Tilting analysis of circular elastic layers interleaving with flexible reinforcements

Bonding with reinforcements can increase the stiffness of elastic layers in the normal direction. The flexibility effect of the reinforcement on the bonded elastic layers of a circular cross-section subjected to pure bending moment is analyzed through a theoretical approach. Based on two kinematics assumptions in the elastic layers, the closed-form solutions of the horizontal displacements in the elastic layers and the reinforcements are solved using the governing equations established by stress equilibrium in the elastic layers and the reinforcements. Through these solved displacements, the tilting stiffness of the bonded elastic layer, the shear stress on the bonding surfaces, and the internal forces of the reinforcements are derived in closed forms.     

Flexure analysis of circular elastic layers bonded between rigid plates

An elastic layer of circular cross-section which is bonded between rigid plates and subjected to pure bending moment is analyzed through a theoretical approach. Based on two kinematic assumptions, the governing equations for the two horizontal displacement functions are established from the equilibrium equations. The horizontal displacements are then solved by satisfying the stress boundary conditions in the elastic layer. Through these solved displacements, the vertical stress in the elastic layer, the shear stress on the bonding surfaces, and the tilting stiffness of the bonded layer are derived in closed-forms and are also compared with the results of finite element analysis.     

Tilting stiffness of elastic layers bonded between rigid plates

A theoretical approach to determine the tilting stiffness of an elastic layer bonded between rigid plates is presented and then applied to derive the formulae of tilting stiffness for layers of infinite-strip, circular and square shapes. Based on two kinematics assumptions, the governing equations for the mean pressure are established from the equilibrium equations and the bulk modulus equation. Satisfying the stress boundary conditions, the pressure functions are solved and the formulae for tilting stiffness are derived. The tilting stiffnesses calculated from these formulae are extremely close to the results obtained from the finite element method for an extensive range of shape factor and Poissons ratio.     

A new formulation for the analysis of elastic layers bonded to rigid surfaces

Elastic layers bonded to rigid surfaces have widely been used in many engineering applications. It is commonly accepted that while the bonded surfaces slightly influence the shear behavior of the layer, they can cause drastic changes on its compressive and bending behavior. Most of the earlier studies on this subject have been based on assumed displacement fields with assumed stress distributions, which usually lead to “average” solutions. These assumptions have somehow hindered the comprehensive study of stress/displacement distributions over the entire layer. In addition, the effects of geometric and material properties on the layer behavior could not be investigated thoroughly. In this study, a new formulation based on a modified Galerkin method developed by Mengi [Mengi, Y., 1980. A new approach for developing dynamic theories for structural elements. Part 1: Application to thermoelastic plates. International Journal of Solids and Structures 16, 1155–1168] is presented for the analysis of bonded elastic layers under their three basic deformation modes; namely, uniform compression, pure bending and apparent shear. For each mode, reduced governing equations are derived for a layer of arbitrary shape. The applications of the formulation are then exemplified by solving the governing equations for an infinite-strip-shaped layer. Closed form expressions are obtained for displacement/stress distributions and effective compression, bending and apparent shear moduli. The effects of shape factor and Poisson’s ratio on the layer behavior are also investigated.     

Compression of solid and annular circular discs bonded to rigid surfaces

Although it is noted in the literature that the presence of a central hole in an elastic layer bonded to rigid surfaces can cause significant drop in its compression modulus, not much attention is given for investigating thoroughly and in detail the influence of the hole on the layer behavior. This paper presents analytical solutions to the problem of the uniform compression of bonded hollow circular elastic layers, which includes solid circular layers as a special case as the radius of hollow section vanishes. The closed-form expressions derived in this study are advanced in the sense that three of the commonly used assumptions in the analysis of bonded elastic layers are eliminated: (i) the incompressibility assumption, (ii) the “pressure” assumption and (iii) the assumption that plane sections remain plane after deformation. Through the use of the analytical solutions derived in the study, the compressive behavior of bonded circular discs is studied. Particular emphasis is given to the investigation of the effects of the existence of a central hole on the compression modulus, stress distributions and maximum stresses/strains in view of three key parameters: radius ratio of the hole, aspect ratio of the disc and Poisson’s ratio of the disc material.     

Mechanics of indentation for piezoelectric thin films on elastic substrate

Frictionless normal indentation problem of rigid flat-ended cylindrical, conical and spherical indenters on piezoelectric film, which is either in frictionless contact with or perfectly bonded to an elastic half-space (substrate), is investigated. Both conducting and insulating indenters are considered. With Hankel transform, the general solutions of the homogeneous governing equations for the piezoelectric layer and the elastic half-space are presented. Using the boundary conditions for a vertical point force or a point electric charge, and the boundary conditions on the film/substrate interface, the Green’s functions can be obtained by solving sets of simultaneous linear algebraic equations. The solution of the indentation problem is obtained by integrating these Green’s functions over the contact area with unknown surface tractions or electric charge distribution, which will be determined from the boundary conditions on the contact surface between the indenter and the film. The solution is expressed in terms of dual integral equations that are converted to a Fredholm integral equation of the second kind and solved numerically. Numerical examples are also presented. The comparison between two film/substrate bonding conditions is made. It shows that the indentation rigidity of the film/substrate system is lower when the film is in frictionless contact with the substrate. The effects of the Young’s modulus and Poisson’s ratio of the elastic substrate, indenter electrical condition and indenter prescribed electric potential on the indentation responses are presented.     

Compression stiffness of circular bearings of laminated elastic material interleaving with flexible reinforcements

The compression stiffness of a circular bearing that consists of laminated elastic layers interleaving with flexible reinforcements is derived in closed form. The effect of bulk compressibility in the elastic layer and the effect of boundary condition at the ends of the bearing are considered. The stiffness of the bearing with monotonic deformation is derived first. Then, the bearings with both ends being free from shear force and the bearings with both ends being bonded to rigid plates are studied. The theoretical solutions to the compression stiffness of the bearings are extremely close to the results obtained by the finite element method, which proves that the displacement assumptions utilized in the theoretical derivation are reasonable.     

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

This paper investigates the plane problem of a frictional receding contact formed between an elastic functionally graded layer and a homogeneous half space, when they are pressed against each other. The graded layer is assumed to be an isotropic nonhomogeneous medium with an exponentially varying shear modulus and a constant Poisson’s ratio. A segment of the top surface of the graded layer is subject to both normal and tangential traction while rest of the surface is devoid of traction. The entire contact zone thus formed between the layer and the homogeneous medium can transmit both normal and tangential traction. It is assumed that the contact region is under sliding contact conditions with the Coulomb’s law used to relate the tangential traction to the normal component. Employing Fourier integral transforms and applying the necessary boundary conditions, the plane elasticity equations are reduced to a singular integral equation in which the unknowns are the contact pressure and the receding contact lengths. Ensuring mechanical equilibrium is an indispensable requirement warranted by the physics of the problem and therefore the global force and moment equilibrium conditions for the layer are supplemented to solve the problem. The Gauss–Chebyshev quadrature-collocation method is adopted to convert the singular integral equation to a set of overdetermined algebraic equations. This system is solved using a least squares method coupled with a novel iterative procedure to ensure that the force and moment equilibrium conditions are satisfied simultaneously. The main objective of this paper is to study the effect of friction coefficient and nonhomogeneity factor on the contact pressure distribution and the size of the contact region.     

Discrete element modeling of the effect of particle size distribution on the small strain stiffness of granular soils

Discrete element modeling was used to investigate the effect of particle size distribution on the small strain shear stiffness of granular soils and explore the fundamental mechanism controlling this small strain shear stiffness at the particle level. The results indicate that the mean particle size has a negligible effect on the small strain shear modulus. The observed increase of the shear modulus with increasing particle size is caused by a scale effect. It is suggested that the ratio of sample size to the mean particle size should be larger than 11.5 to avoid this possible scale effect. At the same confining pressure and void ratio, the small strain shear modulus decreases as the coefficient of uniformity of the soil increases. The Poisson’s ratio decreases with decreasing void ratio and increasing confining pressure instead of being constant as is commonly assumed. Microscopic analyses indicate that the small strain shear stiffness and Poisson’s ratio depend uniquely on the soil’s coordination number.     

Revisiting displacement functions in three-dimensional elasticity of inhomogeneous media

The paper presents a three-dimensional solution to the equilibrium equations for linear elastic transversely isotropic inhomogeneous media. We assume that the material has constant Poisson’s ratios, and its Young’s and shear moduli have the same functional form of dependence on the co-ordinate normal to the plane of isotropy. We show, apparently for the first time, that stresses and displacements in such an inhomogeneous transversely isotropic elastic solid can be represented in terms of two displacement functions which satisfy the second- and fourth-order partial differential equations. We examine and discuss key aspects of the new representation; they include the relationship between the new displacement functions and Plevako’s solution for isotropic inhomogeneous material with constant Poisson’s ratio as well as the application of the new representation to some important classes of three-dimensional elasticity problems. As an example, the displacement function is derived that can be used to determine stresses and displacements in an inhomogeneous transversely isotropic half-space which is subjected to a concentrated force normal to a free surface and applied at the origin (Boussinesq’s problem).     

Macroscopic elastic properties of regular lattices

In this paper we study analytically the elastic properties of the 2-D and 3-D regular lattices consisting of bonded particles. The particle-scale stiffnesses are derived from the given macroscopic elastic constants (i.e. Young's modulus and Poisson's ratio). Firstly a bonded lattice model is presented. This model permits six kinds of relative motion and corresponding forces between each bonded particle pair. By comparing the strain energy distributions between the discrete lattices and the continuum, the explicit relationship between the microscopic and macroscopic elastic parameters can be obtained for the 2-D hexagonal lattice and the 3-D hexagonal close-packed and face-centered cubic structures. The results suggest that the normal stiffness is determined by Young's modulus and the particle size (in 3-D), and that the ratio of the shear to normal stiffness is related to Poisson's ratio. Rotational stiffness depends on the normal stiffness, shear stiffness and particle sizes. Numerical tests are carried out to validate the analytical results. The results in this paper have theoretical implications for the calibration of the spring stiffnesses in the Discrete Element Method.     

Indentation of an elastic layer by a rigid cylinder

The Green’s functions for the indentation of an elastic layer resting on or bonded to a rigid base by a line load are found efficiently and accurately by a combination of contour integration with a series expansion for small arguments. From the form of the equations it is clear that the function is oscillatory when the layer is free to slip over the base, but for the bonded layer, the function simply decays to zero after a single overshoot.The deformation due to pressure distributions of the form of the product of a polynomial with an elliptical (“Hertzian”) term is calculated and the coefficients chosen to match the indentation shape to that of a cylindrical indenter. The resulting pressure distributions behave much as in Johnson’s approximate theory, becoming parabolic instead of elliptical as the ratio b/d of contact width to layer thickness increases, or, for the bonded incompressible (ν = 1/2) layer, becoming bell-shaped for very large b/d.The relation between the approach δ and the contact width b curves has been investigated, and some anomalies in published asymptotic equations noted and, perhaps, resolved.A noticeable feature of our method is that, unlike previous solutions in which the full mixed boundary value problem (given indenter shape / stress-free boundary) has been solved, the bonded incompressible solid causes no problems and is handled just as for lower values of Poisson’s ratio.     

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.

     

Buckling of short beams with warping effect included

A beam theory for the stability analysis of short beam that includes shear deformation and warping of the cross-section is developed. The warping of the cross-section is taken to be an independent kinematics quantity and corresponding force resultants are defined. For the beam subjected to the external loading only at the ends of the beam, equilibrium equations have been obtained by the principle of virtual work. The variations of lateral displacement, rotational angle of the cross-section and the multiplier of the warping shape along the beam axis are solved in closed form and expressed in terms of deformation quantities at the ends of the beam. Based on this beam theory, the lateral stiffness of the beam sustained an axial compression force and a lateral shear force at one end is explicitly derived, from which the equation of the buckling load is established and the buckling load can be solved. When the effect of cross-section warping is neglected, the derived lateral stiffness and buckling load converge to the solutions of the Haringx theory.     

硬夹心矩形夹层板的整体稳定性分析

摘要：本文在Reissner型理论给出的位移模式基础上,修正其软夹心假设,考虑夹心层面内刚度,给出了硬夹心夹层板的几何方程、物理方程,建立了硬夹心夹层板结构在面内纵向载荷作用下的平衡微分方程,并对方程进行了简化,通过理论计算得到了四边简支条件下硬夹心矩形夹层板整体失稳临界载荷的解析解,并分别计算了夹心层材料的弹性模量 、厚度 、泊松比 对硬夹心夹层板临界载荷的影响,结果证明,对于硬夹心夹层结构,夹心层面内刚度对硬夹心夹层板整体失稳临界载荷的影响较大,考虑其面内刚度是必要的。     

曲壁蜂窝夹层板的振动特性研究

曲壁蜂窝具有负刚度特性,可以在大变形过程中吸收能量、抗冲击,并且在冲击过后可以自我恢复而不像传统蜂窝被压溃。本文将曲梁构成的负刚度蜂窝作为芯层,建立夹层板的动力学模型;推导出了曲壁负刚度蜂窝胞元的等效弹性参数,将其周期性排列为蜂窝芯,应用Reddy高阶剪切变形理论、Von-Karman大变形关系和Hamilton原理推导了负刚度蜂窝夹层板的非线性动力学方程;应用Navier法计算了四边简支边界条件下的固有频率。并利用有限元软件ABAQUS建立模型,计算固有频率,与理论计算结果进行比较,结果显示二者的计算结果具有较好的一致性,验证了芯层等效弹性参数及模型的有效性。探讨了在蜂窝胞元具有较高吸能情形下,夹层板在不同芯层厚度、不同芯厚比以及不同胞元曲壁厚度时的固有频率的变化特性。     

A BEM solution to transverse shear loading of composite beams

In this paper a boundary element method is developed for the solution of the general transverse shear loading problem of composite beams of arbitrary constant cross-section. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli with same Poisson’s ratio and are firmly bonded together. The analysis of the beam is accomplished with respect to a coordinate system that has its origin at the centroid of the cross-section, while its axes are not necessarily the principal ones. The transverse shear loading is applied at the shear centre of the cross-section, avoiding in this way the induction of a twisting moment. Two boundary value problems that take into account the effect of Poisson’s ratio are formulated with respect to stress functions and solved employing a pure BEM approach, that is only boundary discretization is used. The evaluation of the transverse shear stresses is accomplished by direct differentiation of these stress functions, while both the coordinates of the shear center and the shear deformation coefficients are obtained from these functions using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The accuracy of the proposed shear deformation coefficients compared with those obtained from a 3-D FEM solution of the ‘exact’ elastic beam theory is remarkable.     

Stress concentration around a hole in a radially inhomogeneous plate

The stress concentration factor around a circular hole in an infinite plate subjected to uniform biaxial tension and pure shear is considered. The plate is made of a functionally graded material where both Young’s modulus and Poisson’s ratio vary in the radial direction. For plane stress conditions, the governing differential equation for the stress function is derived and solved. A general form for the stress concentration factor in case of biaxial tension is presented. Using a Frobenius series solution, the stress concentration factor is calculated for pure shear case. The stress concentration factor for uniaxial tension is then obtained by superposition of these two modes. The effect of nonhomogeneous stiffness and varying Poisson’s ratio upon the stress concentration factors are analyzed. A reasonable approximation in the practical range of Young’s modulus is obtained for the stress concentration factor in pure shear loading.     

具有叉指式电极的1-3型压电复合材料层合板力电耦合行为的三维精确分析

基于三维弹性理论和压电理论,导出了含有1-3型压电复合材料层的有限长矩形层合简支板的静力平衡方程和边界条件,给出了该层合板在叉指式电极和外力共同作用下力电耦合特性的三维精确解.数值算例的计算结果与有限元解进行了对比,取得了很好的一致性.研究了压电矩阵各向异性和刚度矩阵各向异性以及电势等因素对其挠曲面扭率最大值的影响.数值结果表明层合板扭率最大值的绝对值随压电矩阵各向异性系数Rd的增大而增大并随刚度矩阵各向异性系数Rc的减小而增加.     

Nonlinear dynamic response of a functionally graded plate with a through-width surface crack

A nonlinear vibration analysis of a simply supported functionally graded rectangular plate with a through-width surface crack is presented in this paper. The plate is subjected to a transverse excitation force. Material properties are graded in the thickness direction according to exponential distributions. The cracked plate is treated as an assembly of two sub-plates connected by a rotational spring at the cracked section whose stiffness is calculated through stress intensity factor. Based on Reddy’s third-order shear deformation plate theory, the nonlinear governing equations of motion for the FGM plate are derived by using the Hamilton’s principle. The deflection of each sub-plate is assumed to be a combination of the first two mode shape functions with unknown constants to be determined from boundary and compatibility conditions. The Galerkin’s method is then utilized to convert the governing equations to a two-degree-of-freedom nonlinear system including quadratic and cubic nonlinear terms under the external excitation, which is numerically solved to obtain the nonlinear responses of cracked FGM rectangular plates. The influences of material property gradient, crack depth, crack location and plate thickness ratio on the vibration frequencies and transient response of the surface-racked FGM plate are discussed in detail through a parametric study.