A semianalytical and finite-element solution to the unbonded contact between a frictionless layer and an FGM-coated half-plane

2017 ◽  
Vol 24 (2) ◽  
pp. 448-464 ◽  
Author(s):  
Jie Yan ◽  
Changwen Mi ◽  
Zhixin Liu
Keyword(s):  
Integral Equation ◽  
Finite Element ◽  
Singular Integral Equation ◽  
Contact Problem ◽  
Singular Integral ◽  
Material Properties ◽  
Elastic Layer ◽  
Coated Substrate ◽  
Receding Contact ◽  
Contact Size

In this work, we examine the receding contact between a homogeneous elastic layer and a half-plane substrate reinforced by a functionally graded coating. The material properties of the coating are allowed to vary exponentially along its thickness. A distributed traction load applied over a finite segment of the layer surface presses the layer and the coated substrate against each other. It is further assumed that the receding contact between the layer and the coated substrate is frictionless. In the absence of body forces, Fourier integral transforms are used to convert the governing equations and boundary conditions of the plane receding contact problem into a singular integral equation with the contact pressure and contact size as unknowns. Gauss–Chebyshev quadrature is subsequently employed to discretize both the singular integral equation and the force equilibrium condition at the contact interface. An iterative algorithm based on the method of steepest descent has been proposed to numerically solve the system of algebraic equations, which is linear for the contact pressure but nonlinear for the contact size. Extensive case studies are performed with respect to the coating inhomogeneity parameter, geometric parameters, material properties, and the extent of the indentation load. As a result of the indentation, the elastic layer remains in contact with the coated substrate over only a finite interval. Exterior to this region, the layer and the coated substrate lose contact. Nonetheless, the receding contact size is always larger than that of the indentation traction. To validate the theoretical solution, we have also developed a finite-element model to solve the same receding contact problem. Numerical results of finite-element modeling and theoretical development are compared in detail for a number of parametric studies and are found to agree very well with each other.

Frictional contact problem of an anisotropic laterally graded layer loaded by a sliding rigid stamp

2020 ◽  
Vol 234 (10) ◽  
pp. 2024-2041
Author(s):  
Onur Arslan
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Contact Problem ◽  
Computational Methods ◽  
Singular Integral ◽  
Elastic Layer ◽  
Frictional Contact ◽  
Lateral Direction ◽  
Frictional Contact Problem ◽  
Rigid Stamp

This study proposes analytical and computational methods for the solution of the sliding frictional contact problem of an anisotropic laterally graded layer loaded by an arbitrarily shaped rigid stamp. The plane-strain orthotropy prevails in the layer which is bonded to a rigid foundation. Each of four orthotropic stiffness coefficients is exponentially varied through the lateral direction of the elastic layer. The Fourier transformations of the field variables are employed in the formulation. The gradient of a displacement component on the surface is then converted to a singular integral equation of the second kind. The singular integral equation is solved by means of the Gauss–Jacobi quadrature integration techniques, a collocation method, and a recursive integration method for the Cauchy integral considering the flat and triangular stamp profiles. The finite element method solutions of the same contact problems are performed using the augmented Lagrange method which is implemented in virtue of ANSYS design parametric language. An iterative algorithm is additionally utilized for the (incomplete) triangular stamp problem to conveniently reach the solutions for predetermined contact lengths. The convergence and comparative analyses are carried out to elucidate the trustworthiness of the analytical and computational methods proposed. Moreover, the parametric analyses infer that the contact-induced damage risks can be effectively alleviated upon tuning the degree of orthotropy and the lateral heterogeneity of the elastic layer.

On the receding contact between a graded and a homogeneous layer due to a flat-ended indenter

2021 ◽  
pp. 108128652110431
Author(s):  
Rui Cao ◽  
Changwen Mi
Keyword(s):  
Finite Element ◽  
Contact Problem ◽  
Integral Equations ◽  
Singular Integral ◽  
Functional Dependence ◽  
Singular Integral Equations ◽  
Homogeneous Layer ◽  
Cauchy Type ◽  
Receding Contact ◽  
Contact Pressures

This paper solves the frictionless receding contact problem between a graded and a homogeneous elastic layer due to a flat-ended rigid indenter. Although its Poisson’s ratio is kept as a constant, the shear modulus in the graded layer is assumed to exponentially vary along the thickness direction. The primary goal of this study is to investigate the functional dependence of both contact pressures and the extent of receding contact on the mechanical and geometric properties. For verification and validation purposes, both theoretical analysis and finite element modelings are conducted. In the analytical formulation, governing equations and boundary conditions of the double contact problem are converted into dual singular integral equations of Cauchy type with the help of Fourier integral transforms. In view of the drastically different singularity behavior of the stationary and receding contact pressures, Gauss–Chebyshev quadratures and collocations of both the first and the second kinds have to be jointly used to transform the dual singular integral equations into an algebraic system. As the resultant algebraic equations are nonlinear with respect to the extent of receding contact, an iterative algorithm based on the method of steepest descent is further developed. The semianalytical results are extensively verified and validated with those obtained from the graded finite element method, whose implementation details are also given for easy reference. Results from both approaches reveal that the property gradation, indenter width, and thickness ratio all play significant roles in the determination of both contact pressures and the receding contact extent. An appropriate combination of these parameters is able to tailor the double contact properties as desired.

On the contact problem of a moving rigid cylindrical punch sliding over an orthotropic layer bonded to an isotropic half plane

2020 ◽  
Vol 25 (10) ◽  
pp. 1924-1942
Author(s):  
I Çömez ◽  
MA Güler
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Contact Problem ◽  
Half Plane ◽  
Singular Integral ◽  
Integral Transform ◽  
Orthotropic Layer ◽  
Moving Contact ◽  
Contact Width ◽  
Cylindrical Punch

In this study, the frictional moving contact problem for an orthotropic layer bonded to an isotropic half plane under the action of a sliding rigid cylindrical punch is considered. Boundary conditions of the problem include the normal and tangential forces applied to the layer with a cylindrical punch moving on the surface of the layer in the lateral direction at a constant velocity V. It is assumed that the contact area is subjected to the sliding condition where Coulomb’⣙s law is used to relate the tangential traction to the normal traction. Using the Fourier integral transform technique and Galilean transformation, the plane contact problem is reduced to a singular integral equation in which the unknowns are the contact stress and the contact width. The singular integral equation is solved numerically using Gauss–Jacobi integration formulae. Numerical results for the contact widths and the contact stresses are given as a solution.

scholarly journals The Problem of a Finite Strip Compressed Between Two Rough Rigid Stamps

1975 ◽  
Vol 42 (1) ◽  
pp. 81-87 ◽  
Author(s):  
G. D. Gupta
Keyword(s):  
Stress Intensity Factor ◽  
Integral Equation ◽  
Stress Intensity ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Material Properties ◽  
Stress Singularities ◽  
Finite Strip ◽  
Strip Geometry ◽  
Elastostatic Problem

A finite strip compressed between two rough rigid stamps is considered. The elastostatic problem is formulated in terms of a singular integral equation from which the proper stress singularities at the corners are determined. The singular integral equation is solved numerically to determine the stresses along the fixed ends of the strip. The effect of material properties and strip geometry on the stress-intensity factor is presented graphically.

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