ASYMPTOTIC EXPANSIONS OF ZEROS FOR KRAWTCHOUK POLYNOMIALS WITH ERROR BOUNDS

Krawtchouk polynomials are frequently applied in modern physics. Based on the results which were educed by Li and Wong, the asymptotic expansions of Krawtchouk polynomials are improved by using Airy function, and uniform asymptotic expansions are got. Furthermore, the asymptotic expansions of the zeros for Krawtchouk polynomials are again deduced by using the property of the zeros of Airy function, and their corresponding error bounds are discussed. The obtained results give the asymptotic property of Krawtchouk polynomials with their zeros, which are better than the results educed by Li and Wong.     

Torsional-mode transients in variable-thickness shells of revolution

For thin shells of revolution the existence of torsional-vibration modes, uncoupled from bending and extensional modes, has been established[1]. Here a linear second-order differential equation for the uncoupled torsional stress mode is obtained and its solution for impact loading of shells is sought. The mode-superposition method which utilizes the natural modes of vibration predicted by elementary theory, is, in general, not satisfactory for sharp impact loading as many modes are often required for convergence. Hence we employ two novel techniques for solving the impact problems. Firstly a formal asymptotic procedure, based on extensions to geometrical optics, is employed to generate asymptotic wavefront expansions. Rigorous justifications for this formal technique are provided in an appendix. Secondly a transform technique whereby solutions are sought in terms of Bessel functions is discussed and applied to particular impact loading problems. The Bessel function solutions found here can be used to determine the natural frequencies of the shells. Shells both finite and infinite in extent are discussed and reflections at a stress-free end are examined.     

Transient solutions for longitudinally impacted inhomogeneous conical shells

Here we study transient elastic wave propagation in inhomogeneous conical shells. The uniaxial theory is employed and two separate techniques are used for extracting information about the stress field for impact problems. Firstly the formal Karal-Keller method is used enabling us to directly determine asymptotic wavefront expansions for the stress field. A transform technique, based on Eason's [1], is then used to obtain conditions on the physical parameters of the medium which give solutions to the governing equation in terms of Bessel functions and some particular problems are discussed. Previously unknown simple closed-form solutions are obtained. Our results have application to the propagation of waves in filamentary cones of constant wall thickness or in homogeneous cones having an axial temperature gradient.     

Airy stress function for describing the non-linear effect of simply supported plates with movable edges

The general form of the solution of the Airy function for the stress distributions that describe the non-linear effect developed from the large deflection of simply supported plates with movable edges are found by superposition of the Airy functions, which satisfy the large deflection condition and the boundary conditions of the edges. Each term of the Airy function consists of a particular solution and a homogeneous one. The particular solution satisfying the large deflection condition is classified into six cases, depending on the combinations of the modal numbers of the comparison functions. The corresponding homogeneous solution is found to make each Airy function satisfy the boundary condition by using the Fourier series method. The solution is applied to the non-linear analysis of the deflection of the simply supported plates with movable edges under transverse loading, and is verified by comparison with other investigation.     

On one-dimensional shock waves

In this paper uniform asymptotic expansions for the solutions of a system of differential equations are obtained in the domain containing a shock wave. It is shown, in particular, that the function θ(t,x)/ε contained in the expansions and describing the behavior of the solution in the neighborhood of the wave front has, generally speaking, a discontinuity of derivatives at the front. The results are applicable to one-dimensional problems in gas dynamics with low viscosity and heat-conductivity.     

Forced Radial Motions of Nonlinearly Viscoelastic Shells

This paper contains an extensive global treatment of radial motions of compressible nonlinearly viscoelastic cylindrical and spherical shells under time-dependent pressures. It furnishes a variety of conditions on a general class of material properties and on the pressure terms ensuring that there are solutions existing for all times, there are unbounded globally defined solutions, there are solutions that blow up in finite time, and there are solutions having the same period as that of the pressure terms. The shells are described by a geometrically exact 2-dimensional theory in which the shells suffer thickness strains as well as the standard stretching of their base surfaces. Consequently their motions are governed by fourth-order systems of semilinear ordinary differential equations. This work shows that there are major qualitative differences between the nonlinear dynamical behaviors of cylindrical and spherical shells.      

Stress-deformation state of thick shells and plates. Three-dimensional theory (review)

Conclusions In the first part of the present review we surveyed systematically published results of investigations of the stress-deformation state of thick-walled spheres, ellipsoids, cones, circular cylinders, as well as thick slabs, obtained by exact analytic solutions of spatial problems of elasticity theory. Several quantitative results were given of the variation of displacements and stresses with shell or plate thickness, and their comparative analysis was provided, making it possible to establish the validity limits of the corresponding applied theories. We also surveyed systematically published specific results of the spatial stress-deformation state of nearly canonical thick-walled shells, as well as non-thin plates of varying thickness, obtained by effective approximate analytic methods and known exact solutions for the corresponding canonical regions. Especially noted were characteristic mechanical (including boundary) effects on the stress-deformation state of the bodies under consideration. These effects are generated, in particular, by variations in the radius of curvature of the surface, the thickness parameter, the amplitude and frequency of the corrugated surface, material, inhomogeneity, conditions of mechanical contact between layers, the nature of self-balancing loads, and other factors.However, the possibilities of exact and effective approximate analytic solution of boundary value problems of this class in the three dimensional statement are restricted. In the case of shells and plates of mean thickness these results can be substantially supplemented by qualitative and quantitative data, obtained on the basis of analytic solutions in the generalized theory of shells and plates, based on expansions of components of the stress-deformation state in Legendre polynomial series, and making it possible, in principle, to approximate the three-dimensional solution with any required accuracy. This is one of the basic features distinguishing it from the classical and applied theories of shells and plates. The second part of this review will be devoted to systematic and comparative analysis of the results of investigations carried out within the generalized theory of non-thin shells and plates.Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 27, No. 11, pp. 3–27, November, 1991.     

Exact solution for standing monochromatic waves within a wedge

Exact solutions for standing internal waves are obtained in a region bounded by horizontal and inclined planes in the linear, ideal stratified fluid approximation at a constant Brent-V?is?l? frequency. The solutions are expressed in terms of the Macdonald functions and their derivatives. Nonuiform and uniform asymptotics are also constructed in the case of a small slope of the lower plane. The latter asymptotics are expressed in terms of either the Airy function or the Macdonald function. The exact and asymptotic solutions are numerically compared.     

On displacement methods in planar anisotropic elasticity

Two displacement formulation methods are presented for problems of planar anisotropic elasticity. The first displacement method is based on solving the two governing partial differential equations simultaneously/ This method is a recapitulation of the orignal work of Eshelby, Read and Shockley [7] on generalized plane deformations of anisotropic elastic materials in the context of planar anisotropic elasticity.The second displacement method is based on solving the two governing equations separately. This formulation introduces a displacement function, which satisfies a fourth-order partial differential equation that is identical in the form to the one given by Lekhnitskii [6] for monoclinic materials using a stress function. Moreover, this method parallels the traditional Airy stress function method and thus the Lekhnitskii method for pure plane problems. Both the new approach and the Airy stress function method start with the equilibrium equations and use the same extended version of Green's theorem (Chou and Pagano [13], p. 114; Gao [11]) to derive the expressions for stress or displacement components in terms of a potential (stress or displacement) function (see also Gao [10, 11]). It is therefore anticipated that the displacement function involved in this new method could also be evaluated from measured data, as was done by Lin and Rowlands [17] to determine the Airy stress function experimentally.The two different displacement methods lead to two general solutions for problems of planar anisotropic elasticity. Although the two solutions differ in expressions, both of the depend on the complex roots of the same characteristic equation. Furthermore, this characteristic equation is identical to that obtained by Lekhnitskii [6] using a stress formulation. It is therefore concluded that the two displacement methods and Lekhnitskii's stress method are all equivalent for problems of planar anisotropic elasticity (see Gao and Rowlands [8] for detailed discussions).     

Stresses in a deep beam with a central concentrated load

The stress distribution in deep beams has received the attention of mathematicians an research workers since the early days of Wilson, Stokes and Filon.^{1, 2} A general theoretical solution is not possible and particular solutions of the biharmonic equatiion for the Airy stress function have been worked out to satisfy certain loadings. In the case of a deep beam under a symmetrical pressure on the top edge, solutions by (1) Durant and Garwood,³ (2) Chow, Conway and Winter⁴ have been put forward. Systematic photoelastic experimental work appears to be scarce and the object of this article is to present for comparison with existing theories the results of three photoelastic tests on simply supported deep beams carrying a central point load. The experimental stresses and those obtained from the solutions referred to above agreed reasonably well. The normal stresses σ _x calculated from the Wilson-Stokes theory were in error for the case under consideration (the theory is not valid for (H/L)>0.4), and the simple theory of bending is adequate for the maximum tensile stress provided the span exceeds 1.5 times the depth. Results are summarized in figures showing the magnitude and direction of the principal stresses; magnitudes are in a dimensionless from, hence it is possible to adapt them fior any beam of similar depth to span ratio under the same type of loading.     

On solution of problems of dynamics of plates and shells with local structure heterogeneities

A general approach to the solution of nonstationary problems of the theory of shells with local structure heterogeneities is proposed. The approach is based on representation of solutions for the systems of differential motion equations of shell theory in the form of expansions in terms of free forms of vibrations (FFV). Two implementations of the approach, involving FFV expansions of certain members and FFV expansions of the system as a whole, are considered. The capabilities of the approach are illustrated by results for specific problems. In the first implementation, we determine the dynamic stress-strain state of a square plate with three rod supports. A round plate connected to a coaxial circular cylindrical shell is considered as an example of FFV expansion of the system as a whole. Special Design and Engineering Office, S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 10, pp. 46–53, October, 1999     

基于混合变量条形传递函数方法的圆柱壳屈曲分析

提出了一种分析交各向异性圆柱壳和阶梯圆柱壳稳定性问题的混合变量条形传递函数方法。首先基于Fluegge薄壳理论，通过定义广义位移变量和对应的广义力变量，建立了圆柱壳混合变量能量泛函；然后通过引入条形单元，定义混合状态变量和采用传递函数方法对超级壳单元求解，得到具有多种边界条件圆柱壳屈曲问题的半解析解；最后通过位移连续和力平衡条件，可以得到阶梯圆柱壳屈曲问题的解。理论解推导过程表明此方法在引入边界条件和进行阶梯圆柱壳求解时非常方便。算例分析的结果验证了本方法的正确性。     

Fracture analysis of a functionally graded strip with arbitrary distributed material properties

In this paper, the plane elasticity problem for a crack in a functionally graded strip with material properties varying arbitrarily is studied. The governing equation in terms of Airy stress function is formulated and exact solutions are obtained for several special variations of material properties in Fourier transformation domain. A multi-layered model is employed to model arbitrary variations of material properties based on two linear-distributed material softness parameters. The mixed boundary problem is reduced to a system of singular integral equations that are solved numerically. Comparisons with other two existing multi-layered models have been made. Some numerical examples are given to demonstrate the accuracy, efficiency and versatility of the model. Numerical results show that fracture toughness of materials can be greatly improved by graded variation of elastic modulus and the influence of the specific form of elastic modulus on the fracture behavior of FGM is limited.     

Uniform far field asymptotics of internal gravity waves from a source moving in a stratified fluid layer with smoothly varying bottom

The far field asymptotics of internal waves is constructed for the case when a point source of mass moves in a layer of arbitrarily stratified fluid with slowly varying bottom. The solutions obtained describe the far field both near the wave fronts of each individual mode and away from the wave fronts and are expansions in Airy or Fresnel waves with the argument determined from the solution of the corresponding eikonal equation. The amplitude of the wave field is determined from the energy conservation law along the ray tube. For model distributions of the bottom shape and the stratification describing the typical pattern of the ocean shelf eract analytic expressions are obtained for the rays, and the properties of the phase structure of the wave field are analyzed. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 111–120, May–June, 1998. This work was financially supported by the Russian Foundation for Basic Research (project No. 96-01-01120).     

Stress Separation of Interferometrically Measured Isopachics in a Perforated Plate

A method for the stress separation of interferometrically measured isopachics using an Airy stress function is proposed in this study. A Poisson equation that represents the relationship between the sum of principal stresses and an Airy stress function is solved using a finite element method. The Dirichlet boundary condition for solving the Poisson equation is determined by the approximation of an assumed Airy stress function along the boundary of the model. Therefore, the distribution of the Airy stress function is obtained from the measured isopachic contours. Then, the stresses are obtained from the computed Airy stress function. The effectiveness of the proposed method is validated by applying the proposed method to the isopachic contours in a perforated plate obtained by Mach-Zehnder interferometry. Results indicate that stress components around a hole in a plate can be obtained from isopachics by the proposed method.     

Stress-strain state of non-thin plates and shells. Generalized theory (survey)

Conclusions Thus, this part of our survey has presented the main approaches that have been taken to the construction of two-dimensional (in terms of the space coordinates) equations of a generalized theory of plates and shells. The solutions of these equations represent a certain approximation of the solution of the initial three-dimensional problem. They are based on expansion of the sought functions into Fourier series in Legendre polynomials of the thickness coordinate. Studies completed on the basis of the given variants of plate and shell theory were systematized and analyzed. In terms of the method of its construction, the theory involves a regular process of replacing the solution of the three-dimensional problem by the solution (or sequence of solutions) of two-dimensional boundary-value problems or initial-boundary-value problems. Numerical results illustrating the convergence of the successive approximation were presented. It should be noted that to make comparison with the results of classical or applied theories, several of the studies cited here presented solutions of problems for thin plates and shells with allowance only for the initial terms of expansions of the stress and displacement components into base functions (Legendre polynomials).S. P. Timoshenko Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 29, No. 11, pp. 3–34, November, 1993.     

A general nonlinear theory of elastic shells

This paper presents a general nonlinear theory of elastic shells for large deflections and finite strains in reference to a certain natural state. By expanding the displacement components into power series in the coordinate θ³ normal to the undeformed middle surface of shells, the expansions of the Cauchy-Green strain tensors are expressed in terms of these expanded displacement components. Through the modified Hellinger-Reissner variational principle for a three-dimensional elastic continuum, a set of the fundamental shell equations is derived in terms of the expanded Cauchy-Green strain tensors and Kirchhoff stress resultants. The Love-Kirchhoff hypothesis is not assumed and higher order stretching and bending are taken into consideration. For elastic shells of isotropic materials, assuming the strain-energy to be an analytic function of the strain measures, general nonlinear constitutive equations are then derived. Thus, a complete and consistent two-dimensional shell theory incorporating the geometrical and physical nonlinearities is established. The classical theories of shells are directly derivable from the present results by proper truncations of the series.     

Global structural behaviour of thin and moderately thick monoclinic spherical shells using a mixed shear deformation model

Summary The static and dynamic responses of anisotropic spherical shells under a uniformly distributed transverse load are investigated. Analytical solutions using the mixed variational formulation are presented for spherical shells subjected to various boundary conditions. Numerical results of a refined mixed first-order shear deformation theory for natural frequencies, critical buckling, center deflections and stresses are compared with those obtained using the classical shell theory. A variety of simply-supported and clamped boundary conditions are considered and comparisons with the existing literature are made. The sample numerical results presented herein for global structural behaviour of monoclinic spherical shells should serve as references for future comparisons.     

Nonlinear vibration of a rotating laminated composite circular cylindrical shell: traveling wave vibration

In this paper, the large-amplitude (geometrically nonlinear) vibrations of rotating, laminated composite circular cylindrical shells subjected to radial harmonic excitation in the neighborhood of the lowest resonances are investigated. Nonlinearities due to large-amplitude shell motion are considered using the Donnell’s nonlinear shallow-shell theory, with account taken of the effect of viscous structure damping. The dynamic Young’s modulus which varies with vibrational frequency of the laminated composite shell is considered. An improved nonlinear model, which needs not to introduce the Airy stress function, is employed to study the nonlinear forced vibrations of the present shells. The system is discretized by Galerkin’s method while a model involving two degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the forced vibration responses of the two-degrees-of-freedom system. The stability of analytical steady-state solutions is analyzed. Results obtained with analytical method are compared with numerical simulation. The agreement between them bespeaks the validity of the method developed in this paper. The effects of rotating speed and some other parameters on the nonlinear dynamic response of the system are also investigated.