Advertisement

Finite Element Implementation of Structural Constitutive Models

  • Michael S. SacksEmail author

Abstract

It is well established that the highly nonlinear and anisotropic mechanical behaviors of soft tissues are an emergent behavior of the underlying tissue microstructure. Numerical solutions form the cornerstone in the application of constitutive models in contemporary biomechanics. Herein, a structural constitutive model into a finite element framework specialized for membrane tissues. Multiple deformation modes were simulated, including strip biaxial, planar biaxial with two attachment methods, and membrane inflation. Detailed comparisons with experimental data were undertaken to insure faithful simulations of both the macro-level stress–strain insights into adaptations of the fiber architecture under stress, such as fiber reorientation and fiber recruitment. Results indicated a high degree of fidelity and demonstrated interesting microstructural adaptions to stress and the important role of the underlying tissue matrix.

Keywords

Uniaxial Tension Strain Energy Function Fiber Recruitment Fiber Orientation Distribution Stretch Direction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

Funding for this work was supported by FDA contract HHSF223201111595P and NIH/NHLBI Grant NHLBI R01 HL108330 and R01 HL119297-01.

Conflict of Interests

The authors have no conflict of interests to report in this work.

References

  1. ABAQUS. Abaqus user subroutines reference manual. 2011.Google Scholar
  2. Beskos DE, Jenkiins JT. A mechanical model for mammalian tendon. J Appl Mech. 1975;42:755.CrossRefGoogle Scholar
  3. Billiar KL, Sacks MS. A method to quantify the fiber kinematics of planar tissues under biaxial stretch. J Biomech. 1997;30:753–6.PubMedCrossRefGoogle Scholar
  4. Billiar KL, Sacks MS. Biaxial mechanical properties of the native and glutaraldehyde-treated aortic valve cusp: Part II–A structural constitutive model. J Biomech Eng. 2000a;122:327–35.PubMedCrossRefGoogle Scholar
  5. Billiar KL, Sacks MS. Biaxial mechanical properties of the natural and glutaraldehyde treated aortic valve cusp–Part I: experimental results. J Biomech Eng. 2000b;122:23–30.PubMedCrossRefGoogle Scholar
  6. Bischoff JE. Continuous versus discrete (invariant) representations of fibrous structure for modeling non-linear anisotropic soft tissue behavior. Int J Non Linear Mech. 2006;41:167–79.CrossRefGoogle Scholar
  7. Buchanan RM, Sacks MS. Interlayer micromechanics of the aortic heart valve leaflet. Biomech Model Mechanobiol. 2013;3(4):1–4.Google Scholar
  8. Chen H, Liu Y, Slipchenko MN, Zhao X, Cheng JX, Kassab GS. The layered structure of coronary adventitia under mechanical load. Biophys J. 2011;101:2555–62.PubMedPubMedCentralCrossRefGoogle Scholar
  9. Cortes DH, Lake SP, Kadlowec JA, Soslowsky LJ, Elliott DM. Characterizing the mechanical contribution of fiber angular distribution in connective tissue: comparison of two modeling approaches. Biomech Model Mechanobiol. 2010;9:651–8.PubMedPubMedCentralCrossRefGoogle Scholar
  10. Courtney T, Sacks MS, Stankus J, Guan J, Wagner WR. Design and analysis of tissue engineering scaffolds that mimic soft tissue mechanical anisotropy. Biomaterials. 2006;27:3631–8.PubMedGoogle Scholar
  11. Criscione JC, Sacks MS, Hunter WC. Experimentally tractable, pseudo-elastic constitutive law for biomembranes: I. Theory. J Biomech Eng. 2003;125:94–9.PubMedCrossRefGoogle Scholar
  12. Driessen NJ, Mol A, Bouten CV, Baaijens FP. Modeling the mechanics of tissue-engineered human heart valve leaflets. J Biomech. 2007;40:325–34.PubMedCrossRefGoogle Scholar
  13. Fata B, Zhang W, Amini R, Sacks MS. Insights into regional adaptations in the growing pulmonary artery using a meso-scale structural model: effects of ascending aorta impingement. J Biomech Eng. 2014;136:021009.PubMedCrossRefGoogle Scholar
  14. Fung YC. Biomechanics: mechanical properties of living tissues. 2nd ed. New York: Springer; 1993.CrossRefGoogle Scholar
  15. Hansen L, Wan W, Gleason RL. Microstructurally motivated constitutive modeling of mouse arteries cultured under altered axial stretch. J Biomech Eng. 2009;131:101015.PubMedPubMedCentralCrossRefGoogle Scholar
  16. Hariton I, de Botton G, Gasser TC, Holzapfel GA. Stress-driven collagen fiber remodeling in arterial walls. Biomech Model Mechanobiol. 2007;6:163–75.PubMedCrossRefGoogle Scholar
  17. Hollander Y, Durban D, Lu X, Kassab GS, Lanir Y. Experimentally validated microstructural 3D constitutive model of coronary arterial media. J Biomech Eng. 2011;133:031007.PubMedPubMedCentralCrossRefGoogle Scholar
  18. Holzapfel GA, Eberlein R, Wriggers P, Weizascker HW. Large strain analysis of soft biological membranes: formulatin and finite element analysis. Comput Methods Appl Mech Eng. 1996;132:45–61.CrossRefGoogle Scholar
  19. Holzapfel GA, Ogden RW. Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Philos Transact A Math Phys Eng Sci. 2009;367:3445–75.CrossRefGoogle Scholar
  20. Horowitz A, Lanir Y, Yin FC, Perl M, Sheinman I, Strumpf RK. Structural three-dimensional constitutive law for the passive myocardium. J Biomech Eng. 1988;110:200–7.PubMedCrossRefGoogle Scholar
  21. Hughes TJR. The finite element method: linear static and dynamic finite element analysis. New York: Dover; 2000.Google Scholar
  22. Jor JW, Nash MP, Nielsen PM, Hunter PJ. Estimating material parameters of a structurally based constitutive relation for skin mechanics. Biomech Model Mechanobiol. 2011;10:767–78.PubMedCrossRefGoogle Scholar
  23. Joyce EM, Moore JJ, Sacks MS. Biomechanics of the fetal membrane prior to mechanical failure: review and implications. Eur J Obstet Gynecol Reprod Biol. 2009;144 Suppl 1:S121–7.PubMedPubMedCentralCrossRefGoogle Scholar
  24. Kao PH, Lammers S, Tian L, Hunter K, Stenmark KR, Shandas R, Qi HJ. A microstructurally-driven model for pulmonary artery tissue. J Biomech Eng. 2011;133:051002.PubMedPubMedCentralCrossRefGoogle Scholar
  25. Kenedi RM, Gibson T, Daly CH. Biomechanics and related bio-engineering topics. In: Kenedi RM, editor. Bioengineering studies of human skin. Oxford: Pergamon; 1965. p. 147–58.Google Scholar
  26. Lake SP, Barocas VH. Mechanical and structural contribution of non-fibrillar matrix in uniaxial tension: a collagen-agarose co-gel model. Ann Biomed Eng. 2011;39:1891–903.PubMedPubMedCentralCrossRefGoogle Scholar
  27. Lanir Y. A structural theory for the homogeneous biaxial stress-strain relationships in flat collagenous tissues. J Biomech. 1979;12:423–36.PubMedCrossRefGoogle Scholar
  28. Lanir Y. Constitutive equations for fibrous connective tissues. J Biomech. 1983;16:1–12.PubMedCrossRefGoogle Scholar
  29. Marsden JE, Hughes TJR. Mathematical foundations of elasticity. Don Mills: Dover; 1983.Google Scholar
  30. Mirnajafi A, Raymer J, Scott MJ, Sacks MS. The effects of collagen fiber orientation on the flexural properties of pericardial heterograft biomaterials. Biomaterials. 2005;26:795–804.PubMedCrossRefGoogle Scholar
  31. Mitton R. Mechanical properties of leather fibers. J Soc Leather Trades’ Chem. 1945;29:169–94.Google Scholar
  32. Prot V, Skallerud B, Holzapfel G. Transversely isotropic membrane shells with application to mitral valve mechanics. Constitutive modelling and finite element implementation. Int J Numer Methods Eng. 2007;71:987–1008.CrossRefGoogle Scholar
  33. Sacks M. Biaxial mechanical evaluation of planar biological materials. J Elast. 2000;61:199–246.CrossRefGoogle Scholar
  34. Sacks MS. Incorporation of experimentally-derived fiber orientation into a structural constitutive model for planar collagenous tissues. J Biomech Eng. 2003;125:280–7.PubMedCrossRefGoogle Scholar
  35. Soong TT, Huang WN. A stochastic model for biological tissue elasticity in simple elongation. JBiomech. 1973;6:451–8.Google Scholar
  36. Sun W, Sacks MS. Finite element implementation of a generalized Fung-elastic constitutive model for planar soft tissues. Biomech Model Mechanobiol. 2005;4(2-3):190–9.PubMedCrossRefGoogle Scholar
  37. Sun W, Sacks MS, Sellaro TL, Slaughter WS, Scott MJ. Biaxial mechanical response of bioprosthetic heart valve biomaterials to high in-plane shear. J Biomech Eng. 2003;125:372–80.PubMedCrossRefGoogle Scholar
  38. Tong P, Fung YC. The stress-strain relationship for the skin. J Biomech. 1976;9:649–57.PubMedCrossRefGoogle Scholar
  39. Tonge TK, Voo LM, Nguyen TD. Full-field bulge test for planar anisotropic tissues: part II–a thin shell method for determining material parameters and comparison of two distributed fiber modeling approaches. Acta Biomater. 2013;9:5926–42.PubMedCrossRefGoogle Scholar
  40. Waldman SD, Michael Lee J. Boundary conditions during biaxial testing of planar connective tissues. Part 1: dynamic behavior. J Mater Sci Mater Med. 2002;13:933–8.PubMedCrossRefGoogle Scholar
  41. Waldman SD, Sacks MS, Lee JM. Boundary conditions during biaxial testing of planar connective tissues: Part II: Fiber orientation. J Mater Sci Lett. 2002;21:1215–21.CrossRefGoogle Scholar
  42. Wognum S, Schmidt DE, Sacks MS. On the mechanical role of de novo synthesized elastin in the urinary bladder wall. J Biomech Eng. 2009;131:101018.PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2016

Authors and Affiliations

  1. 1.Department of Biomedical EngineeringInstitute for Computational Engineering and Sciences, The University of Texas at AustinAustinUSA

Personalised recommendations