278
18 Theories of Plastic Yield
18.8 Kadashevich–Novozhilov Plasticity Theory
A more general variant of plasticity theory as compared with the Ishlinsky theory
and Prager model was proposed by [3] Yu. I. Kadashevich and V. V. Novozhilov.
The authors accept that the tensors S ij and e ?
ij are related by the ratios of the strain
theory of plasticity
e ij =
1
2G ∗ S ij ,
(18.33)
whereas G ∗ is the function of invariants of the tensor S ij . The following is proposed
as a possible variant of the function G ∗
G
∗
= G
∗ (J
2 (S ij ) ).
For G ∗ = const, the ratio (18.33) coincides with the Ishlinsky theory. A later
paper [8] reveals a physical essence of the tensor S ij by introducing the concept of
micro-stresses.
18.9 Singular Loading Surfaces
By studying the applicability limits of the strain theory of plasticity (p. 260), we
have found that further loading surfaces can be singular. Here we consider the
specifics of a different kind when the initial loading surface defined by the plasticity
condition consists of several smooth surfaces forming when crossed by the rib.
The direction of the normal line to the loading surface on the ribs is not defined.
As to the direction of the plastic strain vector gain, we only know that it lies in the
plane perpendicular to the rib and within the angle limited by normal lines to the
surfaces forming the rib. In the case of additional loading with isotropic hardening,
in the considered case, the loading surface is expanded keeping similarity, and in the
case of translational hardening, the initial surface is displaced in parallel to itself.
In both cases, new specifics are not expressed and smooth loading surfaces remain
smooth. The simplest example of such a process is obtained if we try to generalize
the Saint Venant-Tresca plasticity theory for the case of a hardening material.
Let us assume for simplicity that we know the directions of principal stresses
beforehand. Instead of a non-dimensional space of stresses, the process can be
considered in a three-dimensional sub-space of principal stresses. For σ 1 > σ 2 >
σ 3 , the maximum tangential stress will be
τ max =
σ 1 − σ 3
2
,
and its constancy condition is written as
18 Theories of Plastic Yield
18.8 Kadashevich–Novozhilov Plasticity Theory
A more general variant of plasticity theory as compared with the Ishlinsky theory
and Prager model was proposed by [3] Yu. I. Kadashevich and V. V. Novozhilov.
The authors accept that the tensors S ij and e ?
ij are related by the ratios of the strain
theory of plasticity
e ij =
1
2G ∗ S ij ,
(18.33)
whereas G ∗ is the function of invariants of the tensor S ij . The following is proposed
as a possible variant of the function G ∗
G
∗
= G
∗ (J
2 (S ij ) ).
For G ∗ = const, the ratio (18.33) coincides with the Ishlinsky theory. A later
paper [8] reveals a physical essence of the tensor S ij by introducing the concept of
micro-stresses.
18.9 Singular Loading Surfaces
By studying the applicability limits of the strain theory of plasticity (p. 260), we
have found that further loading surfaces can be singular. Here we consider the
specifics of a different kind when the initial loading surface defined by the plasticity
condition consists of several smooth surfaces forming when crossed by the rib.
The direction of the normal line to the loading surface on the ribs is not defined.
As to the direction of the plastic strain vector gain, we only know that it lies in the
plane perpendicular to the rib and within the angle limited by normal lines to the
surfaces forming the rib. In the case of additional loading with isotropic hardening,
in the considered case, the loading surface is expanded keeping similarity, and in the
case of translational hardening, the initial surface is displaced in parallel to itself.
In both cases, new specifics are not expressed and smooth loading surfaces remain
smooth. The simplest example of such a process is obtained if we try to generalize
the Saint Venant-Tresca plasticity theory for the case of a hardening material.
Let us assume for simplicity that we know the directions of principal stresses
beforehand. Instead of a non-dimensional space of stresses, the process can be
considered in a three-dimensional sub-space of principal stresses. For σ 1 > σ 2 >
σ 3 , the maximum tangential stress will be
τ max =
σ 1 − σ 3
2
,
and its constancy condition is written as