Theoretical Modelling Of Solitary Waves In Soft Bars And Higher Harmonics In Nonlinear Media

With the development of ultrasonic techniques,the influence of elastic nonlinearity on wave propagation in solids has attracted extensive attention.In this dissertation,several theoretical problems of wave propagation in materials and structures involving elastic nonlinearity are investigated.The dissertation consists of two main parts,i.e.the investigation of solitary waves propagating in soft bars under a biasing field(Part 1),and the analysis of higher harmonic generation by material nonlinearity arisen from material micro-damages(Part 2).The major achievements of this dissertation are briefly summarized as follows.In Part 1,the tunability of solitary waves propagating in soft bars is explored.The effective material properties of soft materials can be altered significantly when subjected to biasing fields,such as electric field or pre-stretch.In this context,two cases are considered,1)an electroelastic bar subjected to a biasing longitudinal electric displacement,and 2)a viscoelastic bar subjected to a pre-stretch.An asymptotic analysis is conducted by introducing several asymptotic expansions to simplify the rod governing equations.The boundary conditions on the lateral surface of the rod are satisfied from the asymptotic point of view.For the first case,by the reductive perturbation method,we deduce the far-field equation(i.e.the KdV equation).Then,the leading order of the electroelastic solitary wave solution is presented.Numerical examples are provided to show the influences of the biasing electric displacement and material constants on the solitary waves.It is found that the biasing electric displacement can modulate the velocity of solitary waves with a prescribed amplitude in the electroactive rod.For the second case,following the similar procedure,the KdV-Burgers equation can be formulated,which admits analytical and explicit solutions for kink and kink-like waves in pre-stretched Mooney-Rivlin elastic rods with the consideration of viscous dissipation.We find that the pre-stretch can not only make the kink waves lower and wider,but also change the wave velocity.The competition between the effects of pre-stretch and viscosity on the kink and kink-like waves is also revealed.In Part 2,several simplified theories and simple theoretical models are proposed to work out analytical solutions to higher harmonic generations by material nonlinearity,which can be used to assess material micro-damages.As a starting point,we generally investigate harmonics of plane longitudinal and transverse waves in elastic solids with up to cubic nonlinearity in a one-dimensional setting.Some interesting and useful results for harmonic generation are uncovered.Then,we extend our work to the investigation of wave propagating on a half-space of isotropic incompressible material of cubic nonlinearity.The analytical far-field solution for the cumulative third harmonic surface wave is obtained in a relatively simple and systematic manner.The solution reveals that,in the far field,the resonant third harmonic propagates with the classic Rayleigh wave velocity,whose amplitude increases linearly with the propagation distance.The transmission of the resonant wave from a half-space of nonlinear material into a half-space of linear material is also considered.In a pipe of quadratic material nonlinearity,the analytical solution to the mixing of axisymmetric longitudinal waves and torsional waves are obtained using the shell theory.The resonant waves with difference frequencies propagate in the opposite direction of the corresponding primary wave.The nonlinear shell theory is further simplified to obtain the solution for the cumulative second longitudinal harmonics generated by self-interaction of longitudinal waves in an analytical form.From a practical point of view,some theoretical models to investigate harmonic generation from an inclusion of nonlinear material are also established.By using the continuity conditions of stress and displacement at the interface or using the reciprocity theorem of elastodynamics,the expressions of the reflection waves are obtained,whose amplitudes can provide information of the material constants of the nonlinear media.The reciprocity theorem is proven to have greater utility.As an example,the backscattering of a torsional wave from a small zone of material nonlinearity in a pipe is investigated.The analytical expression of the backscattered wave is obtained by using the reciprocity theorem,whose amplitude is determined by the nonlinearity coefficient and the size of the nonlinear region.Combining the primary wave with a higher frequency wave is proposed to increase the magnitude of the backscattered wave.Using the same method,we also investigate the intersection of two non-collinear waves at a region of quadratic material nonlinearity in an elastic layer in a three-dimensional setting.Based on the mode expansions,the analytical solution to the amplitudes of the Lamb wave and the SH wave are obtained.The theoretical models proposed in this dissertation and the obtained analytical solutions have the potential application in the design of novel acoustic devices and the development of nonlinear ultrasonic techniques for nondestructive evaluation.