Analytical solutions to the precession relaxation of magnetization with uniaxial anisotropy

Based on the Landau-Lifshitz-Gilbert(LLG)equation,the precession relaxation of magnetization is studied when the external field H is parallel to the uniaxial anisotropic field H_(k).The evolution of three-component magnetization is solved analytically under the condition of H=nH_(k)(n=3,1 and 0).It is found that with an increase of H or a decrease of the initial polar angle of magnetization,the relaxation time decreases and the angular frequency of magnetization increases.For comparison,the analytical solution for H_(k)=0 is also given.When the magnetization becomes stable,the angular frequency is proportional to the total effective field acting on the magnetization.The analytical solutions are not only conducive to the understanding of the precession relaxation of magnetization,but also can be used as a standard model to test the numerical calculation of LLG equation.

Hamiltonian system for the inhomogeneous plane elasticity of dodecagonal quasicrystal plates and its analytical solutions

A Hamiltonian system is derived for the plane elasticity problem of two-dimensional dodecagonal quasicrystals by introducing the simple state function. By using symplectic elasticity approach, the analytic solutions of the phonon and phason displacements are obtained further for the quasicrystal plates. In addition, the effectiveness of the approach is verified by comparison with the data of the finite integral transformation method.

Revisiting the analytical solutions for ultimate bearing capacity of pile embedded in rocks

This paper investigates the validity and shortcomings of the existing analytical solution for the ultimate bearing capacity of a pile embedded in a rock mass using the modified HoekeBrown failure criterion.Although this criterion is considered a reference value for empirical and numerical calculations,some limitations of its basic simplifications have not been clarified yet.This research compares the analytical results obtained from the novel discontinuity layout optimization(DLO)method and the numerical solutions from the finite difference method(FDM).The limitations of the analytical solution are considered by comparing different DLO failure modes,thus allowing for the first time a critical evaluation of its scope and conditioning for implementation.Errors of up to 40%in the bearing capacity and unrealistic failure modes are the main issues in the analytical solution.The main aspects of the DLO method are also analyzed with an emphasis on the linearization of the rock failure criterion and the accuracy resulting from the discretization size.The analysis demonstrates DLO as a very efficient and accurate tool to address the pile tip bearing capacity,presenting considerable advantages over other methods.

Bi-Objective Optimization: A Pareto Method with Analytical Solutions

Multiple objectives to be optimized simultaneously are prevalent in real-life problems. This paper develops a new Pareto Method for bi-objective optimization which yields analytical solutions. The Pareto optimal front is obtained in closed-form, enabling the derivation of various solutions in a convenient and efficient way. The advantage of analytical solution is the possibility of deriving accurate, exact and well-understood solutions, which is especially useful for policy analysis. An extension of the method to include multiple objectives is provided with the objectives being classified into two types. Such an extension expands the applicability of the developed techniques.

Analytical solutions for elastic binary nanotubes of arbitrary chirality

Analytical solutions for the elastic properties of a variety of binary nanotubes with arbitrary chirality are obtained through the study of systematic molecular mechanics. This molecular mechanics model is first extended to chiral binary nanotubes by introducing an additional out-of-plane inversion term into the so-called stick-spiral model, which results from the polar bonds and the buckling of binary graphitic crystals. The closed-form expressions for the longitudinal and circumferential Young's modulus and Poisson's ratio of chiral binary nanotubes are derived as functions of the tube diameter. The obtained inversion force constants are negative for all types of binary nanotubes, and the predicted tube stiffness is lower than that by the former stick-spiral model without consideration of the inversion term, reflecting the softening effect of the buckling on the elastic properties of binary nanotubes. The obtained properties are shown to be comparable to available density functional theory calculated results and to be chirality and size sensitive. The developed model and explicit solutions provide a systematic understanding of the mechanical performance of binary nanotubes consisting of III-V and II-VI group elements.

ELASTIC-PLASTIC ANALYTICAL SOLUTIONS FOR AN ECCENTRIC CRACK LOADED BY TWO PAIRS OF ANTI-PLANE POINT FORCES

The improved near crack line analysis method was used to investigate an eccentric cracked plate loaded by two pairs of anti_plane point forces at the crack surface in an elastic_perfectly plastic solid. The analytical solutions of the elastic_plastic stress fields and displacements near the crack line have been found without the assumptions of the small scale yielding. The law that the length of the plastic zone along the crack line is varied with an external loads and the bearing capacity of an eccentric cracked plate are obtained.

THE EXACT ANALYTICAL SOLUTIONS TO THE NEW MODEL OF RESERVOIR FILTRATION PROBLEM

The present study has obtained the new model of the reservoir filtration problem by taking into account the effect of wellbore storage and skin and by making use of the coupled equations of doubled porous media filtration and consequently has got, through various forms of limits, the exact analytical solutions of the three common reservoirs (fissure, homogeneous and the two-layered) pressure distribution under the conditions of three boundaries, i.e., infinite boundary, sealed finite boundary and the finite boundary at constant pressures.

Analytical Solutions to the D-Dimensional Schrodinger Equation with the Eckart Potential

The analytical solutions to the Schrodinger equation with the Eckart potential in arbitrary dimension D is investigated by using the Nikiforov-Uvarov method, and the centrifugal term is treated approximatively with the scheme of Greene and Aldrich. The discrete spectrum is obtained and the wavefunetion is expressed in terms of the Jacobi polynomial or the hypergeometric function. Some special cases of the Eckart potential are discussed for D=3, and the resulting energy equation agrees well with that obtained by other methods.

Analytical solutions for thermal forcing vortices in boundary layer and its applications

Using the Boussinesq approximation, the vortex in the boundary layer is assumed to be axisymmetrical and thermal-wind balanced system forced by diabatic heating and friction, and is solved as an initial-value problem of linearized vortex equation set in cylindrical coordinates. The impacts of thermal forcing on the flow field structure of vortex are analyzed. It is found that thermal forcing has significant impacts on the flow field structure, and the material representative forms of these impacts are closely related to the radial distribution of heating. The discussion for the analytical solutions for the vortex in the boundary layer can explain some main structures of the vortex over the Tibetan Plateau.

Exact analytical solutions to the mean-field model depicting microcavity containing semiconductor quantum wells

By using a two-mode mean-field approximation, we study the dynamics of the microcavities containing semiconductor quantum wells. The exact analytical solutions are obtained in this study. Based on these solutions, we show that the emission from the microcavity manifests periodic oscillation behaviour and the oscillation can be suppressed under a certain condition.

ANALYTICAL SOLUTIONS FOR EQUATIONS OF UNSTEADY FLOW OF NON-NEWTONIAN FLUIDS IN TUBE

This paper presents analytieal solutions to the partial differential equations for unsteady flow of the second-order fluid and Maxwell fluid in tube by using the integral transform method. It can be used to analyse the behaviour of axial velocity and shear stress for unsteady flow of nun-Newtonian visco-elastie fluids in tube, and to provide a theoretical base for the projection of pipe-line engineering.

THE ANALYTICAL SOLUTIONS BASED ON THE CONCEPT OF FINITE ELEMENT METHODS

On the basis of the concept of finite element methods, the rigorous analytical solutions of structural response in terms of the design variables are researched in this paper. The spatial trusses are taken as an example for the solution of the analytical expressions of the explicit displacements which are proved mathematically; then some conclusions are reached that are useful to structural sensitivity analysis and optimization. In the third part of the paper, a generalized geometric programming method is sugguested for the optimal model with the explicit displacement. Finally, the analytical solutions of the displacements of three trusses are given as examples.

ANALYTICAL SOLUTIONS TO STRESS CONCENTRATION PROBLEM IN PLATES CONTAINING RECTANGULAR HOLE UNDER BIAXIAL TENSIONS

The stress concentration problem in structures with a circular or elliptic hole can be investigated by analytical methods. For the problem with a rectangular hole, only approximate results are derived. This paper deduces the analytical solutions to the stress concentration problem in plates with a rectangular hole under biaxial tensions. By using the U-transformation technique and the finite element method, the analytical displacement solutions of the finite element equations are derived in the series form. Therefore, the stress concentration can then be discussed easily and conveniently. For plate problem the bilinear rectangular element with four nodes is taken as an example to demonstrate the applicability of the proposed method. The stress concentration factors for various ratios of height to width of the hole are obtained.

Analytical solutions of transient pulsed eddy current problem due to elliptical electromagnetic concentrative coils

The electromagnetic concentrative coils are indispensable in the functional magnetic stimulation and have potential applications in nondestructive testing. In this paper, we propose a figure-8-shaped coil being composed of two arbitrary oblique elliptical coils, which can change the electromagnetic concentrative region and the magnitude of eddy current density by changing the elliptical shape and/or spread angle between two elliptical coils. Pulsed current is usually the excitation source in the functional magnetic stimulation, so in this paper we derive the analytical solutions of transient pulsed eddy current field in the time domain due to the elliptical concentrative coil placed in an arbitrary position over a half-infinite plane conductor by making use of the scale-transformation, the Laplace transform and the Fourier transform are used in our derivation. Calculation results of field distributions produced by the figure-8-shaped elliptical coil show some behaviours as follows： 1） the eddy currents are focused on the conductor under the geometric symmetric centre of figure-8-shaped coil; 2） the greater the scale factor of ellipse is, the higher the eddy current density is and the wider the concentrative area of eddy current along y axis is; 3） the maximum magnitude of eddy current density increases with the increase of spread angle. When spread angle is 180°, there are two additional reverse concentrative areas on both sides of x axis.

Analytical solutions for constant-rate test in bounded confined aquifers with non-Darcian effect

This paper proposes a simplified analytical solution considering non-Darcian and wellbore storage effect to investigate the pumping flow in a confined aquifer with barrier and recharge boundaries.The mathematical modelling for the pumping-induced flow in aquifers with different boundaries is developed by employing image-well theory with the superposition principle,of which the non-Darcian effect is characterized by Izbash’s equation.The solutions are derived by Boltzmann and dimensionless transformations.Then,the non-Darcian effect and wellbore storage are especially investigated according to the proposed solution.The results show that the aquifer boundaries have non-negligible effects on pumping,and ignoring the wellbore storage can lead to an over-estimation of the drawdown in the first 10 minutes of pumping.The higher the degree of non-Darcian,the smaller the drawdown.

Analytical Solutions to Strain Rates of Reinforced Concrete Simply Supported Slabs under Blast Loads

Based on the Duhamel integral, a couple of analytical solutions are derived to predict the strain rates of concrete and steel reinforcement in reinforced concrete slabs under blast loads and to estimate their variation over depth of a cross-section along the entire length of the member. The analytical approach utilizes the single-degree-of-freedom mode for the analysis of reinforced concrete simply supported one-way panels subjected to blast loads. These analytical solutions can give the strain rate profile for any cross-section at any time and permit variations of strain rate in each time step of numerical iteration method, thus making it possible to directly incorporate strain rate effects into non-linear dynamic response analysis of structural members subjected to blast loads.

Analytical solutions to a compressible boundary layer problem with heat transfer

The problem of momentum and heat transfer in a compressible boundary layerbehind a thin expansion wave was solved by the application of the similarity transformation and theshooting technique. Utilizing the analytical expression of a two-point boundary value problem formomentum transfer, the energy boundary layer solution was represented as a function of thedimensionless velocity, and as the parameters of the Prandtl number, the velocity ratio, and thetemperature ratio.

Analytical solutions of uniform extended dislocations and tractions over a circular area in anisotropic magnetoelectroelastic bimaterials

In this paper, we derive the analytical solutions in a three-dimensional anisotropic magnetoelectroelastic bimaterial space subject to uniform extended dislocations and tractions within a horizontal circular area. By virtue of the Stroh formalism and Fourier transformation, the final expression of solutions in the physical domain contains only line integrals over [0, 2π] rather than infinite integrals. As the reduced cases, the half-space and homogeneous full-space solutions can be directly derived from the present solutions. Also, in terms of material domains, the present solutions can be reduced to the piezoelectric, piezomagnetic, purely elastic materials with different symmetries of material prop- erty. To carry out numerical calculations, Gauss quadrature is adopted. In the numerical examples, the effect of different loading locations on the response at the interface is analyzed. It is shown that, when the magnetic traction or electric dislocation is applied, the physical quantities on the interface may not decrease monotonically as the loading area moves away from the interface. The distributions of different in-plane physical quantities on the upper and lower interfaces under various extended horizontal loadings are compared and the differences are discussed. The work presented in this paper can serve as benchmarks for future numerical studies in related research fields.

ANALYTICAL SOLUTIONS FOR ELASTOSTATIC PROBLEMS OF PARTICLE-AND FIBER-REINFORCED COMPOSITES WITH INHOMOGENEOUS INTERPHASES

By transforming the governing equations for displacement components into Riccati equations, analytical solutions for displacements, strains and stresses for Representive Volume Elements (RVEs) of particle_ and fiber_reinforced composites containing inhomo geneous interphases were obtained. The analytical solutions derived here are new and general for power_law variations of the elastic moduli of the inhomogeneous interphases. Given a power exponent, analytical expressions for the bulk moduli of the composites with inho mogeneous interphases can be obtained. By changing the power exponent and the coefficients of the power terms, the solutions derived here can be applied to inhomogeneous interphases with many different property profiles. The results show that the modulus variation and the thickness of the inhomogeneous interphase have great effect on the bulk moduli of the composites. The particle will exhibit a sort of “size effect”, if there is an interphase.

Analytical solutions for plane problem of functionally graded magnetoelectric cantilever beam

In this paper, an exact analytical solution is presented for a transversely isotropic functionally graded magnetomelectro-elastic （FGMEE） cantilever beam, which is subjected to a uniform load on its upper surface, as well as the concentrated force and moment at the free end. This solution can be applied for any form of gradient distribution. For the basic equations of plane problem, all the partial differential equations governing the stress field, electric, and magnetic potentials are derived. Then, the expressions of Airy stress, electric, and magnetic potential functions are assumed as quadratic polynomials of the longitudinal coordinate. Based on all the boundary conditions, the exact expressions of the three functions can be determined. As numerical examples, the material parameters are set as exponential and linear distributions in the thickness direction. The effects of the material parameters on the mechanical, electric, and magnetic fields of the cantilever beam are analyzed in detail.