The frictional sliding contact problems of rigid parabolic and cylindrical stamps on graded coatings

Graded materials, also known as functionally graded materials (FGMs), are generally two-phase composites with continuously varying volume fractions. Used as coatings and interfacial zones they can reduce thermally and mechanically induced stresses resulting from material property mismatch, increase the bonding strength and provide protection against adverse environments. In this paper, the contact problems of parabolic and cylindrical stamps on graded coatings are considered. The objective of this study is to obtain a series of analytical benchmark solutions for examining the influence of such factors as material inhomogeneity constants the coefficient of friction and various length parameters on the critical stresses that may have a bearing on the fatigue and fracture of the components with graded coatings.     

Exact solution of a compositionally graded piezoelectric layer under uniform stretch, bending and twisting

Electromechanical responses of compositionally graded piezoelectric layers are analysed. The layers consist of polycrystalline piezoelectric ceramics poled along the thickness direction, and thus exhibit material symmetry of a hexagonal crystal in class 6mm. Two cases for layers (i) covered by surface electrodes and (ii) without surface electrodes are considered. Analytical solutions are exact in Saint Venant's sense for both cases. However, solutions are obtained for layers subjected to uniform mechanical loads, including stretch, bending and twisting. Numerical results to show the effects of different compositional gradients are presented.     

Postbuckling of shear deformable FGM cylindrical shells surrounded by an elastic medium

This paper presents a study on the postbuckling response of a shear deformable functionally graded cylindrical shell of finite length embedded in a large outer elastic medium and subjected to axial compressive loads in thermal environments. The surrounding elastic medium is modeled as a tensionless Pasternak foundation that reacts in compression only. The postbuckling analysis is based on a higher order shear deformation shell theory with von Kármán-Donnell-type of kinematic nonlinearity. The thermal effects due to heat conduction are also included and the material properties of functionally graded materials (FGMs) are assumed to be temperature-dependent. The nonlinear prebuckling deformations and the initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the postbuckling response of the shells and an iterative scheme is developed to obtain numerical results without using any assumption on the shape of the contact region between the shell and the elastic medium. Numerical solutions are presented in tabular and graphical forms to study the postbuckling behavior of FGM shells surrounded by an elastic medium of tensionless Pasternak foundation, from which the postbuckling results for FGM shells with conventional elastic foundations are also obtained for comparison purposes. The results reveal that the unilateral constraint has a significant effect on the postbuckling responses of shells subjected to axial compression in thermal environments when the foundation stiffness is sufficiently large.     

Thermoelastic and vibration analysis of functionally graded cylindrical shells

The static response and free vibration of metal and ceramic functionally graded shells are analyzed using the element-free kp-Ritz method. The material properties are assumed to vary continuously along the depth direction. The displacement field is expressed in terms of a set of mesh-free kernel particle functions according to Sander's first-order shear deformation shell theory. The effects of the volume fraction, material property, boundary condition, and length-to-thickness ratio on the shell deflection, axial stress, and natural frequency are examined in detail. Convergence studies of node numbers are performed to verify the effectiveness of the proposed method. Comparisons reveal that the numerical results obtained from the proposed method agree well with those from the classical and finite element methods.     

Contact and thermal analysis of transfer film covered real composite-steel surfaces in sliding contact

For composite-steel surfaces in sliding contact an anisotropic numerical contact algorithm has been developed to study the ‘layer type’ problems. An FE contact analysis was applied to evaluate the contact parameters (real contact area, contact pressure distribution and normal approach). The contact temperature rise was determined by using both a numerical thermal algorithm for stationary and a FE transient thermal technique for ‘fast sliding’ problems.The effect of a continuous transfer film layer (TFL), that had built up during wear of the PEEK matrix material on the steel counterpart, was considered. Its thickness was assumed to be t=1 μm, and its material properties were that of PEEK at room temperature or, in the case of frictional heating, at a temperature of 150°C (i.e. above the glass transition temperature of the polymer matrix).Results are presented for a spherical steel asperity, with/without TFL, sliding over composite surfaces of different fibre orientation, and in addition, for real composite-steel surfaces (based on measured surface roughness data) in sliding contact. The TFL has an effect on the contact parameters especially at higher operating temperatures (i.e. 150°C); it results in the production of a larger contact area and a lower contact pressure distribution. The contact temperature rise is clearly higher if a TFL is present. Due to the low thermal conductivity of PEEK, the TFL is close to the melting state or it even gets molten within a small vicinity of the contact area.     

Nonlinear cylindrical bending analysis of shear deformable functionally graded plates under different loadings using analytical methods

An exact solution is presented for the nonlinear cylindrical bending and postbuckling of shear deformable functionally graded plates in this paper. A simple power law function and the Mori–Tanaka scheme are used to model the through-the-thickness continuous gradual variation of the material properties. The von Karman nonlinear strains are used and then the nonlinear equilibrium equations and the relevant boundary conditions are obtained using Hamilton's principle. The Navier equations are reduced to a linear ordinary differential equation for transverse deflection with nonlinear boundary conditions, which can be solved by exact methods. Finally, by solving some numeral examples for simply supported plates, the effects of volume fraction index and length-to-thickness ratio are studied. It is shown that there is no bifurcation point for simply supported functionally graded plates under compression. The behavior of near-boundary areas predicted by the shear deformation theory and the classical theory is remarkably different.     

An exact solution for buckling of functionally graded circular plates based on higher order shear deformation plate theory under uniform radial compression

In this research, mechanical buckling of circular plates composed of functionally graded materials (FGMs) is considered. Equilibrium and stability equations of a FGM circular plate under uniform radial compression are derived, based on the higher order shear deformation plate theory (HSDT). Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations are established. A buckling analysis of a functionally graded circular plate (FGCP) under uniform radial compression is carried out and the results are given in closed-form solutions. The results are compared with the buckling loads of plates obtained for FGCP based on the first order shear deformation plate theory (FSDT) and classical plate theory (CPT) given in the literature. The study concludes that HSDT accurately predicts the behavior of FGCP, whereas the FSDT and CPT overestimates buckling loads.     

Stress concentration factor due to a circular hole in functionally graded panels under uniaxial tension

The effect of the material property inhomogeneity on the stress concentration factor (SCF) due to a circular hole in functionally graded panels is numerically investigated. The multiple isoparametric finite element formulation is used to simulate the elastostatic boundary value problem. A parametric study is performed by varying the functional form and the direction of the material property gradation. The material property inhomogeneity is characterized by the intrinsic inhomogeneity length scale, modulus ratio and the power-law index. The results from our parametric study showed that the SCF is reduced when Young's modulus progressively increased away from the hole. The angular position of the maximum tensile stress on the surface of the hole remains unaffected by the material property inhomogeneity. The SCF is seen to be most influenced by the power-law index, followed by the variation of the inhomogeneity length scale. The SCF is least affected by the modulus ratio.     

A new algorithm for solving the multi-indentation problem of rigid bodies of arbitrary shapes on a viscoelastic half-space

In this paper the contact problem between rigid indenters of arbitrary shapes and a viscoelastic half-space is considered. Under the action of a normal force the penetration of the indenters changes and a few contact areas appeared. We wish to find the relations which link the pressure distribution, the resultant forces on the indenters and the penetration on the assumption that the surfaces are frictionless. For indenters of arbitrary shapes the problem may be solved numerically by using the matrix inversion method, extended to viscoelastic cases [1]. But when the problem involves a large number of points the matrix inversion method can become very time-consuming. Here the problem is solved using an alternative scheme, called the two-scale iterative method. In this method the local matrix inversion method is used at the micro-scale for each contact area to compute the pressure distribution taking into account interacting effect (the forces on the other contact areas which can be calculated at the macro-scale) between indenters. Two algorithms were proposed. The first algorithm takes into account the distribution of forces on the other contact areas and the second is the approximation of the first algorithm and takes into account the resultant forces on the other contact areas. The method was implemented for a simple configuration of seven spherical indenters, seven spherical-ended cylindrical indenters and seven flat-ended cylindrical indenters as well as for a more complex configuration of 12 randomly positioned indenters of arbitrary shapes: spherical-ended cylindrical, flat-ended cylindrical, conical and cylindrical indenters (finite cylindrical shape with its curved face). This last case is more difficult as the indenting geometry does not have an axisymmetric profile. For all these cases the two-scale iterative method permits to find the pressure distribution and the contact forces versus the penetration. It can be validated by comparing the numerical results to the numerical results obtained with the matrix inversion method.     

A new beam finite element for the analysis of functionally graded materials

A new beam element is developed to study the thermoelastic behavior of functionally graded beam structures. The element is based on the first-order shear deformation theory and it accounts for varying elastic and thermal properties along its thickness. The exact solution of static part of the governing differential equations is used to construct interpolating polynomials for the element formulation. Consequently, the stiffness matrix has super-convergent property and the element is free of shear locking. Both exponential and power-law variations of material property distribution are used to examine different stress variations. Static, free vibration and wave propagation problems are considered to highlight the behavioral difference of functionally graded material beam with pure metal or pure ceramic beams.     

Methods for analysing partial contact between surfaces

This paper reviews methods available for the numerical solution of contact mechanics problems, in particular, those in which the extent of the contact(s) is not known at the start of the analysis. A new objective function is proposed which is generally smooth and which enables contact mechanics problems to be solved by unconstrained minimisation, in spite of the non-linear boundary condition at the interface. The application of the method to plane periodic elastic contacts, and the author's previously published M(x) formulation of such cases, is demonstrated in detail.     

A new algorithm for computing the indentation of a rigid body of arbitrary shape on a viscoelastic half-space

In this paper the contact problem between a rigid indenter of arbitrary shape and a viscoelastic half-space is considered. Under the action of a normal force the penetration of the indenter and the distribution of contact pressure change. We wish to find the relations which link the pressure distribution, the resultant force on the indenter and the penetration on the assumption that the surfaces are frictionless. For indenters of arbitrary shape the problem may be solved numerically by using the Matrix Inversion Method (MIM), extended to viscoelastic case. In this method the boundary conditions are satisfied exactly at specified “matching points” (the mid-points of the boundary elements). It can be validated by comparing the numerical results to the analytic solutions in cases of a spherical asperity (loading and unloading) and a conical asperity (loading only). Finally, the method was implemented for a finite cylindrical shape with its curved face indenting the surface of the half-space. This last example shows the efficiency of the method in case of a prescribed penetration as well as a given normal load history.     

Dynamic analysis of a linear motion guide having rolling elements for precision positioning devices

Linear motion (LM) guides supported by rolling elements can be used to ultra-accurately position precision machines. For accurate positioning, the micro and macro dynamic behavior of the LM guide must be understood, but the research on this subject is rather limited. In this investigation, experiments to reveal the dynamic characteristics of the LM guide were performed and a simplified model to predict the observed dynamics of the LM guide was developed. Several experiments conducted in the present research demonstrated the hysteretic behavior as well as frequency-and force-dependent phenomena of the LM guide. The validity of the proposed modeling method was checked with theoretical and numerical analyses.     

The planar contact problem of interaction between a rigid heat-conductive cylindrical die and an elastic layer in frictional heat emission

The paper presents discussion of a new planar problem of contact interaction between a rigid heat-conductive die of circular cross section and an elastic layer. The apparatus of integral transformation is used to obtain a precise solution of the nonstationary equations of heat conductivity for the layer and the cylindrical die. This method makes it possible to reduce the formulated problem to a system of integral equations with time-variable limits of integration; the structure of the equations is governed by the type of thermophysical conditions on the interaction surface. An algorithm is advanced for solving this type of integral equation; the variations in time of the contact pressure and the interaction area boundaries are explored. Thus, it is possible to make the problem’s mathematical formulation closer to the real distribution of thermoelastic stresses and to estimate more accurately the effect of temperature fields on the value and pattern of the contact-pressure distribution.     

The contact problem for a double-layer cylindrical base with account of friction forces

The contact problem of a double-layer cylindrical base is discussed with account of friction forces. The problem is reduced to an integral equation of the first kind in relation to unknown contact stresses. A direct method of collocation is used to solve it, enabling us to obtain sufficiently accurate solutions for virtually any values of the parameters. The contact stresses and the contact-zone area are calculated for various material and geometrical parameters of the base layers.     

Stress calculation for plastic helical gears under a real transverse contact ratio

Calculation methods of plastic helical gears with a real transverse contact ratio have never been studied in depth. The methods used in some references assume a uniform load distribution along the line of contact or don’t take into consideration the effect of premature mesh between one or more pair of teeth during engagement, which is not in good agreement with numerical results. This paper is focused on analysis of plastic helical gears. The nonuniform load distribution obtained from the proposed mathematical model is based on the calculation of the real contact ratio representing the real number of pairs of teeth in contact. This model also leads to the distribution of the tooth bending stress and the contact stress along the area of contact.     

Analysis and modeling for flexible joint interfaces under micro and macro scale

The flexible joint interfaces with random topography are re-constructed by data point cloud obtained by experimental measurements and a contact model between two rough random surfaces is established. Contact analysis is conducted by FEM considering the elasto-plastic constitution of the material. An exponential relationship model between normal pressure and normal deformation of the joint interfaces is presented by least-square method, and furthermore, the normal contact stiffness expressions of unit area of the joint interfaces are deduced. In order to use micro-scale model to deal with large scale practical engineering interfaces, a cantilever structure connected by bolted joint is taken as a macro scale example, and its dynamic model is established with the pressure distribution on the interface and real contact area obtained by simulations and experiments. Modal test experiments are conducted, and the first three natural frequencies and frequency response functions are obtained to validate the model.     

FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations

In this paper, the generalized differential quadrature (GDQ) method is applied to study the dynamic behavior of functionally graded materials (FGMs) and laminated doubly curved shells and panels of revolution with a free-form meridian. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature a generalization of the Reissner-Mindlin theory, proposed by Toorani and Lakis, is adopted. The governing equations of motion, written in terms of stress resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. Firstly, the differential quadrature (DQ) rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Secondly, the discretization of the system by means of the GDQ technique leads to a standard linear eigenvalue problem, where two independent variables are involved. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the Reissner-Mindlin and the Toorani-Lakis theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed. Finally, different lamination schemes are considered to expand the combination of the two functionally graded four-parameter power-law distributions adopted. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the lamina thickness direction. A two-constituent functionally graded lamina consists of ceramic and metal those are graded through the lamina thickness. A parametric study is performed to illustrate the influence of the parameters on the mechanical behavior of shell and panel structures considered.     

基于TRIZ的新型载车台结构创新设计与计算分析

基于TRIZ理论的技术冲突解决原理,设计了L型单吊点重力自平衡式旋转立体车库。针对其主要结构L型单吊点重力自平衡式载车台进行了力学模型简化,推导出适合该结构力学计算的方法,为载车台的结构设计提供了较为精确的内力计算结果,并应用该计算结果对载车台的垂直框架和水平框架进行了结构设计。