Chapter 10
Mathematical Structural Imperfections
10.1 Mathematical and Physical Theories of Structural
Imperfections
In multiply-connected bodies such as a hollow cylinder, many-valued displacements
are possible. This circumstance was first noted by Weingarten [20] early in the
twentieth century. Timpe [18] studied this phenomenon in detail on flat systems
such as annulus. A more general mathematical theory of structural imperfections
was later developed by Volterra [19] who introduced the term of “distortions”
(distorsioni) for this type of strain, which was later substituted by Love [10] for
“dislocations” (dislocation).
In solid body physics, dislocation theory occurred somewhat later than mathematical theory. The founders [7] of the physical theory of dislocations were Orowan
[15], Taylor [17], Burgers [21] et al. Physically, a dislocation was represented as a
distortion of a correct crystalline lattice of a solid body. These representations were
experimentally proved when electronic microscopes appeared [6]. Another method
for studying dislocations was growing crystals with defects of a structure (image
of a growth spiral in the paraffin crystal at the outlet of a screw-type dislocation is
shown in Fig. 10.2a). It is important to note that a distinctive feature of dislocations,
unlike other defects in crystals, is the significant distortion of regular atom ordering
in the small vicinity of some line piercing the crystal.
Researchers pay the highest attention to one-dimensional (linear) defects whose
size in one direction is much more than the lattice parameter and is comparable with
it in two other directions. Linear defects include dislocations and disclinations.
The simplest type of linear defects are boundary and screw-type dislocations.
A boundary dislocation can be represented (Fig. 10.1) as an introduction (or
removal) of one half-plane in the crystal lattice. In this case, planes surrounding
the defects will not be straight (Fig. 10.1a). They will envelope the boundary of the
implemented (or removed) half-plane so that the lattice structure on crystal faces
will not be distorted and the defects will not be seen. Figure 10.1b gives [6] an
© 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_10
101
Mathematical Structural Imperfections
10.1 Mathematical and Physical Theories of Structural
Imperfections
In multiply-connected bodies such as a hollow cylinder, many-valued displacements
are possible. This circumstance was first noted by Weingarten [20] early in the
twentieth century. Timpe [18] studied this phenomenon in detail on flat systems
such as annulus. A more general mathematical theory of structural imperfections
was later developed by Volterra [19] who introduced the term of “distortions”
(distorsioni) for this type of strain, which was later substituted by Love [10] for
“dislocations” (dislocation).
In solid body physics, dislocation theory occurred somewhat later than mathematical theory. The founders [7] of the physical theory of dislocations were Orowan
[15], Taylor [17], Burgers [21] et al. Physically, a dislocation was represented as a
distortion of a correct crystalline lattice of a solid body. These representations were
experimentally proved when electronic microscopes appeared [6]. Another method
for studying dislocations was growing crystals with defects of a structure (image
of a growth spiral in the paraffin crystal at the outlet of a screw-type dislocation is
shown in Fig. 10.2a). It is important to note that a distinctive feature of dislocations,
unlike other defects in crystals, is the significant distortion of regular atom ordering
in the small vicinity of some line piercing the crystal.
Researchers pay the highest attention to one-dimensional (linear) defects whose
size in one direction is much more than the lattice parameter and is comparable with
it in two other directions. Linear defects include dislocations and disclinations.
The simplest type of linear defects are boundary and screw-type dislocations.
A boundary dislocation can be represented (Fig. 10.1) as an introduction (or
removal) of one half-plane in the crystal lattice. In this case, planes surrounding
the defects will not be straight (Fig. 10.1a). They will envelope the boundary of the
implemented (or removed) half-plane so that the lattice structure on crystal faces
will not be distorted and the defects will not be seen. Figure 10.1b gives [6] an
© 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_10
101