Contact formulation of non-linear problems in the mechanics of shells with their end sections connected by a plane curvilinear rod

Starting from the consistent version of the geometrically non-linear equations of the theory of elasticity for small deformations and arbitrary displacements, a Timoshenko-type model that takes account of shear and compression deformations and also an extended variational Lagrange principle, an improved geometrically non-linear theory of static deformation is constructed for reinforced thin-walled structures with shell elements, the end sections of which are connected by a rod. It is based on the introduction into the treatment of contact forces and torques as unknowns on the lines joining the shells to the rods and it enables all classical and non-classical forms of loss of stability in structures of the class considered to be investigated. An analytical solution of the problem of the stability of a rectangular plate, that is under compression in one direction, supported by a hinge along two opposite edges and joined by a hinge with an elastic rod on one of the other two edges, is found using a simplified version of the linearized equations.     

The stability of a ring under the action of a linear torque,constant along the perimeter

Exact analytical solutions of problems on the static and dynamic forms of the loss of stability of a ring, under the action of a linear torque constant along the perimeter, are found using the consistent equations of the theory of plane curvilinear rods constructed earlier taking account of transverse shears. Two forms of torsion of the ring are examined: the external forces creating a torque remain in the plane of a cross-section of the ring in its initial undeformed state (“dead” forces, case 1) or in its deformed state (“follower” forces, case 2). It is shown that, in the second case, the solution of the static instability problem found is practically identical to the solution of the problem corresponding to the dynamic formulation and is reduced to an examination of the oscillations about the static equilibrium position. In the case of both forms of loading, loss of stability of the ring occurs without deformation of its axial line, with it bending predominantly in the plane of the ring accompanied by a slight distortion. It is established that a study of the forms of loss of stability of the ring for the type of loading considered is only possible using the equations constructed, taking account of transverse shear.     

Bulging of a cylindrical shell with viscoelastic filler during axial compression

The problem of static stability during axial compression of an orthotropic cylindrical shell reinforced by a continuous filler is solved. The filler is considered to be a three-dimensional isotropic linear-viscoelastic body, fixed at its outer surface to the shell. Transverse shear in the shell is taken into account. The loss of stability of the structure is related to the time-dependent development of initial defects, following application of a load, which is smaller than the instant critical load.     

Dynamic and static loss in stability of flexible spherical shells made out of a composite material

The dynamic and static stability of shallow spherical shells which are rectangular in a plane are investigated. It is assumed that the shell is made out of a composite material which is weakly shear resistant and hence the refined theories which allow for transverse shear deformations and rotational inertia are applied. The solutions which were obtained are compared with solutions founded on the basis of the Kirchhoff-Love theory. It is shown that the results which are obtained on the basis of the classical theory are high for both the static and dynamic loss in stability, and are qualitatively different from the results obtained using the refined theory. The solutions were obtained using the Bubnov-Galerkin method in the higher approximations.     

Deformation of cylindrical composite shells under combined dynamic loading

Conclusions A procedure has been shown for calculating the stress-strain state of cylindrical multilayer shells made from composite materials under the combined action of dynamic axial compression and dynamic external pressure, as well as with different variants of combined loading with static and dynamic forces. An investigation has been made of the effect on the mode of the buckled shell surface of the ratio of the application rate of dynamic loads; ranges of loading rates have been established in which stresses predominate caused either by axial compression or external pressure. It has been shown that, as a result of preliminary static loading, a marked change occurs in the initial imperfections of the shell mode which affects subsequent dynamic buckling. To calculate the time when the first defect occurs and its location in the shell body, a procedure has been devised for layer-by-layer strength analysis employing a tensor-polynomial criterion. It was demonstrated that the level of preliminary static loading noticeably affects the time until the first failure of the layer, not only a reduction of this time being possible with an increase in the static loads, but also an increase in it.We should also point out the work in [10] where it is shown that it is possible to weaken the susceptibility of the shell to initial imperfections when internal pressure is applied.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 461–473, May–June, 1981.     

Research in the flexural buckling modes of a cylindrical sandwich shell under the action of a temperature field inhomogeneous across its thickness

A solution to the problem on the stability according to the flexural buckling mode is given for a cylindrical sandwich shell with a transversely soft core of arbitrary thickness. The shell is under the action of a temperature field inhomogeneous across the thickness, and its end faces are fastened in such a way (in the axial direction, the face sections of the external layer are fixed, but of the internal one are free) that an inhomogeneous subcritical stress-strain state arises in the shell across the thickness of its layers. It is shown that, under such conditions, the buckling mode of the shell is mixed flexural. To reveal and investigate this mode, equations of subcritical equilibrium and stability of a corresponding degree of accuracy are needed.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 715–730, November–December, 2004.     

Bending,buckling and free vibration analysis of incompressible functionally graded plates using higher order shear and normal deformable plate theory

In the present study, higher order shear and normal deformable plate theory is developed for analysis of incompressible functionally graded rectangular thick plates. Also, The effect of incompressibility is studied on the static, dynamic and stability responses of thick plate. It is assumed that plate is incompressible and the incompressibility condition is considered in addition to the governing equations for determining the unknowns. Since the plate is thick, higher order shear and normal deformable theory is applied so that the Legendre polynomials are used for expansion of displacement field components in the thickness direction. Also, it is supposed that material properties vary through the thickness based on the power law function. Utilizing the variational approach, governing equations for static, stability and dynamic analysis of plate are derived. Resulted equations are solved analytically for simply supported plates. Finally, the effects of material properties and dimensions on the response of incompressible plates are investigated in details.     

Effect of an elastic core on the dynamic stability of an orthotropic cylindrical shell

A method of determining the regions of dynamic instability of an orthotropic cylindrical shell "bonded" to an elastic cylinder is proposed. An expression for the core reaction is obtained from the coupling conditions for the forces normal to the lateral surface and the radial displacements of the shell and the core at the contact surface. When the reaction is substituted in the system of equations of motion of the shell, the part corresponding to the free vibrations of the cylinder is discarded. The system of equations of motion of the shell is reduced to an equation of Mathieu type, from which transcendental equations for determining the boundaries of the regions of dynamic instability are obtained. These regions are analyzed for various modes of loss of stability and different values of the core modulus of elasticity.     

Stability with the axial compression of an axisymmetrically heated orthotropic cylindrical shell fastened to an elastic cylinder which is pliable with respect to shear

The article gives a solution to the problem of stability with the axial compression of an axisymmetrically heated orthotropic cylindrical shell fastened to an elastic thin-walled cylinder through an intermediate layer. It is assumed that the parameters of the elasticity of the orthotropic shell depend on the temperature, and vary over the thickness of the wall. The intermediate layer is assumed to be isotropic and absolutely rigid in a radial direction, but pliable with respect to axial shear. The thin-walled cylinder is considered to be elastic, isotropic, and unheated.Scientific-Research Institute for Chemical Engineering, Moscow Region. Translated from Mekhanika Polimerov, No. 3, pp. 546–550, May–June, 1970.     

Linearly Coupled,Slowly Varying Oscillators

A dynamical system is considered whose normal frequencies and normal modes vary slowly with time in such a way that two frequencies come into close coincidence. When this occurs the corresponding normal modes undergo a drastic change in their physical properties. Away from coincidence, each normal mode conserves its action. A multiple-time-scale asymptotic procedure is employed to derive equations which describe the mode coupling at coincidence. These equations are solved exactly using parabolic cylinder functions. It is found that in general, action is exchanged between modes at coincidence, but that except for very strong coupling the amount of action exchanged is quite small.     

大位移Timoshenko转轴三维耦合动力学分析

对于高速柔性转轴,综合考虑滑移、弯曲、剪切变形、旋转惯性、陀螺效应和动不平衡等因素,运用Timoshenko旋转梁理论,给出弹性体空间运动的一般性描述,通过Hamilton原理建立弯曲-扭转-轴向三维耦合非线性动力学方程,应用参数摄动方法和假设振型方法进行化简,并用数值模拟分析了轴向刚性滑移、剪切变形、截面尺寸和转速等因素对转轴动力学响应的影响。     

轴向复合载荷作用下变截面悬臂弹性立柱后屈曲分析

基于轴线可伸长弹性杆的几何非线性理论,建立了同时作用端部轴向集中荷载和沿轴线作用分布轴向载荷的变截面弹性悬臂柱的后屈曲控制方程。采用打靶法直接求解了所得强非线性边值问题,给出了截面线性变化的圆截面柱的二次平衡路径及其过屈曲位形曲线。     

The dynamic stability and nonlinear vibration analysis of stiffened functionally graded cylindrical shells

A theoretical model is developed to study the dynamic stability and nonlinear vibrations of the stiffened functionally graded (FG) cylindrical shell in thermal environment. Von Kármán nonlinear theory, first-order shear deformation theory, smearing stiffener approach and Bolotin method are used to model stiffened FG cylindrical shells. Galerkin method and modal analysis technique is utilized to obtain the discrete nonlinear ordinary differential equations. Based on the static condensation method, a reduction model is presented. The effects of thermal environment, stiffeners number, material characteristics on the dynamic stability, transient responses and primary resonance responses are examined.     

Problems of geometric non-linearity and stability in the mechanics of thin shells and rectilinear columns

An analysis of the current state of the geometrically non-linear theory of elasticity and of thin shells is presented in the case of small deformations but large displacements and rotations, the ratios of which are known as the ratios of the non-linear theory in the quadratic approximation. It is shown that they required specific revision and correction by virtue of the fact that, when they are used in the solution of problems, spurious bifurcation points appear. In view of this, consistent geometrically non-linear equations of the theory of thin shells of the Timoshenko type are constructed in the quadratic approximation which enable one to investigate in a correct formulation both flexural as well as previously unknown non-classical forms of loss of stability (FLS) of thin plates and shells, many of which are encountered in practice, primarily in structures made of composite materials with a low shear stiffness. In the case of rectilinear elastic whereas, which are subjected to the action of conservative external forces and are made of an orthotropic material, the three-dimensional equations of the theory of elasticity are reduced to one-dimensional equations by using the Timoshenko model. Two versions of the latter equations are derived. The first of these corresponds to the use of the consistent version of the three-dimensional, geometrically non-linear relations in an incomplete quadratic approximation and the Timoshenko model without taking account of the transverse stretching deformations, and the second corresponds to the use of the three- dimensional relations in the complete quadratic approximation and the Timoshenko model taking account of the transverse stretching deformations. A series of new non-classical problems of the stability of columns is formulated and their analytical solutions are found using the equations which have been derived with the aim of analyzing their richness of content. Among these are problems concerning the torsional, flexural and shear FLS of a column in the case of a longitudinal axial, bilateral transverse and trilateral compression, a flexural-torsional FLS in the case of pure bending and axial compression together with pure bending and, also, a flexural FLS of a column in the case of torsion and a flexural-torsional FLS under conditions of pure shear. Five FLS of a cylindrical shell under torsion are investigated using the linearized neutral equilibrium equations which have been constructed: 1) a torsional FLS where the solution corresponding to it has a zero variability of the functions in the peripheral direction, 2) a purely beam bending FLS that is possible in the case of long shells and is accompanied by the formation of a single half-wave along the length of the shell while preserving the initial circular form of the cross-section, 3) a classical bending FLS, which is accompanied by the formation of a small number of half-waves in the axial direction and a large number of half-waves in a peripheral direction which is true in the case of long shells, 4) a classical bending FLS which holds in the case of short and medium length shells (the third and fourth types of FLS have already been thoroughly studied in the case of isotropic cylindrical shells), 5) a non-classical FLS characterized by the formation of a large number of shallow depressions in the axial as well as in the peripheral directions; the critical value of the torsional moment corresponding to this FLS is practically independent of the relative thickness of the shell. It is established that the well-known equations of the geometrically non-linear theory of shells, which were formulated for the case of the mean flexure of a shell, do not enable one to reveal the first, second and fifth non-classical FLS.     

Geometrically non-linear equations in the theory of momentless shells with applications to problems on the non-classical forms of loss of stability of a cylinder

Geometrically non-linear and linearized equations in the theory of momentless shells are set up based on the kinematic relations in [Paimushin VN, Shalashilin VI. Relations of the theory of deformations in the quadratic approximation and problems of constructing refined versions of the geometrically non-linear theory of laminar structural components. Prikl Mat Mekh 2005; 69(5): 861–81]. The use of these equations, unlike in the case of the well-known equations, enables one to avoid the occurrence of spurious bifurcation points in solving real problems. Non-classical problems of the stability of cylindrical shells under an external pressure, axial compression and torsion are considered, which can be formulated on the basis of the derived equations of the theory of momentless shells. Their exact analytical solutions are found and enable one to estimate the quality of the previously obtained relations [Paimushin VN, Shalashilin VI. Relations of the theory of deformations in the quadratic approximation and problems of constructing refined versions of the geometrically non-linear theory of laminar structural components. Prikl Mat Mekh 2005; 69(5): 861–81] and the richness of content of the equations which have been constructed compared with well-known equations in the mechanics of thin shells. It is established that the majority of the new forms of loss of stability of cylindrical shells which are revealed relate to a number of shear forms, the onset of which is possible before the flexural forms which have been well studied up to now, in the case of small values of the shear modulus of a shell material with a very highly pronounced anisotropy in its properties.     

The non-linear Saint-Venant problem of the torsion,stretching and bending of a naturally twisted rod

Problems on large stretching, torsional and bending deformations of a naturally twisted rod, loaded with end forces and moments, are considered from the point of view of the non-linear three-dimensional theory of elasticity. Particular solutions of the equations of elastostatics are found, which are two-parameter families of finite deformations and which possess the property that, for these deformations, the initial system of three-dimensional non-linear equations reduces to a system of equations with two independent variables. The use of these equations enables one to reduce certain Saint-Venant problems for a naturally twisted rod to two-dimensional non-linear boundary-value problems for a planar domain in the form of the cross-section of a rod. Different formulations of the two-dimensional boundary-value problem for the cross-section are proposed, which differ in the choice of the unknown functions. A non-linear problem of the torsion and stretching of a circular cylinder with helical anisotropy, which is reduced to ordinary differential equations, is considered as a special case.     

Study of wave processes in shells and plates with regard to transverse normal and shear deformation

A variant of the theory of orthotropic plates and cylindrical shells taking account of transverse normal and shear deformation was examined. Independent approximations were adopted for distribution of displacements and stresses over the thickness of the shell. One of the requirements for constructing the theory is physical correctness, which is achieved by utilizing variational methods for formulating the mathematical model. The Reissner principle for dynamic processes was used for derivation of the equations. The elliptical part of the starting differential operator was shown to be symmetrical and positive in the space of the integrate of square functions. We examined the problem of the propagation of axially symmetric harmonic waves in the cylinder using the starting differential equations. These results were compared with those obtained equations derived in elasticity theory. Analysis of induced vibration was carried out for the case of a square plate upon the action of a suddenly applied load.Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, Latvia, October, 1995.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 6. pp. 816–823, November–December, 1995.     

The small free oscillations of a strip

The refined equations of the free oscillations of a rod-strip, constructed previously in a first approximation by reducing the two-dimensional equations to one-dimensional equations by using trigonometric basis functions and satisfying the static boundary conditions on the boundary surfaces are analysed. These equations, the solutions of which are obtained for the case of hinge-supported end sections of the rod, are split into two independent systems of equations. The first of these describe non-classical fixed longitudinal-transverse forms of free oscillations, which are accompanied by a distortion of the plane form of the cross section. It is shown that the oscillation frequencies corresponding to them depend considerably on Poisson's ratio and the modulus of elasticity in the transverse direction, while for a rod of average thickness for the same value of the frequency parameter (the tone) they may be considerably lower than the frequencies corresponding to the classical longitudinal forms of free oscillations, which are performed while preserving the plane form of the cross sections. The second system of equations describes transverse flexural-shear forms of free oscillations, whose frequencies decrease as the transverse shear modulus decreases. They are practically equivalent in quality and content to the similar equations of well-known versions of the refined theories, but, unlike them, when the number of the tone increases and the relative thickness parameter decreases they lead to the solutions of the classical theory of rods.     

弹性介质中受轴向冲击载荷作用的圆柱壳的屈曲临界载荷计算分析

建立并求解了弹性介质中圆柱壳的径向位移控制方程,考虑边界条件及相容条件,得到了应力波传播及反射过程中圆柱壳的动力屈曲分叉条件.通过计算得到了不同时间段屈曲临界载荷与应力波波阵面到达圆柱壳的位置、弹性介质的刚度、壳体未嵌入弹性介质部分的长度与总长之比的关系.数值计算结果表明,弹性介质中的圆柱壳发生轴对称屈曲和非轴对称屈曲趋势一致;嵌入弹性介质部分越深、弹性介质刚度越大圆柱壳越难屈曲;屈曲临界载荷随着弹性介质刚度的增大经历了增长缓慢、增长迅速以及增长较慢3个阶段;应力波反射前波阵面通过分界面后,屈曲仅发生在应力波传播区域,反射波波阵面通过分界面前,临界载荷较小时屈曲先发生在反射端部,随着轴向阶数增大,屈曲覆盖整个圆柱壳区域,反射波波阵面通过分界面后,壳体发生的屈曲始终覆盖整个圆柱壳区域.