We investigate the existence of trapped modes in elastic plates of constant thickness, which possess bends of arbitrary curvature and flatten out at infinity; such trapped modes consist of finite energy localized in regions of maximal curvature. We present both an asymptotic model and numerical evidence to demonstrate the trapping. In the asymptotic analysis we utilize a dimensionless curvature as a small parameter, whereas the numerical model is based on spectral methods and is free of the small-curvature limitation. The two models agree with each other well in the region where both are applicable. Simple existence conditions depending on Poison's ratio are offered, and finally, the effect of energy build-up in a bend when the structure is excited at a resonant frequency is demonstrated.
The dynamic time-harmonic Green's tensor for a transversely isotropic homogeneous linearly elastic solid is studied. First, the representation of the Green's tensor as an integral over the unit sphere is obtained. It consists of three parts: quasi-longitudinal (P), shear-horizontal (SH) and quasi-shear (SV). Then, an exact analytical solution for the SH part in terms of elementary functions is derived. The complete far-eld asymptotic approximation of P and SV parts is obtained next, using the uniform stationary phase method. For the P wave it involves the leading term of the ray series since there is only one arrival of this wave. The wave surface for the SV wave contains conical points and cuspidal edges. The asymptotic description applicable near these singular directions is derived involving the Airy and Bessel functions. The directions close to the points of tangential contact of the SH and SV sheets of the wave surface are also treated. At the end of the paper numerical results in both frequency and time domain are presented. They show that the agreement between the outputs of the asymptotic and direct numerical codes is very good throughout all regions but the former can be orders of magnitude faster.
The following paragraphs and equations contain typographical errors that have no consequence for any other equation or result in the above paper.-The first and second paragraphs of §2:The geometry of the problem is shown in figure 1. We consider a curved plate of constant thickness, 2h, which is made of a homogeneous and isotropic linearly elastic material. The density of the solid is r, and its Lamé constants are l and m. The geometry is two-dimensional, and an orthogonal curvilinear coordinate system (s, h) is adopted, where h is the signed shortest distance from the observation point to the centerline of the waveguide, Kh%h%h, and s is the arclength along the centerline. The shape of the plate is characterized by the angle a between the tangent to the centerline and a fixed line (here we choose this to be the x -axis). Thus the curvature of the centerline is a s ; here, and throughout, the paper subscripts denote partial derivatives with respect to the corresponding variables. We assume that the curvature vanishes at infinity, a s (GN)Z0. In fact, the curvature should decay faster than 1/s at infinity, so that the full angle of the bend, a s Z Ð N
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