面内功能梯度矩形板的近似理论与解答

提出了面内功能梯度矩形板在竖向载荷作用下的近似 理论与解析解. 假设材料常数在面内x轴方向按指数规 律变化.引入了板理论的Reissner-Mindlin假设, 并考虑了板中面上的剪切变形的影响.推导了板在平行于y轴的两边简支, 平行于x轴方向的两边简支或固支情况下中性层法线转角和挠度用Fourier级数表示的解.讨论了退化为Kirchhoff假设下经典薄板理论的解的情况.提供了经典薄板理论在和Reissner-Mindlin假设下的算例并与三维有限元的计算结果进行了比较, 说明了该方法在厚板情况下也是相当精确的.     

功能梯度压电材料圆板的简化理论与解析解

提出了功能梯度圆板在轴对称载荷作用下的简化理论与解析解. 引入了板理论的若干假设(Kirchhoff假设的一部分,Reissner-Mindlin假设和文中提出的假设),并假设材料常数在板厚方向按指数规律变化. 推导了板的周边固定或简支同时又接地情况下中性层法线转角的解和用Fourier-Bessel级数表示的电势解. 这个解在形式上比精确解简单得多,进行数值计算时也相当方便与快捷. 该文给出了板的周边固定、接地情况下的计算结果并进行了讨论,对于理论和方法的正确性作了验证.      

功能梯度与均匀圆板弯曲解的线性转换关系

基于一阶剪切理论, 研究了功能梯度材料圆板与均匀圆板轴对称弯曲解之间的线性转换关系. 通过理论分析和比较 功能梯度材料圆板和均匀圆板在一阶剪切理论下的位移形式的轴对称弯曲控制方程, 发现了功能梯度材料圆板的转角与均匀圆板的转角之间的相似转换关系. 在假设材料性质沿板厚连续变化的情况下, 给出了相似转换系数的解析表达式. 在此基础上, 进一步导出了一阶剪切理论下功能梯度圆板的挠度与经典理论下, 均匀圆板的挠度之间的线性关系. 从而, 可将功能梯度材料圆板在一阶剪切理论下的弯曲问题求解, 转化为相应均匀薄圆板在经典理论下的弯曲问题求解, 以及转换系数的计算问题. 这一方法为功能梯度非均匀中厚度圆板的求解提供了简捷途径, 而且更便于工程应用. 作为例子, 采用上述方法分别求得了周边简支和夹紧条件下, 梯度圆板在均布横向载荷作用下的弯曲解析解, 该解答与Reddy得到的结果完全吻合.      

基于平板振动精确化方程求解动应力集中问题

基于文献[8]给出的平板弯曲振动精确化方程,对含圆孔平板中弹性波散射与动应力集中问题进行了研究.文中给出了分别基于Mindlin板与精确化板方程在不同参数下圆孔动弯矩集中系数的数值结果,并对结果进行了对比分析和讨论.结果表明:在较低频率和薄板情况下,基于文献[8]的方程与基于Mindlin板理论得到的动弯矩结果是基本一致的;在较高频率和厚板情况下,基于文献[8]的方程与基于Mindlin板理论的动弯矩结果相差较大,最大值超出可达16%.由于文献[8]给出的平板振动精确化方程是在没有任何工程假设条件下得到的,因此其分析计算结果更精确一些.      

一种非经典的近似的叠层板弯曲理论

本文利用Hellinger-Reissner变分原理建立复合材料叠层板小变形弯曲的一般理论。这个理论放弃了经典板理论的Kirchhoff-Love假设,考虑了横向剪切变形的影响。利用这个理论计算了一些叠层板的弯曲问题,并与其它方法得到的结果比较,证明这种理论能够获得与精确的弹性解颇为满意一致的结果。     

弹性地基上矩形加肋板自由振动分析的无网格法

应用移动最小二乘无网格法研究弹性地基上矩形加肋板的自由振动问题。假设弹性地基与加肋板紧密接触,以弹簧模拟弹性地基,将弹性地基上的加肋板视为板与肋条组合的结构。基于一阶剪切理论,用无网格伽辽金法推出了板和肋条各自的动能与势能;再通过位移协调条件将两者的能量叠加,得到了弹性地基上整个加肋板的动能与势能。由Hamilton原理导出了弹性地基上加肋板自由振动的控制方程。采用完全转换法引入边界条件,求解自由振动方程,并编制了计算程序,给出了算例。将算例与ABAQUS有限元解及已有文献结果进行了比较分析,其相对误差均在5%以内,验证了该方法计算弹性地基上矩形加肋板结构自振频率的有效性。     

考虑横向拉伸影响的FGM板的静态分析

在三阶剪切变形理论的基础上,添加关于厚度坐标z的幂函数项,并假设板结构的上下表面剪切力为0,提出了一种考虑横向拉伸影响的高阶剪切变形理论。并且研究了简支边界条件下受静态载荷作用的功能梯度材料矩形板的静态弯曲行为。基于虚功原理推导出了功能梯度矩形板的基本方程,利用Navier双三角级数法计算了功能梯度材料矩形板在静态载荷作用下沿厚度方向的位移及应力分布的数值结果。计算结果与三维精确解理论、其他高阶剪切变形理论得到的数值结果进行了比较。对比结果表明,改进的考虑横向拉伸影响的高阶剪切变形理论的正确性和优越性。     

损伤力学中的能量等效假设及其实验验证

大多数损伤模型在构造受损材料本构方程时所采用的应变或应力等效假设在一般情况下并不能满足热力学理论的基本规则.本文提出的能量等效假设,在弹性、塑性和蠕变损伤等普遍情形均证明了其有效性.同时,能量等效假设配合以应变能密度准则,可以很好地描述穿孔铜薄板试件的破断行为.     

用弹性力学方法研究平板稳定性问题

本文用弹性力学的方法研究了面内双向均匀受压和均匀受剪的板状弹性体的稳定性问题.从数学弹性稳定理论的基本方程出发,通过推广胡海昌的位移函数,归结为求解三个非耦合的二阶偏微分方程式,从而减少了解决具体问题所遇到的困难. 文中对四边简支双向均匀受压的矩形板给出了具体算例.计算表明,在薄板情况下得到的临界载荷略低于经典板理论给出的结果.     

粘弹性大挠度圆板的轴对称弯曲

本文探讨粘弹性大挠度圆板的轴对称弯曲的基本方程和求解方法.用半逆解和摄动法分析挠度与膜力,对标准线性固体进行数例计算,并与小挠度理论相比较.全部方程与解答可退化得相应的弹性大挠度板的结果.     

Three-dimensional free vibration analysis of functionally graded piezoelectric annular plates on elastic foundations

Three-dimensional free vibration analysis of functionally graded piezoelectric (FGPM) annular plates resting on Pasternak foundations with different boundary conditions is presented. The material properties are assumed to have an exponent-law variation along the thickness. A semi-analytical approach which makes use of state-space method in thickness direction and one-dimensional differential quadrature method in radial direction is utilized to obtain the influences of the Winkler and shearing layer elastic coefficients of the foundations on the non-dimensional natural frequencies of functionally graded piezoelectric annular plates. The analytical solution in the thickness direction can be acquired using the state-space method and approximate solution in the radial direction can be obtained using the one-dimensional differential quadrature method. Numerical results are given to demonstrate the convergency and accuracy of the present method. The influences of the material property graded index, circumferential wave number and thickness of the annular plate on the dynamic behavior are also investigated. Since three-dimensional free vibration analysis of FGPM annular plates on elastic foundations has not been implemented before, the new results can be used as benchmark solutions for future researches.     

Exact three-dimensional analysis for static torsion of piezoelectric rods

Based on the theory of elasticity, exact analytical and numerical solutions of piezoelectric rods under static torsion are studied. In this paper, direct solution method is used. The main scope is to check the extension of validity of assumptions in previous papers that had been made based on linear distribution of electric potential through the cross section and their influences on deflection and the angle of rotation. Stress and electric induction functions are employed to obtain the exact solution of the static and electrostatic equilibrium equations under torsional loading. It is shown that previous assumptions are valid only in some types of piezoelectric materials, while in other types these assumptions lead to considerable deviations from accurate modeling. The present analytical solutions are compared with three-dimensional finite element analysis results and absolute agreements are found. At the end of this article, torsional rigidity, shape-effects on induced piezoelectric deformation and the range of valid region for linear distribution of electric potential assumption have been studied.     

Stress-deformation state of thick shells and plates. Three-dimensional theory (review)

Conclusions In the first part of the present review we surveyed systematically published results of investigations of the stress-deformation state of thick-walled spheres, ellipsoids, cones, circular cylinders, as well as thick slabs, obtained by exact analytic solutions of spatial problems of elasticity theory. Several quantitative results were given of the variation of displacements and stresses with shell or plate thickness, and their comparative analysis was provided, making it possible to establish the validity limits of the corresponding applied theories. We also surveyed systematically published specific results of the spatial stress-deformation state of nearly canonical thick-walled shells, as well as non-thin plates of varying thickness, obtained by effective approximate analytic methods and known exact solutions for the corresponding canonical regions. Especially noted were characteristic mechanical (including boundary) effects on the stress-deformation state of the bodies under consideration. These effects are generated, in particular, by variations in the radius of curvature of the surface, the thickness parameter, the amplitude and frequency of the corrugated surface, material, inhomogeneity, conditions of mechanical contact between layers, the nature of self-balancing loads, and other factors.However, the possibilities of exact and effective approximate analytic solution of boundary value problems of this class in the three dimensional statement are restricted. In the case of shells and plates of mean thickness these results can be substantially supplemented by qualitative and quantitative data, obtained on the basis of analytic solutions in the generalized theory of shells and plates, based on expansions of components of the stress-deformation state in Legendre polynomial series, and making it possible, in principle, to approximate the three-dimensional solution with any required accuracy. This is one of the basic features distinguishing it from the classical and applied theories of shells and plates. The second part of this review will be devoted to systematic and comparative analysis of the results of investigations carried out within the generalized theory of non-thin shells and plates.Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 27, No. 11, pp. 3–27, November, 1991.     

Accurate equations for laminated composite deep thick shells

This paper derives accurate equations of elastic deformation for laminated composite deep, thick shells. The equations include shells with a pre-twist and accurate force and moment resultants which are considerably different than those used for plates. This is due to the fact that the stresses over the thickness of the shell have to be integrated on a trapezoidal-like cross-section of a shell element to obtain the stress resultants. Numerical results are obtained and showed that accurate stress resultants are needed for laminated composite deep thick shells, especially if the curvature is not spherical. A consistent set of equations of motion, energy functionals and boundary conditions are also derived. These may be used in obtaining exact solutions or approximate ones like the Ritz or finite element methods.     

A refined theory of elastic thick plates for extensional deformation

Based on elasticity theory, various two-dimensional (2D) equations and solutions for extensional deformation have been deduced systematically and directly from the three-dimensional (3D) theory of thick rectangular plates by using the Papkovich–Neuber solution and the Lur’e method without ad hoc assumptions. These equations and solutions can be used to construct a refined theory of thick plates for extensional deformation. It is shown that the displacements and stresses of the plate can be represented by the displacements and transverse normal strain of the midplane. In the case of homogeneous boundary conditions, the exact solutions for the plate are derived, and the exact equations consist of three governing differential equations: the biharmonic equation, the shear equation, and the transcendental equation. With the present theory a solution of these can satisfy all the fundamental equations of 3D elasticity. Moreover, the refined theory of thick plate for bending deformation constructed by Cheng is improved, and some physical or mathematical explanations and proof are provided to support our justification. It is important to note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of nonhomogeneous boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order terms with respect to plate thickness. The correctness of the stress assumptions in the classic plane-stress problems is revised. In an example it is shown that the exact or accurate solutions may be obtained by applying the refined theory deduced herein.     

F E M analysis of delamination buckling in composite plates and shells

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Similarity solution of self-weight consolidation problem for saturated soil

IntroductionA greatdeal of engineeringand environmental projects,such as the tailings reservoir,thehydraulic fill dam,the lake dredging and the estuarine sedimentation,etc.,need to deal withthe problems about the consolidation and sedimentation of recentl…     

The role of convergence in the theory of shells

Similarly to what happens in the finite element method, the concept of convergence can be used for justifying the use of the virtual work and variational theorems in the derivation of the equations of the theory of shells.It is proved that the two-dimensional solution becomes more and more near the three-dimensional ones as the thickness tends to zero, provided the relative values of the bending and membrane stiffness coefficients are not changed when the shell becomes thinner and thinner.Such condition can be respected only if the shell is a generalized one, i.e. if the couple-stresses are not supposed to vanish.The analysis gives a upper bound to the order of magnitude of the distance between the exact and approximate solution and thus provides a powerful method for testing the efficiency and consistency of any particular theory of shells.     

EXACT SOLUTION FOR ORTHOTROPIC MATERIALS WEAKENED BY DOUBLY PERIODIC CRACKS OF UNEQUAL SIZE UNDER ANTIPLANE SHEAR

Orthotropic materials weakened by a doubly periodic array of cracks under far-field antiplane shear are investigated, where the fundamental cell contains four cracks of unequal size. By applying the mapping technique, the elliptical function theory and the theory of analytical function boundary value problems, a closed form solution of the whole-field stress is obtained. The exact formulae for the stress intensity factor at the crack tip and the effective antiplane shear modulus of the cracked orthotropic material are derived. A comparison with the finite element method shows the efficiency and accuracy of the present method. Several illustrative examples are provided, and an interesting phenomenon is observed, that is, the stress intensity factor and the dimensionless effective modulus are independent of the material property for a doubly periodic cracked isotropic material, but depend strongly on the material property for the doubly periodic cracked orthotropic material. Such a phenomenon for antiplane problems is similar to that for in-plane problems. The present solution can provide benchmark results for other numerical and approximate methods.