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Dominant negative Poynting effect in simple shearing of soft tissues

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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.

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References

  1. British Standard BS ISO 8013:2006 Rubber, vulcanized—determination of creep in compression or shear

  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 82:546–559

    Article  MATH  ADS  Google Scholar 

  3. Rivlin RS (1948) Large elastic deformation of isotropic materials iv: further developments of the general theory. Philos Trans R Soc Lond Ser A 241:379–397

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Mihai LA, Goriely A (2011) Positive or negative poynting effect? the role of adscititious inequalities in hyperelastic materials. Proc R Soc Lond A 467:3633–3646

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Destrade M, Murphy JG, Saccomandi G (2012) Simple shear is not so simple. Int J Non-Linear Mech 47:210–214

    Article  Google Scholar 

  6. Horgan CO, Smayda M (2012) The importance of the second strain invariant in the constitutive modeling of elastomers and soft biomaterials. Mech Mater 51:43–52

    Article  Google Scholar 

  7. Janmey PA, McCormick ME, Rammensee S, Leight JL, Georges PC, MacKintosh FC (2007) Negative normal stress in semiflexible biopolymer gels. Nat Mater 6:48–51

    Article  ADS  Google Scholar 

  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 90:403–411

    Article  MATH  Google Scholar 

  9. Horgan CO, Murphy JG (2011) On the normal stresses in simple shearing of fiber-reinforced nonlinearly elastic materials. J Elast 104:343–355

    Article  MATH  MathSciNet  Google Scholar 

  10. Wu MS, Kirchner HOK (2010) Nonlinear elasticity modeling of biogels. J Mech Phys Solids 58:300–310

    Article  MATH  MathSciNet  ADS  Google Scholar 

  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, Vienna

  12. Murphy JG (2013) Transversely isotropic biological, soft tissue must be modelled using both anisotropic invariants. Eur J Mech A/Solids 42:90–96

    Article  MathSciNet  Google Scholar 

  13. Dokos S, Smaill BH, Young AA, LeGrice IJ (2002) Shear properties of passive ventricular myocardium. Am J Physiol Heart Circ Physiol 283:H2650–H2659

    Article  Google Scholar 

  14. Saccomandi G, Beatty MF (2002) Universal relations for fiber-reinforced elastic materials. Math Mech Solids 7:95–110

    Article  MATH  MathSciNet  Google Scholar 

  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 65–108

  16. Merodio J, Ogden RW (2005) Mechanical response of fiber-reinforced incompressible non-linearly elastic solids. Int J Non-Linear Mech 40:213–227

    Article  MATH  Google Scholar 

  17. Vergori L, Destrade M, McGarry P, Ogden RW (2013) On anisotropic elasticity and questions concerning its finite element implementation. Comput Mech 52:1185–1197

    Article  MATH  Google Scholar 

  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 114:536–541

    Article  ADS  Google Scholar 

  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 56:489–497

    Article  Google Scholar 

  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 53:372–387

    Article  Google Scholar 

  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 3:124–129

    Article  Google Scholar 

  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, Berlin

  23. Beatty MF (1989) Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissue. Appl Mech Rev 40:1699–1734

    Article  ADS  Google Scholar 

  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 23:117–132

    Article  Google Scholar 

  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 61:1–48

    Article  MATH  MathSciNet  Google Scholar 

  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, Amsterdam

  27. ADINA R&D Inc (2005) ADINA theory and modeling guide. ADINA R&D Inc, Watertown

  28. ARES Rheometer manual (2006) Rheometrics series user manual. Revision J, TA Instrument-Waters LLC, New Castle

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Acknowledgments

We thank Professor Paul Janmey for providing us with the data of the experiments published in Janmey et al. [7]. JGM would like to thank Professor Alain Goriely for stimulating discussions on this and other topics. The constructive criticism of the anonymous referees is also gratefully acknowledged.

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Authors and Affiliations

  1. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, Galway, Ireland

    M. Destrade & J. G. Murphy

  2. School of Engineering and Applied Science, University of Virginia, Charlottesville, VA , 22904, USA

    C. O. Horgan

  3. Centre for Medical Engineering Research, Dublin City University, Glasnevin, Dublin 9, Ireland

    J. G. Murphy

Authors
  1. M. Destrade
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  2. C. O. Horgan
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  3. J. G. Murphy
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Correspondence to J. G. Murphy.

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Destrade, M., Horgan, C.O. & Murphy, J.G. Dominant negative Poynting effect in simple shearing of soft tissues. J Eng Math 95, 87–98 (2015). https://doi.org/10.1007/s10665-014-9706-5

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  • DOI: https://doi.org/10.1007/s10665-014-9706-5

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