Chapter 26
Eigen Stresses and Dislocations
One only needs to sufficiently heat an object disproportionately in order to
create a state of stress that cannot principally be created or reduced by volume and
surface forces. We are dealing here with eigen stresses. According to Sommerfeld
[50], these arise essentially in three ways: by unevenly heating, by magnetostrictive
and elektrostrictive influences or by developmental stresses (so-called Werdegangsspannungen). The were later explaned by dislocations. In fact, every deviation from
the homogeneity of the base structure of a continuum can be seen as a source of
eigen stresses.
We will explicitly examine the following case: An ideal crystal lattice (constant
temperature and a uniform lattice parameter) may take up the ideal, stress free state
of a mechanical continuum, the so-called stress free ’initial configuration’. The real,
the actual existing lattice now deviates from its ideal base structure because of the
dislocations present everywhere throughout the crystal. The crystal, or the continuum
approximating the crystal is therefore present in a state where it contains eigen
stresses that cannot principally be removed using external forces. There is therefore
principally no displacement vector s = s(x, t), through which the former stress state
can be created from a uniform stress free initial state according to a displacement of its
mass elements. One can indeed create a stress free, in comparison to the ideal initial
configuration strain free state for a sufficiently small environment of every point,
but not, however, for the whole object. For the continuum containing dislocations,
there indeed exists a well-defined strain ε, that once again according to Hooke’s law
(315a) is connected to the stress σ,
∂
∂t
ρ +
3
r =1
∂
∂x r
(ρ v r ) = 0 .
(315a)
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2_26
275
Eigen Stresses and Dislocations
One only needs to sufficiently heat an object disproportionately in order to
create a state of stress that cannot principally be created or reduced by volume and
surface forces. We are dealing here with eigen stresses. According to Sommerfeld
[50], these arise essentially in three ways: by unevenly heating, by magnetostrictive
and elektrostrictive influences or by developmental stresses (so-called Werdegangsspannungen). The were later explaned by dislocations. In fact, every deviation from
the homogeneity of the base structure of a continuum can be seen as a source of
eigen stresses.
We will explicitly examine the following case: An ideal crystal lattice (constant
temperature and a uniform lattice parameter) may take up the ideal, stress free state
of a mechanical continuum, the so-called stress free ’initial configuration’. The real,
the actual existing lattice now deviates from its ideal base structure because of the
dislocations present everywhere throughout the crystal. The crystal, or the continuum
approximating the crystal is therefore present in a state where it contains eigen
stresses that cannot principally be removed using external forces. There is therefore
principally no displacement vector s = s(x, t), through which the former stress state
can be created from a uniform stress free initial state according to a displacement of its
mass elements. One can indeed create a stress free, in comparison to the ideal initial
configuration strain free state for a sufficiently small environment of every point,
but not, however, for the whole object. For the continuum containing dislocations,
there indeed exists a well-defined strain ε, that once again according to Hooke’s law
(315a) is connected to the stress σ,
∂
∂t
ρ +
3
r =1
∂
∂x r
(ρ v r ) = 0 .
(315a)
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2_26
275