scholarly journals Effective Willis constitutive equations for periodically stratified anisotropic elastic media

2011 ◽  
Vol 467 (2130) ◽  
pp. 1749-1769 ◽  
Author(s):  
A. L. Shuvalov ◽  
A. A. Kutsenko ◽  
A. N. Norris ◽  
O. Poncelet
Keyword(s):  
Constitutive Equations ◽  
Low Frequency ◽  
Elastic Media ◽  
Long Wavelength ◽  
Material Parameters ◽  
Effective Elastic Constants ◽  
Matrix Logarithm ◽  
Magnus Series ◽  
The Matrix ◽  
Anisotropic Elastic

A method to derive homogeneous effective constitutive equations for periodically layered elastic media is proposed. The crucial and novel idea underlying the procedure is that the coefficients of the dynamic effective medium can be associated with the matrix logarithm of the propagator over a unit period. The effective homogeneous equations are shown to have the structure of a Willis material, characterized by anisotropic inertia and coupling between momentum and strain, in addition to effective elastic constants. Expressions are presented for the Willis material parameters which are formally valid at any frequency and horizontal wavenumber as long as the matrix logarithm is well defined. The general theory is exemplified for scalar SH motion. Low frequency, long wavelength expansions of the effective material parameters are also developed using a Magnus series, and explicit estimates are derived for the rate of convergence.

scholarly journals Analysis of Wave Propagation in Infinite Piezoelectric Plates

2005 ◽  
Vol 21 (2) ◽  
pp. 103-108 ◽  
Author(s):  
C. Y. Wu ◽  
J. S. Chang ◽  
K. C. Wu
Keyword(s):  
Wave Propagation ◽  
Piezoelectric Materials ◽  
Electric Displacement ◽  
Low Frequency ◽  
Elastic Plates ◽  
Real Form ◽  
Long Wavelength ◽  
Anisotropic Elastic ◽  
Physical Quantities ◽  
Frequency Approximation

ABSTRACTAn analysis is presented for wave propagation in infinite homogeneous elastic plates of piezoelectric materials. The analysis is an extension to the work by Shuvalov [1] on wave propagation in general anisotropic elastic plates. A real form of dispersion equation is provided for a piezoelectric plate subjected to different boundary conditions on the plate surfaces. Perturbation theory [2] is exploited to obtain long-wavelength low-frequency approximation for physical quantities of wave propagation, including wave amplitude, stress, electric potential, electric displacement and velocity.

scholarly journals Tunable characteristics of low-frequency bandgaps in two-dimensional multivibrator phononic crystal plates under prestrain

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Hai-Fei Zhu ◽  
Xiao-Wei Sun ◽  
Ting Song ◽  
Xiao-Dong Wen ◽  
Xi-Xuan Liu ◽  
...  
Keyword(s):  
Phononic Crystal ◽  
Real Life ◽  
Low Frequency ◽  
Acoustic Vibration ◽  
Crystal Plate ◽  
Frequency Noise ◽  
Two Dimensional ◽  
The Matrix ◽  
Noise Frequency ◽  
Realtime Control

AbstractIn view of the influence of variability of low-frequency noise frequency on noise prevention in real life, we present a novel two-dimensional tunable phononic crystal plate which is consisted of lead columns deposited in a silicone rubber plate with periodic holes and calculate its bandgap characteristics by finite element method. The low-frequency bandgap mechanism of the designed model is discussed simultaneously. Accordingly, the influence of geometric parameters of the phononic crystal plate on the bandgap characteristics is analyzed and the bandgap adjustability under prestretch strain is further studied. Results show that the new designed phononic crystal plate has lower bandgap starting frequency and wider bandwidth than the traditional single-sided structure, which is due to the coupling between the resonance mode of the scatterer and the long traveling wave in the matrix with the introduction of periodic holes. Applying prestretch strain to the matrix can realize active realtime control of low-frequency bandgap under slight deformation and broaden the low-frequency bandgap, which can be explained as the multiple bands tend to be flattened due to the localization degree of unit cell vibration increases with the rise of prestrain. The presented structure improves the realtime adjustability of sound isolation and vibration reduction frequency for phononic crystal in complex acoustic vibration environments.

scholarly journals On Randomized Trace Estimates for Indefinite Matrices with an Application to Determinants

2021 ◽  
Author(s):  
Alice Cortinovis ◽  
Daniel Kressner
Keyword(s):  
Large Scale ◽  
Gaussian Process Regression ◽  
Likelihood Estimation ◽  
Random Vectors ◽  
Symmetric Positive Definite ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Trace Estimation ◽  
Scale Matrix ◽  
Existing Result

AbstractRandomized trace estimation is a popular and well-studied technique that approximates the trace of a large-scale matrix B by computing the average of $$x^T Bx$$ x T B x for many samples of a random vector X. Often, B is symmetric positive definite (SPD) but a number of applications give rise to indefinite B. Most notably, this is the case for log-determinant estimation, a task that features prominently in statistical learning, for instance in maximum likelihood estimation for Gaussian process regression. The analysis of randomized trace estimates, including tail bounds, has mostly focused on the SPD case. In this work, we derive new tail bounds for randomized trace estimates applied to indefinite B with Rademacher or Gaussian random vectors. These bounds significantly improve existing results for indefinite B, reducing the number of required samples by a factor n or even more, where n is the size of B. Even for an SPD matrix, our work improves an existing result by Roosta-Khorasani and Ascher (Found Comput Math, 15(5):1187–1212, 2015) for Rademacher vectors. This work also analyzes the combination of randomized trace estimates with the Lanczos method for approximating the trace of f(B). Particular attention is paid to the matrix logarithm, which is needed for log-determinant estimation. We improve and extend an existing result, to not only cover Rademacher but also Gaussian random vectors.

Weak-contrast edge and vertex diffractions in anisotropic elastic media

Wave Motion ◽  
1999 ◽  
Vol 29 (4) ◽  
pp. 363-373 ◽  
Author(s):  
Martin Tygel ◽  
Bjørn Ursin
Keyword(s):  
Elastic Media ◽  
Anisotropic Elastic

scholarly journals The Low-Frequency Array (LOFAR): Opening a New Window on the Universe

2002 ◽  
Vol 199 ◽  
pp. 474-483
Author(s):  
Namir E. Kassim ◽  
T. Joseph W. Lazio ◽  
William C. Erickson ◽  
Patrick C. Crane ◽  
R. A. Perley ◽  
...  
Keyword(s):  
Angular Resolution ◽  
Broad Band ◽  
Low Frequency ◽  
Electromagnetic Spectrum ◽  
Interferometric Imaging ◽  
Very Large Array ◽  
Long Wavelength ◽  
Calibration Techniques ◽  
Long Baselines ◽  
The Universe

Decametric wavelength imaging has been largely neglected in the quest for higher angular resolution because ionospheric structure limited interferometric imaging to short (< 5 km) baselines. The long wavelength (LW, 2—20 m or 15—150 MHz) portion of the electromagnetic spectrum thus remains poorly explored. The NRL-NRAO 74 MHz Very Large Array has demonstrated that self-calibration techniques can remove ionospheric distortions over arbitrarily long baselines. This has inspired the Low Frequency Array (LOFAR)—-a fully electronic, broad-band (15—150 MHz)antenna array which will provide an improvement of 2—3 orders of magnitude in resolution and sensitivity over the state of the art.

scholarly journals The perturbation of electromagnetic fields at distances that are large compared with the object's size

2014 ◽  
Vol 80 (3) ◽  
pp. 865-892 ◽  
Author(s):  
Paul D Ledger ◽  
William R B Lionheart
Keyword(s):  
Magnetic Fields ◽  
Electromagnetic Fields ◽  
Asymptotic Expansions ◽  
Low Frequency ◽  
Perfect Conductor ◽  
Small Object ◽  
Leading Order ◽  
Material Parameters ◽  
Electric And Magnetic Fields

Abstract We rigorously derive the leading-order terms in asymptotic expansions for the scattered electric and magnetic fields in the presence of a small object at distances that are large compared with its size. Our expansions hold for fixed wavenumber when the scatterer is a (lossy) homogeneous dielectric object with constant material parameters or a perfect conductor. We also derive the corresponding leading-order terms in expansions for the fields for a low-frequency problem when the scatterer is a non-lossy homogeneous dielectric object with constant material parameters or a perfect conductor. In each case, we express our results in terms of polarization tensors.

A State Variable Modeling of Plasticity and Necking Under Uniaxial Tension

1992 ◽  
Vol 114 (4) ◽  
pp. 378-383 ◽  
Author(s):  
G. Ferron ◽  
H. Karmaoui Idrissi ◽  
A. Zeghloul
Keyword(s):  
Strain Rate ◽  
Uniaxial Tension ◽  
Plastic Flow ◽  
Constitutive Equations ◽  
Growth Process ◽  
Constant Strain Rate ◽  
Uniaxial Tensile ◽  
Long Wavelength ◽  
State Variable ◽  
Diffusion Controlled

Constitutive equations based on a state variable modeling of the thermo-viscoplastic behavior of metals are discussed, and incorporated in an exact, long-wavelength analysis of the neck-growth process in uniaxial tension. The general formalism is specialized to the case of f.c.c. metals in the range of intragranular, diffusion controlled plastic flow. The model is shown to provide a consistent account of aluminum behavior both under constant strain-rate and creep. Calculated uniaxial tensile ductilities and rupture lives in creep are also compared with experiments.

Constitutive Equations of Power-Law Composites and Failure of Materials Having Multiple Cracks

1994 ◽  
Vol 116 (3) ◽  
pp. 359-366 ◽  
Author(s):  
S. C. Lin ◽  
Y. Hirose ◽  
T. Mura
Keyword(s):  
Power Law ◽  
Constitutive Equations ◽  
Fracture Criterion ◽  
Volume Fraction ◽  
Piecewise Linear ◽  
Pure Shear ◽  
Failure Criteria ◽  
Multiple Cracks ◽  
Critical Volume Fraction ◽  
The Matrix

Based upon the Mori-Tanaka method, the constitutive equations of power-law materials and the failure criteria of multiple cracks materials are investigated. The piecewise linear incremental approach is also employed to analyze the effective stress and strain of the power-law materials. Results are presented for the case of pure shear where the matrix is a power-law material with rigid or void inhomogeneities. For the multiple cracked materials, the Griffith fracture criterion is applied to determine the critical volume fraction which causes the catastrophic failure of a material. The failure criteria of penny shaped, flat ellipsoidal, and slit-like cracked materials are examined and it is found that the volume fraction of cracks and critical applied stress are in linear relation.

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