Conformal Contact Problem Research Articles

A fast method for solving conformal rolling contact problems

For conformal contact situations, some of the assumptions employed in classical contact theories, such as the flatness of the contact area, lose their validity. Existing conformal contact models are challenged by either accuracy or computational efficiency. With the aim of achieving a balanced trade-off between these contrasting requirements, this paper proposes a new simplified method to solve conformal rolling contact problems, which is based on fast non-Hertzian contact model INFCON and FaStrip. Numerical experiments are performed for two typical wheel/rail conformal contact scenarios to assess the performance of the proposed method, taking as the reference the CONTACT method for conformal contacts. The comparative analyses show that the proposed method is in good agreement with the reference method in terms of the determination of the contact stresses, total forces, and the prediction of wear, whereas the traditional half-space approach fails to capture the main features of the conformal contacts. In addition, the high computational efficiency of the proposed method shows the potential for application in vehicle dynamic simulation and damage analysis.

Reprint of: Shakedown in frictional contact of discrete elastic systems: A review

When exposed to cyclic quasi-static loading, elastic bodies in contact may develop a favourable condition where slip ceases after a few cycles, an occurrence commonly known as frictional shakedown. If the amplitude of the cyclic load is greater than a so-called shakedown limit, shakedown cannot occur. In this review paper, the validity of shakedown theorems in the context of conforming contacts with à la Coulomb friction is first discussed. Then, an optimisation method for determining the shakedown limit of elastic discrete three-dimensional systems is reviewed. Finally, an incremental Gauss–Seidel algorithm, extended to three-dimensional systems, is here illustrated in details for the first time. The algorithm allows us to describe the transient response of normal-tangential coupled systems under a given cyclic loading scenario, and to determine their possible shakedown depending on the initial conditions. An example concerning a discrete conforming contact problem, where either coupling or uncoupling conditions can be imposed, is illustrated.

Shakedown in frictional contact of discrete elastic systems: A review

When exposed to cyclic quasi-static loading, elastic bodies in contact may develop a favourable condition where slip ceases after a few cycles, an occurrence commonly known as frictional shakedown. If the amplitude of the cyclic load is greater than a so-called shakedown limit, shakedown cannot occur. In this review paper, the validity of shakedown theorems in the context of conforming contacts with à la Coulomb friction is first discussed. Then, an optimisation method for determining the shakedown limit of elastic discrete three-dimensional systems is reviewed. Finally, an incremental Gauss–Seidel algorithm, extended to three-dimensional systems, is here illustrated in details for the first time. The algorithm allows us to describe the transient response of normal-tangential coupled systems under a given cyclic loading scenario, and to determine their possible shakedown depending on the initial conditions. An example concerning a discrete conforming contact problem, where either coupling or uncoupling conditions can be imposed, is illustrated.

An Analytical Model for Almost Conformal Spherical Contact Problems: Application to Total Hip Arthroplasty with UHMWPE Liner

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.

A Numerical Study on Plane Contact Problem Involving Inclusions

Hertz’s contact theory is widely used in elastic solid contact problems. In this paper, the applicability of Hertz’s contact theory in conformal contact problems was examined. Taking deep groove ball bearing and the outer raceway model as an example, both the FEM analyses and the theoretical studies were conducted. It is found that when the radius coefficient of groove curvature is greater than 0.55, Hertz’s theory can be practically applied for conformal contact. Contact problems of rolling bearings are generally solved by Hertz’s contact theory without considering inclusions and inhomogeneities in the material. In this paper, inclusions are presented in the contact analyses, and the corresponding elastic field distribution of the contact-inclusion model is investigated by using the finite element method.

Approximating the influence coefficients of non-planar elastic solids for conformal contact analysis

The exact contact theory is an efficient alternative to the more general yet computationally expensive Finite Element Method for the detailed study of elastostatic contact problems. For its application in conformal contact problems, the exact contact theory needs to be fed with influence coefficients (ICs) appropriate for non-planar solids. An analytical approximation of the ICs for non-planar solids was proposed in a previous work, avoiding the involved process generally necessary to obtain ICs accurately. This work presents further developments of this approximation, further comparison with numerically obtained ICs, and evaluates the errors incurred when using approximated ICs in conformal contact.

The role of adhesion in contact mechanics.

Adhesive (e.g. van der Waals) forces were not generally taken into account in contact mechanics until 1971, when Johnson, Kendall and Roberts (JKR) generalized Hertz' solution for an elastic sphere using an energetic argument which we now recognize to be analogous to that used in linear elastic fracture mechanics. A significant result is that the load-displacement relation exhibits instabilities in which approaching bodies 'jump in' to contact, whereas separated bodies 'jump out' at a tensile 'pull-off force'. The JKR approach has since been widely used in other geometries, but at small length scales or for stiffer materials it is found to be less accurate. In conformal contact problems, other instabilities can occur, characterized by the development of regular patterns of regions of large and small traction. All these instabilities result in differences between loading and unloading curves and consequent hysteretic energy losses. Adhesive contact mechanics has become increasingly important in recent years with the focus on soft materials (which generally permit larger areas of the interacting surfaces to come within the range of adhesive forces), nano-devices and the analysis of bio-systems. Applications are found in nature, such as insect attachment forces, in nano-manufacturing, and more generally in industrial systems involving rubber or polymer contacts. In this paper, we review the strengths and limitations of various methods for analysing contact problems involving adhesive tractions, with particular reference to the effect of the inevitable roughness of the contacting surfaces.

A spherical conformal contact model considering frictional and microscopic factors based on fractal theory

Spherical conformal contact is widely used in engineering structures. The existing solution of the spherical conformal contact problem always ignores the microscopic characteristics or friction of the spherical surfaces. Therefore, the current paper presents a spherical fractal model to characterize the contact state of spherical pairs considering the microscopic topography of rough spherical surface and the factor of friction. Firstly, a method of characterizing the microscopic topography of rough spherical surface is proposed based on three-dimensional Weierstrass–Mandelbrot function. The fractal contact model of the single asperity is developed by Hertz theory in combination with elasticity. Then, the macroscopic parameters are introduced to construct the contact surface coefficient. The area distribution function under conformal contact region is obtained. Considering the friction factor of the conformal contact region, the microcontact model of the spherical conformal contact is developed based on fractal theory. Finally, the formula between (elastic, elastic-plastic, plastic) contact area, (elastic, elastic-plastic, plastic) contact load and the key parameters (fractal parameters and macro parameters) are derived based on the proposed model. The relationship between the actual contact area and the normal load of the contact region is established. Numerical results show that the proposed model is more accurate for the analysis of the spherical surface contact area and contact load.

Fractal modeling of elastic-plastic contact between three-dimensional rough surfaces

PurposeThis paper aims to propose a semi-analytical model to investigate the elastic-plastic contact between fractal rough surfaces. Parametric studies have been performed to analyze the dependencies between the contact properties and the scale-independent fractal parameters.Design/methodology/approachA modified two-variable Weierstrass-Mandelbrot function has been used to build the geometrical model of rough surfaces. The computation program was developed using software MATLAB R2015a. The results have been qualitatively validated by the existing theoretical and experimental results in the literature.FindingsIn most cases, a nonlinear relation between the load and the displacement of the rigid plane is found. Only under the condition of larger loads, an approximate linear relation can be seen for great D and small G values. (D: fractal dimension and G: fractal roughness).Originality/valueThe contact model of the cylindrical joints (conformal contact) with radial clearance is constructed by using the fractal theory and the Kogut-Etsion elastic-plastic contact model, which includes purely elastic, elastic-plastic and fully plastic contacts. The present method can generate a more reliable calculation result as compared with the Hertz contact model and a higher calculation efficiency as compared with the finite element method for the conformal contact problem.

원형 핀과 구멍의 접촉에서 헤르츠 응력장 가정을 위한 조건

LWR Fuel Development Division, Korea Atomic Energy Research Institute(Receive August 15, 2015; Revised September 20, 2015; Accepted September 21, 2015)Abstract − This paper focuses on the conformal contact problem. A typical example of conformal contact is thecontact between a pin and hole. In particular, this paper focuses on the condition for assuming a contact stressfield to be a Hertzian pressure profile by using well-known classical solutions associated with Hertzian contact.Persson first developed the conformal contact analysis method around half a century ago, but there have beenno significant improvements since then. The present research also adopted this method, but developed new solu-tions from the viewpoint of application to structural design. The analysis began with a comparison between Per-sson°Os conformal contact stress and the Hertzian stress fields. The next step was to check the differences inthe normalized stress values of both. This study used the tolerance for the difference in the peak stresses of Per-sson°Os solution and the Hertz solution to validate the Hertzian assumption. This gave the range for the dif-ference in radii of the pin and hole when the contact force and mechanical properties of the material are specified.The results showed that, at a tolerance of 5%, the Hertzian assumption is valid if half of the contact angle is lessthan 35°ý. In addition, the Hertzian assumption holds even for a relatively long contact length, in contrast to thegeneral incomplete contact problem. This paper discusses these results along with other aspects of the applicationto the design.Keywords − conformal contact(등각 접촉), contact angle(접촉 각), contact length(접촉 길이), Hertzian stressassumption(헤르츠 응력장 가정), contact mechanics(접촉역학)

A Simple Procedure for the Solution of Three-Dimensional Wheel/Rail Conformal Contact Problem

This paper describes a simple and efficient procedure for the treatment of conformal contact conditions with special emphasis on railroad wheel/rail contacts. The general three-dimensional nonconformal contact conditions are briefly reviewed. These nonconformal contact conditions, which are widely used in many applications because of their generality, allow for predicting online one point of contact, provided that the two surfaces in contact satisfy certain geometric requirements. These nonconformal contact conditions fail when the solution is not unique as the result of using conformal surface profiles or surface flatness, situations often encountered in many applications including railroad wheel/rail contacts. In these cases, the Jacobian matrix obtained from the differentiation of the nonconformal contact conditions with respect to the surface parameters suffer from singularity that causes interruption of the computer simulations. The singularities and the fundamental issues that arise in the case of conformal contact are discussed, and a simple and computationally efficient procedure for avoiding such singularities in general multibody systems (MBS) algorithms is proposed. In order to demonstrate the use of the proposed procedure, the wheel climb of a wheelset as the result of an external lateral force is considered as an example. In this example, the wheel and rail profiles lead to conformal contact scenarios that could not be simulated using the nonconformal contact conditions.

Experimental Finite Element Approach for Stress Analysis

This study aims to determining the strain gauge location points in the problems of stress concentration, and it includes both experimental and numerical results. Strain gauges were proposed to be positioned to corresponding locations on beam and blocks to related node of elements of finite element models. Linear and nonlinear cases were studied. Cantilever beam problem was selected as the linear case to approve the approach and conforming contact problem was selected as the nonlinear case. An identical mesh structure was prepared for the finite element and the experimental models. The finite element analysis was carried out with ANSYS. It was shown that the results of the experimental and the numerical studies were in good agreement.

Conformal Contact Between a Rubber Band and Rigid Cylinders

We consider a conformal contact problem between a rubber band and rigid cylinders that involves geometrical and material nonlinearities. The rubber band is assumed to be incompressible, neo-Hookean rubber. From the geometry and elasticity of the band, we present simple formulas to estimate the force–stretch relations and the contact pressure distributions on the cylinder. We show that the theoretical results are in good agreement with those of the finite element analysis when the rubber band is thin enough to be negligible to the bending stiffness. This verifies that the theory can effectively take into account both the material and geometrical nonlinearities of the band under the present conditions.

Conformal Contact Problems of Ball-socket and Ball

This paper focuses attention on non-conformal and almost conformal contact of ball and ball-socket. Two-dimensional finite element models are developed to calculate the normal contact stress distribution and contact area. The effects of geometry dimension and external load on the contact pressure distribution and contact region are presented, respectively. Meanwhile, the results of FEM and solutions of Hertz contact theory are compared. The results indicates that contact state of ball and ball-socket changes from point contact to area contact with the increasing of the dimensionless number-curvature radius coefficient f and the number of f =0.536 (≈0.54) is critical parameter causing the change.

The generalized advancing conformal contact problem with friction, pin loads and remote loading – Case of rigid pin

The advancing, frictional contact problem for a rigid pin indenting an infinite plate with a circular hole is considered. The formulation is general, and considers remotely applied plate-stresses in addition to pin loads. Using the theory of generalized functions, it is found that the governing equation in full sliding is a singular integro-differential equation (SIDE). Partial-slip behavior is governed by an implicit, coupled singular integral equation (SIE) pair. Numerical solutions are presented for both types of problems. It is found that the contact tractions in monotonic loading become independent of the coefficient of friction above a certain threshold value. Finally, problems involving typical ‘fretting-type’ pin loads with and without remote-stresses are also investigated, revealing remarkable effects of the degree of conformality and load path on the steady-state traction distributions.

On the conforming contact problem in a reinforced pin-loaded structure with a non-zero second Dundurs’ constant

Within the framework of two-dimensional linear elasticity, the unilateral frictionless contact between two conformal cylindrical surfaces is governed by an integral equation in which the first and second Dundurs’ constants are involved. In the case of elastic similarity characterized by the zero second Dundurs’ constant, the integral equation is considerably simplified so as to lend itself to a closed-form solution. However, in the case of elastic dissimilarity defined by the non-zero second Dundurs’ constant, the question of obtaining a closed-form solution to the integral equation is a much tougher one. Starting from the integral equation established by To et al. [To, Q.D., He Q.-C., Cossavella, M., Morcant, K., Panait, A., 2007. Closed-form solution for the contact problem of reinforced pin-loaded joints used in glass structures. Int. J. Solids Struct. 44, 3887–3903] for the conformal contact problem originating from a reinforced pin-loaded joint used in tempered glass structures, the present work proposes a new approximate analytical method to solve it in the case of elastic dissimilarity by minimizing an error function. The derived closed-form solution, valid not only for the conformal contact between a pin and an infinite holed plate but also for the one between a pin and a finite holed plate, is shown to be in very good agreement with available numerical results.

Contact Pressure and Shear Stress Analysis on Conforming Contact Problem

Two methods to solve a conforming contact problem are proposed. First method is general and can be applicable to the contact case between elastic arbitrary shape bodies. For verification FEA is performed on the convex-concave sphere contact, and the result of this method is well corresponding to the FEA result. However, the accuracy deteriorates when the mesh aspect ratio is extremely large. This phenomenon is caused by the usage of numerical integration for the calculation of influence coefficient. The second method is devised to avoid this problem, while this improved method is applicable only to the case when the contact area can be considered to be on a cylinder surface. By using this method, the contact pressure can be obtained without the deterioration even in the case of edge load occurring between ball bearing race shoulder and ball. The results of the contact pressure and the shear stress that is necessary for bearing life estimation are compared with the FEA result, which showed well correspondence.

Ball Bearing Fatigue Life Estimation Under the Condition of Contact Ellipse Being Over Shoulder

There are no prediction methods of bearing fatigue life when ball bearing is used under the condition of contact ellipse being over shoulder. To enable this prediction there are two difficulties. The first is contact problem shifts from non-conforming contact problem to conforming contact problem because of large load. The second is there is no practical method for bearing life estimation under the condition like as edge load occurring, though there exist many decision methods about optimal crowning when edge load occurs. For each problem, authors have proposed each solution. Combining those methods, we propose the bearing life ratio estimation method under the condition of contact ellipse being over shoulder. To verify this method, we compare the calculation result with bearing life test result for 50 samples that have different 3 inner ring outer diameters. The both result agree well, therefore the validation of this method was achieved. Ahead of that, we propose improved solution to the first difficulty, and verify the contact pressure and shear stress by comparing FEA result with improved solution when contact ellipse being over shoulder. This solution has been used to estimate above-mentioned bearing life.

Closed-form solution for the contact problem of reinforced pin-loaded joints used in glass structures

In a typical pin-loaded joint used to assemble glass plates, the hole in a glass plate is reinforced by a steel ring glued to the glass plate via a soft resin layer. Thus, the ring is in direct contact with the steel bolt and prevents the glass plate from high stress concentration. This paper proposes an analytical approach to solving the resulting conforming contact problem. The strain and stress fields inside the resin layer are first determined by exploiting the fact that the stiffness of the material constituting it is much smaller than the stiffness of steel and glass. After finding the relevant Green functions for the ring and pin, the frictionless contact between them is then formulated in terms of an integral equation with a Fourier series as the kernel. This integral equation is solved by neglecting the terms of high orders and transforming it into the Cauchy singular integral equation. The derived analytical results for the contact pressure and angle are finally compared to and validated by those obtained by the finite element method.

Normal Force-Displacement Relationship of Spherical Joints With Clearances

The spherical joint with clearances can be modeled as an axi-symmetric quasi-static contact of a sphere in a cavity. The Hertz theory based on the assumption of nonconformal contact is often used to represent the normal force-displacement relationship for the spherical joint with clearances. This assumption limits the application of the theory, especially in the case of occurring large deformation in contact area. The Steuermann theory is effective to solve the conformal contact problem in some cases, but it is strictly dependent on the selecting index n of the polynomial, which is used to represent the contact profiles. In this paper, an approximate contact model of the spherical joints with clearances is developed that is based on using the distributed elastic forces to model the compliant of the surfaces in contact. The new formulation is simple and straightforward, and its validity has been tested by comparison to the finite element results.