494 research outputs found
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 appropriate
to cylinders with material at , and its limiting value at that point, the
solid-cylinder impedance matrix . We show that
is a fundamental material property depending only on the elastic moduli and the
azimuthal order , that is Hermitian and is
negative semi-definite. Explicit solutions for are presented
for monoclinic and higher material symmetry, and the special cases of and
1 are treated in detail. Two methods are proposed for finding , one based on the Frobenius series solution and the other using a
differential Riccati equation with 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 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 has a more closed form
than previous formulas. It significantly improves the efficiency and accuracy
of evaluating 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 (the frequency) and (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 . The resulting dispersion surface may be
characterized through its cuts at constant values of and that
define the passband (real ) 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 . The
pivotal object is the Lyapunov function 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 The derivatives of the functions at
fixed and at fixed and of the function at fixed
are described in detail. The curves at fixed are
shown to be monotonic for real while they may be looped for complex
(i.e. in the stopband areas). The convexity of the closed (first) real
isofrequency curve 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
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