General Power Law Research Articles

Nonlinear stability analysis of stiffened functionally graded material sandwich cylindrical shells with general Sigmoid law and power law in thermal environment using third-order shear deformation theory

This paper investigates analytically nonlinear buckling and postbuckling of functionally graded sandwich circular thick cylindrical shells filled inside by Pasternak two-parameter elastic foundations under thermal loads and axial compression loads. Shells are reinforced by closely spaced functionally graded material (FGM) rings and stringers. The temperature field is taken into account. Two general Sigmoid law and general power law, with four models of stiffened FGM sandwich cylindrical shell, are proposed. Using the Reddy’s third-order shear deformation shell theory (TSDT), stress function, and Lekhnitsky’s smeared stiffeners technique, the governing equations are derived. The closed form to determine critical axial load and postbuckling load-deflection curves are obtained by the Galerkin method. The effects of the face sheet thickness to total thickness ratio, stiffener, foundation, material, and dimensional parameters on critical thermal loads, critical mechanical loads and postbuckling behavior of shells are analyzed. In addition, this paper shows that for thin shells we can use the classical shell theory to investigate stability behavior of shell, but for thicker shells the use of TSDT for analyzing nonlinear stability of shell is necessary and suitable.

Nonlinear stability of functionally graded material (FGM) sandwich cylindrical shells reinforced by FGM stiffeners in thermal environment

In this paper, Donnell’s shell theory and smeared stiffeners technique are improved to analyze the postbuckling and buckling behaviors of circular cylindrical shells of stiffened thin functionally graded material (FGM) sandwich under an axial loading on elastic foundations, and the shells are considered in a thermal environment. The shells are stiffened by FGM rings and stringers. A general sigmoid law and a general power law are proposed. Thermal elements of the shells and reinforcement stiffeners are considered. Explicit expressions to find critical loads and postbuckling load-deflection curves are obtained by applying the Galerkin method and choosing the three-term approximate solution of deflection. Numerical results show various effects of temperature, elastic foundation, stiffeners, material and geometrical properties, and the ratio between face sheet thickness and total thickness on the nonlinear behavior of shells.

De Haas-van Alphen oscillations with non-parabolic dispersions

de Haas-van Alphen oscillation spectrum of two-dimensional systems is studied for general power law energy dispersion, yielding a Fermi surface area of the form $S(E)\propto E^\alpha$ for a given energy $E$. The case $\alpha=1$ stands for the parabolic energy dispersion. It is demonstrated that the periodicity of the magnetic oscillations in inverse field can depend notably on the temperature. We evaluated analytically the Fourier spectrum of these oscillations to evidence the frequency shift and smearing of the main peak structure as the temperature increases.

Love-Wave Propagation in an Inhomogeneous Orthotropic Medium Obeying the Exponential and Generalized Power Law Models

Love-Wave Propagation in an Inhomogeneous Orthotropic Medium Obeying the Exponential and Generalized Power Law Models

Thermomechanical postbuckling analysis of eccentrically stiffened FGM sandwich plates with general Sigmoid and power laws based on TSDT

The buckling and postbuckling behaviors of eccentrically stiffened sandwich plates on elastic foundations subjected to in-plane compressive loads, thermal loads, or thermomechanical loads are presented analytically by using the Reddy’s third-order shear deformation plate theory with von Karman geometrical nonlinearity. Four cases of general Sigmoid and power laws are considered. The material properties of the facesheets, the core layer, and stiffeners are assumed to be temperature-dependent. Theoretical formulations based on the smeared stiffeners technique and third-order shear deformation plate theory are derived. The expressions of thermal parameters are found in the analytical form. Applying the Galerkin method, the expressions for determination of the critical buckling load and analysis of the postbuckling mechanical and thermal load–deflection curves are obtained. The iterative algorithm is presented for the case of temperature-dependent plate material properties. In addition, the influences of thermal element, functionally graded material stiffeners, the facesheet thickness to total thickness ratio, initial imperfection, and foundations are clarified in detail.

Nonlinear thermo-mechanical stability of eccentrically stiffened functionally graded material sandwich doubly curved shallow shells with general sigmoid law and power law according to third-order shear deformation theory

In this paper, the nonlinear analysis of stability of functionally graded material (FGM) sandwich doubly curved shallow shells is studied under thermo-mechanical loads with material properties obeying the general sigmoid law and power law of four material models. Shells are reinforced by the FGM stiffeners and rest on elastic foundations. Theoretical formulations are derived by the third-order shear deformation theory (TSDT) with the von Karman-type nonlinearity taking into account the initial geometrical imperfection and smeared stiffener technique. The explicit expressions for determining the critical buckling load, the post-buckling mechanical, and the thermal load-deflection curves are obtained by the Galerkin method. Two iterative algorithms are presented. The effects of the stiffeners, the thermal element, the distribution law of material, the initial imperfection, the foundation, and the geometrical parameters on buckling and post-buckling of shells are investigated.

Sintering Behavior of TiO2-Supported Model Cobalt Fischer–Tropsch Catalysts under H2 Reducing Conditions and Elevated Temperature

The sintering of model TiO2-supported cobalt catalysts has been studied. The TiO2 supports used were anatase, P25 (85% anatase–15% rutile), and rutile. The catalysts were characterized at each stage of treatment. These treatment stages were calcination, reduction, and sintering. It was found that the TiO2 support does not influence the Co3O4 crystallite and particle size as shown by powder X-ray diffraction (XRD) and transmission electron microscopy (TEM), respectively. The reduction of the cobalt catalysts and bare supports was studied by temperature-programmed reduction (TPR). It was found that the bare supports were not as inert as expected. The supports showed minor reduction as seen in the H2 consumption. All the cobalt catalysts showed a two-step reduction. Sintering of anatase-supported cobalt was shown to be the most substantial with P25- and rutile-supported showing a lower sintering tendency, P25-supported cobalt being the most stable based on TEM measurements. Sintering kinetics based on the generalized power law expression (GPLE) model showed that sintering of anatase-supported cobalt is the most rapid with a large sintering rate constant. Sintering of P25-supported cobalt is the lowest, shown by the lowest sintering rate constant. The study has conclusively shown the effect of the catalyst support phase. The study has also shown that the use of high surface area supports is not necessarily the only answer to preventing sintering. The phase of the catalyst support is also important.

Efficient numerical model for the computation of impedance functions of inclined pile groups in layered soils

This paper introduces a numerical model to obtain the time-harmonic dynamic response of pile foundations in non-homogeneous soils. The model is based on the integral formulation of the elastic problem and the use of Green’s functions for the layered halfspace, and considers the soil as a group of zoned homogeneous, linear, isotropic, viscoelastic layers. The piles, on the other hand, are treated by finite elements as Timoshenko beams. Both formulations are coupled through the required compatibility and equilibrium equations along each pile. After being validated against previous results from the literature, the model is used to study the effects of soil non-homogeneity on the impedance functions of inclined piles and pile groups by considering different soil profiles whose properties vary with depth following a generalized power law. The impedance functions for three representative non-homogeneous soils are presented and compared with the ones of related homogeneous soils. Significant differences appear between the two situations for all studied rake angles. The magnitude of these differences strongly depends on the frequency range considered, specially for the case of pile groups, which shows the necessity of analyzing the problem using soil profiles as close as possible to the actual depth-varying ones.

Springbackward Phenomenon of a Transversely Isotropic Functionally Graded Composite Cylindrical Shell

This study provides an approach to predict the springback phenomenon during post-solidification cooling in a functionally graded hybrid composite cylindrical shell with a transverse isotropic structure. Here, the material properties are given with a general parabolic power-law function. During the theoretical analysis, an appropriate transformation is introduced in the equilibrium equation, which is resulting in a hypergeometrical differential equation. Thermoelastic solutions are obtained and analyzed for a homogeneous, nonhomogeneous, and elastic-plastic state. The solution is validated by applying it to the multilayered functionally graded cylindrical shell using the transfer or propagator matrix method.

Irradiance scintillation of Gaussian beam considering the concept of anisotropy at different scales in weak anisotropic turbulence

In this work, taking the concept of anisotropy at different scales of turbulence cells or eddies into considerations, new analytic expression of the irradiance scintillation index of a Gaussian beam is derived for Gaussian beam propagating through weak anisotropic non-Kolmogorov turbulence. The effective anisotropic factor is introduced to parameterize the asymmetry of turbulence cells or eddies in the horizontal and vertical directions. Also, the finite turbulence inner and outer scales, and the general spectral power law deviated from the classic 11/3 spectral power law behavior for Kolmogorov turbulence are considered in the investigations. The derived expression of irradiance scintillation index for a Gaussian beam can reduce correctly to the previously published results in the special cases of effective anisotropic factor equaling one. Calculations are conducted to analyze the influences of turbulence parameters on the irradiance scintillation index of Gaussian beam under weak anisotropic non-Kolmogorov turbulence.

Analytical Modelling of Elastic-Plastic Interference Fit Joints

The interference fit is a widely process, used to produce a joint assembly of two parts achieved by friction. It is a popular type of joint in machine elements and it is found in many field applications related to the automotive, aerospace, oil and gas and shipbuilding industries. Consequently, a better understanding of its limit behavior together with an accurate evaluation of the residual stresses generated by this process are very important for the design of mechanical components requiring high optimized performance. This work focuses on the analytical development of the residual stresses when the two assembled parts are circular cylinders and deform elasto-plastically under plane strain condition. The constitutive law that governs their material strain hardening behavior is assumed to follow a general power law which also covers the particular cases of elastic perfectly plastic and bilinear hardening. To validate the developed analytical model (AM), the finite element method (FEM) was used and the results showed good agreement between the two approaches. The obtained results show that the stresses increase when the interference value increases causing maximum stress intensity at the inner surface of the two assembled parts to exceed their material yield stress. After hollow shaft plastic collapse, any increase in interference results in a small increase in residual contact pressure with a large increase of the equivalent stresses localized at the hollow shaft inner surface. The method could be used to determine the maximum value of the interference in order to prevent assembly failure.

Irradiance scintillation index for Gaussian beam based on the generalized anisotropic von Karman spectrum

Experiments and theoretical investigations have shown that the atmosphere turbulence exhibits both anisotropic and non-Kolmogorov properties. In this paper, based on the anisotropic generalized von Karman spectrum and the Rytov approximation theory, new expression for the irradiance scintillation index of optical waves is derived for Gaussian beam propagating through weak anisotropic non-Kolmogorov turbulence. Compared with previously published results, it considers simultaneously the asymmetry property of turbulence cells or eddies in the orthogonal xy-plane, the general spectral power law in the range 3–4 instead of constant value of 11/3 for the Kolmogorov turbulence, and the finite turbulence inner and outer scales. Two anisotropic factors are introduced to parameterize the anisotropy of turbulence cells or eddies in horizontal and vertical directions. In the special cases of these two anisotropic factors equaling one and the finite turbulence inner and outer scales equaling separately zero and infinite, the derived expression can reduce correctly to the previously published results. Calculations are performed to analyze the derived results.

Temporal power spectra of plane wave considering finite turbulence inner and outer scales in anisotropic turbulence

Theoretical investigations and experiments have shown that the atmosphere turbulence exhibits both anisotropic and non-Kolmogorov properties compared with traditional atmospheric turbulence which takes the hypotheses of isotropy and homogeneity. There is an increasing interest for modeling atmosphere turbulence with a power spectrum that incorporates anisotropy and general spectral power law deviated from the standard value of 11/3 for the Kolmogorov turbulence. In this study, new analytic expressions for the temporal power spectra of irradiance fluctuations and angle of arrival (AOA) fluctuations have been derived based on the Rytov approximation theory for optical plane wave propagating through weak anisotropic non-Kolmogorov atmosphere turbulence. Compared with the previously published results, the effective anisotropic factor is introduced to the final results to include the anisotropy for different turbulence cells or eddies. The finite turbulence inner and outer scales and the general spectral power law in the range of 3–4 are also considered in the final derived models. The results in this work will help better understand the optical waves’ propagation through anisotropic non-Kolmogorov turbulence.

Empirical Methods for Dynamic Power Law Distributions in the Social Sciences

This paper introduces nonparametric econometric methods that characterize general power law distributions under basic stability conditions. These methods extend the literature on power laws in the social sciences in several directions. First, we show that any stationary distribution in a random growth setting is shaped entirely by two factors — the idiosyncratic volatilities and reversion rates (a measure of cross-sectional mean reversion) for different ranks in the distribution. This result is valid regardless of how growth rates and volatilities vary across different economic agents, and hence applies to Gibrat's law and its extensions. Second, we present techniques to estimate these two factors using panel data. Third, we show how our results offer a structural explanation for a generalized size effect in which higher-ranked processes grow more slowly than lower-ranked processes on average. Finally, we employ our empirical methods using panel data on commodity prices and show that our techniques accurately describe the empirical distribution of relative commodity prices. We also show the existence of a generalized size'' effect for commodities, as predicted by our econometric theory.

A generalization of the power law distribution with nonlinear exponent

The power law distribution is usually used to fit data in the upper tail of the distribution. However, commonly it is not valid to model data in all the range. In this paper, we present a new family of distributions, the so-called Generalized Power Law (GPL), which can be useful for modeling data in all the range and possess power law tails. To do that, we model the exponent of the power law using a non-linear function which depends on data and two parameters. Then, we provide some basic properties and some specific models of that new family of distributions. After that, we study a relevant model of the family, with special emphasis on the quantile and hazard functions, and the corresponding estimation and testing methods. Finally, as an empirical evidence, we study how the debt is distributed across municipalities in Spain. We check that power law model is only valid in the upper tail; we show analytically and graphically the competence of the new model with municipal debt data in the whole range; and we compare the new distribution with other well-known distributions including the Lognormal, the Generalized Pareto, the Fisk, the Burr type XII and the Dagum models.

SU‐G‐206‐17: RadShield: Semi‐Automated Shielding Design for CT Using NCRP 147 and Isodose Curves

Purpose:Computed tomography (CT) exam rooms are shielded more quickly and accurately compared to manual calculations using RadShield, a semi‐automated diagnostic shielding software package. Last year, we presented RadShield's approach to shielding radiographic and fluoroscopic rooms calculating air kerma rate and barrier thickness at many points on the floor plan and reporting the maximum values for each barrier. RadShield has now been expanded to include CT shielding design using not only NCRP 147 methodology but also by overlaying vendor provided isodose curves onto the floor plan.Methods:The floor plan image is imported onto the RadShield workspace to serve as a template for drawing barriers, occupied regions and CT locations. SubGUIs are used to set design goals, occupancy factors, workload, and overlay isodose curve files. CTDI and DLP methods are solved following NCRP 147. RadShield's isodose curve method employs radial scanning to extract data point sets to fit kerma to a generalized power law equation of the form K(r) = ar^b. RadShield's semiautomated shielding recommendations were compared against a board certified medical physicist's design using dose length product (DLP) and isodose curves.Results:The percentage error found between the physicist's manual calculation and RadShield's semi‐automated calculation of lead barrier thickness was 3.42% and 21.17% for the DLP and isodose curve methods, respectively. The medical physicist's selection of calculation points for recommending lead thickness was roughly the same as those found by RadShield for the DLP method but differed greatly using the isodose method.Conclusion:RadShield improves accuracy in calculating air‐kerma rate and barrier thickness over manual calculations using isodose curves. Isodose curves were less intuitive and more prone to error for the physicist than inverse square methods. RadShield can now perform shielding design calculations for general scattering bodies for which isodose curves are provided.

Modified anisotropic turbulence refractive-index fluctuations spectral model and its application in moderate-to-strong anisotropic turbulence.

In this study, the modified anisotropic turbulence refractive-index fluctuations spectral model is derived based on the extended Rytov approximation theory for the theoretical investigations of optical plane and spherical waves propagating through moderate-to-strong anisotropic non-Kolmogorov turbulence. The anisotropic factor which parameterizes the asymmetry of turbulence cells or eddies in the horizontal and vertical directions is introduced. The general spectral power law in the range of 3-4 is also considered compared with the conventional classic value of 11/3 for Kolmogorov turbulence. Based on the modified anisotropic turbulence refractive-index fluctuations spectrum, the analytic expressions of the irradiance scintillation index are also derived for optical plane and spherical waves propagating through moderate-to-strong anisotropic non-Kolmogorov turbulence. They are applicable in a wide range of turbulence strengths and can reduce correctly to the previously published results in the special cases of weak anisotropic turbulence and moderate-to-strong isotropic turbulence. Calculations are performed to analyze the derived models.

Cosmological perturbations in coherent oscillating scalar field models

The fact that fast oscillating homogeneous scalar fields behave as perfect fluids in average and their intrinsic isotropy have made these models very fruitful in cosmology. In this work we will analyse the perturbations dynamics in these theories assuming general power law potentials $V(\phi)=\lambda \vert\phi\vert^{n}/n$. At leading order in the wavenumber expansion, a simple expression for the effective sound speed of perturbations is obtained $c_{\text{eff}}^2 = \omega=(n-2)/(n+2)$ with $\omega$ the effective equation of state. We also obtain the first order correction in $k^2/\omega_{\text{eff}}^2$, when the wavenumber $k$ of the perturbations is much smaller than the background oscillation frequency, $\omega_{\text{eff}}$. For the standard massive case we have also analysed general anharmonic contributions to the effective sound speed. These results are reached through a perturbed version of the generalized virial theorem and also studying the exact system both in the super-Hubble limit, deriving the natural ansatz for $\delta\phi$; and for sub-Hubble modes, exploiting Floquet's theorem.

Heat Conduction: Hyperbolic Self-similar Shock-waves in Solid Medium

Analytic solutions for cylindrical thermal waves in solid medium are given based on the nonlinear hyperbolic system of heat flux relaxation and energy conservation equations. The Fourier-Cattaneo phenomenological law is generalized where the relaxation time and heat propagation coefficient have a general power law temperature dependence. From such laws one cannot form a second order parabolic or telegraph-type equation.We consider the original non-linear hyperbolic system itself with the self-similar Ansatz for the temperature distribution and for the heat flux. As results continuous.

Blood flow through an inclined stenosed artery

The present work is to investigate the effect of the inclination angle on blood flow through the flexible stenosed artery. The mathematical model of blood considered laminar, incompressible, unsteady and fully developed, is represented by the generalized power law model. The numerical Marker and Cell method (MAC) has been utilized to solve the continuity and momentum equations in cylindrical coordinate. The Poisson equation for the pressure has been solved using Successive-Over-Relaxation method (SOR) and the pressure–velocity correction formula has been derived. The effects of inclination angle ( ) and power law index (n) on the flow characteristics of blood such as velocity, pressure drop, and wall shear stress are presented by their representation graphs.