September  2019, 11(3): 301-324. doi: 10.3934/jgm.2019017

Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies

1. 

Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), c\ Nicolás Cabrera, 13-15, Campus Cantoblanco, UAM, 28049 Madrid, Spain

2. 

Real Academia de Ciencias Exactas, Fisicas y Naturales, c\ de Valverde, 22, 28004 Madrid, Spain

3. 

Department of Mechanical and Manufacturing Engineering, University of Calgary. 2500 University Drive NW, Calgary, Alberta, T2N IN4, Canada

* Corresponding author: Víctor Manuel Jiménez Morales

Received  September 2016 Revised  January 2019 Published  August 2019

Fund Project: This work has been partially supported by MINECO Grants MTM2016-76-072-P and the ICMAT Severo Ochoa projects SEV-2011-0087 and SEV-2015-0554. V.M. Jiménez wishes to thank MINECO for a FPI-PhD Position. We would like to thank the referees for their valuable suggestions that have contributed to improve this paper

A Lie groupoid, called material Lie groupoid, is associated in a natural way to any elastic material. The corresponding Lie algebroid, called material algebroid, is used to characterize the uniformity and the homogeneity properties of the material. The relation to previous results in terms of $ G- $structures is discussed in detail. An illustrative example is presented as an application of the theory.

Citation: Víctor Manuel Jiménez Morales, Manuel De León, Marcelo Epstein. Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies. Journal of Geometric Mechanics, 2019, 11 (3) : 301-324. doi: 10.3934/jgm.2019017
References:
[1]

B. A. Bilby, Continuous distributions of dislocations, in Progress in Solid Mechanics, Vol. 1, North-Holland Publishing Co., Amsterdam, 1960, 329–398. Google Scholar

[2]

F. Bloom, Modern Differential Geometric Techniques in the Theory of Continuous Distributions of Dislocations, vol. 733 of Lecture Notes in Mathematics, Springer, Berlin, 1979. Google Scholar

[3]

B. D. Coleman, Simple liquid crystals, Archive for Rational Mechanics and Analysis, 20 (1965), 41-58. doi: 10.1007/BF00250189. Google Scholar

[4]

L. A. Cordero, C. T. Dodson and M. de León, Differential Geometry of Frame Bundles, Mathematics and Its Applications, Springer Netherlands, Dordrecht, 1989, https://books.google.es/books?id=JLSFW8aVzFUC. doi: 10.1007/978-94-009-1265-6. Google Scholar

[5]

M. ElżanowskiM. Epstein and J. Śniatycki, $G$-structures and material homogeneity, J. Elasticity, 23 (1990), 167-180. doi: 10.1007/BF00054801. Google Scholar

[6]

M. Elżanowski and S. Prishepionok, Locally homogeneous configurations of uniform elastic bodies, Rep. Math. Phys., 31 (1992), 329-340. doi: 10.1016/0034-4877(92)90023-T. Google Scholar

[7] M. Epstein, The Geometrical Language of Continuum Mechanics, Cambridge University Press, 2010. doi: 10.1017/CBO9780511762673. Google Scholar
[8]

M. Epstein and M. de León, Geometrical theory of uniform Cosserat media, Journal of Geometry and Physics, 26 (1998), 127–170, http://www.sciencedirect.com/science/article/pii/S0393044097000429. doi: 10.1016/S0393-0440(97)00042-9. Google Scholar

[9]

M. Epstein and M. de León, Continuous distributions of inhomogeneities in liquid-crystal-like bodies, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 457 (2001), 2507–2520, http://rspa.royalsocietypublishing.org/content/457/2014/2507. doi: 10.1098/rspa.2001.0835. Google Scholar

[10]

M. Epstein and M. de León, Unified geometric formulation of material uniformity and evolution, Math. Mech. Complex Syst., 4 (2016), 17-29. doi: 10.2140/memocs.2016.4.17. Google Scholar

[11]

M. EpsteinV. M. Jiménez and M. de León, Material geometry, Journal of Elasticity, 135 (2019), 237-260. doi: 10.1007/s10659-018-9693-2. Google Scholar

[12]

J. D. Eshelby, The force on an elastic singularity, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 244 (1951), 84–112, http://rsta.royalsocietypublishing.org/content/244/877/87. doi: 10.1098/rsta.1951.0016. Google Scholar

[13]

S. FerraroM. de LeónJ. MarreroD. de Diego and M. Vaquero, On the geometry of the hamilton–jacobi equation and generating functions, Archive for Rational Mechanics and Analysis, 226 (2017), 243-302. doi: 10.1007/s00205-017-1133-0. Google Scholar

[14]

V. M. Jiménez, M. de León and M. Epstein, Material distributions, Mathematics and Mechanics of Solids, 1081286517736922.Google Scholar

[15]

V. M. Jiménez, M. de León and M. Epstein, Characteristic distribution: An application to material bodies, Journal of Geometry and Physics, 127 (2018), 19–31, http://www.sciencedirect.com/science/article/pii/S0393044018300378. doi: 10.1016/j.geomphys.2018.01.021. Google Scholar

[16]

V. M. Jiménez, M. de León and M. Epstein, Lie groupoids and algebroids applied to the study of uniformity and homogeneity of cosserat media, International Journal of Geometric Methods in Modern Physics, 15 (2018), 1830003, 60pp. doi: 10.1142/S0219887818300039. Google Scholar

[17]

K. Kondo, Geometry of elastic deformation and incompatibility, 1 (1955), 5–17.Google Scholar

[18]

E. Kr{ö}ner, Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen, vol. 4, 1960, https://books.google.es/books?id=bXCdGwAACAAJ.Google Scholar

[19]

E. Kröner, Mechanics of Generalized Continua, Springer, Heidelberg, 1968.Google Scholar

[20]

R. W. Lardner, Mathematical Theory of Dislocations and Fracture, Mathematical expositions, University of Toronto Press, Toronto, 1974, https://books.google.es/books?id=WZlsAAAAMAAJ. Google Scholar

[21]

K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, vol. 213 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9781107325883. Google Scholar

[22]

J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Dover Publications, Inc., New York, 1994, Corrected reprint of the 1983 original. Google Scholar

[23]

G. A. Maugin, Material Inhomogeneities in Elasticity, vol. 3 of Applied Mathematics and Mathematical Computation, Chapman & Hall, London, 1993 Google Scholar

[24]

I. Moerdijk and J. Mrčun, On integrability of infinitesimal actions, Amer. J. Math., 124 (2002), 567–593, http://muse.jhu.edu/journals/american_journal_of_mathematics/v124/124.3moerdijk.pdf. doi: 10.1353/ajm.2002.0019. Google Scholar

[25]

F. R. N. Nabarro, Theory of Crystal Dislocations, Dover Books on Physics and Chemistry, Dover Publications, New York, 1987, https://books.google.es/books?id=zD5CAQAAIAAJ.Google Scholar

[26]

W. Noll, Materially uniform simple bodies with inhomogeneities, Arch. Rational Mech. Anal., 27 (1967/1968), 1-32. doi: 10.1007/BF00276433. Google Scholar

[27]

J. L. Synge, Principles of Classical Mechanics and Field Theory, no. v. 3, n.o 1 in Handbuch der Physik, Springer, Berlin, 1960, https://books.google.es/books?id=hthAAQAAIAAJ.Google Scholar

[28]

C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, 3rd edition, Springer-Verlag, Berlin, 2004, Edited and with a preface by Stuart S. Antman. doi: 10.1007/978-3-662-10388-3. Google Scholar

[29]

J. N. Valdés, Á. F. T. Villalón and J. A. V. Alarcón, Elementos de la Teoría de Grupoides Y Algebroides, Universidad de Cádiz, Servicio de Publicaciones, Cádiz, 2006, https://books.google.es/books?id=srgOQIoHqfMC.Google Scholar

[30]

C. C. Wang, A general theory of subfluids, Archive for Rational Mechanics and Analysis, 20 (1965), 1-40. doi: 10.1007/BF00250188. Google Scholar

[31]

C. C. Wang, On the geometric structures of simple bodies. A mathematical foundation for the theory of continuous distributions of dislocations, Arch. Rational Mech. Anal., 27 (1967/1968), 33-94. doi: 10.1007/BF00276434. Google Scholar

[32]

C. C. Wang and C. Truesdell, Introduction to Rational Elasticity, Noordhoff International Publishing, Leyden, 1973, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics of Continua. Google Scholar

[33]

A. Weinstein, Lagrangian mechanics and groupoids, Fields Institute Comm., 7 (1996), 207-231. Google Scholar

[34]

A. Weinstein, Groupoids: Unifying internal and external symmetry. A tour through some examples, in Groupoids in Analysis, Geometry, and Physics (Boulder, CO, 1999), vol. 282 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2001, 1–19. doi: 10.1090/conm/282/04675. Google Scholar

show all references

References:
[1]

B. A. Bilby, Continuous distributions of dislocations, in Progress in Solid Mechanics, Vol. 1, North-Holland Publishing Co., Amsterdam, 1960, 329–398. Google Scholar

[2]

F. Bloom, Modern Differential Geometric Techniques in the Theory of Continuous Distributions of Dislocations, vol. 733 of Lecture Notes in Mathematics, Springer, Berlin, 1979. Google Scholar

[3]

B. D. Coleman, Simple liquid crystals, Archive for Rational Mechanics and Analysis, 20 (1965), 41-58. doi: 10.1007/BF00250189. Google Scholar

[4]

L. A. Cordero, C. T. Dodson and M. de León, Differential Geometry of Frame Bundles, Mathematics and Its Applications, Springer Netherlands, Dordrecht, 1989, https://books.google.es/books?id=JLSFW8aVzFUC. doi: 10.1007/978-94-009-1265-6. Google Scholar

[5]

M. ElżanowskiM. Epstein and J. Śniatycki, $G$-structures and material homogeneity, J. Elasticity, 23 (1990), 167-180. doi: 10.1007/BF00054801. Google Scholar

[6]

M. Elżanowski and S. Prishepionok, Locally homogeneous configurations of uniform elastic bodies, Rep. Math. Phys., 31 (1992), 329-340. doi: 10.1016/0034-4877(92)90023-T. Google Scholar

[7] M. Epstein, The Geometrical Language of Continuum Mechanics, Cambridge University Press, 2010. doi: 10.1017/CBO9780511762673. Google Scholar
[8]

M. Epstein and M. de León, Geometrical theory of uniform Cosserat media, Journal of Geometry and Physics, 26 (1998), 127–170, http://www.sciencedirect.com/science/article/pii/S0393044097000429. doi: 10.1016/S0393-0440(97)00042-9. Google Scholar

[9]

M. Epstein and M. de León, Continuous distributions of inhomogeneities in liquid-crystal-like bodies, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 457 (2001), 2507–2520, http://rspa.royalsocietypublishing.org/content/457/2014/2507. doi: 10.1098/rspa.2001.0835. Google Scholar

[10]

M. Epstein and M. de León, Unified geometric formulation of material uniformity and evolution, Math. Mech. Complex Syst., 4 (2016), 17-29. doi: 10.2140/memocs.2016.4.17. Google Scholar

[11]

M. EpsteinV. M. Jiménez and M. de León, Material geometry, Journal of Elasticity, 135 (2019), 237-260. doi: 10.1007/s10659-018-9693-2. Google Scholar

[12]

J. D. Eshelby, The force on an elastic singularity, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 244 (1951), 84–112, http://rsta.royalsocietypublishing.org/content/244/877/87. doi: 10.1098/rsta.1951.0016. Google Scholar

[13]

S. FerraroM. de LeónJ. MarreroD. de Diego and M. Vaquero, On the geometry of the hamilton–jacobi equation and generating functions, Archive for Rational Mechanics and Analysis, 226 (2017), 243-302. doi: 10.1007/s00205-017-1133-0. Google Scholar

[14]

V. M. Jiménez, M. de León and M. Epstein, Material distributions, Mathematics and Mechanics of Solids, 1081286517736922.Google Scholar

[15]

V. M. Jiménez, M. de León and M. Epstein, Characteristic distribution: An application to material bodies, Journal of Geometry and Physics, 127 (2018), 19–31, http://www.sciencedirect.com/science/article/pii/S0393044018300378. doi: 10.1016/j.geomphys.2018.01.021. Google Scholar

[16]

V. M. Jiménez, M. de León and M. Epstein, Lie groupoids and algebroids applied to the study of uniformity and homogeneity of cosserat media, International Journal of Geometric Methods in Modern Physics, 15 (2018), 1830003, 60pp. doi: 10.1142/S0219887818300039. Google Scholar

[17]

K. Kondo, Geometry of elastic deformation and incompatibility, 1 (1955), 5–17.Google Scholar

[18]

E. Kr{ö}ner, Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen, vol. 4, 1960, https://books.google.es/books?id=bXCdGwAACAAJ.Google Scholar

[19]

E. Kröner, Mechanics of Generalized Continua, Springer, Heidelberg, 1968.Google Scholar

[20]

R. W. Lardner, Mathematical Theory of Dislocations and Fracture, Mathematical expositions, University of Toronto Press, Toronto, 1974, https://books.google.es/books?id=WZlsAAAAMAAJ. Google Scholar

[21]

K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, vol. 213 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9781107325883. Google Scholar

[22]

J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Dover Publications, Inc., New York, 1994, Corrected reprint of the 1983 original. Google Scholar

[23]

G. A. Maugin, Material Inhomogeneities in Elasticity, vol. 3 of Applied Mathematics and Mathematical Computation, Chapman & Hall, London, 1993 Google Scholar

[24]

I. Moerdijk and J. Mrčun, On integrability of infinitesimal actions, Amer. J. Math., 124 (2002), 567–593, http://muse.jhu.edu/journals/american_journal_of_mathematics/v124/124.3moerdijk.pdf. doi: 10.1353/ajm.2002.0019. Google Scholar

[25]

F. R. N. Nabarro, Theory of Crystal Dislocations, Dover Books on Physics and Chemistry, Dover Publications, New York, 1987, https://books.google.es/books?id=zD5CAQAAIAAJ.Google Scholar

[26]

W. Noll, Materially uniform simple bodies with inhomogeneities, Arch. Rational Mech. Anal., 27 (1967/1968), 1-32. doi: 10.1007/BF00276433. Google Scholar

[27]

J. L. Synge, Principles of Classical Mechanics and Field Theory, no. v. 3, n.o 1 in Handbuch der Physik, Springer, Berlin, 1960, https://books.google.es/books?id=hthAAQAAIAAJ.Google Scholar

[28]

C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, 3rd edition, Springer-Verlag, Berlin, 2004, Edited and with a preface by Stuart S. Antman. doi: 10.1007/978-3-662-10388-3. Google Scholar

[29]

J. N. Valdés, Á. F. T. Villalón and J. A. V. Alarcón, Elementos de la Teoría de Grupoides Y Algebroides, Universidad de Cádiz, Servicio de Publicaciones, Cádiz, 2006, https://books.google.es/books?id=srgOQIoHqfMC.Google Scholar

[30]

C. C. Wang, A general theory of subfluids, Archive for Rational Mechanics and Analysis, 20 (1965), 1-40. doi: 10.1007/BF00250188. Google Scholar

[31]

C. C. Wang, On the geometric structures of simple bodies. A mathematical foundation for the theory of continuous distributions of dislocations, Arch. Rational Mech. Anal., 27 (1967/1968), 33-94. doi: 10.1007/BF00276434. Google Scholar

[32]

C. C. Wang and C. Truesdell, Introduction to Rational Elasticity, Noordhoff International Publishing, Leyden, 1973, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics of Continua. Google Scholar

[33]

A. Weinstein, Lagrangian mechanics and groupoids, Fields Institute Comm., 7 (1996), 207-231. Google Scholar

[34]

A. Weinstein, Groupoids: Unifying internal and external symmetry. A tour through some examples, in Groupoids in Analysis, Geometry, and Physics (Boulder, CO, 1999), vol. 282 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2001, 1–19. doi: 10.1090/conm/282/04675. Google Scholar

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