Double Contact Problem Research Articles

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

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 frictional receding contact between a graded layer and an orthotropic substrate indented by a rigid flat-ended stamp

This paper solves the frictional receding contact between a functionally graded elastic layer and an orthotropic substrate under the indentation of a rigid flat-ended stamp. Both normal and tangential indentation forces, satisfying Coulomb's friction law, are considered. The shear modulus of the graded layer is assumed to vary exponentially along the transverse dimension, whereas the Poisson's ratio is fixed as a constant. Under the plane strain sliding condition, the governing equations and boundary conditions of the double contact problem are converted into dual singular integral equations with the help of Fourier integral transforms. Gauss-Jacobi quadratures of different kinds and an iterative algorithm are subsequently developed to determine the stationary contact pressure, the receding contact pressure as well as the asymmetric receding contact boundaries. In order to verify and validate the accuracy of semianalytical solutions, the graded finite element modelings are also implemented through defining material properties of the graded layer at the Gaussian integral points of all elements and communicating with the major solver in terms of a user-defined interface. Parametric studies primarily focus on the effects of frictional coefficients, stiffness orthotropy and gradation on the stationary and receding contact properties. The influence of indenter size and thickness ratio between the graded layer and the orthotropic substrate is also investigated. Reasonable agreements of contact pressures and receding contact lengths are identified between the solutions obtained from two independent approaches. Extensive parametric studies provide an effective means of tailoring the double contact properties as desired by the appropriate selection of governing parameters.

Thermoelastic receding contact problem of a layer resting on a half plane with frictional heat generation

The plane steady-state thermoelastic receding double contact problem between a rigid cylindrical punch and a homogenous layer lying on a half plane is investigated in this paper. The frictional rigid insulated punch slides over the layer with a constant velocity and heat flux is generated due to the friction. The speed of the punch is taken so small that the inertia effects may be neglected. The general stress and displacement expressions for the thermoelastic contact problem are derived using the theory of thermoelasticity and Fourier integral transform technique. Applying the boundary conditions, the contact problem is reduced to Cauchy-type singular integral equations of the second kind in which the contact stresses and contact widths are unknown. The singular integral equations are solved numerically using the Gauss-Jacobi integration formula. The effect of thermoelastic parameters, friction coefficient, external load, punch radius on the contact stress, and contact widths are discussed in detail. This is the first study to investigate the thermal effect on the receding contact problem.

On the Receding Contact Between a Homogeneous Elastic Layer and a Half-plane Substrate Coated with Functionally Graded Materials

Mechanical contact threatens the integrity of engineering structures. In particular, receding contact is one of the primary causes that are responsible for the delamination of multilayered elastic structures. This paper aims to analyze the receding contact between a homogeneous elastic layer and a half-plane reinforced by a functionally graded coating. The multilayered structure is indented by a rigid stamp of convex profile. Governing equations and mixed boundary conditions of the double contact problem are converted into a pair of singular integral equations by Fourier integral transforms. The dual integral equations are numerically solved by Gauss–Chebyshev quadrature for the contact pressure and contact length at both interfaces of contact. Taking a circular punch as a case study, the developed algorithm is first validated against classical models available in the literature. Extensive parametric studies are subsequently performed to illustrate the effects of indentation load, geometry and material properties of individual components. Numerical results suggest the possibility of optimizing multilayered structures by the introduction of functionally graded materials (FGMs) as coatings or transitional layers.

A new approach for closed-form analytical solution of two-dimensional symmetric double contacts and the comparison with finite element method

In this paper, a new procedure is developed for the solution of a general two-dimensional uncoupled symmetric double contact problem with smooth contact zones in which the indenter geometry is described by a piecewise biquadratic function. This procedure gives an approximate closed-form solution for any smooth indenter profile. In order to evaluate the accuracy of this approach, it is applied to the symmetric indentation of a flat surface by two rigidly interconnected parabolic indenters and results are compared with the exact unclosed-form solution. Moreover, this procedure is applied to the symmetric indentation of a flat surface by two rigidly interconnected cylinders to compare the results with the finite element solution obtained by the finite element method software, ABAQUS. The results showed that in comparison with the finite element method, this procedure is a fast and highly accurate method with low complexity that makes feasible the possibility of determining approximate closed-form solution for a wide range of indenter geometries with a concavity between two symmetric contact zones; hence it can be useful in practical issues.

On the receding contact between an inhomogeneously coated elastic layer and a homogeneous half-plane

The general double contact problem of an inhomogeneously coated elastic layer indented against a homogeneous half-plane by a rigid punch is investigated. The problem is solved under the assumptions that the contact at both interfaces is frictionless, the three materials possess different shear modulus, and the shear modulus of the functionally graded coating varies exponentially. By the standard method of Fourier integral transforms, the problem is reduced to a system of singular integral equations, which are subsequently transformed into algebraic ones by Gauss–Chebyshev integration formulas. Massive numerical experiments are performed to quantify the influence of stiffness distribution in coating, material and geometric properties of the compound structure, and indentation load on the contact pressure and contact length at both surfaces of contact. The results suggest that in the case of a hard and thick coating large pressure reduction can be achieved for the receding contact at the layer/substrate interface.

Double contact analysis of multilayered elastic structures involving functionally graded materials

This paper analyzes the frictionless double contact problem of a two-layer laminate pressed against a homogeneous half-plane substrate by a rigid punch. The laminate is composed of a homogeneous elastic strip and a functionally graded layer, perfectly bonded along their interface. The mechanical properties of the graded layer are modeled by an exponentially varying shear modulus and constant Poisson’s ratio. Both the governing equations and the boundary conditions of the double contact problem are converted into a pair of singular integral equations by Fourier integral transforms, which are numerically integrated by Chebyshev–Gauss quadrature. The contact pressure and the contact size at both the advancing and the receding contact interface are eventually obtained by an iterative algorithm, developed from the method of steepest descent. Extensive parametric studies suggest that it is possible to control contact stress and contact size by introducing functionally graded materials into multilayered elastic structures.

The axisymmetric stress analysis of double contact problem for functionally graded materials layer with arbitrary graded materials properties

In this paper, the axisymmetric double contact problems for a functionally graded layer indented by a rigid cylindrical indenter and a rigid spherical indenter are considered. A linear multi-layered model is applied to model the functionally graded layer whose shear modulus varies as an arbitrary function. The Hankel integral transform technique and the transfer matrix method are applied to convert the double contact problem to a system of coupled singular integral equations which are solved by the numerical method. The numerical results show that the stiffness ratio and the gradient index have great effect on the contact region and the contact pressure.

Two-dimensional symmetric double contacts of elastically similar materials

The two-dimensional contact problem for an elastic body indenting an elastically similar half plane resulting in double contacts is important for various applications. In this paper, a generic quasi-static two-dimensional symmetric double contact problem with nonsingular end points between two elastically similar half planes, under the constant normal and oscillatory tangential loading, is analyzed. The classical singular integral equations approach is utilized to extract the pressure and shear functions in the contact zones; subsequently boundary conditions at end points are applied and a new side condition is derived and titled “the consistency condition” for symmetric double contacts. This condition is necessary for determining the extent of the contact and stick zones. Next, this analytical approach is applied to the symmetric indentation of a flat surface by two rigidly interconnected wedge-shaped punches.

The axisymmetric double contact problem for a frictionless elastic layer indented by an elastic cylinder

This paper is concerned with the smooth receding contact between an elastic layer and a half space when the layer is compressed by a frictionless semi-infinite elastic cylinder. Upon loading, the contact along the layer-subspace interface shrinks to a circular area, radius of which is unknown. The analysis leads to a system of singular integral equations of the second kind. The integral equations are solved numerically and the contact pressures, extent of contact and the stress intensity factor round the edge of the cylinder are calculated for various material combinations.

The axisymmetric double contact problem for a frictionless elastic layer

The general axisymmetric double contact problem for an elastic layer pressed against a half space by an elastic stamp is considered. The problem is solved under the assumptions that the three materials have different elastic properties, the contact along the interfaces is frictionless and only compressive normal tractions can be transmitted across the interfaces, and, in the case of the elastic stamp, the local radius of curvature of the stamp is large compared to the stamp-layer contact radius. The problem is reduced to a system of singular integral equations in which the contact pressures are the unknown functions. The solution is obtained and extensive numerical results are given for three stamp geometries, namely, rigid and elastic spherical stamps, and a flat-ended rigid cylindrical stamp. The results show that in the case of a flat-ended rigid cylindrical stamp the radius b of the contact area between the layer and the subspace is independent of the magnitude P of the total transmitted load and in all other cases b will depend on P.