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Andrew N. Norris

Publications and source records attributed to Andrew N. Norris.

At least 37 records · Page 2Linked to original sources
arXiv2017

Broadband focusing of underwater sound using a transparent pentamode lens

We report an inhomogeneous acoustic metamaterial lens based on spatial variation of refractive index for broadband focusing of underwater sound. The index gradient follows a modified hyperbolic secant profile designed to reduce aberration and suppress side lobes. The gradient index (GRIN) lens is comprised of transversely isotropic hexagonal microstructures with tunable quasi-static bulk modulus and mass density. In addition, the unit cells are impedance-matched to water and have in-plane shear modulus negligible compared to the effective bulk modulus. The flat GRIN lens is fabricated by cutting hexagonal centimeter scale hollow microstructures in aluminum plates, which are then stacked and sealed from the exterior water. Broadband focusing effects are observed within the homogenization regime of the lattice in both finite element (FEM) simulations and underwater measurements (20-40 kHz). This design approach has potential applications in medical ultrasound imaging and underwater acoustic communications.

physics.class-ph↗
arXiv2017

An inequality for longitudinal and transverse wave attenuation coefficients

Total absorption, defined as the net flux of energy out of a bounded region averaged over one cycle for time harmonic motion, must be non-negative when there are no sources of energy within the region. This passivity condition places constraints on the non-dimensional absorption coefficients of longitudinal and transverse waves, $γ_L$ and $γ_T$, in isotropic linearly viscoelastic materials. Typically, $γ_L, \, γ_T$ are small, in which case the constraints imply that coefficients of attenuation per unit length, $α_L$, $α_T$, must satisfy the inequality ${α_L}/{α_T} \ge { 4c_{T}^3} / { 3c_{L}^3}$ where $c_L$, $c_T$ are the wave speeds. This inequality, which as far as the author is aware, has not been presented before, provides a relative bound on wave speed in terms of attenuation, or {\it vice versa}. It also serves as a check on the consistency of ultrasonic measurements from the literature, with most but not all of the data considered passing the positive absorption test.

physics.class-ph↗
arXiv2016

A high transmission broadband gradient index lens using elastic shell acoustic metamaterial elements

The use of cylindrical elastic shells as elements in acoustic metamaterial devices is demonstrated through simulations and underwater measurements of a cylindrical-to-plane wave lens. Transformation acoustics (TA) of a circular region to a square dictates that the effective density in the lens remain constant and equal to that of water. Piecewise approximation to the desired effective compressibility is achieved using a square array with elements based on the elastic shell metamaterial concept developed in [30]. The size of the elements are chosen based on availability of shells, minimizing fabrication difficulties. The tested device is neutrally buoyant comprising 48 elements of nine different types of commercial shells made from aluminum, brass, copper, and polymers. Simulations indicate a broadband range in which the device acts as a cylindrical to plane wave lens. The experimental findings confirm the broadband quadropolar response from approximately 20 to 40 kHz, with positive gain of the radiation pattern in the four plane wave directions.

physics.class-ph↗
arXiv2015

Enhanced acoustic transmission through a slanted grating

It is known that an acoustic wave incident on an infinite array of aligned rectangular blocks of a different acoustic material exhibits total transmission if certain conditions are met [1] which relate the unique "intromission" angle of incidence with geometric and material properties of the slab. This extraordinary acoustic transmission phenomenon holds for any slab thickness, making it analogous to a Brewster effect in optics, and is independent of frequency as long as the slab microstructure is sub-wavelength in the length-wise direction. Here we show that the enhanced transmission effect is obtained in a slab with grating elements oriented obliquely to the slab normal. The dependence of the intromission angle $θ_i$ is given explicitly in terms of the orientation angle. Total transmission is achieved at incidence angles $\pm θ_i$, with a relative phase shift between the transmitted amplitudes of the $+θ_i$ and $- θ_i$ cases. These effects are shown to follow from explicit formulas for the transmission coefficient. In the case of grating elements that are rigid the results have direct physical interpretation. The analytical findings are illustrated with full wave simulations.

physics.comp-ph↗
arXiv2015

Acoustic integrated extinction

The integrated extinction (IE) is defined as the integral of the scattering cross-section as a function of wavelength. Sohl et al. [1] derived an IE expression for acoustic scattering that is causal, i.e. the scattered wavefront in the forward direction arrives later than the incident plane wave in the background medium. The IE formula was based on electromagnetic results, for which scattering is causal by default. Here we derive a formula for the acoustic IE that is valid for causal and non-causal scattering. The general result is expressed as an integral of the time dependent forward scattering function. The IE reduces to a finite integral for scatterers with zero long-wavelength monopole and dipole amplitudes. Implications for acoustic cloaking are discussed and a new metric is proposed for broadband acoustic transparency.

physics.class-ph↗
arXiv2014

Mechanics of elastic networks

We consider a periodic lattice structure in $d=2$ or $3$ dimensions with unit cell comprising $Z$ thin elastic members emanating from a similarly situated central node. A general theoretical approach provides an algebraic formula for the effective elasticity of such frameworks. The method yields the effective cubic elastic constants for 3D space-filling lattices with $Z=4$, $6$, $8$, $12$ and $14$, the latter being the "stiffest" lattice proposed by Gurtner and Durand (2014). The analytical expressions provide explicit formulas for the effective properties of pentamode materials, both isotropic and anisotropic, obtained from the general formulation in the stretch dominated limit for $Z=d+1$.

cond-mat.mtrl-sci↗
arXiv2014

Tunable cylindrical shell as an element in acoustic metamaterial

Elastic cylindrical shells are fitted with an internal mechanism which is optimized so that, in the quasi-static regime, the combined system exhibits prescribed effective acoustic properties. The mechanism consists of a central mass supported by an axisymmetric distribution of elastic stiffeners. By appropriate selection of the mass and stiffness of the internal mechanism, the shell's effective acoustic properties (bulk modulus and density) can be tuned as desired. Subsonic flexural waves excited in the shell by the attachment of stiffeners are suppressed by including a sufficiently large number of such stiffeners. Effectiveness of the proposed metamaterial is demonstrated by matching the properties of a thin aluminum shell with a polymer insert to those of water. The scattering cross section in water is nearly zero over a broad range of frequencies at the lower end of the spectrum. By arranging the tuned shells in an array the resulting acoustic metamaterial is capable of steering waves. As an example, a cylindrical-to-plane wave lens is designed by varying the bulk modulus in the array according to the conformal mapping of a unit circle to a square.

physics.class-ph↗
arXiv2014

Acoustic scattering from an infinitely long cylindrical shell with an internal mass attached by multiple axisymmetrically distributed stiffeners

A thin infinitely long elastic shell is stiffened by $J$ in number identical lengthwise ribs distributed uniformly around the circumference and joined to a rod in the center. The 2D model of the substructure is a rigid central mass supported by $J$ axisymmetrically placed linear springs. The response of the shell-spring-mass system is quite different from a fluid filled shell or that of a solid cylinder due to the discrete number of contact points which couple the displacement of the shell at different locations. Exterior acoustic scattering due to normal plane wave incidence is solved in closed form for arbitrary $J$. The scattering matrix associated with the normal mode solution displays a simple structure, composed of distinct sub-matrices which decouple the incident and scattered fields into $J$ families. The presence of a springs-mass substructure causes resonances which are shown to be related to the subsonic shell flexural waves, and an approximate analytic expression is derived for the quasi-flexural resonance frequencies. Numerical simulations indicate that the new solution for $J\ge 3$ springs results in a complicated scattering response for plane wave incidence. As the number of springs becomes large enough, the total scattering cross-section is asymptotically zero at low frequencies and slightly increased compared to the empty shell at moderate frequencies due to the added stiffness and mass. It is also observed that the sensitivity to the angle of incidence diminishes as the number of springs is increased. This system can be tuned by selecting the shell thickness, spring stiffness and added mass to yield desired quasi-static effective properties making it a candidate element for graded index sonic crystals.

physics.class-ph↗
arXiv2012

Stable methods to solve the impedance matrix for radially inhomogeneous cylindrically anisotropic structures

A stable approach for integrating the impedance matrix in cylindrical, radial inhomogeneous structures is developed and studied. A Stroh-like system using the time-harmonic displacement-traction state vector is used to derive the Riccati matrix differential equation involving the impedance matrix. It is found that the resulting equation is stiff and leads to exponential instabilities. A stable scheme for integration is found in which a local expansion is performed by combining the matricant and impedance matrices. This method offers a stable solution for fully anisotropic materials, which was previously unavailable in the literature. Several approximation schemes for integrating the Riccati equation in cylindrical coordinates are considered: exponential, Magnus, Taylor series, Lagrange polynomials, with numerical examples indicating that the exponential scheme performs best. The impedance matrix is compared with solutions involving Buchwald potentials in which the material is limited to piecewise constant transverse isotropy. Lastly a scattering example is considered and compared with the literature.

math-ph↗
arXiv2012

Active elastodynamic cloaking

An active elastodynamic cloak destructively interferes with an incident time harmonic in-plane (coupled compressional/shear) elastic wave to produce zero total elastic field over a finite spatial region. A method is described which explicitly predicts the source amplitudes of the active field. For a given number of sources and their positions in two dimensions it is shown that the multipole amplitudes can be expressed as infinite sums of the coefficients of the incident wave decomposed into regular Bessel functions. Importantly, the active field generated by the sources vanishes in the far-field. In practice the infinite summations are clearly required to be truncated and the accuracy of cloaking is studied when the truncation parameter is modified.

math-ph↗
arXiv2012

Special transformations for pentamode acoustic cloaking

The acoustic cloaking theory of Norris (2008) permits considerable freedom in choosing the transformation function f from physical to virtual space. The standard process for defining cloak materials is to first define f and then evaluate whether the materials are practically realizable. In this paper, this process is inverted by defining desirable material properties and then deriving the appropriate transformations which guarantee the cloaking effect. Transformations are derived which result in acoustic cloaks with special properties such as 1) constant density 2) constant radial stiffness 3) constant tangential stiffness 4) power-law density 5) power-law radial stiffness 6) power-law tangential stiffness 7) minimal elastic anisotropy.

cond-mat.mtrl-sci↗
arXiv2012

Source amplitudes for active exterior cloaking

The active cloak comprises a discrete set of multipole sources that destructively interfere with an incident time harmonic scalar wave to produce zero total field over a finite spatial region. For a given number of sources and their positions in two dimensions it is shown that the multipole amplitudes can be expressed as infinite sums of the coefficients of the incident wave decomposed into regular Bessel functions. The field generated by the active sources vanishes in the infinite region exterior to a set of circles defined by the relative positions of the sources. The results provide a direct solution to the inverse problem of determining the source amplitudes. They also define a broad class of non-radiating discrete sources.

math-ph↗
arXiv2012

Employing pre-stress to generate finite cloaks for antiplane elastic waves

It is shown that nonlinear elastic pre-stress of neo-Hookean hyperelastic materials can be used as a mechanism to generate finite cloaks and thus render objects near-invisible to incoming antiplane elastic waves. This approach appears to negate the requirement for special cloaking metamaterials with inhomogeneous and anisotropic material properties in this case. These properties are induced naturally by virtue of the pre-stress. This appears to provide a mechanism for broadband cloaking since dispersive effects due to metamaterial microstructure will not arise.

physics.class-ph↗
arXiv2011

Effective wave numbers for thermo-viscoelastic media containing random configurations of spherical scatterers

The dispersion relation is derived for the coherent waves in fluid or elastic media supporting viscous and thermal effects and containing randomly distributed spherical scatterers. The formula obtained is the generalization of Lloyd and Berry's [Proc. Phys. Soc. Lond. 91, 678-688, 1067], the latter being limited to fluid host media, and it is the three-dimensional counterpart of that derived by Conoir and Norris [Wave Motion 47, 183-197, 2010] for cylindrical scatterers in an elastic host medium.

math-ph↗
arXiv2010

Multiple scattering by cylinders immersed in fluid: high order approximations for the effective wavenumbers

Acoustic wave propagation in a fluid with a random assortment of identical cylindrical scatterers is considered. While the leading order correction to the effective wavenumber of the coherent wave is well established at dilute areal density ($n_0 $) of scatterers, in this paper the higher order dependence of the coherent wavenumber on $n_0$ is developed in several directions. Starting from the quasi-crystalline approximation (QCA) a consistent method is described for continuing the Linton and Martin formula, which is second order in $n_0$, to higher orders. Explicit formulas are provided for corrections to the effective wavenumber up to O$(n_0^4)$. Then, using the QCA theory as a basis, generalized self consistent schemes are developed and compared with self consistent schemes using other dynamic effective medium theories. It is shown that the Linton and Martin formula provides a closed self-consistent scheme, unlike some other approaches.

cond-mat.mtrl-sci↗
arXiv2010

Acoustic metafluids made from three acoustic fluids

Significant reduction in target strength and radiation signature can be achieved by surrounding an object with multiple concentric layers comprised of three acoustic fluids. The idea is to make a finely layered shell with the thickness of each layer defined by a unique transformation rule. The shell has the effect of steering incident acoustic energy around the structure, and conversely, reducing the radiation strength. The overall effectiveness and the precise form of the layering depends upon the densities and compressibilities of the three fluids. Nearly optimal results are obtained if one fluid has density equal to the background fluid, while the other two densities are much greater and much less than the background values. Optimal choices for the compressibilities are also found. Simulations in 2D and 3D illustrate effectiveness of the three fluid shell. The limited range of acoustic metafluids that are possible using only two fluid constituents is also discussed.

cond-mat.mtrl-sci↗
arXiv2010

Wave impedance matrices for cylindrically anisotropic radially inhomogeneous elastic solids

Impedance matrices are obtained for radially inhomogeneous structures using the Stroh-like system of six first order differential equations for the time harmonic displacement-traction 6-vector. Particular attention is paid to the newly identified solid-cylinder impedance matrix ${\mathbf Z} (r)$ appropriate to cylinders with material at $r=0$, and its limiting value at that point, the solid-cylinder impedance matrix ${\mathbf Z}_0$. We show that ${\mathbf Z}_0$ is a fundamental material property depending only on the elastic moduli and the azimuthal order $n$, that ${\mathbf Z} (r)$ is Hermitian and ${\mathbf Z}_0$ is negative semi-definite. Explicit solutions for ${\mathbf Z}_0$ are presented for monoclinic and higher material symmetry, and the special cases of $n=0$ and 1 are treated in detail. Two methods are proposed for finding ${\mathbf Z} (r)$, one based on the Frobenius series solution and the other using a differential Riccati equation with ${\mathbf Z}_0$ as initial value. %in a consistent manner as the solution of an algebraic Riccati equation. The radiation impedance matrix is defined and shown to be non-Hermitian. These impedance matrices enable concise and efficient formulations of dispersion equations for wave guides, and solutions of scattering and related wave problems in cylinders.

cond-mat.mtrl-sci↗