板列弯曲振动及功率流分析的辛空间波传播方法

基于波传播理论,在辛空间下研究了由矩形薄板组成的板列结构的自由波属性以及受迫振动问题.通过将薄板弯曲振动控制方程导入辛对偶体系,得到了薄板波传播参数以及各阶波形的辛解析解.根据波在各板之间的传播、反射以及透射关系和叠加原理得到问题的解.给出了辛空间-波传播框架下各板动能、应变能以及板间功率流的计算表达式.相比传统波传播方法,该方法具有不受边界条件限制以及能够给出波模态辛解析解的特点.以一个三板组合结构为算例,通过与ABAQUS程序得到的有限元参考解进行对比,验证了所提出方法的高效性与精确性.由于完全基于理性推导,不涉及任何试函数的引入,因此该方法也可推广应用于由其他类型板(如中厚板、层合板等)组合的板列结构动力响应分析问题.     

基于非局部应变梯度理论功能梯度纳米板的弯曲和屈曲研究

以纳米机器人等智能器件中的功能梯度纳米板结构为研究对象,基于非局部应变梯度理论,研究了其弯曲和屈曲问题.推导了一般情况下的功能梯度纳米板运动方程,弯曲和屈曲作为其特例可简化而成.分析了非局部尺度参数、材料特征尺度参数、梯度指数、纳米板尺寸等对弯曲挠度和临界屈曲载荷的影响.结果表明:不同高阶连续介质力学理论下的最大挠度都随梯度指数的增大而增大,正方形纳米板挠度较小,且板厚越大,弯曲挠度越小;最大挠度随非局部尺度参数的增大而增大,随材料特征尺度参数的增大而减小.临界屈曲载荷随梯度指数的增大而减小,随板厚、长宽比的增大而增大,随非局部尺度的增大而减小,随材料特征尺度的增大而增大.非局部应变梯度高阶弯曲和屈曲中存在结构软化与硬化机制,两个内特征参数之间具有耦合效应,当非局部尺度大于材料特征尺度时,非局部效应在功能梯度纳米板力学性能中占主导作用;当材料特征尺度大于非局部尺度时,应变梯度效应占主导作用.解析结果还证明了当非局部尺度等于材料特征尺度时,非局部应变梯度理论结果退化为经典结果.     

复杂形状及开孔功能梯度板的三维分析

推导出适应功能梯度材料构件分析的半解析梯度有限元法基本算式,并针对功能梯度构件的材料参数随空间坐标变化的特点,将材料参数纳入到力学方程中进行整体积分计算,从而分析不同情况功能梯度板件问题.该法适应性强而又简洁高效,采用一维离散,给出三维分析结果,是一种解决功能梯度构件力学分析的有效数值方法.用提出的方法分析几种具有复杂形状及开孔的功能梯度板,给出了板件的力学量三维分布形态.     

一种WYL型谱共轭梯度法的全局收敛性

为解决大规模无约束优化问题,该文结合WYL共轭梯度法和谱共轭梯度法,给出了一种WYL型谱共轭梯度法.在不依赖于任何线搜索的条件下,该方法产生的搜索方向均满足充分下降性,且在强Wolfe线搜索下证明了该方法的全局收敛性.与WYL共轭梯度法的收敛性相比,WYL型谱共轭梯度法推广了线搜索中参数σ的取值范围.最后,相应的数值结果表明了该方法是有效的.     

一类含小参数的Hill方程的自由振动

本文研究一类含小参数的Hill方程的初值问题,利用边值问题可解性条件及摄动理论中的伸缩参数法,给出寻求该初值问题近似周期解的方法,并以Mathieu方程为例作了具体计算.     

周期性正弦凸起凹凸板等效刚度的研究

针对构造正交各向异性周期性正弦凸起结构凹凸板的等效刚度问题，根据经典弹性薄板理论，基于对单胞结构力学特性分析和单胞结构在板宏观结构上周期性均匀化分布的特点，推导了正弦凸起凹凸板的等效刚度解析公式.以四边简支周期性正弦凸起结构凹凸板为例，将该文计算结果与有限元模拟结果进行对比，验证了该文等效刚度的合理性和精确性.最后，分析了正弦凸起凹凸板几何参数对等效刚度特性的影响，给出了结构几何参数与等效刚度之间的关系.结果表明:应用该文方法可以有效计算周期性正弦凸起凹凸板的等效刚度；由于凹凸板在构造上的几何结构变化，与基础平板相比其弯曲刚度和抗扭刚度都有明显的提升.该研究结果对凹凸板静力学和动力学的进一步研究以及实际工程应用具有指导意义.     

多层简化应变梯度Timoshenko梁的变分原理分析

材料特征尺寸与其内禀尺寸相当时,材料表现出明显的尺寸效应.基于简化的应变梯度理论,通过半逆法,本文给出多层简化应变梯度Timoshenko梁的变分原理,通过最小总势能原理导出系统的边界条件并对其低阶和高阶边界条件进行讨论,随后给出简支梁系统屈曲载荷和振动频率的Rayleigh(瑞利)解.通过双层梁系统的振动分析算例得到内禀尺寸、长径比等因素对梁系统振动频率的影响.该文构造的Rayleigh解有望对其他数值方法,如有限元法、传递矩阵法等,提供一定的参考和对比.     

面内功能梯度三角形板等几何面内振动分析

基于平面应变理论,利用等几何有限元方法分析了弹性边界条件下面内功能梯度三角形板的面内振动特性.板的材料属性沿厚度方向呈均匀分布,而在面内方向呈任意指数梯度变化.采用非均匀有理B样条(NURBS)基函数对三角形结构进行等几何建模和位移描述,实现了三角形板几何设计和振动分析的无缝衔接.在三角形板边界上引入虚拟弹簧约束并通过调节虚拟弹簧刚度,实现任意边界条件的施加.通过不同的单元细化方案和对比算例,验证了等几何方法的灵活性、准确性和快速收敛性.系统研究了边界条件、材料属性和几何参数对三角形板振动特性的影响.同时给出了弹性边界条件下面内功能梯度三角形板的振动特性解,具有重要参考价值.     

准三维功能梯度微梁的尺度效应模型及微分求积有限元

基于修正的偶应力理论与四参数高阶剪切-法向伸缩变形理论,提出了一种具有尺度依赖性的准三维功能梯度微梁模型,并应用于小尺度功能梯度梁的静力弯曲和自由振动分析中.采用第二类Lagrange方程,推导了微梁的运动微分方程及边界条件.针对一般边值问题,构造了一种融合Gauss-Lobatto求积准则与微分求积准则的2节点16自由度微分求积有限元.通过对比性研究,验证了理论模型以及求解方法的有效性.最后,探究了梯度指数、内禀特征长度、几何参数及边界条件对微梁静态响应与振动特性的影响.结果表明,该文所发展的梁模型及微分求积有限元适用于研究各种长细比的功能梯度微梁的静/动力学问题,引入尺度效应会显著地改变微梁的力学特性.     

非线性规划的不使用导数的解析方法—统一探讨

本文从统一的角度给出了解约束优化问题的修正的解析方法,即用梯度的近似值代替梯度的解析方法,并在较弱的条件下证明了其全局收敛性。本文结果扩大了解析方法的应用范围,为优化算法的实现提供了理论依据.     

Flow and heat transfer of double fractional Maxwell fluids over a stretching sheet with variable thickness

This paper presents research on the fractional boundary layer flow and heat transfer over a stretching sheet with variable thickness. Based on the Caputo operators, the double fractional Maxwell model and generalized Fourier's law are introduced to the constitutive relationships. The governing equations are solved numerically by utilizing the finite difference method. The effects of fractional parameters on the velocity and temperature field are analyzed. The results indicate that the larger is the fractional stress parameter, the stronger is the elastic characteristic. However, fluids show viscous fluid-like behavior for a larger value of fractional strain parameter. Moreover, the numerical solutions are in good agreement with the exact solution and the convergence order can achieve the expected first order. The numerical method in this study is reliable and can be extended to other fractional boundary layer problems over a variable thickness sheet.     

Bending of thick functionally graded plates resting on Winkler–Pasternak elastic foundations

The static response of simply supported functionally graded plates (FGP) subjected to a transverse uniform load (UL) or a sinusoidally distributed load (SL) and resting on an elastic foundation is examined by using a new hyperbolic displacement model. The present theory exactly satisfies the stress boundary conditions on the top and bottom surfaces of the plate. No transverse shear correction factors are needed, because a correct representation of the transverse shear strain is given. The material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of material constituents. The foundation is modeled as a two-parameter Pasternak-type foundation, or as a Winkler-type one if the second parameter is zero. The equilibrium equations of a functionally graded plate are given based on the hyperbolic shear deformation theory of plates presented. The effects of stiffness and gradient index of the foundation on the mechanical responses of the plates are discussed. It is established that the elastic foundations significantly affect the mechanical behavior of thick functionally graded plates. The numerical results presented in the paper can serve as benchmarks for future analyses of thick functionally graded plates on elastic foundations.     

A comparison of strain gradient theories with applications to the functionally graded circular micro-plate

The strain gradient theory of Zhou et al. is re-expressed in a more direct form and the differences with other strain gradient theories are investigated by an application on static and dynamic analyses of FGM circular micro-plate. To facilitate the modeling, the strain gradient theory of Zhou et al. is re-expressed in cylindrical coordinates, and then the governing equation, boundary conditions and initial condition for circular plate are derived with the help of the Hamilton's principle. The present model can degenerate into other models based on the strain gradient theory of Lam et al., the couple stress theory, the modified couple stress theory or even the classical theory, respectively. The static bending and free vibration problems of a simply supported circular plate are solved. The results indicate that the consideration of strain gradients results in an increase in plate stiffness, and leads to a reduction of deflection and an increase in natural frequency. Compared with the reduced models, the present model can predict a stronger size effect since the contribution from all strain gradient components is considered, and the differences of results from all these models are diminishing when the plate thickness is far greater than the material length-sale parameter.     

3D elasticity analytical solution for bending of FG micro/nanoplates resting on elastic foundation using modified couple stress theory

This paper addresses a 3D elasticity analytical solution for static deformation of a simply-supported rectangular micro/nanoplate made of both homogeneous and functionally graded (FG) material within the framework of modified couple stress theory. The plate is assumed to be resting on a Winkler–Pasternak elastic foundation, and its modulus of elasticity is assumed to vary exponentially along thickness. By expanding displacement components in double Fourier series along in-plane coordinates and imposing relevant boundary conditions, the boundary value problem (BVP) of plate system, including its governing partial differential equations (PDEs) of equilibrium are reduced to BVP consisting only ordinary ones (ODEs). Parametric studies are conducted among displacement and stress components developed in the plate and FG material gradient index, length scale parameter, and foundation stiffnesses. From the numerical results, it is concluded that the out-of-plane shear stresses are not necessarily zero at the top and bottom surfaces of plate. The results of this investigation may serve as a benchmark to verify further bending analyses of either homogeneous or FG micro/nanoplates on elastic foundation.     

Effect of a Floating Elastic Plate/Membrane on the Motion Due to a Ring Source in Water With Porous Bed

The velocity potentials due to the presence of a horizontal circular ring of wave sources of timedependent strength in water of finite constant depth with a floating elastic plate or a floating membrane are determined. The uniform bottom is composed of non-dissipative porous medium. The problems are formulated as the initial value problems and the Laplace transform method is used to solve these. For time-harmonic source strength, the steady-state analysis of the potentials reveals the existence of outgoing progressive waves. Graphs for the surface profiles are presented for different values of the tension parameter for the membrane, flexural rigidity of ice and the porous-effect parameter.     

双参数弹性地基上自由边矩形板

本文以迭加法[1]给出在V. Z. Vlazov双参数弹性地基上自由边矩形板的精确解.文中导出了在各种边界条件下的基本解式,迭加这些基本解式,求得了在双参数弹性地基上自由边矩形板的最一般的精确解.它严格满足双参数弹性地基上板的控制微分方程和自由边的边界条件和角点条件.给出了数值结果.计算结果表明:当板的平面尺寸一定,地基深度与板厚度之比H/h=15时,双参数弹性地基与Winkler弹性地基相接近,证明了Winkler地基模式适用于压缩尺寸比较薄的弹性地基.     

样条有限元

本文用三次B样条变分方法解规则区域上板梁组合弹性结构的平衡问题.推导出了适用于各种边界条件的统一计算格式,便于在计算机上实现.与通常有限元相比,具有计算量少、精确度高等显著特点.文中对自然边界条件作为约束条件的影响给予了考虑,并以板的弯曲问题为例说明影响极微.给出了几个数值的例子.     

Three-dimensional exact solution of layered two-dimensional quasicrystal simply supported nanoplates with size-dependent effects

A size-dependent plate model is developed to investigate the elastic responses of the multilayered two-dimensional quasicrystal nanoplates based on the nonlocal strain gradient theory for the first time. A nonlocal stress field parameter and a length scale parameter are taken into account in the new model to capture both stiffness-softening and stiffness-hardening size effects. The exact solution for a single-layer two-dimensional quasicrystal simply supported nanoplate is derived by utilizing the pseudo-Stroh formalism in conjunction with the nonlocal strain gradient theory. Afterward, a dual variable and position method is used to deal with the multilayered case. Numerical examples are presented to study the dependence of size-dependent effect on nanoplate length and the influences of scale parameters on the quasicrystal nanoplate subjected to a z-direction mechanical load on its top surface. The proposed model should be useful to verify various nanoplate theories and other numerical methods.     

Singular boundary integral equations of boundary value problems of elastic dynamics in the case of subsonic running loads

We consider boundary value problems for an elastic medium bounded by a cylindrical surface on which a load with a constant-in-time form is moving at a constant subsonic velocity. This class of problems is a model class for the dynamics of underground structures like transport tunnels and ground transport. The method of generalized functions is developed for the solution of boundary value problems. We construct dynamic analogs of Green formulas and use them to derive singular boundary integral equations, which solve the considered boundary value problems.     

Three-dimensional layerwise-finite element free vibration analysis of thick laminated annular plates on elastic foundation

Using a three-dimensional layerwise-finite element method, the free vibration of thick laminated circular and annular plates supported on the elastic foundation is studied. The Pasternak-type formulation is employed to model the interaction between the plate and the elastic foundation. The discretized governing equations are derived using the Hamilton’s principle in conjunction with the layerwise theory in the thickness direction, the finite element (FE) in the radial direction and trigonometric function in the circumferential direction, respectively. The fast rate of convergence of the method is demonstrated and to verify its accuracy, comparison studies with the available solutions in the literature are performed. The effects of the geometrical parameters, the material properties and the elastic foundation parameters on the natural frequency parameters of the laminated thick circular and annular plates subjected to various boundary conditions are presented.