具有弱界面特性叠层压电球壳的自由振动

针对具有弱界面的叠层压电球壳自由振动,引入两个位移和应力函数,从三维压电弹性理论基本方程出发建立了分别对应于两类振动形式的独立状态方程,并通过球面谐函数展开技术以及近似层合模型将其化为关于径向坐标的常系数状态方程。采用弱界面模型建立状态向量的界面传递关系,与层内传递关系得到球壳内外状态变量的整体传递关系。最后考虑球壳内外边界自由条件, 得到了两类振动形式的频率方程。通过与已有精确解的比较验证了本文解的准确性,数值详细表明弱界面弹性柔性系数的大小对叠层球壳自振频率有较大影响,但弱界面导电性的高低对自振频率的影响不大。     

Characteristics of elastic waves in FGM spherical shells,an analytical solution

In this paper, we investigate the propagation characteristics of elastic guided waves in FGM spherical shells with exponentially graded material in the radial direction. A new separation of variables technique to displacements is proposed to convert the governing equations of the wave motion to the second-order ordinary differential equations with variable coefficients. By further a variables transform technique, these equations are transformed to the Whittaker equations so that analytic solutions can be obtained. For the spherical shell case, by satisfying the traction-free boundary conditions on both the inner and outer surfaces of the shell, we obtain the dispersion equations, which show that both the SH and Lamb-type wave modes are generated in the structure. The calculated dispersion curves in the functionally graded shell demonstrate a clear influence of the gradient coefficient as compared to those of the homogeneous shell, with the Lamb-type waves more sensitive to the gradient coefficient. The mode shapes and the distributions of stresses in the shells for various gradient coefficients are also presented to illustrate their dependence on the gradient coefficient.     

样条状态变量法分析扁球壳的动力响应

基于瞬时变分原理与样条函数理论,建立扁球壳的动力方程,引入样条参数及其对时间的导数作为状态变量,导出状态方程,对空间域,采用样条有限元法,对时间域,采用现代控制论中的状态空间法,此外,建立了一种状态变量的递推计算格式,在计算过程中,不解线性方程组,也不需采用解耦技术,只要作矩阵运算,可直接算出动力响应量,文末给出的数值算例表明本文方法是可靠 与有效的。     

Analytical thermo-elastodynamic solutions for a nonhomogeneous transversely isotropic hollow sphere

Summary  The spherically symmetric dynamic thermoelastic problem for a special nonhomogeneous transversely isotropic elastic hollow sphere is formulated by introduction of a dependent variable and separation of variables technique. The derived solution can be degenerated into that for a homogeneous transversely isotropic hollow sphere, a nonhomogeneous isotropic hollow sphere or a solid sphere. The present method, allow to avoid integral transforms, is suited for a hollow sphere of arbitrary thickness subjected to arbitrary spherical symmetric thermal and mechanical loads, and is convenient in dealing with different boundary conditions of dynamic thermoelasticity . The numerical calculation involved is easy to be performed and its results are also presented. Received 30 October 2001; accepted for publication 21 February 2002 The work was supported by the National Natural Science Foundation of China (No. 10172075 and No. 10002016)     

Symplectic multi-level method for solving nonlinear optimal control problem

By converting an optimal control problem for nonlinear systems to a Hamiltonian system,a symplecitc-preserving method is proposed.The state and costate variables are approximated by the Lagrange polynomial.The state variables at two ends of the time interval are taken as independent variables.Based on the dual variable principle,nonlinear optimal control problems are replaced with nonlinear equations.Furthermore,in the implementation of the symplectic algorithm,based on the 2N algorithm,a multilevel method is proposed.When the time grid is refined from low level to high level,the initial state and costate variables of the nonlinear equations can be obtained from the Lagrange interpolation at the low level grid to improve efficiency.Numerical simulations show the precision and the efficiency of the proposed algorithm in this paper.     

Mode-III interface edge crack between two bonded quarter-planes of dissimilar piezoelectric materials

Summary  The problem of an interface edge crack between two bonded quarter-planes of dissimilar piezoelectric materials is considered under the conditions of anti-plane shear and in-plane electric loading. The crack surfaces are assumed to be impermeable to the electric field. An integral transform technique is employed to reduce the problem under consideration to dual integral equations. By solving the resulting dual integral equations, the intensity factors of the stress and the electric displacement and the energy release rate as well as the crack sliding displacement and the electric voltage across the crack surfaces are obtained in explicit form for the case of concentrated forces and free charges at the crack surfaces and at the boundary. The derived results can be taken as fundamental solutions which can be superposed to model more realistic problems. Received 10 November 2000; accepted for publication 28 March 2001     

多层电磁弹性圆柱壳内波的轴向传播

对多层电磁弹性圆柱壳内波的轴向传播进行了分析。根据柱坐标系下电磁弹性多层结构的几何方程、平衡方程和本构方程,推导出了两个层间变量所满足的状态方程。通过状态方程的解和层间变量连续性条件,得到了多层圆柱体内外表面层间变量的传递关系。最后利用边界条件,导出了波在传播时所满足的频散方程,并求得该结构的模态参数。以一个三层的压电/压磁材料组成的柱壳结构作为数值算例,计算出波在其中轴向传播时的频散关系和模态参数,并对计算结果进行了分析。     

On the Bending, Vibration and Stability of Laminated Rectangular Plates with Transversely Isotropic Layers

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Modeling the response of filled elastomers at finite strains by rigid-rod networks

Summary  A constitutive model is developed for the isothermal response of particle-reinforced elastomers at finite strains. An amorphous rubbery polymer is treated as a network of long chains bridged to permanent junctions. A strand between two neighboring junctions is thought of as a sequence of rigid segments connected by bonds. In the stress-free state, a bond may be in one of two stable conformations: flexed and extended. The mechanical energy of a bond in the flexed conformation is treated as a quadratic function of the local strain, whereas that of a bond in the extended conformation is neglected. An explicit expression is developed for the free energy of a network. Stress–strain relations and kinetic equations for the concentrations of bonds in various conformations are derived using the laws of thermodynamics. In the case of small strains, these relations are reduced to the constitutive equation for the standard viscoelastic solid. At finite strains, the governing equations are determined by four adjustable parameters which are found by fitting experimental data in uniaxial tensile, compressive and cyclic tests. Fair agreement is demonstrated between the observations for several filled and unfilled rubbery polymers and the results of numerical simulation. We discuss the effects of the straining state, filler content, crosslink density and temperature on the adjustable constants. Received 3 January 2001; accepted for publication 12 July 2001     

Stress–strain state of nonthin orthotropic spherical shells of variable thickness

The stress–strain state of an orthotropic spherical shell with thickness varying in two coordinate directions is analyzed. Different boundary conditions are considered, and a refined problem statement is used. A numerical analytic method based on spline-approximation and discrete orthogonalization is developed. The stress–strain state of spherical orthotropic shells with variable thickness is studied     

The propagation of small disturbances in radiative magnetogasdynamics

Summary The non-linearised equations of flow for a perfectly electrically conducting and thermally radiating inviscid gas are analysed using the differential approximation for a grey gas of arbitrary opacity including effects of radiative flux, pressure and energy density. The speeds of propagation of weak discontinuities are established with oblique magnetic field, together with the jump relations for the derivatives of the field variables. For the special case of one-dimensional flows with planar, cylindrical or spherical symmetry the compatibility conditions on the characteristic surfaces associated with the wavefronts are explicitly determined.     

THE NEW SOLUTION FOR THE AXIALLY SYMMETRICAL LAMINAR BOUNDARY LAYER EQUATIONS BETWEEN TWO PARALLEL SPHERICAL SURFACES

The transformations, which are similar to Mangler's transformations, are given in this paper. They change the entrance region flow of axially symmetrical laminar boundary layer between two parallel spherical surfaces into the flow of two-dimensional boundary layer, and simplify the problems. The simplified equattons can be solved by the two-dimensional boundary layer theory and numerical methods. Therefore, a new way is opened up to solve the diffusive laminar fiow in the entrance region between two parallel spherical surfaces.     

The new solution for the axially symmetrical laminar boundary layer equations between two parallel sherical surfaces

The transformations, which are similar to Mangler’s transformations, are given in this paper. They change the entrance region flow of axially symmetrical laminar boundary layer between two parallel spherical surfaces into the flow of two-dimensional boundary layer, and simplify the problems. The simplified equations can be solved by the two-dimensional boundary layer theory and numerical methods. Therefore, a new way is opened up to solve the diffusive laminar flow in the entrance region between two parallel spherical surfaces.     

Driving forces and kinking of an oblique crack in bonded nonhomogeneous materials

Summary  Plane elasticity solutions are presented for the problem of an oblique crack in two bonded media. The material model under consideration consists of a homogeneous half-plane with an arbitrarily oriented crack and a nonhomogeneous half-plane. The Fourier integral transform method is employed in conjunction with the coordinate transformations of field variables in the basic elasticity equations. Formulation of the crack problem results in having to solve a system of singular integral equations for arbitrary crack surface tractions. A crack perpendicular to or along the bonded interface between the homogeneous and nonhomogeneous constituents arises as a limiting case. In the numerical results, the values of mixed-mode stress intensity factors are provided for various combinations of relevant geometric and material parameters of the bonded media. Subsequently, the infinitesimal kinks from the tips of a main crack are presumed, with the corresponding local driving forces being evaluated in terms of the stress intensities of the main crack. The criterion of maximum energy release rate is applied with the aim of making some conjectures concerning the likelihood of kinking and the probable kink direction based on the approximation of local homogeneity and brittleness of the crack-tip behavior. Received 25 September 2001; accepted for publication 13 February 2002     

Acoustic scattering by a sphere in the time domain

A sound pulse is scattered by a sphere leading to an initial–boundary value problem for the wave equation. A method for solving this problem is developed using integral representations involving Legendre polynomials in a similarity variable and Volterra integral equations. The method is compared and contrasted with the classical method, which uses Laplace transforms in time combined with separation of variables in spherical polar coordinates.     

Noncanonical Poisson brackets for elastic and micromorphic solids

This paper investigates the Lagrangian-to-Eulerian transformation approach to the construction of noncanonical Poisson brackets for the conservative part of elastic solids and micromorphic elastic solids. The Dirac delta function links Lagrangian canonical variables and Eulerian state variables, producing noncanonical Poisson brackets from the corresponding canonical brackets. Specifying the Hamiltonian functionals generates the evolution equations for these state variables from the Poisson brackets. Different elastic strain tensors, such as the Green deformation tensor, the Cauchy deformation tensor, and the higher-order deformation tensor, are appropriate state variables in Poisson bracket formalism since they are quantities composed of the deformation gradient. This paper also considers deformable directors to comprise the three elastic strain density measures for micromorphic solids. Furthermore, the technique of variable transformation is also discussed when a state variable is not conserved along with the motion of the body.     

Various analogies for the equilibrium shapes of an elastic thread on two-dimensional surfaces

In accordance with the Kirchhoff analogy, the equilibrium equations of an elastic thread on a plane are equivalent to the equations of motion of a simple pendulum. This analogy is generalized to the case when the thread is situated on a smooth curved surface. The equilibrium equations for the threads in the general case and in the particular cases of flat, cylindrical, and spherical surfaces are derived. For these surfaces the Kirchhoff analogy is generalized to the case of a simple pendulum in an additional force field. There are also considered the electromagnetic and nonholonomic analogies for the equilibrium equations of an elastic thread.     

Bending of functionally graded piezoelectric rectangular plates

By introducing two displacement functions as well as two stress functions, two independent state equations with variable coefficients are derived from the three-dimensional theory equations of piezoelasticity for transverse isotropy. A laminated approximation is used to transform the state equations to those with constant coefficients in each sub-layer. The bending problem of a functionally graded rectangular plate is then analyzed based on the state equations. Numerical results are presented and the effect of material gradient index is discussed. Supported by the National Natural Sciences Foundation of China (No. 10002016).     

Investigation of changes in the spherical shell shape under the action of pulsed loading due to contact interaction with a rigid block

An axisymmetric problem of high strains in a spherical lead shell enclosed into an aluminum “spacesuit” under the action of pulsed loading is considered. The shell straining is described with the use of equations of mechanics of elastoviscoplastic media in Lagrangian variables, and the kinematic relations are determined in the current state metrics. Equations of state are taken in the form of equations of the flow theory with isotropic hardening. The problem is solved numerically by using the variational difference method and the “cross” explicit scheme of integration with respect to time. The influence of the yield stress as a function of the strain rate on changes in the shell shape is studied for different values of loading. The calculated final shape and residual strains are demonstrated to be in good agreement with experimental data.     

A least-squares method for solving the mixed form of the groundwater flow equations

The mixed form of the areal groundwater flow equations is solved with a least-squares finite element procedure (LESFEM). Hydraulic head and x- and y-directed fluxes are state variables. Physical parameters and state variables are approximated using a bilinear basis. Grid refinements and irregular domain boundaries are implemented on rectangular meshes. Residuals are constructed at collocation points for conservation of mass and Darcy's law. Boundary condition residuals are constructed at discrete points along the boundary. The residuals are weighted, squared and summed. A set of algebraic equations is formed by taking the derivatives of the weighted sum of the squares of the residuals with respect to each unknown parameter in the approximation for the state variable and setting them to zero. Proper choice of a potential scaling parameter and residual weights is essential for the effective application of the algorithm. Test problem results demonstrate that the method is effective for both transient and steady state cases. The LESFEM algorithm generates a C°-continuous velocity field. The continuous velocity field and the rectangular mesh simplify the implementation of algorithms that require tracking. In addition, rectangular meshes simplify mesh and boundary generation.