Non-axisymmetric bending of thin annular plates due to circumferentially distributed moments

The non-linear deformation of a thin annular plate subjected to circumferentially distributed bending moments is studied. A von Kármán plate model is adopted to formulate the equations of motion. Free–free boundary conditions have been applied at both inner and outer edges. The matrix formulation of the Differential Quadrature Method is used to discretise and solve the governing equations. Linear analysis predicts that the annular disk deforms axisymmetrically into a spherical dome. However, the proposed non-linear analysis shows that a symmetry breaking bifurcation may occur after which the linear solution becomes unstable and the plate transitions into a non-axisymmetric cylindrical deformation. This is the case when at least one of two parameters reaches a critical value. These parameters are the non-dimensionalised ratio between applied moment and bending stiffness and the ratio between inner and outer radius. Furthermore, it is noted that free-free boundary conditions and circumferentially distributed bending moments do not break the circular symmetry of the annular disk. Hence, the principal axes of curvature in the deformed configuration do not have a preferred orientation. Therefore, the present work describes a shell possessing infinite identical equilibria, having different yet no favoured direction and, hence, links to previous researches on neutrally stable structures.     

Study of bifurcation in a pressurized hyperelastic membrane tube enclosed by a soft substrate

We model a perivascular supported arterial tube as a uniform cylindrical membrane tube enclosed by a soft substrate, and derive the solution bifurcation criterion. We assume the surrounding soft substrate as an elastic foundation with distributed stiffness. We consider the tube to be a neo-Hookean material with isotropic and anisotropic (orthotropic) properties, and study solution bifurcation at a constant axial stretch. In the isotropic case, the surrounding soft substrate can substantially delay the onset of bifurcation through a subcritical jump in circular distension at bifurcation with increasing substrate stiffness. Introduction of anisotropy can significantly change the jump behavior from subcritical to supercritical.     

Cylindrical interface crack in a hollow layered functionally graded cylinder under static torsion

A hollow functionally graded composite cylinder under static torsion, which consists of an inner and outer elastic circular tube with a cylindrical interface crack, is studied in this work. By utilizing Fourier integral transform method, the mixed boundary value problem is reduced to a Cauchy singular integral equation, from which the numerical results of the stress intensity factor are obtained by the Lobatto–Chebyshev quadrature technique. Numerical results demonstrate the coupled effects of geometrical, physical, and functionally graded parameters on the interfacial fracture behavior.     

Asymmetric bifurcations of thick-walled circular cylindrical elastic tubes under axial loading and external pressure

In this paper, we consider bifurcation from a circular cylindrical deformed configuration of a thick-walled circular cylindrical tube of incompressible isotropic elastic material subject to combined axial loading and external pressure. In particular, we examine both axisymmetric and asymmetric modes of bifurcation. The analysis is based on the three-dimensional incremental equilibrium equations, which are derived and then solved numerically for a specific material model using the Adams–Moulton method. We assess the effects of wall thickness and the ratio of length to (external) radius on the bifurcation behaviour.     

Remarks concerning the flexure of a compressible nonlinearly elastic rectangular block

Under consideration is the problem of flexure of compressible nonlinearly elastic rectangular blocks. The discussion is confined to deformations describing the bending of a rectangular block into a sector of a circular cylindrical tube. The predictions based upon the well-known semilinear material model are investigated. Addressed, in particular, are some problems concerning the existence, uniqueness and stability of solutions to specific boundary value problems.     

Bifurcation of rotating thick-walled elastic tubes

The deformation of a circular cylindrical elastic tube of finite wall thickness rotating about its axis is examined. A circular cylindrical deformed configuration is considered first, and the angular speed analysed as a function of an azimuthai deformation parameter at fixed axial extension for an arbitrary form of incompressible, isotropic elastic strain-energy function. This extends the analysis given previously (Haughton and Ogden, 1980) for membrane tubes.Bifurcation from a circular cylindrical configuration is then investigated. Prismatic, axisymmetric and asymmetric bifurcation modes are discussed separately. Their relative importance is assessed in relation to the wall thickness and length of the tube, the magnitude of the axial extension, and the angular speed turning-points. Numerical results are given for a specific form of strain-energy function.Amongst other results it is found that (i) for long tubes, asymmetric modes of bifurcation can occur at low values of the angular speed and before any possible axisymmetric or prismatic modes and (ii) for short tubes, there is a range of values of the axial extension (including zero) for which no bifurcation can occur during rotation.     

Bifurcation of inflated circular cylinders of elastic material under axial loading—I. Membrane theory for thin-walled tubes

Bifurcations of circular cylindrical elastic tubes subjected to inflation combined with axial loading are analysed. Membrane tubes are considered in detail as a background to the more difficult analysis of thickwalled tubes described in the companion paper (Part II). Our results for membranes reinforce and extend those given by R.T. Shield and his co-workers.Two modes of bifurcation are investigated: firstly, a bulging (axisyrmmetric) mode; secondly, a prismatic mode in which the cross-section of the tube becomes non-circular. Necessary and sufficient conditions for the existence of modes of either type are given in respect of an arbitrary (incompressible isotropic) form of elastic strain-energy function. For a closed tube with a fixed axial loading many features of the results have close parallels with recent findings by D.M. Haughton and R.W. Ogden for spherical membranes. On the other hand, some results for tubes with fixed ends have no such parallel. In particular, bifurcation may, under certain conditions, occur before the inflating pressure reaches a maximum. A combination of the two modes is interpreted in terms of bending for a tube under axial compression, and the relative importance of the bending and bulging modes is discussed in relation to the length to radius ratio of the tube. The analytical results are illustrated for specific forms of strain-energy function. Corresponding analysis is given for thick-walled tubes in Part II.     

Circumferential buckling instability of a growing cylindrical tube

A cylindrical elastic tube under uniform radial external pressure will buckle circumferentially to a non-circular cross-section at a critical pressure. The buckling represents an instability of the inner or outer edge of the tube. This is a common phenomenon in biological tissues, where it is referred to as mucosal folding. Here, we investigate this buckling instability in a growing elastic tube. A change in thickness due to growth can have a dramatic impact on circumferential buckling, both in the critical pressure and the buckling pattern. We consider both single- and bi-layer tubes and multiple boundary conditions. We highlight the competition between geometric effects, i.e. the change in tube dimensions, and mechanical effects, i.e. the effect of residual stress, due to differential growth. This competition can lead to non-intuitive results, such as a tube growing to be thinner and yet buckle at a higher pressure.     

波纹管在内压作用下柱失稳临界压力的计算

 讨论了波纹管在内压作用下的柱失稳问题，将波纹管等效成圆 柱壳，证明了该柱壳在内压作用下失稳的临界压力与在轴向均布压力 作用下失稳的临界压力相等，利用作者对波纹管整体弯曲问题的研究 成果确定等效柱壳的抗弯刚度，给出了波纹管在内压作用下柱失稳临 界压力的计算公式. 由对前人实验的观察分析得出波纹管呈弹性柱失 稳的条件为其长细比大于1，在此条件下本文的计算结果和前人的实 验结果相一致.     

On deformations with constant modified stretches

Two plane, non-homogeneous, non-conformal deformations with constant modified stretches (the inflation of a circular cylindrical tube and the bending of a rectangular block into a sector of a circular tube) are shown to be possible in the absence of body forces in isotropic elastic solids that satisfy both the classical pressure-compression inequality and the Baker-Ericksen inequality.     

Shear of rubber tube springs

Rubber tube springs consist basically of cylindrical rubber tubes bonded on their inner and outer curved surfaces to rigid cylindrical tubes. They are widely used as flexible linkages, for example in vehicle suspensions. Rotation of one rigid tube with respect to the other about their common axis subjects the rubber tube to azimuthal shear. Displacement of one rigid tube with respect to the other along their common axis puts the rubber tube into axial shear. Using FEA, we have calculated the stresses set up in both cases, for a long rubber tube of a non-linearly elastic (neo-Hookean) material. The results are compared for the two modes of deformation, and with analytical predictions where available. For a long tube the shear stresses are substantially independent of the end conditions, but the normal stresses are strongly affected, as found previously for sheared rectangular blocks [A.N. Gent, J.B. Suh, S.G. Kelly III, Mechanics of rubber shear springs, Int. J. Nonlinear Mech. 42 (2007) 241-249]. If the end surfaces are stress-free, unexpectedly large normal stresses are generated, even in azimuthal shear. These high tensile stresses are attributed to restraints at the inner and outer cylindrical boundaries that compensate for the absence of stresses on the end surfaces that would be needed to maintain a simple shear deformation. Thus, the boundary conditions affect the stresses everywhere (in contrast to an “end effect” that would diminish away from the ends). Small departures from complete incompressibility are found to lower the internal stresses markedly, and even cause the sign of the stresses to be reversed.     

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.     

One-dimensional deformations of nonlinearly elastic micropolar bodies

We find families of finite deformations of a Cosserat elastic continuum on which the system of equilibrium equations is reduced to a system of ordinary differential equations. These families can be used to describe the expansion, tension, and torsion of a hollow circular cylinder, cylindrical bending of a rectangular slab, straightening of a circular arch, reversing of a cylindrical tube, formation of screw and wedge dislocations in a hollow cylinder, and other types of deformations. In the case of a physically nonlinear material model, the above-listed families of deformations can be used to construct exact solutions of several problems of strong bending of micropolar bodies.     

Surface Forces Driven Wrinkling of an Elastic Half-Space Coated with a Thin Stiff Surface Layer

This paper studies surface instability of a coated semi-infinite linear elastic body interacting with another flat rigid body through surface van der Waals (vdW) forces under plane strain conditions. The emphasis is on the effect of the surface coating layer on the wavelength of surface wrinkling. It is shown that the surface of the coated elastic half-plane is always unstable even in the presence of a very stiff coating layer. However, the numerical results show that the stiff coating layer has a significant effect on the wavelength of the surface instability mode and can effectively prevent the surface from short-wavelength wrinkling. In particular, the surface tangential displacement associated with the surface instability vanishes when the elastic half-plane is incompressible. In this case, the in-plane rigidity of the coating layer has no effect on surface instability while the bending stiffness of the coating layer has an effect on the wave-length of the surface instability mode. Furthermore, the Poisson’s ratio of elastic half-plane has a significant role in the surface instability and the associated wave-length. C. Q. Ru is on leave from the University of Alberta, Edmonton, T6G. 2G8, Canada.     

Stress concentrations associated with circular holes in cylinders and bone in torsion

Experimental studies were undertaken to determine the torsional stress concentration factors (Kt) associated with circular holes in bone. Reflective photoelasticity was used to determine the stress field around a circular hole through one wall of the bone. A single adult sheep femur was used as the torsional model, in which six circular holes were concentrically machined through the posterior cortex. These holes ranged from 10.4 percent to 66.4 percent of the mediolateral bone diameter. From the photoelastic data, a stress concentration curve was developed for bone. The maximum stress location on the boundary of the hole was found to shift from the previously expected 45-deg location. Studies on tubes made of steel and plastic, both coated with photoelastic coating, were also performed. Three different pieces of steel tubing with similar inner to outer diameters were coated with different thicknesses of photoelastic coating. The variation in coating thickness did not appear to influence the stress-concentration factors in steel. TheKt in steel for 10 percent and 20 percent defects agreed with theKt associated with similar defects in bone. A single piece of plastic tubing was used in which six holes from 10 percent to 60 percent of the tube's outer diameter were concentrically machined through one wall. The location of the maximum stress around the boundary of the hole was found to shift, and this agreed with the maximum stress shift found in the bone. Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 8–11.     

The exact solution for the general bending problems of conical shells on the elastic foundation

The general bending problem of conical shells on the elastic foundation (Winkler Medium) is not solved. In this paper, the displacement solution method for this problem is presented. From the governing differential equations in displacement form of conical shell and by introducing a displacement function U(s,θ), the differential equations are changed into an eight-order soluble partial differential equation about the displacement function U(s,θ) in which the coefficients are variable. At the same time, the expressions of the displacement and internal force components of the shell are also given by the displacement function U(s θ). As special cases of this paper, the displacement function introduced by V.S. Vlasov in circular cylindrical shell^[5], the basic equation of the cylindrical shell on the elastic foundation and that of the circular plates on the elastic foundation are directly derived.Under the arbitrary loads and boundary conditions, the general bending problem of the conical shell on the elastic foundation is reduced to find the displacement function U(s,θ).The general solution of the eight-order differential equation is obtained in series form. For the symmetric bending deformation of the conical shell on the elastic foundation, which has been widely usedinpractice,the detailed numerical results and boundary influence coefficients for edge loads have been obtained. These results have important meaning in analysis of conical shell combination construction on the elastic foundation,and provide a valuable judgement for the numerical solution accuracy of some of the same type of the existing problem.     

Towards the solution of finite plane-strain problems for compressible elastic solids

A new approach to the solution of finite plane-strain problems for compressible Isotropie elastic solids is considered. The general problem is formulated in terms of a pair of deformation invariants different from those normally used, enabling the components of (nominal) stress to be expressed in terms of four functions, two of which are rotations associated with the deformation. Moreover, the inverse constitutive law can be written in a simple form involving the same two rotations, and this allows the problem to be formulated in a dual fashion.For particular choices of strain-energy function of the elastic material solutions are found in which the governing differential equations partially decouple, and the theory is then illustrated by simple examples. It is also shown how this part of the analysis is related to the work of F. John on harmonic materials.Detailed consideration is given to the problem of a circular cylindrical annulus whose inner surface is fixed and whose outer surface is subjected to a circular shear stress. We note, in particular, that material circles concentric with the annulus and near its surface decrease in radius whatever the form of constitutive law within the given class. Whether the volume of the material constituting the annulus increases or decreases depends on the form of law and the magnitude of the applied shear stress.     

夹层FGM圆柱壳在扭转载荷作用下的弹性稳定性

采用半解析方法研究了两端简支的功能梯度夹层圆柱壳在端部扭转载荷作用下的弹性稳定性.考虑圆柱壳的里外表层为均匀材料,中间层为材料性质沿厚度方向连续变化的功能梯度材料,并且在界面处的材料性质保持连续. 基于Flügge薄壳理论,建立了位移形式的结构静态屈曲控制方程.根据边界条件将位移表示为三角级数形式,获得包含柱壳端部扭转载荷参数的近似线性代数特征值问题,并通过数值方法求得了表征结构失稳特征的临界载荷. 数值结果表明,临界载荷随着半径与厚度比的增加而减小,随着功能梯度中间层的弹性模量的平均值的增加而增加.      

Bending of rectangular plates carried by vacuum cups

In order to carry thin plates, vacuum cups are frequently used. When the over-hang is large, the deflections and stresses of the plate have considerably large values. In this paper, the rectangular plate hung by circular vacuum cups is treated. The analysis is carried out for the plate, which is subjected to line loads and radial bending moments at the inner circular boundary and free at the outer rectangular boundary. In addition to these boundary conditions, the plate is subjected to different distributed loads on inner and outer domains. First, the general solutions for the deflections on each domain are obtained by using infinite series, which are expressed by the polar coordinate system. The several undetermined constants in these equations are decreased by the conditions of continuity at the inner boundaries. Satisfying the boundary conditions at the finite points on the outer edges of the plate, the deflections and stresses of the plate and the contact pressures between the plate and the vacuum cup are calculated. Typical results are presented in dimensionless graphical form for different parameters and vacuum cup edge conditions.     

P波作用下饱和土中深埋圆形复合式衬砌隧道动应力响应研究

采用波函数展开法,推导出了平面P波入射下饱和土中深埋圆形双层衬砌动应力集中问题的解析解,并将该解退化为弹性介质中双层衬砌对平面P波散射的稳态解,验证了计算结果的正确性。研究结果表明,饱和土中双层衬砌的内边界动应力集中系数值大于外边界;外侧衬砌及内侧衬砌弹性模量对衬砌的动应力集中系数影响较大,在保证外侧衬砌稳定性较好的条件下,内侧衬砌尽可能选择刚度较小的材料;增加外侧衬砌及内侧衬砌的厚度可明显地减小衬砌内的动应力集中系数。