Rigid Flat Punch Research Articles

On two contact problems for a half-plane with static friction

Abstract Exact solutions of two contact problems are constructed for a half-plane, when a finite number of absolutely rigid flat punches with friction are pressed into the half-plane under the influence of different concentrated loads, and when a periodic system of absolutely rigid flat punches with friction is pressed into the half-plane. It is assumed that friction in contact zones is associated with normal contact pressure by the generalized law of dry friction, in which the coefficient of friction depends on the coordinates of the contacting points of the contacting bodies and is directly proportional to the difference between the coordinates of these points and the coordinates of the middle points of the contact zones. The governing equations of the posed problems were obtained in the form of singular integral equations with variable coefficients relative to the contact pressure and their closed solutions were constructed.

Contact problem for an interface crack under action of a moving punch

Nowadays coatings are used in many applications to reduce contact friction, reduce energy losses, improve wear and pitting resistance, and so on in various machine parts. In some cases, coating applications are limited due to coating delamination. The latter is caused by a crack growth due to cyclic loading along a coating interface with the substrate. This paper is focused on modeling and analyzing the behavior of interface cracks between a coating and a substrate. A flat rigid punch subjected to normal and tangential loading is slowly moving along the solid surface. The solid is represented by an elastic half-plane covered by two elastic layers bonded to each other and to the substrate. The elastic materials of the coatings and the substrate are homogeneous but different. The thickness of the layer adjacent to the substrate is considered to be (significantly) smaller than the one above it. This thin layer plays the role of the interface between the other one and the substrate. The interface layer has a crack parallel to its surface and located in the middle of it. The crack surfaces are free from friction and can partially open/close or completely open/close. Some numerical results for the crack stress intensity factors and other solution parameters are presented for cracks small and comparable in size with the punch contact region.

Analysis of stress and deformation of an exponentially graded viscoelastic coated half plane under indentation by a rigid flat punch indenter tip

This paper solves the dynamic contact problem when a rigid flat punch indents into an exponentially graded (FG) viscoelastic coated homogeneous half-plane. A harmonic vertical force is applied to the FG coating, and the solution is obtained for the stress and displacement for both the FG viscoelastic coating and the half-plane using the Helmholtz functions and the Fourier integral transform technique. By applying specific boundary conditions, the contact mechanics problem is converted into a singular integral equation of the first kind. This equation is then solved numerically using the Gauss-Chebyshev integration formulas. The analysis provides detailed insights into how various parameters—such as external excitation frequency, loss factor ratio, Young’s modulus ratio, density ratio, Poisson’s ratio, indentation load, and punch length—affect the dynamic contact stress and dynamic in-plane stress.

Dynamic indentation of viscoelastic orthotropic layer supported by a Winkler–Pasternak foundation

AbstractIn this study, the two-dimensional dynamic contact problem between a rigid flat punch and a viscoelastic orthotropic layer is investigated. The motivation of the study is to provide a better understanding of the vertical vibration of the two-parameter Winkler–Pasternak foundation, which has not yet been investigated. For the contact problem, the mixed boundary conditions on the top and bottom surfaces are transformed into linear equations using the Fourier transform technique and Helmholtz functions. Based on the Gauss–Chebyshev integration formula, the singular integral equation is obtained and solved numerically. As a result of the solutions, the effects of various parameters on the contact stresses are analyzed and examples are given. It was found that the Winkler foundation modulus does not affect the dynamic contact stress, while the Pasternak foundation modulus significantly affects the contact stress.

Viscoelastic amplification of the pull-off stress in the detachment of a rigid flat punch from an adhesive soft viscoelastic layer

The problem of the detachment of a flat indenter from a plane adhesive viscoelastic strip of thickness “b” is studied. For any given retraction speed, three different detachment regimes are found: (i) for very small “b” the detachment stress is constant and equal to the theoretical strength of the interface, (ii) for intermediate values of “b” the detachment stress decays approximately as b−1/2, (iii) for thick layers a constant detachment stress is obtained corresponding to case the punch is detaching from a halfplane. By using the boundary element method a comprehensive numerical study is performed which assumes a linear viscoelastic material with a single relaxation time and a Lennard-Jones force-separation law. Pull-off stress is found to consistently and monotonically increase with unloading rate, but to be almost insensitive to the history of the contact. Due to viscoelasticity, unloading at high enough retraction velocity may allow punches of macroscopic size to reach the theoretical strength of the interface. Finally, a corrective term in Greenwood or Persson theories considering finite size effects is proposed with good agreement between theoretical and numerical results.

Elastic Stress Field beneath a Sticking Circular Contact under Tangential Load

Based on a potential theoretical approach, the subsurface stress field is calculated for an elastic half-space which is subject to normal and uniaxial tangential surface tractions that—in the case of elastic decoupling—correspond to rigid normal and tangential translations of a circular surface domain. The stress fields are obtained explicitly and in closed form as the imaginary parts of compact complex-valued expressions. The stress state in the surface and on the central axis are considered in detail. As, within specific approximations that have been discussed at length in the literature, any tangential contact problem with friction can be understood as a certain incremental series of such rigid translations, the solutions presented here can serve as the basis of very fast superposition algorithms for the analysis of subsurface stress fields in general tangential contact problems with friction. This idea is demonstrated by means of the frictional tangential contact between an elastic half-space and a rigid cylindrical flat punch with rounded corners.

A General Approximate Solution for the Slightly Non-Axisymmetric Normal Contact Problem of Layered and Graded Elastic Materials

We present a general approximate analytical solution for the normal contact of layered and functionally graded elastic materials for almost axisymmetric contact profiles. The solution only requires knowledge of the corresponding contact solution for indentation using a rigid cylindrical flat punch. It is based on the generalizations of Barber’s maximum normal force principle and Fabrikant’s approximation for the pressure distribution under an arbitrary flat punch in an inhomogeneous case. Executing an asymptotic procedure suggested recently for almost axisymmetric contacts of homogeneous elastic media results in a simple approximate solution to the inhomogeneous problem. The contact of elliptical paraboloids and indentation using a rigid pyramid with a square planform are considered in detail. For these problems, we compare our results to rigorous numerical solutions for a general (bonded or unbonded) single elastic layer based on the boundary element method. All comparisons show the quality and applicability of the suggested approximate solution. Based on our results, any compact axisymmetric or almost axisymmetric contact problem of layered or functionally graded elastic materials can be reduced asymptotically to the problem of indenting the material using a rigid cylindrical flat punch. The procedure can be used for different problems in tribology, e.g., within the framework of indentation testing or as a tool for the analysis of local features on a rough surface.

On the imperfect interface of a functionally graded thermoelectric layered structure

This paper investigates the frictionless contact problem of a functionally graded thermoelectric (FGTE) layered half-plane under the action of a rigid flat punch. The punch is an electrical and thermal conductor. The electro-thermo-elastic properties of the FGTE materials vary exponentially with thickness. Especially, the imperfect interface conditions are taken into account. The problem is reduced to a pair of Cauchy singular integral equations using the Fourier integral transform technique, which are numerically solved to yield the surface normal electric current density, the surface normal energy flux, and the surface normal contact pressure. The asymptotic analysis of the kernel function is supplied. The intensity factors are provided to characterize the singularity at the punch ends. The contact behavior of the FGTE coating-TE substrate systems is significantly influenced by the imperfect interface condition and the material property gradient according to numerical results. The design of thermoelectric coating structures may benefit from the theoretical research done here.

Detachment of a Rigid Flat Punch from a Viscoelastic Material

We show that the detachment of a flat punch from a viscoelastic substrate has a relatively simple behaviour, framed between the Kendall’s elastic solution at the relaxed modulus and at the instantaneous modulus, and the cohesive strength limit. We find hardly any dependence of the pull-off force on the details of the loading process, including maximum indentation at preload and loading rate, resulting much simpler than the case of a spherical punch. Pull-off force peaks at the highest speeds of unloading, when energy dissipation is negligible, which seems to be in contrast with what suggested by the theories originated by de Gennes of viscoelastic semi-infinite crack propagation which associated enhanced work of adhesion to dissipation.Graphical abstract

Transient response of a finite thickness strip with thermoelectric effects loaded by an electrode

Thermoelectric (TE) materials exhibiting energy converting and storing capabilities connected with an electrode are increasingly applied in engineering sciences. The contact interfaces of the TE devices are prone to gradual degradation or even failure under thermal shock loading. The investigation of the dynamic contact problem of the TE materials loaded by an electrode is significant for designing reliable TE devices. The transient frictionless contact model of the TE materials with a finite thickness indented by a rigid flat punch is proposed and investigated in this paper. The considered thermal-mechanical coupling problem is converted into the corresponding singular integral equations by the Laplace transform and the Fourier transform techniques, which are solved numerically by a collocation method. Then, the numerical inverse Laplace transform is employed to obtain the time-domain solutions. The time-dependent and thickness-dependent surface normal energy flux and surface normal contact stress, which are significant for the transient contact behavior, are computed and analyzed. The magnitudes of the transient normal energy flux and normal contact stress differ from that in the steady-state. Specifically, the normal contact stress in the transient state is higher than that in the steady-state.

Analytical solution of an orthotropic elasticity for a frictional rigid flat punch on a half plane with an oblique edge crack using a mapping function

A solution of an orthotropic elastic plane problem is derived for a frictional rigid flat punch on a half plane with an oblique edge crack. A mapping function is used for arbitrary configurations. No stress function using a mapping function seems to have been derived. The problem is solved as a Riemann–Hilbert problem. The final exact stress functions are represented by an irrational mapping function as a closed form. Stress components are represented by one complex variable. Arbitrarily shaped half plane problems can be solved by exchanging the mapping function. Therefore, once the analytical solution has been derived, calculating the stress components is simpler than for an isotropic problem. It is easier to use an irrational mapping function than to form a rational mapping function for an isotropic problem. As an example, the stress distributions are shown for a frictional coefficient and an oblique edge crack angle. Four elastic constants are expressed by two independent parameters.

Computational contact mechanics for a medium consisting of functionally graded material coating and orthotropic substrate

This paper presents a semi-analytical method to investigate the frictionless contact mechanics between a functionally graded material (FGM) coating and an orthotropic substrate when the system is indented by a rigid flat punch. From the bottom, the orthotropic substrate is completely bonded to the rigid foundation. The body force of the orthotropic substrate is ignored in the solution, while the body force of the FGM coating is considered. An exponential function is used to define the smooth variation of the shear modulus and density of the FGM coating, and the variation of Poisson’s ratio is assumed to be negligible. The partial differential equation system for the FGM coating and the orthotropic substrate is solved analytically through Fourier transformations. After applying boundary and interface continuity conditions to the mixed boundary value problem, the contact problem is reduced to a singular integral equation. The Gauss–Chebyshev integration method is then used to convert the singular integral equation into a system of linear equations, which are solved using an appropriate iterative algorithm to calculate the contact stress under the rigid flat punch. The parametric analyses presented here demonstrate the effects of normalized punch length, material inhomogeneity, dimensionless press force, and orthotropic material type on contact stresses at interfaces, critical load factor, and initial separation distance between FGM coating and orthotropic substrate. The developed solution procedures are verified through the comparisons made to the results available in the literature. The solution methodology and numerical results presented in this paper can provide some useful guidelines for improving the design of multibody indentation systems using FGMs and anisotropic materials.

The dynamic contact of a viscoelastic coated half-plane under a rigid flat punch

This paper investigates the dynamic contact response of a viscoelastic coated half-plane under a forced rigid punch. This punch, acting on the coating surface, is subjected to a concentrated vertical harmonic force. The coating and half-plane are assumed as viscoelastic materials which can be described by the linear hysteretic damping model. Using the Fourier integral transform technique and the asymptotic method, the dynamic contact problem is reduced to a Cauchy singular integral equation. By using appropriate numerical method, the dynamic contact stress and vertical impedance for the flat punch are calculated. It is found that the dynamic contact behaviors are not only affected by the external excitation frequency but also dependent on the material properties of the coated half-plane.

Thermoelastic contact problem of a magneto-electro-elastic layer indented by a rigid insulating punch

In this study, the plane thermoelastic frictional contact problem between a magneto-electro-elastic (MEE) layer and a rigid cylindrical or flat punch is discussed. Punches are assumed to be perfect electromagnetic and thermal insulators; they slide over the layer with a small constant speed, and due to the friction, heat flux occurs. The general stress, displacement, and electromagnetic expressions for the contact problem are derived using the theory of thermoelasticity. Applying the boundary conditions, the contact problem is reduced to a Cauchy-type singular integral equation, which is then solved numerically using the Gauss-Jacobi integration formula.

Indentation of an elastic half-space by the cylindrical flat punch with a rounded edge under non-monotonic loading

The paper presents a novel approach to solving the contact problem for interaction with friction between a rigid cylindrical flat punch with rounded corners and a linearly elastic half-space. Increasing and decreasing normal loads act on the punch sequentially. Coulomb’s law is used to account for friction. The contact problem is reduced to nonlinear operator equations that must be solved at each step of the discrete loading process. The numerical method is applied. This method includes regularization of the equation, discretization of the regularized equations and the usage of an iterative process to obtain the approximate solution. The calculations were performed for different values of the relative radius of the flat area of the punch base. Distributions of contact traction and configuration of the stick and slip zones are studied at each monotonic stage of the loading process.

Implementation of Dahl\u2019s dynamic friction model to contact mechanics of elastic solids

This study presents an analytical method for the solution of dynamic frictional contact problem between a rigid punch and an isotropic elastic solid. Rigid flat punch moves over the isotropic elastic solid at a constant subsonic speed and friction force develops based on Dahl’s friction law instead of Coulomb’s dry friction law. Dahl’s dynamic friction model is adopted since this model is one of the well-known dynamic friction models in the literature. According to this model, friction force depends only on a displacement rather than the speed of the punch since viscous effects are ignored. Influences of the parameters describing Dahl’s friction model on contact stress at slow and high speed sliding cases are examined. Analytical solution is conducted by means of Galilean and Fourier transformations. Friction force is computed numerically by the use of 4th order Runge–Kutta method for various displacements of the punch. Formulation for the contact problem is reduced to a singular integral equation and normal stress over the surface of elastic half-plane is determined. Obtained contact stresses are compared with those generated through finite element method and results display a high degree of accuracy. The influences of direction of motion, Coulomb’s coefficient of friction, pre-sliding displacement, asperity stiffness and shape factor of hysteresis loop upon contact stresses and stress intensity factors are revealed.

Frictional moving contact problem of a magneto- electro- elastic half plane

In this study, a frictional moving contact problem between a homogeneous magneto-electro-elastic half plane and a conducting rigid flat or cylindrical punch is considered. The punch moves on the half plane in the lateral direction at a subsonic constant velocity V. Applying the Fourier transform and Galilean transformation, the mixed boundary value problem is reduced to Cauchy integral equations in which the unknowns are the contact stress, the contact width, the electric displacement and the magnetic induction on the surface of the half plane. These singular integral equations are solved numerically using the Gauss-Jacobi and Gauss-Chebyshev integration formulas. Numerical results for the contact width, the contact stress, the in-plane stress, the electric displacement, and the magnetic induction on the surface of the half plane are obtained and plotted. This work is the first study that investigates the effect of friction in the moving contact problem of the magneto-electro-elastic half plane.

Oblique surface cracking and crack closure in an orthotropic medium under contact loading

This article presents analytical methods for analysis of an inclined surface crack in an orthotropic medium subjected to contact loading. Governing partial differential equations are derived in accordance with plane theory of orthotropic elasticity. Contact and crack problems are formulated separately utilizing Fourier transformation techniques. The contact problem consists of a rigid flat punch pressing upon an orthotropic half-plane, and is reduced to a singular integral equation of the second kind. The surface crack is assumed to be tilted at an arbitrary angle with respect to the normal of the half-plane surface. Three different crack configurations are considered, which are completely closed, partially-closed, and fully-open cracks. Frictional contact between the crack faces is implemented in the cases of completely closed, and partially-closed cracks. A single singular integral equation is derived for the completely closed crack, whereas each of the problems of partially- and fully-open cracks requires derivation of a set of two singular integral equations. The contact problem and each of the crack problems are solved numerically by means of the expansion-collocation technique. A flowchart is developed and implemented to detect the type of crack closure configuration under a given contact loading. The parametric analyses carried out illustrate the influences of factors such as crack inclination angle, crack face and contact surface friction coefficients, material orthotropy, and relative punch location on crack face contact stresses, degree of crack closure, and mode I and II stress intensity factors.

Contact Stress Analysis for a Functionally Graded Half-Plane at Subsonic, Transonic and Supersonic Sliding Speeds

This paper investigates frictional dynamic contact mechanics of a functionally graded half-plane subjected to moving contact by a rigid flat punch possessing subsonic, transonic and supersonic speeds. The shear modulus along the half-plane is expressed by an exponential function, and the Poisson’s ratio is assumed constant along the graded half-plane. Governing partial differential equations are derived based on the planar theory of elastodynamics. Boundary conditions are applied, and displacement fields in the graded half-plane are determined analytically. Formulation for the contact problem is reduced to a singular integral equation involving Cauchy singularity and a Fredholm kernel. Singular integral equation is solved numerically utilizing a suitable collocation technique. Contact stresses and normalized punch stress intensity factors are calculated for prescribed subsonic, transonic and supersonic speeds of the moving punch. It is expected that the results obtained by this study will help to understand the contact behavior and the surface failure mechanisms of functionally graded materials, especially at transonic and supersonic sliding speeds.

Decohesion of a rigid flat punch from an elastic layer of finite thickness

Interfacial bond strength is a crucial mechanical property of solid adhesive joints that plays an important role in a wide range of applications. A common way of quantifying mechanical strength of an adhered interface is to measure the apparent adhesion strength by finding the maximum pulling force in a tensile bond test and dividing it by the area. Despite the simplicity of this method, there is growing evidence suggesting that the measured quantity is not necessarily an intrinsic attribute of the interface but depends on thickness of the adhesive film and size of the interface. In this work, the critical pull-off force of a flat rigid punch adhered to an elastic film of finite thickness is studied via asymptotic analysis as well as finite element (FE) simulations. The decohesion behavior is found to be governed by two dimensionless parameters η and φ: the former represents the deformation of the interface relative to the interaction distance of interfacial forces, while the latter denotes a correction factor due to finite thickness of the film. As η changes from 0 to infinity, the interface decohesion would take place from uniform detachment (DMT-like) to crack-like propagation (JKR-like). The transition behavior can be well described by an empirical solution, which suggests a new rational procedure to consistently characterize the intrinsic adhesion properties of an adhered interface from tensile bond tests. Finally, the general empirical solution and the adhesion characterization procedure are directly validated by a series of experiments via detaching a stainless-steel flat punch from adhesive Polydimethylsiloxane (PDMS) films. This work not only offers a clear and explicit solution to capture the transition behavior of interface decohesion but also provides a guideline for modulating adhesion performance via geometry rather than chemistry of the adhesives.