Chapter 20
Problem Setting
20.1 Initial Concepts and Definitions
For plastic materials, we will adopt that the lower yield stress must be associated
with shift resistance, as shown later, and the yield stress will mean the maximum
tangential stress before the occurrence of plastic strains (now we speak of macrohomogeneous strain). The occurrence of the yield peak can be explained by the fact
that in some cases it is more complicated to extract dislocation from the Cottrell
cloud formed from homogeneous atoms and vacancies than to move it.
Various phenomena taking place beyond the elastic limit in a general case can be
considered as a result of an abrupt rise in the role of diffusion processes leading
to a substantial change in the mutual location of Cottrell objects [2] relative to
dislocations and, consequently, a slow change in the strength characteristics of
a material (occurrence of slip, occurrence and amalgamation of micro-fractures,
etc.) Vice versa, strain in reaching the yield stress will be represented as (almost)
instantaneous transition from one state of the micro-structure to another one (stable)
in the vicinity of the most stressed point, whereas there must be changes in the
relative location of molecules.
Consequently, relative displacements of molecules must be equal to or more than
distances between molecules. When loading solid bodies, clicks are heard and slip
stripes (surfaces) are formed. If we abstract from dynamic phenomena, we must
admit as follows.
Axiom 20.1 Plastic strain in micro-volumes of a solid body is discrete in space and
instantaneous in time.
Discrete displacements occur as a result of shifts of multiple mobile structural
imperfections in an elastic body. Their presence in any small volume of a body
is deemed reality. It is obvious that defects can be displaced by the value no less
than the distance between atoms and can have final power; the usually suggested
continuity of plastic strains is an unnatural assumption, which is not always justified
© 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_20
303
Problem Setting
20.1 Initial Concepts and Definitions
For plastic materials, we will adopt that the lower yield stress must be associated
with shift resistance, as shown later, and the yield stress will mean the maximum
tangential stress before the occurrence of plastic strains (now we speak of macrohomogeneous strain). The occurrence of the yield peak can be explained by the fact
that in some cases it is more complicated to extract dislocation from the Cottrell
cloud formed from homogeneous atoms and vacancies than to move it.
Various phenomena taking place beyond the elastic limit in a general case can be
considered as a result of an abrupt rise in the role of diffusion processes leading
to a substantial change in the mutual location of Cottrell objects [2] relative to
dislocations and, consequently, a slow change in the strength characteristics of
a material (occurrence of slip, occurrence and amalgamation of micro-fractures,
etc.) Vice versa, strain in reaching the yield stress will be represented as (almost)
instantaneous transition from one state of the micro-structure to another one (stable)
in the vicinity of the most stressed point, whereas there must be changes in the
relative location of molecules.
Consequently, relative displacements of molecules must be equal to or more than
distances between molecules. When loading solid bodies, clicks are heard and slip
stripes (surfaces) are formed. If we abstract from dynamic phenomena, we must
admit as follows.
Axiom 20.1 Plastic strain in micro-volumes of a solid body is discrete in space and
instantaneous in time.
Discrete displacements occur as a result of shifts of multiple mobile structural
imperfections in an elastic body. Their presence in any small volume of a body
is deemed reality. It is obvious that defects can be displaced by the value no less
than the distance between atoms and can have final power; the usually suggested
continuity of plastic strains is an unnatural assumption, which is not always justified
© 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_20
303