The Prandtl-Reuss equations revisited

At the beginning of the last century two different types of constitutive relations to describe the complex behavior of elasto‐plastic material were presented. These were the deformation theory originally developed by Hencky and the Prandtl‐Reuss theory. Whereas the former provides a direct solid‐like relation of stress as function of strain, the latter has been based on an additive composition of elastic and plastic parts of the increments of strains. These in turn were taken as a solid‐ and fluid‐like combination of the de Saint‐Venant/Lévy theory with an incremental form of Hooke's law. Even nowadays this Prandtl‐Reuss theory is still accepted – within the restriction of small elastic deformations, i.e. it is generally stated in most textbooks on plasticity that this theory due to a number of defects can not be applied to large deformations. In the present article it is shown that this restrictive statement may be no longer true. Introducing a specific objective time derivative it could be shown that these defects disappear. At the beginning of the last century two different types of constitutive relations to describe the complex behavior of elastoplastic material were presented. These were the deformation theory originally developed by Hencky and the Prandtl‐Reuss theory. Whereas the former provides a direct solid‐like relation of stress as function of strain, the latter has been based on an additive composition of elastic and plastic parts of the increments of strains. These in turn were taken as a solid‐ and fluid‐like combination of the de Saint‐Venant/Lévy theory with an incremental form of Hooke's law. Even nowadays this Prandtl‐Reuss theory is still accepted within the restriction of small elastic deformations. In the present article it is shown that this restrictive statement may be no longer true. Introducing a specific objective time derivative it could be shown that these defects disappear.