反对称铺设层合板动力问题的Hamilton体系及辛几何解法

基于ＹＮＳ层合格理论，建立反对称铺设层合板动力问题的Ｈａｍｉｌｔｏｎ正则方程，并采用共轭辛正交归一关系给出一对边简支，另一对边为任意支承层合板自振频率的精确解，数值算例讨论了长宽比，铺设角，层数及剪切修正系数的影响。     

Nonlinear vibration analysis of isotropic plate with inclined part-through surface crack

The four modes of vibration of an isotropic rectangular plate with an inclined crack are investigated. It is assumed that the crack remains continuous and its center is located at the center of the plate. The governing nonlinear equation of the transverse vibration of the plate with the plate boundary conditions being simply-supported on all edges is developed. The multiple scale perturbation method is utilized as the solution procedure to find the steady-state frequency response equations for all the four modes of vibration. The equations for the free and forced vibrations are derived and their frequency responses are presented. A special case of large-scale excitation force has also been considered. The parameter sensitivity analysis for the angle of crack, length of crack and the position of the external applied excitation force is performed. It has been shown that according to the aspect ratio of the plate, the vibration modes can have either nonlinear hardening effect or nonlinear softening behavior.     

Free vibration of composite laminated plate with complicated cutout

Abstract

This paper presents the free vibration analysis of a composite laminated square plate with complicated cutout. The problem formulation is based on the higher order shear deformation plate theory HDST C⁰ coupled with a curved quadrilateral p-element. The elements of the stiffness and mass matrices are calculated analytically. The curved edges are accurately represented using the blending function method. A calculation program is developed to determine the fundamental frequencies for different physical and mechanical parameters such as the cutout shape, plate thickness, fiber orientation angle, and boundary conditions. The results obtained show a good agreement with the available solutions in the literature. New results for the fundamentals frequencies of a composite laminated plate with complicated cutout are presented.     

Non-linear design and control optimization of composite laminated plates with buckling and postbuckling objectives

Multiobjective design and control optimization of composite laminated plates is presented to minimize the postbuckling dynamic response and maximize the buckling load. The control objective aims at dissipating the postbuckling elastic energy of the laminate with the minimum possible expenditure of control energy using a closed-loop distributed force. The layer thicknesses and fiber orientations are taken as design variables. The objectives of the optimization problem are formulated based on a shear deformation theory including the von-Karman non-linear effect for various cases of boundary conditions. The non-linear control problem is solved iteratively until an appropriate convergence criterion is satisfied based on Liapunov–Bellman theory. Liapunov function is taken as a sum of positive definite functions with different degrees. Comparative examples for three-layer symmetric and four-layer antisymmetric laminates are given for various cases of edges conditions. Graphical study is carried out to assess the accuracy of results obtained due to the successive iterations. The influences of the boundary conditions, orthotropy ratio, shear deformation, aspect ratio on the laminate optimal design are elucidated.     

FREE VIBRATION OF ANISOTROPIC RECTANGULAR PLATES BY GENERAL ANALYTICAL METHOD

According to the differential equation for transverse displacement function of anisotropic rectangular thin plates in free vibration, a general analytical solution is established. This general solution, composed of the composite solutions of trigonometric function and hyperbolic function, can satisfy the problem of arbitrary boundary conditions along four edges. The algebraic polynomial with double sine series solutions can also satisfy the problem of boundary conditions at four corners. Consequently, this general solution can be used to solve the vibration problem of anisotropic rectangular plates with arbitrary boundaries accurately. The integral constants can be determined by boundary conditions of four edges and four corners. Each natural frequency and vibration mode can be solved by the determinate of coefficient matrix from the homogeneous linear algebraic equations equal to zero. For example, a composite symmetric angle ply laminated plate with four edges clamped has been calculated and discussed.     

Stability of viscoelastic rectangular plate with a piezoelectric layer subjected to follower force

The effects of a piezoelectric layer on the stability of viscoelastic plates subjected to the follower forces are evaluated. The differential equation of motion of the viscoelastic plate with the piezoelectric layer is formulated using the two-dimensional viscoelastic differential constitutive relation and the thin plate theory. The weak integral form of the differential equations and the force boundary conditions are obtained. Using the element-free Galerkin method, the governing equation of the viscoelastic rectangular plate with elastic dilatation and Kelvin–Voigt distortion is derived, subjected to the follower forces coupled with the piezoelectric effect. A generalized complex eigenvalue problem is solved, and the force excited by the piezoelectric layer due to external voltage is modeled as the follower tensile force; this force is used to improve the stability of the non-conservative viscoelastic plate. For the viscoelastic plate with various boundary conditions, the results for the instability type and the critical loads are presented to show the variations in these factors with respect to the location of the piezoelectric layers and the applied voltages. The stability of the viscoelastic plates can be effectively improved by the determination of the optimal location for the piezoelectric layers and the most favorable voltage assignment.     

An analytical solution for buckling of moderately thick functionally graded sector and annular sector plates

In this article, an analytical solution for buckling of moderately thick functionally graded (FG) sectorial plates is presented. It is assumed that the material properties of the FG plate vary through the thickness of the plate as a power function. The stability equations are derived according to the Mindlin plate theory. By introducing four new functions, the stability equations are decoupled. The decoupled stability equations are solved analytically for both sector and annular sector plates with two simply supported radial edges. Satisfying the edges conditions along the circular edges of the plate, an eigenvalue problem for finding the critical buckling load is obtained. Solving the eigenvalue problem, the numerical results for the critical buckling load and mode shapes are obtained for both sector and annular sector plates. Finally, the effects of boundary conditions, volume fraction, inner to outer radius ratio (annularity) and plate thickness are studied. The results for critical buckling load of functionally graded sectorial plates are reported for the first time and can be used as benchmark.     

Solid Circular Plate Clamped along Two Antipodal Edge Arcs and Deflected by a Central Transverse Concentrated Force

A static, purely flexural mechanical analysis is presented for a Kirchhoff solid circular plate, deflected by a transverse central force, and clamped along two antipodal arcs, the remaining part of the boundary being free. By adopting an integral formulation, the contact reaction is assumed to be formed by four equal concentrated forces acting at the support extremities, accompanied by two distributed moments with radial and circumferential axis, respectively. This plate problem is rephrased in terms of a complex-valued Hilbert singular integral equation of the second kind, whose solution is obtained in analytical, integral form. A design chart is presented that reports the plate central deflection as a function of the angular width of the plate supports.     

Permeability anisotropy due to consolidation of compressible porous media

The characterisation of flow through porous media is important for all solid–liquid separation and fluid transport realms. The permeability of porous media can be anisotropic and furthermore, the extent of anisotropy can be increased as a result of an applied compressive force. However, the understanding of how anisotropy develops is incomplete. An overview of research on permeability anisotropy is given and an expression for predicting anisotropy as a function of void ratio is offered. The two underlying assumptions of the proposed model are: flow in different directions occurs within the same network of pores and deformation is primarily due to the compression of the particles in the direction of the applied force rather than due to particle rearrangement. The assumption of network connectivity allows permeability anisotropy to be described as a function of flow path tortuosity only. Results are presented for hydraulic anisotropy measured in lignite that has been upgraded by a compression dewatering method known as mechanical thermal expression. The lignite permeability is shown to be up to eight times greater in the direction perpendicular to compression, suggesting that the rate of dewatering could be significantly increased by choosing the drainage to also be perpendicular to the direction of the applied compressive force. It is illustrated that the proposed anisotropy model can be used to accurately predict the experimentally determined permeability anisotropy ratios for lignite, as well as for other materials including sand, clay and kaolin.     

Time-optimal rendezvous transfer trajectory for restricted cone-angle range solar sails

The advantage of solar sails in deep space exploration is that no fuel consumption is required. The heliocentric distance is one factor influencing the solar radiation pressure force exerted on solar sails. In addition, the solar radiation pressure force is also related to the solar sail orientation with respect to the sunlight direction. For an ideal flat solar sail, the cone angle between the sail normal and the sunlight direction determines the magnitude and direction of solar radiation pressure force. In general, the cone angle can change from 0° to 90°. However, in practical applications, a large cone angle may reduce the efficiency of solar radiation pressure force and there is a strict requirement on the attitude control. Usually, the cone angle range is restricted less more than an acute angle （for example, not more than 40°） in engineering practice. In this paper, the time-optimal transfer trajectory is designed over a restricted range of the cone angle, and an indirect method is used to solve the two point boundary value problem associated to the optimal control problem. Relevant numerical examples are provided to compare with the case of an unrestricted case, and the effects of different maximum restricted cone angles are discussed. The results indicate that （1） for the condition of a restricted cone-angle range the transfer time is longer than that for the unrestricted case and （2） the optimal transfer time increases as the maximum restricted cone angle decreases.     

Linear water wave propagation through multiple floating elastic plates of variable properties

We investigate the problem of linear water wave propagation under a set of elastic plates of variable properties. The problem is two-dimensional, but we allow the waves to be incident from an angle. Since the properties of the elastic plates can be set arbitrarily, the solution method can also be applied to model regions of open water as well as elastic plates. We assume that the boundary conditions at the plate edges are the free boundary conditions, although the method could be extended straightforwardly to cover other possible boundary conditions. The solution method is based on an eigenfunction expansion under each elastic plate and on matching these expansions at each plate boundary. We choose the number of matching conditions so that we have fewer equations than unknowns. The extra equations are found by applying the free-edge boundary conditions. We show that our results agree with previous work and that they satisfy the energy balance condition. We also compare our results with a series of experiments using floating elastic plates, which were performed in a two-dimensional wave tank.     

The Gantry-Tau parallel kinematic machine—kinematic and elastodynamic design optimisation

One of the main advantages of the Gantry-Tau machine is a large accessible workspace/footprint ratio compared to many other parallel machines. The optimal kinematic, elastostatic and elastodynamic design parameters of the machine are still difficult to calculate and this paper introduces an optimisation scheme based on the geometric and functional dependencies to define the workspace and first resonance frequency. This method assumes that each link and universal joint can be described by a mass-spring-damper model and calculates the transfer function from a Cartesian force or torque to Cartesian position or orientation. The evolutionary algorithm based on the complex search method is compared to the gradient-based search function in Matlab integrated optimisation toolbox. Kinematic design obtained by optimisation according to this paper gives a 2D workspace/footprint ratio more than 1.66 and first resonance frequency is more than 50 Hz with components of an existing lab prototype at the University of Agder, Norway.     

NON-AXISYMMETRIC MIXED BOUNDARY VALUE PROBLEM FOR DYNAMIC INTERACTION BETWEEN SATURATED POROUS FOUNDATION AND RECTANGULAR PLATE$^{\bf 1)}$

在同一界面的不同区域具有多种边界条件, 称之为混合边界, 这是一个熟知的力学问题. 对这类问题进行精确分析时, 必须要进行混合边值问题的求解. 而对于一般的三维非轴对称情形, 混合边值问题的求解往往存在数学困难. 本文利用Hilbert定理和双重Fourier变换, 给出了一种求解三维非轴对称混合边值问题的解析方法, 利用该方法对具有混合透水边界的饱和多孔地基上矩形板的振动弯曲进行了解析研究(板与地基接触面为不透水边界, 其余为透水边界). 首先, 基于Kirchhoff理论和Biot多孔介质理论建立矩形板与饱和多孔地基的动力控制方程, 进行耦合求解. 针对板土接触面和非接触面的混合边值问题, 采用双重Fourier变换构造出两对二维对偶积分方程, 以接触应力和接触面孔隙压力为基本未知量, 用Jacobi正交多项式将未知量展开, 再利用Schmidt法对二维对偶积分方程完成求解, 最终推导出板土系统在动力作用下的位移和应力解析式. 通过将本文计算模型退化为单一弹性地基, 与已有研究结果进行对比, 验证了本文方法的正确性和有效性. 最后, 通过数值算例, 对饱和多孔地基上矩形板的动力响应及参数影响做出分析和讨论. 此外, 本文提出的解析法具有一般性, 可广泛应用于复杂接触问题和多场耦合问题的求解.     

饱和多孔地基与矩形板动力相互作用的非轴对称混合边值问题

在同一界面的不同区域具有多种边界条件, 称之为混合边界, 这是一个熟知的力学问题. 对这类问题进行精确分析时, 必须要进行混合边值问题的求解. 而对于一般的三维非轴对称情形, 混合边值问题的求解往往存在数学困难. 本文利用Hilbert定理和双重Fourier变换, 给出了一种求解三维非轴对称混合边值问题的解析方法, 利用该方法对具有混合透水边界的饱和多孔地基上矩形板的振动弯曲进行了解析研究(板与地基接触面为不透水边界, 其余为透水边界). 首先, 基于Kirchhoff理论和Biot多孔介质理论建立矩形板与饱和多孔地基的动力控制方程, 进行耦合求解. 针对板土接触面和非接触面的混合边值问题, 采用双重Fourier变换构造出两对二维对偶积分方程, 以接触应力和接触面孔隙压力为基本未知量, 用Jacobi正交多项式将未知量展开, 再利用Schmidt法对二维对偶积分方程完成求解, 最终推导出板土系统在动力作用下的位移和应力解析式. 通过将本文计算模型退化为单一弹性地基, 与已有研究结果进行对比, 验证了本文方法的正确性和有效性. 最后, 通过数值算例, 对饱和多孔地基上矩形板的动力响应及参数影响做出分析和讨论. 此外, 本文提出的解析法具有一般性, 可广泛应用于复杂接触问题和多场耦合问题的求解.      

ELASTIC DYNAMIC ANALYSIS OF MODERATELY THICK PLATE USING MESHLESS LRPIM

A meshless local radial point interpolation method （LRPIM） for solving elastic dy-namic problems of moderately thick plates is presented in this paper. The discretized system equation of the plate is obtained using a locally weighted residual method. It uses a radial basis function （RBF） coupled with a polynomial basis function as a trial function,and uses the quartic spline function as a test function of the weighted residual method. The shape function has the properties of the Kronecker delta function,and no additional treatment is done to impose essen-tial boundary conditions. The Newmark method for solving the dynamic problem is adopted in computation. Effects of sizes of the quadrature sub-domain and influence domain on the dynamic properties are investigated. The numerical results show that the presented method can give quite accurate results for the elastic dynamic problem of the moderately thick plate.     

Stress concentrations in the particulate composite with transversely isotropic phases

The accurate series solution have been obtained of the elasticity theory problem for a transversely isotropic solid containing a finite or infinite periodic array of anisotropic spherical inclusions. The method of solution has been developed based on the multipole expansion technique. The basic idea of method consists in expansion the displacement vector into a series over the set of vectorial functions satisfying the governing equations of elastic equilibrium. The re-expansion formulae derived for these functions provide exact satisfaction of the interfacial boundary conditions. As a result, the primary spatial boundary-value problem is reduced to an infinite set of linear algebraic equations. The method has been applied systematically to solve for three models of composite, namely a single inclusion, a finite array of inclusions and an infinite periodic array of inclusions, respectively, embedded in a transversely isotropic solid. The numerical results are presented demonstrating that elastic properties mismatch, anisotropy degree, orientation of the anisotropy axes and interactions between the inclusions can produce significant local stress concentration and, thus, affect greatly the overall elastic behavior of composite.     

Large deflection of unsymmetric laminates with mixed boundary conditions

This paper is analytically concerned with non-linear bending of an unsymmetrically laminated angle-ply rectangular plate under lateral load. The plate edges are subjected to the varying rotational constraints. A series solution satisfying the von Karman-type non-linear equations and the required boundary conditions of the plate is presented. In the formulation the edge moments are replaced by an equivalent lateral pressure near the plate edges. Governing equations are reduced to a set of algebraic equations. Numerical results for maximum deflection, bending moment and inplane force of unsymmetric angle-ply plates are graphically presented for various high-modulus materials, aspect ratios, geometries of lamination and boundary conditions. Present results are also compared with available data.     

Application of an anisotropic growth and remodelling formulation to computational structural design

A classical structural optimisation problem consists of a problem-specific objective function which has to be minimised in consideration of particular constraints with respect to design and state variables. In this contribution we adopt a conceptually different approach for the design of a structure which is not based on a classical optimisation technique. Instead, we establish a constitutive micro-sphere-framework in combination with an energy-driven anisotropic microstructural growth formulation, which was originally proposed for the simulation of adaptation and remodelling phenomena in hard biological tissues such as bones.The goal of this contribution is to investigate this anisotropic growth formulation with a special emphasis on its application to structural design problems. To this end, four illustrative three-dimensional benchmark-type boundary value problems are discussed and compared qualitatively with the results obtained by classical structural optimisation strategies. The simulation results capture the densification effects and clearly identify the main load bearing regions. It turns out, that even though making use of this conceptually different growth formulation as compared to the procedures used in a classical structural optimisation context, we identify qualitatively very similar structures or rather regions of densification. Moreover, in contrast to common structural optimisation strategies, which mostly aim to optimise merely the size, shape or topology, our formulation also contains the improvement of the material itself, which—apart from the structural improvement—results in the generation of problem-specific local material anisotropy and textured evolution.     

On the Form of the Contact Reaction in a Solid Circular Plate Simply Supported Along Two Antipodal Edge Arcs and Deflected by a Central Transverse Concentrated Force

A static, purely flexural mechanical analysis is presented for a Kirchhoff solid circular plate, deflected by a transverse central force, and bilaterally supported along two antipodal periphery arcs, the remaining part of the boundary being free. Two kinds of contact reactions are considered, namely the case of distributed reaction force alone, and the situation in which the distributed force is added to a distributed couple of properly selected profile. For both cases this plate problem is formulated in terms of an integral equation of the Prandtl type, coupled with two constraint conditions. The existence of solutions in an appropriate scaled weighted Sobolev space is discussed, and the behaviour of the solution at the endpoints of the support is exhibited. This revised version was published online in June 2006 with corrections to the Cover Date.     

Exact solutions for the stability of viscoelastic rectangular plate subjected to tangential follower force

An exact solution procedure is formulated for the stability analysis of viscoelastic rectangular plate with two opposite edges simply supported and other two edges clamped as well as the viscoelastic rectangular plate with one edge clamped and other three edges simply supported under the action of tangential follower force. Firstly, by assuming the transverse displacement (W) as independent functions which automatically satisfies the simply supported boundary conditions, the governing partial differential equation is reduced to an ordinary differential equation with variable coefficients. Then, by the normalized power series method and applying the boundary conditions yield the eigenvalue problem of finding the roots of a fourth-order characteristic determinant. The results show that the aspect ratio λ and the dimensionless delay time H have great effects on the types of instability and the critical loads of the viscoelastic plates.