Skip to main content
SpringerLink
Log in

A New Mathematical Foundation for Contact Interactions in Continuum Physics

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

The investigation of contact interactions, such as traction and heat flux, that are exerted by contiguous bodies across the common boundary is a fundamental issue in continuum physics. However, the traditional theory of stress established by Cauchy and extended by Noll and his successors is insufficient for handling the lack of regularity in continuum physics due to shocks, corner singularities, and fracture. This paper provides a new mathematical foundation for the treatment of contact interactions. Based on mild physically motivated postulates, which differ essentially from those used before, the existence of a corresponding interaction tensor is established. While in earlier treatments contact interactions are basically defined on surfaces, here contact interactions are rigorously considered as maps on pairs of subbodies. This allows the action exerted on a subbody to be defined not only, as usual, for sets with a sufficiently regular boundary, but also for Borel sets (which include all open and all closed sets). In addition to the classical representation of such interactions by means of integrals on smooth surfaces, a general representation using the distributional divergence of the tensor is derived. In the case where concentrations occur, this new approach allows a description of contact phenomena more precise than before.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambrosio L., Fusco N., Pallara D. (2000) Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford

    MATH  Google Scholar 

  2. Antman S.S. (2005) Nonlinear Problems of Elasticity, 2nd edn. Springer, New York

    MATH  Google Scholar 

  3. Antman S.S., Osborn J.E. (1979) The principle of virtual work and integral laws of motion. Arch. Ration. Mech. Anal. 69, 231–262

    Article  MathSciNet  MATH  Google Scholar 

  4. Banfi C., Fabrizio M. (1979) Sul concetto di sottocorpo nella meccanica dei continui. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 66, 136–142

    MATH  Google Scholar 

  5. Banfi C., Fabrizio M. (1981) Global theory for thermodynamic behaviour of a continuous medium. Ann. Univ. Ferrara 27, 181–199

    MathSciNet  MATH  Google Scholar 

  6. Boussinesq J. (1878) Équilibre d’élasticité d’un sol isotrope sans pesanteur, supportant différents poids. C. R. Math. Acad. Sci. Paris 86, 1260–1263

    MATH  Google Scholar 

  7. Brezis H. (1983) Analyse Fonctionnelle. Théorie et Applications. Masson, Paris

    MATH  Google Scholar 

  8. Cauchy, A.-L. Recherches sur l’équilibre et le mouvement intérieur des corps solides ou fluides, élastiques ou non élastiques. Bull. Soc. Philomath. (1823) 9–13 See also: Oeuvres de Cauchy, Sér. 2, vol. 2, Gauthier-Villars, Paris 1958, 300–304

  9. Cauchy, A.-L. De la pression ou tension dans un corps solide. Ex. de math. 2, 42–56 (1827) See also: Oeuvres de Cauchy, Sér. 2, vol. 7, Gauthier-Villars, Paris 1889, 60–78

  10. Chen G.-Q., Frid H. (1999) Divergence-measure fields and hyperbolic conservation laws. Arch. Ration. Mech. Anal. 147, 89–118

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen G.-Q., Frid H. (2001) On the theory of divergence-measure fields and its applications. Bol. Soc. Bras. Math. 32, 1–33

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen G.-Q., Frid H. (2003) Extended divergence-measure fields and the Euler equations for gas dynamics. Comm. Math. Phys. 236, 251–280

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Chen G.-Q., Torres M. (2005) Divergence-measure fields, sets of finite perimeter, and conservation laws. Arch. Ration. Mech. Anal. 175, 245–267

    Article  MathSciNet  MATH  Google Scholar 

  14. Ciarlet P.G. (1988) Mathematical Elasticity. Vol. 1: Three-Dimensional Elasticity. Amsterdam, North-Holland

    MATH  Google Scholar 

  15. Degiovanni M., Marzocchi A., Musesti A. (1999) Cauchy fluxes associated with tensor fields having divergence measure. Arch. Ration. Mech. Anal. 147, 197–223

    Article  MathSciNet  MATH  Google Scholar 

  16. Degiovanni M., Marzocchi A., Musesti A. (2006) Edge-force densities and second-order powers. Ann. Mat. Pura Appl. (4) 185, 81–103

    Article  MathSciNet  MATH  Google Scholar 

  17. Evans L.C., Gariepy R.F. (1992) Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton

    MATH  Google Scholar 

  18. Federer H. (1969) Geometric Measure Theory. Springer, Berlin

    MATH  Google Scholar 

  19. Flamant A. (1892) Sur la répartition des pressions dans un solide rectangulaire chargé transversalement. C. R. Math. Acad. Sci. Paris 114, 1465–1468

    MATH  Google Scholar 

  20. Fosdick R.L., Virga E.G. (1989) A variational proof of the stress theorem of Cauchy. Arch. Ration. Mech. Anal. 105, 95–103

    Article  MathSciNet  MATH  Google Scholar 

  21. Gilbarg D., Trudinger N.S. (2001) Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  22. Gurtin M.E. The linear theory of elasticity. Handbuch der Physik (Truesdell, C. Ed.), Vol. VIa/2, 1–295, 1972

  23. Gurtin, M.E. Modern continuum thermodynamics. Mechanics Today (Nemat-Nasser, S. Ed.), Vol. 1, New York, 168–213, 1972

  24. Gurtin M.E., Martins L.C. (1976) Cauchy’s theorem in classical physics. Arch. Ration. Mech. Anal. 60, 305–324

    Article  MathSciNet  MATH  Google Scholar 

  25. Gurtin M.E., Mizel V.J., Williams W.O. (1968) A note on Cauchy’s stress theorem. J. Math. Anal. Appl. 22, 398–401

    Article  MathSciNet  MATH  Google Scholar 

  26. Gurtin M.E., Williams W.O. (1967) An axiomatic foundation for continuum thermodynamics. Arch. Ration. Mech. Anal. 26, 83–117

    Article  MathSciNet  MATH  Google Scholar 

  27. Gurtin M.E., Williams W.O. (1971) On the first law of thermodynamics. Arch. Ration. Mech. Anal. 42, 77–92

    Article  MathSciNet  MATH  Google Scholar 

  28. Gurtin M.E., Williams W.O., Ziemer W.P. (1986) Geometric measure theory and the axioms of continuum thermodynamics. Arch. Ration. Mech. Anal. 92, 1–22

    Article  MathSciNet  MATH  Google Scholar 

  29. Infeld L. (1980) An Autobiography 2nd Ed. Chelsea Publishing Company, New York, p. 279

    Google Scholar 

  30. Kellogg O.D. (1929) Foundations of Potential Theory. Springer, Berlin

    Book  MATH  Google Scholar 

  31. Marzocchi A., Musesti A. (2001) Decomposition and integral representation of Cauchy interactions associated with measures. Contin. Mech. Thermodyn. 13, 149–169

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Marzocchi A., Musesti A. (2002) On the measure-theoretic foundations of the second law of thermodynamics. Math. Models Methods Appl. Sci. 12, 721–736

    Article  MathSciNet  MATH  Google Scholar 

  33. Marzocchi A., Musesti A. (2003) Balanced powers in continuum mechanics. Meccanica 38, 369–389

    Article  MathSciNet  MATH  Google Scholar 

  34. Marzocchi A., Musesti A. (2003) The Cauchy stress theorem for bodies with finite perimeter. Rend. Sem. Mat. Univ. Padova 109, 1–11

    MathSciNet  MATH  Google Scholar 

  35. Marzocchi A., Musesti A. (2004) Balance laws and weak boundary conditions in continuum mechanics. J. Elasticity 38, 239–248

    Article  MathSciNet  MATH  Google Scholar 

  36. Noll W. (1958) A mathematical theory of the mechanical behavior of continuous media. Arch. Ration. Mech. Anal. 2, 197–226

    Article  MathSciNet  MATH  Google Scholar 

  37. Noll, W. The foundations of classical mechanics in the light of recent advances in continuum mechanics. The Axiomatic Method with Special Reference to Geometry and Physics (Henkin, L., Suppes, P., Tarski, A. Eds.). North-Holland, 266–281, 1959

  38. Noll, W. The foundations of mechanics. Non Linear Continuum Theories (Truesdell, C., Grioli, G. Eds.). C.I. Conference, M.E., Cremonese 159–200, 1966

  39. Noll W. (1973) Lectures on the foundations of continuum mechanics and thermodynamics. Arch. Ration. Mech. Anal. 52, 62–92

    Article  MathSciNet  MATH  Google Scholar 

  40. Noll, W. Continuum mechanics and geometric integration theory. Categories in Continuum Physics (Lawvere, F.H., Schanuel, S.H. Eds). Lecture Notes in Mathematics, Vol. 1174, Springer, Berlin, 17–29, 1986

  41. Noll W. (1993) The geometry of contact, separation, and reformation of continuous bodies. Arch. Ration. Mech. Anal. 122, 197–212

    Article  MathSciNet  MATH  Google Scholar 

  42. Noll W., Virga E.G. (1988) Fit regions and functions of bounded variation. Arch. Ration. Mech. Anal. 102, 1–21

    Article  MathSciNet  MATH  Google Scholar 

  43. Podio-Guidugli P. (2004) Examples of concentrated contact interactions in simple bodies. J. Elasticity 75, 167–186

    Article  MathSciNet  MATH  Google Scholar 

  44. Rodnay G., Segev R. (2003) Cauchy’s flux theorem in light of geometric integration theory. J. Elasticity 71, 183–203

    Article  MathSciNet  MATH  Google Scholar 

  45. Schuricht F. (1997) A variational approach to obstacle problems for shearable nonlinearly elastic rods. Arch. Ration. Mech. Anal. 140, 103–159

    Article  MathSciNet  MATH  Google Scholar 

  46. Schuricht F. (2002) Variational approach to contact problems in nonlinear elasticity. Calc. Var. Partial Differential Equations 15, 433–449

    Article  MathSciNet  MATH  Google Scholar 

  47. Schuricht, F. Contact problems in nonlinear elasticity. Modeling, analysis, application. Nonlinear Analysis and Applications to Physical Sciences (Benci, V., Masiello, A. Eds.). Springer, Milano, 91–133, 2004

  48. Schuricht F., Mosel H.v.d. (2003) Euler-Lagrange equation for nonlinearly elastic rods with self-contact. Arch. Ration. Mech. Anal. 168, 35–82

    Article  MathSciNet  MATH  Google Scholar 

  49. Segev R. (1986) Forces and the existence of stresses in invariant continuum mechanics. J. Math. Phys. 27, 163–170

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. Segev R. (2000) The geometry of Cauchy’s fluxes. Arch. Ration. Mech. Anal. 154, 183–198

    Article  MathSciNet  MATH  Google Scholar 

  51. Segev R., de Botton G. (1991) On the consistency conditions for force systems. Internat. J. Non-Linear Mech. 26, 47–59

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. Segev R., Rodnay G. (1999) Cauchy’s theorem on manifolds. J. Elasticity 56, 129–144

    Article  MathSciNet  MATH  Google Scholar 

  53. Sikorski R. (1964) Boolean Algebras. 2nd ed. Springer, Berlin

    MATH  Google Scholar 

  54. Šilhavý M. (1985) The existence of the flux vector and the divergence theorem for general Cauchy fluxes. Arch. Ration. Mech. Anal. 90, 195–211

    Article  MathSciNet  MATH  Google Scholar 

  55. Šilhavý M. (1991) Cauchy’s stress theorem and tensor fields with divergence measure in Lp. Arch. Ration. Mech. Anal. 116, 223–255

    Article  MATH  Google Scholar 

  56. Šilhavý M. (2005) Divergence measure fields and Cauchy’s stress theorem. Rend. Sem. Mat. Univ. Padova 113, 15–45

    MathSciNet  MATH  Google Scholar 

  57. Šilhavý M.: Normal traces of divergence measure vectorfields on fractal boundaries. Preprint, Dept. Math., Univ. Pisa, Oct. 2005

  58. Sternberg E., Eubanks R.A. (1955) On the concept of concentrated loads and an extension of the uniqueness theorem in the linear theory of elasticity. J. Ration. Mech. Anal. 4, 135–168

    MathSciNet  MATH  Google Scholar 

  59. Thomson W. (1848): (Lord Kelvin). Note on the integration of the equations of equilibrium of an elastic solid. Cambridge and Dublin Math. J. 3, 87–89

    Google Scholar 

  60. Truesdell C. (1991) A First Course in Rational Continuum Mechanics, Vol. 1, 2nd ed., Academic Press, Boston

    MATH  Google Scholar 

  61. Turteltaub M.J., Sternberg E. (1968) On concentrated loads and Green’s functions in elastostatics. Arch. Ration. Mech. Anal. 29, 193–240

    Article  MathSciNet  MATH  Google Scholar 

  62. Whitney H. (1957) Geometric Integration Theory. Princeton University Press, Princeton

    Book  MATH  Google Scholar 

  63. Williams W.O. (1970) Thermodynamics of rigid continua. Arch. Ration. Mech. Anal. 36, 270–284

    Article  MathSciNet  MATH  Google Scholar 

  64. Williams, W.O. Structure of continuum physics. Categories in Continuum Physics (Lawvere, F.W., Schanuel, S.H. Eds). Lecture Notes in Mathematics, Vol. 1174, Springer, Berlin, 30–37, 1986

  65. Zeidler E. (1988) Nonlinear Functional Analysis and its Applications, Vol. IV: Applications to Mathematical Physics. Springer, New York

    Book  MATH  Google Scholar 

  66. Ziemer W.P. (1983) Cauchy flux and sets of finite perimeter. Arch. Ration. Mech. Anal. 84, 189–201

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität zu Köln, Weyertal 89-90, 50931, Köln, Germany

    Friedemann Schuricht

Authors
  1. Friedemann Schuricht
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Friedemann Schuricht.

Additional information

Communicated by S.S. Antman

This paper is dedicated to Eberhard Zeidler with gratitude on the occasion of his 65th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schuricht, F. A New Mathematical Foundation for Contact Interactions in Continuum Physics. Arch Rational Mech Anal 184, 495–551 (2007). https://doi.org/10.1007/s00205-006-0032-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-006-0032-6

Keywords

Search

Navigation