Manipulating elastic wave propagation in non-linear flexible materials via external stimuli Pavel I. Galich Manipulating elastic wave propagation in non-linear flexible materials via external stimuli Research thesis In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Pavel I. Galich Submitted to the Senate of the Technion - Israel Institute of Technology Elul 5778, August 2018, Haifa The Research Thesis Was Done Under the Supervision of Stephan Rudykh in the Faculty of Aerospace Engineering Acknowledgements The generous financial help of Technion – Israel Institute of Technology and The Irwin and Joan Jacobs Scholarship is gratefully acknowledged. I am deeply thankful to Stephan Rudykh for his professional guidance, for sharing his knowledge and experience with me, and for his dedicated support through my Doctorate studies. I thank Viacheslav (Slava) Slesarenko, Jian Li and Artemii Goshkoderia for useful discussions of my PhD research. I wish to express my gratitude to the Israel Science Foundation for the financial support of my research. Contents Abstract1 List of abbreviations3 1 Preamble4 1.1 Wave propagation in non-linear elastic materials............ 5 1.1.1 Finitely deformed isotropic materials .............. 5 1.1.2 Finitely deformed laminates.................... 5 1.1.3 Finitely deformed fiber-reinforced composites......... 7 1.2 Wave propagation in non-linear electroelastic materials . 9 2 Research methods 11 2.1 Non-linear elasticity ............................ 11 2.1.1 Small amplitude motions superimposed on large deformations 13 2.2 Bloch-Floquet approach for periodic composites . 15 2.2.1 Static finite deformation ...................... 15 2.2.2 Small-amplitude motions superimposed on a deformed state 16 2.3 Non-linear electroelasticity ......................... 16 2.3.1 Electrostatics ............................ 17 2.3.2 Mechanical balance laws...................... 17 2.3.3 Constitutive equations....................... 18 2.3.4 Incremental equations....................... 18 2.3.5 Incremental motions superimposed on finite deformation in the presence of an electric field.................. 19 2.3.6 Plane waves in incompressible DEs subjected to electrome- chanical loading .......................... 20 3 Findings 22 4 Conclusions and discussion 84 4.1 Influence of stiffening on elastic wave propagation in extremely de- formed soft matter: from nearly incompressible to auxetic materials . 85 4.2 Manipulating pressure and shear waves in dielectric elastomers via external electric stimuli........................... 86 4.3 Elastic wave propagation in finitely deformed layered materials . 87 4.4 Shear wave propagation and band gaps in finitely deformed dielec- tric elastomer laminates: long wave estimates and exact solution . 88 4.5 Shear wave propagation in finitely deformed 3D fiber-reinforced composites .................................. 89 Bibliography 91 Abstract Understanding of wave propagation phenomena in solids is pivotal for numerous engineering applications in healthcare, petroleum, military, and aerospace indus- tries. Among them nondestructive material testing, vibration protection of sensi- tive electronics, unwanted noise mitigation, earthquake forecasting, and medical ultrasound diagnostics. The ability of flexible materials to sustain large deforma- tions opens rich prospects for manipulating elastic wave characteristics by defor- mation. Moreover, elastic waves can be tuned through designing micro- or macro- structures, which can be further actively controlled by external stimuli, such as me- chanical loading, electric or magnetic fields. Understanding of how large deforma- tions affect relatively simple isotropic and more complex anisotropic materials is essential for this task. Therefore, in my thesis, I investigated the acoustic properties of the relatively simple initially isotropic non-linear materials and then proceeded with the periodic layered and fiber composites made of these materials. Firstly, I studied the influence of the deformation induced stiffening (intrinsic feature of most soft materials) on the propagation of small-amplitude elastic plane waves. Secondly, I showed that electroelastic isotropic materials (i.e. dielectric elastomers) can be utilized to achieve acoustic functionalities such as decoupling of pressure and shear waves by application of an electric field. Thirdly, I analyzed wave prop- agation in non-linear elastic and electroelastic laminates. In particular, I derived the long wave estimates of the phase and group velocities for the waves propagat- ing in the laminates subject to external mechanical or electrical stimuli. Moreover, I found the material compositions and loading conditions producing wide complete 1 band gaps (frequency ranges where neither pressure nor shear waves can propa- gate) at desired low frequency ranges. Finally, I investigated the anisotropy of elastic wave propagation in finitely deformed fiber composites (FCs). Specifically, by employing rigorous analytical methods of non-linear mechanics, I derived long wave estimates for phase and group velocities of shear waves propagating in non- linear FCs. Next, by utilizing Bloch-Floquet approach in a finite element code, I calculated dispersion relations for shear waves propagating along the fibers in FCs with square arrays of fibers. 2 List of abbreviations FC – Fiber composite P-wave – Pressure (longitudinal) wave S-wave – Shear (transversal) wave BG – Band gap SBG – Shear wave band gap PBG – Pressure wave band gap DE – Dielectric elastomer RVE – Representative Volume Element 3 Chapter 1 Preamble Lightweight composite materials with excellent mechanical properties are of im- mense importance for aerospace engineering [8]; however, in aerospace structures, undesirable vibrations and noise propagate from the source, e.g., an engine, to other parts; this phenomenon requires control, reduction, or even elimination. Therefore, new lightweight materials allowing tunable noise and vibration reduc- tion need to be developed, while maintaining mechanical properties such as stiff- ness, toughness, and damage tolerance. Moreover, better understanding of wave propagation phenomena in flexible materials can benefit nondestructive material testing in composite aerospace structures. Motivated by these requirements for aerospace structural materials, in my PhD research I explored the general field of elastic wave propagation in non-linear isotropic elastic [21] and electroelastic [22] materials and then in layered [19, 23] and fibrous [25] composites made of these materials. 4 1.1 Wave propagation in non-linear elastic materials 1.1.1 Finitely deformed isotropic materials Small amplitude elastic wave propagation in finitely deformed homogeneous isotropic materials was pioneered by Biot [5]. Boulanger and Hayes [6] considered wave propagation in finitely deformed incompressible Mooney-Rivlin materials and de- rived explicit relations for wave velocities. Boulanger et al. [7] extended this work to a broader class of finitely deformed compressible Hadamard materials and first obtained the explicit expressions for the phase velocities of longitudinal (pressure) and transversal (shear) waves. Recently, Destrade and Ogden [15] have revised and generalized the problem of an infinitesimal wave propagation in the finitely deformed hyperelastic materials by application of the invariant theory. More re- cently, I have investigated infinitesimal (linear) wave propagation in finitely de- formed compressible Gent materials, exhibiting pronounced stiffening effects, and obtained closed form expressions for the phase velocities of longitudinal and transver- sal waves [21]. It is worth mentioning that deformation can be utilized not only for the changing of phase and group velocities of elastic waves but also for decou- pling of P- and S-waves in these relatively simple originally isotropic non-linear materials [9, 20]. 1.1.2 Finitely deformed laminates Small amplitude (linear) elastic wave propagation in linear bi-laminates made of isotropic materials was pioneered by by Rytov [43]. He derived explicit disper- 5 sion relations for steady-state P- and S-waves propagating perpendicular and par- allel to the layers. Recently, problem of wave propagation in periodic laminates has been revived in a framework of elastic metamaterials [53, 59]. Hence, I ex- tended classical result of Rytov [43] for non-linear bi-laminates made of hypere- lastic isotropic materials, namely neo-Hookean and Gent materials [19]. This al- lowed me to investigate influence of finite deformations on S- and P-wave band gaps in these laminates. In particular, I found that S-wave band gaps (SBGs) in laminates with compressible neo-Hookean layers are not influenced by defor- mation, because deformation-induced changes in geometry and effective material properties fully compensate each other. Thus, I corrected a misconception that had prevailed in the literature [47], namely that SBGs could be tuned by applica- tion of elastic deformation in compressible neo-Hookean laminates – I rigorously showed, through exact analytical solution, that they could not [19]. However, con- traction or extension of the laminate with Gent layers (exhibiting strong stiffening effects) widens and shifts up SBGs towards higher frequencies due to the stronger effect of deformation-induced change in the layer properties as compared to the change in the layer thicknesses. Moreover, I derived closed-form expressions for the phase and group velocities of long S-waves propagating in any