of stress state. This, in turn, indicates that changes in plastic deformation under loading conditions \(\sigma_i=\mathrm{Const}\), \(I_2S_{ij}=\mathrm{Const}\), and under

under the incompressibility condition of the material \(I_1(S_{ij})=0\), is entirely determined by the influence of the third invariant of the stress deviator. This introduces a certain

Fig. 4

clarity into the question of the role of the third invariant of the stress deviator in the theory of plastic deformations.

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