State-vector equation with damping and vibration analysis of laminates

Based on the modified mixed Hellinger-Reissner(H-R)variational principle for elastic bodies with damping,the state-vector equation with parameters is directionally derived from the principle.A new solution for the harmonic vibration of simply supported rectangular laminates with damping is proposed by using the precise integration method and Muller method.The general solutions for the free vibration of underdamping,critical damp and overdamping of composite laminates are given simply in terms of the linear damping vibration theory.The effect of viscous damping force on the vibration of com- posite laminates is investigated through numerical examples.The state-vector equation theory and its application areas are extended.     

A semi-analytical solution for static and dynamic analysis of plates with piezoelectric patches

A modified mixed variational principle for piezoelectric materials is established and the state-vector equation of piezoelectric plates is deduced directly from the principle. Then the exact solution of the state-vector equation is simply given, and based on the semi-analytical solution of the state-vector equation, a realistic mathematical model is proposed for static analysis of a hybrid laminate and dynamic analysis of a clamped aluminum plate with piezoelectric patches. Both the plate and patches are considered as two three-dimensional piezoelectric bodies, but the same linear quadrilateral element is used to discretize the plate and patches. This method accounts for the compatibility of generalized displacements and generalized stresses on the interface between the plate and patches, and the transverse shear deformation and the rotary inertia of the plate and patches are also considered in the global algebraic equation system. Meanwhile, there is no restriction on the thickness of plate and patches. The model can be also modified to achieve a semi-analytical solution for the transient responses to dynamic loadings and the vibration control of laminated plate with piezoelectric patches or piezoelectric stiffeners.     

Hamiltonian parametric element and semi-analytical solution for smart laminated plates

Based on the Hellinger-Reissner (H-R) mixed variational principle for piezoelectric material, a unified 4-node Hamiltonian isoparametric element of anisotropy piezoelectric material is established. A new semi-analytical solution for the natural vibration of smart laminated plates and the transient response of the laminated cantilever with piezoelectric patch is presented. The major steps of mathematical model are as follows: the piezoelectric layer and host layer of laminated plate are considered as unattached three-dimensional bodies and discretized by the Hamiltonian isoparametric elements. The control equation of whole structure is derived by considering the compatibility of generalized displacements and generalized stresses on the interface between layers. There is no restriction for the side-face geometrical boundaries, the thickness and the number of layers of plate by the use of the present isoparametric element. Present method has wide application area.     

Subregion generalized variational principles for elastic thick plates

Tn this paper, the subregion generalized variational principle for elastic thick plates is proposed. Its main points may be stated as follows:1. Each subregion may be assigned arbitrarily as a potential region or complementary region. The subregion variational principles of potential energy, complementary energy and mixed energy represent three special forms of this principle.2. The number of independent variational variables in each sub-region may be assigned arbitrarily. Any one of the subregions may be assigned as a one-variable-region, two-variable-region or three-variable-region.3. The conjunction conditions of displacements and stresses on each interline of neighbouring subregions may be relaxed. On the basis of this principle the finite element analysis of non-conforming elements for thick plates can be formulated.Different finite element models for thick plates can be obtained by different applications of this principle. In particular,the subregion mixed variational principle for thick plates may be applied to formulating the subregion mixed finite element method for thick plates.     

弹性力学Hamilton方法广义解的适定性

首先引入了Hamilton体系中平面应力弹性力学问题正则方程的Galerkin变分方程,证明了Galerkin变分方程和目前文献中所用的Ritz变分方程的等价性,以及相应广义解的适定性,从而为目前的数值方法提供了理论基础。从证明过程中可以看到广义解实际上是Ritz变分泛函的一个鞍点。     

智能板结构振动特性的半解析法

为智能板结构振动特性的分析提出了一种半解析法.根据压电材料修正后的Hellinger-Re issner变分原理,推导了压电材料的Ham ilton ian等参元.智能板结构的基体板和压电块(压电传感器或驱动器)被看作独立的三维体,并用Ham ilton ian等参元分别离散板和压电块.考虑到板和压电块在连接界面上广义应力和广义位移的连续性,联立板和压电块的方程得到整个结构的方程组.数值实例的分析结果证明了方法的可靠性.     

梯度弹性基础上正交异性薄板的屈曲分析

基于双参数弹性基础模型,研究了梯度弹性基础上正交异性薄板的屈曲问题. 首先,基于能量法与变分原理,给出了梯度弹性基础上正交异性薄板的屈曲控制方程,并得到了梯度弹性基础刚度系数K₁ 与K₂的计算式;进而,通过将位移函数采用三角函数展开的方法,给出了单向压缩载荷作用下、四边简支正交异性弹性基础板屈曲载荷的计算式;在算例中,通过将该文的解退化到单纯的正交异性板,并与经典弹性解比较,证明了理论的正确性;最后,求解了弹性模量在厚度方向上呈幂律分布的梯度基础上的薄板屈曲问题,分析了基础上下表层材料弹性模量比与体积分数指数对屈曲载荷的影响.     

Sensitivity analysis of composite laminated plates with bonding imperfection in Hamilton system

Sensitivity analysis of composite laminated plates with bonding imperfection is carried out based on the radial point interpolation method （RPIM） in a Hamilton system. A set of hybrid governing equations of response and sensitivity quantities is reduced using the spring-layer model and the modified Hellinger-Reissner （H-R） variational principle. The analytical method （AM）, the semi-analytical method （SAM）, and the finite difference method （FDM） are used for sensitivity analysis based on the reduced set of hybrid governing equations. A major advantage of the hybrid governing equations is that the convolution algorithm is avoided in sensitivity analysis. In addition, sensitivity analysis using this set of hybrid governing equations can obtain response values and sensitivity coefficients simultaneously, and accounts for bonding imperfection of composite laminated plates.     

On some general variational principles for creep with applications to thin shells

A constitutive relation for a viscous material subject to small strains but finite rotations is postulated and associated variational theorems are formulated. These are similar to the principle of minimum of potential energy and the Hellinger-Reissner theorem of an elastic solid. The derivation of strain-displacement relations for thin shells subject to small strains but moderately large rotations are given. On this basis a mixed variational principle for thin viscous shells is developed. For the problem of creep collapse of long cylindrical shells under external pressure it is demonstrated that the mixed variational principle may be advantageous compared to other variational theorems. A comparison with the creep collapse theory of Hoff et al. is given.     

Reissner板问题的有限元广义混合法

用一般弹性体的广义混合变分原理,导出了适合Ｒｅｉｓｓｎｅｒ板弯曲问题的广义混合变分原理及其有限元广义混合法。算例说明,该有限元模式的刚度可以改变,比常规位移法的精度高,同时还能克服常规Ｒｅｉｓｓｎｅｒ板位移元用于计算薄板时所出现的“剪切自锁”现象,计算结果稳定,最后分析该法能够克服“剪切自锁”现象的原因。     

考虑剪切效应层合板的Hamilton体系及辛几何方法

基于ＹＮＳ层合板理论,通过对混合能变分原理的修正,建立了层合板问题的Ｈａｍｉｌｔｏｎ正则方程。在辛几何数学框架下,采用共轭辛正交归一关系给出精确解。并与经典层板理论进行了比较。     

The Method of R-Functions in the Solution of Elastic Problems on the Basis of Reissner's Mixed Variational Principle

A method is presented for solving boundary-value elastic problems on the basis of the variational–structural method of R-functions and Reissner's mixed variational principle. A mathematical formulation is given to problems on the deformation of elastic bodies under mixed boundary conditions and bodies interacting with smooth rigid dies. Solutions satisfying all the boundary conditions are proposed. For undetermined components of these solutions, the resolving equations are derived and their properties are studied. A posteriori estimation of numerical solutions is made. As examples, solutions are found to a problem on the stress–strain state of a short cylinder and to a contact problem on a cylinder interacting with a smooth die. A numerical method of solving such problems is analyzed for convergence, and the accuracy of the solutions is estimated.     

Elastic dynamics variational principle for nonlinear elastic body

Based on paper, the variational principle and generalized variational principle of elastic dynamics for elastic body with nonlinear stress-strain relations are introduced in this paper. The generalized instantaneous variational principle is also raised for the mixed harmonious displacement element and the mixed harmonious stress element.     

Mixed variational principle in elasticity theory and canonical systems of equations

The paper proposes a modification of the mixed variational principle from which stationarity conditions are derived in the form of a mixed system of equations resolved for the first derivatives of the displacement and stress components acting in a plane perpendicular to one of the coordinate axes. The variational principle allows decreasing the dimension of the problem of elasticity thus reducing the system of equations to a canonical form. The modified mixed principle helps immediately obtain a canonical system of equations for various applied theories. This possibility is demonstrated with the example of the Timoshenko theory of plates __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 55–62, May 2007.     

A convolution-type semi-analytic DQ approach to transient response of rectangular plates

The convolution-type Gurtin variational principle is known as the only variational principle that is, from the mathematics point of view, totally equivalent to the initial value problem system. In this paper, the equation of motion of rectangular thin plates is first transformed to a new governing equation containing initial conditions by using a convolution method. A convolution-type semi-analytical DQ approach, which involves differential quadrature （DQ） approximation in the space domain and an analytical series expansion in the time domain, is proposed to obtain the transient response solution. This approach offers the same advantages as the Gurtin variational principle and, at the same time, is much simpler in calculation. Numerical results show that it is very accurate yet computationally efficient for the dynamic response of plates.     

Enhanced first-order theory based on mixed formulation and transverse normal effect

An accurate prediction of displacements and stresses for laminated and sandwich plates is presented using an enhanced first-order plate theory based on the mixed variational theorem (EFSDTM) developed in this paper. In the mixed formulation, transverse shear stresses based on an efficient higher-order plate theory (EHOPT) developed by Cho and Parmerter [Cho, M., Parmerter, R.R., 1993. Efficient higher-order composite plate theory for general lamination configurations. AIAA Journal 31, 1299–1306] are utilized and modified to satisfy prescribed lateral conditions, and displacements are assumed to be those of a first-order shear deformation theory (FSDT). Relationships between the modified EHOPT and the FSDT are systematically derived via both the mixed variational theorem and the least-square approximation of difference between in-plane stresses including the transverse normal stress effect. It is shown that the transverse normal stress effect should be considered in predicting the in-plane stresses when the Poisson effect is dominant. The developed EFSDTM preserves the computational advantage of the classical FSDT while allowing for important local through-the-thickness variations of displacements and stresses through the recovery procedure. The accuracy and efficiency of the present theory are assessed by comparing its results with various plate models as well as the three-dimensional exact solutions for thick laminated and sandwich plates.     

Some Theorems in the Theory of Microstretch Thermomagnetoelectroelasticity

The linear dynamic theory of microstretch thermomagnetoelectroelasticity is studied in this paper.First,a reciprocity relation which involves two processes at different instants is established to form the basis of a uniqueness result and a reciprocal theorem.The proof of the reciprocal theorem avoids both using the Laplace transform and incorporating the initial conditions into the equations of motion.The uniqueness theorem is derived with no definiteness assumption on the elastic constitutive coefficients.Then the continuous dependence theorem is discussed upon two external data systems.Finally,the variational principle of Hamilton type which fully characterizes the solution of the mixed boundary-initial-value problem(mixed problem) is obtained.These theorems lay a solid foundation for further theoretical and numerical studies on microstretch thermomagnetoelectroelastic materials.     

GENERALIZED BIHARMONIC OPERATOR AND ITS APPLICATION TO THE BENDING OF ELASTIC THIN PLATES

ＧＥＮＥＲＡＬＩＺＥＤＢＩＨＡＲＭＯＮＩＣＯＰＥＲＡＴＯＲＡＮＤＩＴＳＡＰＰＬＩＣＡＴＩＯＮＴＯＴＨＥＢＥＮＤＩＮＧＯＦＥＬＡＳＴＩＣＴＨＩＮＰＬＡＴＥＳＹｕＺｈｏｎｅｔｔ－ｚｈｉ（俞中直）（Ｄｅｐｔ．ｏｆＥｎｇ．Ｍｅｃｈａｎｉｃｓ,ＤａｌｉａｎＵｎ...     

Analysis of bending, vibration and stability for thin plate on elastic foundation by the multivariable spline element method

I.IntroductionTheplatestructuresonelasticfoundationareofpracticalimportanceinwide-spreadapplicationsinengineering.Basedonrapidprogressofscienceandtechnologyaswellasengineeringconstructions,differentmodelsofeIasticfoundationaredeve1oped,sucha,Winkler,Vlazo…     

The combined energy variational principles in linear elasticity

In this paper, a new kind of mixed energy variational principles in linear elasticity—the combined energy variational principles is presented. First, we define the conjugate body of an elastic body, which is obtained by changing the boundary conditions of the elastic body. Next, we decompose the conjugate body into two component-states, construct functionals of potential energy and complementary energy, respectively, for the component-states and define the additional hybrid energy between the component-states. Thus the functionals of combined energy can be constructed. Three typical combined energy variational principles are demonstrated and the other forms of combined energy variational principles are given. The application of the proposed principles to the calculation of thin plates with complicated boundaries is shown.