Dynamic radial deformations of nonlinear elastic structures. On the influence of constitutive modeling

208 Pages·2017·25.36 MB·English

Dynamic radial deformations of
nonlinear elastic structures
On the influence of constitutive modeling
Author: Damián Aranda-Iglesias
Supervisor: José A. Rodríguez-Martínez
Co-Supervisor: Guadalupe Vadillo
Department of Continuum Mechanics and Structural Analysis
University Carlos III of Madrid
Leganés, June 9, 2017

Dynamic radial deformations of nonlinear
elastic structures
On the influence of constitutive modeling
This dissertation is submitted for the degree of Doctor of Philosophy
Author: Damián Aranda-Iglesias
Supervisor: José A. Rodríguez-Martínez
Co-Supervisor: Guadalupe Vadillo Martín
Members of the Evaluation Committee:
Prof. Ignacio Romero Technical University of Madrid, Spain
Prof. Massimo Ruzzene Georgia Institute of Technology, USA
Prof. Antonino Morassi University of Udine, Italy
University Carlos III of Madrid
Department of Continuum Mechanics
and Structural Analysis
Leganés, June 9, 2017.

A Sara, mi compañera en esta y en todas mis aventuras.
A Paco, Mariló y Concha, nunca estaré suficientemente agradecido.

Acknowledgements
I owe my deepest gratitude to my mentor and friend José A. Rodríguez Martínez.
Thank you for your enthusiasm, your energy, all the opportunities, and your unconditional
encouragement and support. I feel really lucky to have shared this trip with you!
I also want to express my warmest gratitude to my other supervisor Guadalupe
Vadillo Martín, your affable, thoughtful and receptive personality has enlightened some
of the tougher moments. Thanks!
A very special gratitude goes out to Óscar López-Pamies, professor of the Depart-
ment of Civil and Environmental Engineering of the University of Illinois at Urbana-
Champaign, and my supervisor during my research stay. It has been my pleasure to work
with you, thanks for sharing so much wisdom with me and for your support and advice.
Furthermore, I would like to acknowledge the collaboration in this research of Prof.
Volokh and Prof. Rubin from Technion – Israel Institute of Technology. I have learned
a lot working together and your contribution has certainly improved the quality of this
research.
Many thanks to my colleagues at UC3M: Dani, Vila, Mahy, Javi, Vaz, Sara,
Luis, Ángel, Mata, Pablo, Lorena and Óscar. Thanks you for your help, your
friendship and for all the fun we have had in the last years. In a very special way I would
like to mention here Daniel García Gonzalez, Alejandro Mahy Soler and Javier
Vila Morán. I will miss our stimulating and endless discussions guys!
I will always be indebted to my alter ego David Rodríguez Martinez. Thank you
for your help, not only reviewing this document but supporting me at every step.
Last but not least, I would like to thank my family: Paco, Mariló, Concha and
Sara. None of this would have been possible without your daily back up. I will never be
grateful enough

Abstract
The objective of this dissertation is to develop a comprehensive theoretical approach
to the role of the constitutive model on the dynamic radial deformations of nonlinear
elastic structures. Using 1D and 2D models, cylindrical and spherical thick-walled
shells are considered. These geometries are representative of man-made and natural
structures that can be found in a wide variety of engineering applications and biological
systems. Lead-rubber bearings, vibration isolators, peristaltic pumps, rubber bushings,
saccular aneurysms or arteries are examples of nonlinear elastic structures with spherical
and cylindrical geometries that are constantly subjected to all kinds of vibrational and
dynamic loads. The research, which starts by considering isotropic, incompressible
and rate independent constitutive models, is based on the systematic incorporation of
compressibility, viscosity and anisotropy in the description of the mechanical response
of the material. We investigate free and forced vibrations using different initial and
boundary conditions: (1) ab initio elastic stored and kinetic energies, (2) constant radial
pressures, (3) linearly time dependent radial pressures and (4) harmonic time dependent
radial pressures. While the isotropic and incompressible 1D elastic structures subjected
to constant pressure admit an analytical closed-form solution, all the other cases need to
be solved numerically. To this end, we have developed in this work a number of specific
numerical schemes. The overall outcome of this dissertation is to make it plain that
the constitutive model used to describe the mechanical behavior of thick-walled shells
plays a fundamental role in the nonlinear dynamics of such structures. In particular, we
have demonstrated the influence of the constitutive model on: (1) the loss of oscillatory
behavior of the structure, (2) the transition from periodic motions to quasi-periodic and
chaotic motions, (3) the nonlinear resonances of the shells, (4) the propagation of shock
waves within the structure and (5) the onset and development of cavitation instabilities.
Table of contents
Part I Introduction and fundamental concepts of nonlinear
solid mechanics 1
1 Introduction 3
1.1 Research background and motivation . . . . . . . . . . . . . . . . . . . . 3
1.2 General objective and structure of the dissertation . . . . . . . . . . . . . 13
1.3 Originality and quality of the research . . . . . . . . . . . . . . . . . . . 15
2 Nonlinear solid mechanics 17
2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 The concept of stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Balance principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Hyperelastic constitutive models . . . . . . . . . . . . . . . . . . . . . . . 30
Part II 1D finite vibrations of isotropic hyperelastic struc-
tures 33
3 Incompressible isotropic hyperelastic cylindrical structures: periodic,
quasi-periodic and chaotic motion 35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Sample results: autonomous system . . . . . . . . . . . . . . . . . . . . . 42
3.4 Sample results: non-autonomous system . . . . . . . . . . . . . . . . . . 45
3.5 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Incompressible isotropic hyperelastic spherical structures: nonlinear
resonances in idealized aneurysmal lesions 57
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66