11.7: Derivation of the Yield Condition from First Principles (Advanced)

The analysis starts from stating the stress-strain relations for the elastic material, covered in Chapter 3. The general Hook’s law for the isotropic material is

\[\epsilon_{ij} = \frac{1}{E} [(1 + \nu)\sigma_{ij} − \nu \sigma_{kk}\delta_{ij} ] \]

The elastic constitutive equation can also be written in an alternative form, separately for the distortional and dilatational part

\[e_{ij} = \frac{1 + \nu}{E} s_{ij} \quad - \quad \text{ distorsion} \]

\[\epsilon_{kk} = \frac{1 − 2\nu}{E} \sigma_{kk} \quad - \quad \text{ dilatation}\]

The next step is to invoke the basic property of the elastic material that the strain energy density \(\bar{U}\), defined by

\[\bar{U} = \oint \sigma_{ij} d\epsilon_{ij} \]

does not depend on the loading path of the above line integral but only on the final state. Thus, evaluating the strain energy on the proportional (straight) loading path, one gets

\[\bar{U} = \frac{1}{2} \sigma_{ij} \epsilon_{ij} \]

The next step is to prove that the strain energy density can be decomposed into the distortional and dilatational part. This is done by recalling the definition of the stress deviator \(s_{ij}\) and strain deviator \(e_{ij}\)

\[\sigma_{ij} = s_{ij}+ \frac{1}{3} \sigma_{kk}\delta_{ij} \]

\[\epsilon_{ij} = e_{ij} + \frac{1}{3} \epsilon_{kk}\delta_{ij} \]

Introducing Equation (11.4.5) into Equation (11.4.4), there will be four terms in the expression for \(\bar{U}\)

\[2\bar{U} = s_{ij}e_{ij} + s_{ij} \frac{1}{3} \epsilon_{kk}\delta_{ij} + \frac{1}{3} \sigma_{kk}\delta_{ij}e_{ij} + \frac{1}{3} \sigma_{kk}\delta_{ij} \frac{1}{3} \epsilon_{kk}\delta_{ij} \]

Note that \(s_{ij}\delta_{ij} = s_{ii} = 0\) from the definition, Equation (11.4.5). Likewise \(e_{ij}\delta_{ij} = e_{jj} = 0\), also from the definition. Therefore the second and third term of Equation (??) vanish and the energy density becomes

\[\bar{U} = \frac{1}{2} s_{ij}e_{ij} + \frac{1}{6} \sigma_{kk}\epsilon_{ll} = \bar{U}_{dist} + \bar{U}_{dil} \]

Attention is focused on the distortional energy, which with the help of the elasticity law Equation (11.4.2) can be put into the form

\[\bar{U}_{dist} = \frac{1 + \nu}{2E} s_{ij}s_{ij}\]

The product \(s_{ij}s_{ij}\) can be expressed in terms of the components of the stress tensor

\[\begin{align*} s_{ij}s_{ij} &= (\sigma_{ij} − \frac{1}{3} \sigma_{kk}\delta_{ij} )(\sigma_{ij} − \frac{1}{3} \sigma_{kk}\delta_{ij} ) \\[4pt] &= \sigma_{ij}\sigma_{ij} − \frac{1}{3} \sigma_{ij}\sigma_{kk}\delta_{ij} − \frac{1}{3} \sigma_{kk}\delta_{ij}\sigma_{ij} + \frac{1}{9} \sigma_{kk}\sigma_{kk}\delta_{ij}\delta_{ij} \\[4pt] &= \sigma_{ij}\sigma_{ij} − \frac{2}{3} \sigma_{kk}\sigma_{kk} + \frac{1}{3} \sigma_{kk}\sigma_{kk} \end{align*}\]

The final result is

\[\bar{U} = \frac{1 + \nu}{2E} (\sigma_{ij}\sigma_{ij} − \frac{1}{3} \sigma_{kk}\sigma_{kk}) \]

In 1904 the Polish professor Maximilian Tytus Huber proposed a hypothesis that yielding of the material occurs when the distortional energy density reaches a critical value

\[\sigma_{ij}\sigma_{ij} − \frac{1}{3} \sigma_{kk}\sigma_{kk} = C \]

where \(C\) is the material constant that must be determined from tests. The calibration is performed using the uni-axial tension test for which the components of the stress tensor are

\[\sigma_{ij} = \begin{vmatrix} \sigma_{11} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{vmatrix} \]

From Equation (11.4.6) we get

\[\sigma_{11}\sigma_{11} − \frac{1}{3} \sigma_{11}\sigma_{11} = \frac{2}{3} \sigma_{11}\sigma_{11} = C \]

Yielding occurs when \(\sigma_{11} = \sigma_y\) so \(C = \frac{2}{3} \sigma^2_y\). The most general form of the Huber yield condition is

\[(\sigma_{11} − \sigma_{22})^2 + (\sigma_{22} − \sigma_{33})^2 + (\sigma_{33} − \sigma_{11})^2 + 6(\sigma^2_{12} + \sigma^2_{23} + \sigma^2_{31}) = 2\sigma^2_y \]

which was the starting point of the analysis of various special cases in Section 11.4. A similar form of the yield condition for plane stress was derived by von Mises in 1913, based on plastic slip consideration and was later extended to the 3-D case by Hencky. The present form is reformed to in the literature as the Huber-Mises-Hencky yield criterion, called von Mises for short.