Tangential contact problem for a transversely isotropic elastic layer bonded to an elastic foundation

A system consisting of an elastic layer made of a transversely isotropic material bonded to an elastic half-space made of a different transversely isotropic material is considered. An arbitrary tangential displacement is prescribed over a domain S of the layer, while the rest of the layer’s surface is stress-free. The tangential contact problem consists of finding a complete field of stresses and displacements in this system. The generalized-images method developed by the author is used to get an elementary solution to the problem. It is also shown that an integral transform can be interpreted as a sum of generalized images. The case of a circular domain of contact is considered in detail. The results are valid for the case of isotropy as well.     

Wave propagation at the imperfect boundary between transversely isotropic thermodiffusive elastic layer and half-space

The aim of the present investigation is to study surface wave propagation at the imperfect boundary between two transversely isotropic thermodiffusive elastic objects: a layer of finite thickness and a half-space, in the context of the Green–Lindsay theory. The secular equation for surface waves in a compact form is derived after the mathematical model has been developed. The phase velocity and attenuation coefficient are obtained for stiffness (normal and tangential), thermal conductance, and for the case of a welded contact. The dispersion curves for these quantities are presented to depict the effect of stiffness and thermal relaxation times. The amplitudes of displacements, temperature, and concentration at the free plane boundary, as well as the specific loss of energy, are obtained and presented graphically. The special cases are considered and the results are compared with the known ones.     

Contact problem of a functionally graded layer resting on a Winkler foundation

The contact problem for a functionally graded layer supported by a Winkler foundation is considered using linear elasticity theory in this study. The layer is loaded by means of a rigid cylindrical punch that applies a concentrated force in the normal direction. Poisson’s ratio is taken as constant, and the elasticity modulus is assumed to vary exponentially through the thickness of the layer. The problem is reduced to a Cauchy-type singular integral equation with the use of Fourier integral transform technique and the boundary conditions of the problem. The numerical solution of the integral equation is performed by using Gauss–Chebyshev integration formulas. The effect of the material inhomogeneity, stiffness of the Winkler foundation and punch radius on the contact stress, the contact area and the normal stresses are given.     

Explicit dynamics in impact simulation using a NURBS contact interface

In this paper, the impact problem and the subsequent wave propagation are considered. For the contact discretization an intermediate non-uniform rational B-spline (NURBS) layer is added between the contacting finite element bodies, which allows a smooth contact formulation and efficient element-based integration. The impact event is ill-posed and requires a regularization to avoid propagating stress oscillations. A nonlinear mesh-dependent penalty regularization is used, where the stiffness of the penalty regularization increases upon mesh refinement. Explicit time integration methods are well suited for wave propagation problems, but are efficient only for diagonal mass matrices. Using a spectral element discretization in combination with a NURBS contact layer the bulk part of the mass matrix is diagonal.     

An integral formulation for steady-state elastoplastic contact over a coated half-plane

 A boundary-domain integral equation for a coated half-space (elastically isotropic homogeneous substratum, possibly anisotropic coating layer) is developed. The half-space fundamental solution is used, so that the discretization is limited to the potential contact zone (boundary elements), the potentially plastic part of the substratum and the coating layer (domain integration cells). Steady-state elastoplastic analysis is implemented within this framework, for plane-strain conditions, for solving rolling and/or sliding contact problems, where at the moment the contact load comes from either a purely elastic contact analysis or is of Hertz type. The constitutive integration is of implicit type. In order to improve accuracy and computational efficiency, infinite elements are used. Comparison of numerical results with other sources, when available, is satisfactory. The present formulation is also used to compute the contact pressure for an isotropic (or anisotropic) coating on an isotropic homogeneous half-space indented by an elastic punch. Received 29 May 2001     

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

In this paper, the axisymmetric problem of a frictionless double receding contact between a rigid stamp of axisymmetric profile, an elastic functionally graded layer and a homogeneous half space is considered. The graded layer is modelled as a nonhomogeneous medium with an isotropic stress-strain law. Assuming the double contact between the bodies to be frictionless, only compressive normal tractions can be transmitted in each contact area while the rest of the surface is free of tractions. Using an appropriate integral transform, the axisymmetric elasticity equations are converted analytically into a system of singular integral equations where the unknowns are the pressures and the radii of the receding contact area in the two contact zones. The global equilibrium conditions are supplemented to solve the problem. The singular integral equations are solved numerically using orthogonal Chebyshev polynomials. An iterative scheme based on the Newton-Raphson method is employed to obtain the receding contact radii and pressures that satisfy the equilibrium conditions. The main objectives of the paper are to study the effect of the nonhomogeneity parameter, the thickness of the graded layer and the magnitude of the applied load on the contact pressures, the radii of the receding contact zones and the indentation for the case of a spherical rigid punch.     

Thermal stresses in a coating layer. I. General theoretical scheme

The evolution of thermal stresses generated by a thermal perturbation of cylindrical symmetry in a flat infinite isotropic thermoelastic layer of constant (before perturbation) thickness on a half-space rigid substrate is theoretically investigated. The problem is considered in the framework of uncoupled linear thermoelasticity, in the quasi-stationary displacement field approximation. Thermal insulation of the layer is considered in two versions: (a) both its surfaces are adiabatically insulated, and (b) the free surface is adiabatically insulated, whereas the contact surface is kept at constant temperature. The free surface of the layer is assumed to be stress free, and the contact surface is kept on a rigid substrate. The proper equations are solved using suitable Hankel and Fourier transformations. Part I of the paper presents a general theoretical scheme of the problem in a general case (without specification of the perturbation), and it is illustrated by a simple, detailed example (time evolution of stresses at the contact surface of the layer, generated by a surface point instantaneous heat pulse in case a). The analysis of stresses in more realistic cases, modeling realistic situations will be presented in Part II.     

The asymmetric displacement of a rigid spheroidal inclusion embedded in a transversely isotropic medium

Summary An explicit analytical solution is presented for the problem of a rigid spheroidal inclusion embedded in bonded contact with an infinite transversely isotropic elastic medium, where the inclusion is given a constant displacement in a direction perpendicular to the axis of symmetry of the material. The displacement potential representation for the equilibrium of three-dimensional transversely isotropic bodies is used to solve the problem. The loadfeflection relationship for the spheroidal inclusion and its limiting configurations are obtained in closed form. Numerical results are presented to show the effect of both the aspect ratio of the spheroid and the anisotropy on the translational stiffness.With 5 Figures     

燃气轮机拉杆转子考虑接触效应的扭转振动模态分析

燃气轮机转子一般是由多个层叠的轮盘通过拉杆组合而成,各轮盘接触面由磨削加工形成。粗糙度测试结果表明磨削加工的实验拉杆转子轮盘面具有两个不同分形结构的区域。利用结构函数法计算了这种双重分形面的轮廓曲线的分形维数D1、D2和分形粗糙度参数G1、G2 。采用双重分形几何描述接触表面的拓扑结构,并根据赫兹接触理论导出接触微凸体的切向接触刚度。弹塑性双重分形面的切向接触刚度等于所有微观弹性接触点的切向接触刚度的总和。粗糙层是由相接触的微凸体所构成的,其抗扭刚度模化为接触转子轮盘间的一个抗扭弹簧。通过三维有限元模态分析和实验模态分析得到了拉杆转子在不同预紧力下的扭振模态频率。通过上述计算和实验结果识别了粗糙层的抗扭刚度,实验测试结果和理论计算结果相一致,这表明上述接触层抗扭刚度的双重分形模型实合理的,可以有效地考虑接触效应对拉杆转子扭转振动模态频率的影响。     

An analytical solution of the rebound indentation problem for an isotropic linear viscoelastic layer loaded with a spherical punch

A unilateral frictionless axisymmetric contact problem for an isotropic viscoelastic layer attached to a rigid substrate and loaded with a spherical indenter is considered. It is assumed that the indentation protocol is composed of two stages. In the indentation phase, the layer is subjected to displacement loading, while at the end of the first stage, the load is immediately removed and the second stage, called the recovery phase, lasts for a theoretically indefinite time. Under the assumption of time-independent Poisson’s ratio, we derive closed-form analytical expressions for the contact force (in the indentation phase) and for the indentation displacement (in the the recovery phase). The obtained closed-form analytical solution is valid for the indentation phase with an arbitrary monotonic loading displacement and can be used for evaluation of the rebound indentation test for soft biological tissues and originally suggested for assessment of articular cartilage viability.     

Analytical layer-element approach for wave propagation of transversely isotropic pavement

Asphalt pavements have been recognised as transversely isotropic multi-layered structures. In this paper, an analytical layer-element approach is utlised to solve the wave propagation of transversely isotropic multi-layered pavement structures under the falling weight deflectometer impact load. After the application of Fourier–Hankel transform, the Navier's equation for transversely isotropic layer by impulsive force are solved analytically. The global stiffness matrix equation of multilayered structures is further obtained by assembling the interrelated layer-elements, and the actual solution is achieved by numerical inversion of the Fourier–Hankel transform after the solution in the transformed domain is obtained. The layer-element of a single layer and the global stiffness matrix only contain negative exponential functions, which leads to a considerable improvement in computation ef?ciency and stability. Numerical examples are presented to demonstrate the accuracy of this method and to inversitgate the influence of the properties of transversely isotropic elastic materials on the load-displacement responses.     

The rotation of a rigid spheroidal inclusion embedded in a transversely isotropic medium

The exact three-dimensional elasticity solutions are given for two problems related to a rigid spheroidal inclusion embedded in bonded contact with an infinite transversely isotropic elastic medium. The first is of axisymmetric nature in which the inclusion is given a constant rotation about its axis of revolution which coincides with the axis of symmetry of the material. The second problem is asymmetric where the spheroidal inclusion is given a constant rotation about a direction that is perpendicular to the axis of elastic symmetry of the material. The displacement potential representation for the equilibrium of three-dimensional transversely isotropic bodies is used to solve the problem. In both cases, the moment-rotation relationship for the spheroidal inclusion and its limiting configurations are obtained in closed form. Numerical results are presented to show the effect of the aspect ratio of the spheroid on the rotational stiffness.     

An approximate secular equation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer

In this paper, we are interested in the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer. The contact between the layer and the half-space is assumed to be welded. The main purpose of the paper is to establish an approximate secular equation of the wave. By using the effective boundary condition method, an approximate secular equation of fourth order in terms of the dimensionless thickness of the layer is derived. It is shown that this approximate secular equation has high accuracy. From the secular equation obtained, an approximate formula of third order for the velocity of Rayleigh waves is established.     

Rayleigh-type wave propagation on a transversely isotropic viscoelastic layer with yielding and rigid foundations

The present study investigates the propagation of Rayleigh-type wave in a transversely isotropic viscoelastic layer in effect of yielding foundation and rigid foundation in two different cases for two considered models. Numerical computation and graphical demonstration have been carried out for the case when layer is comprised of transversely isotropic viscoelastic Mesaverde clay shale material (Model I) and simply isotropic viscoelastic material (Model II). Closed-form expression of phase velocity and damped velocity for both the cases are deduced analytically. Obtained result is found in well agreement to the established standard results existing in the literature. Significant effect of dilatational viscoelasticity, volume viscoelasticity and yielding parameter on phase and damped velocities for both the considered models has been traced out. The comparative study has been performed to unravel the effect of viscoelasticity over elasticity and anisotropy over isotropy in the context of the present problem. Moreover, comparison of phase and damped velocities for the case of layer with stress-free foundation, layer with rigid foundation and layer with yielding foundation serve as a major highlight of the present work.     

Modelling dilation in an idealised asphalt mixture using discrete element modelling

This paper investigates the use of discrete element modelling (DEM) to simulate the behaviour of a highly idealised bituminous mixture under uniaxial and triaxial compressive creep tests. The idealised mixture comprises single-sized spherical (sand-sized) particles mixed with bitumen and was chosen so that the packing characteristics are known (dense random packing) and the behaviour of the mixture will be dominated by the bitumen and complex aggregate interlock effects will be minimised. In this type of approach the effect of the bitumen is represented as shear and normal contact stiffnesses. A numerical sample preparation procedure has been developed to ensure that the final specimen is isotropic and has the correct volumetrics. Elastic contact properties have been used to investigate the effect of the shear and normal contact stiffnesses on bulk material properties. The bulk modulus was found to be linearly dependent on the normal contact stiffness and independent of the shear contact stiffness. Poisson’s ratio was found to be dependent on only the ratio of the shear contact stiffness to the normal contact stiffness. An elastic contact has been assumed for the compressive normal contact stiffness and a viscoelastic contact for shear and tensile normal contact stiffness to represent the contact behaviour in idealised mixture. The idealised mixture is found to dilate when the ratio of compressive to tensile contact stiffness increases as a function of loading time. Uniaxial and triaxial viscoelastic simulations have been performed to investigate the effect of stress ratio on the rate of dilation with shear strain for the sand asphalt. The numerical results have been validated with experimental data.     

初始椭圆度对螺旋缠绕结构拉伸刚度的影响

为考虑初始椭圆度对螺旋缠绕结构拉伸刚度的影响,建立了一种新的基于椭圆柱面和螺旋带的分析模型,该模型在两者之间建立了一个新的几何关系,并考虑了在椭圆柱面上螺旋带的轴向应变和曲率变化以及柱面和螺旋层之间的接触压力。通过ABAQUS建立一个3D有限元模型验证了理论模型的合理性。在理论模型中,对初始椭圆度进行参数分析得到不同初始椭圆度对椭圆层拉伸刚度的影响,随着管道初始椭圆度的增加,螺旋层拉伸刚度有下降的趋势,但降幅很小。      

The impact of a rigid sphere with an elastic layer of finite thickness

Summary The problem of the impact of a rigid sphere with an elastic isotropic layer is considered in the initial stage of dynamic interaction. The initial stage is characterized by the fact that the velocity of the displacement of the intersection points of the sphere with the upper boundary of the layer is larger than the velocity of longitudinal waves, hence the free surface normal to the contact domain with the body is undisturbed. The method of successive approximations as well as the ray method, according to which the solution behind the fronts of incident and reflected waves is constructed in terms of power series (ray expansions), are used as methods of solution. In the problem under consideration, we used one-term ray expansions whereby the main characteristics of the shock interaction have been obtained, and the possibility of localized damage of the material of the layer at the points lying along the central ray has been examined.     

Torsion of a transversely isotropic elastic layer bonded to a transversely isotropic half-space

Torsional stresses and displacement of a transversely isotropic elastic layer of finite thickness for which torsional shearing forces are prescribed on its boundary surface are considered. The solutions are given for a few particular cases.     

横观各向同性饱和土中单桩竖向振动的简化模型求解

基于饱和多孔介质理论,将土体视为横观各向同性饱和两相介质,利用分离变量法求解了横观各向同性饱和土层的竖向振动,在此基础上建立横观各向同性饱和土中单桩竖向振动的控制方程,借助桩端边界条件和三角函数的正交性对桩竖向振动问题进行了求解,并分析了相关参数对竖向振动的影响。研究表明:桩-土系统存在共振现象,复刚度的变化受激振频率的影响较大;在高频和低频时,各向异性参数对动刚度的影响不同,各向异性参数越大,复刚度的峰值越小。     

Wave theory of high-speed cutting regimes in turning of materials

A theory of high-speed cutting regimes involved in turning of materials has been developed. According to this theory, the cut layer and the blade are considered as acoustic resonators coupled via a nonlinear contact stiffness and excited by the lowest antisymmetric Lamb mode. The turning process is described in terms of a parametric oscillatory system. Using the proposed theory, it is possible to determine optimum turning regimes.