From Waves in Complex Systems to Dynamics of Generalized Continua

The following sections are included:

• PREFACE

• CONTRIBUTORS

• Group Photo

• LAUDATIO FOR PROFESSOR YIH-HSING PAO ON THE OCCASION OF THE INTERNATIONAL SYMPOSIUM ON ENGINEERING MECHANICS 2010, ON THE OCCASION OF HIS 80^TH BIRTHDAY (21/22 May 2010)

• CONTENTS

Phononic crystals (PCs) which consist of periodically arranged media have, in the past two decades, attracted intensive studies due to their renewed properties of band gaps and maneuverable band structures. Recently, Lamb waves in thin plates with PC structures started to receive increasing attention for their potential applications in filters, resonators, and waveguides. This paper presents a review of recent works conducted by the authors and co-workers on this topic. Both theoretical and experimental studies of Lamb waves in two-dimensional (2D) PC plate structures are covered. On the theoretical side, analyses of Lamb waves in 2D phononic plates using the plane wave expansion (PWE) method, finite-difference time-domain (FDTD) method, and finite-element (FE) method are addressed. These methods were applied to determine the frequency ranges of band gaps of Lamb waves and characteristics of the propagating and localized eigenmodes that can exist in the PC plate structures. The theoretical analyses demonstrated the effects of PC based waveguides and resonant cavities. We discuss the influences of geometrical parameters on the guiding and resonance efficiency and the frequencies of waveguide and cavity modes. On the experimental side, we discuss band gaps of Lamb waves in a surface stubbed phononic plate made of aluminum which were demonstrated using laser ultrasonics. Furthermore, we present design and fabrication of a silicon based two-port Lamb wave resonator which utilizes PC reflective gratings to form the cavity. The measured results showed significant improvement of the insertion losses and quality factors of the resonators when the PCs were applied.

The method of reverberation-ray matrix (MRRM) has been developed and successfully applied to analyses of dynamic responses of framed structures and transient wave propagation in layered media. However, attention has been paid mainly to solving various practical problems, and there is a lack of study on the theoretical aspects of the method itself. In this paper, we intend to explore the local phase relations in MRRM based on the theory of first-order linear differential equations with variable coefficients. By using the coordinate transform, a generalized form of the local phase relations is presented, which includes the effect of inhomogeneous terms appearing in the differential equations. These terms may correspond to distributive external loads in a mechanics problem. When the coefficients of the differential equations are constant, the local phase relations will take a simple form. Furthermore, to ensure numerical stability, the phase relations should be rearranged properly, and a modification of the scattering relations is also suggested. These aspects are illustrated by considering the wave propagation in an arbitrarily anisotropic elastic laminate.

A nonlinear acoustic measurement technique is studied for fatigue damage monitoring. An electromagnetic acoustic transducer (EMAT) magnetostrictively couples to a surface-shear wave resonance along the circumference of a rod specimen during rotating bending fatigue of carbon steels. Excitation of the EMAT at the half of the resonance frequency caused the standing wave containing only the second-harmonic component, which was received by the same EMAT to determine the second-harmonic amplitude. Thus measured surface-wave nonlinearity always showed two distinct peaks approximately at 60% and 85% of the total fatigue life. We attribute the earlier peak to the crack nucleation and growth, and the later peak to an increase of free dislocations associated with crack extension in the final stage of life. This non-contact resonance-EMAT measurement can monitor the evolution of the surface-shear-wave nonlinearity throughout the metal's fatigue life and is able to detect the pertinent precursors of the eventual failure.

Acoustoelasticity, or the change in elastic wave speeds with stress, is a well-studied phenomenon for bulk waves. The effect of stress on Lamb waves is not as well understood, although it is clear that anisotropic stresses will produce anisotropy in the Lamb wave dispersion curves. Here the theory of acoustoelastic Lamb wave propagation is considered, and it is shown that, as expected, dispersion curves change anisotropically for an aluminum plate under uniaxial tension. Theoretical predictions are compared to experimental results for the S₀ Lamb wave mode at 250 kHz for a uniaxial load. Theoretical dispersion curves are then used to simulate Lamb waves propagating in a bounded plate subjected to various loading conditions. Signal changes are examined, and the implications for structural health monitoring are discussed.

Extended sources, both physical and synthetic, have a long history in acoustics and seismology. Recently, extended synthetic sources have been used to substantially reduce the computational complexity of reflection seismic waveform inversion (model-driven data fitting). Most proposed synthetic sources for waveform inversion are random in some way. A different, deterministic extended source selection strategy has met with success in application to other continuum-mechanical inverse problems. This paper reports preliminary work on a translation of the method to the reflected wave imaging setting, a basic theoretical framework for analysing its behaviour, and a numerical example. The example shows the very strong tendency of normalized source energy to insert large high angle components into the acoustic field; this tendency is also predicted by the theory. Such high angle waves produce the largest possible discrimination between the responses of two different models, but may not optimally illuminate the subsurface for inversion updates. The paper ends with suggestions for modification of the source selection strategy to better adapt it to inversion applications.

The existing methods of data analysis, either the probability theory or the spectral analysis, are all developed by mathematicians or based on their rigorous mathematical rules. For analyzing data from the real physical world, we have to face the reality of nonstationarity and nonlinearity in the processes. The traditional analysis methods are based on rigorous approach, but cannot fully accommodate these conditions. A new adaptive data analysis was introduced by Huang et al (1998), which was designated by NASA as the Hilbert-Huang Transform (HHT). It consists of Empirical Mode Decomposition (EMD) and the Hilbert Spectral Analysis (HSA) methods; both were introduced recently by Huang et al. (1996, 1998, 1999 and 2003). The method is adaptive, and specifically designed for analyzing data from nonlinear and nonstationary processes. Since its introduction over ten years ago, the HHT has been applied to a wide range of applications, covering (among many others) biology, geophysics, ocean research, engineering, radar and medicine (see, for example, Huang and Attok-Okine, 2005; Huang and Shen, 2005; Huang and Wu, 2008). Yet, up to this time, a rigorous mathematical foundation is still lacking. Under this condition, progresses are still empirically based. An introduction and some of the recent advances on Ensemble Empirical Mode Decomposition (Wu and Huang, 2009), Instantaneous frequency and trend computations, Time-dependent Intrinsic Correlation (Chen, 2010) and its extension to multi-dimensional data (Wu et al. 2009) are summarized and briefly discussed. These advances have made the HHT method much more robust and mature.

Computational Fluid Dynamics (CFD) uses large scale numerical computation to solve problems of fluid flow. Traditionally, it uses either the Eulerian or the Lagrangian coordinate system. These two systems are numerically non-equivalent, but each has its advantages as well as drawbacks. A unified coordinate system (UC) has recently been developed which combines the advantages of both Eulerian and Lagrangian systems, while avoiding their drawbacks. This paper gives an expose of the CFD using the unified coordinates.

Specifically, it will be shown that: (1) Governing equations of fluid flow in any moving coordinates can be written as a system of closed conservation PDEs; consequently, the effects of moving mesh on the flow are fully accounted for. (2) The system of Lagrangian gas dynamics equations is written in conservation PDE form for the first time. (3) The Lagrangian gas dynamics equations in 2-D and 3-D are shown to be theoretically non-equivalent to the Eulerian ones. (4) The UC is superior to both Eulerian and Lagrangian systems in that contact discontinuities are resolved sharply without mesh tangling. (5) Additionally, the UC is superior to the Eulerian as it avoids the tedious and time-consuming task of mesh generation: the mesh in UC is automatically generated by the flow. Many examples are given to demonstrate these properties of the UC.

Recent theorems have established that fully diffuse wave fields permit, on cross-correlating the noisy signal detected at different positions, retrieval of the Green's function, and in particular, retrieval of wave travel times. Application to seismology has led to the creation of maps of seismic surface wave velocity with exceptional resolution. What is perhaps perplexing is that the ambient seismic noise fields that are used for this are rarely fully diffuse; the conditions for the theorems are not satisfied. It is shown here that this may be understood theoretically by a simple asymptotic argument. It is further shown that incompletely diffuse noise fields may permit us to robustly recover more than just travel times, but also ray arrival amplitudes, the ambient field's specific intensity, the strength and density of its scatterers if any, and most importantly attenuation. A theoretical basis for this is presented.

Two scale design optimization for structures and materials with periodic microstructure is studied in the present paper. The aim is to obtain optimal configurations of macro scale structures and microstructure of materials for optimizing structural performance. The link between macro effective material properties and material micro structure is established by a homogenization approach. Structural topology optimization and inverse homogenization is applied to obtain the optimum structural topology design and optimum microstructure. Global structural performance such as minimum compliance design and maximizing fundamental frequency and local performance such as stress concentration are investigated. The proposed method and computational model are validated by a set of numerical examples. A number of issues for further development in two scale design are addressed.

A continuum formulation of lava flow down the mountain side of a volcano is presented. Three regimes are differentiated: Hot, temperate and cold lava. Hot lava is treated as a non-linear viscous heat conducting fluid; temperate lava as a binary solid-fluid frictional mixture and cold lava as a frictional heat conducting granular solid. These regions are separated from one another by mass-exchanging phase change surfaces. At the base, erosion, deposition processes and at the free surface heat transfer by radiation and sensible and latent heat are accounted for. The paper has fundamental character as a model for this kind of gravity flow is presented.

A unified mechanical-physicochemical theory, i.e., a tertiary mixture theory, for modeling a class of biological materials characterized as hydrated-charged-soft tissues has been developed, along with applications to articular cartilage biomechanics studies are presented. Particular emphasis is placed on a linearization procedure for this highly nonlinear constitutive law, and the application of the resultant equations toward determining tissue mechanical behavior and biochemical composition by mechanical indentation. For the first application, the tested specimen is modeled as a homogeneous and isotropic charged-hydrated-soft tissue; in our second application, the specimen will be modeled as a layered laminate and in-plane orthotropic material. In this presentation a descriptive explanation is presented to motivate the need for a tertiary mixture theory to represent the behavior of biological tissues that are comprised of interstitial water (~75%), a porous-permeable charged organic solid matrix of collagen fibers and proteoglycan supra-macromolecules (~25%), and dissolved electrolytes (e.g., Na⁺, Cl^- of negligible%). The viscoelastic behavior of such hydrated soft tissues has two major components: 1) a flow independent component due to the intrinsic macromolecular viscoelasticity; and 2) a flow dependent component due to frictional dissipation from flow through the porous-permeable matrix. A mathematical method of solution is suggested with the introduction of transformation of variables resulting in a new "generalized correspondence principle" between charged-hydrated-soft tissues and linear-isotropic-elastic materials (i.e., elasticity theory). This principle makes the employment of triphasic theory as straightforward as using an elasticity theory.

This paper reviews the theme "Unity in Diversity" that envelopes acoustics, electromagnetics and elastodynamics. The Helmholtz decomposition of vector fields in terms of scalar and vector potentials serves to unify all continuum fields while the constitutive models that have been developed to explain the observed experimental behavior of the fields introduces diversity in wave propagation. This serves as a fertile ground for consideration of various coupled field problems. In this paper, the term coupling is used to refer to the coupling of solenoidal and irrotational fields as well as the coupling of electric and magnetic fields or elastodynamics and electromagnetic fields. It will be clear from this paper that developments during the last quarter century or more from consideration of coupled fields have lead to the development of many exciting applications using piezoelectric, piezomagnetic, magneto-electric, chiral and currently metamaterials. This paper reviews contributions in these areas for applications to sonar and radar coatings, NDE, smart materials and structures, electromagnetic and elastodynamics chirality and electromagnetic and acoustic metamaterials.

The propagation of transient waves in an elastic half-space generated by a moving torque is investigated in this paper. The torque is suddenly applied and moves rectilinearly at a constant speed in the surface. The responses for torques moving at superspeed, transpeed and subspeed relative to the distortional wave speed are analyzed in detail. Closed-form solutions for displacements in the half-space are obtained through the application of the extended Cagniard-de Hoop method and are expressed in terms of elementary functions.

This paper reviews the magnetic force models of magnetizable bodies in a magnetic field as developed in recent decades by comparing their premises, modeling processes and formulae as well as deficiencies, and it discusses their physical significances and scopes of application. Then, a special case of magnetoelastic behavior of ferromagnetic plates subjected to a complex magnetic field is considered. The discrepancies in the magnetoelastic behavior of ferromagnetic plates predicted by different models have been analyzed in detail, and the principal reasons for these differences have been investigated. There are many coexisting models describing the magnetic force exerted on a magnetizable body; they may be attributed to the different approximations and assumptions adopted during the modeling process. Since the distribution of magnetic forces obtained by various models is different in distinct models, the predicted magnetoelastic behavior based on these models is distinct, too. Furthermore, there may in some magnetoelastic problems be a qualitative difference between the predictions of the mechanical behavior and experimental phenomena. The conclusion drawn from this paper may provide a credible theoretical basis for investigating the magnetoelastic behavior of magnetizable elastic structures.

In this paper, the principles and feasibility of 'vibro-wind power' are explored. Vibro- wind denotes the harvesting of energy from the wind as it flows around vibrating structures as an alternative to conventional rotary wind turbines. The basic science involves energy extraction from bodies induced to vibrate due to the nonlinear action of fluid flow and vortices around flexible structures. This research combines research in fluid dynamics, fluid-structure dynamics, nonlinear dynamics and electromechanical energy harvesting to create a new paradigm in energy research as well as to create a potential new energy technology. Two key topics will be addressed: (1) how to take out energy from flow-induced vibrations without quenching the nonlinear fluid-elastic instability, and (2) how to maximize energy conversion from the vibration of a large array of oscillators into electrical energy. A target application is for architectural facades similar to, and as a complement to, solar energy panels.