11.6: Flow Rule

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The simplest form of the associated flow rule for a rigid perfectly plastic material is given by

\[\dot{\epsilon}_{ij} = \dot{\lambda} \frac{\partial F(\sigma_{ij})}{\partial \sigma_{ij}} \]

where the function \(F(\sigma_{ij})\) is defined by Equation (11.3.8), and \(\dot{\lambda}\) is the scalar multiplication factor. Equation (11.3.11) determines uniquely the direction of the strain rate vector, which is always directed normal to the yield surface at a given stress point. In the case of plane stress, the two components of the strain rate vector are

\[\dot{\epsilon}_1 = \dot{\lambda} (2\sigma_1 − \sigma_2) \]

\[\dot{\epsilon}_2 = \dot{\lambda} (2\sigma_2 − \sigma_1) \]

The magnitudes of the components \(\dot{\epsilon}_1\) and \(\dot{\epsilon}_2\) are undetermined, but the ratio, which defines the direction \(\dot{\epsilon}/\epsilon_2\), is uniquely determined.

In particular, under the transverse plain strain \(\dot{\epsilon}_2 = 0\), so \(\sigma_1 = 2\sigma_2\) and \(\sigma_1 = \frac{2}{\sqrt{3}} \sigma_y\).

11.6.1.png — Figure \(\PageIndex{1}\): The strain rate vector is always normal to the yield surface.