Axisymmetric Contact Problem Research Articles

Axisymmetric contact problem for a biphasic cartilage layer with allowance for tangential displacements on the contact surface

A unilateral axisymmetric contact problem for a biphasic cartilage layer indented by a rigid punch is considered. The refined linearized kinetic relationship which takes into account both the radial and tangential displacements of the boundary points of the biphasis cartilage layer is imposed. The obtained analytical solution is valid over long-time periods and can be used for increasing loading conditions. It can be used as it is and as a benchmark for verification of FEM model accuracy.

Axisymmetric contact problem for an elastic half-space and a circular cover plate

The contact interaction problem for a thin circular rigid cover plate and an elastic half-space loaded at infinity by a tensile force directed in parallel to the boundary of the half-space is considered. It is assumed that the cover plate is not resistant to bending deformations. The problem can be reduced to an integral equation of the first kind whose kernel has a logarithmic singularity. The equation is solved approximately by the Multhopp-Kalandia method. The resulting approximate solution is compared with the previously obtained asymptotic solution.

The contact problem for a circular plate with a stress-free end face

An axisymmetric contact problem for a circular elastic plate with a stress-free end face into which two symmetrically arranged punches are imbedded, is considered. The problem is solved using a method developed earlier for bodies of finite sizes, which is based on the generalized orthogonality of homogeneous solutions. The problem reduces to a Fredholm integral equation of the first kind in a function describing the displacement of the surface of the plate outside the punch. These functions are sought in the form of a sum of a Schlömilch series and a power function with a root singularity. The ill-posed infinite system of algebraic equations obtained as a result is regularized by the introduction of a small positive parameter. Since the matrix elements of the system, as well as the contact stresses, are defined by poorly converging numerical and functional series, the technique of summation of the residues of these series is used. The distribution of the contact pressure and the dimensionless imbedding force are found. Examples of the calculation of a plane punch are given.

The contact problem for a two-layer spherical base

Analytical methods for solving problems of the interaction of punches with two-layer bases are described using in the example of the axisymmetric contact problem of the theory of elasticity of the interaction of an absolutely rigid sphere (a punch) with the inner surface of a two-layer spherical base. It is assumed that the outer surface of the spherical base is fixed, that the layers have different elastic constants and are rigidly joined to one anther, and that there are no friction forces in the contact area. Several properties of the integral equation of this problem are investigated, and schemes for solving them using the asymptotic method and the direct collocation method are devised. The asymptotic method can be used to investigate the problem for relatively small layer thicknesses, and the proposed algorithm for solving the problem by the collocation method is applicable for practically any values of the initial parameters. A calculation of the contact stress distribution, the parameters of the contact area, and the relation between the displacement of the punch and the force acting on it is given. The results obtained by these methods are compared, and a comparison with results obtained using Hertz, method is made for the case in which the relative thickness of the layers is large.

Indentation of a round punch into a rough elastic half-space

We consider the axisymmetric contact problem on the indentation of a round punch with plane base into an elastic half-space. Roughness is taken into account in the framework of the well-known model in which the local surface displacements are proportional to some power of the contact pressures at the same point. To study the problem, we use the theory of nonlinear integral equations of Hammerstein type together with the mathematical technique of orthogonal Legendre polynomials and the contraction mapping principle. We obtain an approximate analytic solution of the problem, numerically analyze the results, and reveal typical laws of variation of the main mechanical variables.

Indentation of a smooth axisymmetric punch into a transversely isotropic layer

We consider an axisymmetric contact problem for a transversely isotropic layer on a rigid base. The problem is reduced to a Fredholm integro-differential equation of the first kind for the contact pressure under the punch. The integral equation is solved by the collocation method [1]. Some numerical results are presented.

Axisymmetric contact of a rigid inclusion embedded at the interface of a piezoelectric bimaterial

The axisymmetric contact problem of a rigid inclusion embedded in the piezoelectric bimaterial frictionless interface subjected to simultaneous far-field compression and electric displacement is addressed. With the aid of a robust technique, the coupled governing integral equations of this mixed boundary-value problem are reduced to decoupled Fredholm integral equations with a constraint equation. A useful limiting case for the contact problem of transversely isotropic bimaterials is addressed. The present solution is analytically in agreement with the existing solution for an isotropic bimaterial. Selected numerical results of interest to engineering applications including the radius of separation zone, contact pressure and contact electric displacement are plotted to portray the effects of precompression, piezoelectric coupling and material properties.

Mechanics of axisymmetric wavy surface adhesion: JKR–DMT transition solution

Recent work on the mechanics of detachment of a rigid sphere from an elastic axisymmetric wavy surface in the presence of JKR adhesion has shown that the presence of small-amplitude waviness introduces instabilities into the detachment process which dissipate mechanical energy. These instabilities result in interface toughening and strengthening; both the external work and peak force required for separation of a wavy interface are higher than those for a flat interface. In this paper, we summarize the key dimensionless parameters governing axisymmetric wavy surface adhesion in the JKR regime. We then proceed to derive a solution for the JKR–DMT adhesion transition for the axisymmetric wavy surface contact problem using a Maugis–Dugdale cohesive zone formulation. The phenomenon of interface toughening and strengthening due to the presence of surface waviness is seen to be restricted primarily to the JKR adhesion regime.

The use of flat punch indentation to determine the viscoelastic properties in the time and frequency domains of a soft layer bonded to a rigid substrate

This paper reports a computational study of the indentation of a flat punch into a compressible elastic layer (with Poisson’s ratio varying from 0 to 0.49) bonded to a rigid substrate. Based on the computational results and using Sneddon’s solution [Sneddon IN. The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int J Eng Sci 1965;3:47] and the asymptotic solution [Jaffar MJ. A general solution to the axisymmetric frictional contact problem of a thin bonded elastic layer. Proc Inst Mech Eng C 1997;211:549; Yang FQ. Asymptotic solution to axisymmetric indentation of a compressible elastic thin film. Thin Solid Films 2006;515:2274] as the two limits, a simple expression of the load–depth curve valid for an arbitrary ratio of the indenter radius to the thickness of the layer is obtained. Further, a correlation between indentation load and depth for a rigid flat punch indenting into linearly viscoelastic layers bonded to a rigid substrate is proposed by using the correspondence principle. Several procedures are suggested based on the results reported in this study to determine the viscoelastic properties of the layer in the time or frequency domains. The findings are verified by numerical examples. The results may facilitate the use of depth-sensing indentation tests to characterize the mechanical properties of polymeric films or functional coatings on hard substrates, and some biological materials, e.g. articular cartilage.

The axisymmetric contact problem for an elastic layer loaded with a deformable cover plate

The axisymmetric problem of the contact interaction of an elastic cover plate with an elastic layer, loaded at infinity with a uniform stretching force, directed parallel to the boundaries of the layer, is considered. The cover plate resists stretching but does not resist bending. The contact shearing stress under the cover plate, the displacement of the points of the cover plate and the deformation distortion coefficient of the elastic layer are determined.

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.

Axisymmetric frictionless contact problem of a functionally graded coating with exponentially varying modulus

This paper is concerned with the problem of a functionally graded coated half-space indented by an axisymmetric smooth rigid punch. The shear modulus of the graded coating is assumed to be an exponential function and the Poisson’s ratio is a constant. With the use of Hankel integral transform technique, the axisymmetric frictionless contact problem is reduced to a Cauchy singular integral equation. The contact pressure, contact radius and penetration depth are calculated for various indenters by solving the equations numerically. The results show that these quantities are greatly affected by the gradient of the coating.

Thermoelastic instability in contact problems for rotating solids with heat generation

In this paper we discuss certain cases of thermal instability in axisymmetric contact problems with heat generation and heat exchange. Two solids of revolution pressed against each other are considered. One of the solids rotates about its symmetry axis. A characteristic feature of the problem is that the distance between the centers of gravity of the solids cannot be greater than a certain value

Interaction of a die with a layered elastic foundation

Plane and axisymmetric contact problems for a three-layer elastic half-space are considered. The plane problem is reduced to a singular integral equation of the first kind whose approximate solution is obtained by a modified Multhopp-Kalandiya method of collocation. The axisymmetric problem is reduced to an integral Fredholm equation of the second kind whose approximate solution is obtained by a specially developed method of collocation over the nodes of the Legendre polynomial. An axisymmetric contact problem for an transversely isotropic layer completely adherent to an elastic isotropic half-space is also considered. Examples of calculating the characteristic integral quantities are given.

The Axisymmetric Contact Problem of an Elastic Layer Indented by an Infinite Rigid Punch with a Circular Hole

The Axisymmetric Contact Problem of an Elastic Layer Indented by an Infinite Rigid Punch with a Circular Hole

The solution of the Hertz axisymmetric contact problem

The main terms of the asymptotic form of the solution of the contact problem of the compression without friction of an elastic body and a punch initially in point contact are constructed by the method of matched asymptotic expansions using an improved matching procedure. The condition of unilateral contact is formulated taking account of tangential displacements on the contact surface. An asymptotic solution of the problem for the boundary layer is constructed by the complex potential method. An asymptotic model is constructed, extending the Hertz theory to the case where the surfaces of the punch and elastic body in the vicinity of the contact area are approximated by paraboloids of revolution. The problem of determining the convergence of the contacting bodies from the magnitude of the compressive force is reduced to the problem of calculating the so-called coefficient of local compliance, which is an integral characteristic of the geometry of the elastic body and its fixing conditions.

Contact stresses in rotating bodies with regard for heat generation and convective heat exchange

An axisymmetric contact problem of thermoelasticity for rotating bodies is analyzed with regard for heat generation in the contact region and convective heat exchange between the bodies and the ambient medium. The distribution of stresses in the bodies is determined. Special attention is given to the maximum tangential stresses and the possibility of appearance of tensile stresses.

A finite element analysis and remeshing algorithm for an axisymmetric elasto-plastic contact problem related to indentation measurements

Non-linear quasi-static axisymmetric indentation problem is studied for comparitive computational analysis between elasto-plastic parameters and mechanical behaviour of homogeneous engineering metal materials. Mathematical model, based on deformation theory version of J 2 theory, leads to quasi-static elasto-plastic contact problem with unknown contact domain, depending on the value of a indentation depth. Finite element model with the remeshing algorithm is constructed for simulation of the considered indentation problem. The algorithm permits one to determine the unknown contact zone with high accuracy on each step of indentation. As a result, the indentation curve P( α), corresponding to the deformation curve σ i = σ i( e i), is obtained computationally. It is shown that the monotone indentation curve contains ample information related to elasto-plastic and mechanical properties of a material. In particular, the correlation between the yield stress σ s and the Brinell number H B is estimated.

Weak formulation of axi-symmetric frictionless contact problems with boundary elements: Application to interface cracks

A boundary element formulation of the axi-symmetric elastic problem, including contact conditions for multi-domain problems, is presented in this paper and applied to the analysis of interface cracks. A weak formulation of the equilibrium and compatibility equations has been developed to allow non-conforming discretizations to be used in the contacting boundaries. In order to obtain a high accuracy in the solution (employing continuous linear elements), the singular terms appearing in the axi-symmetric boundary integral equations have been integrated analytically. The numerical analysis of a penny-shaped crack at the interface of two dissimilar materials and of a fibre–matrix debonding crack in the single fibre fragmentation test is presented.

An approximate solution of the axisymmetric contact problem for an elastic sphere

An axisymmetric, fractionally non-linear contact problem for an elastic sphere with a priori unknown boundary of the contactarea is considered. An integral equation for determining the density of the contact pressures is constructed taking account of the shear displacements of the boundary points of the elastic body. An approximate solution, which refines the equations of Hertz' theory, is constructed in the case of a small contact area.