Model of micropolar thin shell oscillations

The paper gives the equations of a general applied two-dimensional theory for the dynamics of micropolar elastic thin shells with independent fields of motion and rotation that completely take into account all rotation-shift and related deformations. Problems on free and forced oscillations of micropolar elastic shells are studied on the basis of this general theory. Special features for the dynamic behavior of shells made of a micropolar elastic material are revealed on the basis of numerical analysis.     

Mathematical model of the dynamics of micropolar elastic thin beams. Free and forced vibrations

A method of hypotheses has been developed to construct a mathematical model of micropolar elastic thin beams. The method is based on the asymptotic properties of the solution ofan initial boundary value problem in a thin rectangle within the micropolar theory of elasticity with independent displacement and rotation fields. An applied model of the dynamics of micropolar elastic thin beams was constructed in which transverse shear strains and related strains are taken into account. The constructed dynamics model was used to solve problems of free and forced vibrations of a micropolar beam. Free vibration frequencies and modes, forced vibration amplitudes, and resonance conditions were determined. The obtained numerical calculation results show the specific features of free vibrations of thin beams. Micropolar thin beams have a free vibration frequency which is almost independent of the thin beam size, but depends only on the physical and inertial properties of the micropolar material. It is shown for the micropolar material that the free vibration frequency values of beams can be readily adjusted and hence a large vibration frequency separation can be achieved, which is important for studying resonance.     

General dynamic theory of micropolar elastic thin plates with free rotation and special features of their natural oscillations

A general applied two-dimensional theory for the dynamics of micropolar elastic thin plates that takes into account rotationally shear and related deformations is developed on the basis of the method of hypotheses with asymptotic confirmation. A problem of natural oscillations of micropolar elastic thin rectangular plates is solved using it. Special features of the dynamic characteristics of elastic thin plates made of a micropolar elastic material are revealed within the framework of numerical analysis.     

Dynamic model of micropolar elastic thin plates with independent fields of displacements and rotations

A general model of dynamic bending of isotropic micropolar elastic thin plates with independent fields of displacements and rotations is presented. The model has been justified asymptotically based on the solutions for special cases subject to simplifying assumptions. The model incorporates transverse shear deformations. Neglecting transverse shear, a model of the dynamics of micropolar elastic thin plates is also constructed. Then, we study free and forced oscillations and derive the natural frequencies, the amplitudes of the forced oscillations and the resonance conditions for micropolar elastic hinge-supported rectangular and circular plates. Finally, the basic characteristic features of micropolar plates are numerically analysed for different values of various elastic and inertial constants of the micropolar material.     

Vibration of cylindrical shells of bimodulus composite materials

A theory is formulated for the small amplitude free vibration of thick, circular cylindrical shells laminated of bimodulus composite materials, which have different elastic properties depending upon whether the fiber-direction strain is tensile or compressive. The theory used is the dynamic, shear deformable (moderately thick shell) analog of the Sanders best first approximation thin shell theory. By means of tracers, the analysis can be reduced to that of various simpler shell theories, namely Love's first approximation, and Donnell's shallow shell theory. As an example of the application of the theory, a closed form solution is presented for a freely supported panel or complete shell. To validate the analysis, numerical results are compared with existing results for various special cases. Also, the effects of the various shell theories, thickness shear flexibility, and bimodulus action are investigated.     

A comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach

Large-amplitude (geometrically non-linear) vibrations of circular cylindrical shells subjected to radial harmonic excitation in the spectral neighbourhood of the lowest resonances are investigated. The Lagrange equations of motion are obtained by an energy approach, retaining damping through Rayleigh's dissipation function. Four different non-linear thin shell theories, namely Donnell's, Sanders-Koiter, Flügge-Lur’e-Byrne and Novozhilov's theories, which neglect rotary inertia and shear deformation, are used to calculate the elastic strain energy. The formulation is also valid for orthotropic and symmetric cross-ply laminated composite shells. The large-amplitude response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of the lowest natural frequency is computed for all these shell theories. Numerical responses obtained by using these four non-linear shell theories are also compared to results obtained by using the Donnell's non-linear shallow-shell equation of motion. A validation of calculations by comparison with experimental results is also performed. Both empty and fluid-filled shells are investigated by using a potential fluid model. The effects of radial pressure and axial load are also studied. Boundary conditions for simply supported shells are exactly satisfied. Different expansions involving from 14 to 48 generalized co-ordinates, associated with natural modes of simply supported shells, are used. The non-linear equations of motion are studied by using a code based on an arclength continuation method allowing bifurcation analysis.     

STRUCTURAL RESPONSE OF LAMINATED COMPOSITE SHELLS SUBJECTED TO BLAST LOADING: COMPARISON OF EXPERIMENTAL AND THEORETICAL METHODS

The results from a theoretical and experimental investigation of the dynamic response of cylindrically curved laminated composite shells subjected to normal blast loading are presented. The dynamic equations of motion for cylindrical laminated shells are derived using the assumptions of Love's theory of thin elastic shells. Kinematically admissible displacement functions are chosen to represent the motion of the clamped cylindrical shell and the governing equations are obtained in the time domain using the Galerkin method. The time-dependent equations of the cylindrically curved laminated shell are then solved by the Runge-Kutta-Verner method. Finite element modelling and analysis for the blast-loaded cylindrical shell are also presented. Experimental results for cylindrically curved laminated composite shells with clamped edges and subjected to blast loading are presented. The blast pressure and strain measurements are performed on the shell panels. The strain response frequencies of the clamped cylindrical shells subjected to blast load are obtained using the fast Fourier transformation technique. In addition, the effects of material properties on the dynamic behaviour are examined. The strain-time history curves show agreement between the experimental and analysis results in the longitudinal direction of the cylindrical panels. However, there is a discrepancy between the experimental and analysis results in the circumferential direction of the cylindrical panels. A good prediction is obtained for the response frequency of the cylindrical shell panels.     

An analytical model to predict the response of fluid-filled shells to impact—a model for blunt head impacts

An approximate analytical model to predict the response of a fluid-filled shell of arbitrary thickness impacting with a solid elastic sphere is proposed and the limits of applicability of the equations developed are discussed. The model is based on combining the Hertzian contact stiffness and the effective local membrane and bending stiffness to derive implicit expressions for global impact characteristics including the duration of impact, the peak force transmitted, peak global acceleration of shell and sphere, and the resultant pressures induced in the fluid. Closed-form explicit expressions are also derived to predict whether the pressure response in the fluid will be hydrostatic or will exhibit large dynamic transients of pressure (and shear strain). It should be noted that the impact of hollow/empty shells with solid spheres, as well as the impact of shells with an elastic half-space, can be straightforwardly treated as limiting cases. The model is of obvious relevance to head impact modelling and selected parametric studies of the response of fluid-filled shells with geometric and material properties about those typical for the human head are given.     

Free vibrations of cylindrical shells with elastic-support boundary conditions

This paper concerns the free vibrations of cylindrical shells with elastic boundary conditions. Based on the Flügge classical thin shell theory, the equations of motion for the cylindrical shells are solved by the method of wave propagations. The wave numbers are obtained by directly solving an eighth order equation. The elastic-support boundary conditions can be arbitrarily specified in terms of 8 independent sets of distributed springs. All the classical homogeneous boundary conditions can be considered as the special cases when the stiffness for each set of springs is equal to either infinity or zero. The present solutions are validated by the results previously given by other researchers and/or obtained using finite element models. The effects on the frequency parameters of elastic restraints are investigated for shells of different geometrical characteristics.     

Effect of elastic end rings on the eigenfrequencies of thin cylinders

The effect of elastic end rings on the eigenfrequencies of thin cylindrical shells is studied by using an exact solution of the linear eigenvalue problem. The out-of-plane and torsional rigidities of the rings are responsible for the overall shell stiffness. Considerable mode interaction exists for modes with low circumferential wave numbers when the mass of the ring is comparable with that of the shell. The hypothetical simply supported and clamped boundary conditions are practically impossible to realize with a finite-mass ring for relatively short and thin shells.     

Large deformation of liquid capsules enclosed by thin shells immersed in the fluid

The deformation of a liquid capsule enclosed by a thin shell in a simple shear flow is studied numerically using an implicit immersed boundary method. We present a thin-shell model for computing the forces acting on the shell middle surface during the deformation within the framework of the Kirchhoff–Love theory of thin shells. This thin-shell model takes full account of finite-deformation kinematics which allows thickness stretching as well as large deflections and bending strains. For hyperelastic materials, the plane-stress assumption is used to compute the hydrostatic pressure and the incompressibility condition yields the thickness strain component and the corresponding change in the thickness. The stresses developing over the cross-section of the shell are integrated over the thickness to yield the stress and moment resultants which are then used to compute the forces acting on the shell middle surface. The immersed boundary method is employed for calculating the hydrodynamics and fluid–structure interaction effects. The location of the thin shell is updated implicitly using the Newton–Krylov method. The present numerical technique has been validated by several examples including an inflation of a spherical shell and deformations of spherical and oblate spheroidal capsules in the shear flow.     

Natural Vibrations of Micropolar Elastic Flexible Plates and Shallow Shells

Acoustical Physics - A mathematical model representing the dynamics of geometrically nonlinear (flexible) micropolar elastic thin plates in Cartesian and curvilinear coordinates is constructed (the...     

Free vibrations of elastic circular toroidal shells

In this paper, the free vibrations of elastic in vacuo circular toroidal shells under different boundary conditions are studied using the linear Sanders thin shell theory. Beam functions are used to describe the motion along the meridional direction whilst trigonometric functions are used to represent the deformation of the cross section. It is shown that both the natural frequencies and the mode shapes can be accurately predicted as long as the employed beam functions satisfy the boundary conditions at the ends of the shells. The dependence of the free vibration characteristics of an elastic toroidal shell upon boundary conditions and toroidal to cross-sectional radius ratio is also illustrated and explained in this paper.     

Structural response of cylindrically curved laminated composite shells subjected to blast loading

This paper is concerned with the theoretical analysis and correlation with the numerical results of the displacement time histories of the cylindrically curved laminated composite shells exposed to normal blast shock waves. The laminated composite shell is clamped at its all edges. The dynamic equation of the cylindrical shell used in this study is valid under the assumptions made in Love's theory of thin elastic shells. The constitutive equations of laminated composite shells are given in the frame of effective modulus theory. The governing equation of the cylindrical shell is solved by the Runge-Kutta method. In addition, a finite element modeling and analysis are presented and compared with the theoretical results. The peak deflections and response frequencies obtained from theoretical and numerical analyses are in agreement. The effects of material properties and geometrical properties are examined on the dynamic behaviour.     

Star models with dark energy

We have constructed star models consisting of four parts: (i) a homogeneous inner core with anisotropic pressure (ii) an infinitesimal thin shell separating the core and the envelope; (iii) an envelope of inhomogeneous density and isotropic pressure; (iv) an infinitesimal thin shell matching the envelope boundary and the exterior Schwarzschild spacetime. We have analyzed all the energy conditions for the core, envelope and the two thin shells. We have found that, in order to have static solutions, at least one of the regions must be constituted by dark energy. The results show that there is no physical reason to have a superior limit for the mass of these objects but for the ratio of mass and radius.     

双层周期加肋有限长圆柱壳声散射精细特征研究

研究了双层周期性加肋有限长圆柱壳在水中的声散射特性. 壳体振动用薄壳理论的Donnell 方程描述,环肋振动用相互独立的薄板纯弯曲振动和平面应力状态下的振动方程描述,忽略弦间流体对环肋轴向力的作用. 数值计算给出远场收发合置情况下的周向目标强度和角度-频率谱图,并据此进行机理分析. 计算结果表明远场散射声场中除壳体弹性贡献外,弦间流体以及环肋与内外壳的相互作用对散射声场的贡献也是很重要的,并且在角度-频率谱中出现了舷间流体引起的流体附加波以及周期环肋引起的Bragg散射等回波精细特征,其中流体附加波是双层加肋圆柱壳声散射最重要的散射精细特征,是以往单层圆柱壳声散射所不具有的现象. 最后通过实验对理论推导进行了验证,实验与理论基本符合. 关键词： 声散射 圆柱壳 环肋 流体附加波     

Dynamic analysis of layered composite shells using nine node degenerate shell elements

The basic objective of the work reported in this paper is to extend a nine-node degenerated shell element developed earlier for stress analysis to the free vibration analysis of thick laminated composites. The nine-noded degenerated shell element is preferable to conventional solid elements for the modeling and analysis of laminated composite shell structures since the shell element works for both thick and thin shells. An enhanced interpolation of the transverse shear strains in the natural coordinates is used in this formulation to produce a shear locking free element and an enhanced interpolation of the membrane strains in the local coordinates is used to produce a membrane locking free element. The interpolation functions used in formulating the assumed strains are based on the Lagrangian interpolation polynomials. Various numerical examples are analyzed and their results are compared with the existing exact solutions where available and the numerical solutions calculated from other shell finite element formulations, to benchmark the current formulation.     

圆柱形水下爆炸实验容器壁部强度设计研究

 基于薄壁壳理论和水下爆炸理论,对圆柱形水下爆炸实验容器在爆炸冲击波作用下弹性范围内的壁部应变进行了理论分析和实验研究。导出了圆柱形水下爆炸实验容器在爆炸冲击波作用下壁部弹性应变与容器直径、壁厚及内部爆炸药量之间的关系,并对计算结果进行了实验验证。实验表明公式求解结果与实验结果具有较好的一致性。     

Characteristics of wave propagation and energy distributions in cylindrical elastic shells filled with fluid

The dispersion behaviour and energy distributions of free waves in thin walled cylindrical elastic shells filled with fluid are investigated. Dispersion curves are presented for a range of parameters and the behaviour of individual branches is explained. A non-dimensional equation which determines the distribution of vibrational energy between the shell wall and the contained fluid is derived and its variation with frequency and material parameters is studied.     

Semi-analytical study of free vibration characteristics of shear deformable filament wound anisotropic shells of revolution

Free vibration characteristics of filament wound anisotropic shells of revolution are investigated by using multisegment numerical integration technique in combination with a modified frequency trial method. The applicability of multisegment numerical integration technique is extended to the solution of free vibration problem of anisotropic composite shells of revolution through the use of finite exponential Fourier transform of the fundamental shell equations. The governing shell equations comprise the full anisotropic form of the constitutive relations, including first-order transverse shear deformation, and all components of translatory and rotary inertia. The variation of the stiffness coefficients along the axis of the shell is also incorporated into the solution method. Filaments are assumed to be placed along the geodesic fiber path on the shell of revolution resulting in the variation of the stiffness coefficients along the axis of the composite shell of revolution with general meridional curvature. Sample solutions have been performed on the effect of the variation of the stiffness coefficients on the free vibration behavior of filament wound truncated conical and spherical shells of revolution.