含孔曲板弹性波散射与动应力分析

基于敞口浅柱壳弹性波动方程及摄动方法，对无限大含孔曲板弹性波散射及动应力问题进行了分析研究，将经典薄板弯曲波动问题的分析解作为本问题的主项，给出了在稳态波下孔洞附近散射波的零阶渐近解。建立了求解含孔曲板弹性波散射与动应力问题的边界积分方程法，利用积分方程法可获得问题的近似分析解。并给出了无限大曲板圆孔附近动应力集中系数的数值结果，且对计算结果进行了分析与讨论。     

The effect of transverse shear deformation on stress concentration factors for shallow shells with a small circular hole

Simplified equations are derived for the analysis of stress concentration for shear-deformable shallow shells with a small hole.General solutions of the equations are obtained,in terms of series,for shallow spherical shells and shallow circular cylindrical shells with asmall circular hole.Approximate explicit solutions and numerical results are obtianed forthe stress concentration factors of shallow circular cylindrical shells with a small hole onwhich uniform pressure is acting.     

正交各向异性平板开孔弹性波的衍射与动应力集中

采用各向异性平板弯曲波动方程及摄动方法，对正交各向异性无限平板开孔弹性波的衍射及动应力集中的问题进行了分析研究，得到了在稳态波作用下此种忆边界条件波动问题的渐近形式分析解。同时采用复变函数方法及保角映射技术，为求解正交各向异性无限平板开孔弹性波的衍射及动力集中问题提供了一种统一规范的分析方法。     

The effect of material and geometry on the non-linear vibrations of orthotropic circular cylindrical shells

The extensive use of circular cylindrical shells in modern industrial applications has made their analysis an important research area in applied mechanics. In spite of a large number of papers on cylindrical shells, just a small number of these works is related to the analysis of orthotropic shells. However several modern and natural materials display orthotropic properties and also densely stiffened cylindrical shells can be treated as equivalent uniform orthotropic shells. In this work, the influence of both material properties and geometry on the non-linear vibrations and dynamic instability of an empty simply supported orthotropic circular cylindrical shell subjected to lateral time-dependent load is studied. Donnell׳s non-linear shallow shell theory is used to model the shell and a modal solution with six degrees of freedom is used to describe the lateral displacements of the shell. The Galerkin method is applied to derive the set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge–Kutta method. The obtained results show that the material properties and geometric relations have a significant influence on the instability loads and resonance curves of the orthotropic shell.     

含孔平板弹性波散射问题的复变函数方法

本文采用平板弯曲波动理论及复变函数方法，对平板开孔弹性波的散射及动应力集中问题进行了分析研究，得到了传播急剧记波时此种平板弯曲波动问题的分析解。若同时采用保角射技术，就为主解平板任意形状开孔弹性波的散射及动应力集中问题提供了一种统一规范的方法。作为算例，本文给出了平板开圆孔和椭圆孔附近的动应力集中系数的数值结果，并对其进行了讨论。     

Scattering of flexural waves and dynamic stress concentrations in Mindlin's thick plates with a cutout

Using the complex variable method and conformal mapping, scattering of flexural waves and dynamic stress concentrations in Mindlin's thick plates with a cutout have been studied. The general solution of the stress problem of the thick plate satisfying the boundary conditions on the contour of cutouts is obtained. Applying the orthogonal function expansion technique, the dynamic stress problem can be reduced into the solution of a set of infinite algebraic equations. As examples, numerical results for the dynamic stress concentration factor in Mindlin's plates with a circular, elliptic cutout are graphically presented in sequence. The project supported by the National Natural Science Foundation of China     

AN ANALYSIS FOR DIFFRACTION OF FLEXURAL WAVES AND DYNAMIC STRESS CONCENTRATIONS IN THICK PLATES WITH A CAVITY

By using the complex variable method and conformal mapping,the diffraction of flexu-ral waves and dynamic stress concentrations in thick plates with a cavity have been studied.A generalsolution of the stress problem of the thick plate satisfying the boundary conditions on the contour of anarbitrary cavity is obtained.By employing the orthogonal function expansion technique,the dynamicstress problem can be reduced to the solution of an infinite algebraic equation series.As an example,the numerical results for the dynamic stress concentration factor in thick plates with a circular,ellipticcavity are graphically presented.The numerical results are discussed.     

Stress Concentration in an Axially Compressed Cylindrical Shell of Medium Thickness with an Elliptic Opening

Stress concentrations at the periphery of elliptic and circular openings are determined experimentally by the photoelastic method for three compressed cylindrical shells of medium thickness made of epoxy resin. The experimental values are compared with numerical values obtained by the theory of thin shells     

Equation in complex variable of axisymmetrical deformation problems for a general shell of revolution

In this paper, the equation of axisymmetrical deformation problems for a general shell of revolution is derived in one complex variable under the usual Love-Kirchhoff assumption. In the case of circular ring shells, this equation may be simplified into the equation given by F.Tölke(1938)^[3], R.A.Clark(1950)^[4]and V.V.Novozhilov(1951)^[5]. When the horizontal radius of the shell of revolution is much larger than the average radius of curvature of meridian curve, this equation in complex variable may be simplified into the equation for slander ring shells. If the ring shell is circular in shape, then this equation can be reduced into the equation in complex variable for slander circular ring shells given by this author (1979)^[6]. If the form of elliptic cross-section is near a circle, then the equation of slander ring shell with near-circle ellipitic cross-section may be reduced to the complex variable equation similar in form for circular slander ring shells.     

Analytic solutions to problem of elliptic hole with two straight cracks in one-dimensional hexagonal quasicrystals

By means of the complex variable function method and the technique of conformal mapping, the anti-plane shear problem of an elliptic hole with two straight cracks in one-dimensional hexagonal quasicrystals is investigated. The solution of the stress intensity factor （SIF） for mode III problem has been found. Under the condition of limitation, both the known results and the SIF solution at the crack tip of a circular hole with two straight cracks and cross crack in one-dimensional hexagonal quasicrystals can be obtained.     

Reissner板弯曲的复变函数分析方法

本文建立了Reissner板弯曲问题的复变函数分析方法,它可以有效地用于分析含一般孔洞板弯曲的应力集中问题。作为应用,文中还给出了一些计算实例。     

The evolution of caustics from holes to cracks

The method of caustics was used for the study of the evolution of stress concentration around a circular hole, which progressively changes in shape and becomes an elliptic hole, tending to an internal crack. The influence of the amount of ellipticity of the holes and their orientation relative to the axis of the applied external loads at infinity on the form of caustics created around the discontinuity was studied, as the elliptic holes tended to become internal cracks. A series of experiments with tension specimens containing small elliptic holes of any ellipticity and orientation was performed. Comparison of experimentally obtained caustics with theory yielded a good agreement of both results. Finally, the use of small elliptic holes drilled all over a biaxial stress field for the determination of the individual principal stresses and the principal directions at the area of the holes was outlined.     

环壳弹性静力及自由振动问题的弱形式求积元分析

基于旋转薄壳理论, 采用求积元法, 建立了求积元法求解环壳问题的单元列式, 并对圆环壳、椭圆环壳的静力及自由振动问题进行了分析. 数值算例与精确解及有限元结果相对比, 证明了求积元法分析此类连续环壳问题的准确和高效性. 同时, 分析结果表明, 椭圆环壳长短轴的比值k对壳体的受力特性及求积元法的收敛率有显著的影响.     

Elastic fields of interacting elliptic inhomogeneities

This paper presents a method for the calculation of two-dimensional elastic fields in a solid containing any number of inhomogeneities under arbitrary far field loadings. The method called pseudo-dislocations method, is illustrated for the solution of interacting elliptic inhomogeneities. It reduces the interacting inhomogeneities problem to a set of linear algebraic equations. Numerical results are presented for a variety of elliptic inhomogeneity arrangements, including the special cases of elliptic holes, cracks and circular inhomogeneities. All these complicated problems can be solved with high accuracy and efficiency.     

Nouvelle approche numerique des solutions de la couche limite laminaire incompressible en regime non-stationnaire autour d'une paroi cylindrique elliptique mise en impulsion brutale

The equations of the unsteady incompressible laminar boundary layers about circular and elliptic cylinders started impulsively from rest are solved, after an original coordinate transformation, with the help of a semi-implicit difference scheme linearly stable without conditions. The knowledge of the phenomenon results only from that of the flow characteristics at the initial time; this makes mandatory the use of an analytical method of the Blasius type. When the ratio of the semi-axes of the elliptic cylinder becomes infinite, the present method gives the solution of the stagnation flow.When the flow presents a separation point in the steady state (circular cylinder), a criterion for stopping computation is proposed. It may be noted that to obtain the stagnation flow, we can continue the computations as far as possible and a steady solution is found with excellent accuracy.     

On the anti-plane shear of an elliptic nano inhomogeneity

The elastic field of an elliptic nano inhomogeneity embedded in an infinite matrix under anti-plane shear is studied with the complex variable method. The interface stress effects of the nano inhomogeneity are accounted for with the Gurtin–Murdoch model. The conformal mapping method is then applied to solve the formulated boundary value problem. The obtained numerical results are compared with the existing closed form solutions for a circular nano inhomogeneity and a traditional elliptic inhomogeneity under anti-plane. It shows that the proposed semi-analytic method is effective and accurate. The stress fields inside the inhomogeneity and matrix are then systematically studied for different interfacial and geometrical parameters. It is found that the stress field inside the elliptic nano inhomogeneity is no longer uniform due to the interface effects. The shear stress distributions inside the inhomogeneity and matrix are size dependent when the size of the inhomogeneity is on the order of nanometers. The numerical results also show that the interface effects are highly influenced by the local curvature of the interface. The elastic field around an elliptic nano hole is also investigated in this paper. It is found that the traction free boundary condition breaks down at the elliptic nano hole surface. As the aspect ratio of the elliptic hole increases, it can be seen as a Mode-III blunt crack. Even for long blunt cracks, the surface effects can still be significant around the blunt crack tip. Finally, the equivalence between the uniform eigenstrain inside the inhomogeneity and the remote loading is discussed.     

Elastica of a variable-arc-length circular curved beam subjected to an end follower force

This paper presents postbuckling behaviors of a variable-arc-length (VAL) circular curved beam subjected to an end follower force. One end of the VAL circular curved beam is hinged while the other end is supported by a frictionless slot, which is fixed horizontally and vertically but is allowed to rotate corresponding to loading direction. When the VAL circular curved beam is deformed, the total arc-length of the circular curved beam varies. Two approaches have been applied for the solution of this problem. The first approach is an elliptic integrals method based on elastica theory, which yields the exact closed-form solution in terms of the first and second kinds of elliptic integrals. For validation of the results, the shooting method is employed for a numerical solution by developing the set of nonlinear governing differential equations together with boundary conditions, and then integrating them by using the fourth-order Runge–Kutta algorithm. The results from both approaches are in very good agreement. From the results, it is found that the VAL circular curved beam subjected to an end follower force can be deformed in many mode shapes. For the first and third modes, the beam exhibits both stable and unstable configurations, whereas for the second mode only an unstable configuration exists. The influences of initial curvature on the critical load and the deformed configurations are highlighted.     

Complex equations of flexible circular ring shells overall-bending in a meridian plane and general solution for the slender ring shells

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采用精确化理论求解厚板任意形开孔动应力集中

摘要：本文基于复变函数与保角映射法,采用平板弯曲振动精确化方程[9],对含任意形开孔平板中弹性波散射与动应力集中问题进行了研究。利用正交函数展开的方法将待解的问题归结为对一组无穷代数方程组的求解。作为算例,计算了自由边界条件下圆孔和椭圆孔的动弯矩集中系数的数值结果,并对板厚与孔径比对动弯矩分布的影响做了分析研究。结果表明：入射波数、平板厚度和椭圆偏心率等参数对动弯矩的分布都有很大的影响。在较低频率和平板较薄的情况下,基于文献[9]的方程与基于Mindlin板的动弯矩结果在数值分布上是基本一致的;在较高频率和平板较厚的情况下,基于文献[9]的方程与基于Mindlin板的动弯矩结果在数值分布计算结果相差较大。由于文献[9]给出的平板振动精确化方程是在没有任何工程假设条件下得到的,因此本文的分析计算结果更精确一些。     

The general solution for axial symmetrical bending of nonhomogeneous circular plates resting on an elastic foundation

In this paper,a new method,the exact analytic method,is presented on the basis of stepreduction method.By this method,the general solution for the bending of nonhomogenouscircular plates and circular plates with a circular hole at the center resting,on an elastfcfoundation is obtained under arbitrary axial symmetrical loads and boundary conditions.The uniform convergence of the solution is proved.This general solution can also be applieddirectly to the bending of circular plates without elastic foundation.Finally,it is onlynecessary to solve a set of binary linear algebraic equation.Numerical examples are givenat the end of this paper which indicate satisfactory results of stress resultants anddisplacements can be obtained by the present method.