Skip to main content
SpringerLink
Log in

A rate dependent directional damage model for fibred materials: application to soft biological tissues

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

On the basis of continuum mechanics, a rate dependent directional damage model for fibred materials with application to soft biological tissues is presented. The structural model is formulated using the concept of internal variables that provides a very general description of materials involving irreversible effects. Continuum damage mechanics is used to describe the softening behavior of soft tissues under large deformation. To account for the rate dependency and to regularize the localization problems associated with strain-softening, a viscous damage mechanism is presented in this paper in decoupled form for matrix and fibers. The numerical implementation in the context of the finite element method is discussed in detail. In order to show the performance of the constitutive model and its algorithmic counterpart some simple examples are included. Results show that the model is able to capture the typical stress–strain behavior observed in fibrous soft tissues and appear to confirm the soundness of the proposed formulation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alastrué V et al (2006) Biomechanical modelling of refractive corneal surgery. ASME J Biomech Eng 128: 150–160

    Article  Google Scholar 

  2. Alastrué V et al (2007a) Assessing the use of the “opening angle method” to enforce residual stresses in patient-specific arteries. Ann Biomed Eng 35: 1821–1837

    Article  Google Scholar 

  3. Alastrué V et al (2007b) Structural damage models for fibrous biological soft tissues. Int J Solids Struc 44: 5894–5911

    Article  MATH  Google Scholar 

  4. Arnoux PJ et al (2002) A visco-hyperelastic with damage for the knee ligaments under dynamic constraints. Comput Methods Biomech Biomed Eng 5: 167–174

    Article  Google Scholar 

  5. Balzani D et al (2006) Simulation of discontinuous damage incorporating residual stress in circumferentially overstretched atherosclerotic arteries. Acta Biomater 2: 609–618

    Article  Google Scholar 

  6. Bazant ZP, Jirasek M (2002) Nonlocal integral formulations of plasticity and damage: survey of progress. J Eng Mech 128: 1119–1149

    Article  Google Scholar 

  7. Bonifasi-Lista C et al (2005) Viscoelastoc properties of the human medial collateral ligament under longitudinal, transverse and shear loading. J Orthopaed Res 23: 67–76

    Article  Google Scholar 

  8. Calvo B et al (2009) On modelling damage process in vaginal tissue. J Biomech 42: 642–651

    Article  MathSciNet  Google Scholar 

  9. Calvo B et al (2007) An uncoupled directional damage model for fibered biological soft tissues. Formulation and computational aspects. Int J Numer Method Eng 69: 2036–2057

    Article  MathSciNet  MATH  Google Scholar 

  10. Calvo B et al (2008) Computational modeling of ligaments at non-physiological situations. Int J Comput Vision Biomech IJV&B 1: 107–115

    Google Scholar 

  11. Chandran PL, Barocas VH (2006) Affine versus non-affine fibril kinematics in collagen networks: theoretical studies of network behavior. ASME J Biomech Eng 128: 259–270

    Article  Google Scholar 

  12. Crisco JJ et al (2002) Strain-rate sensityvity of the rabbit MCL diminishes at traumatic loading rates. J Biomech 35: 1379–1385

    Article  Google Scholar 

  13. Danto MI, Woo SLY (1993) The mechanical properties of skeletally mature rabbit anterior cruciate ligament and patellar tendon over range of strain rates. J Orthopaed Res 11: 58–67

    Article  Google Scholar 

  14. del Palomar AP, Doblare M (2006) On the numerical simulation of the mechanical behaviour of articular cartilage. Int J Numer Meth Engng 67: 244–1271

    Google Scholar 

  15. Flory PJ (1961) Thermodynamic relations for high elastic materials. Trans Faraday Soc 57: 829–838

    Article  MathSciNet  Google Scholar 

  16. Gasser TC, Holzapfel GA (2007) Finite element modeling of balloon angioplasty by considering overstretch of remnant non-diseased tissues in lesions. Comput Mech 40: 47–60

    Article  MATH  Google Scholar 

  17. Govindjee S et al (1995) Anisotropic modelling and numerical simulation of brittle damage in concrete. Int J Numer Method Eng 38: 3611–3633

    Article  MATH  Google Scholar 

  18. Hatami-Marbini H, Picu RC (2009) Heterogeneous long-range correlated deformation of semiflexible random fiber networks. Phys Rev E 80: 046703

    Article  Google Scholar 

  19. Hibbit, Karlsson and Sorensen, Inc. (2008) Abaqus user’s guide, v. 6.8. HKS Inc., Pawtucket

  20. Hokanson J, Yazdami S (1997) A constitutive model of the artery with damage. Mech Res Commun 24: 151–159

    Article  MATH  Google Scholar 

  21. Holzapfel GA (2000) Nonlinear solid mechanics. Wiley, New York

    MATH  Google Scholar 

  22. Holzapfel GA et al (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast 61: 1–48

    Article  MathSciNet  MATH  Google Scholar 

  23. Holzapfel GA, Ogden RW (2009) Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Philos Trans R Soc A 367: 3445–3475

    Article  MathSciNet  MATH  Google Scholar 

  24. Holzapfel GA et al (2002) A layer specific three-dimensional model for the simulation of balloon angioplasty using magnetic resonance imaging and mechanical testing. Ann Biomed Eng 30: 753–767

    Article  Google Scholar 

  25. Humphrey JD (2002) Continuum biomechanics of soft biological tissues. Proc R Soc Lond A 175: 1–44

    Google Scholar 

  26. Johnson GA et al (1996) A single integral finite strain viscoelastic model of ligaments and tendons. ASME J Biomech Eng 118: 221–226

    Article  Google Scholar 

  27. Ju JW (1989) On energy-based coupled elastoplastic damage theories: Constitutive modeling and computational aspects. Int J Solids Struct 25: 803–833

    Article  MATH  Google Scholar 

  28. Marsden JE, Hughes TJR (1994) Mathematical foundations of elasticity. Dover, New York

    Google Scholar 

  29. Natali AN et al (2003) A transverselly isotropic elasto-damage constitutive model for the periodontal ligament. Comput Methods Biomech Biomed Eng 6: 329–336

    Article  Google Scholar 

  30. Natali AN et al (2005) Anisotropic elasto-damage constitutive model for the biomechanical analysis of tendons. Med Eng Phys 27: 209–214

    Article  Google Scholar 

  31. Nedjar B (2001) Elastoplastic-damage modelling including the gradient of damage: formulation and computational aspects. Int J Solids Struct 38: 5421–5451

    Article  MATH  Google Scholar 

  32. Ogden RW (1996) Non-linear elastic deformations. Dover, New York

    Google Scholar 

  33. Oliver J (1996) Modeling strong discontinuities in solid mechanics via strain softening constitutive equations. Part I: Fundamentals. Int J Numer Method Eng 39: 3575–3601

    Article  MATH  Google Scholar 

  34. Oliver J (1996) Modeling strong discontinuities in solid mechanics via strain softening constitutive equations. Part II: Numerical simulation. Int J Numer Meth Eng 39: 3602–3624

    Google Scholar 

  35. Peña E et al (2006) A three-dimensional finite element analysis of the combined behavior of ligaments and menisci in the healthy human knee joint. J Biomech 39(9): 1686–1701

    Article  Google Scholar 

  36. Peña E et al (2008) On finite strain damage of viscoelastic fibred materials. Application to soft biological tissues. Int J Numer Method Eng 74: 1198–1218

    Article  MATH  Google Scholar 

  37. Peña E et al (2007) Computational modelling of diarthrodial joints. Physiological, pathological and pos-surgery simulations. Arch Comput Methods Eng 14(1): 47–91

    Article  MathSciNet  MATH  Google Scholar 

  38. Peña E, Doblare M (2009) An anisotropic pseudo-elastic approach for modelling Mullins effect in fibrous biological materials. Mech Res Commun 36: 784–790

    Article  MATH  Google Scholar 

  39. Peña E et al (2008) Application of the natural element method to finite deformation inelastic problems in isotropic and fiber-reinforced biological soft tissues. Comput Methods Appl Mech Eng 197: 1983–1996

    Article  MATH  Google Scholar 

  40. Peña E et al (2008) On modelling nonlinear viscoelastic effects in ligaments. J Biomech 41: 2659–2666

    Article  Google Scholar 

  41. Peña E et al (2009) On the Mullins effect and hysteresis of fibered biological materials: a comparison between continuous and discontinuous damage models. Int J Solids Struct 46: 1727–1735

    Article  MATH  Google Scholar 

  42. Pence TJ et al (2008) On the computation of stress in affine versus nonaffine fibril kinematics within planar collagen network models. ASME J Biomech Eng 130: 041009

    Article  Google Scholar 

  43. Pinsky PM, Datye V (1991) A microstructurally-based finite element model of the incised human cornea. J Biomech 10: 907–922

    Article  Google Scholar 

  44. Pioletti DP et al (1999) Strain rate effect on the mechanical behavior of the anterior cruciate ligament-bone complex. Med Eng Phys 25: 95–100

    Article  Google Scholar 

  45. Puso MA, Weiss JA (1998) Finite element implementation of anisotropic quasilinear viscoelasticity. ASME J Biomech Eng 120: 162–170

    Article  Google Scholar 

  46. Rodríguez JF et al (2008) Finite element implementation of a stochastic three dimensional finite-strain damage model for fibrous soft tissue. Comput Methods Appl Mech Eng 197: 946–958

    Article  MATH  Google Scholar 

  47. Schechtman H, Bader DL (2002) Fatigue damage of human tendons. J Biomech 35: 347–353

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  49. Simo JC, Ju JW (1987) Strain- and stress-based continuum damage models. I. Formulation. Int J Solids Struct 23: 821–840

    Article  MATH  Google Scholar 

  50. Simo JC, Ju JW (1987) Strain- and stress-based continuum damage models. II. Computational aspects. Int J Solids Struct 23: 841–870

    Article  MATH  Google Scholar 

  51. Simo JC et al (1985) Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Comput Methods Appl Mech Eng 51: 177–208

    Article  MathSciNet  MATH  Google Scholar 

  52. Spencer AJM (1971) Theory of invariants. In: Continuum physics. Academic Press, New York, pp 239–253

  53. Valanis KC (1985) On the uniqueness of solution of the initial value problem in softening materials. J Appl Mech 52: 649–653

    Article  MathSciNet  MATH  Google Scholar 

  54. Weiss JA et al (1996) Finite element implementation of incompressible, transversely isotropic hyperelasticity. Comput Methods Appl Mech Eng 135: 107–128

    Article  MATH  Google Scholar 

  55. Woo SLY et al (1990) The effects of strain rate on the properties of the medial collateral ligament in skeletally inmatura and mature rabbits: a biomechanical and histological study. J Orthopaed Res 8: 712–721

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Group of Structural Mechanics and Materials Modeling, Aragón Institute of Engineering Research, University of Zaragoza, Zaragoza, Spain

    E. Peña

  2. CIBER-BBN, Centro de Investigación Biomédica en Red en Bioingeniería, Biomateriales y Nanomedicina, Zaragoza, Spain

    E. Peña

Authors
  1. E. Peña
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Peña.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Peña, E. A rate dependent directional damage model for fibred materials: application to soft biological tissues. Comput Mech 48, 407–420 (2011). https://doi.org/10.1007/s00466-011-0594-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-011-0594-5

Keywords

Search

Navigation