These so-called Prandtl-Reuss equations rapidly were accepted in science and application. First problems emerged, when after World War II these relations were transferred to application within large deforma-tions. The objectivity of the incorporated rates was questioned.

The Prandtl-Reuss equations can be solved by making the substitution σ cosθ 3 2Y xx =− (8.4.17) in the plastic region. In what follows, use is made of the trigonometric relations θ θ θ π θ θ θ π sin 2 1 cos 2 3 6 cos sin 2 3 cos 2 1 6 sin ⎟= + ⎠ ⎞ ⎜ ⎝ ⎛ − …

In the following, we use the Prandtl-Reuss equation. = p 2 ε σ ′σ ′λ . p . with the stress rate can be determined based on the hardening rule which will be discussed in the next chapter.

Abstract-An Eulerian finite element formulation is presented for problems of large elastic-plastic flow. The method is based on Hill’s variational principle for incremental deformations, and is ideally suited to isotropically hardening Prandtl-Reuss materials.

The full elastic-plastic stress-strain relations are now, using Hooke’s law: These expressions are called the Prandtl-Reuss equations. If the first, elastic, terms are neglected, they are known as the Lévy-Mises equations. The magnitude of the plastic straining is determined by the multiplier dλ .

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.

Prandtl-Reuss Relations in terms of Effective Parameters Using the Prandtl-Reuss (Levy-Mises) flow rule 8.4.1, and the definitions 8.6.10-11 for effective stress and effective plastic strain, one can now express the plastic

A generalized constitutive equation of Prandtl-Reuss type plastic materials is derived. Von Mises yield condition and Levy-St. Venant flow rule are assumed for isotropic and non-workhardening material and the rate type constitutive equation of elastic materials is modified to include the plastic deformation. The obtained equa­

This set of equations describes the deformation of an elasto-plastic body occupying a bounded domain d ( d = 2 or 3) which is fixed along a part of its boundary , under the action of a body force with density f and a surface traction g along = .

We study the n-dimensional wave equation with an elasto-plastic nonlinear stress-strain relation. We investigate the case of heterogeneous materials, i.e. x-dependent parameters that are periodic at the scale >0. We study the limit !0 and derive the plasticity equations for the homogenized material. We prove the well-posedness for the original and