Skip to main content
SpringerLink
Log in

Probabilistic mechanics of fibrous structures

  • Original Papers
  • Published:
Zeitschrift für angewandte Mathematik und Physik ZAMP Aims and scope Submit manuscript

Abstract

In this paper the response behaviour of fibrous structures to a given uniaxial load is treated on the basis of probabilistic mechanics. For the illustration of the theoretical analysis, a cellulose network with specified properties is employed. The significant relations, which link the microscopic variables to the intermediate ones (mesoscopic) and thus to the macroscopic quantities are material functionals or operators. An operator for the given cellulose structure is derived in an explicit form. As a result of the analysis the internal stresses due to the given load are numerically evaluated.

Zusammenfassung

Das Verhalten einer Faserstruktur die einer axialen Belastung unterworfen ist, wird auf Grund der stochastischen Mechanik untersucht. Zur Veranschaulichung der theoretischen Behandlung wird eine zellulose Struktur mit spezifischen Eigenschaften benutzt. Die wichtigsten Relationen welche die mikroskopischen, dazwischen liegenden (mesoskopischen) und letzthin die makroskopischen Variablen verbindet, sind die Material-Operatoren. Es wird eine explizite Form eines solchen Operators für die zellulose Struktur abgeleitet. Die innere Spannungsverteilung, die wesentlich von der durchschnittlichen makroskopischen abweicht, wird sodann als ein Ergebnis der Untersuchung angezeigt.

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. H. Corte,Paper and board, in Composite materials, Ed. L. Holliday, Elsevier, New York 1966.

    Google Scholar 

  2. A. H. Nissan,The rheological behaviour of hydrogen bonded solids, Parts 1 and 2, Trans. Faraday Soc.53, 2048–2053 (1957).

    Google Scholar 

  3. R. W. Perkins,A theory of mechanical response of fiber networks, Wood and Fiber5, 26–33 (1973).

    Google Scholar 

  4. D. R. Axelrad,Micromechanics of solids, PWN-Elsevier Publ. Co. (1978).

  5. K. Rezai,The determination of surface deformations by holographic-electro-optical processing, Ph. D. Thesis, McGill Univ., Montreal 1981.

    Google Scholar 

  6. D. R. Axelrad,Foundations of the probabilistic mechanics of discrete media, Pergamon Press, Oxford 1984.

    Google Scholar 

  7. D. R. Axelrad,Mechanical models of relaxation phenomena, Adv. in Molec. Relax. Processes2, 41 (1970).

    Google Scholar 

  8. Yu. N. Rabotnov,Creep problems in structural members, North-Holland Publ. Co., Amsterdam 1969.

    Google Scholar 

  9. D. R. Axelrad, D. Atack and J. W. Provan,Microrheology of fibrous systems, Rheol. Acta12, 170 (1973).

    Google Scholar 

  10. D. R. Axelrad, S. Basu,Mechanical relaxation theory of fibrous structures, Adv. in Molec. Relax. Processes11, 165 (1977).

    Google Scholar 

  11. D. R. Axelrad,Theory of bond failure in hydrogen-bonded solids, Adv. in Molec. Relax. Processes15, 51 (1979).

    Google Scholar 

  12. Frey-Wyssling, Biochem. Bio-Phys. Acta18, 166 (1955).

    Google Scholar 

  13. J. Jones, J. Polymer Sci.32, 371 (1958).

    Google Scholar 

  14. C. Y. Liang,I. Hydrogen bonds in native cellulose, J. Polymer Sci., Vol.37 (1959).

  15. W. C. Hamilton and J. A. Ibers,Hydrogen bonding in solids, Benjamin Publ. Co., New York 1968.

    Google Scholar 

  16. W. Bollmann,Crystal defects and crystalline interfaces, Springer-Verlag, New York 1970.

    Google Scholar 

  17. J. Yvon,Correlations and entropy in classical statistical mechanics, Pergamon Press, Oxford 1969.

    Google Scholar 

  18. D. R. Axelrad and K. Rezai,Determination of surface displacements by holographic electro optical processing, J. Appl. Opt.,21, 2001 (1982).

    Google Scholar 

  19. L. A. Santaló,Integral geometry and geometric probability, Addison-Wesley Publ. Co. Reading, MA. (1976).

    Google Scholar 

  20. C. Mack,An exact formula for Q k (n),the probable number of k-aggregates in a random distribution of points, Phyl. Mag., 7th ser.39, 778 (1948).

    Google Scholar 

  21. O. Kallmes, H. Corte and G. Bernier,The structure of paper, II. The statistical geometry of a multiplanar fiber network, Tappi44, 519 (1961).

    Google Scholar 

  22. V. Ya. Andreichenko,Structure and mechanical properties of the paper sheet, (1). Mathematical description of the sheet structure, VNII Tsellyul-Bumazh. Prom. No. 58, p. 80 (1971).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Micromechanics Research Laboratory, McGill University, Montreal, P. Q., Canada

    D. R. Axelrad & K. Rezai

  2. Physics Division, Pulp and Paper Research Institute of Canada, Canada

    D. Atack

Authors
  1. D. R. Axelrad
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Rezai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. Atack
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Axelrad, D.R., Rezai, K. & Atack, D. Probabilistic mechanics of fibrous structures. Z. angew. Math. Phys. 35, 497–513 (1984). https://doi.org/10.1007/BF00945071

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00945071

Keywords

Search

Navigation