Spatial variability of the dynamic response in periodic and non-homogeneous elastic media.

Abstract

The dynamics of elastic media and structures are of great practical interest due to their varied application in areas from ranging manufacturing to medicine. Surface structures like plates or shells or assemblies composed of constituent surface structures display a rich variety of spatial variability in their response to dynamic loading. In fact, the spatial variations observed in the dynamic response of vibrating plates, called Chladni’s patterns, initiated the race to formulate the mathematics describing the dynamics of plates. Given the importance of structures such as beams, plates, and shells in practical applications, the spatial variability of their dynamic response is studied here in the context of (i) wave propagation in thin surface structures with randomly varying properties and (ii) modal vibrations of finite assemblies composed of beams, plates and shells.
In the first half of this work, the wave propagation of flexural waves in thin elastic plates and shells with non-uniform properties is studied. Particularly, the effect of spatially correlated random variation of thickness is considered. However, the results are shown to be generalisable to variations in other material and geometric properties. Initially, planar wavefronts undergo random focusing, and interesting branching structures, which lead to locations of large amplitudes, are observed in the path of propagation of the wavefront. A connection to the phenomenon called branched flows, hitherto unobserved in elasticity, is established. A scaling law is associated with the expected location of high amplitudes typified branched flows. In this work, it is established that this is observed in flexural waves supported by thin beams and shells too. This is despite the fact that markedly different equations of motion govern flexural elastic wave propagation in surface structures as compared to the other kinds of waves that show branched flows such as electromagnetic and shallow water waves. The emergence of branched flows and the associated scaling law is demonstrated theoretically from the analysis of partial, ordinary and stochastic differential equations. The phenomenon under question exists in the ray optics limit. Hence, the wave dynamics problem is approximated as a ray propagation problem using the WKB/eikonal approximation and solved numerically using a finite difference scheme. Finally, the full wave elastodynamics problem is solved using finite element simulations on high-performance clusters. This captures some of the subtleties, such as dispersion, that are missed by the ray approximation. The theoretical and numerical analyses are all found to be in excellent agreement. These results suggest a certain universality of branched flows in wave dynamics regardless of the physics involved.
Subsequently, the modal vibration of pillared plates is considered. These structures have attracted some interest as metamaterials. This has led to Bloch-Floquet-based analyses of their wave propagation characteristics. However, a study of the modal vibration of finite pillared plates is missing from the literature. There is a need to understand the modal vibration of such structures which can be understood as an abstraction of certain industrial assemblies such as fin-less heat exchangers. From numerical experiments, it is observed that small displacements of the based plate confer a spatial phase relationship to the tip deflection of the identical pillars, reminiscent of Chladni’s patterns, within the degenerate frequency band corresponding to the resonance frequency of the pillars. This observation is leveraged to construct a Rayleigh quotient-based approximation of the modal vibration of such pillared plates. These are compared with finite element simulations which are found to be in excellent agreement. The use of the Hungarian algorithm, from combinatorics, to match modes from Rayleigh quotient-based approximation and those from FE simulations is discussed.
Finally, the modal vibration of tube-and-fin banks is studied. These are critical constituent elements of some heat exchangers. These structures are composed of numerous identical metallic tubes that are decorated with equally spaced thin metal plates or “fins”. The fins increase the surface area available for heat exchange and are usually arranged perpendicular to the tube axes. Heat exchangers are often mounted onto engines that inevitably provide base excitation which excites lower vibrational modes of tube-and-fin banks. Therefore, understanding the modal vibration of tube-and-fin banks is of interest in designing heat exchangers. Based on some numerical experiments and physical arguments, it is ascertained that the lower vibrational mode shapes of tube-and-fin banks can be constructed as twisting and bending modes of some dynamically equivalent rod and beam respectively. Using this insight, Rayleigh quotient-based models of these mode shapes are set up which agree well with FE simulations. Moreover, these models are found to be very computationally efficient as compared to FE simulations. The role of the hexagonal and square lattice arrangement of tubes on fins is considered. The packaging of this proposed model as a graphical user interface for deployment in the industry is also discussed.

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Identifiers

ORCID for Kevin Jose: orcid.org/0000-0002-1268-4476

ORCID for Neil Ferguson: orcid.org/0000-0001-5955-7477

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