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Continuum Mechanics and Thermodynamics

, Volume 28, Issue 1–2, pp 379–390 | Cite as

Material description of fluxes in terms of differential forms

  • Salvatore FedericoEmail author
  • Alfio Grillo
  • Reuven Segev
Original Article

Abstract

The flux of a certain extensive physical quantity across a surface is often represented by the integral over the surface of the component of a pseudo-vector normal to the surface. A pseudo-vector is in fact a possible representation of a second-order differential form, i.e. a skew-symmetric second-order covariant tensor, which follows the regular transformation laws of tensors. However, because of the skew-symmetry of differential forms, the associated pseudo-vector follows a transformation law that is different from that of proper vectors, and is named after the Italian mathematical physicist Gabrio Piola (1794–1850). In this work, we employ the methods of Differential Geometry and the representation in terms of differential forms to demonstrate how the flux of an extensive quantity transforms from the spatial to the material point of view. After an introduction to the theory of differential forms, their transformation laws, and their role in integration theory, we apply them to the case of first-order transport laws such as Darcy’s law and Ohm’s law.

Keywords

Differential Geometry Differential form Flux Material Spatial Darcy’s law Ohm’s law 

Mathematics Subject Classification

53A45 74A05 

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References

  1. 1.
    Ateshian G.A., Weiss J.A.: Anisotropic hydraulic permeability under finite deformation. J. Biomech. Eng. 132, 111004 (2010)CrossRefGoogle Scholar
  2. 2.
    Auffray N., Dell’Isola F., Eremeyev V.A., Madeo A., Rosi G.: Analytical continuum mechanics à la Hamilton–Piola: least action principle for second gradient continua and capillary fluids. Math. Mech. Solids 20, 375–417 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    dell’Isola, F., Madeo, A., Seppecher, P.: Boundary conditions at fluid-permeable interfaces in porous media: a variational approach. Int. J. Solids Struct. 46 (2009)Google Scholar
  4. 4.
    dell’Isola, F., Maier, G., Perego, U., Andreaus, U., Esposito, R., Forest, S. (eds.): The Complete Works of Gabrio Piola, vol. I, Springer, Berlin (2014)Google Scholar
  5. 5.
    Dorfmann, A., Ogden, R.W.: Nonlinear electroelasticity. Acta Mech. (2005)Google Scholar
  6. 6.
    Edmiston, J., Steigmann, D.J.: Analysis of nonlinear electroelastic membranes. In: Ogden, R.W., Steigmann, D.J. (eds.) Mechanics and Electrodynamics of Magneto- and Electro-Elastic Materials, CISM Courses and Lectures No. 527, International Centre for Mechanical Sciences, pp. 153–180. Springer, Berlin (2011)Google Scholar
  7. 7.
    Epstein M.: The Geometrical Language of Continuum Mechanics. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  8. 8.
    Epstein M., El zanowski M.: Material Inhomogeneities and Their Evolution. Springer, Berlin (2007)Google Scholar
  9. 9.
    Eringen A.C., Maugin G.A.: Electrodynamics of Continua I. Springer, Berlin (1990)CrossRefGoogle Scholar
  10. 10.
    Federico S.: Covariant formulation of the tensor algebra of non-linear elasticity. Int. J. Non-Linear Mech. 47, 273–284 (2012)ADSCrossRefGoogle Scholar
  11. 11.
    Federico, S.: Porous materials with statistically oriented reinforcing fibres. In: Dorfmann, L., Ogden, R.W. (eds.) Nonlinear Mechanics of Soft Fibrous Materials, CISM Courses and Lectures No. 559, International Centre for Mechanical Sciences, pp. 49–120. Springer, Berlin (2015)Google Scholar
  12. 12.
    Federico S.: Some remarks on metric and deformation. Math. Mech. Solids 20, 522–539 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Federico S., Grillo A.: Elasticity and permeability of porous fibre-reinforced materials under large deformations. Mech. Mater. 44, 58–71 (2012)CrossRefGoogle Scholar
  14. 14.
    Felsager, B.: Geometry, particles and fields. Springer, Berlin, Germany (1989)Google Scholar
  15. 15.
    Grillo A., Zingali G., Borrello G., Federico S., Herzog W., Giaquinta G.: A multiscale description of growth and transport in biological tissues. Theor. Appl. Mech. 34, 51–87 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Grillo, A., Federico, S.,Wittum, G.:Growth, mass transfer, and remodeling in fiber-reinforced,multi-constituent materials. J. Non-Linear Mech. 47, 388–401 (2012)Google Scholar
  17. 17.
    Madeo, A., dell’Isola, F., Ianiro, N., Sciarra, G.: A variational deduction of second gradient poroelasticity II: an application to the consolidation problem. J. Mech. Mater. Struct. 3, 607–625 (2008)Google Scholar
  18. 18.
    Marsden J.E., Hughes T.J.R.: Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliff (1983)zbMATHGoogle Scholar
  19. 19.
    Maugin G.A.: Configurational Forces: Thermomechanics, Physics, Mathematics, and Numerics. CRC Press, Boca Raton (2011)zbMATHGoogle Scholar
  20. 20.
    Maugin G.A.: Continuum Mechanics Through the Twentieth Century: A Concise Historical Perspective. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  21. 21.
    Noll, W.: Materially uniform simple bodies with inhomogeneities. Arch. Rat. Mech. Anal. 27, 1–32 (1967)Google Scholar
  22. 22.
    Rudin W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York (1976)zbMATHGoogle Scholar
  23. 23.
    Sciarra G., Dell’Isola F., Ianiro N., Madeo A.: A variational deduction of second gradient poroelasticity I: General theory, journal of mechanics of materials and structures. J. Mech. Mater. Struct. 3, 507–526 (2008)CrossRefGoogle Scholar
  24. 24.
    Segev R.: Notes on metric independent analysis of classical fields. Math. Methods Appl. Sci. 36, 497–566 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tomic A., Grillo A., Federico S.: Poroelastic materials reinforced by statistically oriented fibres—numerical implementation and application to articular cartilage. IMA J. Appl. Math. 79, 1027–1059 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Trimarco C.: A Lagrangian approach to electromagnetic bodies. Tech. Mech. 22, 175–180 (2002)Google Scholar
  27. 27.
    Trimarco, C., Maugin, G.A.: Material mechanics of electromagnetic bodies. In: Kienzler, R., Maugin, G.A. (eds.) Configurational Mechanics of Materials, CISM Courses and Lectures No. 427, International Centre for Mechanical Sciences, pp. 129–172. Springer, Berlin (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Salvatore Federico
    • 1
    Email author
  • Alfio Grillo
    • 2
  • Reuven Segev
    • 3
  1. 1.Department of Mechanical and Manufacturing EngineeringThe University of CalgaryCalgaryCanada
  2. 2.DISMA - Department of Mathematical Sciences “G.L. Lagrange”Politecnico di TorinoTorinoItaly
  3. 3.Department of Mechanical EngineeringBen Gurion UniversityBeer-ShevaIsrael

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