Abstract
The theory of continuous distributions of dislocations and other material defects, when formulated in terms of differential forms, is shown to comprise also the discrete, or singular, counterpart, in which defects are concentrated on lower dimensional regions, such as surfaces, lines, and points. The mathematical tool involved in this natural transition is the theory of de Rham currents, which plays in regard to differential forms the same role as the theory of Schwartz distributions plays with respect to ordinary functions. After a review of the main mathematical aspects, the theory is illustrated with a profusion of examples and applications.
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Notes
- 1.
Available at https://archive.org/details/lecturesdelivere00clarrich, page 43.
- 2.
On this point, it is pertinent to mention the works of the philosopher of science Mario Bunge [1]. Bunge, though, warns us about the dangers of ‘explicatio obscurum per obscurius’.
- 3.
See [3], p. 6.
- 4.
Ibid., p. 8.
- 5.
- 6.
By virtue of the point-wise vector-space character of \(\varLambda ^p(T^*_X{\mathscr M})\), with \(X \in {\mathscr M}\).
- 7.
In his momentous article [20], Schwartz describes the continuity of a distribution T(⋅) as follows: ‘Si une suite de fonctions ϕ i, ont leurs noyaux contenus dans un compact fixe et si elles convergent uniformément vers 0, ainsi que chacune de leurs dérivées, alors les T(ϕ i) convergent vers 0’. In other words, the requirement placed on the continuity of the functional T is stronger than the mere uniform convergence of the functions, since it entails that each of the sequences of derivatives must also converge uniformly to zero. When generalizing this idea to manifolds, Schwartz and de Rham demand that the supports must all be contained in a single compact set within the domain of a chart. The derivatives are taken with respect to the chart coordinates.
- 8.
See [4], p. 1.
- 9.
Ibid.
- 10.
The existence of this maximal set is proved as a theorem in [4], p. 35.
- 11.
The justification for this statement can be found in a theorem [4] stating that every closed current is homologous to a differential form.
- 12.
A good example in a body \(\mathscr B\) is a loop, or a curve whose ends are not in \(\mathscr B\).
- 13.
See [9], p. 357.
- 14.
For example, \(\mathscr T\) could be a compact submanifold of \(\mathscr M\), or a submanifold such as \(\mathscr S\) in Example 8.
- 15.
See [10], p. 813. The allusion to Kirchhoff’s law of electrical circuits, where the algebraic sum of electrical currents at a node is zero, is very pertinent.
- 16.
Intuitively speaking, each line comes out, as it were, from \(\mathscr M\). This condition guarantees that the restriction to \({\mathscr S}_i\) of each compactly supported form in \(\mathscr M\) is also compactly supported in \({\mathscr S}_i\).
- 17.
Although the cylindrical coordinate system is not well-defined on the z-axis, this fact is not of significance for this example.
- 18.
See, e.g., [22], p. 492.
- 19.
The assumption that these two 1-forms are closed is made explicitly to concentrate on the role of the discontinuity hypersurface \(\mathscr T\), thus ignoring explicitly the possibility of existence of smooth dislocations.
- 20.
A chiral molecule can exist in two isomeric varieties, known as enantiomers. They are mutual mirror images.
- 21.
See, e.g. [19].
- 22.
It is remarkable that n = 3 is the maximum dimension for which all forms are automatically decomposable. This can be used as a somewhat banal argument for our space to be 3-dimensional, but nor more banal than the acoustic argument according to which wave fronts propagate sharply only in odd dimensional spaces. In a 2-dimensional world, we would not be able to communicate by sharp signals.
- 23.
For an illuminating presentation of these ideas, see [21].
- 24.
Following [14].
- 25.
The neologism ‘distriation’ is meant to suggest the disruption of the striated structure implied by the filaments in compatible fascicles.
- 26.
A more rigorous limiting process is presented in another chapter of this volume authored by Kupferman and Olami [13].
- 27.
See, e.g., [12].
- 28.
For simplicity of the exposition, we will assume that \(F{\mathscr M}\) is globally trivializable.
- 29.
More precisely, in a given trivialization of the bundle, at each point \(X \in {\mathscr M}\), Φ(X) amounts to a left translation of the fibre \(\pi ^{-1}(X) = GL(n,{\mathbb R})\) by some element of \(GL(n,{\mathbb R})\).
- 30.
This is the coordinate expression of the map defined in Remark 6.
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Epstein, M., Segev, R. (2020). Regular and Singular Dislocations. In: Segev, R., Epstein, M. (eds) Geometric Continuum Mechanics. Advances in Mechanics and Mathematics(), vol 43. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-42683-5_5
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