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.Comment: 39 pages, 2 figure

Nonlinear shear wave interaction at a frictional interface: Energy dissipation and generation of harmonics

Analytical 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.Comment: 17 pages, 10 figure

Effective speed of sound in phononic crystals

A new formula for the effective quasistatic speed of sound $c$ in 2D and 3D periodic materials is reported. The approach uses a monodromy-matrix operator to enable direct integration in one of the coordinates and exponentially fast convergence in others. As a result, the solution for $c$ has a more closed form than previous formulas. It significantly improves the efficiency and accuracy of evaluating $c$ for high-contrast composites as demonstrated by a 2D example with extreme behavior.Comment: 4 pages, 1 figur

Analytical formulation of 3D dynamic homogenization for periodic elastic systems

Homogenization 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 (Willis 1997) 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 (Shuvalov 2011) which also reduces to Willis form but with different effective moduli.Comment: 24 pages, 4 figure

Spectral properties of a 2D scalar wave equation with 1D-periodic coefficients: application to SH elastic waves

The paper provides a rigorous analysis of the dispersion spectrum of SH (shear horizontal) elastic waves in periodically stratified solids. The problem consists of an ordinary differential wave equation with periodic coefficients, which involves two free parameters $\omega$ (the frequency) and $k$ (the wavenumber in the direction orthogonal to the axis of periodicity). Solutions of this equation satisfy a quasi-periodic boundary condition which yields the Floquet parameter $K$. The resulting dispersion surface $\omega (K,k)$ may be characterized through its cuts at constant values of $K, k$ and $\omega$ that define the passband (real $K$) and stopband areas, the Floquet branches and the isofrequency curves, respectively. The paper combines complementary approaches based on eigenvalue problems and on the monodromy matrix $\mathbf{M}$. The pivotal object is the Lyapunov function $\Delta (\omega ^{2},k^{2}) \equiv 1/2\mathrm{trace}\mathbf{M}=\cos K$ which is generalized as a function of two variables. Its analytical properties, asymptotics and bounds are examined and an explicit form of its derivatives obtained. Attention is given to the special case of a zero-width stopband. These ingredients are used to analyze the cuts of the surface $\omega (K,k).$ The derivatives of the functions $\omega (k)$ at fixed $K$ and $\omega (K)$ at fixed $k$ and of the function $K(k)$ at fixed $\omega$ are described in detail. The curves $\omega (k)$ at fixed $K$ are shown to be monotonic for real $K,$ while they may be looped for complex $K$ (i.e. in the stopband areas). The convexity of the closed (first) real isofrequency curve $K(k)$ is proved thus ruling out low-frequency caustics of group velocity. The results are relevant to the broad area of applicability of ordinary differential equation for scalar waves in 1D phononic (solid or fluid) and photonic crystals.Comment: 35 pages, 4 figure