# American Institute of Mathematical Sciences

doi: 10.3934/jgm.2022001
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## The evolution equation: An application of groupoids to material evolution

 1 Universidad de Alcalá (UAH), Departamento de Física y Matemáticas, Av. de León, 4A, 28805 Alcalá de Henares, Madrid, Spain 2 Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C\Nicolás Cabrera, 13-15, Campus Cantoblanco, UAM, 28049 Madrid, Spain 3 Real Academia de Ciencias Exactas, Fisicas y Naturales, C/de Valverde 22, 28004 Madrid, Spain

*Corresponding author: Víctor Manuel Jiménez

To the memory of Kirill Mackenzie

Received  August 2021 Early access January 2022

The aim of this paper is to study the evolution of a material point of a body by itself, and not the body as a whole. To do this, we construct a groupoid encoding all the intrinsic properties of the material point and its characteristic foliations, which permits us to define the evolution equation. We also discuss phenomena like remodeling and aging.

Citation: Víctor Manuel Jiménez, Manuel de León. The evolution equation: An application of groupoids to material evolution. Journal of Geometric Mechanics, doi: 10.3934/jgm.2022001
##### References:
 [1] H. Brandt, Über eine Verallgemeinerung des Gruppenbegriffes, Math. Ann., 96 (1927), 360-366.  doi: 10.1007/BF01209171. [2] A. R. Carotenuto, A. Cutolo, S. Palumbo and M. Fraldi, Growth and remodeling in highly stressed solid tumors, Meccanica, 54 (2019), 1941-1957.  doi: 10.1007/s11012-019-01057-5. [3] B. D. Coleman, Simple liquid crystals, Arch. Rational Mech. Anal., 20 (1965), 41-58.  doi: 10.1007/BF00250189. [4] M. de León, M. Epstein and V. M. Jiménez, Material Geometry: Groupoids in Continuum Mechanics, World Scientific, Singapore, 2021. doi: 10.1142/12168. [5] C. Ehresmann, Catégories topologiques et catégories différentiables, in Colloque Géom. Diff. Globale (Bruxelles, 1958), Centre Belge Rech. Math., Louvain, 1959, 137–150. [6] C. Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, in Séminaire Bourbaki, Vol. 1, Soc. Math. France, Paris, 1995, 153–168. [7] C. Ehresmann, Les prolongements d'une variété différentiable. V. Covariants différentiels et prolongements d'une structure infinitésimale, C. R. Acad. Sci. Paris, 234 (1952), 1424-1425. [8] C. Ehresmann, Sur les connexions d'ordre supérieur, in Dagli Atti del V Congresso dell'Unione Matematica Italiana, 1956, 344–346. [9] M. Epstein, The Geometrical Language of Continuum Mechanics, Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511762673. [10] M. Epstein, Laminated uniformity and homogeneity, Mach. Res. Comm., 93 (2018), 66-69.  doi: 10.1016/j.mechrescom.2017.05.004. [11] M. Epstein, Mathematical characterization and identification of remodeling, growth, aging and morphogenesis, J. Mech. Phys. Solids, 84 (2015), 72-84.  doi: 10.1016/j.jmps.2015.07.009. [12] 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. [13] M. Epstein and M. Elżanowski, Material Inhomogeneities and Their Evolution. A Geometric Approach, Interaction of Mechanics and Mathematics, Springer, Berlin, 2007. doi: 10.1007/978-3-540-72373-8. [14] V. M. Jiménez, M. de León and M. Epstein, Characteristic distribution: An application to material bodies, J. Geom. Phys., 127 (2018), 19-31.  doi: 10.1016/j.geomphys.2018.01.021. [15] V. M. Jiménez, M. de León and M. Epstein, Lie groupoids and algebroids applied to the study of uniformity and homogeneity of Cosserat media, Int. J. Geom. Methods Mod. Phys., 15 (2018), 60pp. doi: 10.1142/S0219887818300039. [16] V. M. Jiménez, M. de León and M. Epstein, Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies, J. Geom. Mech., 11 (2019), 301-324.  doi: 10.3934/jgm.2019017. [17] V. M. Jiménez, M. de León and M. Epstein, Material distributions, Math. Mech. Solids, 25 (2020), 1450-1458.  doi: 10.1177/1081286517736922. [18] K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9781107325883. [19] J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Dover Publications, Inc., New York, 1994. [20] W. Noll, On the Continuity of the Solid and Fluid States, Ph.D thesis, Indiana University, 1954. [21] J. Pradines, Théorie de Lie pour les groupoïdes différentiables. Relations entre propriétés locales et globales, C. R. Acad. Sci. Paris Sér. A-B, 263 (1966), A907–A910. [22] E. K. Rodriguez, A. Hoger and A. D. McCulloch, Stress-dependent finite growth in soft elastic tissues, J. Biomechanics, 27 (1994), 455-467.  doi: 10.1016/0021-9290(94)90021-3. [23] D. J. Saunders, The Geometry of Jet Bundles, London Mathematical Society Lecture Note Series, 142, Cambridge University Press, Cambridge, 1989.  doi: 10.1017/CBO9780511526411. [24] P. Stefan, Accessible sets, orbits, and foliations with singularities, Proc. London Math. Soc. (3), 29 (1974), 699-713.  doi: 10.1112/plms/s3-29.4.699. [25] H. J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc., 180 (1973), 171-188.  doi: 10.1090/S0002-9947-1973-0321133-2. [26] C. H. Turner, On Wolff's law of trabecular architecture, J. Biomechanics, 25 (1992), 1-9.  doi: 10.1016/0021-9290(92)90240-2. [27] 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. [28] C.-C. Wang, A general theory of subfluids, Arch. Rational Mech. Anal., 20 (1965), 1-40.  doi: 10.1007/BF00250188. [29] C. C. Wang and C. Truesdell, Introduction to Rational Elasticity, Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics of Continua, Noordhoff International Publishing, Leyden, 1973. [30] A. Weinstein, Groupoids: Unifying internal and external symmetry. A tour through some examples, in Groupoids in Analysis, Geometry, and Physics (Boulder, CO, 1999), Contemp. Math., 282, Amer. Math. Soc., Providence, RI, 2001, 1–19. doi: 10.1090/conm/282/04675.

show all references

##### References:
 [1] H. Brandt, Über eine Verallgemeinerung des Gruppenbegriffes, Math. Ann., 96 (1927), 360-366.  doi: 10.1007/BF01209171. [2] A. R. Carotenuto, A. Cutolo, S. Palumbo and M. Fraldi, Growth and remodeling in highly stressed solid tumors, Meccanica, 54 (2019), 1941-1957.  doi: 10.1007/s11012-019-01057-5. [3] B. D. Coleman, Simple liquid crystals, Arch. Rational Mech. Anal., 20 (1965), 41-58.  doi: 10.1007/BF00250189. [4] M. de León, M. Epstein and V. M. Jiménez, Material Geometry: Groupoids in Continuum Mechanics, World Scientific, Singapore, 2021. doi: 10.1142/12168. [5] C. Ehresmann, Catégories topologiques et catégories différentiables, in Colloque Géom. Diff. Globale (Bruxelles, 1958), Centre Belge Rech. Math., Louvain, 1959, 137–150. [6] C. Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, in Séminaire Bourbaki, Vol. 1, Soc. Math. France, Paris, 1995, 153–168. [7] C. Ehresmann, Les prolongements d'une variété différentiable. V. Covariants différentiels et prolongements d'une structure infinitésimale, C. R. Acad. Sci. Paris, 234 (1952), 1424-1425. [8] C. Ehresmann, Sur les connexions d'ordre supérieur, in Dagli Atti del V Congresso dell'Unione Matematica Italiana, 1956, 344–346. [9] M. Epstein, The Geometrical Language of Continuum Mechanics, Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511762673. [10] M. Epstein, Laminated uniformity and homogeneity, Mach. Res. Comm., 93 (2018), 66-69.  doi: 10.1016/j.mechrescom.2017.05.004. [11] M. Epstein, Mathematical characterization and identification of remodeling, growth, aging and morphogenesis, J. Mech. Phys. Solids, 84 (2015), 72-84.  doi: 10.1016/j.jmps.2015.07.009. [12] 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. [13] M. Epstein and M. Elżanowski, Material Inhomogeneities and Their Evolution. A Geometric Approach, Interaction of Mechanics and Mathematics, Springer, Berlin, 2007. doi: 10.1007/978-3-540-72373-8. [14] V. M. Jiménez, M. de León and M. Epstein, Characteristic distribution: An application to material bodies, J. Geom. Phys., 127 (2018), 19-31.  doi: 10.1016/j.geomphys.2018.01.021. [15] V. M. Jiménez, M. de León and M. Epstein, Lie groupoids and algebroids applied to the study of uniformity and homogeneity of Cosserat media, Int. J. Geom. Methods Mod. Phys., 15 (2018), 60pp. doi: 10.1142/S0219887818300039. [16] V. M. Jiménez, M. de León and M. Epstein, Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies, J. Geom. Mech., 11 (2019), 301-324.  doi: 10.3934/jgm.2019017. [17] V. M. Jiménez, M. de León and M. Epstein, Material distributions, Math. Mech. Solids, 25 (2020), 1450-1458.  doi: 10.1177/1081286517736922. [18] K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9781107325883. [19] J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Dover Publications, Inc., New York, 1994. [20] W. Noll, On the Continuity of the Solid and Fluid States, Ph.D thesis, Indiana University, 1954. [21] J. Pradines, Théorie de Lie pour les groupoïdes différentiables. Relations entre propriétés locales et globales, C. R. Acad. Sci. Paris Sér. A-B, 263 (1966), A907–A910. [22] E. K. Rodriguez, A. Hoger and A. D. McCulloch, Stress-dependent finite growth in soft elastic tissues, J. Biomechanics, 27 (1994), 455-467.  doi: 10.1016/0021-9290(94)90021-3. [23] D. J. Saunders, The Geometry of Jet Bundles, London Mathematical Society Lecture Note Series, 142, Cambridge University Press, Cambridge, 1989.  doi: 10.1017/CBO9780511526411. [24] P. Stefan, Accessible sets, orbits, and foliations with singularities, Proc. London Math. Soc. (3), 29 (1974), 699-713.  doi: 10.1112/plms/s3-29.4.699. [25] H. J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc., 180 (1973), 171-188.  doi: 10.1090/S0002-9947-1973-0321133-2. [26] C. H. Turner, On Wolff's law of trabecular architecture, J. Biomechanics, 25 (1992), 1-9.  doi: 10.1016/0021-9290(92)90240-2. [27] 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. [28] C.-C. Wang, A general theory of subfluids, Arch. Rational Mech. Anal., 20 (1965), 1-40.  doi: 10.1007/BF00250188. [29] C. C. Wang and C. Truesdell, Introduction to Rational Elasticity, Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics of Continua, Noordhoff International Publishing, Leyden, 1973. [30] A. Weinstein, Groupoids: Unifying internal and external symmetry. A tour through some examples, in Groupoids in Analysis, Geometry, and Physics (Boulder, CO, 1999), Contemp. Math., 282, Amer. Math. Soc., Providence, RI, 2001, 1–19. doi: 10.1090/conm/282/04675.
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