Abstract
The 1950s foundational literature on rational mechanics exhibits two somewhat distinct paradigms to the representation of continuous distributions of defects in solids. In one paradigm, the fundamental objects are geometric structures on the body manifold, e.g., an affine connection and a Riemannian metric, which represent its internal microstructure. In the other paradigm, the fundamental object is the constitutive relation; if the constitutive relations satisfy a property of material uniformity, then it induces certain geometric structures on the manifold. In this paper, we first review these paradigms, and show that they are equivalent if the constitutive model has a discrete symmetry group (otherwise, they are still consistent; however, the geometric paradigm contains more information). We then consider bodies with continuously distributed edge dislocations, and show, in both paradigms, how they can be obtained as homogenization limits of bodies with finitely many dislocations as the number of dislocations tends to infinity. Homogenization in the geometric paradigm amounts to a convergence of manifolds; in the constitutive paradigm it amounts to a Γ-convergence of energy functionals. We show that these two homogenization theories are consistent, and even identical in the case of constitutive relations having discrete symmetries.
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Notes
- 1.
Strictly speaking, the intrinsic condition is that \(\mathcal {G}_p\) is discrete for some \(p\in \mathcal {M}\) (and therefore for every \(p\in \mathcal {M}\)). By locally flat, we mean that the curvature tensor vanishes; globally flat implies also a trivial holonomy. Note that the term flat has a different interpretation in [36], where it describes a curvature- and torsion-free connection.
- 2.
This proposition is a more general version of [36, Proposition 11.8].
- 3.
Actually, any map \(\phi :T_p\mathcal {M}\to \mathcal {M}\) with ϕ(0) = p whose differential at the origin is the identity will do.
- 4.
The estimate (11) does not appear in this corollary explicitly; it follows from its fourth part, using the fact a small triangle on \(\mathcal {M}\) with edges that are Levi-Civita geodesics is, to leading order, Euclidean (this follows from standard triangle comparison results).
- 5.
In [19, Section 3.2], choosing θ = o(1), d = o(1∕n) implies, in the notation of [19], n −1 ≪ D ≪ 1, which then implies L ∞ convergence (see the proof of [19, Proposition 2]). The general case is very similar, since we are only considering minuscule pieces of the manifolds, in which the only geometry that plays a role is the structure of the singular points (everything else is uniformly close to the trivial Euclidean plane). See also [20, Section 2.3.2, Example 2].
- 6.
The quasiconvexity assumption is natural from a variational point of view, as it guarantees the existence of an energy minimizer of the functional; see also Remark 6.
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Acknowledgements
This project was initiated in the Oberwolfach meeting “Material Theories” in July 2018. RK was partially funded by the Israel Science Foundation (Grant No. 1035/17), and by a grant from the Ministry of Science, Technology and Space, Israel and the Russian Foundation for Basic Research.
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Epstein, M., Kupferman, R., Maor, C. (2020). Limits of Distributed Dislocations in Geometric and Constitutive Paradigms. 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_8
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