Advertisement

From Stress–Strain Relations to Growth and Remodeling Theories: A Historical Reflection on Microstructurally Motivated Constitutive Relations

  • J. D. HumphreyEmail author

Abstract

As noted early on by Y.C. Fung, one of the greatest needs in biomechanics is formulation of constitutive relations for tissues that experience multiaxial loading. Although most investigators today seek to glean ideas on constitutive formulations from the latest papers, there is often much to learn from the earliest papers wherein truly original ideas can be found. In this Chapter, I provide a brief historical reflection on the formulation of constitutive relations for cardiovascular tissues, with particular focus on contributions by Y. Lanir. In this way, we can recall seminal works upon which much of our field has been built as well as see how past work continues to influence constitutive formulations, even in frontier areas such as soft tissue growth and remodeling.

Keywords

Residual Stress Collagen Fiber Constitutive Relation Strain Energy Function Total Strain Energy 
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

Acknowledgments

It is my privilege to contribute to this volume celebrating the 70th birthday of Professor Yoram Lanir and his many important contributions to the field of biomechanics. I have not only learned much from his many papers and lectures, I was also most fortunate to work alongside him in the mid-1980s, while I was a postdoctoral fellow, in the laboratory of Professor Frank Yin at Johns Hopkins University during Professor Lanir’s mini-sabbatical. Iremember well performing biaxial stretching tests together on excised pariet al pericardium and learning many “tricks of the trade” that Professor Lanir shared as we designed and performed new experiments. These ideas ranged from different practical means of attaching thin specimens to the biaxial stretching device and keeping them submerged in the physiologic bathing solution to novel theoretical ideas for changing the basic boundary value problem in ways to yield increased information. Professor Lanir had, with Professor Y.C. (Bert) Fung, reported the first careful biaxial testing experiments on soft tissues (skin) in the 1970s while he was a postdoctoral fellow, hence it was a great opportunity for me to learn from his experiences. Moreover, Professor Lanir’s seminal papers on microstructurally motivated constitutive relations for planar soft tissues had appeared relatively recently, in the late 1970s and early 1980s, and he had novel ideas of how to extract information on the constitutive parameters by designing experiments based on the constitutive theory, not simply the ease of measurement. Such lessons in both practical and theoretical approaches continue to serve me well.

I also remember working with Professor Lanir during that period to study the stability of rubber-like materials under biaxial dead loading. To do this, we designed and built in short order a simple biaxial system for loading thin sheets of rubber. In hindsight, our haste to design the setup actually lead to a somewhat dangerous experiment to apply relatively large dead loads to the sample, which we expected to be (and was) unstable or at least metastable. Nevertheless, the real lesson for me was in the design process and again I remember this time fondly. I thus am delighted to offer my warmest congratulations to Professor Lanir on the occasion of his 70th birthday. I thus thank Professors Sacks and Kassab for the invitation to contribute.

Finally, my continuing ability to study the fascinating field of biomechanics has been made possible by generous funding, primarily from the National Institutes of Health and National Science Foundation, for which I am grateful. I also acknowledge the wonderful contributions by so many excellent graduate students, postdoctoral fellows, and colleagues who continue to be a blessing to me.

References

  1. Baek S, Rajagopal KR, Humphrey JD. A theoretical model of enlarging intracranial fusiform aneurysms. J Biomech Eng. 2006;128:142–9.PubMedCrossRefGoogle Scholar
  2. Cardamone L, Valentin A, Eberth JF, Humphrey JD. Origin of axial prestress and residual stress in arteries. Biomech Model Mechanobiol. 2009;8:431–46.PubMedPubMedCentralCrossRefGoogle Scholar
  3. Choi HS, Vito RP. Two-dimensional stress-strain relationship for canine pericardium. J Biomech Eng. 1990;112:153–9.PubMedCrossRefGoogle Scholar
  4. Choung CJ, Fung YC. On residual stress in arteries. J Biomech Eng. 1986;108:189–92.CrossRefGoogle Scholar
  5. Cowin SC, Hegedus DH. Bone remodeling I. Theory of adaptive elasticity. J Elast. 1976;6:313–26.CrossRefGoogle Scholar
  6. Demer LL, Yin FCP. Passive biaxial mechanical properties of isolated canine myocardium. JPhysiol. 1983;339:615–30.PubMedPubMedCentralCrossRefGoogle Scholar
  7. Dingemans KP, Teeling P, Lagendijk JH, Becker AE. Extracellular matrix of the human aortic media: an ultrastructural histochemical study of the adult aortic media. Anat Rec. 2000;258:1–14.PubMedCrossRefGoogle Scholar
  8. Eberth JF, Cardamone L, Humphrey JD. Altered mechanical properties of carotid arteries in hypertension. J Biomech. 2011;44:2532–7.PubMedPubMedCentralCrossRefGoogle Scholar
  9. Ferruzzi J, Vorp DA, Humphrey JD. On constitutive descriptors for the biaxial mechanical behavior of human abdominal aorta and aneurysms. J R Soc Interface. 2011a;8:435–50.CrossRefGoogle Scholar
  10. Ferruzzi J, Collins MJ, Yeh AT, Humphrey JD. Mechanical assessment of elastin integrity in fibrillin-1 deficient carotid arteries: implications for Marfan syndrome. Cardiovasc Res. 2011b;92:287–95.PubMedPubMedCentralCrossRefGoogle Scholar
  11. Fung YC. Elasticity of soft tissues in simple elongation. Am J Physiol. 1967;28:1532–44.Google Scholar
  12. Fung YC. Biomechanics: mechanical properties of living tissues. New York: Springer; 1981.CrossRefGoogle Scholar
  13. Green AE, Adkins JE. Large elastic deformations and non-linear continuum mechanics. Oxford: Oxford University Press; 1970.Google Scholar
  14. Green AE, Zerna W. Theoretical elasticity. Oxford: Oxford University Press; 1960.Google Scholar
  15. Holzapfel GA. Nonlinear solid mechanics: a continuum approach for engineering. New York: Wiley; 2000.Google Scholar
  16. Holzapfel GA, Gasser TC, Ogden RW. A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast. 2000;61:1–48.CrossRefGoogle Scholar
  17. Horowitz A, Lanir Y, Yin FCP, Perl M, Sheinman I, Strumpf RK. Structural three-dimensional constitutive law for the passive myocardium. J Biomech Eng. 1988;110:200–7.PubMedCrossRefGoogle Scholar
  18. Humphrey JD. Cardiovascular solid mechanics: cells, tissues, and organs. New York: Springer; 2002.CrossRefGoogle Scholar
  19. Humphrey JD. Continuum biomechanics of soft biological tissues. Proc R Soc Lond A. 2003;459:3–46.CrossRefGoogle Scholar
  20. Humphrey JD. Vascular adaptation and mechanical homeostasis at tissue, cellular, and sub-cellular levels. Cell Biochem Biophys. 2008;50:53–78.PubMedCrossRefGoogle Scholar
  21. Humphrey JD, Rajagopal KR. A constrained mixture model for growth and remodeling of soft tissues. Math Models Methods Appl Sci. 2002;12:407–30.CrossRefGoogle Scholar
  22. Humphrey JD, Yin FCP. On constitutive relations and finite deformations of passive cardiac tissue. I. A pseudostrain-energy function. J Biomech Eng. 1987;109:298–304.PubMedCrossRefGoogle Scholar
  23. Humphrey JD, Vawter DL, Vito RP. Pseudoelasticity of excised visceral pleura. J Biomech Eng. 1987;109:115–20.PubMedCrossRefGoogle Scholar
  24. Humphrey JD, Strumpf RK, Yin FCP. Determination of a constitutive relation for passive myocardium: I. A new functional form. J Biomech Eng. 1990a;112:333–9.PubMedCrossRefGoogle Scholar
  25. Humphrey JD, Strumpf RK, Yin FCP. Determination of a constitutive relation for passive myocardium: II. Parameter estimation. J Biomech Eng. 1990b;112:340–6.PubMedCrossRefGoogle Scholar
  26. Humphrey JD, Eberth JF, Dye WW, Gleason RL. Fundamental role of axial stress in compensatory adaptations by arteries. J Biomech. 2009;42:1–8.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. Lanir Y, Fung YC. Two-dimensional mechanical properties of rabbit skin. II. Experimental results. J Biomech. 1974;7:171–82.PubMedCrossRefGoogle Scholar
  30. Nevo E, Lanir Y. Structural finite deformation model of the left ventricle during diastole and systole. J Biomech Eng. 1989;111:342–9.PubMedCrossRefGoogle Scholar
  31. Oden JT. Finite elements of nonlinear continua. New York: McGraw-Hill; 1972.Google Scholar
  32. Rodriguez EK, McCulloch AD, Hoger A. Stress-dependent finite growth in soft elastic tissues. JBiomech. 1994;27:455–67.PubMedCrossRefGoogle Scholar
  33. Sacks MS. Biaxial mechanical evaluation of planar biological materials. J Elast. 2000;61:199–246.CrossRefGoogle Scholar
  34. Skalak R. Growth as a finite displacement field. In: Carlson DE, Shield RT, editors. Proceed IUTAM symposium finite elasticity. The Hague: Martinus Nijhoff; 1981. p. 347–55.Google Scholar
  35. Taber LA. Biomechanics of growth, remodeling, and morphogenesis. Appl Mech Rev. 1995;48:487–545.CrossRefGoogle Scholar
  36. Thompson DW. On growth and form. New York: Cambridge University Press; 1999.Google Scholar
  37. Timoshenko SP. History of strength of materials. New York: Dover Publications; 1983.Google Scholar
  38. Treolar LRG. Physics of rubber elasticity. Oxford: Clarendon Press; 1975.Google Scholar
  39. Truesdell C, Noll W. The nonlinear field theories of mechanics. In: Flugge S, editor. Handbuch der Physik, vol. III/3. Berlin: Springer; 1965.Google Scholar
  40. Truesdell C, Toupin RA. The classical field theories. In: Flugge S, editor. Handbuch der Physik, vol. III/1. Berlin: Springer; 1960.Google Scholar
  41. Valentin A, Cardamone L, Baek S, Humphrey JD. Complementary vasoactivity and matrix remodeling in arterial adaptations to altered flow and pressure. J Roy Soc Interface. 2009;6:293–306.CrossRefGoogle Scholar
  42. Vawter DL, Fung YC, West JB. Elasticity of excised dog lung parenchyma. J Appl Physiol. 1978;45:261–9.PubMedGoogle Scholar
  43. Wilson JS, Baek S, Humphrey JD. Importance of initial aortic properties on the evolving regional anisotropy, stiffness, and wall thickness of human abdominal aortic aneurysms. J R Soc Interface. 2012;9:2047–58.CrossRefGoogle Scholar
  44. Yin FCP, Strumpf RK, Chew PH, Zeger SL. Quantification of the mechanical properties of noncontracting canine myocardium under simultaneous biaxial loading. J Biomech. 1987;20:577–89.PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2016

Authors and Affiliations

  1. 1.Department of Biomedical EngineeringYale UniversityNew HavenUSA

Personalised recommendations