11.4: Yield Condition

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

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