电活性聚合物球壳的力电不稳定性

应用连续介质力学有限变形理论,分析了不可压电活性聚合物球壳在外加电场及内压作用下发生非对称变形的力电不稳定性问题。文中给出了不同外加电场下球壳的变形曲线和应力分布曲线, 结果表明对壁厚小于临界壁厚值的薄壁球壳,当内压大于临界内压值时,球壳可以产生不稳定的非对称变形。文中求得了球壳发生不稳定变形的临界壁厚及临界内压,探讨了外加电场对两个临界值的影响规律,同时讨论了外加电场对球壳中应力分布的影响。     

Dynamic elastic and plastic buckling of complete spherical shells

A theoretical investigation is undertaken into the dynamic instability of complete spherical shells which are loaded impulsively and made from either linear elastic or elastic-plastic materials. It is shown that certain harmonics grow quickly and cause a shell to exhibit a wrinkled shape which is characterized by a critical mode number. The critical mode numbers are similar for spherical and cylindrical elastic shells having the same R/h ratios and material parameters, but may be larger or smaller in an elastic-plastic spherical shell depending on the values of the various parameters. Threshold velocities are also determined in order to obtain the smallest velocity that a shell can tolerate without excessive deformation. The threshold velocities for the elastic and elastic-plastic spherical shells are larger than those which have been published previously for cylindrical shells having the same R/h ratios and material parameters.     

Effect of thickness inhomogeneities in internally pressurized elastic-plastic spherical shells

The behaviour of elastic-plastic spherical shells under internal pressure is investigated numerically for thickness-to-radius ratios ranging from cases of thin shells to very thick shells. The shells under consideration are made of strain-hardening elastic-plastic material with a smooth yield-surface. Attention is restricted to axisymmetric deformations, and results are presented for initial thickness inhomogeneities in various axisymmetric shapes. For smooth thickness-variations in the shape of the critical bifurcation mode, the reduction in maximum pressure is studied together with the distribution of deformations in the final collapse mode. Also, the possibility of flow localization due to more localized, initially thin regions on a spherical shell is investigated.     

Electromechanical responses and instability of electro-active polymer cylindrical shells

Based on the theories of finite deformation elasticity, electromechanical responses and instability of an incompressible electro-active polymer (EAP) cylindrical shell, which is subjected to an internal pressure and a static electric field, are studied. Deformation curves and distribution of stresses are obtained. It is found that an internal pressure together with an electric field may cause the unstable non-monotonic deformation of the shell. It is also shown that a critical thickness for the shell exists, and the shell may undergo the unstable deformation if its thickness is less than this critical value. In addition, the effects of the electric field, axial stretch, thickness, and internal pressure on the instability of the shell are discussed.     

A numerical analysis of spherical indentation response of thin hard films on soft substrates

In this paper, finite element simulations of spherical indentation of a thin hard film deposited on a soft substrate are carried out. The primary objective of this work is to understand the operative mode of deformation of the film corresponding to various stages of indentation. The transition from contact dominant behaviour to that governed by flexure of the film on the plastically yielding substrate is investigated from analysis of the load versus displacement curve as well as the stress distribution in the film. It is found that onset of bending deformation in the film occurs when the contact radius is about 0.2–0.3 of the film thickness. Further, distinct membrane stresses arise in the film for indentation depth greater than half the film thickness. The implications of these results on indentation fracture of the film are briefly discussed. Finally, the effects of substrate yield strength and presence of residual stresses on the indentation response are examined.     

Vibration analysis of nonhomogeneous orthotropic cylindrical shells including combined effect of shear deformation and rotary inertia

This paper is the result of an investigation on the vibration of non-homogeneous orthotropic cylindrical shells, based on the shear deformation theory. Assume that the Young’s moduli, shear moduli and density of the orthotropic material are continuous functions of the coordinate in the thickness direction. The basic equations of non-homogeneous orthotropic cylindrical shells with the shear deformation and rotary inertia are derived in the framework of Donnell-type shell theory. The ends of a non-homogeneous orthotropic cylindrical shell are considered as simply supported. The basic equations are reduced to the sixth-order algebraic equation for the frequency using the Galerkin method. Solving this algebraic equation, the lowest values of non-dimensional frequency parameters for non-homogeneous orthotropic cylindrical shells with and without shear deformation and rotary inertia are obtained. Calculations, effects of shear stresses and rotary inertia, orthotropy, non-homogeneity and shell geometry parameters on the lowest values of non-dimensional frequency parameter are described. The results are verified by comparing the obtained values with those in the existing literature.     

Non-linear axisymmetric response of functionally graded shallow spherical shells under uniform external pressure including temperature effects

This paper presents an analytical approach to investigate the non-linear axisymmetric response of functionally graded shallow spherical shells subjected to uniform external pressure incorporating the effects of temperature. Material properties are assumed to be temperature-independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Equilibrium and compatibility equations for shallow spherical shells are derived by using the classical shell theory and specialized for axisymmetric deformation with both geometrical non-linearity and initial geometrical imperfection are taken into consideration. One-term deflection mode is assumed and explicit expressions of buckling loads and load-deflection curves are determined due to Galerkin method. Stability analysis for a clamped spherical shell shows the effects of material and geometric parameters, edge restraint and temperature conditions, and imperfection on the behavior of the shells.     

内压椭球壳塑性变形的发生部位与扩展过程分析

对内压椭球的就发布作了图形描述，指出椭球壳上一点的应力状态只与壳体的轴比λ和该点在壳体上的位置有关，并进一步分析了内压椭球壳的塑性变形的发生部位及其扩展过程。     

Relationships of the Timoshenko-type theory of thin shells with arbitrary displacements and strains

A new modified version of the Timoshenko theory of thin shells is proposed to describe the process of deformation of thin shells with arbitrary displacements and strains. The new version is based on introducing an unknown function in the form of a rotation vector whose components in the basis fitted to the deformed mid-surface of the shell are the components of the transverse shear vector and the extensibility in the transverse direction according to Chernykh. For the case with the shell mid-surface fitted to an arbitrary non-orthogonal system of curvilinear coordinates, relationships based on the use of true stresses and true strains in accordance with Novozhilov are obtained for internal forces and moments. Based on these relationships, a problem of static instability of an isotropic spherical shell experiencing internal pressure is solved. The shell is considered to be made either of a linear elastic material or of an elastomer (rubber), which is described by Chernykh’s relationships.     

Optimal design of axisymmetric plastic shallow shells of von Mises material

An optimal design technique developed earlier for axisymmetric plates and circular cylindrical shells is accommodated for shallow spherical shells subjected to uniform transverse pressure. Material of the shells is assumed to be rigid-plastic obeying the von Mises yield condition and the associated deformation law. The post-yield behaviour of the shells is taken into account. The weight minimization is performed under the condition that the maximal deflection of the shell of variable thickness coincides with the deflection of the reference shell of constant thickness. The problem is transformed into a non-linear boundary value problem which is solved numerically.     

Global optimum design of externally pressurized isogrid stiffened cylindrical shells with added T-rings

PANDA2 is a code for the minimum-weight design of perfect and imperfect elastic stiffened panels and shells made of composite laminates and subjected to multiple sets of in-plane loads, edge moments, normal pressure, and temperature. The scope of PANDA2 is increased to include global optimization and the capability to handle isogrid stiffening. The enhanced program is used to find global optimum designs of internally T-isogrid and internally T-ring stiffened perfect and imperfect isotropic cylindrical shells under uniform external pressure. For the cases studied, it is found that for the perfect optimized shells the isogrid stiffening is important but the rings are not, whereas the opposite holds for the optimized shells with an initial general buckling modal imperfection of amplitude equal to one per cent of the shell radius     

Nonlinear bending response and buckling of ring-stiffened cylindrical shells under pure bending

In this paper, the nonlinear bending response of finite length cylindrical shells with stiffening rings is investigated by using a modified Brazier approach. The basic assumptions for the present study are that the deformation of a shell subjected to pure bending can be simplified into a two-stage process. One is that the shell ovalizes but its axis remains straight; the other is that the bending of the shell is regarded as a beam with nonuniform ovalization. The nonlinear bending response is derived by applying the minimum potential energy principle and the corresponding critical moment, associated with local buckling, is determined by employing the Seide–Weingarten approximation. Numerical results are shown and compared with those obtained from other methods, which demonstrates that the assumptions used in the present study are reasonable.     

Postbuckling of internal pressure loaded FGM cylindrical shells surrounded by an elastic medium

This paper presents a study on the postbuckling response of a functionally graded cylindrical shell of finite length embedded in a large outer elastic medium and subjected to internal pressure in thermal environments. The surrounding elastic medium is modeled as a tensionless Pasternak foundation that reacts in compression only. The postbuckling analysis is based on a higher order shear deformation shell theory with von Kármán–Donnell-type of kinematic nonlinearity. The thermal effects due to heat conduction are also included and the material properties of functionally graded materials (FGMs) are assumed to be temperature-dependent. The nonlinear prebuckling deformations and the initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the postbuckling response of the shells and an iterative scheme is developed to obtain numerical results without using any assumption on the shape of the contact region between the shell and the elastic medium. Numerical solutions are presented in tabular and graphical forms to study the postbuckling behavior of FGM shells surrounded by an elastic medium of tensionless elastic foundation of the Pasternak-type, from which results for conventional elastic foundations are obtained as comparators. The results reveal that the unilateral constraint has a significant effect on the postbuckling response of shells subjected to internal pressure in thermal environments when the foundation stiffness is sufficiently large.     

Influences of large deformation and rotary inertia on wave propagation in piezoelectric cylindrically laminated shells in thermal environment

Influences of large deformation (geometrical non-linear) and rotary inertia on wave propagation in a long, piezoelectric cylindrically laminated shell in thermal environment is presented in this paper. Nonlinear dynamic governing equations of piezoelectric cylindrically laminated shells are derived by means of Hamilton’s principle. The wave propagation modes are obtained by solving an eigenvalue problem. Numerical examples show that the characteristics of wave propagation in piezoelectric cylindrically laminated shells are relates to the large deformation, rotary inertia and thermal environment of the piezoelectric cylindrically laminated shells. The effect of large deformation, rotary inertia and thermal load on wave propagation in the piezoelectric cylindrically laminated shells is discussed by comparing with the result from the small deformation (geometrical linear shell theory). This method may be used to investigate wave propagation in various laminated material, layers numbers and thickness of piezoelectric cylindrically laminated shells under large deformation. The results carried out can be used in the ultrasonic inspection techniques and structural health monitoring.     

Forced axisymmetric motions of viscoelastic cylindrical shells

The isothermal response of a viscoelastic cylindrical shell, of finite length, to arbitary axisymmetric surface forces, initial conditions, and boundary conditions is considered within the linear theory of thin shells. The problem is formulated with the effects of shear deformation and rotatory inertia included; the viscoelastic properties are assumed to be isotropic and homogeneous. The response is first found formally in terms of a causal Green's function. It is then shown that when Poisson's ratio is constant, the causal Green's function can be expanded in a series of orthonormal spatial eigenfunctions of an associated elastic shell eigenvalue problem. The resulting solution for the general problem is an eigenfunction series with Laplace transformed time-dependent coefficients. The general solution is applied to predicting the motion of a uniform, simply-supported cylindrical shell, initially quiescent, which is subjected to a step pressure moving with constant velocity. For this example, the relaxation function of the shell material in uniaxial extension is taken to be that of a standard linear solid. The motions predicted by simpler shell models, namely, shells with bending only and without bending, are also considered for comparison. Here, the absolute values of the Fourier coefficients in the shell displacement series go to zero faster than the inverse of the first or second power of positive integers when bending is excluded or included, respectively. Numerical results are presented for a moderately long and relatively thick, nearly elastic, cylindrical shell.     

Nonlinear buckling and postbuckling of heated functionally graded cylindrical shells under combined axial compression and radial pressure

The nonlinear large deflection theory of cylindrical shells is extended to discuss nonlinear buckling and postbuckling behaviors of functionally graded (FG) cylindrical shells which are synchronously subjected to axial compression and lateral loads. In this analysis, the non-linear strain-displacement relations of large deformation and the Ritz energy method are used. The material properties of the shells vary smoothly through the shell thickness according to a power law distribution of the volume fraction of the constituent materials. Meanwhile, by taking the temperature-dependent material properties into account, various effects of external thermal environment are also investigated. The non-linear critical condition is found by defining the possible lowest point of external force. Numerical results show various effects of the inhomogeneous parameter, dimensional parameters and external thermal environments on non-linear buckling behaviors of combine-loaded FG cylindrical shells. In addition, the postbuckling equilibrium paths are also plotted for axially loaded pre-pressured FG cylindrical shells and there is an interesting mode jump exhibited.     

Radial oscillations of nonhomogeneous,thick-walled cylindrical and spherical shells subjected to finite deformations

The infinitesimal breathing motions of long cylindrical tubes and hollow spherical shells of arbitrary wall thickness subjected to a finite deformation field caused by uniform internal and/or external pressures are investigated. A neo-Hookean material with a material constant varying continuously along the radial direction is used. The shell is first subjected to finite static deformations and is then exposed to a secondary dynamic displacement field. Based on the theory of small deformations superposed on large deformations, closed form expressions are obtained for the frequency of small oscillations about the highly prestressed state. Frequency versus initial deformation parameter curves are given for several nohomogeneity functions and for various wall thicknesses.     

Non-linearities in rotation and thickness deformation in a new third-order thickness deformation theory for static and dynamic analysis of isotropic and laminated doubly curved shells

A geometrically non-linear theory is developed for shells of generic shape allowing for third-order thickness and shear deformation and rotary inertia by using eight parameters; geometric imperfections are also taken into account. The geometrically non-linear strain–displacement relationships are derived retaining full non-linear terms in all the 8 parameters, i.e. in-plane and transverse displacements, rotations of the normal and thickness deformation parameters; these relationships are presented in curvilinear coordinates, ready to be implemented in computer codes. Higher order terms in the transverse coordinate are retained in the derivation so that the theory is suitable also for thick laminated shells. Three-dimensional constitutive equations are used for linear elasticity. The theory is applied to circular cylindrical shells complete around the circumference and simply supported at both ends to study initially static finite deformation. Both radially distributed forces and displacement-dependent pressure are used as load and results for different shell theories are compared. Results show that a 6 parameter non-linear shell theory is quite accurate for isotropic shells. Finally, large-amplitude forced vibrations under harmonic excitation are investigated by using the new theory and results are compared to other available theories. The new theory with non-linearity in all the 8 parameters is the only one to predict correctly the thickness deformation; it works accurately for both static and dynamics loads.     

Axially compressed buckling of pressured multiwall carbon nanotubes

This paper studies axially compressed buckling of an individual multiwall carbon nanotube subjected to an internal or external radial pressure. The emphasis is placed on new physical phenomena due to combined axial stress and radial pressure. According to the radius-to-thickness ratio, multiwall carbon nanotubes discussed here are classified into three types: thin, thick, and (almost) solid. The critical axial stress and the buckling mode are calculated for various radial pressures, with detailed comparison to the classic results of singlelayer elastic shells under combined loadings. It is shown that the buckling mode associated with the minimum axial stress is determined uniquely for multiwall carbon nanotubes under combined axial stress and radial pressure, while it is not unique under pure axial stress. In particular, a thin N-wall nanotube (defined by the radius-to-thickness ratio larger than 5) is shown to be approximately equivalent to a single layer elastic shell whose effective bending stiffness and thickness are N times the effective bending stiffness and thickness of singlewall carbon nanotubes. Based on this result, an approximate method is suggested to substitute a multiwall nanotube of many layers by a multilayer elastic shell of fewer layers with acceptable relative errors. Especially, the present results show that the predicted increase of the critical axial stress due to an internal radial pressure appears to be in qualitative agreement with some known results for filled singlewall carbon nanotubes obtained by molecular dynamics simulations.     

Stability and limit states of elastoplastic spherical shells under static and dynamic loading

The problem of elastoplastic deformation, buckling, and postcritical behavior of spherical shells is solved using a finite element method and a cross-type explicit scheme of time integration. Stability problems for hemispherical shells under external pressure and compression between rigid plates are considered. The influence of holes and boundary conditions on shell deformation is investigated. It is shown that the calculation results are in good agreement with experimental data.