Quasi-static Formula Research Articles

Effect of thermal treatment on the mode I fracture toughness of granite under dynamic and static coupling load

In order to study the effects of temperature and loading rate on the dynamic fracture behaviour of granite under a given preload, dynamic fracture tests of pure mode-I were carried out on cracked straight-through Brazilian disc (CSTBD) specimens with a dynamic and static coupling test device based on the split Hopkinson pressure bar (SHPB) system. The experiments were performed at different loading rates and specimens were heat-treated at different temperatures. The thermal damage of specimens increased with ascending treatment temperature in general except at 100 °C, and the samples almost lost their load-bearing capacity after heat treatment at 800 °C. Based on the one-dimensional stress wave theory, under the condition of dynamic stress balance, the dynamic fracture toughness of the granite under a given static preload was obtained by substituting the average force on both ends of the CSTBD specimen into the quasi-static formula of the stress intensity factor of the central cracked disk and combining the crack initiation time measured by the strain gauge method. The results showed that the dynamic fracture toughness increases linearly with the loading rate after the same heat treatment, and decreases with increasing treatment temperature in general under the same loading rate, although the fracture toughness values of heat treatment at 100–200 °C and 400–600 °C are very close to each other. Partial correlation analysis was used to prove that the effect of loading rate on the dynamic fracture toughness of granite is greater than that of temperature. A high-speed camera was utilized to record the failure process of specimens. Most of the specimens were separated into two roughly identical halves, while a small number of specimens were broken into several blocks due to multiple crack growth. In addition, triangular crushed zones of different sizes are observed at the loading ends of all specimens. The results obtained in this paper are of great value for the analysis of the fracture characteristics and induced cracking of deep rocks.

A Quasi-Analytical Tool for the Characterization of Transmission Lines at High Frequencies [EM Programmer's Notebook

In this article, we present a freely accessible software tool that allows for fast characterization of dynamic phenomena in a wide variety of transmission lines that include characteristic impedance, effective dielectric constant, and losses, such as radiation into space and surface waves. For printed transmission lines, the radiation effects are of particular importance when the transverse dimensions of the transmission lines become significant in terms of wavelength. Generally, dispersion and losses of the line due to these dynamic phenomena are predicted by full-wave simulations as quasi-static formula do not suffice. The presented software tool, freely downloaded from http://terahertz.tudelft.nl, is capable of accurately analyzing the most widely used transmission lines at high frequencies.

High-frequency homogenization for layered hyperbolic metamaterials

We propose an analytical approach for calculation of the homogenized dielectric functions ${\ensuremath{\epsilon}}_{\ensuremath{\parallel}}(\ensuremath{\omega})$ and ${\ensuremath{\epsilon}}_{\ensuremath{\perp}}(\ensuremath{\omega})$ of one-dimensional periodic metal-dielectric structure. The obtained formulas are valid at high frequencies near the points of topological transition from an elliptic to hyperbolic regime. The proposed method of high-frequency homogenization takes into account rapidly varying electromagnetic fields within the metallic component of a unit cell, in particular, the evanescent character of the plasmonic mode and oscillatory behavior of the waveguidelike modes. Our results show good correspondence to the exact solution of the Rytov's dispersion equation and significant deviation from the widely used quasistatic formulas obtained by spatial averaging along the direction of periodicity $z$ of $\ensuremath{\epsilon}(z)$ and $1/\ensuremath{\epsilon}(z)$. The quasistatic approach ignores $z$ dependence of the fields that leads to its limited applicability near the frequency of topological transition.

Accurate 1D forward and inverse modeling of high-frequency helicopter-borne electromagnetic data

Frequency-domain helicopter-borne electromagnetic (HEM) data are commonly interpreted using quasistatic approximations in forward and inverse solutions. At high-frequencies ([Formula: see text]), the accuracy of this approach is often insufficient. Implementation of a full solution of the forward problem that includes displacement currents in both subsurface and air, however, may cause singularity problems during numerical evaluation of the secondary field integral using Hankel or Laplace transforms, particularly at high frequencies. These effects can be reduced by a wavenumber shift during numerical evaluation, a transformation of the integral by partial integration, or a combination of both. Based on these corrections, the resulting forward responses obtained with various fast Hankel transforms or numerical Laplace transforms are nearly identical. The inversion of HEM data also requires some modification. For a half-space inversion, the parameter sets, such as look-up tables, linear or polynomial approximations have to be modified for frequencies above some 10 kHz. The quasistatic formulas used in multi-layer inversion procedures have to be replaced by the accurate formulas. Fortunately, singularity problems are restricted to the forward calculation as the derivatives of the Jacobian matrix are not affected. Accurate high-frequency forward and inverse modeling enables a successful usage of helicopter-borne electromagnetics for near-surface investigations and expands the range of applications in environmental and geotechnical issues.

Piecewise static Hamiltonian for an atom in a strong laser field

We show that it is possible to use a piecewise constant Hamiltonian to describe the main features of the dynamics of an atom interacting with a laser field. In particular we show that using this approximation we are able to give a good description of the ionization signal, of the HHG spectra and of the attosecond pulses generated by the radiating electron. Finally, we give an explicit formula to evaluate the ionization rate in the time dependent laser field. This formula, which is a generalization of the Landau formula for the ionization rate of an atom in a static electric field, fairly well reproduces the numerical ionization rates for a broad range of laser frequency and intensity. The main advantage of this formula is that it can be used well beyond the limits of the quasi-static formula for the ionization rate of an atom.

Lorentz model for metamaterials: Optical frequency resonance circuits

We illustrate the strong connection between optics and high-frequency electronics by presenting circuit calculations for resonant metamaterials in the optical frequency range. An $RLC$ circuit model is considered. Equations for impedances are derived and used to calculate the resonance positions and quality factors of a variety of samples. These calculations are not quasistatic and include interactions between sites. The calculated resonance values show good agreement with simulations and experiments. Single-site quasistatic formulas are derived from the more general equations.

A finite element analysis for using Brazilian disk in split Hopkinson pressure bar to investigate dynamic fracture behavior of brittle polymer materials

An experimental method of using the Brazilian disk specimen in the split Hopkinson pressure bar (SHPB) apparatus is proposed to perform dynamic fracture testing of brittle materials. The average force on the specimen, which is obtained according to the one-dimensional elastic wave propagation theory, is adopted to determine the dynamic stress intensity factor (DSIF) by extending the quasi-static formula of the stress intensity factor (SIF) to impact loading condition. A finite element model for the Brazilian disk-SHPB dynamic fracture testing system is established. The numerical analysis confirms the validity of the one-dimensional experimental measuring principle. The effects of rise time of incident stress pulse and specimen geometry are discussed. It is shown that there exists a matching relation between the specimen and the pressure bar system.

An exact TEM calculation of loss in a stripline of arbitrary dimensions

Based on a modification of a conformal mapping originally introduced by R.H.T. Bates (1956), an exact expression for the quasi-static conductive attenuation constant alpha /sub c/, of a general stripline with arbitrary dimensions is obtained. The method is based on a quasi-static formula for alpha /sub c/ which assumes a transverse electromagnetic (TEM) mode and is thus restricted to the case where the skin depth is small compared to the strip thickness. The final formula for alpha /sub c/ involves an integral of only a smoothly varying function, since an analytic extraction of a term representing the singular behavior of the charge density is performed in the analysis. A comparison of the results obtained using the present method and the approximate formulas of S.B. Cohn (1955) shows that a maximum error of about 5-6% occurs in the Cohn formulas at a point where the two approximate formulas are joined. >

The thin-substrate approximation for reflection from the end of a slab-loaded parallel-plate waveguide with application to microstrip patch antennas

Integrals arising in the Wiener-Hopf solution for wide microstrip antennas are approximated in closed form for the case of an electrically thin substrate. The resulting expressions are used to obtain simple, closed-form expressions for resonant frequency and unloaded radiation Q for rectangular microstrip patch antennas which are valid when commonly used quasi-static formulas no longer hold. Numerical comparisons show significant shifts in resonant frequencies as compared with existing theories.

Note on an E-Plane Waveguide Step with Simultaneous Change of Media (Short Papers)

A simple quasi-static formula for the discontinuity susceptance of a 2:1 E-plane waveguide step with a simultaneous change of media is found using the theory of singular integral equations. An important difference from previous solutions by this method comes from the determination of the constants in the solution.

L'écoulement dans une tuyère de fusée accélérée (2e note)

The effect of rocket acceleration on the internal nozzle flow is investigated further by taking into account the effects of total mass variation. This leads to a time-dependant problem which is solved numericaly with the aid of the quasi-static formulas found in the first paper and based on a constant acceleration. These formulae are assumed to be valid for short time intervals. Acceleration effects could be noticeable during the final period of combustion.

Dynamic Potentials in Gyrotropic Plasma

Wave equations are derived for the scalar and vector potentials of guided electromagnetic waves propagating in a laterally bounded magnetized plasma along the axis of the static magnetic field. Solutions are given in the Coulomb and the Lorentz gauge. These solutions are used in a critical examination of the conceptual basis of the quasistatic theory because they allow convenient partitioning of the high-frequency electric field into a lamellar and a rotational part. In the quasistatic approximation the electric field is considered to be entirely lamellar, i.e., the gradient of a scalar potential that is not consistent with Maxwell's equations. To avoid inconsistencies the problem is formulated so that the dynamic scalar potential replaces the quasistatic potential. The rotational part of the electric field is readily obtained as the time derivative of the vector potential. Exact quantitative comparison of two parts of the electric field is made possible in this way and the relative importance of each part is evaluated for a few typical cases of interest. There exist some waves with dispersion characteristics that can not be obtained with the quasistatic formula although the condition that the phase velocity be smaller than the velocity of light presumed to be necessary for the applicability of the quasistatic approximation is satisfied. In these waves the rotational part of the electric field is comparable or greater than the lamellar part. Conversely, in modes with dispersion characteristics that can be well approximated with the quasistatic formula even when the phase velocity exceeds the velocity of light the lamellar part of the electric field is much larger than the rotational part. It can be inferred that the essential element of the quasistatic theory is the assumption that the high-frequency electric field is predominantly lamellar rather than the condition that the phase velocity be smaller than the velocity of light and related criteria.