Chapter 15
Plasticity Theory of
Henky–Nadai–Ilyushin
15.1 Laws of Active Elastic–Plastic Deformation
Assume that a body element is subject to simple strain and the intensity of stresses
σ i (t) changes with time t. Let us fix two moments in time t 1 and t 2 , so that t 2 > t 1 . If
σ i (t 2 ) > σ i (t 1 ), the element strain is called active. Active strain is called the loading
process. If σ i (t 2 ) < σ i (t ∗ ) at t ∗ < t 2 , the strain is called passive. In this case, they
say that there is a process of unloading.
One of the primary provisions 1 of the strain theory of plasticity is an assumption
that for active strain, plastic strain rises beyond the elastic limit. Based on this
hypothesis and using equivalents of theses 1 − 3 from the previous paragraph,
the following primary laws of the theory of low elastic-elastic deformations are
formulated.
1. Hardening law. The intensity of stresses is universal and is a function (() of
strain intensity that does not depend on the type of stressed state:
σ i = i ).
(15.1)
A possible form of this universal function for hardening material is shown in
Fig. 15.1. It shows the diagram for active loading and a trajectory section in the
case of unloading.
2. Law of volumetric strain elasticity. The volumetric strain of a body complies with
Hooke’s law:
σ 0 = Kε 0 .
(15.2)
1 This assertion is true for simple loading of a hardening material, but in general it is incorrect for
complex loading.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_15
195
Plasticity Theory of
Henky–Nadai–Ilyushin
15.1 Laws of Active Elastic–Plastic Deformation
Assume that a body element is subject to simple strain and the intensity of stresses
σ i (t) changes with time t. Let us fix two moments in time t 1 and t 2 , so that t 2 > t 1 . If
σ i (t 2 ) > σ i (t 1 ), the element strain is called active. Active strain is called the loading
process. If σ i (t 2 ) < σ i (t ∗ ) at t ∗ < t 2 , the strain is called passive. In this case, they
say that there is a process of unloading.
One of the primary provisions 1 of the strain theory of plasticity is an assumption
that for active strain, plastic strain rises beyond the elastic limit. Based on this
hypothesis and using equivalents of theses 1 − 3 from the previous paragraph,
the following primary laws of the theory of low elastic-elastic deformations are
formulated.
1. Hardening law. The intensity of stresses is universal and is a function (() of
strain intensity that does not depend on the type of stressed state:
σ i = i ).
(15.1)
A possible form of this universal function for hardening material is shown in
Fig. 15.1. It shows the diagram for active loading and a trajectory section in the
case of unloading.
2. Law of volumetric strain elasticity. The volumetric strain of a body complies with
Hooke’s law:
σ 0 = Kε 0 .
(15.2)
1 This assertion is true for simple loading of a hardening material, but in general it is incorrect for
complex loading.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_15
195