Skip to main content
SpringerLink
Log in

A mathematical model of the spine based on mixture theory of directed curves

Eine allgemeine Theorie für eine Mischung von richtungsorientierten Kurven und ein biomechanisches Modell der Wirbelsäule

  • Contributed Papers
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Summary

A general nonlinear theory of a mixture of interacting directed curves is developed. Constitutive equations are presented for a class of viscoelastic materials with fading memory. A model of the ligamentous spine as a composite or mixture of directed curves is proposed. Suitable constraints are imposed which yield a simple nonlinear theory governing the response of the spine regarded as a composite rod which can undergo bending, twisting, axial extension and cross-sectional shear deformation.

Zusammenfassung

Die Arbeit befaßt sich mit der Entwicklung einer allgemeinen Theorie für eine Mischung von richtungsorientierten Kurven. Stoffgleichungen für einen einfachen viskoelastischen Werkstoff mit verschwindendem Gedächtnis werden hergeleitet. Um die Anwendung der allgemeinen Theorie zu beweisen, wird eine Mischung spezieller orientierter Kurven als ein mathematisches Modell der Wirbelsäule vorgeschlagen. Sodann wird auf Grund einer einfachen nichtlinearen Theorie gezeigt, daß es genügt, das mechanische Verhalten der Wirbelsäule durch ein aus zwei Komponenten zusammengesetzten Stab, mit Längs- und Schubdehnung, Biegungs- und Verdrehungsdeformation, darzustellen.

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. Antman, S.: The Theory of Rods. Handbuch der Physik, Vol. VIa/2. Berlin-Heidelberg-New York: Springer. 1972.

    Google Scholar 

  2. Cohen, H.: A Nonlinear Theory of Elastic Directed Curves. Int. J. of Engng. Sci.4, 511–524 (1966).

    Google Scholar 

  3. Whitman, A. B., andC. N. De Silva: A Dynamical Theory of Elastic Directed Curves. ZAMP20, 200–212 (1969).

    Google Scholar 

  4. Whitman, A. B., andC. N. DeSilva: Dynamics and Stability of Elastic Cosserat Curves. Int. J. Solids Structures6, 411–422 (1970).

    Google Scholar 

  5. Whitman, A. B., andC. N. DeSilva: Stability in a Linear Theory of Elastic Rods. Acta Mechanica15, 295–308 (1972).

    Google Scholar 

  6. DeSilva, C. N., andA. B. Whitman: A Thermodynamical Theory of Directed Curves. J. Math. Phys.12, 1603–1609 (1971).

    Google Scholar 

  7. Craine, R. E., A. E. Green, andP. M. Naghdi: A Mixture of Viscous Elastic Materials with Different Constituent Temperatures. Quart. J. Mech. Appl. Math.23, 171–184 (1969).

    Google Scholar 

  8. DeSilva, C. N. andP. Kaloni: The Balance Equations of Reacting Oriented Continua. J. de Mechanique8, 543–548 (1969).

    Google Scholar 

  9. Green, A. E., andP. M. Naghdi: A Theory of Mixtures. Arch. Rat. Mech. Anal.24, 243–263 (1967).

    Google Scholar 

  10. Green, A. E., andP. M. Naghdi: On Basic Equations for Mixtures. Quart. J. Mech. Appl. Math.22, 427–438 (1969).

    Google Scholar 

  11. Green, A. E., andP. M. Naghdi: The Flow of Fluid Through an Elastic Solid. Acta Mechanica9, 329–340 (1970).

    Google Scholar 

  12. Green, A. E., andP. M. Naghdi: Entropy Inequalities for Mixtures. Quart. J. Mech. Appl. Math.24, 473–485 (1971).

    Google Scholar 

  13. Green, A. E., andN. Laws: Global Properties of Mixtures. Arch. Ratl. Mech. Anal.43, 45–61 (1971).

    Google Scholar 

  14. DeSilva, C. N., andM. Sioshansi: Mixure Theory of Directed Surfaces and a Model of the Cornea. Acta Mechanica19, 89–108 (1974).

    Google Scholar 

  15. Coleman, B. D., andW. Noll: An Approximation Theorem for Functionals with Application to Continuum Mechanics. Arch. Ratl. Mech. Anal.6, 355–370 (1960).

    Google Scholar 

  16. Coleman, B. D.: Thermodynamics of Materials with Memory. Arch. Ratl. Mech. Anal.17, 1–46 (1964).

    Google Scholar 

  17. Rizzi, M. A.: Die menschliche Haltung—klinische und biomechanische Betrachtungen. Zeit. für Präventivmed.17, 341–348 (1973a).

    Google Scholar 

  18. Rizzi, M. A.: Untersuchungsmethoden zur Beurteilung der Haltung. Zeit. für Präventivmed. (to appear. 1973b).

  19. Steindler, A: Kinesiology of the Human Body. Springfield, Ill.: Charles C Thomas. 1955.

    Google Scholar 

  20. Rasch, P. J., andR. K. Burke: Kinesiology and Applied Anatomy. Philadelphia: Lea and Febiger. 1971.

    Google Scholar 

  21. Chacon, R. V., andR. S. Rivlin: Representational Theorems in the Mechanics of Materials with Memory. ZAMP15, 444–459 (1964).

    Google Scholar 

  22. Roberts, V. L., E. L. Stech, andC. T. Terry: Review of Mathematical Models which Describe Human Response to Acceleration. A.S.M.E. Paper No. 66-WA/BHF-13 (1966).

  23. Orne, D., andY. K. Liu: A Mathematical Model of Spinal Response to Impact. J. Biomechanics4, 49–71 (1971).

    Google Scholar 

  24. von Gierke, H. E.: Biodynamic Models and their Applications. J. Acoust. Soc. Am.50, 1397–1413 (1971).

    Google Scholar 

  25. Hirsch, C.: The Reaction of Intervertebral Discs to Compression Forces. J. Bone Jt. Surg.37-A, 1188–1196 (1965).

    Google Scholar 

  26. Moffat, C. A., andR. H. Howard: The Investigation of Vertebral Injury Sustained During Aircrew Ejection. Technology Inc., San Antonio, Texas. Q. Prog. Rep. No. 2, NASA Contract NAS 2-5062 (1968).

    Google Scholar 

  27. Whitman, A. B., andC. N. DeSilva: An Exact Solution in a Nonlinear Theory of Rods. J. Elasticity4 (1974).

Download references

Author information

Authors and Affiliations

  1. Rheumatologie und Physikalische Medizin, Am Schanzengraben, Zürich, Switzerland

    M. A. Rizzi

  2. Department of Mechanical Engineering Sciences, Wayne State University, Detroit, Mich., USA

    A. B. Whitman

  3. Institut für Biomedizinische Technik, Eidgenössische Technische Hochschule, Zürich, Switzerland

    C. N. DeSilva

Authors
  1. M. A. Rizzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. B. Whitman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. N. DeSilva
    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

Rizzi, M.A., Whitman, A.B. & DeSilva, C.N. A mathematical model of the spine based on mixture theory of directed curves. Acta Mechanica 21, 241–260 (1975). https://doi.org/10.1007/BF01181057

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01181057

Keywords

Search

Navigation