On the Principal Directions of Stresses and Strains
Abstract Generated abstract
This paper examines the commonly used assumption in small elastic-plastic deformation theory that the principal axes of the stress and strain tensors coincide. It formulates the orientation of the stress axes relative to the strain axes through direction cosines and treats the deformation work under simple monotonic loading as a constrained variational problem, with orthogonality and normalization conditions imposed by Lagrange multipliers. Solving the resulting Euler equations for the general case of distinct principal stresses and strains, the author finds that the minimizing orientation is the identity orientation. The analysis therefore interprets coincidence of principal stress and strain directions as a necessary condition for minimum work in the elastic-plastic deformation process, rather than merely an empirical or unexplained assumption.
Full Text
On the Principal Directions of Stresses and Strains
G. A. Doshchinsky
(Presented by the scientific seminar of the Department of Strength of Materials)
One of the propositions of the theory of small elastic-plastic deformations is the assumption that the principal axes of the stress and strain tensors coincide. This proposition is interpreted in different ways in the educational and scientific literature. In some cases it is regarded as a necessary initial hypothesis, in others—as an approximate consequence of experimental observations; most often this proposition is presented without any justification at all. At the same time, it can be shown that the coincidence of the principal directions for stresses and strains (in particular, for the conditions of simple loading of an isotropic body) is a manifestation of a known physical regularity that determines the conditions for the natural course of an equilibrium process—namely, that the coincidence of the principal stresses and strains is a necessary condition for the minimum work of the process of elastic-plastic deformation.
As is known from the theory of deformations, the deformed state of an element can be determined by three mutually orthogonal principal components of strain: \(e_1, e_2, e_3\). In exactly the same way, the stressed state is determined by three principal stresses: \(\sigma_1, \sigma_2, \sigma_3\).
The relations between stresses on different planes, as well as the relations between strain components in different directions, do not depend on the properties of materials, but are determined entirely by statics or geometry.
For a certain state of a body undergoing deformation and characterized by the value of the loading parameter \(\lambda\), let us isolate an elementary cube with faces perpendicular to the principal directions of strain in this state. Let the direction \(de_1\) coincide with the \(x\)-axis, \(de_2\) with the \(y\)-axis, and \(de_3\) with the \(z\)-axis. Suppose that at this arbitrarily chosen instant the directions of the principal stresses do not coincide with the principal directions of strain, and that the rotated axes \(\sigma_1, \sigma_2, \sigma_3\) form with the \(x\)-axis angles whose cosines are respectively equal to \(\alpha_1, \beta_1, \gamma_1\); with the \(y\)-axis, \(\alpha_2, \beta_2, \gamma_2\); and with the \(z\)-axis, \(\alpha_3, \beta_3, \gamma_3\) (Fig. 1). Then, on the faces of the indicated cube, one can show the stresses:
\[ \sigma_x = \sigma_1 \cdot \alpha_1^2 + \sigma_2 \cdot \beta_1^2 + \sigma_3 \cdot \gamma_1^2, \]
\[ \sigma_y = \sigma_1 \cdot \alpha_2^2 + \sigma_2 \cdot \beta_2^2 + \sigma_3 \cdot \gamma_2^2, \]
\[ \sigma_z = \sigma_1 \cdot \alpha_3^2 + \sigma_2 \cdot \beta_3^2 + \sigma_3 \cdot \gamma_3^2. \]
Moreover, there will also be shear stresses on the faces; but since the faces of the cube are perpendicular to the principal directions of strain at the given stage and, consequently, do not become skewed, the increment of the work of deformation is determined only by the expression
\[ dA=\sigma_x\,de_1+\sigma_y\,de_2+\sigma_z\,de_3. \]
The work of deformation when the loading parameter changes from \(0\) to \(\lambda\) is
\[ A=\int_{0}^{\lambda}\left[(\sigma_1\alpha_1^2+\sigma_2\beta_1^2+\sigma_3\gamma_1^2)e'_1 +(\sigma_1\alpha_2^2+\sigma_2\beta_2^2+\sigma_3\gamma_2^2)e'_2+\right. \]
\[ \left. +(\sigma_1\alpha_3^2+\sigma_2\beta_3^2+\sigma_3\gamma_3^2)e'_3\right]\,d\lambda, \]
where all the quantities entering are assumed to vary monotonically with the change of the loading parameter \(\lambda\).
Fig. 1. \(\alpha_1^*=\arc\cos\alpha_1,\ \beta_1^*=\ars\cos\beta_1,\ \ldots\ldots,\ \gamma_3^*=\arc\cos\gamma_3,\)
The cosines of the angles determining the mutual orientation of the axes of the principal stresses and strains will be related to one another by certain relations. Therefore we arrive at the following problem of the calculus of variations—to determine the functions:
\[ u_1(\lambda),\quad u_2(\lambda),\ldots u_n(\lambda), \]
which minimize the integral
\[ I=\int F[\lambda;\ u_1(\lambda);\ u_2(\lambda)\ldots u_n(\lambda)]\,d\lambda, \]
where the sought functions, in addition to the boundary conditions and continuity conditions, must satisfy certain additional requirements over the entire interval of integration. If the additional conditions are represented by a system of \(m\) constraint equations \((m<n)\)
\[ \Phi_i[\lambda;\ u_1(\lambda);\ u_2(\lambda);\ldots u_n(\lambda)]=0,\qquad i=1\div m), \]
then the sought functions (that is, the conditions giving a minimum to the integral \(I\)) are determined by the system of equations
\[ \frac{d}{d\lambda}\left[\frac{\partial F_1}{\partial u_i(\lambda)'}\right] -\frac{\partial F_1}{\partial u_i(\lambda)}=0, \qquad (i=1\div n), \]
where
\[ F_1=F+\sum \varphi_i(\lambda)\cdot \Phi_i, \qquad (i=1\div m) \]
and \(m\) is the number of constraint equations.
The constraint equations in the present case are a system of equations that determine the direction and orthogonality of the axes of the principal stresses:
\[ \begin{aligned} \alpha_i^2+\beta_i^2+\gamma_i^2&=1 &&(i=k)\\ \alpha_i\cdot \alpha_k+\beta_i\cdot \beta_k+\gamma_i\cdot \gamma_k&=0 &&(i\ne k) \end{aligned} \qquad (i,k=1,2,3). \]
The auxiliary function \(F_1\), in expanded form, has the form
\[ \begin{aligned} F_1={}&(\sigma_1\cdot \alpha_1^2+\sigma_2\cdot \beta_1^2+\sigma_3\cdot \gamma_1^2)e_1' +(\sigma_1\cdot \alpha_2^2+\sigma_2\cdot \beta_2^2+\sigma_3\cdot \gamma_2^2)e_2'\\ &+(\sigma_1\cdot \alpha_3^2+\sigma_2\cdot \beta_3^2+\sigma_3\cdot \gamma_3^2)e_3' +\varphi_1(\alpha_1^2+\beta_1^2+\gamma_1^2-1)\\ &+\varphi_2(\alpha_2^2+\beta_2^2+\gamma_2^2-1) +\varphi_3(\alpha_3^2+\beta_3^2+\gamma_3^2-1)\\ &+\varphi_4(\alpha_1\cdot \alpha_2+\beta_1\cdot \beta_2+\gamma_1\cdot \gamma_2) +\varphi_5(\alpha_2\cdot \alpha_3+\beta_2\cdot \beta_3+\gamma_2\cdot \gamma_3)\\ &+\varphi_6(\alpha_1\cdot \alpha_3+\beta_1\cdot \beta_3+\gamma_1\cdot \gamma_3), \end{aligned} \]
where \(\varphi_i(\lambda)\) are undetermined functional Lagrange multipliers.
In an active process of simple loading, for any character of deformation (elastic or plastic), the conditions determining the minimum of the work of deformation will reduce to the following system of equations:
\[ 1.\quad \frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \alpha_1'}\right) -\frac{\partial F_1}{\partial \alpha_1} = 2\sigma_1\cdot \alpha_1\cdot e_1' +2\varphi_1\cdot \alpha_1 +\varphi_4\cdot \alpha_2 +\varphi_6\cdot \alpha_3 =0. \]
\[ 2.\quad \frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \beta_1'}\right) -\frac{\partial F_1}{\partial \beta_1} = 2\sigma_2\cdot \beta_1\cdot e_1' +2\varphi_1\cdot \beta_1 +\varphi_4\cdot \beta_2 +\varphi_6\cdot \beta_3 =0. \]
\[ 3.\quad \frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \gamma_1'}\right) -\frac{\partial F_1}{\partial \gamma_1} = 2\sigma_3\cdot \gamma_1\cdot e_1' +2\varphi_1\cdot \gamma_1 +\varphi_4\cdot \gamma_2 +\varphi_6\cdot \gamma_3 =0. \]
\[ 4.\quad \frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \alpha_2'}\right) -\frac{\partial F_1}{\partial \alpha_2} = 2\sigma_1\cdot \alpha_2\cdot e_2' +2\varphi_2\cdot \alpha_2 +\varphi_4\cdot \alpha_1 +\varphi_5\alpha_3 =0. \]
\[ 5.\quad \frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \beta_2'}\right) -\frac{\partial F_1}{\partial \beta_2} = 2\sigma_2\cdot \beta_2\cdot e_2' +2\varphi_2\cdot \beta_2 +\varphi_4\cdot \beta_1 +\varphi_5\cdot \beta_3 =0. \]
\[ 6.\quad \frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \gamma_2'}\right) -\frac{\partial F_1}{\partial \gamma_2} = 2\sigma_3\cdot \gamma_2\cdot e_2' +2\varphi_2\cdot \gamma_2 +\varphi_4\cdot \gamma_1 +\varphi_5\cdot \gamma_3 =0. \]
\[ 7.\quad \frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \alpha_3'}\right) -\frac{\partial F_1}{\partial \alpha_3} = 2\sigma_1\cdot \alpha_3\cdot e_3' +2\varphi_3\cdot \alpha_3 +\varphi_5\cdot \alpha_2 +\varphi_6\cdot \alpha_1 =0. \]
\[ 8.\quad \frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \beta_3'}\right) -\frac{\partial F_1}{\partial \beta_3} = 2\sigma_2\cdot \beta_3\cdot e_3' +2\varphi_3\cdot \beta_3 +\varphi_5\cdot \beta_2 +\varphi_6\cdot \beta_1 =0. \]
\[ 9.\quad \frac{d}{d\lambda}\left(\frac{\partial F_1}{\partial \gamma_3'}\right) -\frac{\partial F_1}{\partial \gamma_3} = 2\sigma_3\cdot \gamma_3\cdot e_3' +2\varphi_3\cdot \gamma_3 +\varphi_5\cdot \gamma_2 +\varphi_6\cdot \gamma_1 =0. \]
\[ 10.\quad \alpha_1^2+\beta_1^2+\gamma_1^2=1. \]
\[ 11.\quad \alpha_2^2+\beta_2^2+\gamma_2^2=1. \]
\[ 12.\quad \alpha_3^2+\beta_3^2+\gamma_3^2=1. \]
\[ 13.\quad \alpha_1\cdot \alpha_2+\beta_1\cdot \beta_2+\gamma_1\cdot \gamma_2=0. \]
\[ 14.\quad \alpha_2\cdot \alpha_3+\beta_2\cdot \beta_3+\gamma_2\cdot \gamma_3=0. \]
\[ 15.\quad \alpha_1\cdot \alpha_3+\beta_1\cdot \beta_3+\gamma_1\cdot \gamma_3=0. \]
We indicate the main stages of the solution. Eliminating \(\varphi_1,\ \varphi_2,\ \varphi_3\) from equations \((1 \div 9)\), we obtain the system:
\[ \begin{aligned} \varphi_4\left(\frac{\alpha_2}{2\alpha_1}-\frac{\beta_2}{2\beta_1}\right) +\varphi_6\left(\frac{\alpha_3}{2\alpha_1}-\frac{\beta_3}{2\beta_1}\right) &=e'_1(\sigma_2-\sigma_1),\\ \varphi_4\left(\frac{\alpha_2}{2\alpha_1}-\frac{\gamma_2}{2\gamma_1}\right) +\varphi_6\left(\frac{\alpha_3}{2\alpha_1}-\frac{\gamma_3}{2\gamma_1}\right) &=e'_1(\sigma_3-\sigma_1),\\ \varphi_4\left(\frac{\alpha_1}{2\alpha_2}-\frac{\beta_1}{2\beta_2}\right) +\varphi_5\left(\frac{\alpha_3}{2\alpha_2}-\frac{\beta_3}{2\beta_2}\right) &=e'_2(\sigma_2-\sigma_1),\\ \varphi_4\left(\frac{\alpha_1}{2\alpha_2}-\frac{\gamma_1}{2\gamma_2}\right) +\varphi_5\left(\frac{\alpha_3}{2\alpha_2}-\frac{\gamma_3}{2\gamma_2}\right) &=e'_2(\sigma_3-\sigma_1),\\ \varphi_5\left(\frac{\alpha_2}{2\alpha_3}-\frac{\beta_2}{2\beta_3}\right) +\varphi_6\left(\frac{\alpha_1}{2\alpha_3}-\frac{\beta_1}{2\beta_3}\right) &=e'_3(\sigma_3-\sigma_1),\\ \varphi_5\left(\frac{\alpha_2}{2\alpha_3}-\frac{\gamma_2}{2\gamma_3}\right) +\varphi_6\left(\frac{\alpha_1}{2\alpha_3}-\frac{\gamma_1}{2\gamma_3}\right) &=e'_3(\sigma_3-\sigma_1), \end{aligned} \]
which, with the aid of relations \(10 \div 15\), can be simplified:
\[ \begin{aligned} \varphi_4\cdot\gamma_3-\varphi_6\cdot\gamma_2 &=2\alpha_1\cdot\beta_1\cdot e'_1(\sigma_1-\sigma_2),& \varphi_4\cdot\beta_3-\varphi_5\cdot\beta_1 &=2\alpha_2\cdot\gamma_2\cdot e'_2(\sigma_1-\sigma_3),\\ \varphi_4\cdot\beta_3-\varphi_6\cdot\beta_2 &=2\alpha_1\cdot\gamma_1\cdot e'_1(\sigma_3-\sigma_1),& \varphi_5\cdot\gamma_1-\varphi_6\cdot\gamma_2 &=2\alpha_3\cdot\beta_3\cdot e'_3(\sigma_2-\sigma_1),\\ \varphi_4\cdot\gamma_3-\varphi_6\cdot\gamma_1 &=2\alpha_2\cdot\beta_2\cdot e'_2(\sigma_2-\sigma_1),& \varphi_6\cdot\beta_1-\varphi_6\cdot\beta_2 &=2\alpha_3\cdot\gamma_3\cdot e'_3(\sigma_1-\sigma_3). \end{aligned} \]
From this system of equations one can express the values \(\varphi_4,\ \varphi_5,\ \varphi_6\):
\[ \varphi_4=-2e'_1(\sigma_1\cdot\alpha_1\cdot\alpha_2+\sigma_2\cdot\beta_1\cdot\beta_2+\sigma_3\cdot\gamma_1\cdot\gamma_2), \]
\[ \varphi_5=-2e'_2(\sigma_1\cdot\alpha_2\cdot\alpha_3+\sigma_2\cdot\beta_2\cdot\beta_3+\sigma_3\cdot\gamma_2\cdot\gamma_3), \]
\[ \varphi_6=-2e'_3(\sigma_1\cdot\alpha_3\cdot\alpha_1+\sigma_2\cdot\beta_3\cdot\beta_1+\sigma_3\cdot\gamma_3\cdot\gamma_1), \]
and the successive elimination of \(\varphi_4,\ \varphi_5,\ \varphi_6\) leads to equations of the type
\[ (\sigma_1\cdot\alpha_1\cdot\alpha_2+\sigma_2\cdot\beta_1\cdot\beta_2+\sigma_3\cdot\gamma_1\cdot\gamma_2)(e'_1-e'_2)=0, \]
\[ (e'_1\cdot\alpha_1\cdot\beta_1+e'_2\cdot\alpha_2\cdot\beta_2+e'_3\cdot\alpha_3\cdot\beta_3)(\sigma_1-\sigma_2)=0. \]
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 \(\sigma_1\ne\sigma_2\ne\sigma_3\), we find:
\[ \varphi_4=0,\qquad \varphi_5=0,\qquad \varphi_6=0. \]
Substituting the values \(\varphi_4,\ \varphi_5,\ \varphi_6\) into equations \((1 \div 9)\), we obtain:
\[ \alpha_1(\sigma_1\cdot e'_1+\varphi_1)=0,\qquad \alpha_2(\sigma_1\cdot e'_2+\varphi_2)=0,\qquad \alpha_3(\sigma_1\cdot e'_3+\varphi_3)=0. \]
\[ \beta_1(\sigma_2\cdot e'_1+\varphi_1)=0,\qquad \beta_2(\sigma_2\cdot e'_2+\varphi_2)=0,\qquad \beta_3(\sigma_2\cdot e'_3+\varphi_3)=0. \]
\[ \gamma_1(\sigma_3\cdot e'_1+\varphi_1)=0,\qquad \gamma_2(\sigma_3\cdot e'_2+\varphi_2)=0,\qquad \gamma_3(\sigma_3\cdot e'_3+\varphi_3)=0. \]
Adding to this the system of constraint equations \((10 \div 15)\) and solving jointly, for the general case \(\sigma_1\ne\sigma_2\ne\sigma_3\) we find:
\[ \alpha_1=1,\qquad \alpha_2=0,\qquad \alpha_3=0. \]
\[ \beta_1=0,\qquad \beta_2=1,\qquad \beta_3=0. \]
\[ \gamma_1=0,\qquad \gamma_2=0,\qquad \gamma_3=1. \]
The solution obtained indicates that coincidence of the principal directions of stresses and strains is a necessary condition for the minimum work of the deformation process.