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

M. Rhimi, Sami El-Borgi, N. Lajnef

Research output: Contribution to journalArticle

30 Citations (Scopus)

### Abstract

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.

Original language English 787-798 12 Mechanics of Materials 43 12 https://doi.org/10.1016/j.mechmat.2011.08.013 Published - Dec 2011 Yes

### Fingerprint

singular integral equations
traction
Integral equations
Substrates
Newton-Raphson method
integral transformations
punches
half spaces
indentation
Indentation
Elasticity
polynomials
inhomogeneity
elastic properties
Polynomials
profiles

### Keywords

• Axisymmetric double receding contact
• Hankel transform
• Singular integral equations

### ASJC Scopus subject areas

• Materials Science(all)
• Instrumentation
• Mechanics of Materials

### Cite this

In: Mechanics of Materials, Vol. 43, No. 12, 12.2011, p. 787-798.

Research output: Contribution to journalArticle

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AB - 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.

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KW - Singular integral equations

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