Three-dimensional Contact Problems Research Articles

Explicit frictional stick–slip dynamics of elastic contact problem incorporating the LuGre model

Dynamical contact problem with friction is of continuous concern and have been investigated over the years. The effective and accurate description of the development of the friction force is the key to this problem. However, the stick–slip phenomenon, as typical dynamical behaviour, has rarely been considered when developing contact algorithms for frictional contact problems of elastic bodies. The conventionally used Coulomb’s model, with differentiating the static and kinetic coefficient of friction, can lead to inaccurate results due to errors generated when switch between the stick or slip states. In this work, a three-dimensional mortar-based frictional contact algorithm is proposed in an explicit scheme. The proposed algorithm incorporates the LuGre model to account for dynamical frictional behaviours such as stick–slip. The effectiveness and accuracy of the proposed algorithm are validated through several simplified two-dimensional cases. The sticking time ratio and the evolution of the global coefficient of friction with time of a three-dimensional contact problem with a randomly rough contact interface are investigated as an application.

Peridynamic contact models for fracture analysis based on the micro-beam bond

Contact problems are ubiquitous in engineering systems. Establishing contact models to predict the crack propagation and fracture is significant. Therefore, the present study proposes two peridynamic contact models for two-dimensional (2D) and three-dimensional (3D) contact problems based on the micro-beam bond and the Hertz theory, using the relative position of particles and the contact micromoduli to calculate the contact forces. Firstly, the strain is generated on the contact surface when two bodies come into contact. Simultaneously, the contact stiffness is obtained by the Hertz theory, and the contact forces are calculated according to the relationship between the strain of contact surface and the contact stiffness. Next, based on the peridynamic theory, the contact horizon and contact micromoduli are defined for the calculation of contact forces. Finally, the contact micromoduli for 2D and 3D contact problems are derived by equating the contact forces obtained from both theories. Additionally, this study defines the correction coefficients to reduce the calculation error of the proposed models and investigates the influence of different contact horizons on the calculation of contact forces. To validate the effectiveness of the proposed contact models, three numerical examples are compared with the results from the finite element method or experiments. It is demonstrated that the proposed contact models accurately calculate the contact forces and predict the crack propagation.

A generalization of B-differentiable equations method for elastoplastic contact problems with dual mortar discretization

In this study, a novel method based on the B-differentiable equations (BDEs) is proposed to solve the three-dimensional (3D) static and dynamic elastoplastic contact problems with friction. The contact conditions are formulated as the BDEs system, which can precisely satisfy Coulomb’s law of friction. Based on the mortar segment-to-segment contact model, the contact constraints are enforced by dual Lagrange multipliers for nonconforming meshes of the contact boundary. The non-smooth and nonlinear system equations composed of equilibrium and contact equations are simultaneously solved by the B-differentiable damping Newton method (BDNM), which has global convergence property. In each iteration, firstly, some contact displacements and contact forces, called substituted contact variables, are expressed by others according to the different contact states and contact conditions. The equilibrium equation is then solved, where the stiffness matrix is updated by substituting the contact variables. At last, the substituted contact variables will be restored. In the proposed method, the contact flexibility matrix is no longer needed, which saves the time for calculating the contact flexibility matrix due to changes in the stiffness matrix resulting from the update of stresses in the iteration step. By comparing with the results obtained by finite element commercial software ANSYS, the accuracy of the proposed algorithm is verified, and the property of convergence and efficiency are demonstrated for the elastoplastic contact problems by several numerical examples.

Three-dimensional frictional contact within the framework of couple stress elasticity

This work investigates a three-dimensional size-dependent frictional contact problem of a rigid spherical punch sliding on an elastic half-space. The size-dependent material behaviors are governed by the couple stress theory, and the corresponding frequency response functions, which include two additional length-related elastic parameters, are obtained. Based on the conjugate gradient method and the discrete convolution-fast Fourier transform algorithm, a semi-analytical model is established to predict the frictional size-dependent contact behaviors and then the detailed parameter analysis is performed. The transverse effects of the couple stress, which are induced due to the considerations of the two additional elastic parameters in couple stress theory, are investigated in the contact problems for the first time. And the dependences of the size-dependent contact behavior on the coupling effects of the friction and the two additional elastic parameters are presented.

FFT-Based Numerical Method for Nonlinear Elastic Contact

In theoretical research pertaining to sealing, a contact model must be used to obtain the leakage channel. However, for elastoplastic contact, current numerical methods require a long calculation time. Hyperelastic contact is typically simplified to a linear elastic contact problem, which must be improved in terms of calculation accuracy. Based on the fast Fourier transform, a numerical method suitable for elastoplastic and hyperelastic frictionless contact that can be used for solving two-dimensional and three-dimensional (3D) contact problems is proposed herein. The nonlinear elastic contact problem is converted into a linear elastic contact problem considering residual deformation (or the equivalent residual deformation). Results from numerical simulations for elastic, elastoplastic, and hyperelastic contact between a hemisphere and a rigid plane are compared with those obtained using the finite element method to verify the accuracy of the numerical method. Compared with the existing elastoplastic contact numerical methods, the proposed method achieves a higher calculation efficiency while ensuring a certain calculation accuracy (i.e., the pressure error does not exceed 15%, whereas the calculation time does not exceed 10 min in a 64 × 64 grid). For hyperelastic contact, the proposed method reduces the dependence of the approximation result on the load, as in a linear elastic approximation. Finally, using the sealing application as an example, the contact and leakage rates between complicated 3D rough surfaces are calculated. Despite a certain error, the simplified numerical method yields a better approximation result than the linear elastic contact approximation. Additionally, the result can be used as fast solutions in engineering applications.

Solving three-dimensional contact problems for foundation design in green building

Design of foundations on an elastic base is carried out using the solution of three-dimensional problems of contact interaction. Improving the accuracy of engineering calculations is necessary to ensure economic efficiency and increase energy savings in green building. The problems of indentation of punches with a flat base bounded by doubly connected close to polygonal contact areas are researched in the present work. Small parameter method is used to obtain explicit analytical expressions for the contact pressure distribution and the punch displacement dependence in a simplified form, which is convenient for engineering practice. The found load-displacement dependence satisfies the known inequalities that are valid for an arbitrary contact domain. Also a numerical-analytical method is in consideration. It uses the simple layer potential expansion and successive approximations for the problems accounting roughness of the elastic half-space. Roughness coefficient is considered as a parameter of regularization of the integral equation for the smooth contact problem. The results of both methods coincide with sufficient accuracy.

A new approach for solving three-dimensional elasto-plastic wheel-rail normal contact problems

A new approach for solving three-dimensional elasto-plastic wheel-rail normal contact problems

Convergence and accuracy problems of the conventional TCA model – Critical analysis and novel solution for crowned spur gears

In this work the quasi-static model of the three-dimensional geometrical non-conjugate contact problem for two [Formula: see text] surfaces is studied. The set of contact equations is formulated by using a new parameterisation that enables to reduce the conventional system of five nonlinear equations with five unknown position and contact parameters to just two nonlinear equations with two changeable parameters. The novel model is computationally efficient and demonstrates increased accuracy and stability of the numerical solution, compared to the conventional model described by Litvin, which suffers from convergence problems and requires a high computational effort. The new model is implemented to spur gear with crowned tooth surfaces to parametrically estimate the susceptibility to diverse misalignments of the contact pressure, transmission error and path of contact.

A study of the radiation thermal boundary conditions influence in three-dimensional thermomechanical contact problems

Any body, by the simple fact of having a certain temperature on its surface, emits heat in the form of electromagnetic radiation. The heat flux emitted is proportional to the fourth power of the temperature of the solid, increasing the inherent non-linearity of the thermomechanical problems. This energy transfer, for systems at high temperatures, has greater relevance than the possible exchange of heat between bodies in contact by conduction or convection processes with a surrounding fluid in the interstitial areas. For this reason, this work presents a robust formulation to study three-dimensional thermomechanical contact problems where radiation boundary and interstitial conditions are simulated. The Boundary Element Method is used to compute the thermomechanical influence coefficients, and a double iterative procedure is used to ensure the achievement of the thermomechanical contact conditions. The final non-linear thermomechanical system of equations is solved using a Newton-Raphson procedure. The proposed formulation is validated with a comparison with some benchmark problems found in the literature and afterward applied to solve different three-dimensional thermomechanical contact problems where the thermomechanical variables are greatly affected by radiation boundary and interstitial conditions.

Strong form mesh-free hp-adaptive solution of linear elasticity problem

We present an algorithm for hp-adaptive collocation-based mesh-free numerical analysis of partial differential equations. Our solution procedure follows a well-established iterative solve–estimate–mark–refine paradigm. The solve phase relies on the Radial Basis Function-generated Finite Differences (RBF-FD) using point clouds generated by advancing front node positioning algorithm that supports variable node density. In the estimate phase, we introduce an Implicit-Explicit (IMEX) error indicator, which assumes that the error relates to the difference between the implicitly obtained solution (from the solve phase) and a local explicit re-evaluation of the PDE at hand using a higher order approximation. Based on the IMEX error indicator, the modified Texas Three Step marking strategy is used to mark the computational nodes for h-, p- or hp-(de-)refinement. Finally, in the refine phase, nodes are repositioned and the order of the method is locally redefined using the variable order of the augmenting monomials according to the instructions from the mark phase. The performance of the introduced hp-adaptive method is first investigated on a two-dimensional Peak problem and further applied to two- and three-dimensional contact problems. We show that the proposed IMEX error indicator adequately captures the global behaviour of the error in all cases considered and that the proposed hp-adaptive solution procedure significantly outperforms the non-adaptive approach. The proposed hp-adaptive method stands for another important step towards a fully autonomous numerical method capable of solving complex problems in realistic geometries without the need for user intervention.

Contact analysis and performance evaluation of ring type traveling wave ultrasonic motors based on a surface contact model

A surface contact model of ring type traveling wave ultrasonic motors is proposed in this paper in order to describe the circular three-dimensional contact and friction problem between the stator and rotor accurately. Differing from previous contact models of traveling wave ultrasonic motors, the flexibility of the rotor, which results in the irregular shape of contact area and inhomogeneous distribution of contact pressure along radial direction, is taken into account in the proposed model. The normal contact between the stator and rotor is analyzed with finite element method and the tangential force transmission on the contact surface is investigated with analytical method. The contact pressures and the tangential stresses on the contact surface are obtained. Mechanical performances of the motor are also studied based upon the proposed model. Finally, torque-speed curves of a prototype motor are measured and compared with the calculated results of the proposed model, and good agreements are obtained. In addition, the proposed surface contact model is compared with previous line contact models. It is found that the calculated results of the surface contact model are closer to measured results than those of line contact models. The comparison results verify the accuracy of the proposed surface contact model.

A virtual element method for 3D contact problems with non-conforming meshes

The Virtual Element Method is ideal for problems with non-conforming meshes since nodes can be inserted at the virtual element boundary to obtain a conforming mesh structure, without changing the ansatz of the element. The contact conditions can be employed on different enforcing strategies. This feature can be explored also for three-dimensional contact problems. In this work, we introduce a new framework, including a remeshing strategy, which leads to conforming meshes. Various numerical examples in 3D will be shown, including standard cases like the contact patch test and Hertzian contact.

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.

A singularity analysis in Keer’s elastic indentation problem

A non-axisymmetric three-dimensional frictionless contact problem for an elastic half-space and a cylindrical indenter with the face in a wedge form is considered under the assumption of a circular region of contact. Explicit formulas are derived for the critical indenter displacement and contact force that are needed to ensure full contact over the entire contact region. It is shown that the contact pressure beneath the indenter maintains a logarithmic singularity, which is produced by the wedge edge and is similar to the singularity of the contact pressure in the corresponding two-dimensional contact problem for a wedge rigid indenter.

A Numerical Model for Solving Three-Dimensional Rolling Contact Problems with Elastic Coating Layers

In this work, a numerical model is proposed for three-dimensional rolling contact problems with one or two elastic layers, and the tangential contact solution is emphasized. Previous works on this topic have mostly been two-dimensional, in which only longitudinal creepage has been considered. With the three-dimensional model presented in this work, all possible creepages, such as the longitudinal, lateral and spin creepages are taken into account. In order to improve the calculation efficiency, the conjugate gradient method and the FFT technique are employed. The influence coefficients for displacement and stress are obtained from the corresponding frequency response functions. The numerical results are validated against existing results and good agreement can be found. The effects of the different layers’ thicknesses and elastic moduli under different creepage combinations on the traction distribution and stick/slip results are investigated. It can be seen that by adjusting the layer parameters the traction and stick/slip results can be modified significantly, and it may, therefore, be very useful information for improving the rolling contact fatigue and mitigating wear problems in various mechanical systems.Graphical

Периодические контактные и смешанные задачи теории упругости (обзор)

Results are reviewed collected in the investigations of periodic contact and mixed problems of the plane, axially symmetric and spatial elasticity theory. Among mixed problems, cut (crack) problems are focused integral equations of which are connected with those for contact problems. The periodic contact problems stimulate research of the discrete contact of rough (wavy) surfaces. Together with classical elastic domains (half-plane, half-space, plane and full space), we consider periodic problems for cylinder, layer, cone and spatial wedge. Most publications including fun-damental ones by Westergaard and Shtaerman deals with plane periodic problems of the elasticity theory. Here, one can mention approaches based on complex variable functions, Fourier series, Green’s functions and potential func-tions. A fracture mechanics approach to the plane periodic contact problem was developed. Methods and approaches are considered which allow us to take friction forces, adhesion and wear into account in the periodic contact. For spatial periodic and doubly periodic contact and properly mixed problems, we describe such methods as the localiza-tion method, the asymptotic methods, the method of nonlinear boundary integral equations, the fast Fourier trans-form. The half-space is the simplest model for elastic solids. But for the simplest straight-line periodic punch system, some three-dimensional contact problems (normal contact or tangential contact for shifted cohesive coatings) turn out to be incorrect because their integral equations contain divergent series. Considering three-dimensional periodic problems, I.G. Goryacheva disposes circular punches in special way (circular orbits, polar coordinated are used for centers of the punches), in this case one can prove convergence of the series in the integral equation (it is important that the punches are circular). For the periodic problems for an elastic layer, V.M. Aleksandrov has shown that the series in integral equations converge but the kernels become more complicated. In the present paper, we demonstrate that for the straight-line periodic punch system of arbitrary form the contact problem for a half-space turns out to be correct in case of more complicated boundary conditions. Namely, it can be sliding support or rigid fixation of a half-plane on the half-space boundary, the half-plane boundary should be parallel to the straight-line (the punch system axis) for arbitrary finite distance between the parallel lines. On this way, for sliding support, the kernel of the period-ic problem integral equation kernel is free of integrals, it consists of single convergent series (normal contact, the kernel is given in two equivalent forms). We consider classical percolation (how neighboring contact domains pene-trate one to another, investigated by K.L. Johnson, V.A. Yastrebov with co-authors) for the three-dimensional periodic contact amplification as well as percolation for the straight-line punch system. A similar approach is suggested for the case of periodic tangential contact (coatings system cohesive with a half-space boundary shifted along its axis or perpendicular to it). Here, one can separate out unique solutions of auxiliary problems because the line of changing boundary conditions on the half-space boundary can provoke non-uniqueness. The method proposed opens possibility to consider more complicated three-dimensional periodic contact problems for straight-line punch systems with changing boundary conditions inside the period.

Evolution of adhesive wear and friction in elastic-plastic spherical contact

A finite element (FE) model is presented for the evolution of fracture and friction in elastic-plastic adhesive spherical contact. The model is based on a realistic criterion of ductile fracture in a three-dimensional contact problem and the failed elements are deleted. Full stick contact condition is used at the contact interface to model a strong adhesion there. Fracture evolution is presented allowing the process of material removal in the form of a wear particle. The evolution of friction with increasing tangential loading as well as sliding inception are presented. Predicted wear coefficient and static and dynamic friction coefficients are compared with experimental results and possible reasons for discrepancies between the predicted and experimental results are discussed.

Non-linear interface thermal conditions in three-dimensional thermoelastic contact problems

In this paper, a new and robust formulation to solve three-dimensional thermomechanical contact problems under non-linear interface thermal contact (ITC) conditions is presented. This methodology allows us to consider non-linear thermal contact conductance, including the presence of thermal interface materials (TIM), and convective conditions at the interstitial contact region. The Boundary Element Method (BEM) is assumed in order to compute the thermoelastic influence coefficients. The interface thermomechanical contact methodology is based on an augmented Lagrangian formulation where an efficient iterative Uzawa type scheme of resolution is proposed. The methodology is validated by comparison with benchmark solutions and later on, some additional examples are presented to study the influence of several ITC conditions, i.e., the presence of TIM doped with carbon nanotubes (CNT) and/or interstitial boundary conditions, in the thermoelastic contact variables (i.e., the contact area, the contact tractions, the temperature or the thermal flux distributions) of this kind of problems.

Fretting wear with variable coefficient of friction in gross sliding conditions

The two- and three-dimensional contact problems with fretting wear in the gross-slip regime are considered from a unified point of view in the framework of linear elasticity. It is assumed that the linear wear is proportional to the accumulated dissipated energy due to friction with a variable Coulomb coefficient. Based on a uniform approximation for the contact pressure, an asymptotic model for the evolution of the contact zone is constructed for arbitrary initial gap between the surfaces of contacting elastically similar bodies. In the case of Hertzian contact geometry, the model equations are derived in the explicit form.

Prediction and analysis of material removal characteristics for robotic belt grinding based on single spherical abrasive grain model

Comprehensive study of the microscopic material removal mechanism remains an open challenge facing the robotic belt grinding of complex geometries. In the present paper, a new material removal rate (MRR) model is developed underlying the motion trajectory of single spherical abrasive grain by taking into consideration the elastic deformation of the heterogeneous contact wheel. In this model, the contact pressure distribution at the contact wheel–workpiece interface is determined by converting the three-dimensional contact problem into a one-dimensional linear spring model with respect to the large deformation of the contact wheel. In addition, the combined modulus and Poisson's ratio of the heterogeneous contact wheel are further considered based on the geometric relation of the Young's modulus and Poisson's ratio. Compared with the existing MRR model, the proposed MRR model can significantly reduce both the root mean square error (RMSE) and mean absolute percentage error (MAPE) values from 2.401 to 1.725, and 18.426% to 14.942%, respectively. Particularly, an optimization strategy from the perspective of process parameters is implemented to balance the grinding quality and efficiency.