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.
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References
Baek S, Rajagopal KR, Humphrey JD. A theoretical model of enlarging intracranial fusiform aneurysms. J Biomech Eng. 2006;128:142–9.
Cardamone L, Valentin A, Eberth JF, Humphrey JD. Origin of axial prestress and residual stress in arteries. Biomech Model Mechanobiol. 2009;8:431–46.
Choi HS, Vito RP. Two-dimensional stress-strain relationship for canine pericardium. J Biomech Eng. 1990;112:153–9.
Choung CJ, Fung YC. On residual stress in arteries. J Biomech Eng. 1986;108:189–92.
Cowin SC, Hegedus DH. Bone remodeling I. Theory of adaptive elasticity. J Elast. 1976;6:313–26.
Demer LL, Yin FCP. Passive biaxial mechanical properties of isolated canine myocardium. JPhysiol. 1983;339:615–30.
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.
Eberth JF, Cardamone L, Humphrey JD. Altered mechanical properties of carotid arteries in hypertension. J Biomech. 2011;44:2532–7.
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.
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.
Fung YC. Elasticity of soft tissues in simple elongation. Am J Physiol. 1967;28:1532–44.
Fung YC. Biomechanics: mechanical properties of living tissues. New York: Springer; 1981.
Green AE, Adkins JE. Large elastic deformations and non-linear continuum mechanics. Oxford: Oxford University Press; 1970.
Green AE, Zerna W. Theoretical elasticity. Oxford: Oxford University Press; 1960.
Holzapfel GA. Nonlinear solid mechanics: a continuum approach for engineering. New York: Wiley; 2000.
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.
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.
Humphrey JD. Cardiovascular solid mechanics: cells, tissues, and organs. New York: Springer; 2002.
Humphrey JD. Continuum biomechanics of soft biological tissues. Proc R Soc Lond A. 2003;459:3–46.
Humphrey JD. Vascular adaptation and mechanical homeostasis at tissue, cellular, and sub-cellular levels. Cell Biochem Biophys. 2008;50:53–78.
Humphrey JD, Rajagopal KR. A constrained mixture model for growth and remodeling of soft tissues. Math Models Methods Appl Sci. 2002;12:407–30.
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.
Humphrey JD, Vawter DL, Vito RP. Pseudoelasticity of excised visceral pleura. J Biomech Eng. 1987;109:115–20.
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.
Humphrey JD, Strumpf RK, Yin FCP. Determination of a constitutive relation for passive myocardium: II. Parameter estimation. J Biomech Eng. 1990b;112:340–6.
Humphrey JD, Eberth JF, Dye WW, Gleason RL. Fundamental role of axial stress in compensatory adaptations by arteries. J Biomech. 2009;42:1–8.
Lanir Y. A structural theory for the homogeneous biaxial stress-strain relationships in flat collagenous tissues. J Biomech. 1979;12:423–36.
Lanir Y. Constitutive equations for fibrous connective tissues. J Biomech. 1983;16:1–12.
Lanir Y, Fung YC. Two-dimensional mechanical properties of rabbit skin. II. Experimental results. J Biomech. 1974;7:171–82.
Nevo E, Lanir Y. Structural finite deformation model of the left ventricle during diastole and systole. J Biomech Eng. 1989;111:342–9.
Oden JT. Finite elements of nonlinear continua. New York: McGraw-Hill; 1972.
Rodriguez EK, McCulloch AD, Hoger A. Stress-dependent finite growth in soft elastic tissues. JBiomech. 1994;27:455–67.
Sacks MS. Biaxial mechanical evaluation of planar biological materials. J Elast. 2000;61:199–246.
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.
Taber LA. Biomechanics of growth, remodeling, and morphogenesis. Appl Mech Rev. 1995;48:487–545.
Thompson DW. On growth and form. New York: Cambridge University Press; 1999.
Timoshenko SP. History of strength of materials. New York: Dover Publications; 1983.
Treolar LRG. Physics of rubber elasticity. Oxford: Clarendon Press; 1975.
Truesdell C, Noll W. The nonlinear field theories of mechanics. In: Flugge S, editor. Handbuch der Physik, vol. III/3. Berlin: Springer; 1965.
Truesdell C, Toupin RA. The classical field theories. In: Flugge S, editor. Handbuch der Physik, vol. III/1. Berlin: Springer; 1960.
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.
Vawter DL, Fung YC, West JB. Elasticity of excised dog lung parenchyma. J Appl Physiol. 1978;45:261–9.
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.
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.
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.
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Humphrey, J.D. (2016). From Stress–Strain Relations to Growth and Remodeling Theories: A Historical Reflection on Microstructurally Motivated Constitutive Relations. In: Kassab, G., Sacks, M. (eds) Structure-Based Mechanics of Tissues and Organs. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-7630-7_7
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DOI: https://doi.org/10.1007/978-1-4899-7630-7_7
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