×

Semiholonomic second order connections associated with material bodies. (English) Zbl 1338.53049

Summary: The thermomechanical behavior of a material is expressed mathematically by means of one or more constitutive equations representing the response of the body to the history of its deformation and temperature. These settings induce a set of connections which can express local properties. We replace two of them by a second order connection and prove that the holonomity of this connection classifies our materials.

MSC:

53C05 Connections (general theory)
70F20 Holonomic systems related to the dynamics of a system of particles
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
74A20 Theory of constitutive functions in solid mechanics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] M. Epstein and M. El\Dzanowski, Material Inhomogeneities and Their Evolution, Interaction of Mechanics and Mathematics, Springer, Berlin, Germany, 2007.
[2] M. Epstein, The Geometrical Language of Continuum Mechanics, Cambridge University Press, Cambridge, UK, 2010. · Zbl 1206.53002
[3] I. Kolá\vr, P. W. Michor, and J. Slovák, Natural Operations in Differential Geometry, Springer, Berlin, Germany, 1993. · Zbl 0782.53013
[4] J. Virsik, “On the holonomity of higher order connections,” Cahiers de Topologie et Geometrie Differentielle Categoriques, vol. 12, pp. 197-212, 1971. · Zbl 0223.53026
[5] P. Va, “Transformations of semiholonomic 2- and 3-jets and semiholonomic prolongation of connections,” Proceedings of the Estonian Academy of Sciences, vol. 59, no. 4, pp. 375-380, 2010. · Zbl 1213.53019
[6] G. Atanasiu, V. Balan, N. Brînzei, and M. Rahula, Differential Geometric Structures: Tangent Bundles, Connections in Bundles, Exponential Law in the Jet Space, Librokom, Moscow, Russia, 2010. · Zbl 1247.43005
[7] M. Rahula, P. Va, and N. Voicu, “Tangent structures: sector-forms, jets and connections,” Journal of Physics: Conference Series, vol. 346, Article ID 012023, 2012.
[8] M. Doupovec and W. M. Mikulski, “Reduction theorems for principal and classical connections,” Acta Mathematica Sinica, English Series, vol. 26, no. 1, pp. 169-184, 2010. · Zbl 1186.53036
[9] P. Va, “On the Ehresmann prolongation,” Annales Universitatis Mariae Curie-Skłodowska A, vol. 61, pp. 145-153, 2007. · Zbl 1136.53024
[10] M. Rahula, New Problems in Differential Geometry, vol. 8, World Scientific Publishing, River Edge, NJ, USA, 1993. · Zbl 0795.53002
[11] W. M. Mikulski, “Natural transformations transforming functions and vector fields to functions on some natural bundles,” Mathematica Bohemica, vol. 117, no. 2, pp. 217-223, 1992. · Zbl 0810.58004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.