GEOMETRIC REPRESENTATION OF PATTERNS OF ELASTIC-PLASTIC DEFORMATION

G. A. DOSHCHINSKY

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

The basis of the theory of elastic-plastic deformation is a series of hypotheses which, in the most general form given by A. A. Ilyushin [1], are represented by the following propositions:

1. The linear invariants of the stress and strain tensors are proportional

\[ \sigma = K\Theta . \tag{1} \]

1. The directing tensors of stresses and strains coincide

\[ (\overline{D}_s) = (\overline{D}_e). \tag{2} \]

1. The stress intensity is a definite function of the strain intensity, independent of the type of stressed state,

\[ \sigma_i = \Phi(e_i). \tag{3} \]

These propositions have received practically sufficient experimental confirmation. The best agreement with experiment of the indicated hypotheses occurs when the properties of materials are isotropic and the conditions of simple loading are observed, i.e., loading in which all components of the stress tensor increase proportionally to a single parameter \(\lambda\). The considerable development of the theory of plasticity, especially in recent times, has made it possible to obtain a large number of practically important solutions.

However, the mathematical theory of plasticity, being constructed on purely analytical categories, lacks visual clarity. This circumstance, to a certain extent, makes it difficult to improve the theory, requires special mathematical preparation from those wishing to study the foundations of the theory, and sometimes leads to differing interpretations of the basic concepts.

Below we consider the possibility of a visual geometric representation of the principal dependencies that occur in the process of elastic-plastic deformation (the most studied case of loading is analyzed—simple loading).

The interpretation presented includes the usual method of graphical representation in stress space of the conditions of pro-

of elasticity and plasticity, supplemented by certain geometric relations. In this case, the rather intangible concepts—the intensity of stresses and strains—are replaced by certain new categories. The latter, however, does not lead to any substantial change in the relations characteristic of elastic-plastic deformation and reflected by the initial hypotheses of the theory noted above. In the exposition a more general case is considered, free from hypothesis (1), i.e., the possibility of the existence of residual volumetric deformation is assumed.

One of the initial premises of the theory of elastic-plastic deformation is the assumption that the directions of the principal stresses and strains coincide. For a geometric reflection of this condition, we combine the systems of orthogonal coordinates \(\sigma_1, \sigma_2, \sigma_3\) and \(e_1, e_2, e_3\), denoting the axes respectively by 1, 2, 3. Without imposing any conditions on the relations of the principal stresses and strains, we shall regard all axes as equivalent. The axis of symmetry of the stress and strain spaces \(x\) forms equal angles \(\varphi\) with the coordinate axes 1, 2, 3,

\[ \cos \varphi = \frac{1}{\sqrt{3}} . \tag{4} \]

Let us introduce the concept of mean strain as a quantitative measure of strain for the general case. Assuming, for plastic materials, equal resistance to tension and compression, i.e., taking into account only the absolute values of the strains, we shall calculate the mean values as the root-mean-square deviation from the initial undeformed state

\[ e_{\mathrm{cp}} = \sqrt{\frac{\sum_{i=1}^{i=n} e_i^2}{n-1}}, \]

where \(n\) is the number of measurements.

Strain in the general case is characterized by three measurements. The behavior of a material under elastic-plastic deformation is established on the basis of an analysis of its behavior in some initial experiment. Since in equivalent states the denominator can be reduced, in what follows we shall call the quantity

\[ e_{\mathrm{cp}} = \sqrt{e_1^2 + e_2^2 + e_3^2}. \tag{5} \]

the mean strain. Similarly, as a measure of the state of stress, we shall take the mean stress

\[ \sigma_{\mathrm{cp}} = \sqrt{\sigma_1^2 + \sigma_2^2 + \sigma_3^2}. \tag{6} \]

The strain components under elastic-plastic deformation may be expressed in terms of the stresses as follows:

\[ e_1 = \frac{1}{E(\lambda)} \left[\sigma_1 - \mu(\lambda)(\sigma_2+\sigma_3)\right], \qquad e_2 = \frac{1}{E(\lambda)} \left[\sigma_2 - \mu(\lambda)(\sigma_3+\sigma_1)\right], \]

\[ e_3 = \frac{1}{E(\lambda)} \left[\sigma_3 - \mu(\lambda)(\sigma_1+\sigma_2)\right], \]

where \(E(\lambda)\) and \(\mu(\lambda)\) are characteristics of the material at the given stage of deformation \([2, 3, 4, 6]\).

As was shown earlier \([5]\), the quantity \(e_{cp}\) proves to be a sufficiently reliable general measure of deformation, independent of the type of stressed state. The condition of equivalence of states obtained on the basis of this criterion, expressed in stresses (7), agrees well with experimental data

\[ \sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2-k(\lambda)(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1)}=\sigma_s . \tag{7} \]

Using the usual method of spatial interpretation of the conditions of strength and plasticity, it can be shown that the equivalent states of the material in terms of stresses are determined by points located on the surface of an ellipsoid of revolution with longitudinal axis \(x\)

\[ \frac{x^2}{a^2}+\frac{y^2+z^2}{b^2}=1, \tag{8} \]

where

\[ a=\frac{\sigma_s}{\sqrt{1-k(\lambda)}};\qquad b=\frac{\sqrt{2}\cdot\sigma_s}{\sqrt{2+k(\lambda)}};\qquad \]

\[ k(\lambda)=\frac{4\mu(\lambda)-2\mu(\lambda)^2}{1+2\mu(\lambda)^2}. \]

The surface determining the equivalent states in terms of deformation will be a surface of complete polar symmetry—a sphere with radius

\[ R=e_{\text{plast.}}\sqrt{1+2\mu(\lambda)^2}. \tag{9} \]

The presence of complete symmetry of the determining surface is, in this case, a reflection of the initial assumption in the theory concerning the complete isotropy of the material.

The mean stress \(\sigma_{cp}\) and the mean deformation \(e_{cp}\) are purely quantitative measures of stress and deformation. In order to give some qualitative characteristic of the stressed and deformed state, let us introduce the concept of vector quantities of the resultant deformation \(\bar e_p\) and the resultant stress \(\bar\sigma_p\), defined as the geometric sum of the principal deformations or stresses

\[ \bar e_p=\bar e_1+\bar e_2+\bar e_3;\qquad \bar\sigma_p=\bar\sigma_1+\bar\sigma_2+\bar\sigma_3. \tag{10} \]

The direction of the components of the principal deformations and stresses, when their relative values are taken into account, will determine a certain orientation of the vectors \(\bar e_p\) and \(\bar\sigma_p\) in the coordinate-axis system 1, 2, 3. This, in turn, will make it possible to characterize the qualitative aspect of the stressed and deformed state. Thus, for example, coincidence of \(\bar e_p\) with the axis of symmetry of the space \(x\) in the positive or negative octant will indicate that only volumetric deformation is taking place—compaction or loosening of the material. Any deviation

\(\overline e_p\) from the \(x\)-axis will indicate the presence of shear deformation and characterize the ratio of the changes in volume and shape change. The same may also be said with respect to the vector \(\overline \sigma_p\).

The vectors \(\overline \sigma_p\) and \(\overline e_p\), drawn from the origin of coordinates \(1, 2, 3\), do not determine the stress and deformation in any particular direction, but are understood here only as quantities whose resolution along the axes \(1, 2, 3\) gives the values of the principal stresses and deformations¹. The moduli of the vectors \(\overline \sigma_p\) and \(\overline e_p\) are equal to the values of the mean stress (5) and the mean deformation (6).

The direction cosines of these vectors will be:

\[ \begin{aligned} &\text{for } \overline \sigma_p && \cos \alpha'=\frac{\sigma_1}{\sigma_p}, \qquad \cos \beta'=\frac{\sigma_2}{\sigma_p}, \qquad \cos \gamma'=\frac{\sigma_3}{\sigma_p},\\[6pt] &\text{for } \overline e_p && \cos \alpha''=\frac{e_1}{e_p}, \qquad \cos \beta''=\frac{e_2}{e_p}, \qquad \cos \gamma''=\frac{e_3}{e_p}. \end{aligned} \tag{11} \]

Between the numerical values of \(\overline \sigma_p\) and \(\overline e_p\) there exists a relation that may be found from the initial tension diagram. To establish the connection with the tension diagram that determines the mechanical properties of the material, let us find the values of the mean deformation and mean stress in equivalent states. In doing so we shall proceed from the generalized flow curve

\[ \sigma_{\mathrm{eq}}=\Phi(e_{\mathrm{eq}}) \tag{12} \]

\[ \sigma_{\mathrm{eq}}= \sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2-k(\lambda)(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1)} ; \]

\[ e_{\mathrm{eq}}=\frac{e_{\mathrm{avg}}}{\sqrt{1+2\mu(\lambda)^2}}, \]

considered earlier [6] and in good agreement with experiment.

For a certain stressed state in which

\[ \sigma_1=\sigma_1; \qquad \sigma_2=m\sigma_1; \qquad \sigma_3=n\sigma_1, \]

the mean stress is

\[ \sigma_p=\sqrt{\sigma_1^2+m^2\sigma_1^2+n^2\sigma_1^2} =\sigma_1\sqrt{1+m^2+n^2}. \tag{13} \]

The magnitude of the stress \(\sigma_1\), equivalent to the tensile stress \(\sigma\) in a tension test, may be expressed from the condition of equivalence in stresses (7)

\[ \sqrt{\sigma_1^2+m^2\sigma_1^2+n^2\sigma_1^2 -k(\lambda)(m\sigma_1^2+n\sigma_1^2+mn\sigma_1^2)} =\sigma, \]

whence

\[ \sigma_1= \frac{\sigma}{\sqrt{1+m^2+n^2-k(\lambda)(m+n+mn)}} . \tag{14} \]

The tensile stress \(\sigma\), equivalent to the prescribed value \(\sigma_{cp}\), is determined from expressions (13) and (14) as:

\[ \sigma = \frac{\sigma_{cp}}{A}, \tag{15} \]

where

\[ A = \sqrt{\frac{1 + m^{2} + n^{2}}{1 + m^{2} + n^{2} - k(\lambda)(m + n + mn)}} . \]

\(A\) is the coefficient of the type of stress state (for simple loading, when using the incompressibility condition, \(A=\mathrm{const}\)). The quantity \(e_{cp}\), corresponding to the value \(\sigma_{cp}\) from the generalized diagram \(\sigma\)—\(e\), can be found from expressions (12)

\[ e = \frac{e_{cp}}{\sqrt{1 + 2\mu(\lambda)^2}}, \qquad e_{cp} = e \cdot \sqrt{1 + 2\mu(\lambda)^2}. \tag{16} \]

Thus one may say that the quantities \(\sigma_{cp}\) and \(e_{cp}\) (taking into account (15), (16) and the tensile diagram \(\sigma = E(\lambda)\cdot e,\ \mu=\mu(\lambda)\)) are related, for any stress state, by the dependence

\[ \frac{\sigma_{cp}}{A} = E(\lambda)\frac{e_{cp}}{\sqrt{1 + 2\mu(\lambda)^2}}, \tag{17} \]

where \(E(\lambda)\) is the secant modulus of the tensile diagram. For \(m=n=0\), dependence (17) reduces to the tensile diagram, allowing, from the data of a tensile test of the material \(\sigma=f_1(e),\ \mu(\lambda)=f_2(e)\), the magnitude \(e_{cp}\) to be found for a prescribed value \(\sigma_{cp}\). The possibility of solving such a problem (even if only by successive approximations) follows from the good agreement of the original equation (12) with experimental data.

Let us determine the mutual disposition of the vectors \(\bar{\sigma}_p\) and \(\bar{e}_p\) in the combined coordinate system \(\sigma_1,\ \sigma_2,\ \sigma_3\) and \(e_1,\ e_2,\ e_3\). In this case the ratio of the projections of \(\bar{\sigma}_p\) and \(\bar{e}_p\) onto the coordinate axes will correspond to the dependences between the components of the stressed and deformed states, i.e., will give the physical equations.

The equation of a plane passing through two directions and the point of their intersection \(x_1;\ y_1;\ z_1\) has the form:

\[ (x-x_1)(\cos\beta_1\cdot\cos\gamma_2-\cos\beta_2\cdot\cos\gamma_1) + (y-y_1)(\cos\gamma_1\cdot\cos\alpha_2 \]

\[ -\cos\gamma_2\cdot\cos\alpha_1) + (z-z_1)(\cos\alpha_1\cdot\cos\beta_2-\cos\alpha_2\cdot\cos\beta_1)=0. \]

The direction cosines of the axis of symmetry of the space and of the vector of the resultant strain are expressed by relations (4) and (11). Substituting these values and taking into account that the point of intersection is the origin of coordinates, we have:

\[ x\left(\frac{e_3}{e_p\sqrt{3}}-\frac{e_2}{e_p\sqrt{3}}\right) + y\left(\frac{e_1}{e_p\sqrt{3}}-\frac{e_3}{e_p\sqrt{3}}\right) + \tag{18} \]

\[ + z\left(\frac{e_2}{e_p\sqrt{3}}-\frac{e_1}{e_p\sqrt{3}}\right)=0. \]

After cancellations, the equation of the plane passing through the vector of the resultant deformation and the axis of symmetry of the space takes the form:

\[ x(e_3-e_2)+y(e_1-e_3)+z(e_2-e_1)=0. \tag{19} \]

The direction cosines of the vector of the resultant stress and of the axis of symmetry of the stress space are given by relations (4) and (11). The point of intersection of these directions is also the origin of coordinates. Therefore, on the basis of equation (18), we obtain:

\[ x(\sigma_3-\sigma_2)+y(\sigma_1-\sigma_3)+z(\sigma_2-\sigma_1)=0. \tag{20} \]

From the analytical expressions defining the law of similarity of the deviators (hypothesis 2), the following relations follow:

\[ \sigma_3-\sigma_2=2G(e_3-e_2), \qquad \sigma_1-\sigma_3=2G(e_1-e_3), \]

\[ \sigma_2-\sigma_1=2G(e_2-e_1). \]

Substituting into equation (20) and canceling by \(2G\), we obtain:

\[ x(e_3-e_2)+y(e_1-e_3)+z(e_2-e_1)=0 \]

This means that the planes: the axis of symmetry of the stress space—the vector of the resultant stress and the axis of symmetry of the strain space—the vector of the resultant deformation, coincide.

In the same way it can be shown that the planes: the axis of symmetry of the \(x\)-space—the vector of the resultant stress and the axis of symmetry of the \(x\)-space—the vector of the resultant elastic deformation \(\varepsilon_p\) (defined below) prove to be coincident.

The components of the total (elastic-plastic) deformation consist of two parts: elastic and residual, i.e., analytically they are represented in the form of algebraic sums:

\[ e_1=\varepsilon_1+\delta_1; \qquad e_2=\varepsilon_2+\delta_2; \qquad e_3=\varepsilon_3+\delta_3. \tag{21} \]

This notation does not imply any particular relations among the components of deformation, but reflects only the fact of the existence of elastic and residual deformation. Therefore the value of the resultant total deformation can be determined in the following way:

\[ e_{cp}=\sqrt{(\varepsilon_1+\delta_1)^2+(\varepsilon_2+\delta_2)^2+(\varepsilon_3+\delta_3)^2}= \]

\[ =\sqrt{(\varepsilon_1^2+\varepsilon_2^2+\varepsilon_3^2)+(\delta_1^2+\delta_2^2+\delta_3^2)+2(\varepsilon_1\delta_1+\varepsilon_2\delta_2+\varepsilon_3\delta_3)}. \tag{22} \]

For a “purely elastic” deformation, in which no residual changes are observed along any of the principal directions \((\delta_1=\delta_2=\delta_3=0)\),

\[ e_{cp}=\sqrt{\varepsilon_1^2+\varepsilon_2^2+\varepsilon_3^2}=\varepsilon_{cp}. \]

The vector \(\overline{\varepsilon}_p\) with modulus \(\varepsilon_p\) shall be called the vector of the resultant elastic deformation. Its direction is determined by the cosines:

\[ \cos\alpha^*=\frac{\varepsilon_1}{\varepsilon_p};\quad \cos\beta^*=\frac{\varepsilon_2}{\varepsilon_p};\quad \cos\gamma^*=\frac{\varepsilon_3}{\varepsilon_p}. \tag{23} \]

The directed quantity \(\overline{\delta}_p\) with modulus

\[ \delta_p=\delta_{cp}=\sqrt{\delta_1^2+\delta_2^2+\delta_3^2} \]

will analogously be called the vector of the resultant plastic deformation. On the basis of the theorem on the projection of a resultant, taking (21) into account, we arrive at the expression

\[ \overline{e}_p=\overline{\varepsilon}_p+\overline{\delta}_p. \tag{24} \]

For large deformations, when the components of the elastic deformation may be neglected, i.e., when one may put

\[ \varepsilon_1=\varepsilon_2=\varepsilon_3=0, \]

on the basis of (22) and (24) we obtain

\[ \overline{e}_p=\overline{\delta}_p. \]

Analogously, for “purely elastic” deformation \(\overline{e}_p=\overline{\varepsilon}_p\).

Proceeding from (19) and (24), one may conclude that the geometric representation of the law of proportionality of tangential stresses to the corresponding shears (or of the law of similarity of deviators) is the arrangement of the vectors \(\overline{\sigma}_p\), \(\overline{e}_p\), \(\overline{\varepsilon}_p\), \(\overline{\delta}_p\) in one plane passing through the axis of symmetry of the space \(x\).

To establish the mutual orientation of the vectors \(\overline{\sigma}_p\) and \(\overline{e}_p\), it is now sufficient to determine the angle between them. The equation of a straight line passing through the origin of coordinates and making the angles \(\alpha,\beta,\gamma\) with the coordinate axes has the form:

\[ \frac{x}{\cos\alpha}=\frac{y}{\cos\beta}=\frac{z}{\cos\gamma}. \]

For the vector of the resultant deformation,

\[ \frac{e_1}{\cos\alpha''}=\frac{e_2}{\cos\beta''}=\frac{e_3}{\cos\gamma''}, \]

and for the resultant stress,

\[ \frac{\sigma_1}{\cos\alpha'}=\frac{\sigma_2}{\cos\beta'}=\frac{\sigma_3}{\cos\gamma'}. \]

The angle between two straight lines is determined by the relation

\[ \cos\psi=\cos\alpha'\cdot\cos\alpha''+\cos\beta'\cdot\cos\beta''+\cos\gamma'\cdot\cos\gamma''. \]

Therefore the angle \(\psi\) between the vectors \(\overline{\sigma}_p\) and \(\overline{e}_p\) can be determined in the following way:

\[ \cos \psi = \frac{\sigma_1 e_1+\sigma_2 e_2+\sigma_3 e_3}{\sigma_p e_p}, \]

whence

\[ \psi=\arccos \frac{\sigma_1^2+\sigma_2^2+\sigma_3^2-2\mu(\lambda)(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1)} {E(\lambda)\cdot \sigma_p\cdot e_p}. \tag{25} \]

In particular, in shear \(\overline{\sigma}_p\) and \(\overline{e}_p\) coincide in direction \((\psi=0)\). Since the vectors \(\overline{\sigma}_p\) and \(\overline{e}_p\) are located in one plane with the axis of symmetry of the space, another expression can also be given for the angle \(\psi\)

\[ \psi=\psi''-\psi', \tag{26} \]

where \(\psi'\) and \(\psi''\) are the deviations of the vectors \(\overline{\sigma}_p\) and \(\overline{e}_p\) from the axis of symmetry \(x\).

The direction of the vector \(\overline{e}_p\) in deformation space makes it possible to establish the relation of this characteristic of deformation to the change in volume. If small quantities of higher order are neglected, then the magnitude of the relative change in volume, as is known, is determined by the algebraic sum of the strains

\[ \Theta=e_1+e_2+e_3. \tag{27} \]

The projections of the principal strain components onto the axis of symmetry of the space are respectively expressed as:

\[ n_{p(x)}e_1=e_1\cdot \cos \varphi=-\frac{e_1}{\sqrt{3}}; \qquad n_{p(x)}e_2=e_2\cdot \cos \varphi=\frac{e_2}{\sqrt{3}}; \]

\[ n_{p(x)}e_3=e_3\cdot \cos \varphi=\frac{e_3}{\sqrt{3}}. \]

On the basis of the theorem on the projection of the resultant, it follows that the change in volume is proportional to the magnitude of the projection of the resultant strain vector onto the axis of symmetry of the space

\[ n_{p(x)}\overline{e}_p= \frac{e_1+e_2+e_3}{\sqrt{3}} = \frac{\Theta}{\sqrt{3}} = e_p\cdot \cos \psi''. \]

Invariance of volume during deformation is characterized by the sliding of the end of the vector \(\overline{e}_p\) along a plane perpendicular to the \(x\)-axis. If the change in volume is neglected, this plane will pass through the origin of coordinates.

Similarly, the elastic change in volume can be determined through the projection of the resultant elastic strain onto the \(x\)-axis

\[ n_{p(x)}\overline{\varepsilon}_p= \frac{\varepsilon_1+\varepsilon_2+\varepsilon_3}{\sqrt{3}} = \frac{\Theta_{\mathrm{el}}}{\sqrt{3}} = \varepsilon_p\cdot \cos \psi^*. \]

The deviation of \(\overline{e}_p\) from the axis of symmetry of the space

\[ \psi''=\arccos \frac{\Theta}{e_p\sqrt{3}}. \tag{28} \]

for the vector of the resultant elastic deformation

\[ \psi^*=\arccos \frac{\Theta_{ypr}}{\varepsilon_p\sqrt{3}}. \]

The projection of the resultant stress \(\bar{\sigma}_p\) onto the \(x\)-axis is

\[ n_{p(x)}\bar{\sigma}_p=\frac{\sigma_1+\sigma_2+\sigma_3}{\sqrt{3}} =\sqrt{3}\cdot\sigma_0=\sqrt{3}\cdot p, \]

and the deviation of this vector from the axis of symmetry of the space is

\[ \psi'=\arccos \frac{\sigma_1+\sigma_2+\sigma_3}{\sigma_p\sqrt{3}}. \tag{29} \]

Consequently, the regularities of volumetric deformation can be characterized by the change in the ratios of the projections of the resultant stress \(\bar{\sigma}_p\) and the resultant deformation \(\bar e_p\) onto the axis of symmetry of the space,

\[ p=K(\lambda)\cdot\Theta,\qquad n_{p(x)}\bar{\sigma}_p=3K(\lambda)\cdot n_{p(x)}\bar e_p . \]

Thus, all the elements and relations necessary for describing the process of elastic-plastic deformation acquire a clear geometric form. On this basis modeling is also possible. Elastic-plastic deformation under any stressed state proceeds in such a way that the behavior of the material can be clarified from the following geometric picture.

In the simplest test (tension), the characteristics of the mechanical properties of the given material during deformation are determined in the form of the initial diagrams

\[ \sigma=f_1(e),\qquad \mu(\lambda)=f_2(e). \]

The stress intensity under any stressed state is characterized by the vector \(\bar{\sigma}_p\). If a stressed state is specified in which all components increase proportionally to one parameter, then in the coordinate system \(\sigma_1,\sigma_2,\sigma_3\) the direction of the vector \(\bar{\sigma}_p\) remains unchanged, while its magnitude changes in proportion to the loading parameter \(\lambda\). To each value of \(\sigma_p\) there corresponds a definite value, in magnitude, of \(e_p\), according to the properties of the material at the given stage. In the system of combined coordinates \((\sigma_1,\sigma_2,\sigma_3)\), \((e_1,e_2,e_3)\), the vectors \(\bar{\sigma}_p\) and \(\bar e_p\) issue from a common origin. In the loading process \(\bar e_p\) deviates from \(\bar{\sigma}_p\) by the angle \(\psi\), remaining in one plane passing through \(\bar{\sigma}_p\) and an axis equally inclined to the coordinate axes. The projections of the vector \(\bar e_p\) onto the coordinate axes determine the values of the principal components of deformation, and onto the axis of symmetry of the space—the magnitude of the volumetric deformation. The projections of the constituent vectors \(\bar e_p=\bar{\varepsilon}_p+\bar{\delta}_p\) onto the coordinate axes give the values of the elastic and residual parts of the deformation components (Fig. 1).

In equivalent states under different force actions, the vectors \(\bar e_p\) touch the surface of the common sphere (9), and \(\bar{\sigma}_p\) touches the ellipsoid (8). Fig. 2 shows a number of successive stages of deformation\(^{1}\).

Note. It should be noted that the present work, like the earlier one [6], presupposes the following subdivision.

1. Simple loading is loading under which the components of the stress state change proportionally to a single parameter. The change in the components of strain is then determined by the properties of the real material.

2. Simple deformation is deformation with a change in the components of strain proportional to a single parameter. The change in stresses is determined by the properties of the real material.

For a volume-incompressible material (neglecting the change in volume under elastic-plastic deformation), and also within the elastic range, simple loading corresponds to simple deformation.

Fig. 1

Fig. 2

REFERENCES

1. Ilyushin A. A. Plasticity, Gostekhizdat, 1948.

2. Rabotnov Yu. N. Strength of Materials, Moscow State University Press, 1950.

3. Rzhanitsyn A. R. Analysis of Structures Taking into Account the Plastic Properties of Materials, Moscow, 1954.

4. Zhukov A. M. On Poisson’s ratio in the plastic region. Izv. AS USSR, Department of Technical Sciences, No. 12, 1954.

5. Dotsinsky G. A. Theory of the limiting elastic state. Izv. Tomsk Polytechnic Institute, vol. 85, 1957.

6. Dotsinsky G. A. On the theory of elastic-plastic deformation. Izv. Tomsk Polytechnic Institute, vol. 85, 1957.

7. Maisel V. M. Reference Manual on Machine Building, vol. 1 (Mathematics), DNTVU, 1937.

1. In other words, they have only a geometric meaning. ↩