The Prandtl‐Reuss equations revisited

Abstract

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 Hookes 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.

The Multiplicative Decomposition of the Deformation Gradient in Plasticity—Origin and Limitations

“…Several steps towards a more general theory to account also for a hardening of the material or to reflect the Bauschinger effect-to mention only a few examples-have been undertaken shortly after. 17 In this respect, we refer to the recently published paper (Bruhns 2014b) which has been devoted to the development of the PrandtlReuss theory. This paper also contains some remarks about an extension of these relations towards larger or even finite deformations.…”

“…We refer e.g. to Bruhns (2014b). 22 Alternatively, the sets (E − E p , E p , α, κ) as well as (S, E p , α, κ) may be useful for certain specific purposes.…”

“…(3.51) could be integrated to give a really elastic relation. We therefore refer to Bruhns (2014b) and the numerous references mentioned therein. 20 Its formulation from a purely mechanical point of view has been developed later by Naghdi and coworkers and others.…”

“…At that time the main focus of the different considerations was on the mathematical description of the basics of plasticity within the general frame of continuum mechanics, and the combination of the solid-like elastic behavior of most materials below a certain threshold and the fluid-like plastic behavior beyond this limit. Accordingly in a second contribution (Bruhns 2014b) the significant input of Ludwig Prandtl and Endre Reuss to formulate a unified rate relation for elasticplastic behavior was underlined. These so-called Prandtl-Reuss equations and several amendments related with the effort to establish more realistic descriptions of several phenomena observed during experiments of plastic flow like the hardening of the material and the Bauschinger effect were rapidly accepted in science and application.…”

Two‐level models of polycrystalline elastoviscoplasticity: Complex loading under large deformations

“…The spin tensor, which was called a "logarithmic spin", included in the corresponding expression for the corotational rate was defined. Works of this group of researchers [13][14][15][16][17] have shown the benefits of the proposed approach. In particular, the equivalence of hypoelastic and hyperelastic laws was shown, as well as absence of energy dissipation for an isotropic elastic body in the closed cycles of deformations.…”