# 11.4: Yield Condition

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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

\[\sigma^2_{11} − \sigma_{11} \sigma_{22} + \sigma^2_{22} + 3\sigma^2_{12} = \sigma^2_y \]

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

\[\sigma^2_1 − \sigma_1\sigma_2 + \sigma^2_2 = \sigma^2_y \]

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

\[\bar{\epsilon} = \int \bar{\dot{\epsilon}} dt \]