非均匀复合材料板中剪切波传播的研究

基于弹性界层中弹性波干涉理论,采用有效介质法,研究了剪切波在非均匀、纤维随机分布复合材料板中的传播,得到了非均匀弹性介质内的有效波数。通过满足弹性有界层的上、下边界条件,得到了非均匀界层中的频散方程。作为特例,绘出了不同参数下板中的前四阶频散曲线。可以看出,非均匀弹性有界层中的频散曲线和均匀界层中的有很大不同。最后分析了纤维和基体特性比、纤维的体积份数以及板厚与纤维半径比对频散曲线的影响。     

A 3-D Ritz solution for free vibration of circular/annular functionally graded plates integrated with piezoelectric layers

This paper addresses three-dimensional (3-D) free vibration characteristic of thick circular/annular functionally graded (FG) plates with surface-bonded piezoelectric layers on the basis of 3-D Ritz solution. Three displacement components along with electrical potential field of the plate are expressed by a set of Chebyshev polynomials multiplied by geometry boundary functions. Both open-circuit and closed-circuit surface conditions are taken into account. The mechanical properties of the FG plates are assumed to vary continuously through the thickness of the plate and obey either exponent or power law distribution of the volume fraction of the constituents. The effect of thickness-to-radius ratio, inner-to-outer radius ratio, piezo-to-host thickness ratio and gradient index on the natural frequencies of coupled piezoelectric FG circular/annular plates is investigated for different electrical and mechanical boundary conditions. It is observed that, unlike isotropic homogeneous circular/annular plates, frequency parameters of their piezoelectric coupled FG counterparts significantly increase with an enhancement in the host plate thickness to radius ratio. Results also show that the frequency parameters for open-circuit condition are higher than those for closed-circuit condition.     

内置竖向插板加强型管节点静力强度研究

焊接圆钢管节点主管的径向刚度一般远小于支管的轴向刚度,因此在静力作用下发生破坏的部位常位于主管表面靠近焊缝处。为了提高管节点的承载能力,可采用在靠近节点部位的主管内部设置竖向插板的方法提高管节点的静力强度。利用有限元方法对33个加强与未加强的T节点的静力破坏过程进行了模拟和分析,调查了节点区域和内置插板在静力加载过程中的应力演变过程。通过模型的参数研究发现:插板的长度对节点的静力强度影响较大,随着插板长度的增加,节点的静力强度提高,但长度在达到一定的数值后提高效果不再明显;插板的厚度对节点的静力强度的提高效果不大,但插板的厚度不宜过小,防止插板先于节点产生失稳破坏而起不到显著的加强效果。最后,对24个不同尺寸的T型节点分别进行了内置竖向插板加强前和加强后的有限元模拟,研究了推荐尺寸的插板对节点承载力的提高效果。     

An exact analytical solution for freely vibrating piezoelectric coupled circular/annular thick plates using Reddy plate theory

Based on third-order shear deformation plate theory of Reddy, the authors aim to provide an exact analytical solution for free vibration analysis of thick circular/annular plates, both upper and lower surfaces of which are in contact with a piezoelectric layer. Natural frequencies are determined by the solution of the coupled electromechanical governing equations for a combination of free, soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the plate. The electrodes on each piezoelectric layer are assumed to be short-circuited. The Maxwell electrostatics equation is satisfied by adopting a half-sine distribution of the electric potential in the transverse direction of the piezoelectric layers. A comparison of the present exact natural frequencies for piezoelectric coupled circular/annular plates with different boundary conditions is made with previously published results obtained by the Mindlin plate theory and 3-D modified finite element method. The effects of plate parameters such as host thickness to radius ratios, inner to outer radius ratios and piezoelectric to host thickness ratios on the natural frequencies of laminated circular/annular plates are investigated for different combinations of boundary conditions. Results obtained by the present exact closed-form solutions can be served as benchmark data for investigators to validate their numerical and analytical methods in the future.     

Analytical layerwise free vibration analysis of circular/annular composite sandwich plates with auxetic cores

In the present research, free vibration of circular and annular sandwich plates with auxetic (negative Poisson’s ratio) cores and isotropic/orthotropic face sheets is investigated for different combinations of the boundary conditions. To ensure that the results are accurate and reliable, a global–local layerwise plate theory is employed instead of the traditional equivalent single-layer theories. The governing equations are derived based on Hamilton’s principle and solved using a Taylor transform whose center is located at the outer radius of the plate. Due to this hint, the resulting semi-analytical solution can be employed for both circular and annular sandwich plates. After investigation of vibration behavior of a single-layer annular auxetic plate, a comprehensive parametric study including evaluation of effects of the auxeticity for sandwich plates with isotropic and orthotropic face sheets, symmetric and asymmetric layups, different core to sheet thickness, radius to thickness, and inner to outer radius ratios, and various boundary conditions, is carried out. Results show that unlike the single-layer auxetic plates that exhibit a transition state, the auxeticity may considerably increase the natural frequencies and rigidities of the circular/annular sandwich plates, especially when the boundary conditions induce higher rigidity in the plate or when the fibers are along the radial direction. Accuracy of results of the employed layerwise theory and the proposed semi-analytical solution is verified by comparing the results with those of the three-dimensional theory of elasticity extracted from the ABAQUS software.     

Elastic response of functionally graded circular plates under a drop-weight

The three-dimensional impact behaviours of functionally graded (FG) circular plates were studied under a drop-weight. The functionally graded circular plate was composed of ceramic (SiC) and metal (Al) phases and the through-thickness mechanical properties through the region between the metal and ceramic layers vary continuously according to a power-law distribution of the volume fraction of the ceramic. The through-thickness material properties of the FG circular plate were determined using the Mori–Tanaka scheme. The effects of layer number and compositional gradient exponent as well as impactor velocity and plate radius on the elastic impact response of the FG circular plates were investigated. The compositional gradient exponent, impactor velocity and plate radius played an important role on the impact response of the FG circular plates, whereas the layer number through the plate thickness had a minor effect. In addition, the failure strains in all layers were determined using Tamura–Tomota–Ozowa (TTO) model in order to predict the damage regions in each layer through the thickness of FG circular plates.     

Vibration of turbomachinery rotating blades made-up of functionally graded materials and operating in a high temperature field

Summary. The vibration of turbomachinery rotating blades made-up of functionally graded materials and operating in a temperature field is considered. In this context, the blade is modeled as a thin-walled beam mounted on a rigid hub at a presetting angle, rotating with a constant angular velocity, and exposed to a steady temperature field of a prescribed gradient through the blade wall thickness. The effect of the initial twist of the blade is also taken into consideration. Results are presented for two constituents, metal-ceramic based materials that are assumed to be temperature-dependent and graded in the thickness direction according to a simple power law distribution. Numerical results highlighting the effects of the volume fraction in conjunction with those of the presetting and pretwist angles, temperature gradient, rotating speed and hub radius are presented, and pertinent conclusions are outlined.     

Thermal stability analysis of circular functionally graded sandwich plates of variable thickness using pseudo-spectral method

In the present study, the thermal stability of laminated functionally graded (FGM) circular plates of variable thickness subjected to uniform temperature rise based on the first-order shear deformation plate theory is presented. Furthermore, two models for FGM plates with variable thickness, corresponding with two manufacturing methods, are proposed. The laminated FGM plate with variable thickness is considered as a sandwich plate constituted of a homogeneous core of variable thickness and two constant thickness FGM face sheets whose material properties are assumed to be graded in the thickness direction according to a simple power law. In order to determine the distribution of the prebuckling thermal load along the radius, the membrane equation is solved using the shooting method. Subsequently, employing the pseudo-spectral method that makes use of Chebyshev polynomials, the stability equations are solved numerically to evaluate the critical temperature rise. The results demonstrate that the thermal stability is significantly influenced by the thickness variation profile, aspect ratio, the volume fraction index, and the core-to-face sheet thickness ratio.     

Transition of hertzian fracture to flexure fracture produced in glass plates by impact

The strength and fracture of a glass plate in a three-dimensional state of stress produced by the transverse impact of a spherical body is investigated. The plate is supported by a ring foundation whose radius is comparable to the thickness of the plate. A general impact theory is developed to determine the critical stresses in the plate as a function of impact velocity, plate thickness, and support span.

Impact experiments were carried out to measure the impact velocity required to fracture a glass plate specimen as a function of support span. Hertzian fracture was observed at a short span of support whereas flexure fracture was observed at a larger support span. The impact velocity of fracture was measured to be increasing with increasing support span and its corresponding critical stress of fracture was also determined.     

An Equation for Stress Concentration Factor in Countersunk Holes

A detailed three-dimensional finite element stress analysis was conducted on straight-shank and countersunk rivet holes in a plate subjected to tension loading. The study included a wide range of plate width to radius, thickness to radius, countersunk depth to thickness ratios and countersunk angles(θ_c). The stress concentration is maximum at or near the countersunk edge. The stress concentration depends on countersunk depth, plate thickness and width and it is nearly independent of the countersunk angle for 80° ≤ θ_c ≤ 120°. Using the finite element results and limiting conditions, an equation for stress concentration factor is developed and verified.     

Properties of the Anisimkin Jr.' modes in quartz plates

Dispersion curves, surface displacements, and displacement profiles over the plate thickness are numerically calculated for acoustic plate modes, propagating in fused and Y-rotated, X-cut quartz samples with thickness h/lambda in the range 0-4.5 (h, thickness; lambda, wavelength). The Anisimkin Jr.' modes with velocity v(n) close to that of longitudinal bulk wave v(L) and the dominant longitudinal displacement u1 distributed uniformly through the plate thickness are found in quartz crystals with cut angles mu = 120 degrees-140 degrees and plate thickness h/lambda = 0-0.36, 0.94-1.78, 2.32-3.08, and 3.64-4.44. The same modes in fused quartz are not found, except in a narrow region near zero thickness.     

Optimization of a circular thin-film piezoelectric actuator lying on a clamped multilayered elastic plate

A system consisting of a circular multilayered thin-film elastic plate and a piezoelectric actuator, which is generally used for ultrasound generation in air, is studied in this paper. Effects of the electrode dimension of a circular thinfilm piezoelectric actuator lying on a clamped multilayered elastic plate are discussed theoretically, while the first-order theory of asymmetrically laminated piezoelectric plates with consideration of coupled extension and flexure of the reference plane is used. Numerical results show that the deflection of the elastic plate can be optimized by adjusting the radius of the top electrode.     

随机结构反应概率密度演化分析的切球选点法

发展了随机结构反应概率密度演化分析中随机参数空间的切球选点法。密度演化方法是一类直接获取随机结构动力反应概率密度函数及其演化过程的有效方法。在多个随机变量时，随机变量空间中的离散代表点选点规则直接关系到密度演化方法的精度和效率。本文构造了平面内等半径相切圆圆心分布定位的算法，以此为基础，建立了三维空间中等半径相切球球心坐标定位的计算公式。从而给出随机变量空间中的离散代表点及其赋得概率。计算表明，基于空间切球法的选点规则具有良好的精度和效率，在2个和3个随机变量情况下是较为理想的选点方法。     

Vibration and damping analysis of a three-layered composite annular plate with a viscoelastic mid-layer

The natural frequencies and modal loss factors of the three-layered annular plate with a viscoelastic core layer and two polar orthotropic laminated face layers are considered. The discrete layer annular finite element is employed to derive the equations of motion for the three-layered annular plate. The viscoelastic material in the central layer is assumed to be incompressible, and the extensional and shear moduli are described by the complex quantities. Complex eigenvalued problems are then solved, and the frequencies and modal loss factors of the composite plate are extracted. The results of the symmetric and non-symmetric composite annular plates are both presented. The effects of material properties, radius to thickness ratio, stacking sequences and thickness of face layers, and thickness of the viscoelastic core layer are discussed.     

Critical response of shielded plates subjected to hypervelocity impact

A ballistic limit equation for hypervelocity impact on thin plates is derived analytically. This equation applies to cases of impulsive impact on a plate that is protected by a multi-shock shield, and is valid in the range of velocity above 6 km/s. Experimental tests were conducted at the NASA Johnson Space Center on square aluminum plates. Comparing the center deflections of these plates with the theoritical deflections of a rigid-plastic plate subjected to a blast load, one determines the dynamic yield strength of the plate material. The analysis is based on a theory for the expansion of the fragmented projectile and on a simple failure criterion. Curves are presented for the critical projectile radius verus the projectile velocity, and for the critical plate thickness versus the velocity. These curves are in good agreement with curves that have been generated empirically.     

Consideration of stiffness and mass effects of relatively thicker electrodes with Mindlin plate theory

Mindlin plate theory has been widely used in the high-frequency vibrations of piezoelectric crystal plates with emphasis on its applications in crystal resonator analysis and design. The plate equations were derived without considering the effect of electrodes from the beginning. But continuing efforts have been made to include the mechanical effect, or the mass loading, through the consideration of the mass ratio of the electrodes and crystal blank. Such a consideration has been effective for relatively thin electrodes before, but the ever-increasing mass ratio has been pressing further improvement to take into account relatively thicker electrodes. To extend Mindlin plate equations for these applications, we derive the plate equations systematically with the approximation of displacements in electrodes with those in the crystal blank. As a result, both mass and stiffness effects of electrodes are considered through ratios of the thickness, density, and elastic constants of the electrodes to those of the crystal blank, respectively, and the plate equations are modified accordingly. A practical design of the electrodes and crystal blank are analyzed to demonstrate the necessity of such modifications to Mindlin plate equations.     

Ballistic limit for shielded plates subjected to hypervelocity impact

A new method of determining ballistic limits for hypervelocity impact is proposed. This method applies to cases of impulsive impact on a plate protected by a multi-shock shield, which corresponds to projectile velocities in the range above 6 km s⁻¹. It is shown, by using experimental and analytical results, that the plate will not fail, provided the impact parameters are bounded by certain critical values. A simple equation that relates the critical values is established. Curves are presented for the critical projectile radius versus the projectile velocity, and for the critical plate thickness versus the velocity. These curves are in good agreement with curves that have been generated empirically.     

层板弯曲与振动的近似分析

横向剪切变形对复合材料层合板弯曲与振动的影响甚大。在本文的近似分析中,假定板在弯曲时横向位移沿整个板厚为常量。横向剪切应变沿各层厚度方向也分别为常量,但各层不同。文中以特殊正交各向异性层合板为例,采用两种不同的方法建立了各层剪切应变间的关系,推演了层合板横向弯曲与振动的微分方程组及边界条件。算例表明,即使层合板的跨——厚比很小,用本文两种分析方案计算位移、应力及固有频率,都仍具有较高的精度。      

Tri-axial deformation of a plastic-rigid solid

The paper deals with the deformation of an ideal plastic solid that is initially rigid, i.e., elastic deformations are neglected. In addition, the solid flows by means of the mechanism of extended slip, for which the rotation-rate vector field remains continuous and the strain-rate tensor is solenoidal. The Tresca yield criterion applies to such a solid and with an associated flow-rule is represented in a manner that includes both bi-axial and tri-axial states of strain. Two new theorems are proved, and a second-order partial differential equation is derived for the first invariant of the stress tensor (hydrostatic pressure); the analogue of a similar published equation for the bi-axial strain case. To illustrate the methodology, the above theory is applied to the tri-axial problem of a thick metal plate clamped round a circle and deflected by means of pressure. It is shown, from the exact solution, that the errors due to the use of the approximate membrane formula for a clamped thin plate are small, even for a 6mm thick plate clamped on a circle 100mm in diameter. Surprisingly, there is an initial thickening of the plate and it is shown that, regardless of the plate thickness, the pressure passes through a maximum at a deflection equal to \({1/\sqrt{3}}\) of the radius of the clamping circle.     

Roll-formability of cryo-rolled ultrafine aluminium sheet

Ultrafine-grain aluminium sheet was produced by rolling at cryogenic (CR) and at room temperature (RTR). Commercial purity aluminium plate was reduced in 30 passes from an initial material thickness of 10 mm to a final thickness of 2 mm (80% reduction). Tensile stress and strength were significantly increased while total elongation was drastically reduced. It was found that despite the low tensile elongation both materials are able to accommodate high localised strains in the neck leading to a high reduction in area. The formability of the material was further investigated in bending operations. A minimum bending radius of 6 mm (CR) and 5 mm (RTR) was found and pure bending tests showed homogeneous forming behaviour for both materials. In V-die bending the cryo-rolled material showed strain localisations across the final radius and kinking of the sample. It has been found that even if the total elongation in tension is close to zero leading to early failure in V-die bending, ultra-fine grained and low ductile sheet metals can be roll formed to simple section shapes with small radii using commercial roll forming equipment.