Visco-hyperelastic law for finite deformations: a frequency analysis Authors Mathieu Charlebois BioMedical Engineering Department, Faculty of Medicine McGill University Hamid Motallebzadeh BioMedical Engineering Department, Faculty of Medicine McGill University W. Robert J. Funnell Department of Biomedical Engineering and Otolaryngology – Head & Neck Surgery, Faculty of Medicine McGill University Original Paper

First Online: 11 September 2012 Received: 15 March 2012 Accepted: 21 August 2012 DOI :
10.1007/s10237-012-0435-2

Cite this article as: Charlebois, M., Motallebzadeh, H. & Funnell, W.R.J. Biomech Model Mechanobiol (2013) 12: 705. doi:10.1007/s10237-012-0435-2
Abstract Some biological tissues are repeatedly stimulated under cyclic loading, and this stimulation can be combined with large pressures, thus leading to large deformations. For such applications, visco-hyperelastic models have been proposed in the literature and used in finite-element studies. An extensively used quasi-linear model (QLVH), which assumes linear evolution equations, is compared with a nonlinear model (NLVH), which assumes a multiplicative split of the deformation gradient. The comparison is made here using sets of simulations covering a large frequency range. Lost and stored energies are computed, and the additional parameter of the NLVH model is set to two values found in the literature (NLVH-2 and NLVH-30 models). The predicted behaviour is very similar for all models at small strains, with each time constant (and corresponding viscous modulus) being associated with a damping peak and a stored-energy increase. When the strain amplitude is increased, the ratio of lost to stored energy increases for the QLVH model, but decreases for the NLVH models. The NLVH-30 model also displays a shift of the peak damping towards higher frequencies. Before reaching a steady state, all models display a decay of energy independent of the frequency, and the additional parameter of the NLVH model permits the modelling of complex types of evolution of the damping. In conclusion, this study compares the behaviour of two viscous hyper-elastic laws to allow an informed choice between them.

Keywords Middle ear Vocal folds Viscoelasticity Large deformations Nonlinearity Frequency response

References Aernouts J, Dirckx J (2011) Static versus dynamic gerbil tympanic membrane elasticity: derivation of the complex modulus. Biomech. Model. Mechanobiol. 11(6): 829–840

CrossRef Athanasiou K, Natoli R (2008) Introduction to continuum biomechanics. Synth Lect Biomed Eng 3(1): 1–206

CrossRef Cheng T, Dai C, Gan R (2007) Viscoelastic properties of human tympanic membrane. Ann Biomed Eng 35(2): 305–314

CrossRef Ciambella J, Destrade M, Ogden RW (2009) On the ABAQUS FEA model of finite viscoelasticity. Rubber Chem Technol 82(2): 184–193

CrossRef Ciambella J, Paolone A, Vidoli S (2010) A comparison of nonlinear integral-based viscoelastic models through compression tests on filled rubber. Mech Mater 42(10): 932–944

CrossRef Decraemer W, Maftoon N, Dirckx J, Funnell WRJ (2011) The effects of the asymmetric shape of the tympanic membrane on the asymmetry of ossicular displacements for positive and negative static pressures. ARO Midwinter Meeting

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

Govindjee S, Reese S (1997) A presentation and comparison of two large deformation viscoelasticity models. J Eng Mater Technol 119(5): 2859–2868

Holzapfel G (2000) Nonlinear solid mechanics: a continuum approach for engineering, chapter 2. Wiley, Chichester

Holzapfel G, Gasser T (2001) A viscoelastic model for fiber-reinforced composites at finite strains: continuum basis, computational aspects and applications. Comput Methods Appl Mech Eng 190(34): 4379–4403

CrossRef Ladak HM, Funnell WRJ, Decraemer WF, Dirckx JJJ (2006) A geometrically nonlinear finite-element model of the cat eardrum. J Acoust Soc Am 119(5): 2859–2868

CrossRef Lejeunes S, Boukamel A, Meo S (2010) Finite element implementation of nearly-incompressible rheological models based on multiplicative decompositions. Comput Struct 89(3-4): 411–421

CrossRef Lin R, Brocks W, Betten J (2006) On internal dissipation inequalities and finite strain inelastic constitutive laws: theoretical and numerical comparisons. Int J Plast 22(10): 1825–1857

MATH CrossRef Maas SA, Ellis BJ, Ateshian GA, Weiss JA (2012) FEBio: finite elements for biomechanics. J Biomech Eng 134(1): 011005-1–011005-10

CrossRef Martins P, Natal Jorge R, Ferreira A (2006) A comparative study of several material models for prediction of hyperelastic properties: application to silicone-rubber and soft tissues. Strain 42(3): 135–147

CrossRef Mase G, Mase G (1999) Continuum mechanics for engineers, chapter 9. CRC Press, Boca Raton

CrossRef Nguyen TD, Jones RE, Boyce BL (2007) Modeling the anisotropic finite-deformation viscoelastic behavior of soft fiber-reinforced composites. Int J Solids Struct 44(25-26): 8366–8389

MATH CrossRef Nguyen TD (2007) A comparison of a nonlinear and quasilinear viscoelastic anisotropic model for fibrous tissues. IUTAM Symp Cell Mol Tissue Mech 16(1): 19–29

Park S, Schapery R (1999) Methods of interconversion between linear viscoelastic material functions part I–A numerical method based on Prony series. Int J Solids Struct 36(11): 1653–1675

MATH CrossRef Patzák B, Bittnar Z (2001) Design of object oriented finite element code. Adv Eng Softw 32(10–11): 759–767

MATH CrossRef Peña E, Calvo B, Martínez M, Doblaré M (2007) An anisotropic visco-hyperelastic model for ligaments at finite strains: formulation and computational aspects. Int J Solids Struct 44(3-4): 760–778

MATH CrossRef Puso MA, Weiss JA (1998) Finite element implementation of anisotropic quasi-linear viscoelasticity using a discrete spectrum approximation. J Biomech Eng 120(1): 62–70

CrossRef Qi L, Funnell WRJ, Daniel SJ (2008) A nonlinear finite-element model of the newborn middle ear. J Acoust Soc Am 124(1): 337–347

CrossRef Reese S, Govindjee S (1998) A theory of finite viscoelasticity and numerical aspects. Int J Solids Struct 35(26–27): 3455–3482

MATH CrossRef Stach B (2008) Clinical audiology: an introduction. Cengage Learning, Clifton Park

Stuhmiller J (1989) Use of modeling in predicting tympanic membrane rupture. Ann Otol Rhinol Laryngol Suppl 140: 53–60

Taylor Z, Comas O, Cheng M, Passenger J, Hawkes D, Atkinson D, Ourselin S (2009) On modelling of anisotropic viscoelasticity for soft tissue simulation: numerical solution and gpu execution. Med Image Anal 13(2): 234–244

CrossRef Wineman A (2009) Nonlinear viscoelastic solids—a review. Math Mech Solids 14(3): 300–366

MathSciNet MATH CrossRef Zemlin WR (1998) Speech and hearing science: anatomy and physiology. 4th edn. Allyn and Bacon, Boston

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

Zhang K, Siegmund T, Chan RW (2006) A constitutive model of the human vocal fold cover for fundamental frequency regulation. J Acoust Soc Am 119(2): 1050–1062

CrossRef