ON THE COINCIDENCE OF THE DIRECTING TENSORS OF STRESSES AND STRAINS IN THE MATHEMATICAL THEORY OF PLASTICITY

G. A. Doshchinsky

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

One of the initial premises of the mathematical theory of plasticity is the assumption of the coincidence of the directing tensors of stresses and strains [1], [2].

\[ (\bar D_s) = (\bar D_\varepsilon). \]

Often this condition is represented in the form of two separate propositions: 1. The principal directions of the stressed and deformed states coincide. 2. The maximum tangential stresses are proportional to the maximum shears. However, little attention is paid to the justification of these initial propositions. In some cases they are regarded as necessary initial hypotheses, in others as a consequence of experimental observations.

At the same time it can be shown that these propositions are in direct connection with the well-known minimal principle determining the conditions for the course of natural equilibrium processes, namely, they are necessary conditions for an extremum of the work of the process of elastic-plastic deformation. Previously, in a similar way, the question of the coincidence of principal stresses and strains [4] was already considered; it is part of the general solution given below.

The solution carried out has a direct bearing on the conditions of simple loading, but in the same form it is also fulfilled in the general case of loading if, in the solution, the principal strains are replaced by increments of the principal elongations. Therefore the conclusions acquire a general character, provided the isotropy of the material is preserved.

The stressed state in the neighborhood of a certain point of a body is completely determined by three orthogonal principal stresses: $\sigma_1$, $\sigma_2$, $\sigma_3$. For a completely isotropic body, moreover one behaving identically in tension and compression, the direction of the vectors $\sigma_1$, $\sigma_2$, $\sigma_3$ relative to the material particles of the body may have no significance in solving individual problems. Combining the stresses $\sigma_1$, $\sigma_2$, $\sigma_3$ with some arbitrary coordinate system $1, 2, 3$, we introduce the vector $\sigma$, whose projections on the axes $1, 2, 3$ would be equal to the scalar values corresponding to the magnitudes of the stresses $\sigma_1$, $\sigma_2$, $\sigma_3$. The modulus of such a vector may give an idea of the degree of stress of the body at the given point, and its direction in the coordinate system $1, 2, 3$ may characterize the type of stressed state. In particular, for an equal deviation of $\sigma$ from the axes $1, 2, 3$ we shall be dealing with a hydrost—

Fig. 1.

by uniform all-round tension (hydrostatic pressure in the negative octant of the axes 1, 2, 3). We shall denote the cosines of the angles between \(\sigma\) and the coordinate axes 1, 2, 3 by \(\eta_1, \eta_2, \eta_3\), respectively (Fig. 1).

The deformed state at the point of the body under consideration may be characterized by three principal elongations: \(e_1, e_2, e_3\). In analogy with the preceding, in the coordinate space \(x, y, z\) we introduce the deformation vector \(e\), with projections on the coordinate axes corresponding to the quantities \(e_1, e_2, e_3\), and with cosines of the angles with the axes \(x, y, z\) equal to \(\Theta_1, \Theta_2, \Theta_3\). The natural initial unstressed state of the body corresponds to the coincidence of the origins of the coordinate systems 1, 2, 3 and \(x, y, z\).

Considering the deformation process with the axes \(x, y, z\) connected with the directions of the principal elongations, one may, at an arbitrary instant, represent over a certain time interval the infinitesimal increments of the principal elongations in the form:
\(de_2=d(e\cdot\Theta_2)=de\cdot\Theta_2+d\Theta_2\cdot e\),
\(de_1=d(e\cdot\Theta_1)=de\cdot\Theta_1+d\Theta_1\cdot e\),
\(de_3=d(e\cdot\Theta_3)=de\cdot\Theta_3+d\Theta_3\cdot e\).
Expressing in general form the relation between stresses and deformations, one may suppose that at this instant the directions of the principal stresses do not coincide with the directions of the principal elongations. In this case the stresses \(\sigma_1, \sigma_2, \sigma_3\) form with the axis \(X\) angles whose cosines are respectively equal to \(\alpha_1, \beta_1, \gamma_1\); with the axis \(Y\)—\(\alpha_2, \beta_2, \gamma_2\); and with the axis \(Z\)—\(\alpha_3, \beta_3, \gamma_3\). Assuming that a monotonic deformation process is associated with a monotonic change of stresses, and isolating an elementary cube with faces perpendicular to the principal directions of deformation, we express the stresses on its faces:

\[ \begin{aligned} \sigma_x&=\sigma_1\cdot\alpha_1^2+\sigma_2\cdot\beta_1^2+\sigma_3\cdot\gamma_1^2 =\sigma(\eta_1\cdot\alpha_1^2+\eta_2\cdot\beta_1^2+\eta_3\cdot\gamma_1^2),\\ \sigma_y&=\sigma_1\cdot\alpha_2^2+\sigma_2\cdot\beta_2^2+\sigma_3\cdot\gamma_2^2 =\sigma(\eta_1\cdot\alpha_2^2+\eta_2\cdot\beta_2^2+\eta_3\cdot\gamma_2^2),\\ \sigma_z&=\sigma_1\cdot\alpha_3^2+\sigma_2\cdot\beta_3^2+\sigma_3\cdot\gamma_3^2 =\sigma(\eta_1\cdot\alpha_3^2+\eta_2\cdot\beta_3^2+\eta_3\cdot\gamma_3^2). \end{aligned} \]

There will also be shear stresses on the faces of such a cube. But since the faces of the cube are perpendicular to the principal directions of deformation and therefore do not undergo shear, the increment of the work of deformation will be determined only by the expression

\[ dA=\sigma_x\cdot de_1+\sigma_y\cdot de_2+\sigma_z\cdot de_3 . \]

The work of deformation when the loading parameter \(\lambda\) changes is

\[ A=\int_0^\lambda \sigma\,[\eta_1\cdot\alpha_1^2+\eta_2\cdot\beta_1^2+\eta_3\cdot\gamma_1^2)(e'\cdot\Theta_1+\Theta_1'\cdot e)+ \]

\[ +(\eta_1\cdot\alpha_2^2+\eta_2\cdot\beta_2^2+\eta_3\cdot\gamma_2^2)(e'\cdot\Theta_2+\Theta_2'\cdot e)+(\eta_1\cdot\alpha_3^2+\eta_2\cdot\beta_3^2+ \]

\[ +\eta_3\cdot\gamma_3^2)(e'\cdot\Theta_3+\Theta_3'\cdot e)]\,d\lambda, \]

where all the quantities entering here are assumed to be monotonic functions of the loading parameter \(\lambda\).

Since the cosines of the angular quantities entering here are related to one another by definite relations, determining the conditions under which the work of the deformation process is extremal leads to the following problem of the calculus of variations: determine the functions \(U_1(\lambda)\); \(U_2(\lambda);\ldots U_n(\lambda)\) giving an extremum to the integral

\[ I=\int_{\lambda_1}^{\lambda_2} F[\lambda;\ U_1(\lambda);\ U_2(\lambda);\ \ldots U_n(\lambda)]\,d\lambda, \]

where the required functions, besides the boundary conditions and continuity conditions, must satisfy certain additional requirements-

relative to the entire interval of integration. If the additional conditions are represented by a system of \(m\) constraint equations \((m<n)\)

\[ \Phi_i[\lambda;\, U_1(\lambda);\, U_2(\lambda);\, \ldots;\, U_n(\lambda)], \]

then the unknown functions (i.e., the conditions giving an extremum of the integral \(I\)) are determined by the system of equations

\[ \frac{d}{d\lambda}\left[\frac{\partial F_1}{\partial U_i'(\lambda)}\right] - \frac{\partial F_1}{\partial U_i(\lambda)} =0, \]

where \(F_i=F+\sum \varphi_i(\lambda)\cdot \Phi_i\;(i=1 \div m)\), and by the \(m\) constraint equations. In the case under consideration, the constraint equations are represented by a system of equations determining the mutual orientation and orthogonality of the axes of the principal stresses and strains,

\[ \alpha_i^2+\beta_i^2+\gamma_i^2=1 \quad (i=\kappa) \]

\[ \alpha_i\cdot \alpha_\kappa+\beta_i\cdot \beta_\kappa+\gamma_i\cdot \gamma_\kappa=0 \quad (i\ne \kappa) \]

\[ (i,\ \kappa=1,2,3) \]

and by relations for the cosines of the angles \(\eta\) and \(\Theta\). The auxiliary function \(F_1\) takes the form:

\[ \begin{aligned} F_1={}& \sigma(\eta_1\cdot \alpha_1^2+\eta_2\cdot \beta_1^2+\eta_3\cdot \gamma_1^2)(e'\cdot \Theta_1+\Theta_1'\cdot e) +\sigma(\eta_1\cdot \alpha_2^2+\eta_2\cdot \beta_2^2+{}\\ &+\eta_3\cdot \gamma_2^2)(e'\cdot \Theta_2+\Theta_2'\cdot e) +\sigma(\eta_1\cdot \alpha_3^2+\eta_2\cdot \beta_3^2+\gamma_3^2\cdot \eta_3)(e'\cdot \Theta_3+\Theta_3'\cdot e)+{}\\ &+\varphi_1(\alpha_1^2+\beta_1^2+\gamma_1^2-1) +\varphi_2(\alpha_2^2+\beta_2^2+\gamma_2^2-1) +\varphi_3(\alpha_3^2+\beta_3^2+\gamma_3^2-1)+{}\\ &+\varphi_4(\alpha_1\cdot \alpha_2+\beta_1\cdot \beta_2+\gamma_1\cdot \gamma_2) +\varphi_5(\alpha_2\cdot \alpha_3+\beta_2\cdot \beta_3+\gamma_2\cdot \gamma_3)+{}\\ &+\varphi_6(\alpha_1\cdot \alpha_3+\beta_1\cdot \beta_3+\gamma_1\cdot \gamma_3) +\varphi_7(\eta_1^2+\eta_2^2+\eta_3^2-1) +\varphi_8(\Theta_1^2+\Theta_2^2+\Theta_3^2-1), \end{aligned} \]

where \(\varphi_i(\lambda)\) are undetermined functional Lagrange multipliers. The conditions determining the extremum of the work of the deformation process are now represented by the following system of equations:

\[ \text{1. }\quad \frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \alpha_1'}\right) - \frac{\partial F_1}{\partial \alpha_1} = 2\sigma\eta_1\alpha_1(e'\cdot \Theta_1+\Theta_1'\cdot e) +2\varphi_1\cdot \alpha_1+\varphi_4\cdot \alpha_2+\varphi_6\cdot \alpha_3=0 \]

\[ \text{2. }\quad \frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \beta_1'}\right) - \frac{\partial F_1}{\partial \beta_1} = 2\sigma\eta_2\beta_1(e'\cdot \Theta_1+\Theta_1'\cdot e) +2\varphi_1\cdot \beta_1+\varphi_4\cdot \beta_2+\varphi_6\cdot \beta_3=0 \]

\[ \text{3. }\quad \frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \gamma_1'}\right) - \frac{\partial F_1}{\partial \gamma_1} = 2\sigma\eta_3\gamma_1(e'\cdot \Theta_1+\Theta_1'\cdot e) +2\varphi_1\cdot \gamma_1+\varphi_4\cdot \gamma_2+\varphi_6\cdot \gamma_3=0 \]

\[ \text{4. }\quad \frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \alpha_2'}\right) - \frac{\partial F_1}{\partial \alpha_2} = 2\sigma\eta_1\alpha_2(e'\cdot \Theta_2+\Theta_2'\cdot e) +2\varphi_2\cdot \alpha_2+\varphi_4\cdot \alpha_1+\varphi_5\cdot \alpha_3=0 \]

\[ \text{5. }\quad \frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \beta_2'}\right) - \frac{\partial F_1}{\partial \beta_2} = 2\sigma\eta_2\beta_2(e'\cdot \Theta_2+\Theta_2'\cdot e) +2\varphi_2\cdot \beta_2+\varphi_4\cdot \beta_1+\varphi_5\cdot \beta_3=0 \]

\[ \text{6. }\quad \frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \gamma_2'}\right) - \frac{\partial F_1}{\partial \gamma_2} = 2\sigma\eta_3\gamma_2(e'\cdot \Theta_2+\Theta_2'\cdot e) +2\varphi_2\cdot \gamma_2+\varphi_4\cdot \gamma_1+\varphi_5\cdot \gamma_3=0 \]

\[ \text{7. }\quad \frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \alpha_3'}\right) - \frac{\partial F_1}{\partial \alpha_3} = 2\sigma\eta_1\alpha_3(e'\cdot \Theta_3+\Theta_3'\cdot e) +2\varphi_3\cdot \alpha_3+\varphi_5\cdot \alpha_2+\varphi_6\cdot \alpha_1=0 \]

\[ \text{8. }\quad \frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \beta_3'}\right) - \frac{\partial F_1}{\partial \beta_3} = 2\sigma\eta_2\beta_3(e'\cdot \Theta_3+\Theta_3'\cdot e) +2\varphi_3\cdot \beta_3+\varphi_5\cdot \beta_2+\varphi_6\cdot \beta_1=0 \]

\[ \text{9. }\quad \frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \gamma_3'}\right) - \frac{\partial F_1}{\partial \gamma_3} = 2\sigma\eta_3\gamma_3(e'\cdot \Theta_3+\Theta_3'\cdot e) +2\varphi_3\gamma_3+\varphi_5\cdot \gamma_2+\varphi_6\cdot \gamma_1=0 \]

\[ \begin{aligned} 10.\quad &\frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \eta_1'}\right)-\frac{\partial F_1}{\partial \eta_1} = \sigma \alpha_1^2(e'\cdot \Theta_1+\Theta_1'\cdot e) +\sigma \alpha_2^2(e'\cdot \Theta_2+\Theta_2'\cdot e)+{}\\ &\hspace{4.9cm} +\sigma \alpha_3^2(e'\cdot \Theta_3+\Theta_3'\cdot e)+2\varphi_7\cdot \eta_1=0 \end{aligned} \]

\[ \begin{aligned} 11.\quad &\frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \eta_2'}\right)-\frac{\partial F_1}{\partial \eta_2} = \sigma \beta_1^2(e'\cdot \Theta_1+\Theta_1'\cdot e) +\sigma \beta_2^2(e'\cdot \Theta_2+\Theta_2'\cdot e)+{}\\ &\hspace{4.9cm} +\sigma \beta_3^2(e'\cdot \Theta_3+\Theta_3'\cdot e)+2\varphi_7\cdot \eta_2=0 \end{aligned} \]

\[ \begin{aligned} 12.\quad &\frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \eta_3'}\right)-\frac{\partial F_1}{\partial \eta_3} = \sigma \gamma_1^2(e'\cdot \Theta_1+\Theta_1'\cdot e) +\sigma \gamma_2^2(e'\cdot \Theta_2+\Theta_2'\cdot e)+{}\\ &\hspace{4.9cm} +\sigma \gamma_3^2(e'\cdot \Theta_3+\Theta_3'\cdot e)+2\varphi_7\cdot \eta_3=0 \end{aligned} \]

\[ \begin{aligned} 13.\quad &\frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \Theta_1'}\right)-\frac{\partial F_1}{\partial \Theta_1} = \left[\sigma e(\eta_1\alpha_1^2+\eta_2\beta_1^2+\eta_3\gamma_1^2)\right]-{}\\ &\hspace{4.9cm} -\sigma e'(\eta_1\alpha_1^2+\eta_2\beta_1^2+\eta_3\gamma_1^2)+2\varphi_8\cdot \Theta_1=0 \end{aligned} \]

\[ \begin{aligned} 14.\quad &\frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \Theta_2'}\right)-\frac{\partial F_1}{\partial \Theta_2'} = \left[\sigma e(\eta_1\alpha_2^2+\eta_2\beta_2^2+\eta_3\gamma_2^2)\right]-{}\\ &\hspace{4.9cm} -\sigma e'(\eta_1\alpha_2^2+\eta_2\beta_2^2+\eta_3\gamma_2^2)+2\varphi_8\cdot \Theta_2=0 \end{aligned} \]

\[ \begin{aligned} 15.\quad &\frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \Theta_3'}\right)-\frac{\partial F_1}{\partial \lambda} = \left[\sigma e(\eta_1\alpha_3^2+\eta_2\beta_3^2+\eta_3\gamma_3^2)\right]-{}\\ &\hspace{4.9cm} -\sigma e'(\eta_1\alpha_3^2+\eta_2\beta_3^2+\eta_3\gamma_3^2)+2\varphi_8\cdot \Theta_3=0 \end{aligned} \]

\[ \begin{aligned} 16.\quad &\frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \sigma'}\right)-\frac{\partial F_1}{\partial \sigma} = (\eta_1\alpha_1^2+\eta_2\beta_1^2+\eta_3\gamma_1^2)(e'\cdot \Theta_1+\Theta_1'\cdot e)+{}\\ &\hspace{4.2cm} +(\eta_1\alpha_2^2+\eta_2\beta_2^2+\eta_3\gamma_2^2)(e'\cdot \Theta_2+\Theta_2'\cdot e)+{}\\ &\hspace{4.2cm} +(\eta_1\alpha_3^2+\eta_2\beta_3^2+\eta_3\gamma_3^2)(e'\cdot \Theta_3+\Theta_3'\cdot e)=0 \end{aligned} \]

\[ \begin{aligned} 17.\quad &\frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial e'}\right)-\frac{\partial F_1}{\partial e}=\\ &\hspace{1.0cm} \left[\sigma\cdot \Theta_1(\eta_1\alpha_1^2+\eta_2\beta_1^2+\eta_3\gamma_1^2) +\sigma\cdot \Theta_2(\eta_1\alpha_2^2+\eta_2\beta_2^2+\eta_3\gamma_2^2)+\right.\\ &\hspace{1.0cm} \left.+\sigma\cdot \Theta_3(\eta_1\alpha_3^2+\eta_2\beta_3^2+\eta_3\gamma_3^2)\right] -\left[\sigma\cdot \Theta_1'(\eta_1\alpha_1^2+\eta_2\beta_1^2+\eta_3\gamma_1^2)+\right.\\ &\hspace{1.0cm} \left.+\sigma\cdot \Theta_2'(\eta_1\alpha_2^2+\eta_2\beta_2^2+\eta_3\gamma_2^2) +\sigma\cdot \Theta_3(\eta_1\alpha_3^2+\eta_2\beta_3^2+\eta_3\gamma_3^2)\right]=0 \end{aligned} \]

\[ \begin{array}{ll} 18.\quad \alpha_1^2+\beta_1^2+\gamma_1^2=1, &22.\quad \alpha_2\alpha_3+\beta_2\beta_3+\gamma_2\gamma_3=0,\\[2mm] 19.\quad \alpha_2^2+\beta_2^2+\gamma_2^2=1, &23.\quad \alpha_1\alpha_3+\beta_1\beta_3+\gamma_1\gamma_3=0,\\[2mm] 20.\quad \alpha_3^2+\beta_3^2+\gamma_3^2=1, &24.\quad \eta_1^2+\eta_2^2+\eta_3^2=1,\\[2mm] 21.\quad \alpha_1\alpha_2+\beta_1\beta_2+\gamma_1\gamma_2=0, &25.\quad \Theta_1^2+\Theta_2^2+\Theta_3^2=1. \end{array} \]

We shall indicate the main points in solving this system of equations. Eliminating \(\varphi_1,\varphi_2,\varphi_3\) from \((1\div 9)\), we obtain a system of 6 equations of the form:

\[ \varphi_4(\alpha_2\beta_1-\alpha_1\beta_2)+\varphi_6(\alpha_3\beta_1-\alpha_1\beta_3) =2\sigma(e'\cdot \Theta_1+\Theta_1'\cdot e)(\eta_1-\eta_2)\alpha_1\beta_1 \]

\[ \varphi_4(\alpha_2\gamma_1-\alpha_1\gamma_2)+\varphi_6(\alpha_3\gamma_1-\alpha_1\gamma_3) =2\sigma(e'\cdot \Theta_1+\Theta_1'\cdot e)(\eta_1-\eta_3)\alpha_1\gamma_1 \]

\[ \varphi_4(\alpha_1\beta_2-\beta_1\alpha_2)+\varphi_5(\alpha_3\beta_2-\alpha_2\beta_3) =2\sigma(e'\cdot \Theta_2+\Theta_2'\cdot e)(\eta_1-\eta_2)\alpha_2\beta_2 \]

\[ \varphi_4(\alpha_1\gamma_2-\alpha_2\gamma_1)+\varphi_5(\alpha_3\gamma_2-\alpha_2\gamma_3) =2\sigma(e'\cdot \Theta_2+\Theta_2'\cdot e)(\eta_1-\eta_3)\alpha_2\gamma_2 \]

\[ \varphi_5(\alpha_2\beta_3-\alpha_3\beta_2)+\varphi_6(\alpha_1\beta_3-\alpha_3\beta_1) =2\sigma(e'\cdot \Theta_3+\Theta_3'\cdot e)(\eta_1-\eta_2)\alpha_3\beta_3 \]

\[ \varphi_5(\alpha_2\gamma_3-\alpha_3\gamma_2)+\varphi_6(\alpha_1\gamma_3-\alpha_3\gamma_1) =2\sigma(e'\cdot\Theta_3+\Theta_3'\cdot e)(\eta_1-\eta_3)\alpha_3\gamma_3, \]

which, with the aid of the relations existing between the cosines of the angles under a coordinate transformation, may be simplified to the form [3].

\[ \begin{aligned} \varphi_4\gamma_3-\varphi_6\gamma_2&=2\sigma e_1'(\eta_2-\eta_1)\alpha_1\beta_1, & \varphi_4\beta_3-\varphi_5\beta_1&=2\sigma e_2'(\eta_3-\eta_1)\alpha_2\gamma_2, \\ \varphi_4\beta_3-\varphi_6\beta_2&=2\sigma e_1'(\eta_1-\eta_3)\alpha_1\gamma_1, & \varphi_5\gamma_1-\varphi_6\gamma_2&=2\sigma e_3'(\eta_1-\eta_2)\alpha_3\beta_3, \\ \varphi_4\gamma_3-\varphi_5\gamma_1&=2\sigma e_2'(\eta_1-\eta_2)\alpha_2\beta_2, & \varphi_5\beta_1-\varphi_6\beta_2&=2\sigma e_3'(\eta_3-\eta_1)\alpha_3\gamma_3 . \end{aligned} \]

Hence, using the relations for the cosines of the angles [3], one can express the values of the multipliers

\[ \begin{aligned} \varphi_4&=2\sigma e_2'(\eta_1\alpha_1\alpha_2+\eta_2\beta_1\beta_2+\eta_3\gamma_1\gamma_2),\\ \varphi_5&=2\sigma e_3'(\eta_1\alpha_2\alpha_3+\eta_2\beta_2\beta_3+\eta_3\gamma_2\gamma_3),\\ \varphi_6&=2\sigma e_1'(\eta_1\alpha_1\alpha_3+\eta_2\beta_1\beta_3+\eta_3\gamma_1\gamma_3), \end{aligned} \]

and also obtain a series of equations of the type

\[ (\sigma_1-\sigma_2)(\Theta_1\alpha_1\beta_1+\Theta_2\alpha_2\beta_2+\Theta_3\alpha_3\beta_3), \]

or

\[ (e_1'-e_2')(\eta_1\alpha_1\alpha_2+\eta_2\beta_1\beta_2+\eta_3\gamma_1\gamma_2). \]

Leaving aside the particular solutions \(\sigma_1=\sigma_2=\sigma_3\) and \(e_1'=e_2'=e_3'\), which do not correspond to the general case under consideration, \(\sigma_1\ne\sigma_2\ne\sigma_3\), \(e_1'\ne e_2'\ne e_3'\), we obtain \(\varphi_4=\varphi_5=\varphi_6=0\).

Substitution of the values \(\varphi_4,\varphi_5,\varphi_6\) simplifies equations \((1\div 9)\) to the form

\[ \begin{array}{lll} \alpha_1(\sigma_1e_1'+\varphi_1)=0 & \alpha_2(\sigma_1e_2'+\varphi_2)=0 & \alpha_3(\sigma_1e_3'+\varphi_3)=0 \\[4pt] \beta_1(\sigma_2e_1'+\varphi_1)=0 & \beta_2(\sigma_2e_2'+\varphi_2)=0 & \beta_3(\sigma_2e_3'+\varphi_3)=0 \\[4pt] \gamma_1(\sigma_3e_1'+\varphi_1)=0 & \gamma_2(\sigma_3e_2'+\varphi_2)=0 & \gamma_3(\sigma_3e_3'+\varphi_3)=0 . \end{array} \]

Adding the system of equations \((18\div 23)\) and solving jointly for the conditions \(\sigma_1\ne\sigma_2\ne\sigma_3\), \((e_1'\ne e_2'=e_3')\), we obtain:

\[ \begin{array}{lll} \alpha_1=1 & \alpha_2=0 & \alpha_3=0\\ \beta_1=0 & \beta_2=1 & \beta_3=0\\ \gamma_1=0 & \gamma_2=0 & \gamma_3=1 . \end{array} \]

Such values of the cosines of the angles determining the mutual orientation of the axes of the principal directions of stresses and strains correspond to the coincidence of these axes.

Consequently, the coincidence of the principal directions of stresses and strains is one of the necessary conditions ensuring an extremum of the work of the deformation process.

Substituting the values of the cosines \(\alpha_1,\beta_1,\ldots,\gamma_3\) into equations \((10\div 17)\), we substantially simplify the latter to the form:

\[ \begin{array}{rl} 10. & \sigma e'\Theta_1+\sigma e\Theta_1' + 2\varphi_7\cdot\eta_1=0\\[3pt] 11. & \sigma e'\Theta_2+\sigma e\Theta_2' + 2\varphi_7\cdot\eta_2=0\\[3pt] 12. & \sigma e'\Theta_3+\sigma e\Theta_3' + 2\varphi_7\cdot\eta_3=0\\[3pt] 13. & e\eta_1\sigma' + \eta_1\sigma e' + 2\varphi_8\cdot\Theta_1=0\\[3pt] 14. & e\eta_2\sigma' + \eta_2\sigma e' + 2\varphi_8\cdot\Theta_2=0\\[3pt] 15. & e\eta_3\sigma' + \eta_3\sigma e' + 2\varphi_8\cdot\Theta_3=0 . \end{array} \]

1. \[ e'(\eta_1\Theta_1+\eta_2\Theta_2+\eta_3\Theta_3) +e(\eta_1\Theta'_1+\eta'_2\Theta_2+\eta_3\Theta'_3)=0 \]

2. \[ \sigma'(\eta_1\Theta_1+\eta_2\Theta_2+\eta_3\Theta_3) +\sigma(\eta'_1\Theta_1+\eta'_2\Theta_2+\eta'_3\Theta_3)=0 \]

Multiplying 10–12, 16 by \(\sigma\), and 13–15, 17 by \(e\), we obtain:

\[ \begin{array}{ll} 10. & e'_1=-\dfrac{2\varphi_7}{\sigma^2}\sigma_1 \qquad 13. \quad \sigma'_1=-\dfrac{2\varphi_8}{e^2}e_1 \\[1.2em] 11. & e'_2=-\dfrac{2\varphi_7}{\sigma^2}\sigma_2 \qquad 14. \quad \sigma'_2=-\dfrac{2\varphi_8}{e^2}e_2 \\[1.2em] 12. & e'_3=-\dfrac{2\varphi_7}{\sigma^2}\sigma_3 \qquad 15. \quad \sigma'_3=-\dfrac{2\varphi_8}{e^2}e_3 . \end{array} \]

Whence, when using the Lode parameters \(\mu\) and \(\nu\), the equality of which corresponds to similarity of the deviators, it follows that:

\[ \nu_{e'}=\frac{2e'_2-e'_1-e'_3}{e'_1-e'_3} =\frac{2\sigma_2-\sigma_1-\sigma_3}{\sigma_1-\sigma_3} =\mu_\sigma , \]

\[ \mu_{\sigma'}=\frac{2\sigma'_2-\sigma'_1-\sigma'_3}{\sigma'_1-\sigma'_3} =\frac{2e_2-e_1-e_3}{e_1-e_3} =\nu_e . \]

For simple loading \(\sigma_1=\sigma'_1\cdot\lambda;\ \sigma_2=\sigma'_2\cdot\lambda;\ \sigma_3=\sigma'_3\lambda\),
or simple deformation \(e_1=e'_1\cdot\lambda=e_2;\ =e'_2\cdot\lambda\),
we arrive at similarity of the stress and strain deviators (respectively, proportionality between shear strains and tangential stresses).

Consequently, similarity of the deviators is a necessary condition ensuring an extremum of the work of the deformation process.

Equations 16, 17 lead to the trivial conditions

\[ e=\mathrm{const};\quad \sigma=\mathrm{const}. \]

In the solution carried out, the directions of the principal elongations were associated with the coordinate system used: \(X, Y, Z\), which corresponds to the conditions of simple loading—deformation. It is not difficult to imagine that the solution remains valid also in the general case of loading, when the instantaneous directions of the principal velocities (directions of the principal increments of deformation) are combined with the moving coordinate system: \(X, Y, Z\). In the general case, the necessary conditions for an extremum of the work of the deformation process are determined by the similarity of the deviators of stresses and increments of deformations (strain rates—as is assumed by the flow theory). For an isotropic elastic body, the conditions considered determine the maximum of the increment of the potential. Taking anisotropy into account is associated with imposing additional conditions.

REFERENCES

1. A. A. Ilyushin. Plasticity. Gostekhizdat, 1948.
2. V. V. Sokolovskii. Theory of Plasticity. GITTL, 1950.
3. Reference Manual on Mechanical Engineering. Vol. 1. (Mathematics). Edited by Maisel V. M., 1937.
4. G. A. Doshchinskii. On the principal directions of stresses and strains. Izv. TPI, vol. 157, 1970.