Abstract
We show how some differential geometric structures associated with a concept of a homogeneous space appear naturally in a kinematic model of continuously distributed defects in an elastic crystal solid and discuss how one can use them to describe the defectiveness of such a continuum.
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References
1.
Davini
C.
A proposal for a continuum theory of defective crystals . Arch Rat Mech Anal 1986 ; 96 : 295 –317 .
2.
Parry
GP.
Group properties of defective crystal structures . Math Mech Solids 2003 ; 8 : 515 –538 .
3.
Elżanowski
M
Preston
S.
On continuously defective elastic crystals . Miskolc Math Notes 2013 ; 14 (2 ): 659 –670 .
4.
Yavari
A
Goriely
A.
Riemann–Cartan geometry of nonlinear disclination mechanics . Math Mech Solids 2013 ; 18 (1 ): 91 –102 .
5.
Elżanowski
M.
Torsion and curvature in continuously defective solid crystals . Atti Accad Pelor Pericol Classe Sci Fis Mat Nat 2021 ; 99 (11 ) A4 : 1 –9 .
6.
Elżanowski
M
Parry
GP.
Connection and curvature in crystals with non-constant dislocation density . Math Mech Solids 2019 ; 24 (6 ): 1714 –1725 .
7.
Kobayashi
S.
Transformation Groups in Differential Geometry . Berlin : Springer-Verlag , 1995 .
8.
Olver
PJ.
Equivalence, Invariants, and Symmetry . Cambridge : Cambridge University Press , 1995 .
9.
Kobayashi
S
Nomizu
K.
Foundations of Differential Geometry , Vols. I and II . New York : John Wiley & Sons, Inc ., 1996 .
10.
Elżanowski
M
Parry
GP
. A kinematics of defects in solid crystals . In:
Segev
R
Epstein
M
(eds.) Geometric Continuum Mechanics . Cham : Birkhäuser , 2020 .
11.
Epstein
M.
The Geometrical Language of Continuum Mechanics . Cambridge : Cambridge University Press , 2010 .
12.
Lee
EH.
Elastic–plastic deformation at finite strains . J Appl Mech 1969 ; 36 : 1 –6 .
13.
Parry
GP
Šilhavý
M.
Elastic invariants in the theory of defective crystals . Proc R Soc Lond A 1999 ; 455 : 4333 –4346 .
14.
Elżanowski
M
Parry
GP.
Material symmetry in a theory of continuously defective crystals . J Elasticity 2004 ; 74 : 215 –237 .
15.
Poor
WA.
Differential Geometric Structures . New York : McGraw-Hill , 1981 .
16.
Wang
HC.
On invariant connections over a principal fiber bundle . Nagoya Math J 1958 ; 13 : 1 –19 .