STUDY OF THE PATTERNS OF PLASTIC DEFORMATION DURING ROTATION OF THE PRINCIPAL AXES OF THE STRESS TENSOR

G. A. Doshchinskii, N. V. Pogozheva
(Submitted by the scientific seminar of the Department of Strength of Materials)

The influence of rotation of the axes of the stress tensor on the plastic state of materials is, in the theory of plasticity, almost unstudied, despite the large number of investigations of plastic deformation with implementation of loading programs of the type \(P—M\) and \(P—M—q\) [1], [2], [3], and others. The rotation of the principal axes of stresses carried out in the cited works was accompanied by changes in other characteristics of the stressed state as well, which did not make it possible to isolate, in pure form, the role of rotation of the principal axes in the process of plastic deformation and, consequently, to establish the patterns of the latter associated with this rotation.

The stressed state of the material under the plane scheme realized in the experiments is completely determined by two scalar values of the principal stresses and one angle of rotation \(\alpha\) of the principal axes. The condition of constancy of the principal stresses corresponds to invariance of the second and third invariants of the stress deviator

\[ I_2(S_{ij})=\mathrm{const}, \qquad I_3(S_{ij})=\mathrm{const}. \tag{1} \]

The first invariant of the stress deviator (taking into account the generally accepted assumption that hydrostatic pressure has no influence on the process of change of form) is equal to zero.

\[ I_1(S_{ij})=0 \tag{2} \]

Conditions (1), in turn, determine the invariance of the stress intensity and of the type of stressed state

\[ \sigma_i=\sqrt{\sigma_1^2+\sigma_2^2-\sigma_1\cdot\sigma_2} =\frac{1}{2}\sqrt{I_2(S_{ij})}=\mathrm{const}, \tag{3} \]

\[ \operatorname{tg}\omega_s= \frac{1}{\sqrt{3}}\, \frac{I_3(S_{ij})}{[2I_2(S_{ij})]^{3/2}}. \]

Consequently, complex loading realized under condition (3) makes it possible to reveal the patterns of plastic behavior of materials caused by change only in the angle of rotation of the axes of the stress tensor. The values of all the other loading parameters remain unchanged in this case.

The realization of complex loading of this kind was carried out in \(P—q—M\) experiments on thin-walled tubular specimens [4], [5], [6], [7], [8]. The angle of rotation of the principal axes in these works lay in the range \(\alpha=0\div90^\circ\). As a result of these experiments, at various \(\sigma_i=\mathrm{Const}\)

there was observed, in the course of rotation of the axes, an increment of plastic strain approximately proportional to the angle of rotation (or to the arc of the loading trajectory). The degree of the increment of plastic strain was greater the smaller the realized ratios of the principal stresses, \(\sigma_2/\sigma_1\).

The aim of the investigations described in this article is to trace the character of the increment of plastic strains at large angles of rotation of the axes \((\alpha = 0 \div 180^\circ)\), and also to study the vector properties of the increments of plastic strains under these loading conditions. In the experiments, copper tubes \((D = 35\ \mathrm{mm},\ t = 1.5\ \mathrm{mm})\) and steel (St. 35, St. 45) tubular specimens \((D = 30\ \mathrm{mm},\ t = 1\ \mathrm{mm})\) were used.

On the basis of the results of experiments carried out according to the procedure described in [5], [6], graphs were plotted of the increment of the intensity of plastic strain as a function of the angle of rotation of the axes (Figs. 1, 2, 3), showing the approximate proportionality of the increment of plastic strain to the angle of rotation of the axes, which had also been noted earlier [5], [6], [7], [8].

However, in these experiments, owing to the large angle of rotation, a new circumstance was revealed, namely that the proportionality noted occurs only over a portion bounded by an angle of rotation of the axes of \(90^\circ\). Further rotation of the principal axes shows a significant slowing of the increment of plastic strain. The angle \(\alpha = 90^\circ\) is a break point in the curves \(\Delta Э_i = f(a)\).

Fig. 1

On some copper specimens, after carrying out a rotation of the principal stress axes through \(180^\circ\) and complete unloading of the specimen, repeated loading was performed according to the same program. In this case a continuation of the increment of plastic strains is found, the regularity of which is similar to the regularity of the increment during the first rotation, but its degree in the latter case is considerably lower (Fig. 2). This phenomenon requires more detailed study and explanation.

Fig. 2

Graphical differentiation of the obtained dependences \(\Delta Э_i = f(a)\) makes it possible to conclude that the modulus of the vector of strain increments remains constant throughout the entire loading process.

To study the vector properties of increments of plastic deformation, let us represent geometrically the loading process in the \(S_1—S_2—S_3\) space of A. A. Ilyushin’s five-dimensional space. Upon reaching the curve \(\sigma_i = \mathrm{const}\) by simple loading \(P—q\), the stress vector \(S\) is located entirely in the \(S_1—S_2\) plane. When the specimen is twisted by a moment \(M\), a third component of the stress vector \(S_3\) appears, and it

Fig. 3

begins to describe a conical surface with its vertex at the origin of coordinates (the cone angle being determined by the imposed ratio of the principal stresses \(\frac{\sigma_2}{\sigma_1}=m\)). The loading trajectory will be represented by a circle described by the end of the vector \(S\). Graphical differentiation of the experimental curves \(\mathcal{E}_j=f_j(\alpha)\) \((j=1,2,3)\) makes it possible to determine the three components of the vector of increments of plastic strains at each point of the loading trajectory. From the expression for the scalar product of the stress vector and the strain-increment vector,

\[ \bar S \cdot d\bar{\mathcal{E}} = |S| \cdot |d\mathcal{E}| \cdot \cos \chi \tag{4} \]

the angular divergence between the vectors for the combined deviatoric subspaces of stresses and strains is determined by the relation

\[ \cos \chi = \frac{\bar S \cdot d\bar{\mathcal{E}}}{|S|\cdot |d\mathcal{E}|} = \frac{S_1\cdot d\mathcal{E}_1+S_2\cdot d\mathcal{E}_2+S_3\cdot d\mathcal{E}_3} {\sqrt{S_1^2+S_2^2+S_3^2}\cdot \sqrt{d\mathcal{E}_1^2+d\mathcal{E}_2^2+d\mathcal{E}_3^2}} . \tag{5} \]

Substitution into this formula of the strain increments \(d\mathcal{E}_1, d\mathcal{E}_2, d\mathcal{E}_3\), obtained by graphical differentiation, as well as of the corresponding values \(S_1, S_2, S_3\), was used to determine the angular divergence \(\chi\), which proved to be practically constant over the entire length of the loading trajectory. However, the noted constancy of the angular divergence between the vectors \(\bar S\) and \(d\bar{\mathcal{E}}\) does not yet fully determine the orientation of the vector of the plastic-strain increment, since the loading process is reflected not by a plane but by a spatial geometrical picture. To clarify fully the vector properties of the plastic-strain increment, spatial representations of the loading processes were constructed. A schematic depiction of such a representation of the process in the space \(S_1(\mathcal{E}_1), S_2(\mathcal{E}_2), S_3(\mathcal{E}_3)\) is shown in Fig. 4, a. In constructing representations of the processes and analyzing them, it became clear that the vectors of plastic-strain increments lie in planes tangent to the conical surface at the given loading point. The picture of the process obtained on the development of the conical surface (Fig. 4, b) confirms that the strain-increment vectors are constant in magnitude, while the direc-

make with the directions of the corresponding strain vectors a certain angle \(\chi\), approximately constant along the entire length of the loading trajectory:

\[ |d\bar{Э}| \approx \mathrm{Const}, \quad \chi \approx \mathrm{Const}. \tag{6} \]

A comparison of the mean values of the angle shows their closeness to the angle \(\chi^{*}\), determined for the corresponding conditions when the type of stress state is changed. An analogous comparison of the values of the moduli of the increments of plastic strains with the moduli obtained in loadings for the same materials and the same value \(\Э_i = \mathrm{Const}\), but with a change in the type of stress state, leads to the conclusion

\[ |d\bar{Э}| = |d\bar{Э}|^{*}. \]

Adoption of the conditions

\[ \chi = \chi^{*}, \quad |d\bar{Э}| = |d\bar{Э}|^{*} \tag{7} \]

makes it possible to conclude that the transition from a process of simple loading to a complex one, at the same stress intensity, by a change in the type of stress state with fixed principal axes, or by a rotation of the principal axes with fixed values \(\sigma_1 = \mathrm{Const}\), \(\sigma_2 = \mathrm{Const}\), is accompanied by an increase in plastic strains, the vector and scalar properties of which in both cases are expressed by common regularities. This commonality, apparently, is determined by a single cause—the anisotropy of the hardening process.

Fig. 4

LITERATURE

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