Skip to main content
SpringerLink
Log in

Rheology and Nonlinear Elasticity

  • Chapter
  • First Online:
Continuum Mechanics Through the Twentieth Century

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 196))

  • 2560 Accesses

Abstract

This first specialized chapter deals with the awaited generalization to mechanical behaviours that deviate from linear elasticity and standard Newtonian viscous fluids, that is, elasticity in large deformations and the rheology of complex fluids. These extensions were kindled by the mechanics of rubber elasticity and artificial fabrics and of fluids with high viscosity and visco-plastic response. It happens that the same scientists were involved in these two lines as a result of a required focus on the bases of continuum mechanics, in particular the theory of finite deformations in a rational geometric background, and the need to account for complex flow features in some fluids. Ronald Rivlin, with his incommensurable contributions, is the great hero in this adventure. Other scientists whose work was seminal are initially E. Bingham, M. Reiner, L.G.R. Treloar, P. J. Flory, M.A. Mooney, and F.D. Murnaghan, and more recently J.G. Oldroyd, A.E. Green, J.L. Ericksen, C.A. Truesdell, B.D. Coleman, and W. Noll. The survey includes the models of neo-Hookean materials, Mooney-Rivlin materials, Rivlin-Ericksen fluids, and unsuccessful attempts such as those of Reiner-Rivlin fluids and hypoelasticity. Appropriately introduced tools have been those of Rivlin-Ericksen tensors, Oldroyd and Jaumann time derivatives, and invariant representations of scalar and tensorvalued functions. Through Rivlin and his co-workers the whole carries a strong print of British applied mathematics although Italian and Russian contributions to nonlinear elasticity cannot be overlooked. The mechanics of soft living tissues has now become the best field of application of these developments.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Bannes HA, Hutton JF, Walter K (1989) An introduction to rheology. Elsevier, UK

    Google Scholar 

  • Bell JF (1973) The experimental foundations of solid mechanics. In: Flügge S, Truesdell CA (eds) Handbuch der Physik, vol VIa/1. Springer, Berlin, pp 1–813

    Google Scholar 

  • Bingham EC (1922) Fluidity and plasticity. McGraw Hill, New York

    Google Scholar 

  • Bouasse H, Carriére Z (1903) Courbes de traction du caoutchouc vulcanisé. Ann Fac Sci Toulouse 5:257–283

    Article  MATH  Google Scholar 

  • Carroll MM, Hayes MA (2006) In memory of Ronald S. Rivlin, Mathematics and Mechanics of Solids 11:103–112

    Google Scholar 

  • Coleman BD, Gurtin ME (1967) Thermodynamics with internal variables. J Chem Phys 47:597–613

    Article  Google Scholar 

  • Coleman BD, Markovitz H, Noll W (1966) Viscometric flows of non-Newtonian fluids: theory and experiment. Springer, NewYork

    Book  Google Scholar 

  • Coleman BD, Noll W (1961) Foundations of linear viscoelasticity. Rev Mod Phys 33:239–249

    Article  MathSciNet  MATH  Google Scholar 

  • Ericksen JL, Rivlin RS (1954) Large elastic deformations of homogeneous anisotropic materials. J Rat Mech Anal 3:281–301

    MathSciNet  MATH  Google Scholar 

  • Eringen AC (1962) Nonlinear theory of continuous media. McGraw Hill, New York

    Google Scholar 

  • Eringen AC (1967) Mechanics of continua. Wiley, New York

    MATH  Google Scholar 

  • Eringen AC, Maugin GA (1990) Electrodynamics of continua. Springer, New York

    Book  Google Scholar 

  • Fung YC (1993) Biomechanics: mechanical properties of living tissues. Springer, New York

    Google Scholar 

  • Giesekus H (1984) Phänomenologische Rheologie: Ein Einführung. Springer, Berlin

    Google Scholar 

  • Goldenblatt II (1962) Some problems of the mechanics of deformable media. Noordhoff, Groningen (English translation from the Russian)

    Google Scholar 

  • Green AE, Adkins JE (1970) Large elastic deformations, 2nd edn. Oxford University Press, Oxford

    MATH  Google Scholar 

  • Green AE, Rivlin RS (1957) The mechanics of non-linear materials with memory I. Arch Rat Mech Anal 1:1–21

    Article  MathSciNet  MATH  Google Scholar 

  • Huilgol RR, Phan-Thien N (1997) Fluid mechanics of viscoelasticity. Elsevier, Amsterdam

    Google Scholar 

  • Jaumann G (1911) Geschlossenes system physikalischer und chemischer differenzialgesetze, Sitzber. Akad Wiss Wien (Ha) 120:385–530

    MATH  Google Scholar 

  • Larson RG (1988) Constitutive equations for polymeric melts and solutions. Butterworth, Boston

    Google Scholar 

  • Lodge AS (1964) Elastic liquids: an introductory vector treatment of finite-strain polymer rheology. Academic, New York

    Google Scholar 

  • Luri’e AI (1980) Nonlinear theory of elasticity. Nauka, Moscow (in Russian)

    Google Scholar 

  • Maugin GA (1992) The thermomechanics of plasticity and fracture. Cambridge University Press, UK

    Book  MATH  Google Scholar 

  • Maugin GA (1999) The thermomechanics of nonlinear irreversible behaviors: an introduction. World Scientific, Singapore

    MATH  Google Scholar 

  • Maugin GA, Drouot R (1983) Internal variables and the thermodynamics of macromolecule solutions. Int J Eng Sci 21:705–724

    Article  MathSciNet  MATH  Google Scholar 

  • Mooney MA (1940) A theory of large elastic deformation. J Appl Phys 11:582–592

    Article  MATH  Google Scholar 

  • Mullins L (1947) Effect of stretching on the properties of rubber. J Rubber Res 16:275–282

    Google Scholar 

  • Mullins L (1969) Softening of rubber by deformation. Rubber Chem Technol 42:339–362

    Article  Google Scholar 

  • Murnaghan FD (1937) Finite deformation of an elastic solid. Amer J Math 59:235–260

    Article  MathSciNet  MATH  Google Scholar 

  • Murnaghan FD (1951) Finite deformation of an elastic solid. Wiley, New York (Unfairly criticized by Truesdell in his review of this book reprinted in Truesdell, 1984, pp 148–150)

    Google Scholar 

  • Noll W (1955) On the continuity of the solid and fluid states. J Rat Mech Anal 4:13–81

    Google Scholar 

  • Noll W (1958) A mathematical theory of the mechanical behavior of continuous media. Arch Rat Mech Anal 2:197–226

    Article  MATH  Google Scholar 

  • Novozhilov VV (1953) Foundations of the nonlinear theory of elasticity. Graylock Press, Rochester (English translation from the 1948 Russian edition)

    MATH  Google Scholar 

  • Ogden RW (1972) Large deformation isotropic elasticity, Part I. Proc R Soc A326:565–585; Part II, ibid A328:567–583

    Google Scholar 

  • Ogden RW (1982) Elastic deformations of rubberlike solids. In: Hopkins HG, Sewell MJ (eds) Mechanics of solids (The Rodney Hill 60th anniversary volume). Pergamon Press, Oxford, pp 499–537

    Google Scholar 

  • Ogden RW (2003) Nonlinear elasticity, anisotropy, material stability and residual stresses in soft tissues. In: Holzapfel GA, Ogden RW (eds) Biomechanics of soft tissues in cardiovascular systems (CISM lecture no. 441). Springer, Wien, pp 65–108

    Google Scholar 

  • Ogden RW, Roxburg DG (1999) A pseudo-elastic model forth Mullins effect in filled rubber. Proc R Soc Lond A455:2861–2878

    Google Scholar 

  • Oldroyd (1950) On the formulation of rheological equations of state. Proc R Soc A200:523–541

    Google Scholar 

  • Reiner M (1945) A mathematical theory of dilatancy. Am J Math 67:350–362

    Article  MathSciNet  MATH  Google Scholar 

  • Rivlin RS (1948) Large elastic deformation of isotropic materials, Part I. Philos Trans R Soc Lond A240:459–490; Part II, ibid A241:379–397

    Google Scholar 

  • Rivlin RS (1956) Solution of some problems in the exact theory of visco-elasticity. Ration Mech Anal 5:179–188

    MathSciNet  MATH  Google Scholar 

  • Rivlin RS (1984) Forty years of non-linear continuum mechanics. In: Proceedings of the Conference on Ninth International Congress of Rheology. pp 2783–2811

    Google Scholar 

  • Rivlin RS (1996) Autobiographical postcript. In: Barenblat GI, Joseph DD (eds) Collected papers of R.S. Rivlin. Springer, New York, p 22

    Google Scholar 

  • Signorini A (1943) Trasfomazioni termoelastiche finite, Memoria 1. Ann di Mat Pura ed Applicata 22(4):33–143

    Article  MathSciNet  MATH  Google Scholar 

  • Signorini A (1949) Trasfomazioni termoelastiche finite, Memoria 2. Ann di Mat Pura ed Applicata 30:1–72

    Article  MathSciNet  MATH  Google Scholar 

  • Signorini A (1955) Trasfomazioni termoelastiche finite, Memoria 3. Ann di Mat Pura ed Applicata 39:147–201

    Article  MathSciNet  MATH  Google Scholar 

  • Schowalter WR (1976) Non-Newtonian fluid mechanics. Pergamon Press, London

    Google Scholar 

  • Spencer AJM (1971) Theory of invariants. In: Eringen AC (ed) Continuum physics, Vol 1, Chap. 3. Academic Pres, New York

    Google Scholar 

  • Spencer AJM (1972) Deformation of fibre-reinforced materials. Clarendon Press, Oxford

    Google Scholar 

  • Spencer AJM (1976) Continuum mechanics. Longman, Harlow

    Google Scholar 

  • Tanner RI, Walter K (1999) Rheology: an historical perspective. Elsevier, UK

    Google Scholar 

  • Treloar LGR (1946) The elasticity of a network of long-chain molecules. Trans Faraday Soc 42:83–94

    Article  MathSciNet  Google Scholar 

  • Treloar LRG (2005) The physics of rubber elasticity, 3rd edn. Clarendon Press, Oxford

    Google Scholar 

  • Truesdell CA (1952) The mechanical foundations of elasticity and fluid dynamics. J Rat Mech Anal 1(1):125–300

    MathSciNet  MATH  Google Scholar 

  • Truesdell CA (1974) The meaning of viscometry in fluid dynamics. Annu Rev Fluid Mechanics 6:111–146. Academic Press, New York

    Google Scholar 

  • Truesdell CA (1984) An idiot’s fugitive essays on science. Springer, New York

    Book  MATH  Google Scholar 

  • Vyalov SS (1986) Rheological fundamentals of soil mechanics. Mir Publishers, Moscow (translation from the Russian)

    MATH  Google Scholar 

  • Weissenberg K (1949) Abnormal substances and abnormal phenomena of flow. In: Proceedings of the International Congress of Rheology (1948). North-Holland, Amsterdam, pp I-29–I-46

    Google Scholar 

  • Zener C (1948) Elasticity and anelasticity of metals. The University of Chicago Press, Chicago

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Jean Le Rond d’Alembert, Université Pierre et Marie Curie, Place Jussieu 4, 75252, Paris Cedex 05, France

    Gérard A. Maugin

Authors
  1. Gérard A. Maugin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gérard A. Maugin .

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Maugin, G.A. (2013). Rheology and Nonlinear Elasticity. In: Continuum Mechanics Through the Twentieth Century. Solid Mechanics and Its Applications, vol 196. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6353-1_3

Download citation

  • DOI: https://doi.org/10.1007/978-94-007-6353-1_3

  • Published:

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-6352-4

  • Online ISBN: 978-94-007-6353-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation