Elasticity and Hereditariness
This chapter collects the lecture notes of the module “Elasticity and Hereditatiness of Lipid Bilayers” delivered at CISM in July 2016. Such material is based primarily on three papers coauthored by this lecturer, and which have been contributing to shed light on the mechanical behavior of lipid bilayers. In particular, the breakthrough from this research is that the underlying nonlinear elastic response of lipid bilayers is fully determined as long as the membrane energy is obtained. Bending and saddle splay rigidities are shown here to be directly obtainable from the membranal response, as well as the line tension, holding together domains in which lipids are in different phases. The power law hereditariness of lipid membranes strikingly shown through rheometric tests has been analyzed in this work through a suitable energetics obtained by the author and coworkers and penalizing small perturbations of ground configurations of such systems.
KeywordsLipid Membrane Total Potential Energy Line Tension Elastic Case Spontaneous Curvature
The author wishes to thank the organizer of this course, David Steigmann, for his invitation to contribute to this course. The other lecturers are also acknowledged for the nice and extended discussions that allowed for exchange of ideas on the topic of this course.
The author is extremely grateful to Giuseppe Zurlo (National University of Galway, Ireland), formerly his Ph.D. student, for the very extensive discussions and long-standing collaboration from his early days in 2002. His key contribution to this research has had huge impact in its assessment and development. Timothy J. Healey (Cornell University) and Roberto Paroni (University of Sassari, Italy) also gratefully acknowledged for the very extended discussions on the early stages of the 2008 work.
Grateful acknowledgements go to Massimiliano Zingales (University of Palermo), Kaushik Dayal (Carnegie Mellon University) as collaborators on key aspects related to the hereditary response of lipid bilayers. Massimiliano Fraldi (University of Napoli-Federico II) is also gratefully ackowledged for his illuminating remarks and insights on biological tissues and biomechanics, as well as Valentina Piccolo (University of Trento), a graduate student working with myself and other people on various topics, who also provided new perspectives on the applications of Fractional Analysis to lipid membranes and helped a lot to edit this work.
The author is grateful to the financial support provided by (i) the NSF Grant no.DMS-0635983 of the Center for Nonlinear Analysis, Carnegie Mellon University, (ii) for the direct financial support of the Dept. of Mechanical Engineering and Materials Science-MEMS of the University of Pittsburgh for, and also to (iii) the support of the EU Grant “INSTABILITIES” ERC-2013-ADG Instabilities and nonlocal multiscale modelling of materials held by Prof. Davide Bigoni from the University of Trento.
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