ON THE DEFORMATION OF SIMPLE SHEAR

G. A. DOSHCHINSKII, V. I. MAKSAK

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

In many technological problems of the theory of plasticity one encounters a deformed state that is called simple shear. The deformation of simple shear occurs in metal cutting, hole punching, certain types of stamping, and in a number of other technological operations.

In the simplest case, simple shear can be realized by twisting a thin-walled tube. Characteristic of this type of deformation is the continuous rotation of the axes of the stress tensor relative to the material particles of the body, i.e., conditions of complex loading are created.

The results of comparing experimental tensile and torsional curves show a certain discrepancy between them. At small deformations this discrepancy is insignificant and lies within the limits of experimental error. Thus, in experiments [1] on the tension and torsion of tubular specimens made of steel 3, the flow curve under torsion, taken in the coordinates $\sigma_i - e_i$, lies somewhat lower than under tension, while in experiments [2] with specimens made of steel 20 it lies somewhat higher.

At large deformations it is impossible to obtain tensile and torsional curves under a homogeneous stress state. Therefore attempts were made to analyze the stress state in the neck of a stretched specimen [3] and under torsion of a solid cylinder [4]. In this it turned out that, in the coordinates $\sigma_i - e_i$, the curve under torsion is located considerably lower than under tension. However, the construction of these curves was carried out with recalculation of the inhomogeneous stressed and deformed states, which is inevitably associated with certain errors.

Below an attempt is set forth to give a qualitative explanation of the nature and magnitude of the deviations of the flow curves under tension and torsion.

The deformation of simple shear is defined by the following relations between displacements [5]:

\[ u_1 = \gamma \cdot y, \qquad u_2 = u_3 = 0, \tag{1} \]

where $u_i$ are the displacement components,

$\gamma$ is the proportionality coefficient (shear).

To describe the deformed state at small deformations, one may use nine parameters that are linear combinations of displacement derivatives [6]:

\[ e_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right); \qquad \omega_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i}\right). \tag{2} \]

The geometric meaning of the quantities \(e_{ij}\) is clear. The quantities \(\omega_{ij}\) are proportional to the rotation of an infinitesimal volume surrounding the point at which they are defined. The rotation itself of an infinitesimal volume with respect to the axes \(x, y\) is expressed as:

\[ \operatorname{tg}\psi_x= \frac{\omega_x} {\sqrt{(1+e_{yy})(1+e_{zz})-\frac{1}{4}\gamma_{yz}^{2}}}, \]

\[ \operatorname{tg}\psi_y= \frac{\omega_y} {\sqrt{(1+e_{xx})(1+e_{zz})-\frac{1}{4}\gamma_{xz}^{2}}}, \tag{3} \]

\[ \operatorname{tg}\psi_z= \frac{\omega_z} {\sqrt{(1+e_{xx})(1+e_{yy})-\frac{1}{4}\gamma_{xy}^{2}}}. \]

As applied to simple shear, of the parameters (2) only two will be nonzero,

\[ \gamma_{xy}=\gamma;\qquad \omega_z=-\frac{\gamma}{2}. \tag{4} \]

Infinitesimal elements, according to (3), undergo the rotation

\[ \operatorname{tg}\psi_z=-\frac{\gamma}{\sqrt{4-\gamma^2}}. \tag{5} \]

In [7] an approximate law relating stresses and deformations is given, which in vector form may be written as follows:

\[ |d\bar{\Э}_p|=k\bigl(|\bar p|-|\bar p_s|\bigr)|d\bar T|+\Phi'(|\bar p|)\,d\bar N, \tag{6} \]

where

\[ |\bar{\Э}|=\sqrt{\frac{3}{2}\,e_i};\qquad |\bar p|=\sqrt{\frac{2}{3}\,\sigma_i}, \]

\(k\) is the proportionality coefficient, constant for the given material,

\(\Phi(|\bar p|)\) is a function determined from a tensile test. The quantity \(|d\bar T|\) can be determined as

\[ |d\bar T|=|\bar p|\,d\psi, \tag{7} \]

where

\[ d\psi=\frac{d\gamma}{\sqrt{4-\gamma^2}} \tag{8} \]

is obtained by differentiating (5). The quantity \(|d\bar N|\) is equal to \(|d\bar p|\), the increment of the stress vector; therefore (6) takes the form

\[ |d\bar{\Э}_p|=k\bigl(|\bar p|-|\bar p_s|\bigr)|\bar p|\,d\psi +\Phi'(|\bar p|)|d\bar p|. \tag{9} \]

After integrating (9) we have:

\[ |\bar{\Э}_p|= k\int_{|\bar p_s|}^{|\bar p|} \bigl(|\bar p|-|\bar p_s|\bigr)|\bar p|\,d\psi +\bigl[\Phi(|\bar p|-\Phi(|\bar p_s|)\bigr]. \tag{10} \]

The second term determines the plastic deformation that would have been obtained as a result of simple loading. The first term ...

gives the value of the plastic strain that appeared as a result of the rotation of the material particles. Therefore, the flow curve in torsion will lie below the curve in tension. The order of their ratio will also depend on the magnitude of the first term.

For small deformations, in (8) the quantity $\gamma^2$ may be neglected. Since $d\gamma = |d\overline{\mathcal E}^{p}|/\sqrt{2}$, replacing $|d\overline{\mathcal E}^{p}|$ by $\Phi'(|\overline p|)\,|d\overline p|$, (8) can be rewritten in the following form

\[ de_i=\frac{d\gamma}{2}=\frac{|d\overline{\mathcal E}^{p}|}{2\sqrt{2}} =\frac{\Phi'(|\overline p|)\,|d\overline p|}{2\sqrt{2}} . \tag{11} \]

Denoting the first term in (10) by $|\overline{\mathcal E}^{p}_{\psi}|$, and taking (11) into account, it may be rewritten in the form

\[ |\overline{\mathcal E}^{p}_{\psi}|= \frac{k}{2\sqrt{2}} \int_{|\overline p_s|}^{|\overline p|} \bigl(|\overline p|-|\overline p_s|\bigr)\,|\overline p|\,\Phi'(|\overline p|)\,|d\overline p| . \tag{12} \]

From the known curve in tension $|\overline{\mathcal E}^{p}|=\Phi(|\overline p|)$, the value of the integral (12) can be determined by numerical methods or, by approximating $\Phi(|\overline p|)$, for example, by a power function, by direct integration

\[ |\overline{\mathcal E}^{p}|=\Phi(|\overline p|)=B\cdot |\overline p|^{\beta}. \tag{13} \]

Substituting (13) into (12) and integrating, we obtain

\[ |\overline{\mathcal E}^{p}_{\psi}|= \frac{B\beta k}{2\sqrt{2}(\beta+1)(\beta+2)} \left[(\beta+1)|\overline p|^{\beta+2} -(\beta+2)|\overline p_s|\,|\overline p|^{\beta+1} +|\overline p_s|\right]. \tag{14} \]

For steel 45 [7], $B \simeq 1.2\cdot 10^{-8}$; $\beta \simeq 3.9$; $k=2.86\cdot 10^{-4}$; $|\overline p_s|=29.4\ \text{kg}/\text{mm}^2$. For the last experimental point $|\overline p|=53.8\ \text{kg}/\text{mm}^2$. Substituting these data into (14), one can calculate the magnitude of the additional plastic strain that would be obtained in torsion of a tubular specimen made of this material. It turns out that $|\overline{\mathcal E}^{p}_{\psi}|\simeq 0.0019$, which corresponds to a strain intensity $e_i\simeq 0.00155$. The value $|\overline p|=53.8\ \text{kg}/\text{mm}^2$ corresponds to a strain $e_i\simeq 8\%$. At the same value of $|\overline p|$ in torsion, the strains will be greater by $0.16\%$, which is about half of the elastic strain, and the tension and torsion curves will practically coincide. In stresses these curves will differ even less.

For large deformations all the expressions become more complicated, and in (12) the integration cannot be carried through to the end. It therefore seems expedient to analyze qualitatively, at least in a first approximation, the order of the divergence of the tension and torsion curves at large deformations, i.e., the order of magnitude of the additional plastic strain that arises as a result of the rotation of particles in simple shear.

The results of experiments [8] on complex loading with constant stress intensity at large deformations showed that the increment of plastic strains in the region of complex loading is small in comparison with the preliminary strain. Thus, in torsion of a specimen [8] up to the value $\sigma_i=83\ \text{kg}/\text{mm}^2$, he obtained a shear of about $0.83$, which corresponds to a rotation of the material particles relative to the stress vector through an angle $\psi\simeq 25^\circ$. The subsequent complex loading $\sigma_i=\mathrm{Const}=83\ \text{kg}/\text{mm}^2$ caused a rotation of the stress vector relative to the particles through $90^\circ$ and an increment of plastic strain

on the order of 0.4%. Therefore, one may suppose that, at the first stage of loading in simple shear, a rotation of the stress vector by 25° caused an additional deformation of no more than 0.4%. This value of the additional deformation will have little effect on the flow curves in tension and torsion at large deformations.

To verify these assumptions, the following experiments were carried out. On the basis of the method for studying large deformations [9], experimental flow curves in pure torsion in the coordinates $\sigma_i — e_i$ were obtained for steel 3, ShKh15, EI1T, and brass from tests of three pairs of solid specimens for each material, as well as flow curves in compression for three cylindrical specimens of each material. In Fig. 1, in coordinates of true stresses and strains, the results of torsion tests are plotted with light circles, and the mean values of the data for three specimens in compression with dark circles. It is seen that, for each material, all the experimental points fall very close to a single curve.

Fig. 1

The investigations carried out make it possible to draw the following conclusions. Deformation by simple shear is a complex loading with rotation of the axes of the stress tensor.

Therefore, in principle, the torsion and tension–compression diagrams cannot coincide; however, their deviations in strain, in accordance with the previously established regularities of the increment of plastic deformation upon rotation of the axes [7, 8, 9], are so small that they lie within the limits of experimental error.

With a sufficient degree of accuracy it may be noted that, for both small and large deformations in tension–compression and torsion of isotropic metals, the flow curves in the coordinates $\sigma_i — e_i$ coincide.

References

1. G. A. Doshchinskii. On the foundations of the theory of elastic-plastic deformation. Dissertation for the degree of Candidate of Technical Sciences. Tomsk. 1958.

2. A. M. Zhukov. Plastic deformations of steel under complex loading. Izv. AN SSSR, OTN, No. 11, 1954.

3. N. N. Davidenkov, N. I. Spiridonova. Analysis of the state of stress in the neck of a tensile specimen. Zavodskaya laboratoriya, No. 6, 1945.

4. N. N. Davidenkov. Determination of stresses in plastic torsion. Zavodskaya laboratoriya, No. 6, 1948.

5. A. Nadai. Plasticity and Fracture of Solids. IL, 1954.

6. V. V. Novozhilov. Theory of Elasticity. Sudpromgiz. 1958.

7. V. I. Maksak. Plastic deformation of steel under complex loading. Izv. TPI, vol. 139, 1966.

8. V. I. Maksak, G. A. Doshchinskii. Investigation of large plastic deformations under complex loading. Izv. TPI, the present collection.

9. V. I. Maksak, G. A. Doshchinskii. Methodology and investigation of large plastic deformations under simple loading. Izv. TPI, the present collection.