Effect Of Higher-order Terms Research Articles

Effect of T-stress on the fracture of a four point bend specimen

Recent studies have shown that considering the effect of higher order terms and constant term in Williams Series in the fracture mechanics problems leads to better results. These terms play a significant role in determining the stress and strain fields and the energy around the crack tip which in turn can affect the fracture of the specimen predicted by different fracture criteria. In this paper, the effect of T-stress on the crack growth initiation angle in brittle materials and in elastic state is studied. For this purpose, a four point anti-symmetrical bend specimen, capable of creating mixed mode, is considered. The problem is analyzed with the aid of finite element method (FEM). The results are validated comparing them with the existing results in the literature. Finally, the effect of specimen geometry on the T-stress value and effect of T-stress on different fracture criteria will be assessed.

A model for the flow of gas mixtures in adsorption dominated dual-porosity reservoirs incorporating multi-component matrix diffusion—Part II numerical algorithm and application examples

In part one of this study a matrix-fracture transfer function was presented for multi-species gas migration in dual-porosity media. This second part presents the numerical implementation of this transfer function. Since the transfer function proposed in Part I is of an integral form with history-dependency and the resultant flow equations become a set of integro-(partial) differential equations, an algorithm is developed here for these equations. The algorithm is formulated using a step-by-step finite difference procedure combined with a fully implicit quadrature scheme. The quadrature scheme is based on the modified trapezoidal method and is designed for variable time-step size. The transfer function and the algorithm developed are incorporated into the commercial gas reservoir simulator SIMED II, and three examples are illustrated here. Example 1 demonstrates the performance and feasibility of the algorithm in applications, and Example 2 discusses the effect of higher order terms as well as its influence on flow profiles. The impact of mutual-diffusivity to flow profiles and well performance is illustrated in Example 3.

Exact relativistic tritium β-decay endpoint spectrum in a hadron model

We present the relativistic calculation of the \ensuremath{\beta}-decay of tritium in a hadron model. The elementary particle treatment (EPT) of the transition $^{3}\mathrm{H}$ $\ensuremath{\rightarrow}$$^{3}\mathrm{He}$ $+{e}^{\ensuremath{-}}+{\overline{\ensuremath{\nu}}}_{e}$ is performed in analogy with the description of the \ensuremath{\beta}-decay of neutron. The effects of higher order terms of hadron current and nuclear recoil are taken into account in this formalism. The relativistic Kurie function is derived and presented in a simple form suitable for the determination of neutrino masses from the shape of the endpoint spectrum. A connection with the commonly used Kurie function is established.

Delbrück scattering in combined Coulomb and laser fields

We study Delbr\"uck scattering in a Coulomb field in the presence of a laser field. The amplitudes are calculated in the Born approximation with respect to the Coulomb field and exactly in the parameters of the laser field having arbitrary strength, spectral content, and polarization. The case of high initial photon energy is investigated in detail for a monochromatic circularly polarized laser field. It is shown that the angular distribution of the process substantially differs from that for Delbr\"uck scattering in a pure Coulomb field. The value of the cross section under discussion may exceed the latter at realistic laser parameters that essentially simplify the possibility of the experimental observation of the phenomenon. The effect of high order terms in the quantum intensity parameter $\ensuremath{\chi}$ of the laser field is found to be very important already at relatively small $\ensuremath{\chi}$.

HIGH BIT RATE DENSE DISPERSION MANAGED OPTICAL COMMUNICATION SYSTEMS WITH DISTRIBUTED AMPLIFICATION

In this paper we have investigated optical pulse propa- gation in a dense dispersion managed (DM) optical communication system operating at a speed of 100 Gb/s and more taking into account of the effects of third order dispersion, intra-pulse Raman scattering and self steepening. Using perturbed variational formulation, we have obtained several ordinary differential equations for various pulse pa- rameters. These equations have been solved numerically to identify launching criteria in the first DM cell of the system. Full numerical simulation of the nonlinear Schrodinger equation has been employed to identify effects of higher order terms on pulse propagation and to in- vestigate the intra-pulse interaction. The roles played by these higher order linear and nonlinear effects have been identified. It has been found that the shift of the pulse centre due to intra-pulse Raman scat- tering increases with the increase in the distance of propagation and average dispersion. We have noticed that for higher value of average dispersion pulses travel less distance before collision than for lower average dispersion.

Micromagnetic Study of the Complex Transverse Susceptibility of Uniaxial Ferromagnets With Quartic Anisotropy

A micromagnetic model based on critical curves formalism for the complex transverse susceptibility (TS) of particulate media, for very low values of the frequency of the ac field, is presented. The effect of higher order terms in the uniaxial anisotropy energy series expansion and the effect of the amplitude of excitatory ac field are taken into account. When second order term is present in the uniaxial anisotropy energy ground state can change from easy-axis to easy-cone type and a complex energy profile is generated. It is shown that additional Barkhausen jumps driven by higher-order anisotropy constants can give rise to an imaginary part of TS

Dissipative Waves in a Relaxing Gas Exhibiting Mixed Nonlinearity

We consider a weakly dissipative quasilinear system of PDEs, governingthe unsteady one-dimensional motion of a relaxing gas, and investigatethe effects ofhigher order terms and the source term, present in the differentialsystem, on the wave motion associated with it. The undisturbed stateof the medium, which is assumed to be uniform and at restadmits mixed nonlinearity, i.e., the quadratic nonlinearity parameterΓ inherent in the system changes sign depending on the basestate. The method of multiple scales is employed to determine theevolution equation governing small perturbations to the state whereΓ is of O(∈), and the effects of nonlinearity arenoticeable over times of order O(∈−2).

F(R) Cosmology in the First Order Formalism

In the present work we consider those theories that are obtained from a Lagrangian density ℒ T (R) = f(R)√{-g} + ℒ M , that depends on the curvature scalar and a matter Lagrangian that does not depend on the connection, and apply Palatini's method to obtain the field equations. We start with a brief discussion of the field equations of the theory and apply them to a cosmological model described by the FRW metric. Then, we introduce an auxiliary metric to put the resultant equations into the form of GR with cosmological constant and coupling constant that are curvature depending. We show that we reproduce known results for the quadratic case. We find relations among the present values of the cosmological parameters q 0, H 0, $$\mathop {(G/G)}\limits^ \circ _0$$ and $$\mathop {(G/G)}\limits^{ \circ \circ } _0 $$ . Next we use a simple perturbation scheme to find the departure in angular diameter distance with respect to General Relativity. Finally, we use the observational data to estimate the order of magnitude of what is essentially the departure of f(R) from linearity. The bound that we find for f″ (0) is so huge that permit almost any f(R). This is in the nature of things: the effect of higher order terms in f(R) are strongly suppressed by power of Planck's time 8πG 0. In order to improve these bounds more research on mathematical aspects of these theories and experimental consequences is necessary.

CP and flavour in effective type I string models

Effective type I string models allow stabilization of the dilaton and moduli fields with only a single gaugino condensate. We show that, as well as breaking supersymmetry, the stabilization can spontaneously break CP. We find that this source of CP violation hints strongly at a natural solution to the supersymmetric CP and flavour problems. Even though the CP violation generates physical phases in the Yukawa couplings, all the supersymmetry-breaking terms are found to be automatically real and given by the U(1) charges of the associated Yukawa couplings. These can be chosen to have a structure (degenerate or non-universal) which suppresses FCNCs and EDMs. We examine the phenomenological implications, including the generation of the μ-term, and the effect of higher-order terms.

Wedge corner stress behaviour of bonded dissimilar materials

A contour integral, based on Betti’s reciprocal theorem, is used in conjunction with the finite element method to evaluate the magnitude of the wedge corner stress intensities associated with the higher order terms of the singular stress field near the interface corner of a bi-material joint. It is shown that using a different auxiliary field can eliminate the dependence of the wedge corner stress intensity on the integration path observed by [W.C. Carpenter, Int. J. Fracture 73 (1995) 93–108]. Finite element analysis of a typical joint geometry is used to demonstrate the path-independence of the magnitude of the stress intensities evaluated using the proposed method, and to show the effects of higher order terms on the stress state near the interface corner.

GEOMETRIC STIFFNESS AND STABILITY OF RIGID BODY MODES

The objective of this study is to examine the effect of geometric stiffness forces on the stability of elastic and rigid body modes. A simple rotating beam model is used to demonstrate the effect of axial forces and dynamic coupling between the modes of displacement on the rigid body motion. The effect of longitudinal deformation due to bending is systematically introduced to the dynamic equations using the principle of virtual work. The effect of higher order terms in the inertia forces as the result of including longitudinal displacement caused by bending deformation is examined using several models. One of these models is a linear model is which the effect of longitudinal displacement due to bending is neglected in formulating the inertia forces, but this effect is considered when the elastic forces are formulated. This model shows unstable behavior at high values of the angular velocity of the beam. Three different beam models are then developed in order to examine the effect of geometric stiffness forces. In the first model, called theconsistent complete model(CCM), the effect of longitudinal displacement caused by bending is included in formulating both the inertia and elastic forces. In the second model, called theconsistent incomplete model(CIM), the effect of longitudinal displacement due to bending is neglected in formulating both the elastic and inertia forces. In the third model, thesecond inconsistent model(SIM), the effect of longitudinal displacement due to bending is included in formulating the inertia forces, but this effect is neglected when the elastic forces are formulated. Numerical results obtained in this investigation demonstrate that the three models lead to a stable solution at high values of angular velocities. These results also demonstrate that including the effect of longitudinal displacement due to bending in the inertia forces is not the only approach that can be used to maintain the beam stability at high values of angular velocity. The effect of geometric stiffness forces on the stability of rigid body modes of a translating and rotating beam model is also examined in this paper.

A New Technique Based on the Transformation of Variables for Nonlinear Drift and Rossby Vortices

The quasi-two-dimensional nonlinear equations for drift and Rossby vortices have some stationary multipole solutions, and especially the dipole vortex solution is called modon. These solutions are valid only in the lowest order where the fluid velocity has a stream function. In order to investigate features of the multipole solutions more accurately, the effect of the higher order terms, for example the polarization drift in a plasma or the Coriolis force in a rotating planet, needs to be considered. It is shown that the higher order analysis through a new technique based on a transformation of variables is much easier than a straightforward iteration. The solutions in this analysis are obtained by inverse transformation to the original coordinates, where the profiles of potentials are distorted by the effects of higher order terms.

Goldstone theorem and diquark confinement beyond rainbow-ladder approximation

The quark Dyson-Schwinger equation and meson Bethe-Salpeter equation are studied in a truncation scheme that extends the rainbow-ladder approximation such that, in the chiral limit, the isovector, pseudoscalar meson remains massless. Repulsive contributions, which appear at higher order in the Bethe-Salpeter kernel, eliminate the bound state pole in the quark-quark (diquark) channel that is present in rainbow-ladder approximation. The net effect of higher order terms on the meson bound state masses is small.

Surface anisotropy and spin-reorientation transitions in ultrathin magnetic films

A micromagnetic theory is presented for spin-reorientation transitions in ultrathin magnetic films with normal surface anisotropy. Both discrete and continuum models are investigated and coincidence is found between the results derived from them. Two continuous transitions in spin orientation, from uniform and normal to canting, and then to uniform and in-plane, are observed as the film thickness is increased. The effects of higher-order terms in the expansions of the anisotropies compared to the dominant terms are also considered. A method for evaluating the surface anisotropy is formulated. The relation between the present micromagnetic theory and the phenomenological theory is discussed.

Application of Stochastic Finite-Element Method to Deflection Analysis of Pavement Structures

The effect of random variations in layer properties of a pavement-subgrade system on surface deflections is investigated. This is done by using a stochastic finite-element framework based on the perturbation method. The Taylor's expansion scheme is used. The variations of the surface deflections are related to the variations of layer moduli via a coefficient matrix, thereby allowing an investigation of the sensitivity of measured deflections to the variation of pavement and subgrade moduli. The analyses confirm that surface deflections are not very sensitive to variations of modulus within the pavement structure. Using the Boussinesq-Odemark approximation, it is concluded that, for the sample problem analyzed, the effect of higher-order terms on variation estimates of displacement is negligible when the coefficient of variation of subgrade modulus is less than 15 percent. The effect of inertia on the coefficient of variation of displacement due to variations in layer moduli is also briefly addressed. It is demonstrated that the variation in displacement under the load is sensitive to both the loading frequency and the equivalent mass of the pavement-subgrade system, particularly in the vicinity of the natural frequency of the system.

Application of Stochastic Finite-Element Method to Deflection Analysis of Pavement Structures

The effect of random variations in layer properties of a pavement-subgrade system on surface deflections is investigated. This is done by using a stochastic finite-element framework based on the perturbation method. The Taylor's expansion scheme is used. The variations of the surface deflections are related to the variations of layer moduli via a coefficient matrix, thereby allowing an investigation of the sensitivity of measured deflections to the variation of pavement and subgrade moduli. The analyses confirm that surface deflections are not very sensitive to variations of modulus within the pavement structure. Using the Boussinesq-Odemark approximation, it is concluded that, for the sample problem analyzed, the effect of higher-order terms on variation estimates of displacement is negligible when the coefficient of variation of subgrade modulus is less than 15 percent. The effect of inertia on the coefficient of variation of displacement due to variations in layer moduli is also briefly addressed. It i...

Improved two-point function approximations for design optimization

Two-point approximations are developed by utilizing both the function and gradient information of two data points. The objective of this work is to build a high-quality approximation to realize computational savings in solving complex optimization and reliability analysis problems. Two developments are proposed in the new twopoint approximations: 1) calculation of a correction term by matching with the previous known function value and supplementing it to the first-order approximation for including the effects of higher order terms and 2) development of a second-order approximation without the actual calculation of second-order derivatives by using an approximate Hessian matrix. Several highly nonlinear functions and structural examples are used for demonstrating the new two-point approximations that improved the accuracy of existing first-order methods. EVELOPMENTof new and improved constraint approximations has been an active area of research in the mathematical optimization community over the last two decades.13 More accurate function approximations reduce the finite element analysis cost and increase the search domain in each iteration of an optimization run. Recent developments in constraint approximations addressed methods for eigenvalues, dynamic response, static forces, structural reliability, etc. Some of the typical methods are summarized in the following. Kirsch4 developed a scaling factor for a stiffness matrix and approximated the displacements, stresses, and forces with respect to sizing and topology variables. Vanderplaats and Salajegheh5 used two levels of approximations for discrete sizing and shape design of structures, first for solving the continuous problem and next the discrete one using a dual theory. Canfield6 developed an approach for eigenvalues by approximating the modal strain and kinetic energies independently. Thomas et al.7 presented an approximation for the frequency response of damped structures by using the concept of Ref. 6. A multivariate spline approximation was developed by Wang and Grandhi8 making use of the previously generated exact finite element analyses. Similarly, a Hermite approximation for n-dimensional problems used several data points and had the property of reproducing the function and gradient values at the known data points.9 Canfield10 used multipoint data in building an approximate Hessian matrix and constructed a second-order approximation for improving the accuracy. Toropov et al.11 approximated functions as a linear combination or products of variables, and the coefficients of the polynomial were computed by applying a least squared approach on multiple data points. Fadel et al. 12 considered intermediate variables in terms of exponentials and they were computed by matching the approximate function gradients with the previous point's exact values. This approach did not compare the function values of the previous point. Wang and Grandhi13 used a single exponential for all of the variables and calculated it by matching the approximate function with the previous point exact value, and the gradient information of the previous point was not utilized. The proposed paper develops an improved two-point approximation utilizing both function and gradient information of two data points. In the new two-point approximation, two developments are proposed: 1) calculation of a correction term based on two function values and

The contribution of high order Zernike modes to wavefront tilt

The contribution of high order Zernike modes on average phase gradient is investigated. The resulting temporal power spectra are compared to data collected using the tip/tilt servo at the Sydney University Stellar Interferometer (SUSI). The low frequency part of the power spectrum follows a − 2 3 power law which is in agreement with other similar work. Due to the effect of higher order terms, the high frequency power law for average phase gradient becomes − 11 3 rather than the − 17 3 predicted for the ‘tilt’ component of the Zernike representation of the wavefront.

Effect of Higher Order Terms on Near-Tip Field of Cracks Immediately after Beginning of Rapid Propagation. Measurement by Interferometry.

A method of interferometry is applied to study the near-tip field of cracks which are propagating in PMMA specimens at a speed of about 550 m/s. The interference fringes around the cracks are photographed with a holographic optical system at 18 μs or at 73 μs after the crack initiation. The interference fringes are measured and analyzed in the directions of ±72 degrees from the propagation direction of the crack in order to compare the measurement results obtained through interferometry with those through the caustic method. It is found that the effect of higher order terms appears in the near-tip field at 18 μs after the crack initiation, but disappears at 73 μs, and the singular field is developed as far as 9 mm from the tip. Consequently, it can be concluded that, in the directions of ±72 degrees from the propagation direction, the effect of higher order terms appears within several tens of microseconds after the beginning of rapid crack propagation where crack speed is more than it few hundred m/s. These results are in good agreement with those obtained through the caustic method.

Effects of regular pumping on photon statistics in masers and lasers.

In laser systems, regular pumping always reduces the photon noise. In micromaser systems, however, regular pumping may lead to photon noise reduction as well as photon noise increase depending on the operation conditions. The effect of higher-order terms in the expansion of ln(1+pu) with respect to the pumping regularity parameter p is discussed. These terms have a significant effect on the photon statistics in micromaser systems. Their effect in laser systems is quite small compared with the second order and can be neglected.