non-Hertzian Contact Problems Research Articles

Generalisation of the linear theory of rolling contact to a single double-elliptic contact region and its application to solve non-Hertzian contact problems using extended FASTSIM

Most contact models addressing the solution of the tangential problem assume an elliptical shape of the contact patch which is derived from the Hertz theory. In particular, the linear theory of rolling contact developed by Kalker is based on this assumption. However, it is well known that in many practical cases the shape of the contact region formed between the rail and a railway wheel can be far from elliptic. Recently, the Simple Double-Elliptic Contact (SDEC) shape has been proposed as an approximation of a wide variety of non-elliptic patches. In this paper, a formulation of the linear theory of rolling contact for a SDEC shape is proposed. The coefficients of the original linear theory for elliptic patches are maintained and four new coefficients are introduced to consider the effect of non-ellipticity. The coefficients of the theory are derived numerically using software CONTACT. Then, an extension of FASTSIM for a SDEC region is introduced and the obtained creep force curves under various conditions are compared to existing formulations of FASTSIM, using results from software CONTACT as a reference. The comparisons show that the proposed model is as fast as FASTSIM and provides more accurate solutions than existing versions of FASTSIM.

A simplified model for solving wheel-rail non-Hertzian normal contact problem under the influence of yaw angle

Wheelset yaw motion significantly affects wheel-rail non-Hertzian contact behaviour, but currently, few wheel-rail normal contact models consider this issue. In this paper, a simplified wheel-rail contact model is built to study the wheel-rail non-Hertzian normal contact problem considering the yaw angle. First, three basic assumptions for the approximate calculation of this problem are proposed and are validated by using results calculated by Kalker's variational method (KVM), and the approximate expression of the wheel-rail non-Hertzian contact problem considering the yaw angle is obtained based on these assumptions. Then, the approximate expression is applied to the MKP (Modified Kik–Piotrowski) contact model, and an improved MKP model that considers the wheelset yaw angle is proposed, i.e., the MKP-YAW model. Compared with the results calculated by KVM, the computational accuracy of the proposed MKP-YAW model in calculating the wheel-rail contact considering the yaw angle is similar to that of the MKP contact model in calculating the wheel-rail contact without considering the yaw angle.

A non-Hertzian wheel-rail contact model considering wheelset yaw and its application in wheel wear prediction

To obtain a relatively accurate wheel-rail contact solution with less time consuming, some so-called fast-accurate methods based on the concept of virtual penetration were provided by researchers. When dealing with the non-Hertzian contact problem with a sharply changing lateral relative curvature, it may be incorrect to calculate the penetration by the Hertzian contact parameters of the initial contact point. This paper proposes a method to get a proper penetration by considering the change of local relative curvature of wheel-rail profiles based on the semi-Hertzian method. Furthermore, the wheelset yaw angle is considered by using the ‘Trace line method’ to get a twisty contact patch which is verified by the program CONTACT. Finally, the proposed method is applied for the wheel wear prediction of a high-speed train. Field tests are carried out to determine the wear coefficient for wheel-rail contact of the high-speed train and to validate the wheel wear prediction model.

An adaptive mesh refinement approach for solving non-Hertzian elastic contact problems

Semi-analytical methods are a common way of solving non-hertzian contact problems when designing mechanical components. These methods require of the discretization of the domain into a set of pressure elements and their accuracy and computational cost are related to the number of elements in which the domain is discretized. But, while the accuracy increases as the pressure element mesh is refined, the computational cost increases quadratically with the number of pressure elements. So in the great majority of the cases, a commitment between accuracy and computational cost must be achieved. In this work, a new approach has been developed to improve the performance of semi-analytical methods for solving contact problems. This approach uses an adaptive mesh refinement strategy, based on the quadtree decomposition of the domain. As a result, the computational cost decreases, while the accuracy of the method remains constant.

A Fast Correction for Elastic Quarter-Space Applied to 3D Modeling of Edge Contact Problems

Applying the Hertz theory to some non-Hertzian contact problems can produce acceptable results. Nevertheless, including the influence of free surfaces requires numerical methods, many of which are based on the Boussinesq–Cerruti solution. This paper presents a new approach, which is better capable of releasing quarter-space free surfaces from shear and normal internal stresses without engendering any increase in calculation times. The mirrored pressure for shear correction is multiplied by a correction factor (ψ), which accounts for the normal load. The expression ψ is derived from the Hetényi correction process, and the resulting displacements show an enhanced correspondence with validation finite element method models; with an imposed fluctuating pressure, the maximum edge displacement error was −21.90% for a shear load correction (Poisson coefficient ν=0.3), and introducing the ψ factor reduced the deviation to −9.55%, while for ν of 0.15, the maximum error was −11.30%, which was reduced to +0.60% with the ψ factor. This study introduces the factor ψ in a 3D elastic contact algorithm. The resulting calculation scheme is then able to simulate any point or line contact problems. Compared with coincident ends and sharp edge contact validation values, the model shows high conformity levels.

Computationally efficient method for the solution of a singular integro-differential equation

A computationally efficient method is presented for the solution of a singular integro-differential equation which is the governing equation for a non-Hertzian contact problem in plane elasticity. The efficiency of this solution method is realized by formulating the elasticity problem in the complex domain and selecting an approximating function which is sectionally holomorphic and expandable by a series of Legendre polynomials.

Validation of the boundary element flexibility approach for elastic contact analysis

This paper discusses the validity and accuracy of the results obtained using the boundary element flexibility approach for solving elastic contact problems. The method is such that the size of the flexibility matrices, which represent the relationship between tractions and displacements at the contact nodes depends only on the actual number of nodes in contact. Therefore, the approach is computationally very efficient. Moreover the contact analysis and the formation of the boundary element matrices can be carried out separately. Thus it is easy to implement the flexibility contact approach into any standard boundary element code. This paper presents a new contact criterion for the solution and carries out a series of comparisons with results obtained using other techniques. Although constant elements were employed in all cases, it is possible to extend the same approach to higher order elements if required. The flexibility approach is validated against Hertzian and non-Hertzian contact problems, including numerical solutions. The results are shown to be in agreement and the accuracy of the solution is investigated.

Analysis of a Transversely Isotropic Half Space Under Normal and Tangential Loadings

This paper analyzes the response of a transversely isotropic half space subjected to various distributions of normal and tangential contact stresses on its surface. Both the interior displacement and stress fields are given in closed form. Among them, rectangular patch solutions are constructed for application to solutions to non-Hertzian contact problems.

Solution of the non-conformal unbonded contact problems by the incremental convex programming method

The main objective of this paper is to exploit an incremental convex programming procedure in the framework of a finite element model to solve the non-conformal unbonded contact problems. The developed model neither needs the insertion of any special artificial element “gap element”, nor requires the determination of a flexibility matrix. The model is flexible enough to deal with any non-Hertzian contact problem, whether the surfaces are coated or noncoated.

A contribution to the calculation of the contact stress distribution between two elastic bodies of revolution with non-elliptical contact area

A numerical method is presented for calculating the contact area and the normal stress distribution between two elastic bodies of revolution with parallel axes. For the discretization of the Fredhom integral equation the contact area is divided into strips perpendicular to the axes of revolution. The stress distribution in each strip is assumed to be semi-elliptical in the strip direction and constant in the other direction. The contact area is calculated iteratively by satisfying the contact condition and the stress condition. In addition for a given total normal force an additional iteration is required to improve the value obtained for the rigid-body approach. The contact area and the stress distribution between two ellipsoids are compared with Hertzian solutions. To investigate the non-Hertzian contact problem a number of cases of wheel-rail contact are considered.

Singularity Consideration in Numerical Solution of Contact Stress Problems

The paper gives an account of different approaches to deal with the weak singularity in numerical methods of contact stress problems when the methods are based on the fundamental equation relating the elastic displacement with pressure. Singularity consideration in a new method to simultaneously determine the shape of the contact area and the pressure distribution, particularly in non-Hertzian contact problems, has been dealt with using elliptic functions. Necessity of proper analysis of singularity is discussed and the final results when compared with Hertz solution have been shown to be satisfactory.