Stress boundary conditions for plate bending

The determination of the appropriate boundary conditions for a two-dimensional theory of elastic flat plates (and shells) consistent with the expected order of accuracy of the theory is both critical and challenging. The reciprocal theorem of elasticity will be applied in a novel way to obtain the appropriate stress boundary conditions for plate bending accurate to all order (with respect to the usual dimensionless thickness parameter) for plates of general edge geometry and loading. Kirchhoff’s two contracted stress boundary conditions are shown to be consistent with a leading term (thin plate) approximation theory, but the more general results obtained herein are needed for higher order theories.     

A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory

A size-dependent Kirchhoff micro-plate model is developed based on the strain gradient elasticity theory. The model contains three material length scale parameters, which may effectively capture the size effect. The model can also degenerate into the modified couple stress plate model or the classical plate model, if two or all of the material length scale parameters are taken to be zero. The static bending, instability and free vibration problems of a rectangular micro-plate with all edges simple supported are carried out to illustrate the applicability of the present size-dependent model. The results are compared with the reduced models. The present model can predict prominent size-dependent normalized stiffness, buckling load, and natural frequency with the reduction of structural size, especially when the plate thickness is on the same order of the material length scale parameter.     

VIBRATION ANALYSIS OF MICROSCALE PLATES BASED ON MODIFIED COUPLE STRESS THEORY

A non-classical Kirchhoff plate model is developed for the dynamic analysis of microscale plates based on the modified couple stress theory in which an internal material length scale parameter is included. Unlike the classical Kirchhoff plate model, the newly developed model can capture the size effect of microscale plates. Two boundary value problems of rectangular micro- plates are solved and the size effect on the lowest two natural frequencies is investigated. It is shown that the natural frequencies of the microscale plates predicted by the current model are size-dependent when the plate thickness is comparable to the material length scale parameter.     

考虑表面效应的双层圆板的自由振动分析

考虑微生化传感器中谐振器的结构特点,基于Kirchhoff薄板理论与表面弹性理论推导了考虑表面效应的双层圆板的自由振动方程.使用伽辽金法得到了近似解.分析了硬化与软化表面效应与表面残余应力对双层圆板固有频率的影响.结果表明,与已有简化的单层圆板模型相比,现有考虑表面效应的双层板模型会得到与之不同的固有频率.随着板厚与上...     

基于应变梯度有限元的单层石墨烯振动研究

建立了单层石墨烯等效非局部薄板的一种新的有限元模型,并运用有限元法分析不同边界条件下单层石墨烯振动的小尺度效应。给出了基于弹性应变梯度理论下Kirchhoff板的振动方程。发展了一种4节点24自由度的板单元,用于离散化求解考虑微纳结构尺度效应的高阶微分方程。在研究四边简支板振动时,考虑应变梯度的非局部弹性有限元数值计算结果与理论分析结果相一致。用有限元方法研究了不同尺寸、振动波长、振动模态阶数、边界条件类型以及非局部参数的单层石墨烯振动。     

面内变刚度薄板弯曲问题的挠度?弯矩耦合神经网络方法

发展了一种求解面内变刚度功能梯度薄板弯曲问题的神经网络方法. 面内变刚度薄板弯曲问题的偏微分控制方程为一复杂的4阶偏微分方程, 传统的基于强形式的神经网络解法在求解该偏微分方程时可能会遇到难以收敛、边界条件难以处理的情况. 本文基于Kirchhoff薄板弯曲理论, 提出了一种直角坐标系下任意面内变刚度薄板弯曲问题的神经网络解法. 神经网络模型包含挠度网络与弯矩网络, 分别用于预测薄板的挠度与弯矩, 从而将求解4阶偏微分方程转换为求解一系列二阶偏微分方程组, 通过对挠度、弯矩试函数的构造可使得神经网络计算结果严格满足边界条件. 在误差的反向传播中, 根据本文提出的误差函数公式计算训练误差并结合Adam优化算法更新模型的内部参数. 求解了不同边界条件、形状的面内变刚度薄板弯曲问题, 并将所得计算结果与理论解、有限元解进行对比. 研究表明, 本文模型对于求解面内变刚度薄板弯曲问题具备适应性, 虽然模型中的弯矩网络收敛较挠度网络要慢, 但本文方法在试函数的构造上更为简单、适应性更强.      

桁架板等效刚度分析

桁架材料的连续介质等效模型的研究已有相当基础,而工程中桁架材料往往以类板结构形式出现,其变形表现出明显的弯曲特征。将类板桁架材料采用弯曲板模型模拟,研究合理的方法确定等效板模型的刚度具有重要意义。本文在基于Kirchhoff假定的小挠度薄板弹性理论框架下,研究了类板桁架材料的等效弯曲薄板模型,提出了确定薄板模型等效刚度的基于Dirichlet位移边界条件的代表体元法,给出了确定各刚度系数所对应的代表体元的边界位移形式。具体计算了几种典型形式桁架板的等效刚度,并采用有限元离散模型和实验技术分析了桁架板在一定的边界约束和荷载作用下的响应,并与等效板模型的分析结果进行了对比。结果表明,在响应分析中,具有等效刚度的薄板模型可准确模拟类板桁架材料;连续介质板等效刚度计算的积分法不能给出准确的桁架板等效刚度,而基于Dirichlet位移边界条件的代表体元法获得的等效板的刚度具有很高的精度。     

Deflection of sandwich plates in terms of corresponding Kirchhoff plate solutions

Summary This study presents exact relationships between the deflections of isotropic sandwich plates and their corresponding Kirchhoff plates. The governing equilibrium equations for the sandwich plates are derived on the basis of the Reissner-Mindlin shear deformation plate theory. The considered plates are either (i) simply supported, of general polygonal shape and under any transverse loading condition or (ii) simply supported and clamped circular plates under axisymmetric loading. As the relationships are exact under the assumptions used in the plate theories, one may obtain exact deflection solutions of sandwich plates if the Kirchhoff plate solutions are exact. The relationships should also be useful for the development of approximate formulas for plates with other shapes, boundary and loading conditions, and may serve to check numerical deflection values computed from sandwich plate analysis software.     

A new microstructure-dependent Saint–Venant torsion model based on a modified couple stress theory

In this paper a new modified couple stress model is developed for the Saint–Venant torsion problem of micro-bars of arbitrary cross-section. The proposed model is derived from a modified couple stress theory and has only one material length scale parameter. Using a variational procedure the governing differential equation and the associated boundary conditions are derived in terms of the warping function. This is a fourth order partial differential equation representing the analog of a Kirchhoff plate having the shape of the cross-section and subjected to a uniform tensile membrane force with mixed Neumann boundary conditions. Since the fundamental solution of the equation is known, the problem could be solved using the direct Boundary Element Method (BEM). In this investigation, however, the Analog Equation Method (AEM) solution is applied and the results are cross checked using the Method of Fundamental Solutions (MFS). Several micro-bars of various cross-sections are analyzed to illustrate the applicability of the developed model and to reveal the differences between the current model and an existing one which, however, contains two additional constants related to the microstructure. Moreover, useful conclusions are drawn from the micron-scale torsional response of micro-bars, giving thus a better insight in the gradient elasticity approach of the deformable bodies.     

Asymptotic frequencies of various damped nonlocal beams and plates

A striking difference between the conventional local and nonlocal dynamical systems is that the later possess finite asymptotic frequencies. The asymptotic frequencies of four kinds of nonlocal viscoelastic damped structures are derived, including an Euler–Bernoulli beam with rotary inertia, a Timoshenko beam, a Kirchhoff plate with rotary inertia and a Mindlin plate. For these undamped and damped nonlocal beam and plate models, the analytical expressions for the asymptotic frequencies, also called the maximum or escape frequencies, are obtained. For the damped nonlocal beams or plates, the asymptotic critical damping factors are also obtained. These quantities are independent of the boundary conditions and hence simply supported boundary conditions are used. Taking a carbon nanotube as a numerical example and using the Euler–Bernoulli beam model, the natural frequencies of the carbon nanotubes with typical boundary conditions are computed and the asymptotic characteristics of natural frequencies are shown.     

A homogenized Reissner–Mindlin model for orthotropic periodic plates: Application to brickwork panels

This paper describes a new procedure for the homogenization of orthotropic 3D periodic plates. The theory of Caillerie [Caillerie, D., 1984. Thin elastic and periodic plates. Math. Method Appl. Sci., 6, 159–191.] – which leads to a homogeneous Love–Kirchhoff model – is extended in order to take into account the shear effects for thick plates. A homogenized Reissner–Mindlin plate model is proposed. Hence, the determination of the shear constants requires the resolution of an auxiliary 3D boundary value problem on the unit cell that generates the periodic plate. This homogenization procedure is then applied to periodic brickwork panels.A Love–Kirchhoff plate model for linear elastic periodic brickwork has been already proposed by Cecchi and Sab [Cecchi, A., Sab, K., 2002b. Out-of-plane model for heterogeneous periodic materials: the case of masonry. Eur. J. Mech. A-Solids 21, 249–268 ; Cecchi, A., Sab, K., 2006. Corrigendum to A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork [Int. J. Solids Struct., vol. 41/9–10, pp. 2259–2276], Int. J. Solids Struct., vol. 43/2, pp. 390–392.]. The identification of a Reissner–Mindlin homogenized plate model for infinitely rigid blocks connected by elastic interfaces (the mortar thin joints) has been also developed by the authors Cecchi and Sab [Cecchi A., Sab K., 2004. A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork. Int. J. Solids Struct. 41/9–10, 2259–2276.]. In that case, the identification between the 3D block discrete model and the 2D plate model is based on an identification at the order 1 in the rigid body displacement and at the order 0 in the rigid body rotation.In the present paper, the new identification procedure is implemented taking into account the shear effect when the blocks are deformable bodies. It is proved that the proposed procedure is consistent with the one already used by the authors for rigid blocks. Besides, an analytical approximation for the homogenized shear constants is derived. A finite elements model is then used to evaluate the exact shear homogenized constants and to compare them with the approximated one. Excellent agreement is found. Finally, a structural experimentation is carried out in the case of masonry panel under cylindrical bending conditions. Here, the full 3D finite elements heterogeneous model is compared to the corresponding 2D Reissner–Mindlin and Love–Kirchhoff plate models so as to study the discrepancy between these three models as a function of the length-to-thickness ratio (slenderness) of the panel. It is shown that the proposed Reissner–Mindlin model best fits with the finite elements model.     

基于各向异性修正偶应力理论的Reddy型层合板的自由振动

论文基于各向异性修正偶应力理论建立了只含一个尺度参数的Reddy型复合材料层合板的自由振动模型.同见诸于文献的细观尺度Kirchhoff薄板偶应力模型相比,论文提出的新模型能够更精确的预测细观尺度下的中、厚层合板的自振频率.基于Hamilton原理推导了细观尺度下Reddy型复合材料层合板的运动微分方程以及边界条件,并以正交铺设的四边简支复合材料层合方板为例进行了解析求解,分析了尺度参数对自振频率的影响并对比了Kirchhoff、Mindlin和Reddy等三种板模型计算结果的异同.算例结果表明论文所给出的模型能够捕捉到复合材料层合板自由振动问题的尺度效应.另外,在细观尺度下Kirchhoff板模型所预测的自振频率相对于Mindlin板模型和Reddy板模型总是过高,且越接近厚板三者的差别就越大,这与经典理论中三种板模型的对比情况是一致的.     

Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory

In this study, non-linear free vibration of micro-plates based on strain gradient elasticity theory is investigated. A general form of Mindlin’s first-strain gradient elasticity theory is employed to obtain a general Kirchhoff micro-plate formulation. The von Karman strain tensor is used to capture the geometric non-linearity. The governing equations of motion and boundary conditions are obtained in a variational framework. The Homotopy analysis method is employed to obtain an accurate analytical expression for the non-linear natural frequency of vibration. For some specific values of the gradient-based material parameters, the general plate formulation can be reduced to those based on some special forms of strain gradient elasticity theory. Accordingly, three different micro-plate formulations are introduced, which are based on three special strain gradient elasticity theories. It is found that both geometric non-linearity and size effect increase the natural frequency of vibration. In a micro-plate having a thickness comparable with the material length scale parameter, the strain gradient effect on increasing the non-linear natural frequency is higher than that of the geometric non-linearity. By increasing the plate thickness, the strain gradient effect decreases or even diminishes. In this case, geometric non-linearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for micro-plates with some specific thickness to length scale parameter ratios, both geometric non-linearity and size effect have significant role on increasing the frequency of non-linear vibration.     

A Bending-Gradient model for thick plates,Part II: Closed-form solutions for cylindrical bending of laminates

In the first part (Lebée and Sab, 2010a) of this two-part paper we have presented a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff–Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called Bending-Gradient plate theory is an extension to arbitrarily layered plates of the Reissner–Mindlin plate theory which appears as a special case when the plate is homogeneous. Moreover, we demonstrated that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner–Mindlin model. In this paper, the Bending-Gradient theory is applied to laminated plates and its predictions are compared to those of Reissner–Mindlin theory and to full 3D (Pagano, 1969) exact solutions. The main conclusion is that the Bending-Gradient gives good predictions of deflection, shear stress distributions and in-plane displacement distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity.     

Formulation of Problems in the General Kirchhoff—Love Theory of Inhomogeneous Anisotropic Plates

In this paper we study the procedure of reducing the three-dimensional problem of elasticity theory for a thin inhomogeneous anisotropic plate to a two-dimensional problem in the median plane. The plate is in equilibrium under the action of volume and surface forces of general form. À notion of internal force factors is introduced. The equations for force factors (the equilibrium equations in the median plane) are obtained from the thickness-averaged three-dimensional equations of elasticity theory. In order to establish the relation between the internal force factors and the characteristics of the deformed middle surface, we use some prior assumptions on the distribution of displacements along the thickness of the plate. To arrange these assumptions in order, the displacements of plate points are expanded into Taylor series in the transverse coordinate with consideration of the physical hypotheses on the deformation of a material fiber being originally perpendicular to the median plane. The well-known Kirchhoff—Love hypothesis is considered in detail. À closed system of equations for the theory of inhomogeneous anisotropic plates is obtained on the basis of the Kirchhoff—Love hypothesis. The boundary conditions are formulated from the Lagrange variational principle.     

Direct approach-based analysis of plates composed of functionally graded materials

The classical plate theory can be applied to thin plates made of classical materials like steel. The first theory allowing the analysis of such plates was elaborated by Kirchhoff. But this approach was connected with various limitations (e.g., constant material properties in the thickness direction). In addition, some mathematical inconsistencies like the order of the governing equation and the number of boundary conditions exist. During the last century many suggestions for improvements of the classical plate theory were made. The engineering direction of improvements was ruled by applications (e.g., the use of laminates or sandwiches as the plate material), and so new hypotheses for the derivation of the governing equations were introduced. In addition, some mathematical approaches like power series expansions or asymptotic integration techniques were applied. A conceptional different direction is connected with the direct approach in the plate theory. This paper presents the extension of Zhilin’s direct approach to plates made of functionally graded materials. The second author was supported by DFG grant 436RUS17/21/07.     

The Stress State of Multilayered Physically Nonlinear Plates

A theory of and a solution to stationary and nonstationary problems for multilayered unsoldered plates are constructed. They are shown to converge within the framework of the kinematic Kirchhoff model with physical nonlinearity allowed for. The methods of variational iterations and finite differences are used for stress–strain analysis of a two-layered unsoldered plate of constant thickness with allowance for structural and physical nonlinearities. A wide class of problems is considered. In addition, the complex vibrations of two-layered plates are studied for arbitrary boundary conditions and for different gaps between layers     

Variational analysis of gradient elastic flexural plates under static loading

Gradient elastic flexural Kirchhoff plates under static loading are considered. Their governing equation of equilibrium in terms of their lateral deflection is a sixth order partial differential equation instead of the fourth order one for the classical case. A variational formulation of the problem is established with the aid of the principle of virtual work and used to determine all possible boundary conditions, classical and non-classical ones. Two circular gradient elastic plates, clamped or simply supported at their boundaries, are analyzed analytically and the gradient effect on their static response is assessed in detail. A rectangular gradient elastic plate, simply supported at its boundaries, is also analyzed analytically and its rationally obtained boundary conditions are compared with the heuristically obtained ones in a previous publication of the authors. Finally, a plate with two opposite sides clamped experiencing cylindrical bending is also analyzed and its response compared against that for the cases of micropolar and couple-stress elasticity theories.     

Kirchhoff and thomson-tait transformations in the classical theory of plates

The transformation of the torque into the transverse force is considered; this transformation is traditional in the educational literature [1] and was proposed by Kirchhoff [2] and Thomson and Tait [3] to match the order of the differential equation of the classical theory of plates with the number of boundary conditions. It is shown that this transformation is not universal and its mathematical and physical justification depends on the conditions of the plate fixation and loading. It is shown that this justification is absent for the most widely used problems of bending of a rectangular plate freely supported and fixed on the contour.     

Boundary Element Analysis of Raft Foundations on Piles

The boundary element method is used in the formulation of models for the analysis of raft foundations on piles. Two models are considered: a Kirchhoff plate on a layered elastic half-space and a Kirchhoff plate on a Winkler soil. The plates are modelled using conforming boundary elements and the piles by using linear finite elements. Mindlin's solution is used as influence function within the half-space while Boussinesq's solution, a precursor although a particular case of Mindlin's solution, is used to derive the deflections of the soil surface. The models are used in the analysis of some raft foundations on piles and the results and relative merits are discussed.