Abstract
For high-order continuum mechanics and classical field theories configurations are modeled as sections of general fiber bundles and generalized velocities are modeled as variations thereof. Smooth stress fields are considered and it is shown that three distinct mathematical stress objects play the roles of the traditional stress tensor of continuum mechanics in Euclidean spaces. These objects are referred to as the variational hyper-stress, the traction hyper-stress and the non-holonomic stress. The properties of these three stress objects and the relations between them are studied.
Similar content being viewed by others
References
Aifantis, E.C.: On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30, 1279–1299 (1992)
Askes, H., Aifantis, E.C.: Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int. J. Solids Struct. 48, 1962–1990 (2011)
Bertram, A.: Compendium on Gradient Materials (2017). http://www.ifme.ovgu.de/ifme_media/FL/Publikationen/Compendium+on+Gradient+Materials_May+2017.pdf
Binz, E., Śniatycki, J., Fischer, H.: Geometry of Classical Fields. North-Holland, Amsterdam (1988)
de León, M., Rodrigues, P.R.: Generalized Classical Mechanics and Field Theory: A Geometrical Approach of Lagrangian and Hamiltonian Formalisms Involving Higher Order Derivatives. North-Holland, Amsterdam (1985)
dell’Isola, F., Seppecher, P., Madeo, A.: How contact interactions may depend on the shape of Cauchy cuts in \(N\)th gradient continua: approach “á la d’Alembert”. Z. Angew. Math. Phys. 63, 1119–1141 (2012)
dell’Isola, F., Seppecher, P., della Corte, A.: The postulations á la d’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results. Proc. R. Soc. A 471, 20150415 (2015)
Fosdick, R.: A generalized continuum theory with internal corner and surface contact interactions. Contin. Mech. Thermodyn. 28, 275–292 (2016)
Giachetta, G., Mangiarotti, L., Sardanashvily, G.: Advanced Classical Field Theory. World Scientific, Singapore (2009)
Gotay, M.J., Isenberg, J., Marsden, J.E., Montgomery, R.: Momentum maps and classical fields. Part I: Covariant field theory (2003). arXiv:physics/9801019 [math-ph]
Hirsch, M.: Differential Topology. Springer, Berlin (1976)
Kijowski, J., Tuczyjew, W.M.: A Symplectic Framework for Field Theories. Lecture Notes in Physics, vol. 107. Springer, Berlin (1979)
Kupferman, R., Olami, E., Segev, R.: Stress theory for classical fields. Math. Mech. Solids (2017). https://doi.org/10.1177/1081286517723697
Lewis, A.: Notes on Global Analysis. Appendix F: Multilinear Algebra (2014). http://www.mast.queensu.ca/~andrew/teaching/math942/pdf/1appendixf.pdf
Michor, P.W.: Manifolds of Differentiable Mappings. Shiva, Chandigarh (1980)
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)
Mindlin, R.D.: Second gradient of strain and surface tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965)
Noll, W.: The foundations of classical mechanics in the light of recent advances in continuum mechanics. In: Henkin, L., Suppes, P., Tarski, A. (eds.) The Axiomatic Method, with Special Reference to Geometry and Physics, pp. 266–281. North-Holland, Amsterdam (1959)
Noll, W., Virga, E.G.: On edge interactions and surface tension. Arch. Ration. Mech. Anal. 111, 1–31 (1990)
Palais, R.S.: Foundations of Global Non-Linear Analysis. Benjamin, Elmsford (1968)
Podio-Guidugli, P.: Cauchy’s construction for flat complex bodies. J. Elast. 118, 101–107 (2015)
Ru, C.Q., Aifantis, E.C.: A simple approach to solve boundary-value problems in gradient elasticity. Acta Mech. 101, 59–68 (1993)
Saunders, D.J.: The Geometry of Jet Bundles. London Mathematical Society: Lecture Notes Series, vol. 142. Cambridge University Press, Cambridge (1989)
Segev, R.: Forces and the existence of stresses in invariant continuum mechanics. J. Math. Phys. 27, 163–170 (1986)
Segev, R.: Notes on metric independent analysis of classical fields. Math. Methods Appl. Sci. 36, 497–566 (2013)
Segev, R.: Geometric analysis of hyper-stresses. Int. J. Eng. Sci. 120, 100–118 (2017)
Segev, R., DeBotton, G.: On the consistency conditions for force systems. Int. J. Non-Linear Mech. 26, 47–59 (1991)
Segev, R., Rodnay, G.: Cauchy’s theorem on manifolds. J. Elast. 56(2), 129–144 (1999)
Segev, R., Śniatycki, J.: On jets, almost symmetric tensors, and traction hyper-stresses. Math. Mech. Complex Syst. 6, 101–124 (2018)
Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)
Toupin, R.A.: Theories of elasticity with couple-stress. Arch. Ration. Mech. Anal. 17, 85–112 (1964)
Acknowledgements
Both authors are grateful to BIRS for sponsoring the Banff Workshop on Material Evolution, June 11–18, 2017, which led to this collaboration. R.S.’s work has been partially supported by H. Greenhill Chair for Theoretical and Applied Mechanics and the Pearlstone Center for Aeronautical Engineering Studies at Ben-Gurion University.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Segev, R., Śniatycki, J. Hyper-stresses in \(k\)-Jet Field Theories. J Elast 135, 457–483 (2019). https://doi.org/10.1007/s10659-018-9691-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-018-9691-4