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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.

The complex physics behind electroadhesion-based tactile displays poses an enormous modeling challenge since not only the fingerpad structure with multiple non-linear layers, but also the roughness at the microscopic scale play a decisive role. To investigate tactile perception, a potential model should also offer the possibility to extract mechanical stimuli at the sites of the relevant mechanoreceptors. In this paper, we present a two-scale approach that involves a finite element model (FEM) at the macroscopic scale and a simple bearing area model (BAM) that accounts for the measured roughness on the papillary ridges. Both separate scales couple in an iterative way using the concept of an equivalent air gap. We show that the electroadhesion-induced changes in friction and contact area predicted by the proposed model are in qualitative agreement with recent experimental studies. In a simple example, we demonstrate that the model can readily be extended by a neural dynamics model to investigate the tactile perception of electroadhesion.

We present a formulation of the boundary element method (BEM) for simulating the tangential contact with an elastic half-space coated with an elastic layer with different elastic properties. We use the fast Fourier-transform-based formulation of BEM, while the fundamental solution is determined directly in the Fourier space. Numerical tests are validated by comparison with available asymptotic analytical solutions for a very thin and a very thick layer, as well as with FEM calculations for layers with finite thickness.

In the present work, a recently extended version of the method of dimensionality reduction (MDR) for layered elastic media is applied for the first time using a semi-analytical approach. It is based on a priori knowledge of the cylindrical flat punch solution which is determined numerically using the boundary element method (BEM). We consider arbitrary indenters of revolution producing a circular area of contact with bonded and unbonded layers of arbitrary thickness. The proposed method reduces the contact solution to the numerically efficient evaluation of simple one-dimensional integrals. We further show that the solution of JKR-adhesive contacts with layers and contacts with linear-viscoelastic layers is straightforward using the well-known MDR formalisms. A specific focus has been devoted to study the thickness effect in different application examples. Comparisons with the literature and finite element simulations show very good agreement with the proposed method.

Two numerical methods are proposed to improve accuracy of the numerical calculation of fretting wear in the framework of the Method of Dimensionality Reduction (MDR). Due to the singularity of the transformation equations, instabilities appear at the border between the stick and slip regions after many transformations from the one-dimensional to the three-dimensional contact and back. In these two methods, the transformation equations are reformulated to weaken the singularity of the integrals and a stable simulation of fretting wear is realized even with the wear models which go beyond the classical Archard law. With an example of dual-oscillation, we show the change in the worn profile of a parabolic indenter as well as the stress distribution on the contacting surface during the oscillating cycles under the Archard’s law of wear and Coulomb’s law of friction.

Electrovibration is one of the key technologies in surface haptics. By inducing controlled electrostatic forces, the friction within a sliding contact between the human finger and a capacitive screen is modulated, which in turn gives effective tactile feedback to the user. Such powerful haptic displays can be built into mobile phones, tablets, navigation devices, games consoles and many other devices of consumer electronics. However, due to the layered structure and complex material of human skin, the underlying contact mechanical processes have not yet been fully understood. This work provides new continuum-based approaches to macroscopic modeling of the electro-adhesive frictional contact. A solution of pure normal contact between a human finger and a rigid, smooth plane under electroadhesion is derived by applying Shull's compliance method in the extended regime of large deformations. Based on these results and assuming pressure-controlled friction, a model for the sliding electro-adhesive contact is developed, which adequately predicts the friction force and coefficient of friction over the whole range of relevant voltages and applied normal forces. The experimentally observed area reduction caused by the tangential force is incorporated in a more empirical than profound contact mechanical way. This effect is studied with the help of a two-dimensional finite element model of the fingertip, assuming non-linear elastic material for the skin tissue. Although the simulations are restricted to non-adhesive tangential contacts, they show a significant reduction of the contact area, which is caused by large deformations of the non-linear elastic material around the distal phalanx. This result indicates that adhesion is only of secondary importance for the area reduction.

The stress state in the volume of contacting bodies may essentially influence the material behavior. For evaluating various modes of inelastic behavior and/or failure, such as plastic deformation, crack initiation, and propagation or fatigue, the complete stress tensor beneath the contact interface may be of importance. For many geotechnical and biomechanical applications, the hydrostatic pressure gradient beneath the contact is of interest as well. However, most theories for normal and tangential contact provide only few stress components in the contact surface. In the present paper, we show that the full stress state in the half-space can be easily found for axisymmetric bodies. We provide expressions in form of one-dimensional integrals for all components of the stress tensor and the hydrostatic pressure gradient inside the half-space. In terms of numerical complexity, the proposed method can be advantageous to other elaborate methods.

Due to its high relevance for designing ball joints in mechanical engineering and (artificial) hip joints in biomechanics, the almost conformal elastic contact between a sphere and a spherical cup represents an important contact problem of current research. As no closed-form analytical solution to the problem has been found to date, full computational methods such as the finite element method are needed for analysis. However, they often require incredibly long, unacceptable calculation times, making parameter studies hardly practicable. For this reason, approximate analytical and semi-analytical models are applied, capable of predicting quantities of interest with sufficient accuracy. In the present work, a very simple model based on a radially directed Winkler foundation is presented, which provides (approximate) closed-form analytical solutions for both the pressure distribution and the dependencies between macroscopic contact quantities such as normal force and indentation depth. To ensure an optimal mapping of a specific contact problem, only the foundation modulus must be defined in a suitable way. As an example, the proposed model has been successfully adapted to adequately simulate the frictionless normal contact for hard-on-soft hip implants. For this purpose, the foundation modulus was approximated with the aid of a finite element analysis instead of adopting it from already well-established models, as the latter produce clearly erroneous results for large liner thicknesses and large Poisson’s ratios. By a comparison with extensive parameter studies of finite element simulations, it is demonstrated that the proposed model provides acceptable results for all commonly used hard-on-soft hip implants. On this basis, the influence of geometrical changes of the femoral head and the acetabular cup on the maximum pressure as well as the half-contact angle is discussed, and consequences on the wear behavior are deduced.

In the present work, we analyze a device for measuring the dynamic response of rubber in sliding contact. The principle of the device is to measure excited oscillations of a mechanical system with a sliding contact between a rubber roller and a rigid surface. Depending on the contact properties, varying oscillation amplitudes are measured. The goal is to determine dynamic response properties of rubber from oscillating tests. For this sake, an analytical model is introduced in which the contact problem and system dynamics are considered in detail. Analytical and numerical results obtained for this model are compared with some experimental data and discussed both on a qualitative and quantitative level.

Wear of a cylindrical punch composed by two different materials alternatively distributed in annular forms is studied with the method of dimensionality reduction (MDR). The changes in surface topography and pressure distribution during the wear process is obtained and validated by the boundary element method (BEM). The pressure in each annular ring approaches a constant in a stationary state where the surface topography does not change any more. Furthermore, in an easier manner, using direct integration, the limiting profile in a steady wear state is theoretically calculated, as well as the root mean square (RMS) of its surface gradient, which is closely related to the coefficient of friction between this kind of surface and an elastomer. The dependence on the wear coefficients and the width of the annular areas of two phases is obtained.