Analysis using elastic properties is only applicable when the stress remains below the yield stress. However yield stress is a single value while the stress state is a tensor. The simplest case is for a material in a uni-axial stress state where the material yields when
σ x x = σ y \sigma_{xx} = \sigma_y σ xx = σ y When there is a general (multi-axial) stress state there are a number
of possible yield (or failure) criteria.
Principal stresses
The general stress tensor is
σ = [ σ x x σ x y σ x z σ y x σ y y σ y z σ z x σ z y σ z z ] \sigma =
\begin{bmatrix}
\sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
\sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
\sigma_{zx} & \sigma_{zy} & \sigma_{zz} \\
\end{bmatrix} σ = σ xx σ y x σ z x σ x y σ yy σ zy σ x z σ yz σ zz The principal stresses are derived by rotating the stress tensor so that the shear stresses are zero resulting in a set of principal stresses
σ = [ σ I 0 0 0 σ I I 0 0 0 σ I I I ] \sigma =
\begin{bmatrix}
\sigma_{I} & 0 & 0 \\
0 & \sigma_{II} & 0 \\
0 & 0 & \sigma_{III} \\
\end{bmatrix} σ = σ I 0 0 0 σ II 0 0 0 σ III where
σ I > σ I I > σ I I I \sigma_I > \sigma_{II} > \sigma_{III} σ I > σ II > σ III Maximum principal stress – Lamé
The simplest of these if that the material yields when the maximum principal stress reaches the yield stress.
σ I = σ y \sigma_I = \sigma_y σ I = σ y This criteria is applicable to brittle materials.
Maximum shear stress – Tresca
From Mohr's circle, based on the principal stresses σ I \sigma_I σ I σ I I \sigma_{II} σ II σ I I I \sigma_{III} σ III
τ y = m a x ( σ I − σ I I 2 , σ I I − σ I I I 2 , σ I − σ I I I 2 ) \tau_y = max\left(\dfrac{\sigma_I - \sigma_{II}}{2}, \dfrac{\sigma_{II} - \sigma_{III}}{2}, \dfrac{\sigma_I - \sigma_{III}}{2}\right) τ y = ma x ( 2 σ I − σ II , 2 σ II − σ III , 2 σ I − σ III ) leading to a yield criterion
m a x ( ( σ I − σ I I ) , ( σ I I − σ I I I ) , ( σ I − σ I I I ) ) = σ y max\left((\sigma_I - \sigma_{II}), (\sigma_{II} - \sigma_{III}), (\sigma_I - \sigma_{III})\right) = \sigma_y ma x ( ( σ I − σ II ) , ( σ II − σ III ) , ( σ I − σ III ) ) = σ y Effective stress – von Mises
Using the principal stresses an effective (distortional) stress can be defined as
σ e = ( σ I − σ I I ) 2 + ( σ I I − σ I I I ) 2 + ( σ I I I − σ I ) 2 2 \sigma_e = \sqrt{\dfrac{(\sigma_I - \sigma_{II})^2 + (\sigma_{II} - \sigma_{III})^2 + (\sigma_{III} - \sigma_I)^2}{2}} σ e = 2 ( σ I − σ II ) 2 + ( σ II − σ III ) 2 + ( σ III − σ I ) 2 The von Mises yield criterion is then
( σ I − σ I I ) 2 + ( σ I I − σ I I I ) 2 + ( σ I I I − σ I ) 2 2 = σ y \sqrt{\dfrac{(\sigma_I - \sigma_{II})^2 + (\sigma_{II} - \sigma_{III})^2 + (\sigma_{III} - \sigma_I)^2}{2}} = \sigma_y 2 ( σ I − σ II ) 2 + ( σ II − σ III ) 2 + ( σ III − σ I ) 2 = σ y As with the Tresca criterion a hydrostatic state of stress σ I = σ I I = σ I I I \sigma_I = \sigma_{II} = \sigma_{III} σ I = σ II = σ III