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Computational Mechanics

, Volume 52, Issue 4, pp 959–969 | Cite as

At least three invariants are necessary to model the mechanical response of incompressible, transversely isotropic materials

  • M. Destrade
  • B. Mac Donald
  • J. G. Murphy
  • G. Saccomandi
Original Paper

Abstract

The modelling of off-axis simple tension experiments on transversely isotropic nonlinearly elastic materials is considered. A testing protocol is proposed where normal force is applied to one edge of a rectangular specimen with the opposite edge allowed to move laterally but constrained so that no vertical displacement is allowed. Numerical simulations suggest that this deformation is likely to remain substantially homogeneous throughout the specimen for moderate deformations. It is therefore further proposed that such tests can be modelled adequately as a homogenous deformation consisting of a triaxial stretch accompanied by a simple shear. Thus the proposed test should be a viable alternative to the standard biaxial tests currently used as material characterisation tests for transversely isotropic materials in general and, in particular, for soft, biological tissue. A consequence of the analysis is a kinematical universal relation for off-axis testing that results when the strain-energy function is assumed to be a function of only one isotropic and one anisotropic invariant, as is typically the case. The universal relation provides a simple test of this assumption, which is usually made for mathematical convenience. Numerical simulations also suggest that this universal relation is unlikely to agree with experimental data and therefore that at least three invariants are necessary to fully capture the mechanical response of transversely isotropic materials.

Keywords

Biomechanics Constitutive laws  Finite element 

References

  1. 1.
    Rivlin RS (1997). In: Barenblatt GI, Joseph DD (eds) Collected papers of R.S. Rivlin, vol 1. Springer, New YorkGoogle Scholar
  2. 2.
    Spencer AJM (1972) Deformations of fibre-reinforced materials. Oxford University Press, OxfordzbMATHGoogle Scholar
  3. 3.
    Pagano NJ, Halpin JC (1968) Influence of end constraint in the testing of anisotropic bodies. J Compos Mater 2:18–31CrossRefGoogle Scholar
  4. 4.
    Marín JC, Cañas J, París F, Morton J (2002) Determination of G\(_{12}\) by means of the off-axis tension test. Part I: review of gripping systems and correction factors. Compos A 33:87–100CrossRefGoogle Scholar
  5. 5.
    Xiao Y, Kawai M, Hatta H (2010) An integrated method for off-axis tension and compression testing of unidirectional composites. J Compos Mater 45:657–669CrossRefGoogle Scholar
  6. 6.
    Moon H, Truesdell C (1974) Interpretation of adscititious inequalities through the effects pure shear stress produces upon an isotropic elastic solid. Arch Ration Mech Anal 55:1–17MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Rajagopal KR, Wineman AS (1987) New universal relations for nonlinear isotropic elastic materials. J Elast 17:75–83MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mihai LA, Goriely A (2012) Positive or negative Poynting effect? The role of adscititious inequalities in hyperelastic materials. Proc R Soc A (to appear)Google Scholar
  9. 9.
    Destrade M, Murphy JG, Saccomandi G (2012) Simple shear is not so simple. Int J Nonlinear Mech 47:210–214CrossRefGoogle Scholar
  10. 10.
    Rivlin RS (1948) Large elastic deformations of isotropic materials. IV. Further developments of the general theory. Philos Trans R Soc Lond A 241:379–397MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gent AN, Suh JB, Kelly SG III (2007) Mechanics of rubber shear springs. Int J Nonlinear Mech 42:241–249CrossRefGoogle Scholar
  12. 12.
    Horgan CO, Murphy JG (2010) Simple shearing of incompressible and slightly compressible isotropic nonlinearly elastic materials. J Elast 98:205–221MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Humphrey JD, Yin FCP (1987) A new constitutive formulation for characterizing the mechanical behavior of soft tissues. Biophys J 52:563–570CrossRefGoogle Scholar
  14. 14.
    Humphrey JD, Strumpf RK, Yin FCP (1990) Determination of a constitutive relation for passive myocardium: I. New Funct Form J Biomech Eng 112:333–339Google Scholar
  15. 15.
    Horgan CO, Murphy JG (2012) On the modeling of extension-torsion experimental data for transversely isotropic biological soft tissues. J Elast 108:179–191MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wenk JF, Ratcliffe MB, Guccione JM (2012) Finite element modeling of mitral leaflet tissue using a layered shell approximation. Med Biol Eng Comput 50:1071–1079CrossRefGoogle Scholar
  17. 17.
    Ning X, Zhu Q, Lanir Y, Margulies SS (2006) A transversely isotropic viscoelastic constitutive equation for brainstem undergoing finite deformation. J Biomech Eng 128:925–933CrossRefGoogle Scholar
  18. 18.
    Destrade M, Gilchrist MD, Prikazchikov DA, Saccomandi G (2008) Surface instability of sheared soft tissues. J Biomech Eng 130:0610071–0610076CrossRefGoogle Scholar
  19. 19.
    Holzapfel GA, Gasser TC, Ogden RW (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast 61:1–48MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Holzapfel GA, Ogden RW, Gasser TC (2006) Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J R Soc Interface 3:15–35CrossRefGoogle Scholar
  21. 21.
    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, Wien, pp 65–108Google Scholar
  22. 22.
    Holzapfel GA, Ogden RW (2009) On planar biaxial tests for anisotropic nonlinearly elastic solids. A continuum mechanical framework. Math Mech Solids 14:474–489CrossRefzbMATHGoogle Scholar
  23. 23.
    Ogden RW, Saccomandi G, Sgura I (2004) Fitting hyperelastic models to experimental data. Comput Mech 34:484–502CrossRefzbMATHGoogle Scholar
  24. 24.
    Nì Annaidh A, Bruyère K, Destrade M, Gilchrist MD, Maurini C, Otténio M, Saccomandi G (2012) Automated estimation of collagen fibre dispersion in the dermis and its contribution to the anisotropic behaviour of skin. Ann Biomed Eng 40:1666–1678Google Scholar
  25. 25.
    Moulton MJ, Creswell LL, Actis RL, Myers KW, Vannier MW, Szabo BA, Pasque MK (1995) An inverse approach to determining myocardial material properties. J Biomech 28:935–948 Google Scholar
  26. 26.
    Yeoh OH, Fleming PD (1997) A new attempt to reconcile the statistical and phenomenological theories of rubber elasticity. J Polym Sci B 35:1919–1931CrossRefGoogle Scholar
  27. 27.
    Yamamoto E, Hayashi K, Yamamoto N (1999) Mechanical properties of collagen fascicles from the rabbit patellar tendon. J Biomech Eng 121:124–131CrossRefGoogle Scholar
  28. 28.
    Van Kerckhoven R, Kalkman EAJ, Saxena PR, Schoemaker RG (2000) Altered cardiac collagen and associated changes in diastolic function of infarcted rat hearts. Cardiovasc Res 46:316–323CrossRefGoogle Scholar
  29. 29.
    Guo D-L, Chen B-S, Liou N-S (2007) Investigating full-field deformation of planar soft tissue under simple-shear tests. J Biomech 40:1165–1170CrossRefGoogle Scholar
  30. 30.
    Atkin RJ, Fox N (1980) An introduction to the theory of elasticity. Longman, LondonzbMATHGoogle Scholar
  31. 31.
    Dokos S, Smaill BH, Young AA, LeGrice IJ (2002) Shear properties of passive ventricular myocardium. Am J Physiol Heart Circ Physiol 283:H2650–H2659Google Scholar
  32. 32.
    Horgan CO, Murphy JG (2011) Simple shearing of soft biological tissues. Proc R Soc A 467:760–777MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • M. Destrade
    • 1
    • 2
  • B. Mac Donald
    • 3
  • J. G. Murphy
    • 3
    • 1
  • G. Saccomandi
    • 4
  1. 1.School of Mathematics, Statistics, and Applied MathematicsNational University of Ireland GalwayGalwayIreland
  2. 2.School of Mechanical and Materials EngineeringUniversity College DublinDublin 4Ireland
  3. 3.Centre for Medical Engineering ResearchDublin City UniversityDublin 9Ireland
  4. 4.Dipartimento di Ingegneria IndustrialeUniversità degli Studi di PerugiaPerugiaItaly

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