利用振动模态分析确定对称复合层板结构的材料特性

针对周边简支和周边固支含弯曲·扭曲耦合的对称复合材料层板,本文首先推导了利用振动模态分析进行层板５个弯曲刚度常数评价的理论公式,并利用振动测试得到的有限个离散点的挠度信息对振动模态进行近似处理,最后利用最小二乘法求解层板的弯曲刚度。在数值算例中,本文对两种边界条件下三种对称复合材料层板的弯曲刚度进行了评价,同时考察了测试点数目对弯曲刚度常数评价误差的影响。     

Finite volume analysis of laminated composite plates

A numerical procedure for analysis of general laminated plates under transverse load is developed utilizing the Mindlin plate theory, the finite volume discretization, and a segregated solution algorithm. The force and moment balance equations with the laminate constitutive relations are written in the form of a generic transport equation. In order to obtain discrete counterparts of the governing equations, the plate is subdivided into N control volumes by a Cartesian numerical mesh. As a result, five sets of N linear equations with N unknowns are obtained and solved using the conjugate gradient method with preconditioning. For the method validation, a number of test cases are designed to cover thick and thin laminated plates with aspect ratio (width to thickness) from 4 to 100. Simply supported orthotropic, symmetric cross‐ply, and angle‐ply laminated plates under uniform and sinusoidal pressure loads are solved, and results are compared with available analytical solutions. The shear correction factor of 5/6 is utilized throughout the procedure, which is consistent with test cases used in the reviewed literature. Comparisons of the finite volume method results for maximum deflections at the center of the plate and the Navier solutions obtained for aspect ratios 10, 20, and 100 shows a very good agreement. Copyright © 2016 John Wiley & Sons, Ltd.     

On the use of Levy's method for symmetrically laminated composite plates

For rectangular plates with two opposite edges simply supported and no bending twisting coupling, Levy's method can yield exact solutions for static deflections, free vibrations and buckling loads. However, three distinct solutions are obtained, depending on the relative magnitude of the plate rigidities, which in turn are functions of material properties, laminate thickness and lay-up. In this communication a stiffness invariant formulation is used to identify those laminates which fall under the three cases defined.     

Nonlinear bending analysis of orthotropic rectangular plates by the method of differential quadrature

The behavior of thin, rectangular, orthotropic elastic plates, with immovable edges and undergoing large deflections, is investigated by the numerical technique of differential quadrature. Approximate results are obtained, using the Newton-Raphson method and, alternatively, a finite-difference-based method to solve the nonlinear systems of equations. Bending stresses, membrane stresses, and deflections are calculated for plates with fully clamped and simply supported flexural edge conditions under uniform pressure loading. Results are compared with existing analytical, numerical, and experimental ones. The present method gives good accuracy and is computationally efficient.     

Impact analysis of laminated composite plates and shells by super finite elements

A super finite element method that exhibits coarse-mesh accuracy is used to predict the transient response of laminated composite plates and cylindrical shells subjected to non-penetrating impact by projectiles. The governing equations are based on the classical theories of thin laminated plates and shells taking into account the von Karman kinematics assumptions for moderately large deflections. A non-linear Hertzian-type contact law accounting for curvatures of the colliding bodies is adopted to calculate the impact force . The theoretical basis of the present finite element model is verified by analysing impact-loaded laminated composite plate and shell structures that have previously been studied through analytical or other numerical procedures. The predictive capability of the present numerical approach is successfully demonstrated through comparisons between experimentally-measured and computed force-time histories for impact of carbon fibre-reinforced plastic (CFRP) plates. The current computational model offers a relatively simple and efficient means of predicting the structural impact response of laminated composite plates and shells.     

轴向运动大挠度板的非线性动力学行为

该文研究了受周期激励轴向运动大挠度板横向振动的稳定性及分岔现象。在von Kàrmàn非线性大挠度板理论基础上,利用达朗贝尔原理建立系统的动力学模型。通过Galerkin截断,将时间变量和空间变量、位移函数和应力函数耦合在一起的偏微分方程离散,得到系统运动常微分方程。利用数值方法分析板随轴向运动速度、外激励力幅值、长宽比和轴向拉力变化时的运动分岔行为。利用最大Lyapunov指数和Poincaré映射图识别系统的动力学行为。结果发现,当板的某些参数变化时,系统出现分岔现象。不同参数时,系统呈现周期运动、倍周期运动、概周期运动,甚至混沌运动。     

基于内变量损伤模型的复合材料板的后屈曲分析

基于Talreja张量内变量损伤模型,建立了复合材料单层板平面应力问题的损伤本构关系,进而导出了具初始挠度的考虑损伤效应的复合材料单层板的非线性压屈平衡方程。然后,将未知函数在空间上应用有限差分法离散,在时间上等分为小的时间段,且所有非线性项被线性化,整个问题采用迭代法求解。数值结果表明,损伤将导致结构刚度不断削弱,板的后屈曲变形将随着时间的增加而增加。     

Design of plates with given rigidities from a minimum number of layers and materials

The problem of designing a plate with a given set of rigidities using a minimum number of layers, which, as it has been established, is equal to four, has been considered. We consider the cases where flexural rigidities, rigidities in the plate plane, and asymmetric rigidities and where only longitudinal and flexural rigidities, which are the physical characteristics of the plane, are given. It has been proved that in both cases the ranges of longitudinal and flexural rigidities are equal (thus the question of the role of asymmetric rigidities is solved). __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 79, No. 5, pp. 165–174, September–October, 2006.     

The large deflection dynamic plastic response of rectangular plates

By assuming a kinematically admissible, time-dependent velocity field and considering the global equilibrium of all the forces acting on each rigid segment during large deflection of a plate, a complete theoretical investigation is undertaken herein to trace the large deflection dynamic plastic response (including the transient phase) of simply-supported or fully-clamped rectangular plates. This procedure deduces two modifying factors ƒ₁ and ƒ₂ which reflect the effect of the membrane forces and are employed to formulate the governing equations in the case of large deflections. The present prediction of the final plate deflection coincides excellently with the corresponding experimental results for deflections up to 5–10 times the plate thickness. This theory greatly improves the estimates based on the bending-only theory and provides a new way to trace the transient phase of dynamically loaded plates when the effect of membrane forces is significant.     

An integral-equation formulation of nonlinear deformation in a stack of buffered plates

An integral-equation formulation has been derived for nonlinear deformation in a stack of buffered Kirchhoff plates. The plates are assumed to follow a nonlinear bending moment-curvature law and the buffer material to follow the generalized Hooke’s law. By employing the recently derived special Green’s function for multilayers with interfacial membrane and flexural rigidities as the kernel, the integral-equation formulation only involves the surface loading area (for application to an indentation problem) and the portion of plates undergoing nonlinear deformation. Based on the integral equation, an efficient and accurate boundary element method has been derived to numerically solve the cylindrical indentation problem of the material with a bilinear flexural bending law for the plates. Numerical examples are presented to show a progressive damage process of yielding across a stack of plates as well as to demonstrate the validity and accuracy of the present integral-equation formulation.     

Lee plate equations for electroded quartz crystal plates with the consideration of electrode density and stiffness

Lee plate equations for high-frequency vibrations of piezoelectric plates have been established and improved over the last decades with the sole objective of obtaining the accurate prediction of frequency and mode shapes to aid crystal resonator design. The latest improvement includes extra terms related to derivatives of the flexural displacement to adjust the accuracy and consider electrodes for practical applications. As part of the efforts to make the equations applicable for resonator design with the improved frequency accuracy and consideration of electrodes, we derived Lee plate equations for electroded plates by changing the integration limits in the dimension reduction procedure to signify the dominant role of the crystal plate. By modifying the density terms in plate equations to include the contribution of both electrode stiffness and density, the accuracy of the thickness-shear vibration frequency of electroded plates is improved for commonly used electrode materials.     

Flexural vibrations of poroelastic plates

Summary The governing equations of flexural vibrations of thin, fluid-saturated poroelastic plates are derived in detail. The plate material obeys Biot's theory of poroelasticity with one degree of porosity, while the plate theory employed is the one due to Kirchhoff. These governing equations are compared with the corresponding ones for thermoelastic plates and a poroelastic-thermoelastic analogy for flexural plate dynamics is established in the frequency domain. The dynamic response of a rectangular, simply supported, poroelastic plate to harmonic load is obtained analytically-numerically and the effects of inertia as well as of porosity and permeability on the response is assessed.     

组合楼盖竖向承重体系中楼板变形及内力分析

在组合楼盖体系中,当组合梁作为横向承重构件时,其变形对于楼板的内力和变形有很大影响。采用薄板理论对组合楼盖竖向承重体系中的单向混凝土板进行分析,建立控制方程,运用单三角级数法得到其挠度函数,进而求得曲率和弯矩的计算式并进行简化。通过与有限元结果对比,验证了简化公式的准确性。参数分析结果表明,随板相对刚度增大和梁上荷载增大,板内横向曲率和弯矩减小。即使在单向板中,计算板内弯矩也需同时考虑两个方向的曲率。     

On the flexural response of moderately thick bimodular laminated plates on elastic foundations

An exact flexural analysis of rectangular simply supported bimodular laminated plates attached to Winkler-Pasternak elastic foundations and subjected to single sinusoidal heap loading is described. A computer implementation of the analysis is used to generate numerical results for the primary quantities of interest, e.g. deflections, etc., which are presented in dimensionless graphical format. The effects of thickness ratio and the magnitudes of the foundation constants on plate response are highlighted, particularly in respect of single and two-layer cross-ply plates made of aramid- and polyester-rubbers.     

Piezoelectric ceramic rectangular transducers in flexural vibration

Based on the equivalent elastic method and coupled vibration theory, an analytic method is presented to study the flexural vibration of rectangular transducers consisting of piezoelectric ceramic thin plates. By introducing a mechanical coupling coefficient, the flexural vibration of the piezoelectric ceramic rectangular thin plate is reduced to two simple, one-dimensional flexural vibrations of narrow piezoelectric ceramic strips. The resonance frequency equations for the piezoelectric ceramic rectangular thin-plate transducers in flexural vibration are derived under the free and simply supported boundary conditions analytically. The relationship between the resonance frequency and the flexural vibrational order, the geometrical shape, and the dimensions of the piezoelectric ceramic rectangular thin-plate transducer is analyzed. It is demonstrated that the one-dimensional vibrational theory for the flexural vibration of a narrow piezoelectric ceramic strip and the stripe-mode flexural vibrational theory for the piezoelectric ceramic rectangular thin plate can be derived directly from the theory obtained in this paper. Experimental results show that the measured resonance frequencies of the piezoelectric ceramic rectangular thin-plate transducers in flexural vibration under free-boundary conditions are in good agreement with the calculated results. The method presented in this paper can be used in the resonance frequency analysis of vibrating systems in coupled vibration.     

Thermomechanical bending analysis of functionally graded sandwich plates with both functionally graded face sheets and functionally graded cores

The bending response of functionally graded material (FGM) sandwich plates subjected to thermomechanical loads is investigated using a four-variable refined plate theory. A new type of FGM sandwich plate, namely, both FGM face sheets and an FGM hard core, is considered. Containing only four unknown functions, the governing equations are deduced based on the principle of virtual work and then these equations are solved via the Navier approach. Analytical solutions are obtained to predict the deflections and stresses of simply supported FGM sandwich plates. Benchmark comparisons of the solutions obtained for a degradation model (functionally graded face sheets and homogeneous cores) with ones computed by several other theories are conducted to verify the accuracy and efficiency of the present approach. The influences of volume fraction distribution, geometrical parameters, and thermal load on dimensionless deflections and normal and shear stresses of the FGM sandwich plates are studied.     

Two-way bending behavior of 3-D GFRP sandwich panels with through-thickness fiber insertions

This paper presents the details of a research program that was conducted to evaluate the two-way bending behavior of 3-D glass fiber reinforced polymer (GFRP) sandwich panels. The panels consist of GFRP skins with a foam core and through-thickness fiber insertions. While the behavior of these panels under one-way bending is relatively well understood the behavior under two-way bending has not yet been investigated. An experimental program was conducted to evaluate the effect of the fiber insertion pattern and the panel thickness on the two-way bending behavior under the effect of a concentrated load. The experimental results were used to verify a non-linear, static finite element model which was used to introduce a simplified method to predict the behavior. The measured and predicted responses indicate that at lower deflections the panel behavior is dominated by plate bending action while for higher deflections membrane action dominates. The finite element analysis was extended to study the effect of different parameters which were not tested in the experimental program. The parametric study indicates that increasing the relative flexural or shear rigidities of the panel alters the behavior towards the plate bending mechanism thereby reducing the percentage of load carried by membrane action.     

Non-homogeneous response of cross-ply laminated elastic plates using a higher-order theory

A consistent formulation for the bending of cross-ply laminated composite plates that possess non-homogeneous elastic properties is presented. Based on a third-order shear deformation plate theory, the governing equations are obtained using the principle of virtual work. With the help of the small parameter method, a wide variety of results are presented for the symmetric and antisymmetric analysis of non-homogeneous rectangular laminated plates. The influence of non-homogeneity, lamination schemes, aspect ratio and material anisotropy on the deflections and stresses is investigated. The new results for non-homogeneous response of composite plates should serve as bench marks for future comparisons.     

Flexural vibration of a lithium niobate piezoelectric plate with a ferroelectric inversion layer

Abstract

Two-dimensional equations for flexural motions of a lithium niobate piezoelectric plate are derived from three-dimensional equations. The plate has a ferroelectric inversion layer where the piezoelectric constants reverse signs. The equations show an unconventional behavior that a uniform electric field along the plate thickness can produce bending. This offers many possibilities for new designs of devices. Waves in unbounded plates are examined for the accuracy of the equations. A piezoelectric energy harvester based on such a plate is analyzed as an example of the application of the equations derived in vibrations of finite plates.     

A finite element for the nonlinear analysis of laminated circular plates

Recently, a new finite element for modelling axisymmetric circular plates was developed.¹ The elements were based on Mindlin's shear-deformable plate theory, and unlike most conventional plate elements, these new elements could be stacked on top of one another to model laminated plates. The elements assured continuity of the displacements between the layers, but not continuity of the traction vectors. Neither interlaminar slip nor debonding between the layers was considered.

In this paper, these new plate elements are incorporated into a Von Karman-type plate theory. The elements may be used to model problems where the membrane stresses play an important role in determining the plate's deflection. For example, it is well-known that for a homogeneous, isotropic plate the membrane stresses begin to affect the deflections when the deflections reach about half the thickness of the plate. For laminated plates, regardless of the magnitude of the deflections, the membrane stresses are present in each layer to balance the interlaminar shear stresses. In this paper, the behavior of these membrane stresses and their influence on the deflections of laminated circular plates are investigated.