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

  • J. D. HumphreyEmail author


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.


Residual Stress Collagen Fiber Constitutive Relation Strain Energy Function Total Strain Energy 
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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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© Springer Science+Business Media, LLC 2016

Authors and Affiliations

  1. 1.Department of Biomedical EngineeringYale UniversityNew HavenUSA

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