Skip to main content

A discrete-time approach to the formulation of constitutive models for viscoelastic soft tissues

Biomechanics and Modeling in Mechanobiology volume 3, pages 85–97 (2004)Cite this article

Abstract

This paper presents a novel approach to constitutive modeling of viscoelastic soft tissues. This formulation combines an anisotropic strain energy function, accounting for preferred material directions, to define the elastic stress–strain relationship, and a discrete time black-box dynamic model, borrowed from the theory of system identification, to describe the time-dependent behavior. This discrete time formulation is straightforwardly oriented to the development of a recursive time integration scheme that calculates the current stress state by using strain and stress values stored at a limited number of previous time instants. The viscoelastic model and the numerical procedure are assessed by implementing two numerical examples, the simulation of a uniaxial tensile test and the inflation of a thin tube. Both simulations are performed using parameter values based on previous experiments on preserved bovine pericardium. Parameters are then adjusted to investigate the sensitivity of the model. The hypotheses the model relies upon are discussed and the main limitations are stated.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

43,34 €

Price includes VAT (Finland)
Tax calculation will be finalised during checkout.

Fig. 1
Fig. 2a,b
Fig. 3
Fig. 4
Fig. 5a,b
Fig. 6a,b
Fig. 7a,b
Fig. 8

References

  • Barbenel J, Zioupos P, Fisher J (1986) The mechanical properties of bovine pericardium. In: Heart valve engineering. ImechE, London, pp 99–102

    Google Scholar 

  • Bergström JS, Boyce MC (1998) Constitutive modeling of the large strain time-dependent behaviour of elastomers. J Mech Phys Solids 46:931–954

    Article  Google Scholar 

  • Billings SA (1980) Identification of nonlinear systems: a survey. IEEE Proc 127D:272–285

    Google Scholar 

  • Billings SA, Fakhouri Y (1977) Identification of nonlinear systems using the Wiener model. Electron Lett 13:502–504

    Google Scholar 

  • Bingham DN, DeHoff PH (1979) A constitutive equation for the canine anterior cruciate ligament. ASME J Biomech Eng 101:15–22

    Google Scholar 

  • Choi HS, Vito RP (1990) Two-dimensional stress–strain relationship for canine pericardium. ASME J Biomech Eng 112:153–159

    CAS  Google Scholar 

  • Coleman BD, Noll W (1961) Foundations of linear viscoelasticity. Rew Mod Phys 3:239–249

    Article  Google Scholar 

  • Fung YC (1967) Elasticity of soft tissues in simple elongation. Am J Physiol 213:1532–1544

    CAS  PubMed  Google Scholar 

  • Fung YC (1993) Biomechanics: mechanical properties of living tissues. 2nd edn. Springer, Berlin Heidelberg New York

    Google Scholar 

  • Fung YC, Tong P (2001) Classical and computational solid mechanics. World Scientific, Singapore

    Google Scholar 

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

    Google Scholar 

  • Green AE, Rivlin RS, Spencer AJM (1959) The mechanics of nonlinear materials with memory: Part 2. Arch Rat Mech Anal 3:82–90

    Google Scholar 

  • Holzapfel GA, Gasser TC (2001) A viscoelastic model for fiber-reinforced composites at finite strains: continuum basis, computational aspects and applications. Comp Meths Appl Mech Eng 190:4379–4403

    Article  Google Scholar 

  • Holzapfel GA, Gasser TC, Stadler M (2002) A structural model for the viscoelastic behavior of arterial walls: continuum formulation and finite element analysis. Eur J Mech A-Solid 21:441–463

    Article  Google Scholar 

  • Huber N, Tsakmakis C (2000) Finite deformation viscoelasticity laws. Mech Mat 32:1–18

    Article  Google Scholar 

  • Johnson GA, Liversay GA, Woo SL-Y, Rajagopal KR (1996) A single integral finite strain viscoelastic model of ligaments and tendons. ASME J Biomech Eng 118:221–226

    Google Scholar 

  • Otto P, Wernstedt J (1978) Development and application of suboptimal test signal sequences for identifying nonlinear systems. In: Proceedings of 7th IFAC Congress, Helsinki, pp 1927–1933

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

    CAS  Google Scholar 

  • Quaglini V (2000) Characterization of mechanical properties and identification of constitutive parameters for soft tissues. PhD Thesis, Politecnico di Milano, M Res

  • Quaglini V, Previdi F, Contro R, Bittanti S (2002) A discrete-time nonlinear Wiener model for the relaxation of soft biological tissues. Med Eng Phys 24:9–19

    Article  PubMed  Google Scholar 

  • Reid GJ (1985) Linear systems fundamentals. Continuous and discrete, classic and modern, 2nd edn. McGraw-Hill, Singapore

    Google Scholar 

  • Rivlin RS (1966) The fundamental equations of nonlinear continuum mechanics. In: Pai S-I (ed) Dynamics of fluids and plasmas (Burgers Anniversary volume). Academic, New York

    Google Scholar 

  • Sacks M (2000) Biaxial mechanical evaluation of planar biological materials. J Elasticity 61:199–246

    Article  Google Scholar 

  • Shoemaker PA, Schneider D, Lee MC, Fung YC (1986) A constitutive model for two-dimensional soft tissues and its applications to experimental data. J Biomech 19:695–702

    Article  CAS  PubMed  Google Scholar 

  • Simo JC (1987) On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Comput Meths Appl Mech Eng 60:153–173

    Article  Google Scholar 

  • Sjöberg J, Zhang Q, Ljung L, Benveniste A, Delyon B, Glorennec P-Y, Hjalmarsson H, Juditsky A (1995) Nonlinear black-box modeling in system identification: a unified overview. Automatica 31:1691–1724

    Article  Google Scholar 

  • Sorenson HW (1980) Parameter estimation. Principles and problems. Marcel Dekker Inc., New York

    Google Scholar 

  • Spencer AJM (1992) Continuum theory of the mechanics of fibre-reinforced composites. Springer, Berlin Heidelberg New York

    Google Scholar 

  • Taylor RL, Pister KS, Goudreau GL (1970) Thermomechanical analysis of viscoelastic solids. Int J Num Meth Eng 2:45–59

    CAS  PubMed  Google Scholar 

  • Truesdell C, Noll W (1965) The non-linear field theories of mechanics. In: Flugge S (ed) Handbuck der Physik vol III/3. Springer, Berlin Heidelberg New York

  • Vena P, Contro R (1999) A viscoelastic model for anisotropic biological tissues in finite strain. In: Wunderlich W (ed) Proceedings of the European conference on computational mechanics 1999, TU Munchen

  • Wang JL, Parnianpour M, Shirazi-Adl A, Engin AE (1997) Failure criterion of collagen fiber: viscoelastic behavior simulated by using load control data. Theor Appl Fract Mech 27:1–12

    Article  CAS  Google Scholar 

  • Weiss J, Maker BN, Govindjee S (1996) Finite element implementation of incompressible, transversely isotropic iperelasticity. Comp Meths Appl Mech Eng 135:107–128

    Article  Google Scholar 

  • Young JT, Vaishnav RN, Patel DJ (1977) Nonlinear anisotropic viscoelastic properties of canine arterial segments. J Biomech 10:549–559

    Article  CAS  PubMed  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the Italian National Council of Research (CNR) for the financial support within the “PFMSTA II” project.

Author information

Affiliations

  1. Department of Structural Engineering, Laboratory of Biological Structure Mechanics (LaBS), Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133, Milano, Italy

    V. Quaglini, P. Vena & R. Contro

Authors
  1. V. Quaglini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Vena
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Contro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. Vena.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Quaglini, V., Vena, P. & Contro, R. A discrete-time approach to the formulation of constitutive models for viscoelastic soft tissues. Biomech Model Mechanobiol 3, 85–97 (2004). https://doi.org/10.1007/s10237-004-0050-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10237-004-0050-y

Keywords

  • Viscoelastic Model
  • Strain Energy Function
  • Kirchhoff Stress
  • Time Integration Scheme
  • Bovine Pericardium