This paper introduces tools on fibre geometry towards the framework of mechanics of microstructured continuum. The material is modelled by an appropriate bundle for which the associated connection and metric are induced from the Euclidean space by a smooth transformation represented by a fibre morphism from the bundle to Euclidean space. Furthermore, the general kinematic structure of the theory includes macroscopic and microscopic fields in a multiscaled approach, including large transformation. Defects appear in this geometrical point of view by an induced curvature, torsion and non-metricity tensor in the induced geometry. Special attention is given to transport along a finite path in order to extend the standard infinitesimal analysis of torsion and curvature to a macroscopical point of view. Both theoretical and numerical analysis may be handled without additional difficulties. Accordingly, several examples of transformation involving the distribution of material defects are exhibited and analysed.
CartanÉ. Sur une généralisation de la notion de courbure de riemann et les espaces à torsion. C R Acad Sci Paris1922; 174: 593–595.
2.
CosseratECosseratF.Sur la théorie des corps déformables. Paris: Dunod, 1909.
3.
BurgersJBurgersW.Dislocations in crystal lattices. In EirichFR (ed.) Rheology. New York: Academic Press, 1956, pp. 141–199.
4.
VolterraV.Sur l’équilibre des corps élastiques multiplement connexes. Ann Sci École Norm Sup (3)1907; 24: 401–517.
5.
WeylH.Reine Infinitesimalgeometrie. Math Z1918; 2(3–4): 384–411.
6.
CartanÉ. Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie). Ann Sci École Norm Sup (3)1923; 40: 325–412.
7.
CartanÉ. Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) (Suite). Ann Sci École Norm Sup (3)1924; 41: 1–25.
8.
CartanÉ. Sur les variétés à connexion affine, et la théorie de la relativité généralisée (deuxième partie). Ann Sci École Norm Sup (3)1925; 42: 17–88.
9.
KondoK.Geometry of elastic deformation and incompatibility. Memoirs of the unifying study of the basic problems in engineering sciences by means of geometry. Vol. I. Tokyo: Gakujutsu Bunken Fukyu-Kai, 1955.
10.
BilbyBABulloughRSmithE.Continuous distributions of dislocations: A new application of the methods of non-Riemannian geometry. Proc R Soc Lond A1955; 231: 263–273.
11.
KrönerE.Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch Rat Mech Anal1960; 4: 273–334.
12.
KrönerE. Physics of defects. In: Continuum Theory of Defects, Less houches, session xxxv, 1980. Amsterdam: North-Holland, 1981.
13.
KatanaevM.Geometric theory of defects. Phys Usp2005; 48(7): 675–701.
14.
NollW. Materially uniform simple bodies with inhomogeneities. Arch Rat Mech Anal 1967/68; 27: 1–32.
15.
RakotomananaLR.Contribution à la modélisation géométrique et thermodynamique d’une classe de milieux faiblement continus. Arch Rat Mech Anal1998; 141(3): 199–236.
16.
EpsteinMSegevR.Geometric aspects of singular dislocations. Math Mech Solids2014; 19(4): 337–349.
17.
LeeELiuD.Finite-strain elastic–plastic theory with application to plane-wave analysis. J Appl Phys1967; 38: 19–27.
18.
YavariAGorielyA.Weyl geometry and the nonlinear mechanics of distributed point defects. Proc R Soc Lond Ser A Math Phys Eng Sci2012; 468(2148): 3902–3922.
19.
ReinaCSchlömerkemperAContiS.Derivation of as the continuum limit of crystalline slip. J Mech Phys Solids2016; 89: 231–254.
20.
YavariAGorielyA.Riemann-Cartan geometry of nonlinear dislocation mechanics. Arch Rat Mech Anal2012; 205(1): 59–118.
21.
KleinertH.Multivalued Fields in Condensed Matter, Electromagnetism, and Gravitation. Hackensack, NJ: World Scientific, 2008.
22.
RakotomananaLR.Covariance and Gauge Invariance in Continuum Physics: Application to Mechanics, Gravitation, and Electromagnetism (Progress in Mathematical Physics, Vol. 73). Cham: Birkhäuser/Springer, 2018.
23.
KleinertH.Nonholonomic mapping principle for classical and quantum mechanics in spaces with curvature and torsion. Gen Relativ Gravitat2000; 32(5): 769–839.
24.
FuthazarGLe MarrecLRakotomananaL.Covariant gradient continua applied to wave propagation within defective material. Arch Appl Mech2014; 84(9–11): 1339–1356.
25.
PeshkovIRomenskiEDumbserM.Continuum mechanics with torsion. Contin Mech Thermodyn2019; 31(5): 1517–1541.
26.
PeshkovIRomenskiE.A hyperbolic model for viscous Newtonian flows. Contin Mech Thermodyn2016; 28(1–2): 85–104.
27.
DumbserMPeshkovIRomenskiE.A unified hyperbolic formulation for viscous fluids and elastoplastic solids. In: Theory, Numerics and Applications of Hyperbolic Problems. II (Springer Proceedings in Mathematics and Statistics, Vol. 237). Cham: Springer, 2018, pp. 451–463.
28.
EringenACSuhubiES. Nonlinear theory of simple micro-elastic solids. I. Internat J Engrg Sci1964; 2: 189–203.
29.
SuhubiESEringenAC. Nonlinear theory of micro-elastic solids. II. Internat J Engrg Sci1964; 2: 389–404.
30.
MindlinRD.Micro-structure in linear elasticity. Arch Rat Mech Anal1964; 16: 51–78.
31.
PolizzottoC.A second strain gradient elasticity theory with second velocity gradient inertia – part I: Constitutive equations and quasi-static behavior. Int J Solids Struct2013; 50(24): 3749–3765.
32.
ForestS.Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J Eng Mech2009; 135(3): 117–131.
33.
GrammenoudisPTsakmakisC.Micromorphic continuum. Part I: Strain and stress tensors and their associated rates. Int J Non-Lin Mech2009; 44(9): 943.
34.
GhibaIDNeffPMadeoA, et al. The relaxed linear micromorphic continuum: existence, uniqueness and continuous dependence in dynamics. Math Mech Solids2015; 20(10): 1171–1197.
35.
MadeoANeffPGhibaID, et al. Band gaps in the relaxed linear micromorphic continuum. ZAMM Z Angew Math Mech2015; 95(9): 880–887.
36.
MadeoABarbagalloGColletM, et al. Relaxed micromorphic modeling of the interface between a homogeneous solid and a band-gap metamaterial: new perspectives towards metastructural design. Math Mech Solids2018; 23(12): 1485–1506.
37.
NeffPForestS.A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. J Elasticity2007; 87(2–3): 239–276.
38.
YavariAMarsdenJEOrtizM.On spatial and material covariant balance laws in elasticity. J Math Phys2006; 47(4): 042903.
39.
YavariAMarsdenJE.Covariant balance laws in continua with microstructure. Rep Math Phys2009; 63(1): 1–42.
40.
YavariAGorielyA.Riemann-Cartan geometry of nonlinear disclination mechanics. Math Mech Solids2013; 18(1): 91–102.
41.
EpsteinMElżanowskiM.Material inhomogeneities and their evolution: A geometric approach. In: Interaction of Mechanics and Mathematics. Berlin: Springer, 2007.
42.
RougéeP.Mécanique des grandes transformations, (Mathématiques & Applications (Berlin) [Mathematics & Applications], Vol. 25). Berlin: Springer-Verlag, 1997.
43.
WangCC. On the geometric structures of simple bodies. A mathematical foundation for the theory of continuous distributions of dislocations. Arch Rat Mech Anal 1967/68; 27: 33–94.
44.
IlievBZ. Fibre bundle formulation of nonrelativistic quantum mechanics. I. Introduction. The evolution transport. J Phys A2001; 34(23): 4887–4918.
45.
RomanoG.Continuum mechanics on manifolds. Lecture Notes, University of Naples Federico II, Naples, Italy, 2007; pp. 1–695.
46.
EpsteinM.The Elements of Continuum Biomechanics. New York: John Wiley & Sons, 2012.
47.
BradlynBReadN.Low-energy effective theory in the bulk for transport in a topological phase. Phys Rev B2015; 91: 125303.
48.
LychevSKoifmanKPetrenkoA, et al. Modeling and optimization for oriented growing solids. In: 2020 15th International Conference on Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy’s Conference) (STAB), pp. 1–4.
49.
HéleinF.Manifolds obtained by soldering together points, lines, etc. arXiv preprint arXiv:0904.4616, 2009.
50.
KolevBDesmoratR.Objective rates as covariant derivatives on the manifold of riemannian metrics. arXiv preprint arXiv:2106.01126, 2021.
51.
NakaharaM.Geometry, Topology and Physics. (2nd ed.) ( Graduate Student Series in Physics). Bristol: Institute of Physics, 2003.
52.
EpsteinM.The Geometrical Language of Continuum Mechanics. Cambridge: Cambridge University Press, 2010.
53.
EpsteinM.Differential Geometry: Basic Notions and Physical Examples. Cham: Springer, 2014.
54.
BaoDChernSSShenZ.An Introduction to Riemann-Finsler Geometry (Graduate Texts in Mathematics, Vol. 200). New York: Springer-Verlag, 2000.
55.
BucataruIMironR.Applications to dynamical systems. In: Finsler-Lagrange Geometry. Bucharest: Editura Academiei Române, 2007.
BejancuA.Finsler Geometry and Applications ( Ellis Horwood Series: Mathematics and Its Applications). New York: Ellis Horwood, 1990.
58.
KondoK.Non-riemannian and finslerian approaches to the theory of yielding. Int J Eng Sci1963; 1(1): 71–88.
59.
SaczukJ.On the role of the Finsler geometry in the theory of elasto-plasticity. Rep Math Phys1997; 39(1): 1–17.
60.
FuMFSaczukJStumpfH.On fibre bundle approach to a damage analysis. Internat J Engrg Sci1998; 36(15): 1741–1762.
61.
StumpfHSaczukJ.A generalized model of oriented continuum with defects. ZAMM Z Angew Math Mech2000; 80(3): 147–169.
62.
ClaytonJD.On Finsler geometry and applications in mechanics: Review and new perspectives. Adv Math Phys2015; 11: 828475.
63.
ClaytonJD.Generalized Finsler geometric continuum physics with applications in fracture and phase transformations. Z Angew Math Phys2017; 68(1): 9.
64.
ClaytonJ.Finsler geometry of nonlinear elastic solids with internal structure. J Geom Phys2017; 112: 118–146.
65.
YajimaTNagahamaH.Connection structures of topological singularity in micromechanics from a viewpoint of generalized Finsler space. Ann Phys2020; 532(12): 2000306.
66.
BucataruIMironR.Nonlinear connections for nonconservative mechanical systems. Rep Math Phys2007; 59(2): 225–241.
67.
YajimaTNagahamaH.KCC-theory and geometry of the Rikitake system. J Phys A2007; 40(11): 2755–2772.
68.
LeKCStumpfH.On the determination of the crystal reference in nonlinear continuum theory of dislocations. Proc R Soc Lond Ser A Math Phys Eng Sci1996; 452(1945): 359–371.
69.
EringenAC.Microcontinuum Field Theories. I. Foundations and Solids. New York: Springer-Verlag, 1999.
70.
ObukhovYNPonomarievVNZhytnikovVV.Quadratic Poincaré gauge theory of gravity: A comparison with the general relativity theory. Gen Relativ Gravitat1989; 21(11): 1107–1142.
71.
LeKStumpfH.Nonlinear continuum theory of dislocations. Int J Eng Sci1996; 34(3): 339–358.
72.
Choquet-BruhatYDeWitt-MoretteCDillard-BleickM.Analysis, Manifolds and Physics (2nd ed.). Amsterdam: North-Holland, 1982.
