Time-harmonic Displacements Research Articles

Scattering of elastic waves by a cluster of cylindrical inclusions: multiple scattering analysis and equivalent scatterer modeling

Abstract The two-dimensional scattering of elastic waves by a cluster of aligned circular cylindrical elastic inclusions embedded in a circular cylindrical domain of an infinitely extended elastic medium is analysed numerically. The analysis is based on a semi-analytical method where the scattered waves are expressed as the eigenfunction expansions of the time-harmonic displacement potentials and the expansion coefficients are determined by a numerical collocation technique. The wave fields within and around the cluster are demonstrated for different spatial arrangements of inclusions in the cluster. It is also examined if the scattering characteristics of the cluster can be simulated by a single equivalent scatterer which has the effective homogenized properties of the matrix/inclusion composite. As a result, it is shown that for sufficiently small inclusion radius-wavelength ratios, the scattering characteristics of the cluster are relatively insensitive to the spatial inclusion arrangements therein, and can be reasonably simulated by those of the equivalent scatterer.

Thermoelastic damping in a thin circular transversely isotropic Kirchhoff–Love plate due to GN theory of type III

This research deals with the study of thermoelastic damping in transversely isotropic thin circular Kirchhoff–Love plate. Mathematical model is formed for time-harmonic displacement and temperature fields due to the GN theory of thermoelasticity of type III. The model is solved to obtain the expressions for thermoelastic damping, displacement, and temperature fields for circumferential surface wave modes for simply supported and clamped plate resonators. The numerical results for both the simply supported and clamped boundary conditions for an axisymmetric circular plate are illustrated. It is observed that the damping of vibrations is considerably influenced by surface wave modes. The thickness of the plate also has major effects on the vibrations of resonators. The computer-simulated results to illustrate the effect of circumferential surface wave modes on thermoelastic damping for the GN III theory of thermoelasticity have been depicted graphically.

Reconstruction of shear wave speed in layered media using time-harmonic excitation and pulse-echo imaging

Recent advances in high-frequency ultrasound imaging have opened new possibilities for quantitative characterization of skin properties. Noninvasive, accurate measurements of elasticity for multiple layers (e.g., stratum corneum, epidermis, dermis) are potentially beneficial for development and validation of new skin treatments. Here, pulse-echo ultrasound imaging using linear arrays is used to measure time-harmonic tissue displacements induced by a mechanical shaker. It is shown that reconstruction of time-harmonic displacement amplitude and phase in induced elastic waves is feasible for frame rates exceeding the Nyquist frequency and for echoes acquired using multiple transmits per frame. Measured wavefields are inverted using spatial Fourier analysis, numerical solution of the Helmholtz equation by finite difference methods, and direct observation of shear wavelengths. These approaches are validated using a 5 MHz diagnostic imaging array with a clinical scanner (Zonare z.one, L8-1 array) and multlilayered phantoms with varying elastic properties and echogenicity. Further verification of these methods is performed by comparing with finite-element simulations employing a linear elasticity model. Feasibility for measurement of skin layer elastic properties using high-frequency ultrasonic imaging is demonstrated using a 50 MHz linear array and a small-animal scanner (Vevo 2100, MS700 array) for measurements on porcine skin.Recent advances in high-frequency ultrasound imaging have opened new possibilities for quantitative characterization of skin properties. Noninvasive, accurate measurements of elasticity for multiple layers (e.g., stratum corneum, epidermis, dermis) are potentially beneficial for development and validation of new skin treatments. Here, pulse-echo ultrasound imaging using linear arrays is used to measure time-harmonic tissue displacements induced by a mechanical shaker. It is shown that reconstruction of time-harmonic displacement amplitude and phase in induced elastic waves is feasible for frame rates exceeding the Nyquist frequency and for echoes acquired using multiple transmits per frame. Measured wavefields are inverted using spatial Fourier analysis, numerical solution of the Helmholtz equation by finite difference methods, and direct observation of shear wavelengths. These approaches are validated using a 5 MHz diagnostic imaging array with a clinical scanner (Zonare z.one, L8-1 array) and multlilaye...

Revisiting the Jones Eigenproblem in Fluid-Structure Interaction

The Jones eigenvalue problem first described by D.S. Jones in 1983 concerns unusual modes in bounded elastic bodies: time-harmonic displacements whose tractions and normal components are both identically zero on the boundary. This problem is usually associated with a lack of unique solvability for certain models of fluid-structure interaction. The boundary conditions in this problem appear, at first glance, to rule out {\it any} non-trivial modes unless the domain possesses significant geometric symmetries. Indeed, Jones modes were shown to not be possible in most $C^\infty$ domains (see article by T. Harg\'e 1990). However, we should in this paper that while the existence of Jones modes sensitively depends on the domain geometry, such modes {\it do} exist in a broad class of domains. This paper presents the first detailed theoretical and computational investigation of this eigenvalue problem in Lipschitz domains. We also analytically demonstrate Jones modes on some simple geometries.

Acoustic wave propagation in a channel bifurcated by an elastic partition

Linear wave propagation in a narrow channel that is axially partitioned by a flexible membrane is examined. The enclosed fluid is excited by the time harmonic displacement in the channel cross-section. The axial variation in the acoustic impedance of the partition gives rise to the generation of evanescent modes in the channel. The effect of these evanescent modes on the vibration of the membrane is of particular interest. It is shown that in the limit of high channel aspect ratio one can model these modes by an effective source distribution along the surface of the membrane. The asymptotic analysis of the source distribution is presented. (NSF Grant 0841392)

Acoustic streaming in channel bifurcated by an elastic partition

Fluid motion in a narrow channel partitioned along its long axis by a flexible membrane is examined. The enclosed fluid is excited by the time harmonic displacement in the channel cross-section. The acoustic streaming that ensues and its impact of microfluidic transport is of particular interest in this work. A three-dimensional regularized Stokeslet based analysis is presented for fluid motion in the low streaming Reynolds number regime. The effect of frequency modulation on fluid motion is examined.

Elastodynamics of radially inhomogeneous spherically anisotropic elastic materials in the Stroh formalism

A method for solving elastodynamic problems in radially inhomogeneous elastic materials with spherical anisotropy is presented, i.e. materials having c ijkl = c ijkl ( r ) in a spherical coordinate system { r , θ , ϕ }. The time-harmonic displacement field u ( r , θ , ϕ ) is expanded in a separation of variables form with dependence on θ , ϕ described by vector spherical harmonics with r -dependent amplitudes. It is proved that such separation of variables solution is generally possible only if the spherical anisotropy is restricted to transverse isotropy (TI) with the principal axis in the radial direction, in which case the amplitudes are determined by a first-order ordinary differential system. Restricted forms of the displacement field, such as u ( r , θ ), admit this type of separation of variables solution for certain lower material symmetries. These results extend the Stroh formalism of elastodynamics in rectangular and cylindrical systems to spherical coordinates.

Modeling the quasicompressional wave field of a rectangular transducer in a transversely isotropic solid.

A fast model for simulating the transient quasicompressional wave field of a rectangular ultrasonic transducer directly coupled to a transversely isotropic elastic half-space of general orientation is developed. The so-called two-tier asymptotic approach and the uniform stationary phase method are used to derive the high-frequency asymptotics of time-harmonic displacements. Then, transient fields are modeled by means of harmonic synthesis. In geometrical regions, the formulas involve elementary and inside boundary layers, well-known special functions (Fresnel integral and generalized Fresnel integral), and are applicable in the radiating near field. The asymptotics elucidate the physics in terms of various arrivals and give explicit dependence of the radiated waves upon model parameters. The asymptotic code is tested against a direct numerical solution. It is at least a thousand times faster but describes accurately both arrival times and amplitudes of various pulses radiated by the transducer.

Radiation and scattering of waves on an elastic half-space; A non-commutative matrix Wiener-Hopf problem

Many problems in linear elastodynamics; or dynamic fracture mechanics; can be reduced to Wiener-Hopf functional equations defined in some region of a complex transform plane. Apart from a few simple cases; the inherent coupling between shear and compressional body motions gives rise to coupled systems of equations; and so the resulting Wiener-Hopf kernels are of matrix form. The key step in the solution of a Wiener-Hopf equation; which is to decompose the kernel into a product of two factors with particular analyticity properties; can be accomplished explicitly for scalar kernels. However; apart from special matrices which yield commutative factorizations; no procedure has yet been devised to exactly factorize general matrix kernels. It is the aim of this article to show that a new procedure for obtaining approximate factors of matrix kernels is applicable to the class of matrix kernels found in elasticity. This is performed; for ease of exposition; by way of a simple but non-trivial example: a linear elastic half-space; occupying the region y > 0; where ( x; y; z) are cartesian coordinates; has a free boundary on x > 0; y = 0; and has imposed time-harmonic displacements on x < 0; y = 0. Via the substitution of a scalar function in the kernel by its Padé approximant; an approximate solution to this boundary value problem is obtained explicitly for three different forcing cases. The approximate factorization technique described herein is simple to apply; is shown to converge to the exact result as the Padé number increases; and a maximum error bound is easily obtained. Numerical evaluation of the explicit results reveals that convergence to the exact solution is extremely rapid (i.e. only a small number of poles and zeros are required in the Padé approximant); which is confirmed by a global energy balance calculation; and so numerical calculations are extremely rapid. The outgoing cylindrically spreading compressional and shear wave coefficients are evaluated for several values of Poisson's ratio; v; and the outgoing Rayleigh wave is determined for all values of v (< 0.5).

On quasistatic problems of cracks in a non-homogeneous elastic layer

Simple solutions are given for plane and anti-plane problems of cracks in strips when the strip is made of two or more different elastic media. Cases considered are when time harmonic displacements are applied to the strip sides with a stationary crack, or when fixed displacements are applied to the strip sides and the crack moves uniformly. These results generalise in some respects results given recently by Matcynski [1], [2], [3].