Two-dimensional contact mechanics of functionally graded materials with arbitrary spatial variations of material properties

A multi-layered model for frictionless contact analysis of functionally graded materials (FGMs) with arbitrarily varying elastic 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 linear function while the Poisson’s ratio is assumed to be a constant. With the model, the frictionless contact problem of a functionally graded coated half-space is investigated. By using the transfer matrix method and Fourier integral transform technique, the problem is reduced to a Cauchy singular integral equation. The contact pressure, contact region and indentation are calculated for various indenters by solving the equations numerically.     

Fracture analysis of functionally graded coatings: plane deformation

A new multi-layered model for fracture analysis of functionally graded materials (FGMs) with arbitrarily varying elastic moduli under plane deformation has been developed. In this model, the FGM is divided into several sub-layers and in each sub-layer the shear modulus is assumed to be a linear function of the depth while the Poisson's ratio is assumed to be a constant. With this new model, an interface crack problem of a functionally graded coating bonded to a homogeneous half-plane under normal and shear loading is investigated. Employment of transfer matrix method and Fourier integral transform technique reduces the problem to a system of Cauchy singular integral equations which are solved numerically. Stress intensity factors of an interface crack are obtained for the cases of the shear modulus varying in an exponential manner and in a sinusoidal manner. Comparison of the present new model to other existing models shows that the new one is more efficient.     

Axisymmetric frictionless contact of functionally graded materials

The main interest of this study is a new method to solve the axisymmetric frictionless contact problem of functionally graded materials (FGMs). 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 a series of sub-layers with shear modulus varying linearly in each sub-layer and continuous at the sub-interfaces. With this model, the axisymmetric frictionless contact problem of a functionally graded coated half-space is investigated. By using the transfer matrix method and Hankel integral transform technique, the problem is reduced to a Cauchy singular integral equation. The contact pressure, contact region and indentation are calculated for various indenters by solving the equations numerically. An erratum to this article can be found at     

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.     

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。     

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.     

The axisymmetric partial slip contact problem of a graded coating

In this paper, the axisymmetric contact problem with the partial slip condition for a functionally graded coated half-space which is indented by a rigid spherical indenter is considered. The material properties are assumed to vary along the thickness of the coating. A series of linear functions of the thickness are used to model the functionally graded coating with the arbitrarily varying shear modulus. The contact problem is formulated in terms of a set of Cauchy singular integral equations by employment of Hankel integral transforms technique and transfer matrix method. By using the uncoupled solution and the coupled solution,the coupled equations are solved. The effect of the coating’s gradient on the normal and radial tractions in the whole contact region is presented. The results show that the contact tractions and the size of the stick zone can be altered by adjusting the gradient of the coating. This may have potential applications in the resistance of contact deformation and damage.     

Thermoelastic instability of functionally graded materials with interaction of frictional heat and contact resistance

Both of the frictional heat and thermal contact resistance have a grave responsibility for the localized high temperature (hot spots) at the contact region, which is known as one of the most dangerous appearances in the brakes systems. In this paper, we study the thermoelastic instability (TEI) of a functionally graded material (FGM) half-plane sliding against a homogeneous half-plane at the in-plane direction. The interaction of the frictional heat and thermal contact resistance is taken into account in the TEI analysis. The material properties of the FGM half-plane are supposed to follow the exponential function along the thickness direction. The coupled TEI problem of FGMs is solved by using the perturbation method. The frictionally excited TEI of FGMs is also considered by neglecting the effect of the thermal contact resistance. The results show that the thermal contact resistance, sliding speed and gradient index have significant influence on the TEI. It is found that the variation of the gradient index of FGMs can increase the critical sliding speed and critical heat flux, and therefore improve the TEI of the sliding system.     

Sliding frictional contact between a rigid punch and a laterally graded elastic medium

Analytical and computational methods are developed for contact mechanics analysis of functionally graded materials (FGMs) that possess elastic gradation in the lateral direction. In the analytical formulation, the problem of a laterally graded half-plane in sliding frictional contact with a rigid punch of an arbitrary profile is considered. The governing partial differential equations and the boundary conditions of the problem are satisfied through the use of Fourier transformation. The problem is then reduced to a singular integral equation of the second kind which is solved numerically by using an expansion–collocation technique. Computational studies of the sliding contact problems of laterally graded materials are conducted by means of the finite element method. In the finite element analyses, the laterally graded half-plane is discretized by quadratic finite elements for which the material parameters are specified at the centroids. Flat and triangular punch profiles are considered in the parametric analyses. The comparisons of the results generated by the analytical technique to those computed by the finite element method demonstrate the high level of accuracy attained by both methods. The presented numerical results illustrate the influences of the lateral nonhomogeneity and the coefficient of friction on the contact stresses.     

功能梯度材料涂层半空间的轴对称光滑接触问题

求解了功能梯度材料涂层半空间的轴对称光滑接触问题,其中梯度层剪切模量按照线性变化,利用Hankel积分变换方法求解微分方程,将问题化为具有Cauchy型奇异核的积分方程.数值方法求解表明:功能梯度材料涂层半空间在刚性柱形压头和球形压头作用下,接触表面分布应力,接触半径以及最大压痕受材料梯度效应的影响较大.     

功能梯度条共线Griffith裂纹反平面剪切冲击

研究正交各向异性功能梯度条中多个共线Griffith裂纹的反平面剪切冲击问题．材料两个方向的剪切模量假定按比例同时以特定的梯度变化．采用Laplace和Fourier变换及引进位错密度函数将问题化为求解Cauchy奇异积方程，进而化为代数方程数值求解．考查材料非均匀性、正交性和功能梯度条高度对裂尖动态断裂特性的影响．动应力强度因子的数值结果显示：增加剪切模量的梯度和(或)增加垂直于裂纹面方向的剪切模量，可以抑制动应力强度因子的幅度；若功能梯度条较薄，增大条形域的高度也可抑制裂纹扩展．     

Modeling method for a crack problem of functionally graded materials with arbitrary properties—piecewise-exponential model

A new model, piecewise-exponential model (PE model), is developed to investigate the crack problem of the functionally graded materials (FGMs) with arbitrary properties. In the PE model, the functionally graded material is divided into some nonhomogeneous layers along the gradient direction of the properties, with each layer’s properties varying exponentially. By this way, the real material properties can be approached by a series of exponential functions. Since the real material properties are used on both surfaces of each nonhomogeneous layer, the nature of continuously varying properties of FGMs can be approached accurately. The influences of the local nonhomogeneity on the crack-tip fields can be fully considered. By using the new model, the fracture problem of a functionally graded strip with arbitrary properties and a crack vertical to the free surfaces is studied. The integral transform method, the theory of residues and the theory of singular integral equation are applied. Some representative samples with different kinds of nonhomogeneous properties are analyzed and the corresponding stress intensity factors (SIFs) are presented. It is shown that the PE mode is effective for investigating the crack problems of the FGMs with arbitrary properties.     

Stochastic finite element nonlinear free vibration analysis of piezoelectric functionally graded materials beam subjected to thermo-piezoelectric loadings with material uncertainties

In this paper, second order statistics of large amplitude free flexural vibration of shear deformable functionally graded materials (FGMs) beams with surface-bonded piezoelectric layers subjected to thermopiezoelectric loadings with random material properties are studied. The material properties such as Young’s modulus, shear modulus, Poisson’s ratio and thermal expansion coefficients of FGMs and piezoelectric materials with volume fraction exponent are modeled as independent random variables. The temperature field considered is assumed to be uniform and non-uniform distribution over the plate thickness and electric field is assumed to be the transverse components E _z only. The mechanical properties are assumed to be temperature dependent (TD) and temperature independent (TID). The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics. A C ⁰ nonlinear finite element method (FEM) based on direct iterative approach combined with mean centered first order perturbation technique (FOPT) is developed for the solution of random eigenvalue problem. Comparison studies have been carried out with those results available in the literature and Monte Carlo simulation (MCS) through normal Gaussian probability density function.     

Analytical solution to continuous contact problem for a functionally graded layer loaded through two dissimilar rigid punches

A continuous contact problem of functionally graded layer resting on an elastic semi-infinite plane, which is loaded with through two different blocks is addressed in this study. The elasticity theory and integral transformation techniques are used in solution of the problem. The problem is reduced to a system of singular integral equations, and solved numerically by the aid of appropriate Gauss–Chebyshev integration formula. It is assumed that the elastic semi-infinite homogeneous plane is isotropic and all surfaces are frictionless and continuous. The shear modulus and the mass density of the FG layer vary exponentially along the thickness direction.     

DYNAMIC STRESS FIELD AROUND THE MODE Ⅲ CRACK TIP IN AN ORTHOTROPIC FUNCTIONALLY GRADED MATERIAL

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｒｅｃｅｎｔｙｅａｒｓ,ｇｒｅａｔａｔｔｅｎｔｉｏｎｓｈａｖｅｂｅｅｎｐａｉｄｔｏｔｈｅｒｅｓｅａｒｃｈｏｆＦｕｎｃｔｉｏｎａｌｌｙＧｒａｄｅｄＭａｔｅｒｉａｌｓ(ＦＧＭ).Ｆｒｏｍｔｈｅｖｉｅｗｐｏｉｎｔｓｏｆａｐｐｌｉｅｄｍｅｃｈａｎｉｃｓ,ＦＧＭａｒｅｎｏｎ＿ｈｏｍｏｇｅｎｅｏｕｓｓｏｌｉｄｓ.Ｔｈｅｎｏｎ＿ｈｏｍｏｇｅｎｅｉｔｙｏｆＦＧＭｈａｓａｇｒｅａｔｉｎｆｌｕｅｎｃｅｏｎｔｈｅｉｒｍｅｃｈａｎｉｃａｌｂｅｈａｖｉｏｒ,ｅｓｐｅｃｉａｌｌｙｗｈｅｎｔｈｅｃｏｍｐ…     

The wave propagation and dynamic response of rectangular functionally graded material plates with completed clamped supports under impulse load

In this paper, the wave propagation and dynamic response of the rectangular FGM plates with completed clamped supports under impulse load are analyzed. The effective material properties of functionally graded materials (FGMs) for the plate are assumed to vary continuously through the plate thickness and be distributed according to a volume fraction power law along the plate thickness. Considering the effects of transverse shear deformation and rotary inertia, the governing equations of the wave propagation in the functionally graded plate are derived by using the Hamilton’s principle. A complete discussion of dispersion of the FGM plates is given. Using the dispersion relation and integral transforms, exact integral solutions for the FGM plates under impulse load are obtained. The influence of volume fraction distributions on wave propagation and dynamic response of the FGM plates is analyzed.     

The Anti-plane Dynamic Fracture Analysis of Functionally Graded Materials with Arbitrary Spatial Variations of Material Properties

Anti-plane dynamic fracture analysis is presented for functionally graded materials (FGM) with arbitrary spatial variations of material properties. The FGM with the material properties varying continuously in an arbitrary manner is modeled as a multi-layered medium with the elastic modulus and mass density varying linearly in each sub-layer and continuous at the interfaces between two adjacent sub-layers. With this linearly inhomogeneous multi-layered model, the problem of a crack in a graded interfacial zone bonded to two homogeneous half-spaces or in a coating bonded to a homogeneous half-space subjected to the anti-plane shear impact load is investigated. Laplace and Fourier transforms and transfer matrix are applied to reduce the associated mixed boundary value problem to a Cauchy singular integral equation which is solved numerically in the Laplace transformed domain. The dynamic stress intensity factors (DSIF) are obtained by using the numerical technique of Laplace inversion.     

Two-dimensional thermoelastic contact problem of functionally graded materials involving frictional heating

The two-dimensional thermoelastic sliding frictional contact of functionally graded material (FGM) coated half-plane under the plane strain deformation is investigated in this paper. A rigid punch is sliding over the surface of the FGM coating with a constant velocity. Frictional heating, with its value proportional to contact pressure, friction coefficient and sliding velocity, is generated at the interface between the punch and the FGM coating. The material properties of the coating vary exponentially along the thickness direction. In order to solve the heat conduction equation analytically, the homogeneous multi-layered model is adopted for treating the graded thermal diffusivity coefficient with other thermomechanical properties being kept as the given exponential forms. The transfer matrix method and Fourier integral transform technique are employed to convert the problem into a Cauchy singular integral equation which is then solved numerically to obtain the unknown contact pressure and the in-plane component of the surface stresses. The effects of the gradient index, Peclet number and friction coefficient on the thermoelastic contact characteristics are discussed in detail. Numerical results show that the distribution of the contact stress can be altered and therefore the thermoelastic contact damage can be modified by adjusting the gradient index, Peclet number and friction coefficient.     

功能梯度材料涂层平面裂纹分析

研究粘接于均质基底材料上功能梯度涂层平面裂纹问题. 假设功能梯度材料剪切模量的倒数为坐标的线性函数，而泊松比为常数. 采用Fourier变换和传递矩阵法将该混合边值问题化为奇异积分方程组，通过数值求解获得 应力强度因子. 考察了材料梯度变化形式、结构几何尺寸和材料梯度参数对裂纹应力强度因子的影响，发现 功能梯度材料涂层尺寸、裂纹长度以及材料梯度参数均对应力强度因子有显著影响.     

FRACTURE ANALYSIS OF A FUNCTIONALLY GRADED STRIP UNDER PLANE DEFORMATION

In this paper the plane elasticity problem for a functionally graded strip containing a crack is considered. It is assumed that the reciprocal of the shear modulus is a linear function of the thickness-coordinate, while the Possion's ratio keeps constant. By utilizing the Fourier transformation technique and the transfer matrix method, the mixed boundary problem is reduced to a system of singular integral equations that are solved numerically. The influences of the geometric parameters and the graded parameter on the stress intensity factors and the strain energy release rate are investigated. The numerical results show that the graded parameters, the thickness of the strip and the crack size have significant effects on the stress intensity factors and the strain energy release rate.