16,414 research outputs found
Effective wave numbers for thermo-viscoelastic media containing random configurations of spherical scatterers
No full textThe 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. London 91, 678–688 (1967)], 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.Peer reviewe
Effective shear speed in two-dimensional phononic crystals
No full textThe quasistatic limit of the antiplane shear-wave speed ('effective speed') c in 2D periodic lattices is studied. Two new closed-form estimates of c are derived by employing two different analytical approaches. The first proceeds from a standard background of the plane wave expansion (PWE). The second is a new approach, which resides in x-space and centers on the monodromy matrix (MM) introduced in the 2D case as the multiplicative integral, taken in one coordinate, of a matrix with components being the operators with respect to the other coordinate. On the numerical side, an efficient PWE-based scheme for computing c is proposed and implemented. The analytical and numerical findings are applied to several examples of 2D square lattices with two and three high contrast components, for which the new PWE and MM estimates are compared with the numerical data and with some known approximations. It is demonstrated that the PWE estimate is most efficient in the case of densely packed stiff inclusions, especially when they form a symmetric lattice, while in general it is the MM estimate that provides the best overall fitting accuracy.Peer reviewe
Integral identities for reflection, transmission and scattering coefficients
No full textSeveral integral identities related to acoustic scattering are presented. In each case the identity involves the integral over frequency of a physical quantity. For instance, the integrated transmission loss, a measure of the transmitted acoustic energy through an inhomogeneous layer, is shown to have a simple expression in terms of spatially averaged physical quantities. Known identities for the extinction cross section and for the acoustic energy loss in a slab with a rigid backing, are shown to be special cases of a general procedure for finding such integral identities.Peer reviewe
Elastodynamics of radially inhomogeneous spherically anisotropic elastic materials in the Stroh formalism
No full textA method is presented for solving elastodynamic problems in radially inhomogeneous elastic materials with spherical anisotropy, i.e. materials such that cijkl = cijkl(r) in a spherical coordinate system. The time harmonic displacement field u is expanded in a separation of variables form with dependence on 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 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 admit this type of separation of variables solutions for certain lower material symmetries. These results extend the Stroh formalism of elastodynamics in rectangular and cylindrical systems to spherical coordinates.Peer reviewedReceived July 29, 2011; accepted September 16, 2011; published online December 23, 2011. Print publication date February 2012. Manuscript dated September 14, 2011
Mechanics of elastic networks
No full textWe 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 three-dimensional space-filling lattices with Z=4, 6, 8, 12 and 14, the last being the ‘stiffest’ lattice proposed by Gurtner & Durand (Gurtner & Durand 2014 Proc. R. Soc. A 470, 20130611. (doi:10.1098/rspa.2013.0611)). The analytical expressions provide explicit formulae 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.Peer reviewe
Nonlinear shear wave interaction at a frictional interface: Energy dissipation and generation of harmonics
No full textAnalytical and numerical modelling of the nonlinear interaction of shear wave with a frictional interface is presented. The system studied is composed of two homogeneous and isotropic elastic solids, brought into frictional contact by remote normal compression. A shear wave, either time harmonic or a narrow band pulse, is incident normal to the interface and propagates through the contact. Two friction laws are considered and their influence on interface behavior is investigated : Coulomb's law with a constant friction coefficient and a slip-weakening friction law which involves static and dynamic friction coefficients. The relationship between the nonlinear harmonics and the dissipated energy, and their dependence on the contact dynamics (friction law, sliding and tangential stress) and on the normal contact stress are examined in detail. The analytical and numerical results indicate universal type laws for the amplitude of the higher harmonics and for the dissipated energy, properly non-dimensionalized in terms of the pre-stress, the friction coefficient and the incident amplitude. The results suggest that measurements of higher harmonics can be used to quantify friction and dissipation effects of a sliding interface.Peer reviewe
On the quasistatic effective elastic moduli for elastic waves in three-dimensional phononic crystals
No full textEffective elastic moduli for 3D solid-solid phononic crystals of arbitrary anisotropy and oblique lattice structure are formulated analytically using the plane-wave expansion (PWE) method and the recently proposed monodromy-matrix (MM) method. The latter approach employs Fourier series in two dimensions with direct numerical integration along the third direction. As a result, the MM method converges much quicker to the exact moduli in comparison with the PWE as the number of Fourier coefficients increases. The MM method yields a more explicit formula than previous results, enabling a closed-form upper bound on the effective Christoffel tensor. The MM approach significantly improves the efficiency and accuracy of evaluating effective wave speeds for high-contrast composites and for configurations of closely spaced inclusions, as demonstrated by three-dimensional examples.Peer reviewe
Analytical formulation of 3D dynamic homogenization for periodic elastic systems
No full textHomogenization of the equations of motion for a three dimensional periodic elastic system is considered. Expressions are obtained for the fully dynamic effective material parameters governing the spatially averaged fields by using the plane wave expansion (PWE) method. The effective equations are of Willis form [1] with coupling between momentum and stress and tensorial inertia. The formulation demonstrates that the Willis equations of elastodynamics are closed under homogenization. The effective material parameters are obtained for arbitrary frequency and wavenumber combinations, including but not restricted to Bloch wave branches for wave propagation in the periodic medium. Numerical examples for a 1D system illustrate the frequency dependence of the parameters on Bloch wave branches and provide a comparison with an alternative dynamic effective medium theory [2] which also reduces to Willis form but with different effective moduli.Peer reviewedReceived by the journal publisher November 28, 2011; accepted January 13, 2012; published online April 25, 2012. Manuscript dated January 16, 2012
Hyperelastic cloaking theory: Transformation elasticity with pre-stressed solids
No full textTransformation elasticity, by analogy with transformation acoustics and optics, converts material domains without altering wave properties, thereby enabling cloaking and related effects. By noting the similarity between transformation elasticity and the theory of ncremental motion superimposed on finite pre-strain it is shown that the constitutive parameters of transformation elasticity correspond to the density and moduli of small-on-large theory. The formal equivalence indicates that transformation elasticity can be achieved by selecting a particular finite (hyperelastic) strain energy function, which for isotropic elasticity is semilinear strain energy. The associated elastic transformation is restricted by the requirement of statically quilibrated pre-stress. This constraint can be cast as trF = constant, where F is the deformation gradient, subject to symmetry constraints, and its consequences are explored both analytically and through numerical examples of cloaking of anti-plane and in-plane wave motion.Peer reviewe
Multiple scattering by cylinders immersed in fluid: high order approximations for the effective wavenumbers
No full textAcoustic 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 (n0) of scatterers, in this paper the higher order dependence of the coherent wavenumber on n0 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 n0, to higher orders. Explicit formulas are provided for corrections to the effective wavenumber up to O(n04). 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 other approaches.Peer reviewe
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