Axial loading of elliptical-section bonded rubber blocks

Closed-form expressions for the small axial deflection and stress distribution of axially loaded rubber blocks of elliptical cross-section, whose ends are bonded to rigid plates, are derived using a superposition approach. The governing equations and conditions are satisfied exactly, based upon the classical theory of elasticity. Easily calculable expressions are derived for the corresponding apparent Young’s modulus and the modified apparent Young’s modulus in forms analogous to those previously given for blocks of circular cross-section.     

Mechanics of rubber shear springs

FEA calculations have been carried out for a model rubber shear spring, consisting of a block of a highly elastic material, bonded between two rigid parallel plates and sheared by displacing one of the plates parallel to the other in its own plane. The block was prevented from deforming in the perpendicular direction, and thus was deformed in plane strain. Stress distributions along the bond-line and the center-line are reported and compared with those expected from the theory of large elastic deformations. Unexpected tensile stresses were found to develop in the interior of the sheared block. They are attributed to the absence on the end surfaces of the stresses needed to maintain a simple shear, causing a pronounced change in the reference pressure—a consequence that is usually overlooked. Because the internal stresses are governed by the boundary conditions, they were strongly affected by the shape of the end surfaces. In addition, they were reduced markedly by assigning values to Poisson's ratio slightly lower than 0.5, thus allowing some volume expansion of the rubber. Strain energy release rates were also evaluated for growth of a crack along the bond-line, starting at the edges, and compared with those reported previously by Lindley and Teo [Energy for crack growth at the bonds of rubber springs, Plast. Rubber Mat. Appl. 4 (1979) 29-37], Muhr et al. [A fracture mechanics study of natural rubber-to-metal bond failure, J. Adhes. Sci. Technol. 10 (1996) 593-616], Gregory and Muhr [Stiffness and fracture analysis of bonded rubber blocks in simple shear, in: D. Boast, V.A. Coveny (Eds.), Finite Element Analysis of Elastomers, Professional Engineering Publications, Bury St. Edmunds, UK, 1999, pp. 265-274] and Gough and Muhr [Initiation of failure of rubber close to bondlines, in: Proceedings of the International Rubber Conference, Maastricht, Netherlands, June 2005, IOM Communications Ltd., London, 2005, pp. 165-174]. They confirm that a long crack at the compression edge will grow faster than one at the tension edge, but the results for short cracks were inconclusive.     

Frictional moving contact problem between a conducting rigid cylindrical punch and a functionally graded piezoelectric layered half plane

Meccanica - In this study, a frictional moving contact problem between an electrically conducting rigid cylindrical punch and a functionally graded piezoelectric material (FGPM) layer bonded to a...     

Frictional contact problem of a rigid stamp and an elastic layer bonded to a homogeneous substrate

In this study, the frictional contact problem for a layer bonded to a homogeneous substrate is considered according to the theory of elasticity. The layer is indented by a rigid cylindrical stamp which is subjected to concentrated normal and tangential forces. The friction between the layer and the stamp is taken into account. The problem is reduced to a singular integral equation of the second kind in which the contact pressure function and the contact area are the unknown by using integral transform technique and the boundary conditions of the problem. The singular integral equation is solved numerically using both the Jacobi polynomials and the Gauss?CJacobi integration formula, considering equilibrium and consistency conditions. Numerical results for the contact pressures, the contact areas, the normal stresses, and the shear stresses are given, for both the frictional and the frictionless contacts.     

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.     

Winkler地基模型上功能梯度材料涂层的摩擦接触问题

研究Winker地基模型上功能梯度材料涂层在一刚性圆柱形冲头作用下的摩擦接触问题。功能梯度材料涂层表面作用有法线向和切线向集中作用力。假设材料非均匀参数呈指数形式变化,泊松比为常量,利用Fourier积分变换技术将求解模型的接触问题转化为奇异积分方程组,再利用切比雪夫多项式对所得奇异积分方程组进行数值求解。最后,给出了功能梯度材料非均匀参数、摩擦系数、Winker地基模型刚度系数及冲头曲率半径对接触应力分布和接触区宽度的影响情况。     

Fretting contact of two elastic solids with graded coatings under torsion

In this paper, the fretting contact problem for two elastic solids with graded coatings is investigated. We assume a conventional axisymmetric Hertzian contact takes place between two elastic solids under the action of the normal pressure. The application of the torque produces an annulus of slip. It is assumed that the surface shear traction within the contact area is limited by Coulomb’s friction law and the torsion angel was produced within the central adhesion zone as a rigid body. The linear multi-layer model is used to model the functionally graded coating with arbitrarily varying shear modulus. This model divides the coating into a series of sub-layers with the elastic modulus varying linearly in each sub-layer and continuous on the sub-interfaces. By using the transfer matrix method and Hankel integral transform technique, this problem is formulated as the solution of the Cauchy singular integral equations. The contact tractions are calculated by solving the equations numerically. The results show that the appropriate gradual variation of the shear modulus can significantly alter the contact tractions. Therefore, graded coatings may have potential applications in improving the resistance to fretting contact damage at the contact surfaces.     

Indentation of a viscoelastic rubber covered roll by a rigid plane surface

Deformations of a viscoelastic rubberlike layer bonded to a rigid cylinder and indented by a rigid plane surface are studied by the finite element method. The constitutive relation assumed for the viscoelastic rubber is that proposed by Boltzman. Some of the assumptions made to simplify the work are that the roll cover is rotating at a uniform angular speed, steady state has reached, the deformations of the rubberlike layer are infinite-simal and the effect of inertia forces is negligible. Results presented include the pressure distribution at the contact surface and the stress distribution near the bond surface.     

Two-dimensional sliding frictional contact of functionally graded materials

A multi-layered model for sliding frictional contact analysis of functionally graded materials (FGMs) with arbitrarily varying shear modulus under plane strain-state deformation has been developed. Based on the fact that an arbitrary curve can be approached by a series of continuous but piecewise linear curves, the FGM is divided into several sub-layers and in each sub-layers the shear modulus is assumed to be a linear function while the Poisson's ratio is assumed to be a constant. In the contact area, it is assumed that the friction is one of Coulomb type. With this model the fundamental solutions for concentrated forces acting perpendicular and parallel to the FGMs layer surface are obtained. Then the sliding frictional contact problem of a functionally graded coated half-space is investigated. The transfer matrix method and Fourier integral transform technique are employed to cast the problem to a Cauchy singular integral equation. The contact stresses and contact area are calculated for various moving stamps by solving the equations numerically. The results show that appropriate gradual variation of the shear modulus can significantly alter the stresses in the contact zone.     

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.     

Compression analysis of rectangular elastic layers bonded between rigid plates

An elastic layer bonded between two rigid plates has higher compression stiffness than the elastic layer without bonding. While the finite element method can be applied to calculate the stiffness, the compression stiffness of bonded rectangular layers derived through a theoretical approach in this paper provides a convenient way for parametric study. Based on two kinematics assumptions, the governing equation for the mean pressure is derived from the equilibrium equations. Using the approximate shear boundary condition, the mean pressure is solved and the compression stiffness of the bonded rectangular layer is then established in an explicit single-series form. Through the solved pressure, the horizontal displacements are derived from the corresponding equilibrium equations, from which the shear stress on the bonding surface can be found. It is found that the effect of the rectangular aspect on the compression stiffness is significant only when Poisson’s ratio is near 0.5. For the smaller Poisson’s ratio, the compression stiffness of the rectangular layer can be approximated by the formula for the infinite-strip layer of the same shape factor.     

胶结土石混合体力学特性的块石形状效应细观机理分析

考虑块石形状为球体、正方体和长方体三种情况,通过正方体与球体相比较来探究块石不同棱角度对胶结土石混合体力学特性的影响,通过长方体与正方体相比较来探究块石不同球度对胶结土石混合体力学特性的影响。首先,基于不规则颗粒三维离散元精细模拟技术实现了正方体和长方体块石数值模型的建立;然后建立含石量为30%和80%的块石形状分别为球体、正方体和长方体的胶结土石混合体三维离散元随机结构模型;最后,对土石混合体大三轴试验进行颗粒流数值模拟,获得了不同含石量、不同块石形状下胶结土石混合体的强度特征和变形特征,并分别就低、高两种含石量下块石形状对土石混合体力学特性影响的细观机理进行了深入地分析。结果表明：块石含量和形状均会显著影响胶结土石混合体的力学特性,并且两者间具有复杂的交互作用;微裂纹、块石颗粒平均旋转量、应变能和摩擦功等的演化规律能够很好地从细观水平上反映块石形状影响的作用机理。     

Elastic layers bonded to flexible reinforcements

Elastic layers bonded to reinforcing sheets are widely used in many engineering applications. While in most of the earlier applications, these layers are reinforced using steel plates, recent studies propose to replace “rigid” steel reinforcement with “flexible” fiber reinforcement to reduce both the cost and weight of the units/systems. In this study, a new formulation is presented for the analysis of elastic layers bonded to flexible reinforcements under (i) uniform compression, (ii) pure bending and (iii) pure warping. This new formulation has some distinct advantages over the others in literature. Since the displacement boundary conditions are included in the formulation, there is no need to start the formulation with some assumptions (other than those imposed by the order of the theory) on stress and/or displacement distributions in the layer or with some limitations on geometrical and material properties. Thus, the solutions derived from this formulation are valid not only for “thin” layers of strictly or nearly incompressible materials but also for “thick” layers and/or compressible materials. After presenting the formulation in its most general form, with regard to the order of the theory and shape of the layer, its applications are demonstrated by solving the governing equations for bonded layers of infinite-strip shape using zeroth and/or first order theory. For each deformation mode, closed-form expressions are obtained for displacement/stress distributions and effective layer modulus. The effects of three key parameters: (i) shape factor of the layer, (ii) Poisson’s ratio of the layer material and (iii) extensibility of the reinforcing sheets, on the layer behavior are also studied.     

Plane contact problem for periodic laminated composite involving frictional heating

Summary The stationary problem of a rigid thermally insulated punch sliding over the boundary surface of a periodic two-layered thermoelastic half-space is considered. The heat generated in the contact area is assumed to be caused by frictional forces. The problem is formulated within the framework of thermoelasticity with microlocal parameters, and it is reduced to a system of two integral equations, which is solved numerically. The effects connected with the composite structure are analyzed.     

大变形软材料接触摩擦的在线测试装备与技术分析

我国高档密封件与液压件产品几乎全部依赖进口,国家重大技术装备及国防装备等配套零部件行业的许多问题与摩擦学有十分重要的关系,基础件摩擦学又是国际上的研究热点与发达国家的竞争高地. 在本文中阐述了国内外关键机械零部件的接触界面原位在线测试的研究进展,及其在大变形软材料密封系统中的应用,综述了摩擦润滑中迁移状态实时在线观测技术、聚合物密封界面在线观测技术和国内在摩擦过程的微区域原位研究等进展,还分析了特殊与极端工况条件下大变形聚合物密封的性能测试、可靠性寿命分析以及多工况联合测试的技术进展,分析和讨论了高性能密封件行业的基础共性难题,最后进行了总结与展望.      

具有弹性层的两圆管间夹入弹性薄膜时的回转接触

应用弹性接触理论及Fourier变换,考虑接触域的粘着及微小滑状态,对具有弹性层的两圆筒间夹入弹性薄膜的非对称回转接触问题进行了0研究,讨论了接触宽度,弹性层厚度及摩擦系数等对弹性薄膜传送速度和圆筒公称圆周速度之比的影响。     

Effect of anisotropy on thermoelastic contact problem

This paper is concerned with the stationary plane contact of an insulated rigid punch and a half-space which is elastically anisotropic but thermally conducting. The frictional heat generation inside the contact region due to the sliding of the punch over the half-space surface and the heat radiation outside the contact region are taken into account. With the help of Fourier integral transform, the problem is reduced to a system of two singular integral equations. The equations are solved numerically by using Gauss-Jacobi and trapezoidal-rule quadratures. The effects of anisotropy and thermal effects are shown graphically.     

解三维摩擦接触问题的一个二阶锥线性互补法

针对三维摩擦接触问题的求解,给出了一种基于参变量变分原理的二阶锥线性互补法. 首先,基于三维Coulomb摩擦锥在数学表述上属于二阶锥的事实,利用二阶锥规划对偶理论,建立了三维Coulomb摩擦接触条件的参变量二阶锥线性互补模型,它是二维Coulomb摩擦接触条件参变量线性互补模型在三维情形下的自然推广;随后,利用参变量变分原理与有限元方法,建立了求解三维摩擦接触问题的二阶锥线性互补法. 较之于将三维Coulomb摩擦锥进行显式线性化的线性互补法,该方法无需对三维Coulomb摩擦锥进行线性化,因而在保证精度的前提下所解问题的规模要小很多. 最后通过算例展示了该方法的特点.      

Effect of friction on subsurface stresses in sliding line contact of multilayered elastic solids

A numerical integral scheme based on Fourier transformation approach is employed to investigate the effect of friction on subsurface stresses arising from the two-dimensional sliding contact of two multilayered elastic solids. The analysis incorporates bonded and unbonded interface boundary conditions between the coating layers. Two line contact problems are presented. The first one is the contact problem between a rigid cylinder and a two-layer half space and the second one is the indentation of a multilayered elastic half-space by a flat rigid punch. The effects of the surface coating on the contact pressure distribution and subsurface stress field are presented and discussed.     

块石形状对土石混合体力学行为影响的颗粒流模拟

考虑块石形状为球体、正方体和长方体三种情况,通过正方体与球体相比较来探究块石棱角度不同对土石混合体力学特性的影响,通过长方体与正方体相比较来探究块石球度不同对土石混合体力学特性的影响。首先,提出特定形状块石三维离散元精细建模的方法;接着建立含石量为30%和80%的块石形状分别为球体、正方体和长方体的土石混合体三维颗粒流数值模型;然后,对土石混合体大三轴试验进行颗粒流模拟,获得了不同含石量及不同块石形状的土石混合体试样的宏观力学特征;最后,详细分析了块石形状对土石混合体力学行为影响的细观机理。结果表明,块石形状会影响土石混合体的力学行为,其影响的程度与含石量密切相关;配位数、块石颗粒平均旋转量和摩擦功的演化规律能够很好地从细观水平上反映块石形状的作用机理。