Solid Circular Plate Clamped along Two Antipodal Edge Arcs and Deflected by a Central Transverse Concentrated Force

A static, purely flexural mechanical analysis is presented for a Kirchhoff solid circular plate, deflected by a transverse central force, and clamped along two antipodal arcs, the remaining part of the boundary being free. By adopting an integral formulation, the contact reaction is assumed to be formed by four equal concentrated forces acting at the support extremities, accompanied by two distributed moments with radial and circumferential axis, respectively. This plate problem is rephrased in terms of a complex-valued Hilbert singular integral equation of the second kind, whose solution is obtained in analytical, integral form. A design chart is presented that reports the plate central deflection as a function of the angular width of the plate supports.     

冰载荷下不锈钢复合板挠度特性分析

以复合板中面的挠度响应作为不锈钢复合板抗冲击性能的评价指标,基于能量法和经典层合板理论,考虑层间结构参数设计,通过横向载荷下的弯曲平衡微分方程,建立冰载荷下不锈钢复合板挠度响应简化解析模型。该分析模型将整个动态响应分析过程分为冰载荷计算分析和动力学方程求解两个阶段。分析了冰载荷模型的面倾角、冲击速度和碰撞位置对冰载荷的影响,确定极端工况参数,汇总接触面的节点力数据;分析了层厚比对挠度响应的影响规律;基于LS-DYNA有限元仿真以及数值算例分析,对比挠度响应仿真结果和解析计算值,验证了本文简化解析模型的准确性,研究结果对不锈钢复合板抗冲击性能分析和评估具有一定的参考价值。     

Stress analysis in rolling contact problem of a finite thickness FGM layer

The rolling contact problem of a non-homogeneous layer is considered here. The graded layer possesses a variable elastic modulus with an exponential distribution. The Poisons ratio is assumed to be constant. A rigid cylindrical indenter is rolling over the surface of the graded layer with a constant velocity. First, the Navier equations of equilibrium are solved in the Fourier domain. Later, the boundary and the continuity conditions are satisfied in order to extract the governing singular integral equations. The numerical solution of the integral equations is provided by means of the Gauss–Chebyshev integration method. Finally, the sensitivity of the solution is analyzed for the effective parameters namely: the stiffness ratio, the layer thickness and the coefficient of friction. The results indicate that a minimum value of the coating thickness is required to alleviate the severe stress gradients in the critical locations. If the coating thickness decreases by a 50% then the Von Mises stress will increases about 20%. Also, a softening graded layer can result in a lower stress level over the interface which may enhance the bonding strength.

     

Analytical solution to a mixed boundary value elastic problem of a roller-guided panel of laminated composite

This study presents an analytical solution to elastic field in a roller-guided panel of symmetric cross-ply laminated composite material. The mixed boundary value two-dimensional plane stress elasticity problem is formulated in terms of a single displacement potential function. This reduces the problem to the solution of a single fourth order partial differential equation of equilibrium as the other equilibrium equation is satisfied automatically. The solution is obtained in terms of an infinite Fourier series. To present some numerical results, a panel of glass/epoxy laminated composite is considered and different components of stress and displacement at different sections of the panel are presented graphically. To justify the present analytical solution, it is compared with the finite element solution obtained by using the commercial software ANSYS. It is found that the two solutions agree well with each other. This ensures that the formulation developed in this study based on the displacement potential approach can be used to obtain analytical solution of an elastic field in structural elements of laminated composite under any mode of boundary conditions prescribed in terms of either stress, displacement or any combination of these.     

基于能量变分法的钢底板波形钢腹板组合箱梁畸变效应分析

为合理分析钢底板波形钢腹板梯形箱梁的畸变效应,按各板件面内外抗弯刚度不变的原则将全截面等效为钢材,利用圣维南原理考虑顶底板对波形钢腹板的约束作用,修正畸变扇性坐标分布模式,基于能量变分法建立畸变控制微分方程。与已有文献及有限元进行对比分析,并研究腹板俯角和波形钢腹板厚度变化对畸变翘曲正应力的影响。结果表明,本文解析解与文献解及ANSYS解均吻合较好;基于圣维南原理修正后的扇形坐标分布模式更合理;利用本文等效方法亦可分析传统波形钢腹板组合箱梁的畸变效应;腹板俯角的设置有利于减小畸变翘曲正应力;波形钢腹板厚度变化对腹板与底板交接处的畸变翘曲正应力影响显著。     

Stress intensity factor for orthotropic three layered material with crack under arbitrary anti-plane shear

A three-layered orthotropic material with a center crack is subjected to an arbitrary anti-plane shear loading. The problem is formulated as a mixed boundary value problem by using the Fourier integral transform method. This gives a Fredholm integral equation of the second kind. The integral equation is solved numerically and anti-plane shear stress intensity factors are analyzed in terms of the product of shear modulus, material orthotropy for each layer and the thickness of the outer layer and the order of the loading polynomial.     

角点悬空弯曲厚矩形板的边界积分法

在边界积分法中引用了拟基本系统矩形板,在该拟基本系统与实际系统之间应用功的互等定理,得到一挠曲面方程的积分表达式,只要对此表达式进行极简单的积分便可得到该挠曲面方程,这比直接求解Reissner挠度控制方程要简单,边界积分法的求解过程概念清晰,计算伊始便给出了挠曲面方程的总体表达式.以Reissner厚板理论为基础,应用边界积分法研究了角点悬空厚矩形板的弯曲问题,给出了在集中荷载作用下两邻边固定另两邻边自由且角点悬空弯曲厚矩形板的封闭解析解,并给出了相应的数据和图表以供工程上的应用和参考.     

Fracture analysis of a weak-discontinuous interface in a symmetrical functionally gradient composite strip loaded by anti-plane impact

The anti-plane impact fracture analysis was performed for a weak-discontinuous interface in a symmetrical functionally gradient composite strip. A new bi-parameter exponential function was introduced to simulate the continuous variation of material properties. Using Laplace and Fourier integral transforms, we reduced the problem to a dual integral equation and obtained asymptotic analytical solution of crack-tip stress field. Based on the numerical solution of the second kind of Fredholm integral equation transformed from the dual integral equation, the effects of the two non-homogeneity parameters on DSIF were discussed. It was indicated that the relative stiffness of the interface and the general stiffness of the whole structure are two important factors affecting the impact fracture behavior of the weak-discontinuous interface. The greater the relative stiffness of the interface is, the higher the value of the dynamic stress intensity factor will be. The greater the general stiffness of the whole structure is, the shorter the time for DSIF to arrive at the peak value and then to stabilize to the steady one. If the general stiffness of the whole structure is great enough, there will be an oscillation between the peak and steady values of DSIF.     

SH波在压电材料条中垂直界面裂纹处的散射

研究了SH波在压电材料条中裂纹处的散射.压电材料条两侧涂有相同梯度参数的两个半无限大功能梯度材料,裂纹垂直于界面.通过Fourier变换,利用边界条件把问题转化为柯西核奇异积分方程,然后利用Chebyshev多项式对奇异积分方程进行数值求解.通过数值计算,分析讨论了压电条的几何参数和SH波频率对标准动应力强度因子的影响.     

Cracked elastic layer under compressive mechanical loads

We consider boundary value problem in which an elastic layer containing a finite length crack is under compressive loading. The crack is parallel to the layer surfaces and the contact between crack surfaces are either frictionless or with adhesive friction or Coulomb friction.Based on fourier integral transformation techniques the solution of the formulated problems is reduced to the solution of a singular integral equation, then, using Chebyshev’s orthogonal polynomials, to an infinite system of linear algebraic equations. The regularity of these equations is established. The expressions for stress and displacement components in the elastic layer are presented. Based on the developed analytical algorithm, extensive numerical investigations have been conducted.The results of these investigations are illustrated graphically, exposing some novel qualitative and quantitative knowledge about the stress field in the cracked layer and their dependence on geometric and applied loading parameters. It can be seen from this study that the crack tip stress field has a mode II type singularity.     

Stress Analysis for an Elastic Semispace with Surface and Graded Layer Coatings under Induced Torsion

In this paper, the axisymmetric torsional problem of a coating structure consisting of a surface coating, a functionally graded layer and a substrate under a rigid cylindrical punch is investigated. The coating and substrate are homogeneous materials with distinct physical properties while the intermediate layer is inhomogeneous with its shear modulus changing exponentially along the thickness direction. The Hankel integral transform technique is employed to reduce the torsional problem to a singular integral equation with a Cauchy kernel. The circumferential shear stress and displacement fields in the coating structure are calculated by solving the integral equation numerically. The results show that the stiffness ratio has significant effect on the distribution of the circumferential stress and displacement at the interface.     

A receding contact axisymmetric problem between a functionally graded layer and a homogeneous substrate

In this paper, we consider the axisymmetric problem of a frictionless receding contact between an elastic functionally graded layer and a homogeneous half-space, when the two bodies are pressed together. The graded layer is modeled as a nonhomogeneous medium with an isotropic stress–strain law and is subjected over a part of its top surface to normal tractions while the rest of it is free of tractions. Since the contact between the two bodies is assumed to be frictionless, then only compressive normal tractions can be transmitted in the contact area. Using Hankel transform, the axisymmetric elasticity equations are converted analytically into a singular integral equation in which the unknowns are the contact pressure and the receding contact radius. The global equilibrium condition of the layer is supplemented to solve the problem. The singular integral equation is solved numerically using orthogonal Chebychev polynomials and an iterative scheme is employed to obtain the correct receding contact length that satisfies the global equilibrium condition. The main objective of the paper is to study the effect of the material nonhomogeneity parameter and the thickness of the graded layer on the contact pressure and on the length of the receding contact.     

TORSIONAL VIBRATIONS OF RIGID CIRCULAR PLATE ON TRANSVERSELY ISOTROPIC SATURATED SOIL

An analytical method was presented for the torsional vibrations of a rigid disk resting on transversely isotropic saturated soil. By Hankel transform, the dynamic governing differential equations for transversely isotropic saturated poroelastic medium were solved. Considering the mixed boundary-value conditions, the dual integral equations of torsional vibrations of a rigid circular plate resting on transversely isotropic saturated soil were established. By appropriate transform, the dual integral equations were converted into a Fredholm integral equation of the second kind. Subsequently, the dynamic compliance coefficient, the torsional angular amplitude of the foundation and the contact shear stress were expressed explicitly. Selected examples were presented to analyse the influence of saturated soil's anisotropy on the foundation's vibrations.     

Acoustic reflection from a baffled elastic/piezoelectric plate

A finite difference/boundary integral procedure to determine the acoustic reflected pressure from a fluid-loaded bi-laminated plate is described. The bi-laminate is composed of a piezoelectric layer and an elastic layer in contact with the fluid, and is held by an acoustically hard baffle. In the numerical model, the fluid pressure at fluid/solid interface is replaced by a continuum of point sources weighted by the normal acceleration of the elastic plate, and the governing equation system is solved in the solid domain. With the normal acceleration found, the reflected pressure in the fluid is determined by an integral expression involving the Green's function. It is demonstrated that an appropriate applied voltage potential across the piezoelectric layer has the effect of cancelling either the reflected or scattered pressure of the plate at any chosen field points in the fluid. Project supported by the National Natural Science Foundation of China (No. 10172039).     

Analytical solution of rectangular plate with in-plane variable stiffness

The bending problem of a thin rectangular plate with in-plane variable stiffness is studied. The basic equation is formulated for the two-opposite-edge simply supported rectangular plate under the distributed loads. The formulation is based on the assumption that the flexural rigidity of the plate varies in the plane following a power form, and Poisson’s ratio is constant. A fourth-order partial differential equation with variable coefficients is derived by assuming a Levy-type form for the transverse displacement. The governing equation can be transformed into a Whittaker equation, and an analytical solution is obtained for a thin rectangular plate subjected to the distributed loads. The validity of the present solution is shown by comparing the present results with those of the classical solution. The influence of in-plane variable stiffness on the deflection and bending moment is studied by numerical examples. The analytical solution presented here is useful in the design of rectangular plates with in-plane variable stiffness.     

A receding contact plane problem between a functionally graded layer and a homogeneous substrate

In this paper, we consider the plane problem of a frictionless receding contact between an elastic functionally graded layer and a homogeneous half-space, when the two bodies are pressed together. The graded layer is modeled as a nonhomogeneous medium with an isotropic stress–strain law and over a certain segment of its top surface is subjected to normal tractions while the rest of this surface is free of tractions. Since the contact between the two bodies is assumed to be frictionless, then only compressive normal tractions can be transmitted in the contact area. Using integral transforms, the plane elasticity equations are converted analytically into a singular integral equation in which the unknowns are the contact pressure and the receding contact half-length. The global equilibrium condition of the layer is supplemented to solve the problem. The singular integral equation is solved numerically using Chebychev polynomials and an iterative scheme is employed to obtain the correct receding contact half-length that satisfies the global equilibrium condition. The main objective of the paper is to study the effect of the material nonhomogeneity parameter and the thickness of the graded layer on the contact pressure and on the length of the receding contact.     

层状横观各向同性地基上异性矩形薄板的弯曲解析解

选用更具广泛性的层状横观各向同性弹性地基模型,来分析四边自由各向异性矩形地基板的弯曲解析解。先基于直角坐标下横观各向同性体的静力胡海昌通解,借助双重傅里叶变换及矩阵传递法,获得层状横观各向同性地基的静力位移场和应力场;然后将异性薄板的弯曲控制方程,与基于层状横观各向同性弹性地基的位移解建立的板与地基变形协调方程相结合,先按对称性分解,再用三角级数法,得出层状横观各向同性弹性地基上四边自由各向异性矩形薄板的弯曲解析解,包括地基反力、板的挠度及板的内力的解析表达式。克服了数值法的弊端,取消了对地基反力的假设,且避免了矩阵指数函数的计算;同时考虑了地基的层状性及板和地基的各向异性,从而得到板的内力及地基反力更切实际的分布规律。算例结果与文献的有限元结果吻合良好,证明本文方法是切实可行的。     

润滑状态下球-盘点接触摩擦副的低速轻载滑滚特性研究

选用镀Cr膜的玻璃盘和GCr15球作为摩擦副,在NGY-6纳米润滑膜测量仪上开展球-盘点接触摩擦副在润滑状态下的低速轻载滑滚特性试验,研究不同接触应力、钢球转速、滑滚比等参数对摩擦副的摩擦系数及对应油膜厚度的影响规律.结果表明:当接触应力和钢球转速一定时,摩擦系数随滑滚比的增大逐渐增加后达到稳定状态,当滑滚比较大时,滑滚比的变化对油膜厚度几乎没有影响;当滑滚比一定时,摩擦系数随接触应力的增大逐渐增大,随钢球转速的增大逐渐减小,油膜厚度随接触应力的增大逐渐减小,随钢球转速的增大逐渐增大.摩擦副在弹流润滑状态下,摩擦系数的增加幅度随接触应力的变化较小,而在薄膜润滑状态下,其增加幅度变大.摩擦副在薄膜润滑状态下,当滑滚比在0.10~0.50变化时,摩擦系数和油膜厚度基本处于稳定状态.     

Interrelation of creep and relaxation for nonlinearly viscoelastic materials: application to ligament and metal

Creep and stress relaxation are known to be interrelated in linearly viscoelastic materials by an exact analytical expression. In this article, analytical interrelations are derived for nonlinearly viscoelastic materials which obey a single integral nonlinear superposition constitutive equation. The kernel is not assumed to be separable as a product of strain and time dependent parts. Superposition is fully taken into account within the single integral formulation used. Specific formulations based on power law time dependence and truncated expansions are developed. These are appropriate for weak stress and strain dependence. The interrelated constitutive formulation is applied to ligaments, in which stiffness increases with strain, stress relaxation proceeds faster than creep, and rate of creep is a function of stress and rate of relaxation is a function of strain. An interrelation was also constructed for a commercial die-cast aluminum alloy currently used in small engine applications.     

Scattering of water waves by thin vertical plate submerged below ice-cover surface

The present paper is concerned with scattering of water waves from a vertical plate, modeled as an elastic plate, submerged in deep water covered with a thin uniform sheet of ice. The problem is formulated in terms of a hypersingular integral equation by a suitable application of Green's integral theorem in terms of difference of potential functions across the barrier. This integral equation is solved by a collocation method using a finite series involving Chebyshev polynomials. Reflection and transmission coefficients are obtained numerically and presented graphically for various values of the wave number and ice-cover parameter.