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Biomechanics and Modeling in Mechanobiology

, Volume 12, Issue 2, pp 201–213 | Cite as

An anisotropic elastic-viscoplastic damage model for bone tissue

  • J. J. SchwiedrzikEmail author
  • P. K. Zysset
Original Paper

Abstract

A new anisotropic elastic-viscoplastic damage constitutive model for bone is proposed using an eccentric elliptical yield criterion and nonlinear isotropic hardening. A micromechanics-based multiscale homogenization scheme proposed by Reisinger et al. is used to obtain the effective elastic properties of lamellar bone. The dissipative process in bone is modeled as viscoplastic deformation coupled to damage. The model is based on an orthotropic ecuntric elliptical criterion in stress space. In order to simplify material identification, an eccentric elliptical isotropic yield surface was defined in strain space, which is transformed to a stress-based criterion by means of the damaged compliance tensor. Viscoplasticity is implemented by means of the continuous Perzyna formulation. Damage is modeled by a scalar function of the accumulated plastic strain \({D(\kappa)}\) , reducing all element s of the stiffness matrix. A polynomial flow rule is proposed in order to capture the rate-dependent post-yield behavior of lamellar bone. A numerical algorithm to perform the back projection on the rate-dependent yield surface has been developed and implemented in the commercial finite element solver Abaqus/Standard as a user subroutine UMAT. A consistent tangent operator has been derived and implemented in order to ensure quadratic convergence. Correct implementation of the algorithm, convergence, and accuracy of the tangent operator was tested by means of strain- and stress-based single element tests. A finite element simulation of nano- indentation in lamellar bone was finally performed in order to show the abilities of the newly developed constitutive model.

Keywords

Bone Constitutive model Viscoplasticity Damage 

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References

  1. Bushby AJ, Ferguson VL, Boyde A (2004) Nanoindentation of bone: comparison of specimens tested in liquid and embedded in polymethylmethacrylate. J Mater Res 19: 249–259CrossRefGoogle Scholar
  2. Carnelli D, Gastaldi D, Sassi V, Contro R, Ortiz C, Vena P (2010) A finite element model for direction-dependent mechanical response to nanoindentation of cortical bone allowing for anisotropic post-yield behavior of the tissue. J Biomech Eng 132(8): 081008CrossRefGoogle Scholar
  3. Carosio A, Willam K, Etse G (2000) On the consistency of viscoplastic formulations. Int J Solids Struct 37(48–50): 7349–7369zbMATHCrossRefGoogle Scholar
  4. Chaboche JL (2008) A review of some plasticity and viscoplasticity constitutive theories. Int J Plast 24: 1642–1693zbMATHCrossRefGoogle Scholar
  5. Charlebois M, Jirasek M, Zysset P (2010) A nonlocal constitutive model for trabecular bone softening in compression. Biomech Model Mechanobiol 9: 597–611CrossRefGoogle Scholar
  6. Cowin SC (1979) On the strength anisotropy of bone and wood. J Appl Mech 46(4): 832–838zbMATHCrossRefGoogle Scholar
  7. Cowin S, Mehrabadi M (1995) Anisotropic symmetries of linear elasticity. Appl Mech Rev 48: 247–285CrossRefGoogle Scholar
  8. Etse G, Carosio A (1999) Constitutive equations and numerical approaches in rate dependent material formulations, MECOMGoogle Scholar
  9. Fondrk MT, Bahniuk EH, Davy DT (1999) A damage model for nonlinear tensile behavior of cortical bone. J Biomech Eng 121: 533–541CrossRefGoogle Scholar
  10. Fratzl P, Weinkamer R (2007) Nature’s hierarchical materials. Prog Mater Sci 52(8): 1263–1334CrossRefGoogle Scholar
  11. Garcia D (2006) Elastic plastic damage laws for cortical bone. Ph.D. thesis, Ecole Polytechnique Federale de LausanneGoogle Scholar
  12. Garcia D, Zysset P, Charlebois M, Curnier A (2009) A three-dimensional elastic plastic damage constitutive law for bone tissue. Biomech Model Mechanobiol 8(2): 149–165CrossRefGoogle Scholar
  13. Green AE, Naghdi PM (1965) A general theory of an elastic-plastic continuum. Arch Ration Mech Anal 18(4): 251–281MathSciNetzbMATHCrossRefGoogle Scholar
  14. Gross T (2010) The effects of heterogeneous mineralization on the elastic and yield properties of human cancellous bone. Diploma Thesis, Vienna University of TechnologyGoogle Scholar
  15. Gupta H, Zioupos P (2008) Fracture of bone tissue: the ’hows’ and the ’whys’. Med Eng Phys 30(10): 1209–1226CrossRefGoogle Scholar
  16. Gupta HS, Wagermaier W, Zickler GA, Raz-Ben Aroush D, Funari SS, Roschger P, Wagner HD, Fratzl P (2005) Nanoscale deformation mechanisms in bone. Nano Lett 5(10): 2108–2111CrossRefGoogle Scholar
  17. Gupta H, Wagermaier W, Zickler G, Hartmann J, Funari S, Roschger P, Wagner H, Fratzl P (2006) Fibrillar level fracture in bone beyond the yield point. Int J Fract 139: 425–436zbMATHCrossRefGoogle Scholar
  18. Gupta HS, Fratzl P, Kerschnitzki M, Benecke G, Wagermaier W, Kirchner HO (2007) Evidence for an elementary process in bone plasticity with an activation enthalpy of 1 eV. J R Soc Interface 4(13): 277–282CrossRefGoogle Scholar
  19. Hansma P, Fantner G, Kindt J, Thurner P, Schitter G, Turner P, Udwin S, Finch M (2005) Sacrificial bonds in the interfibrillar matrix of bone. J Musculoskelet Neuronal Interact 5(4): 313–315Google Scholar
  20. Hellmich C, Ulm FJ (2002) Are mineralized tissues open crystal foams reinforced by crosslinked collagen?—some energy arguments. J Biomech 35(9): 1199–1212CrossRefGoogle Scholar
  21. Hengsberger S, Kulik A, Zysset P (2002) Nanoindentation discriminates the elastic properties of individual human bone lamellae under dry and physiological conditions. Bone 30(1): 178–184CrossRefGoogle Scholar
  22. Keyak JH, Rossi SA (2000) Prediction of femoral fracture load using finite element models: an examination of stress- and strain-based failure theories. J Biomech 33(2): 209–214CrossRefGoogle Scholar
  23. Kuhn HW, Tucker AW (1951) Nonlinear programming. In: Proceedings of 2nd Berkeley symposium. University of California Press, pp 481–492Google Scholar
  24. Lucchini R, Carnelli D, Ponzoni M, Bertarelli E, Gastaldi D, Vena P (2011) Role of damage mechanics in nanoindentation of lamellar bone at multiple sizes: experiments and numerical modeling. J Mech Behav Biomed Mater 4(8): 1852–1863CrossRefGoogle Scholar
  25. Maghous S, Dormieux L, Barthèlèmy J (2009) Micromechanical approach to the strength properties of frictional geomaterials. Eur J Mech A Solids 28(1): 179–188MathSciNetzbMATHCrossRefGoogle Scholar
  26. Mazza G (2008) Anisotropic elastic properties of vertebral bone measured by microindentation. Diploma thesis, Politecnico di MilanoGoogle Scholar
  27. Natali A, Carniel E, Pavan P (2008) Constitutive modelling of inelastic behaviour of cortical bone. Med Eng Phys 30(7): 905–912CrossRefGoogle Scholar
  28. Nemat-Nassar S, Mori M (1993) Micromechanics: overall properties of heterogeneous materials. Elsevier Science Publishers, The NetherlandsGoogle Scholar
  29. Perzyna P. (1966) Fundamental problems in viscoplasticity. Elsevier, The Netherlands, pp 243–377Google Scholar
  30. Ponthot JP (1995) Radial return extensions for visco-plasticity and lubricated friction. In: SMIRT-13 international conference on structural mechanics and reactor technologyGoogle Scholar
  31. Poon B, Rittel D, Ravichandran G (2008) An analysis of nanoindentation in linearly elastic solids. Int J Solids Struct 45(24): 6018–6033zbMATHCrossRefGoogle Scholar
  32. Rakatomanana RL, Curnier A, Leyvraz PF (1991) An objective elastic plastic model and algorithm applicable to bone mechanics. Eur J Mech A Solids 10(3): 327–342Google Scholar
  33. Reisinger A, Pahr D, Zysset P (2010) Sensitivity analysis and parametric study of elastic properties of an unidirectional mineralized bone fibril-array using mean field methods. Biomech Model Mechanobiol 9: 499–510CrossRefGoogle Scholar
  34. Reisinger A, Pahr D, Zysset P (2011) Elastic anisotropy of bone lamellae as a function of fibril orientation pattern. Biomech Model Mechanobiol 10: 67–77CrossRefGoogle Scholar
  35. Shih CF, Lee D (1978) Further developments in anisotropic plasticity. J Eng Mater Technol 100(3): 294–302CrossRefGoogle Scholar
  36. Simo JC, Ju JW (1987) Strain- and stress-based continuum damage models—I. Formulation. Int J Solids Struct 23(7): 821–840zbMATHCrossRefGoogle Scholar
  37. Spiesz EM (2011) Experimental and computational micromechanics of mineralized tendon and bone. Ph.D. thesis, Vienna University of TechnologyGoogle Scholar
  38. Spiesz EM, Reisinger AG, Roschger P, Zysset PK (2011) Experimental validation of a multiscale model of mineralized collagen fibers at two levels of hierarchy. Osteoporos Int 22: 561–666CrossRefGoogle Scholar
  39. Sugawara Y, Kamioka H, Honjo T, Tezuka K, Takano-Yamamoto T (2005) Three-dimensional reconstruction of chick calvarial osteocytes and their cell processes using confocal microscopy. Bone 36(5): 877–883CrossRefGoogle Scholar
  40. Tai K, Ulm FJ, Ortiz C (2006) Nanogranular origins of the strength of bone. Nano Lett 6(11): 2520–2525CrossRefGoogle Scholar
  41. Voyiadjis G, Peters R (2010) Size effects in nanoindentation: an experimental and analytical study. Acta Mech 211: 131–153zbMATHCrossRefGoogle Scholar
  42. Weiner S, Arad T, Sabanay I, Traub W (1997) Rotated plywood structure of primary lamellar bone in the rat: orientations of the collagen fibril arrays. Bone 20(6): 509–514CrossRefGoogle Scholar
  43. Weiner S, Wagner HD (1998) The material bone: structure-mechanical function relations. Annu Rev Mater Sci 28(1): 271–298CrossRefGoogle Scholar
  44. Weiner S, Traub W, Wagner H (1999) Lamellar bone: structure-function relations. J Struct Biol 126(3): 241–255CrossRefGoogle Scholar
  45. Wolfram U, Wilke HJ, Zysset PK (2010) Rehydration of vertebral trabecular bone: influences on its anisotropy, its stiffness and the indentation work with a view to age, gender and vertebral level. Bone 46(2): 348–354CrossRefGoogle Scholar
  46. Yeni Y, Dong X, Fyhrie D, Les C (2004) The dependence between the strength and stiffness of cancellous and cortical bone tissue for tension and compression: extension of a unifying principle. Bio-Med Mater Eng 14(3): 303–310Google Scholar
  47. Zhang J, Niebur GL, Ovaert TC (2008) Mechanical property determination of bone through nano- and micro-indentation testing and finite element simulation. J Biomech 41(2): 267–275CrossRefGoogle Scholar
  48. Zhang J, Michalenko MM, Kuhl E, Ovaert TC (2010) Characterization of indentation response and stiffness reduction of bone using a continuum damage model. J Mech Behav Biomed Mater 3(2): 189–202CrossRefGoogle Scholar
  49. Zinkiewicz OC, Valliapan S, King IP (1969) Elastoplastic solutions of engineering problems initial stress, finite element approach. Int J Num Methods Eng 1: 75–100CrossRefGoogle Scholar
  50. Zysset PK (1994) A constitutive law for trabecular bone. Ph.D. thesis, Ecole Polytechnique Federale de LausanneGoogle Scholar
  51. Zysset PK, Curnier A (1995) An alternative model for anisotropic elasticity based on fabric tensors. Mech Mater 21(4): 243–250CrossRefGoogle Scholar
  52. Zysset PK, Guo EX, Hoffler EC, Moore KE, Goldstein SA (1999) Elastic modulus and hardness of cortical and trabecular bone lamellae measured by nanoindentation in the human femur. J Biomech 32(10): 1005–1012CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institute of Lightweight Design and Structural BiomechanicsVienna University of TechnologyViennaAustria
  2. 2.Institute of Surgical Technology and BiomechanicsUniversity of BernBernSwitzerland

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