Contribution Of Higher Order Terms Research Articles

On the effects of higher order stress terms in pure mode III loading of bi‐material notches

AbstractThe effects of higher order terms (HOTs) on the stress distribution at the vicinity of the sharp corner of two bonded dissimilar materials were investigated under pure out‐of‐plane (mode III) loading. To achieve this aim, finite element (FE) analysis in conjunction with the over‐deterministic technique were performed for two specimens, each containing four different notch opening angles. The coefficients of singular and non‐singular stress terms were then computed from the displacement field determined from the FE analysis. The elastic stresses obtained directly from the FE analysis were compared with those reconstructed from the truncated series expansions, in order to validate the accuracy of the calculated coefficients. The numerical results show that although the contribution of HOTs in the out‐of‐plane stress field is less than that of the in‐plane notch problems, ignoring their effects can lead to considerable errors in the stress field estimation. The magnitude of numerical error depends on the geometry and loading conditions of the bi‐material notch.

Averaged strain energy density estimated rapidly from the nodal stresses by FEM for cracks under mixed mode loadings including the T-stress contribution

The present contribution reviews a recently proposed method to rapidly estimate the averaged SED at the tip of short as well as long cracks under in-plane I+II and long cracks under out-of-plane I+III mixed mode loadings. Short cracks are distinguished from long cracks by considering that the stress fields within the control volume of short cracks are no longer governed solely by the stress intensity factors (SIFs), but the contribution of higher order terms, and primarily the T-stress, becomes significant to estimate the averaged SED. According to the proposed method, the averaged SED is calculated using the linear elastic nodal stresses evaluated by FEM either at the crack tip, to account for the SIFs contribution, and at selected FE nodes of the crack free edges, to include the T-stress contribution. The advantage of the proposed approach is two-fold: coarse FE meshes can be adopted; moreover, geometrical modelling the control volume is no longer necessary. To validate the approach, cracked plates subjected to in-plane I+II mixed mode loading as well as bars weakened by circumferential outer cracks subjected to out-of-plane mixed mode I+III loading have been analysed. A comparison between approximate values of the averaged SED according to the nodal stress approach and those derived directly from the FE strain energy adopting very refined FE meshes has been successfully performed.

A third and fourth order perturbation theory for dipolar hard spheres

In this work, we develop and analyze a third order and a fourth order perturbation theory for non-polarizable dipolar hard spheres. The electrostatic potential is split into a short-ranged and a long-ranged part. A perturbation expansion is applied for the short-ranged potential where the contribution of higher order terms is approximated through a [2,1]-Padé approximation for the third order theory and by a [2,2]-Padé approximation for the fourth order theory. Rodgers and Weeks [J. Chem. Phys. 131, 244108 (2010)] developed the Local Molecular Field (LMF) theory for the long-ranged contribution of the electrostatic energy. The LMF theory requires the relative permittivity of the considered fluid. We apply a perturbation theory for the relative permittivity and show that the LMF is then in very good agreement with the results for the long-ranged energy from molecular simulations. The short-ranged contribution to the Helmholtz energy from the third order and from the fourth order perturbation theory is, for densities ρ* ≤ 1 and for dipole moments of μ*2 ≤ 1, in very good agreement with the data from molecular simulations. For larger dipole densities (up to μ*2 ≤ 4 for ρ* ≲ 1), we observe significantly improved results for the fourth order perturbation theory compared to the third order perturbation theory.

Rapid finite element evaluation of the averaged strain energy density of mixed‐mode (I + II) crack tip fields including the T‐stress contribution

AbstractThe strain energy density (SED) averaged over a material dependent control volume has been demonstrated to control fracture and fatigue behaviour of different materials in many engineering design situations. A method to rapidly calculate the averaged SED at the tip of long cracks under in‐plane mixed mode (I + II) loading has been recently proposed. It was based on the peak stresses evaluated from finite element (FE) analyses, according to the peak stress method (PSM). The aim of this work is to extend this FE approach to short cracks. In the present paper short cracks are distinguished from long cracks by considering that the stress fields within the control volume of short cracks are no longer governed solely by the stress intensity factors (SIFs), but the contribution of higher order terms, and primarily the T‐stress, becomes significant to estimate the averaged SED. Therefore the averaged SED is calculated using the linear elastic nodal stresses evaluated by FEM either at the crack tip, to account for the SIF contribution, and at selected FE nodes of the crack free edges, to include the T‐stress contribution. The proposed approach is referred to as nodal stress approach. The advantage of the nodal stress approach is two‐fold: there is no need of mesh refinements in the close neighbourhood of the points of singularity, so that coarse FE meshes can be adopted; moreover, geometrical modelling of the control volume in FE models is no longer necessary. Infinite plates weakened by central small cracks subjected to mixed mode I + II loading and a lap joint geometry have been analysed taking into consideration different crack lengths, mode mixities and average finite element sizes of the employed meshes. A comparison between approximate values of the averaged SED according to the nodal stress approach and those derived directly from the FE strain energy adopting very refined FE meshes has been successfully performed within a range of applicability.

Rescattering effects for theC12(p,2p)B11reaction at 400 MeV/u

Background: Single-particle occupancies for a wide range of magic and near-magic nuclei are significantly below the values predicted by independent-particle shell-model calculations.Purpose: We aim in this paper to address the question: To what extent does the extracted structure information have uncertainties due to approximations involved in the reaction formalism and assumed dynamic model?Method: The proton knockout reaction $^{12}\mathrm{C}{(p,2p)}^{11}\mathrm{B}$ at 400 MeV/u incident energy is analyzed using the few-body Faddeev--Alt-Grassberger-Sandhas reaction framework. A two-body model is used for $^{12}\mathrm{C}$ which involves an inert $^{11}\mathrm{B}({\frac{3}{2}}^{\ensuremath{-}})$ core and a valence proton. The contributions of higher order terms beyond the single scattering approximation are studied, in particular the ones corresponding to the rescattering between $^{12}\mathrm{C}$ constituents.Results: Total cross section and core transverse momentum distribution are calculated. We have found that rescattering terms between the composite nucleus constituents contribute dominantly to the absorptive distortion.Conclusions: The accurate treatment of these rescattering terms in the reaction formalism is crucial if one aims to extract reliable structure information from the data. We have also found that work needs to be performed in the reaction theory in order to incorporate a complete structure description of the projectile nucleus and to bridge the proton knockout results with those obtained from electron scattering.

Contribution of higher order terms in electron-acoustic solitary waves with vortex electron distribution

The basic equations describing the nonlinear electron-acoustic waves in a plasma composed of a cold electron fluid, hot electrons obeying a trapped/vortex-like distribution, and stationary ions, in the long-wave limit, are re-examined through the use of the modified PLK method. Introducing the concept of strained coordinates and expanding the field variables into a power series of the smallness parameter e, a set of evolution equations is obtained for various order terms in the perturbation expansion. The evolution equation for the lowest order term in the perturbation expansion is characterized by the conventional modified Korteweg–deVries (mKdV) equation, whereas the evolution equations for the higher order terms in the expansion are described by the degenerate(linearized) mKdV equation. By studying the localized traveling wave solution to the evolution equations, the strained coordinate for this order is determined so as to remove possible secularities that might occur in the solution. It is observed that the coefficient of the strained coordinate for this order corresponds to the correction term in the wave speed. The numerical results reveal that the contribution of second order term to the wave amplitude is about 20 %, which cannot be ignored.

Acoustic oscillations in the early universe

Acoustic oscillations in the baryonic-photon plasma leave an imprint on both the cosmic microwave background radiation (CMBR) and the power spectrum of the mass distribution in the universe. This effect is well known and is currently applied to determine the value for quantities such as the percentage of dark matter and dark energy in the universe. This study examines the contributions of higher order terms in the equations for the radiation-baryonic acoustic oscillations prior to radiation decoupling from matter. The energy and momentum conservation equations are combined with the Einstein field equations for a Robertson Walker metric containing the perturbations. An attempt is made to consider the effect of the higher order perturbations on the observed power spectrum and on the anisotropy of the CMBR. The sensitivity of these higher order terms to adiabatic, instead of non-adiabatic, boundary conditions on the fluctuations outside the particle horizon is considered. [Work supported by KSA.]

Higher order nonlinear effects on wave structures in a four-component dusty plasma with nonisothermal electrons

The higher order solutions of dust acoustic wave in dusty plasma consisting of positively charged warm adiabatic dust, negatively charged cold dust, and nonisothermally distributed electrons are studied. The Schamel–KdV equation is derived using reductive perturbation method (RPM). RPM is further extended to include the contributions of higher order terms and a generalized KdV equation is derived to observe the deviation from isothermality. Effects of nonisothermal parameter, mass and charge ratio, ratio of ion to electron temperatures, and ratio of dust to ion temperatures have been thoroughly studied. By using the renormalization method of Kodama and Taniuti [J. Phys. Soc. Jpn. 45, 298 (1978)], authors have also discussed characteristics of the dressed solitons.

Improved linearity performance of AlGaN/GaN MISHFET over conventional HFETs: An optimization study for wireless infrastructure applications

In this work an extensive study on the linearity distortion behaviour of AlGaN/GaN MISHFET is performed and compared with those of conventional HFET structures. The contribution of higher order terms in a Taylor series expansion of small signal drain current is considered by evaluating higher order transconductance coefficients. Linearity figures of merit such as input intercept power, intermodulation distortion and other higher order harmonics have been investigated for both the structures. The impact of gate biasing on the linearity is also examined. The influence of critical technology parameters such as gate length, doping density, dopant layer thickness and gate insulator thickness is investigated to optimize the MISHFET structure for better linearity characteristics. Improved linearity performance has been observed for insulated gate structure proving its superiority over conventional HFETs for RF wireless and low noise applications. The results obtained reveal that by careful optimization of technology parameters, a suitably designed MISHFET architecture is more linear than its conventional counterparts.

Fine-structure measurements in high-Ln=17and 20 Rydberg states of barium

Fine-structure intervals in $n=17$ and 20 barium Rydberg levels with $6\ensuremath{\le}L\ensuremath{\le}11$ have been measured precisely with the resonant excitation Stark ionization spectroscopy microwave technique. The measured intervals were analyzed using the long-range polarization model, including allowance for the large nonadiabatic effects of the ${\text{Ba}}^{+}$ $5d$ level and the contributions of higher-order terms. The analysis determined improved values in atomic units of both dipole and quadrupole polarizabilities of ${\text{Ba}}^{+}$, ${\ensuremath{\alpha}}_{1}=123.88(5)$ and ${\ensuremath{\alpha}}_{2}=4420(250)$.

Hedge's Ultimate Numerical Technique (HUNT) for Steady State Diffusion Problem

A more realistic numerical technique hereafter known as Hegde's Ultimate Numerical Technique (HUNT) is developed and demonstrated on a one dimensional and a two dimensional steady state diffusion problem of heat transfer. The available numerical methods developed are based on finite difference technique neglecting the contribution of higher order terms in Taylor series expansion of the function leading to an approximation and the error in the solution. In the present effort of the HUNT, the optimization of the partial derivatives leads to the elimination of the error and justifies the stability and the convergence of the solution. The HUNT procedure based on the interface theory developed by the author, is capable of providing the ultimate optimum solution to all the partial derivatives considered as decision vectors. Even though the HUNT is demonstrated on one dimensional and two dimensional steady state diffusion equations, it does not require rigorous efforts to apply it to three dimensional problems of fluid flow and heat transfer. As pilot exercises the HUNT is demonstrated on a one dimensional circular fin and a two dimensional plate to obtain the temperature distribution. The result is compared with the analytical method and the finite volume method for which the results are available in the literature. To the knowledge of the authors, HUNT is both different and a unique example of its kind.

The modified reductive perturbation method as applied to Boussinesq equation: strongly dispersive case

In this work, we extended the application of “the modified reductive perturbation method” to Boussinesq equation for strongly dispersive case and tried to obtain the contribution of higher order terms in the perturbation expansion. It is shown that the first order term in the perturbation expansion is governed by the non-linear Schrödinger equation and the second order term is governed by the linearized Schrödinger equation with a non-homogeneous term. In the long-wave limit, a travelling wave type of solution to these equations is also given.

Higher order approximations in reductive perturbation method: strongly dispersive waves

Contribution of higher order terms in the perturbation expansion for the strongly dispersive ion-plasma waves is examined through the use of modified reductive perturbation method developed early by us. It is shown that the lowest order term in the expansion is governed by the nonlinear Schrödinger equation while the second-order term is governed by the linear Schrödinger equation. For the small wave number region a set of solution is presented for the evolution equations.

Contributions of higher order terms to nonlinear waves in fluid-filled elastic tubes: strongly dispersive case

In the present work, employing the nonlinear equations of an incompressible, isotropic and elastic thin tube and the approximate equations of an incompressible inviscid fluid, and then utilizing the modified reductive perturbation technique presented by us [15] the amplitude modulation of weakly nonlinear waves is examined. It is shown that the first order term in the perturbation expansion is governed by a nonlinear Schrödinger equation and the second order term is governed by the linearized Schrödinger equation with a nonhomogeneous term. In the longwave limit a travelling wave type of solution to these equations are also given.

Contribution of Higher Order Terms in Nonlinear Ion-Acoustic Waves: Strongly Dispersive Case

Contribution of higher order terms in the perturbation expansion for the strongly dispersive ion-plasma waves is examined through the use of modified reductive perturbation method developed by us [J. Phys. Soc. Jpn. 68 (1999) 1833]. In the analysis it is shown that the lowest order term in the expansion is governed by the nonlinear Schrodinger equation while the second order term is governed by the linear Schrodinger equation. For the small wave number region a set of solution is presented for the evolution equations.

The Korteweg–de Vries–Burgers hierarchy in fluid-filled elastic tubes

In this work, contribution of higher order terms in modified reductive perturbation method is studied for the propagation of weakly nonlinear waves in fluid-filled elastic tubes. The basic set of equation of fluid and equation of tube is reduced to the Korteweg–de Vries–Burgers equation for the first order displacement component in the radial direction and a linear Korteweg–de Vries–Burgers inhomogeneous equation for the second order displacement component in the radial direction. Dynamical processes of the solitary waves have been numerically analyzed by solving the Korteweg–de Vries–Burgers equation for the first order and the linearized KdV–Burgers equation with an inhomogeneous equation for the second order using pseudo-spectral method.

On the contribution of higher order terms to solitary waves in fluid filled elastic tubes

In the present work, employing the nonlinear equations of an incompressible, isotropic and elastic thin tube and the approximate equations of an incompressible inviscid fluid, the propagation of weakly nonlinear waves, in such a medium is studied through the use of the modified multiple scale expansion method. It is shown that the evolution of the lowest order (first order) term in the perturbation expansion may be described by the Korteweg-de Vries equation. The governing equation for the second order terms and the localized travelling wave solution for these equations are also obtained. The applicability of the present model to flow problems in arteries is discussed.

Competition between standard and exotic double beta decays

We discuss the contributions of higher order terms in weak Hamiltonian to the standard two-neutrino double beta decay. The formalism for the unique first forbidden transitions has been developed, and it is shown that they can alter the two-electron energy spectrum. Yet, their effect is too small to screen the detection of exotic neutrinoless double beta decays, which are candidates for testing the physics beyond the standard model.

Multiple Beam Wedge Plate Lateral Shear Interferometry in Transmission

Abstract It is shown that a multiple beam wedge plate interferometer can be used in transmission mode to display accurately the lateral aberration curve of a collimating lens. Experimental results showing the sharpened fringes and a comparison with the computed lateral aberration curves are given. Further, for larger shear, some new features are observed; these are explained by considering contributions of higher order terms in the Taylor's series expansion of the path function.

Reflection multiple beam wedge plate shear interferometry for lens testing

It is shown that a multiple beam wedge plate shear interferometry can be used to represent the lateral aberration curve of a lens more accurately. For larger shear, some new artifacts appear in the interferogram which have been explained by considering contributions of higher order terms in the Taylor’s series expansion of the path Lunction.