On The Singular Integral Solution To Mechanical Properties Of Receding Contact Involving Functionally Graded Materials

Receding contact is a kind of contact problem with smaller final contact size with increasing contact force.Receding contact has not received enough attention for a long time due to its rarity.However,experimental results have revealed that receding contact is a key reason for coating peeling and shedding which widely exist in civil engineering structures and mechanical parts.Therefore,a suitable mechanical model is necessary to be established for a multilayered elastic structure.Theoretical,experimental and computational tools should also be promptly developed.Functionally graded materials(FGMs)are classified as a new kind of materials whose properties and functions are graded along one or more spatial dimensions.FGMs are an ideal transition media between two materials with different properties.They are also commonly used as coatings to obtain some specific properties,such as thermal barrier coating of turbine blade.On this basis,the control mechanism and influence of elasticity modulus gradation on the mechanical properties of multilayered structure are explored.After the homogeneous elastic medium subjected to a receding contact is coated with a graded layer,the deterministic role of geometric and mechanical parameters of the graded layer and other system factors on the receding contact stress and length are quantitatively analyzed.Be constrained by the difficulty of the theoretical calculation and the complexity of the problem,the shear modulus of the functionally graded layer is assumed to vary exponentially along the thickness direction.The effect of gravity and friction is also assumed to be negligible.By using the classical plane theory of elasticity,the integral transform theory and the numerical integration method of singular integral equation,the basic equations of elasticity theory and the mixed boundary conditions in three typical plane models all containing a functionally graded layer,a homogenous layer and a homogenous half-plane are reduced to a Cauchy singular integral equation(or a pair of Cauchy singular integral equations).They are subsequently discretized into a set of nonlinear algebraic equations with respect to the receding contact stress and length by the use of Gauss type numerical quadratures.Finally,an efficient numerical iterative algorithm is developed to solve the mechanical properties of the receding contact with high accuracy.The following problems are consecutively investigated:(1)By using Fourier integral transform method,explicit expressions of displacements and stresses in the functionally graded layer,homogenous layer and homogenous half-plane are obtained in the transformed Fourier space.These expressions are applicable to the plane contact problem of any multilayered structures consisting of a functionally graded layer,a homogenous layer and/or a homogenous half-plane.(2)With the above expressions of displacement and stressed,the receding contact stress and length between a homogenous layer with a functionally graded layer on top and a homogenous half-plane are solved.In this process,two kinds of integral equations are solved numerically.One is improper integral and the other is Cauchy type singular integral equation(or a pair of Cauchy singular integral equations).In addition,the effects of indentation force,structural and mechanical parameters on the mechanical properties of receding contact are quantitatively calculated by dimensional analysis and extensive numerical examples.(3)With the expressions of displacement and stressed in Fourier space and the results on the above model,the receding contact problem between a homogenous layer with a functionally graded layer on bottom and a homogenous half-plane is solved.The effects of various factors on the receding contact in the model are analyzed under the application of both surface forces and rigid indenters.(4)The solution method is further improved,and the receding contact problem between a homogenous layer and a homogenous half-plane coated with a functionally graded layer is solved.The receding contact stress and length in all three-layered models are compared and analyzed.Based on the theoretical derivation,the numerical solution of singular integral equations and extensive parametric analysis,the main research results are summarized as follow.(1)When the external forces are equal,the peak value of receding contact stress between the homogenous layer and homogenous half-plane with coating is positively correlated with the length of receding contact.In the rest models,the peak value of receding contact stress is inversely correlated with the length of receding contact.(2)The magnitude of indentation force or the size of the indenter have more influence on the distribution of receding contact stress,while the distribution pattern of the indentation force and the shape of symmetrical convex indenter have less impact.(3)Reducing the stiffness of the homogeneous layer or increasing the stiffness of the homogeneous half-plane can effectively reduce the receding contact damage.(4)When the functionally graded layer is located between the external loads and the receding contact interface,increasing the thickness of functionally graded coating can enhance the performance to resist the damage of receding contact stress.The relationship between the thickness of the functionally graded layer and the receding contact length also depends on the relative hardness between the coating and the homogenous half-plane.The thick and soft coating leads to a longer receding contact whereas the thick and hard coating leads to a shorter receding contact.(5)The effect of functionally graded layer on the receding contact length lacks a monotonical trend,which depends on the position of functionally graded layer in the three-layered model.If the functionally graded layer is resting on the top of the homogenous layer and the upper contact length is fixed,the receding contact length is positively related to the hardness of functionally graded layer.Otherwise there is no simple relation between them.If the functionally graded layer is attached to the bottom of the homogenous layer,the receding contact length firstly decreases and then increases with the increase of the hardness of the functionally graded layer.If the functionally graded layer is coated on the half-plane boundary,the receding contact length is negatively related to the hardness of the functionally graded layer.For the first time,this dissertation proposed three-layered receding contact structures containing a functionally graded layer.The regulating effect of the indentation force,structural and mechanical parameter on the receding contact stress and length is systematically analyzed.The high precision computation on improper integrals and a Cauchy type singular integral(or a pair of Cauchy singular integral equations)was achieved by establishing and solving the mechanic model and analyzing a large number of parameters.The research results have important reference significance on the analysis of interlayer stripping and receding behavior in multilayered civil engineering and mechanical structures.They are also valuable references to the receding contact and fracture mechanics of FGMs.