Journal of Engineering Mathematics

, Volume 95, Issue 1, pp 87–98 | Cite as

Dominant negative Poynting effect in simple shearing of soft tissues

Article

Abstract

We identify three distinct shearing modes for simple shear deformations of transversely isotropic soft tissue which allow for both positive and negative Poynting effects (that is, they require compressive and tensile lateral normal stresses, respectively, in order to maintain simple shear). The positive Poynting effect is that usually found for isotropic rubber. Here, specialisation of the general results to three strain-energy functions which are quadratic in the anisotropic invariants, linear in the isotropic strain invariants and consistent with the linear theory suggests that there are two Poynting effects which can accompany the shearing of soft tissue: a dominant negative effect in one mode of shear and a relatively small positive effect in the other two modes. We propose that the relative inextensibility of the fibres relative to the matrix is the primary mechanism behind this large negative Poynting effect.

Keywords

Elasticity Modelling Poynting effect Simple shear Soft tissue Transverse isotropy 

References

  1. 1.
    British Standard BS ISO 8013:2006 Rubber, vulcanized—determination of creep in compression or shearGoogle Scholar
  2. 2.
    Poynting JH (1909) On pressure perpendicular to the shear planes in finite pure shears, and on the lengthening of loaded wires when twisted. Proc R Soc Lond Ser A 82:546–559MATHCrossRefADSGoogle Scholar
  3. 3.
    Rivlin RS (1948) Large elastic deformation of isotropic materials iv: further developments of the general theory. Philos Trans R Soc Lond Ser A 241:379–397MATHMathSciNetCrossRefADSGoogle Scholar
  4. 4.
    Mihai LA, Goriely A (2011) Positive or negative poynting effect? the role of adscititious inequalities in hyperelastic materials. Proc R Soc Lond A 467:3633–3646MATHMathSciNetCrossRefADSGoogle Scholar
  5. 5.
    Destrade M, Murphy JG, Saccomandi G (2012) Simple shear is not so simple. Int J Non-Linear Mech 47:210–214CrossRefGoogle Scholar
  6. 6.
    Horgan CO, Smayda M (2012) The importance of the second strain invariant in the constitutive modeling of elastomers and soft biomaterials. Mech Mater 51:43–52CrossRefGoogle Scholar
  7. 7.
    Janmey PA, McCormick ME, Rammensee S, Leight JL, Georges PC, MacKintosh FC (2007) Negative normal stress in semiflexible biopolymer gels. Nat Mater 6:48–51CrossRefADSGoogle Scholar
  8. 8.
    Destrade M, Gilchrist MD, Motherway J, Murphy JG (2012) Slight compressibility and sensitivity to changes in Poisson’s ratio. Int J Numer Methods Eng 90:403–411MATHCrossRefGoogle Scholar
  9. 9.
    Horgan CO, Murphy JG (2011) On the normal stresses in simple shearing of fiber-reinforced nonlinearly elastic materials. J Elast 104:343–355MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Wu MS, Kirchner HOK (2010) Nonlinear elasticity modeling of biogels. J Mech Phys Solids 58:300–310MATHMathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Spencer AJM (1984) Constitutive theory for strongly anisotropic solids. In Spencer AJM (ed) Continuum theory of the mechanics of fibre-reinforced composites. CISM Courses and Lectures Series No. 282. Springer-Verlag, ViennaGoogle Scholar
  12. 12.
    Murphy JG (2013) Transversely isotropic biological, soft tissue must be modelled using both anisotropic invariants. Eur J Mech A/Solids 42:90–96MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dokos S, Smaill BH, Young AA, LeGrice IJ (2002) Shear properties of passive ventricular myocardium. Am J Physiol Heart Circ Physiol 283:H2650–H2659CrossRefGoogle Scholar
  14. 14.
    Saccomandi G, Beatty MF (2002) Universal relations for fiber-reinforced elastic materials. Math Mech Solids 7:95–110MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Ogden RW (2003) Nonlinear elasticity, anisotropy, material stability and residual stresses in soft tissue. In Biomechanics of soft tissue in cardiovascular systems. CISM Courses and Lectures Series No. 441. Springer, Vienna, pp 65–108Google Scholar
  16. 16.
    Merodio J, Ogden RW (2005) Mechanical response of fiber-reinforced incompressible non-linearly elastic solids. Int J Non-Linear Mech 40:213–227MATHCrossRefGoogle Scholar
  17. 17.
    Vergori L, Destrade M, McGarry P, Ogden RW (2013) On anisotropic elasticity and questions concerning its finite element implementation. Comput Mech 52:1185–1197MATHCrossRefGoogle Scholar
  18. 18.
    Gennisson J-L, Catheline S, Chaffaõ S, Fink M (2003) Transient elastography in anisotropic medium: application to the measurement of slow and fast shear wave speeds in muscles. J Acoust Soc Am 114:536–541CrossRefADSGoogle Scholar
  19. 19.
    Papazoglou S, Rump J, Braun J, Sack I (2006) Shear wave group velocity inversion in MR elastography of human skeletal muscle. Magn Reson Med 56:489–497CrossRefGoogle Scholar
  20. 20.
    Sinkus R, Tanter M, Catheline S, Lorenzen J, Kuhl C, Sondermann E, Fink M (2005) Imaging anisotropic and viscous properties of breast tissue by magnetic resonance-elastography. Magn Reson Med 53:372–387CrossRefGoogle Scholar
  21. 21.
    Morrow DA, Haut Donahue TL, Odegard GM, Kaufman KR (2010) Transversely isotropic tensile material properties of skeletal muscle tissue. J Mech Behav Biomed Mater 3:124–129CrossRefGoogle Scholar
  22. 22.
    Truesdell C, Noll W (1965) The non-linear field theories of mechanics. In: Flugge S (ed) Encyclopedia of Physics, vol III/3, 3rd edn. Springer-Verlag, BerlinGoogle Scholar
  23. 23.
    Beatty MF (1989) Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissue. Appl Mech Rev 40:1699–1734CrossRefADSGoogle Scholar
  24. 24.
    Feng Y, Okamoto RJ, Namani R, Genin GM, Bayly PV (2013) Measurements of mechanical anisotropy in brain tissue and implications for transversely isotropic material models of white matter. J Mech Behav Biomed Mater 23:117–132CrossRefGoogle Scholar
  25. 25.
    Holzapfel GA, Gasser TC, Ogden RW (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast 61:1–48MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Le Tallec P (1994) Numerical methods for nonlinear three-dimensional elasticity. In: Ciarlet PG, Lions JL (eds) Handbook of Numerical Analysis, vol III. Elsevier, AmsterdamGoogle Scholar
  27. 27.
    ADINA R&D Inc (2005) ADINA theory and modeling guide. ADINA R&D Inc, WatertownGoogle Scholar
  28. 28.
    ARES Rheometer manual (2006) Rheometrics series user manual. Revision J, TA Instrument-Waters LLC, New CastleGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Applied MathematicsNational University of Ireland GalwayGalwayIreland
  2. 2.School of Engineering and Applied ScienceUniversity of VirginiaCharlottesvilleUSA
  3. 3.Centre for Medical Engineering ResearchDublin City UniversityDublin 9Ireland

Personalised recommendations