Abstract
We review various mathematical constructions relevant to the kinematical model of defective crystals that Davini proposed in 1986. Partly, the motivation for this is the need to place quantities that are useful in phenomenological theories of inelastic behaviour (which are many and rather varied) in a general mathematical framework. Partly, too, simple assumptions regarding defective crystal symmetries are inadequate, so a re-evaluation of those assumptions is necessary. Also, as motivation, we take the current effort in continuum mechanics to rationalize the connection between continuum and discrete models of materials, and so review results which elucidate the rigorous connection between continuous and discrete structures in the context of Davini’s model.
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Notes
- 1.
Let G be a three dimensional Lie group, with commutator (x, y) ≡x −1 y −1 xy. Let G ≡ G 0 and define G 1 ≡ (G, G 0), the group generated by elements of the form (x, y), x ∈ G, y ∈ G 0. Define G k ≡ (G, G k−1) inductively, k ≥ 1. G is called nilpotent if and only if G k is the trivial group {0} for sufficiently large k. For three dimensional nilpotent groups,
, where e is a temporary notation for the group identity 0.Let \(\mathbf {\mathfrak {g}}\) be the Lie algebra corresponding to a Lie group G, with Lie bracket \([{\mathbf {{x}}},{\mathbf {{y}}}], {\mathbf {{x}}}, {\mathbf {{y}}}\in \mathbf {\mathfrak {g}}\). Let \(\mathbf {\mathfrak {g}}\equiv \mathbf {\mathfrak {g}}_0\) and define \(\mathbf {\mathfrak {g}}_1 \equiv \left [\mathbf {\mathfrak {g}},\mathbf {\mathfrak {g}}_0\right ]\), the subspace generated by elements of the form \([{\mathbf {{x}}}, {\mathbf {{y}}}],{\mathbf {{x}}}\in \mathbf {\mathfrak {g}}, {\mathbf {{y}}} \in \mathbf {\mathfrak {g}}_0\). Define \(\mathbf {\mathfrak {g}}_k\equiv \left [\mathbf {\mathfrak {g}}, \mathbf {\mathfrak {g}}_{k-1}\right ]\) inductively, k ≥ 1. \(\mathbf {\mathfrak {g}}\) is called nilpotent if and only if \(\mathbf {\mathfrak {g}}_k\) is the trivial subspace {0} for sufficiently large k. For three dimensional nilpotent algebras,
.A Lie group is nilpotent if and only if the corresponding Lie algebra is nilpotent (Gorbatsevich, Onishchik, Vinberg [19]).
- 2.
For the motivation of this assumption see [13].
- 3.
A vector field on a manifold M is considered complete if the corresponding flow is globally defined.
- 4.
Given a Lie group G and its closed subgroup G x, the quotient space G∕G x is called a homogeneous manifold if it admits a structure of smooth manifold.
- 5.
The process of composing the actions of one-parameter subgroups to obtain the whole group G acting on the corresponding base space is only valid if the group G is connected [35].
- 6.
The reductivity of a homogeneous space G∕G 0 is usually defined by requiring that there exists a vector space \(\frak {D}\subset \frak {g}\) such that the algebra \(\frak {g}=\frak {g}_0\oplus \frak {D}\) and such that it is invariant under the adjoint action of the subgroup G 0. The condition \([\frak {g}_0,\frak {D}]\subset \frak {D}\) implies the invariance of the distribution \(\frak {D}\) under the adjoint action of the group G 0, but not vice versa. However, when the group G is a connected Lie group, both conditions are equivalent.
Note that not all homogeneous spaces are reductive [44]. Note also that it is not necessarily easy to determine if a given homogeneous space is indeed reductive as it requires finding a vector space complement to \(\frak {g}_0\) in \(\frak {g}\) (among all possible complements) which satisfies the said condition of invariance.
- 7.
See for example [24].
- 8.
The equivariance of the one-form Π is the direct consequence of the assumption that the homogeneous manifold G∕G 0 is reductive as shown in [13].
- 9.
More detailed presentation can be found in [13].
- 10.
To confirm this isomorphism select
$$\displaystyle \begin{aligned}\begin{pmatrix}0&0&0\\ 0&0&-1\\ 0&1&0\end{pmatrix},\: \begin{pmatrix}0&-1&0\\ 1&0&0\\ 0&0&0\end{pmatrix},\: \begin{pmatrix}0&0&-1\\ 0&0&0\\ 1&0&0\end{pmatrix}\end{aligned}$$as a basis of \(\frak {so}(3)\) [2].
, where e is a temporary notation for the group identity 0.
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We are very grateful to the EPSRC for support provided via Research Grant EP/M024202/1.
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Elżanowski, M.Z., Parry, G.P. (2020). A Kinematics of Defects in Solid Crystals. 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_7
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