Axisymmetric Contact Research Articles

Axisymmetric contact of two different power-law graded elastic bodies and an integral equation with two Weber–Schafheitlin kernels

Summary This article analyzes the axisymmetric contact problem of two elastic inhomogeneous bodies whose Young moduli are power functions of depth and the exponents are not necessarily the same. It is shown that the model problem is equivalent to an integral equation with respect to the pressure distribution whose kernel is a linear combination of two Weber–Schafheitlin integrals. The pressure is expanded in terms of the Jacobi polynomials, and the expansion coefficients are recovered by solving an infinite system of linear algebraic equations of the second kind. The coefficients of the system are represented through Mellin convolution integrals and computed explicitly. The Hertzian and Johnson–Kendall–Robertson adhesive models are employed to determine the contact radius, the displacement of distant points of the contacting bodies, the pressure distribution and the elastic normal displacement of surface points outside the contact circular zone. The effects of the exponents of the Young moduli and the surface energy density on the pressure distribution and the displacements are numerically analyzed.

Adhesion of graded elastic materials: A full self-consistent model and its application

A full self-consistent model (FSCM) of the axisymmetric adhesive contact between a rigid punch with an arbitrary surface shape and a power-law graded elastic half-space is developed. The self-consistent equation between the surface gap and the surface interaction (e.g., the Lennard–Jones force law) involves a nonlinear singular integral, posing a great challenge to numerical calculations. By applying the properties of Gauss’s hypergeometric function, the integral singularity is eliminated in the numerical calculation through Riemann–Stieltjes integral. Case studies for power-law punch profiles are performed and the self-consistent equation can be expressed in a dimensionless form with three dimensionless parameters, namely a shape index, a gradient exponent, and a new generalized Tabor number. The FSCM results are obtained by solving the self-consistent equation through the surface central gap control method and Newton–Raphson iterative method. For large generalized Tabor numbers, the force–displacement curves are ‘S-shaped’ and condense to the extended JKR limit in the high-load branch. As the generalized Tabor number decreases, a continuous transition from the extended JKR model to the Bradley model for the adhesion of power-law graded materials is obtained. It is found that the pull-off force of a graded material usually depends on the three dimensionless parameters, but for some cases of the shape index, it is not sensitive to the gradient exponent when the generalized Tabor number is fixed. Asymptotic solutions are derived to predict the unstable jump points, which coincide well with the FSCM predictions. The FSCM is applied to validate the extended Maugis–Dugdale (M–D) model of graded materials and it is found that the accuracy of the original M–D-n-k model using the maximum strength condition to determine the cohesive stress is limited. By introducing the rigid-limit-consistency condition of the pull-off force to determine the cohesive stress, the M–D-n-k model is improved and its predictions show good consistency with the FSCM results.

A numerical method based on Polonsky-Keer's conjugate gradient iterative scheme for solving axisymmetric contact of layered half-space

Axisymmetric contact analysis of layered half-space is of considerable interest in many tribological applications including protective hard coating, metal-matrix composites, etc. Polonsky-Keer's iterative scheme based on conjugate gradient method (CGM) is powerful for a general three-dimensional contact problem using rectangular meshes, but would inevitably lead to excessive computational burden and inaccurate boundary discretization for axisymmetric contact. In this paper, Polonsky-Keer's CG-based iterative scheme is adapted for axisymmetric indentation of a layered elastic half-space by taking advantage of annular ring discretization, where the elementary solutions for the surface deflection and subsurface stresses can be determined with assistance of the Hankel transforms. The present method can not only effectively circumvent the discretization error, but is also versatile and flexible to solve the contact between the layered materials and an arbitrary axisymmetric indenter. Parametric studies are provided to validate the present solutions, and the comparisons of the average relative errors for different types of elements are conducted to demonstrate the excellent convergence and accuracy of the proposed numerical scheme.

Axisymmetric adhesive contact of multi-layer couple-stress elastic structures involving graded nanostructured materials

Graded nanostructured (GNS) materials has attracted widespread attention in the recent years. The gradually changed microstructures and mechanical properties from the surface to the interior layer in graded nanostructured materials can improve the mechanical properties of the surface. The motivation of this research was to explore the effect of size effect, adhesion and gradient on the mechanical properties in contact between GNS coating and spherical indenter. The multi-layer couple-stress elastic structures is used to simulate the GNS coating with arbitrarily varying material properties. Based on the Maugis-Dugdale theory, the axisymmeric adhesive contact problem for multi-layer couple-stress elastic coated substrate system indented by spherical punch is converted to singular integral equations. Numerically calculation are conducted to obtain the influence of the characteristic material length, adhesion parameter and gradient index on the indentation, surface stress, surface displacement, total attraction force. The present results should be helpful for the understanding of the micro contact mechanics of GNS coating.

Mathematical Modeling of the Contact Interaction of the Fuel Element Section, Including up to 100 Pellets, Taking into Account Creep

An algorithm for solving axisymmetric contact problems of thermoelasticity taking into account creep processes is considered. To account for the contact interaction of bodies, the mortar method was used, and the modified Jacobi method was used to solve the resulting system of linear equations. For the numerical solution of the problem simulating the creep process, time discretization based on the implicit Euler method was applied, and the Newton method was used to linearize the resulting system of equations. An algorithm with automatic step selection based on obtaining an estimate of the local error of the method is proposed. The results of applying the proposed algorithm to a demonstration problem simulating thermomechanical processes in a section of a fuel element comprising from 1 to 100 fuel pellets are presented.

An innovative and efficient solution for axisymmetric contact problem between structure and half-space

In this paper, with displacement increment as variables, the scaled boundary finite element equations of the axisymmetric plate and half-space are respectively derived based on the principle of virtual work. The dual vector is introduced to solve the boundary stiffness matrix. Linear complementary contact conditions are formulated as B-differentiable equations. With discrete, static assumptions of small displacement, B-differentiable equations system was formed by combining the contact conditions and the stiffness matrix of the plate-foundation. The equations can be iteratively solved by B-differentiable Newton method. In numerical experiments, the contact stress, slip and contact status of each contact node pair are compared with the results of the augmented Lagrangian contact algorithm based on FEM. When adopting different load types, thickness-to-radius ratio, plate-foundation stiffness ratio and friction coefficient, the axisymmetric contact problems between plate and half-space are discretized by 120 one-dimensional first-order elements. The applicability, accuracy and effectiveness of B-SBFEM for axisymmetric contact problems are verified by numerical examples.

Influence of Wear Profile Geometry on Critical Plane Fatigue Crack Initiation Criteria in Plane and Axisymmetric Elastic Fretting Contacts

The subsurface elastic stress fields in plane and axisymmetric contacts with friction under oscillating tangential loading are calculated via a very robust, high-precision method, which operates with appropriate superpositions of analytic solutions for the respective Hertzian contact problems. Based on the stress fields, two critical plane fatigue crack initiation criteria—the Smith-Watson-Topper (SWT) parameter and the Findley parameter—are evaluated for three types of contact profile geometries: (unworn) parabolic contact, the partial slip limiting wear profile of an initially parabolic contact, and truncated parabolic contact. Appropriate scaling laws are introduced to formulate a general solution in terms of non-dimensional variables. The crack initiation criteria are determined in the full subsurface loading plane. It is found that the truncated profile—which may originate from sliding wear—has practically the same local distribution of crack initiation criteria as the unworn profile, despite the (weak) stress singularity at the edge of the flat face. The partial slip limiting wear profile, on the other hand, exhibits a strong edge stress singularity at the boundary of the permanent stick zone, the crack initiation criteria are drastically increased (and theoretically infinite). Also, while for the unworn and truncated profiles high values of the crack initiation criteria are extremely localized around “hotspots” at the surface, for the partial slip limiting wear profile they reach much deeper into the subsurface material. This offers a new explanation for the dominance of fatigue failure in the partial slip regime of fretting. The differences between plane and axisymmetric cases are generally small. The SWT parameter is generally more localized than the Findley parameter.

The axisymmetric contact in couple-stress elasticity taking into account adhesion

In this paper, the spherical indenter contact problem for the couple-stress elasticity by considering the adhesion is investigated. The Maugis–Dugdale adhesive theory is considered to describe the adhesion effect. The mixed boundary problem is solved by using the Hankel transform technical and singular integral equation method. The influences of the adhesion parameter, the characteristic material length parameter and Poisson’s ratio on the pull-off force, the relation between the applied force and indentation depth, the contact stresses and the normal displacement are discussed. The results predicted the importance of size-effect in the adhesive contact. The present work provides a fundamental basis for probing the mechanical properties in the microstructure materials.

FFT-BASED IMPLEMENTATION OF THE MDR TRANSFORMATIONS FOR HOMOGENEOUS AND POWER-LAW GRADED MATERIALS

It is shown how the Abel transform solution to the general axisymmetric normal contact problem for homogeneous and power-law graded elastic materials, which is paramount for the solution of different classes of tribological problems with the help of the method of dimensionality reduction (MDR), can be written in terms of explicit convolutions. These can be very efficiently evaluated with the 1D Fast Fourier Transform (FFT), which reduces the numerical complexity of the transformations from the order of N2 for the standard matrix-vector-multiplication (MVM) algorithm to the order of N. Convergence and performance of the proposed method are studied in detail. As an illustrative example a fretting wear simulation based on the new implementation is shown, the results of which are compared to the standard MVM implementation.

Adhesive Boundary Element Method Using Virtual Crack Closure Technique

In this study, a new adhesive contact model is built upon a boundary element method (BEM) model developed by Pohrt and Popov (2015). The strain energy release rate (SERR) on the edge of the bonding interface is evaluated using Virtual Crack Closure Technique (VCCT) which is shown to have better accuracy and weaker mesh-size dependency than the closed-form SERR formula derived by Pohrt and Popov. A composite delamination criterion is proposed for crack nucleation and propagation. Numerical results predicted by the present model are in good agreement with the analytical solutions of two classic problems, namely, the axisymmetric parabolic contact and the sinusoidal waviness contact in the plane strain condition. The model of Pohrt and Popov can achieve a similar accuracy for the axisymmetric parabolic contact where the mesh grid is non-conforming to the crack front. Once the conforming mesh grid is used, the accuracy of their model is significantly deteriorated, especially at high work of adhesion and high mesh density. In both BEM models, however, the crack nucleation is found to be mesh-dependent which may be solved by introducing an upper limit for the tensile normal traction.

ABOUT THE ANALYTICAL SOLUTION OF THE EQUATION OF IMPACT FORCE OF TWO ELASTIC BODIES

A nonlinear differential equation of the force of direct central impact of elastic bodies of revolution, which have a singular point on the boundary contact surface, where its curvature is infinite, is compiled. To determine the coefficients of the equation and the order of its power nonlinearity, the well-known solution of the axisymmetric contact problem of the theory of elasticity, constructed by I. Ya. Shtaermann, is used. In the formulation of the dynamic problem, the classical assumptions of the theory of quasi-static impact proposed by H. Hertz were also used. The constituted equation of impact force is reduced to the Bernoulli equation and its closed analytical solution is constructed, which is expressed in terms of the Ateb-sine. Analytical time dependences of the impact force and the convergence of the centers of mass of elastic bodies are obtained. Compact formulas have been derived for calculating the maxima of these quantities, as well as the durations of the process of compression and impact of bodies. Compact approximations of Ateb-sine by elementary functions are proposed. Thanks to these approximations, it was possible to obtain a fairly simple analytical sweep in time of a fast-flowing mechanical process. Traditionally, in other works such a scan was obtained by numerical solution of the corresponding integral equations that determine the force of an impact. Examples of calculations are given in which the influence of various factors on the main characteristics of a body impact with a small initial velocity is investigated. The limitation on the collision rate is due to the elastic formulation of the problem, where the possibility of plastic deformations is excluded. As a result of this formulation, the need to determine the rate of recovery rate has disappeared, for it is equal to one. Comparison of numerical results is carried out, to which the obtained analytical solutions and the numerical integration of the impact force equation on a computer lead. Small divergences of the results confirmed the accuracy of the derived calculation formulas. Numerical results relate to the impact of a steel body on a fixed rubber half-space, the analogue of which is observed in practice when falling pieces of mineral raw materials on the rolls of a vibration classifier lined with rubber.

Finite element analysis of indentation contact of double piezoelectric spheres

As a result of the limited research works on the contact between piezoelectric materials, this research seeks to delve into this area, by using the finite element method. A two-dimensional finite element model of nanoindentation response of piezoelectric materials is established via this method. The axisymmetric contact behavior of the two transversely isotropic piezoelectric spheres under the condition of friction-free sliding is analyzed, and the relationship between indentation displacement, electric potential and contact force is obtained, which are linear and quadratic respectively. The results show that the variation trend of the stress field and electric field at the contact boundary presents a non-monotonic relationship with an increase in the distance from the indenter to the center of the loading axis. In addition, the results of finite element numerical simulation are in good agreement with the numerical verification results. For semi-infinite materials, the model can be used for accurate contact analysis, which guides verifying the extraction technology of contact characteristics of finite piezoelectric materials. The results obtained in this paper are helpful to understand the microscopic contact mechanics of piezoelectric materials.

A Simple Semi-Analytic Contact Mechanical Model for Tangential and Torsional Fretting Wear of Axisymmetric Contacts

Fretting wear of axisymmetric contacts is considered within the framework of the Hertz–Mindlin approximation and the Archard law for the linear wear. If the characteristic time scale for the wear is much larger than the duration of a single fretting oscillation, the profile change due to wear during one fretting cycle can be neglected for the contact problem as a zero-order approximation. This allows to give an exact contact solution during each fretting cycle, depending on the current worn profile, and thus for the explicit statement of an ordinary integro-differential equation system for the time-evolution of the fretting profile, which can be easily solved numerically. The proposed method gives the same results as a known, contact mechanically more rigorous simulation procedure that also operates within the framework of the Hertz–Mindlin approximation, but works significantly faster than the latter one. Tangential and torsional fretting wear are considered in detail. A comparison of the numerical prediction for the evolution of the worn profile in partial slip torsional fretting of a rubber ball on abrasive paper shows good agreement with experimental results from the literature.

A Simple Semi-Analytic Contact Mechanical Model for Tangential and Torsional Fretting Wear of Axisymmetric Contacts

Fretting wear of axisymmetric contacts is considered within the framework of the Hertz–Mindlin approximation and the Archard law for the linear wear. If the characteristic time scale for the wear is much larger than the duration of a single fretting oscillation, the profile change due to wear during one fretting cycle can be neglected for the contact problem as a zero-order approximation. This allows to give an exact contact solution during each fretting cycle, depending on the current worn profile, and thus for the explicit statement of an ordinary integro-differential equation system for the time-evolution of the fretting profile, which can be easily solved numerically. The proposed method gives the same results as a known, contact mechanically more rigorous simulation procedure that also operates within the framework of the Hertz–Mindlin approximation, but works significantly faster than the latter one. Tangential and torsional fretting wear are considered in detail. A comparison of the numerical prediction for the evolution of the worn profile in partial slip torsional fretting of a rubber ball on abrasive paper shows good agreement with experimental results from the literature.

Explicit transformation between non-adhesive and adhesive contact problems by means of the classical Johnson-Kendall-Roberts formalism.

The classic Johnson-Kendall-Roberts (JKR) contact theory was developed for frictionless adhesive contact between two isotropic elastic spheres. The advantage of the classical JKR formalism is the use of the principle of superposition of solutions to non-adhesive axisymmetric contact problems. In the recent years, the JKR formalism has been extended to other cases, including problems of contact between an arbitrary-shaped blunt axisymmetric indenter and a linear elastic half-space obeying rotational symmetry of its elastic properties. Here the most general form of the JKR formalism using the minimal number of a priori conditions is studied. The corresponding condition of energy balance is developed. For the axisymmetric case and a convex indenter, the condition is reduced to a set of expressions allowing explicit transformation of force-displacement curves from non-adhesive to corresponding adhesive cases. The implementation of the developed theory is demonstrated by presentation of a two-term asymptotic adhesive solution of the contact between a thin elastic layer and a rigid punch of arbitrary axisymmetric shape. Some aspects of numerical implementation of the theory by means of Finite-Element Method are also discussed. This article is part of a discussion meeting issue 'A cracking approach to inventing new tough materials: fracture stranger than friction'.

Численное решение задачи контактного взаимодействия элементов твэла с помощью mortar-метода и метода декомпозиции области

The paper presents algorithms for solving axisymmetric contact interaction problems for several thermoelastic bodies using unmatched meshes. We employed the finite element method to obtain numerical solutions to problems of thermal conductivity and the theory of elasticity. We took contact interaction into account by applying the mortar method and the method of domain decomposition. The mortar method requires solving an ill-conditioned system of linear algebraic equations with a zero block at the main diagonal. To solve it numerically, we used a modified method of successive over-relaxation (MSSOR), which makes it possible to reduce solving the system of equations for all contacting bodies to sequentially solving systems of equations for each body separately. We showcase our algorithm results by solving an example problem simulating thermomechanical processes in a nuclear fuel element. We analyse the features of the stress-strain state in the structure and compare the results obtained using the mortar method and the domain decomposition method. The computational domain in the problem considered consisted of 10 nuclear fuel pellets and a cladding section. The analysis results showed that the quantitative stress-strain state properties in a system of bodies obtained by the two methods are quite close to each other. This confirms the fact that these algorithms may be correctly applied to solving similar problems

Indentation method accounting for thickness effect of viscoelastic layer

AbstractIndentation tests have been widely used to evaluate the mechanical properties of industrial and biological materials in many studies. In this study, we obtained an analytical solution for the axisymmetric contact problem for a viscoelastic layer bonded to a rigid substrate and indented by a flat‐ended cylindrical punch using the elastic–viscoelastic correspondence principle. The analytical solution was obtained from a method using an infinite system of simultaneous equations instead of a conventional method using the second kind of Fredholm equation. Relaxation indentation tests were conducted for silicon rubber, polyurethane gel (a soft material used for elastography), and articular cartilage from a bovine femoral head. The articular cartilage was tested as obtained and then tested again after treatment by collagenase. The viscoelastic models in this study were a three‐element model (standard linear solid model) and a five‐element model (generalized Maxwell model), and the viscoelastic parameters for the models were determined by fitting the analytical solution to results for applied load as it varied over time during relaxation indentation tests. The results suggested that the five‐element model was more appropriate than the three‐element model for reproducing the relaxation indentation behavior of silicon rubber, polyurethane gel, and articular cartilage. Furthermore, the viscoelastic parameters for treated articular cartilage clearly decreased compared with those for unaltered articular cartilage. These results suggest that the method for evaluating the time‐dependent mechanical properties of articular cartilage based on analytical solutions of relaxation indentation tests could be a useful tool for evaluating the state of degenerated cartilage.

Influence of running-in in dry and wet conditions on the interfacial adhesion of tire tread rubber

To study the running-in effects on the adhesion properties of tire tread compounds, an approach is proposed based on theoretical analysis of experimental load-distance curves. The approach uses the Maugis model of adhesion in an axisymmetric elastic contact. Classical AFM tests with a ball of 450 nm glued to the probe are performed for two different surfaces of the tire tread rubber after running-in in dry and wet friction regimes and for the original rubber surface. It is shown that running-in in wet conditions decreases the surface energy of rubber. Also, running-in in both wet and dry conditions leads to softening of a surface layer of rubber.

Axisymmetric thermoelastic contact of an FGM-coated half-space under a rotating punch

This paper investigates the axisymmetric thermoelastic contact of a functionally graded material (FGM) coated half-space. A rigid insulated punch rotates on the surface of the FGM coating with a constant angular velocity. The frictional heating is generated within the contact region. The thermoelastic properties of the coating vary exponentially along the thickness direction. By using the Hankel integral transform, the problem is reduced to Cauchy singular integral equations, which are solved numerically to obtain the normal contact stress, radial stress and surface temperature. The effects of the friction coefficient, angular velocity, gradient index, coating thickness and punch geometry on the surface thermoelastic contact characteristics are investigated and discussed in detail.

Modeling of the interaction of cylindrical bodies with complex surface properties

The axisymmetric contact problem of interaction between cylindrical bodies with complex surface properties (elastic properties or shapes) is described. For this problem, its mathematical model is constructed. It is mixed integral equation. An analytical solution for the problem of finding contact stresses under the bush is obtained. It allow one to make efficient calculations even when the coating nonuniformity and bush internal profile described by rapidly changing functions.