# 11.4: Yield Condition

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

From the previous section, the uniaxial yield condition under tension/compression in the x-direction is

$\sigma_{11} = \pm \sigma_y$

In the general 3-D, all six components of the stress tensor contribute to yielding of the material. The von Mises yield condition takes the form

$\frac{1}{2} [(\sigma_{11} − \sigma_{22})^2 + (\sigma_{22} − \sigma_{33})^2 + (\sigma_{33} − \sigma_{11})^2 ] + 3(\sigma^2_{12} + \sigma^2_{23} + \sigma^2_{31})] = \sigma^2_y$

or in a short-hand notation

$F(\sigma_{ij}) = \sigma_y \nonumber$

The step-by-step derivation of the above equation is given in the next section. Here, several special cases are considered.

## Principle coordinate system

All non-diagonal components of the stress tensor vanish, $$\sigma_{12} = \sigma_{23} = \sigma_{31} = 0$$. Then, Equation (11.3.8) reduces to

$(\sigma_1 − \sigma_2)^2 + (\sigma_2 − \sigma_3)^2 + (\sigma_3 − \sigma_1)^2 = 2\sigma^2_y$

where $$\sigma_{1}$$, $$\sigma_{2}$$ $$\sigma_{3}$$ are principal stresses. The graphical representation of Equation (??) is the open ended cylinder normal to the octahedral plane, Figure ($$\PageIndex{1}$$).

The equation of the straight line normal to the octahedral plane and passing through the origin is

$\sigma_1 + \sigma_2 + \sigma_3 = 3p$

where $$p$$ is the hydrostatic pressure. Since the hydrostatic pressure does not have any effect on yielding, the yield surface is an open cylinder.

## Plane stress

Substituting $$\sigma_{13} = \sigma_{23} = \sigma_{33} = 0$$ in Equation (11.3.8), the plane stress yield condition becomes

$\sigma^2_{11} − \sigma_{11} \sigma_{22} + \sigma^2_{22} + 3\sigma^2_{12} = \sigma^2_y$ Figure $$\PageIndex{1}$$: Representation of the von Mises yield condition in the space of principal stresses.

In particular, in pure shear $$\sigma_{11} = \sigma_{22} = 0$$ and $$\sigma_{12} = \sigma_y/\sqrt{3}$$. In the literature $$\sigma_y/\sqrt{3} = k$$ is called the yield stress in shear corresponding to the von Mises yield condition. In the principal coordinate system $$\sigma_{12} = 0$$ and the yield condition takes a simple form

$\sigma^2_1 − \sigma_1\sigma_2 + \sigma^2_2 = \sigma^2_y$ Figure $$\PageIndex{2}$$: The von Mises ellipse in the principal coordinate system.

The graphical representation of Equation (??) is the ellipse shown in Figure ($$\PageIndex{2}$$). Several important stress states can be identified in Figure ($$\PageIndex{2}$$).

• Point 1 and 2: Uniaxial tension, $$\sigma_{1} = \sigma_{2} = \sigma_{y}$$
• Point 7 and 11: Uniaxial compression, $$\sigma_{1} = \sigma_{2} = -\sigma_{y}$$
• Point 3: Equi-biaxial tension, $$\sigma_{1} = \sigma_{2}$$
• Point 9: Equi-biaxial compression, $$-\sigma_{1} = -\sigma_{2}$$
• Points 2, 4, 8 and 10: Plain strain, $$\sigma_{1} = \frac{2}{\sqrt{3}}\sigma_{y}$$
• Points 6 and 12: Pure shear, $$\sigma_{1} = -\sigma_{2}$$

The concept of the plane strain will be explained in the section dealing with the flow rule.

## Equivalent stress and equivalent strain rate

In the finite element analysis the concept of the equivalent stress $$\bar{\sigma}$$ or the von Mises stress is used. It is defined by in terms of principal stresses

$\bar{\sigma} = \frac{1}{2} [(\sigma_{11} − \sigma_{22})^2 + (\sigma_{22} − \sigma_{33})^2 + (\sigma_{33} − \sigma_{11})^2 ]$

The equivalent stress $$\bar{\sigma} (\sigma_{ij})$$ is the square root of the left hand side of Equation (11.3.8). Having defined the equivalent stress, the energy conjugate equivalent strain rate can be evaluated from

$\bar{\sigma} \bar{\dot{\epsilon}} = \sigma_{ij} \dot{\epsilon_{ij}}$

and is given by

$\bar{\dot{\epsilon}} = \left\{ \frac{2}{9} [(\dot{\epsilon}_{11} − \dot{\epsilon}_{22})^2 + (\dot{\epsilon}_{22} − \dot{\epsilon}_{33})^2 + (\dot{\epsilon}_{33} − \dot{\epsilon}_{11})^2 ] \right\}^{1/2}$

The equivalent strain is obtained from integrating in time the equivalent strain rate

$\bar{\epsilon} = \int \bar{\dot{\epsilon}} dt$

This page titled 11.4: Yield Condition is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Tomasz Wierzbicki (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.