Gareth P. Parry firstname.lastname@example.org
Two-dimensional defective crystals with non-constant dislocation density and unimodular solvable group structure
Parry, Gareth P.; Zyskin, Maxim
Maxim Zyskin email@example.com
In Parry and Zyskin  we outlined mathematical methods which seemed to be necessary in order to discuss crystal structures with non-constant dislocation density tensor (ddt). This was part of a programme to investigate the geometry of continuously defective crystals and the symmetries of associated discrete structures -- one can think of the programme as an attempt to generalize the use of crystallographic groups as material symmetries in non-linear elasticity theory, for perfect crystals, to deal with the case where defects are present.
The methods used rely on the following fact: when the ddt is non-constant, (given technical assumptions), there is a Lie group that acts on the set of material points, and the dimension of the group is strictly greater than that of the ambient space in which the crystal resides. So there is a non-trivial isotropy group associated with the group action.
We develop ideas, and recap the requisite mathematical apparatus, in the context of Davini's model of defective crystals, then focus on a particular case where the ddt is such that a solvable three dimensional Lie group acts on a two dimensional crystal state. We construct the corresponding discrete structures too.
The paper is an extension of , where the analogous group was nilpotent.
Parry, G. P., & Zyskin, M. (2018). Two-dimensional defective crystals with non-constant dislocation density and unimodular solvable group structure. Journal of Elasticity, 1-39. https://doi.org/10.1007/s10659-018-9680-7
|Journal Article Type||Article|
|Acceptance Date||May 25, 2018|
|Online Publication Date||Jun 28, 2018|
|Publication Date||Jun 28, 2018|
|Deposit Date||Jun 6, 2018|
|Publicly Available Date||Jun 28, 2018|
|Journal||Journal of Elasticity|
|Peer Reviewed||Peer Reviewed|
|Keywords||Crystals; Defects; Lie groups|
|Copyright Statement||Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by/4.0|
Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by/4.0
You might also like
Connection and curvature in crystals with non-constant dislocation density
Discrete structures in continuum descriptions of defective crystals
On symmetries of crystals with defects related to a class of solvable groups (S1)
On symmetries of crystals with defects related to a class of solvable groups (S2)
Reconciliation of local and global symmetries for a class of crystals with defects