# American Institute of Mathematical Sciences

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. [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. [3] B. D. Coleman, Simple liquid crystals, Archive for Rational Mechanics and Analysis, 20 (1965), 41-58.  doi: 10.1007/BF00250189. [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. [5] M. Elżanowski, M. Epstein and J. Śniatycki, $G$-structures and material homogeneity, J. Elasticity, 23 (1990), 167-180.  doi: 10.1007/BF00054801. [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. [7] M. Epstein, The Geometrical Language of Continuum Mechanics, Cambridge University Press, 2010.  doi: 10.1017/CBO9780511762673. [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. [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. [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. [11] M. Epstein, V. M. Jiménez and M. de León, Material geometry, Journal of Elasticity, 135 (2019), 237-260.  doi: 10.1007/s10659-018-9693-2. [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. [13] S. Ferraro, M. de León, J. Marrero, D. 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. [14] V. M. Jiménez, M. de León and M. Epstein, Material distributions, Mathematics and Mechanics of Solids, 1081286517736922. [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. [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. [17] K. Kondo, Geometry of elastic deformation and incompatibility, 1 (1955), 5–17. [18] E. Kr{ö}ner, Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen, vol. 4, 1960, https://books.google.es/books?id=bXCdGwAACAAJ. [19] E. Kröner, Mechanics of Generalized Continua, Springer, Heidelberg, 1968. [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. [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. [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. [23] G. A. Maugin, Material Inhomogeneities in Elasticity, vol. 3 of Applied Mathematics and Mathematical Computation, Chapman & Hall, London, 1993 [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. [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. [26] W. Noll, Materially uniform simple bodies with inhomogeneities, Arch. Rational Mech. Anal., 27 (1967/1968), 1-32.  doi: 10.1007/BF00276433. [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. [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. [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. [30] C. C. Wang, A general theory of subfluids, Archive for Rational Mechanics and Analysis, 20 (1965), 1-40.  doi: 10.1007/BF00250188. [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. [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. [33] A. Weinstein, Lagrangian mechanics and groupoids, Fields Institute Comm., 7 (1996), 207-231. [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.

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. [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. [3] B. D. Coleman, Simple liquid crystals, Archive for Rational Mechanics and Analysis, 20 (1965), 41-58.  doi: 10.1007/BF00250189. [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. [5] M. Elżanowski, M. Epstein and J. Śniatycki, $G$-structures and material homogeneity, J. Elasticity, 23 (1990), 167-180.  doi: 10.1007/BF00054801. [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. [7] M. Epstein, The Geometrical Language of Continuum Mechanics, Cambridge University Press, 2010.  doi: 10.1017/CBO9780511762673. [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. [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. [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. [11] M. Epstein, V. M. Jiménez and M. de León, Material geometry, Journal of Elasticity, 135 (2019), 237-260.  doi: 10.1007/s10659-018-9693-2. [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. [13] S. Ferraro, M. de León, J. Marrero, D. 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. [14] V. M. Jiménez, M. de León and M. Epstein, Material distributions, Mathematics and Mechanics of Solids, 1081286517736922. [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. [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. [17] K. Kondo, Geometry of elastic deformation and incompatibility, 1 (1955), 5–17. [18] E. Kr{ö}ner, Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen, vol. 4, 1960, https://books.google.es/books?id=bXCdGwAACAAJ. [19] E. Kröner, Mechanics of Generalized Continua, Springer, Heidelberg, 1968. [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. [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. [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. [23] G. A. Maugin, Material Inhomogeneities in Elasticity, vol. 3 of Applied Mathematics and Mathematical Computation, Chapman & Hall, London, 1993 [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. [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. [26] W. Noll, Materially uniform simple bodies with inhomogeneities, Arch. Rational Mech. Anal., 27 (1967/1968), 1-32.  doi: 10.1007/BF00276433. [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. [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. [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. [30] C. C. Wang, A general theory of subfluids, Archive for Rational Mechanics and Analysis, 20 (1965), 1-40.  doi: 10.1007/BF00250188. [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. [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. [33] A. Weinstein, Lagrangian mechanics and groupoids, Fields Institute Comm., 7 (1996), 207-231. [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.
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