Investigations on the dynamic influence of the contact angle on frictional sliding processes between rough surfaces using NURBS and mortar-based augmented Lagrangian method

https://doi.org/10.1016/j.triboint.2021.106889Get rights and content

Highlights

  • NURBS based Augmented Lagrangian contact model coupled with the mortar method.

  • Sliding contact process with large contact angles and large deformations.

  • Robust numerical behavior.

  • Pronounced dynamics of the resulting contact forces.

  • Difference between the local coefficient of friction and a globally interpreted coefficient of friction.

Abstract

The paper deals with the influence of contact kinematics on the coefficient of friction (COF) during sliding processes on a mesoscopic scale (asperity). The primary emphasis here is on the influence of contact angles, which is particularly significant for rough surfaces with steep asperities. The experimental setup here is sliding contact undergoing large deformations between topographies with a single asperity and multiple asperities. Since this question is numerically highly challenging, a frictional contact model using NURBS basis functions and mortar-based augmented Lagrangian method was developed here. During the process, the local normal force and frictional force change their magnitudes and directions highly dynamically. Due to large contact angles, a global interpretation of the COF differs from the locally assumed COF.

Introduction

Friction is a phenomenon that is not yet fully understood for lubricated, but especially for dry contacts. This is also reflected in the fact, that in most practical applications the over 300-year-old Amontons' laws, also known as Coulomb's law, is still applied. It states that the magnitude of frictional forces is proportional to the magnitude of normal forces. The dimensionless quotient between the forces is the so-called coefficient of friction (COF), often symbolized by μ, which has to be measured experimentally on tribometers in general. A main reason for the requirement of such complex measurements is the absence of the possibility to calculate the COF for rough surfaces in a macroscopic scale, where a strong interaction between the occurring forces, elastic and plastic deformations and surface changes due to wear mechanisms must be taken into account. Since all the mechanisms act on the different scales and differ significantly from each other, a scaling of the problem is practically very complex.

In the literature numerous fundamental works on the modeling of mechanical contacts on different scales are available. The Prandtl-Tomlinson model [1,2] was originally developed for an atomic scale. In this model, a body with one degree of freedom is moved over a sinusoidal potential with a given wavelength, whereby the inertial force, velocity-dependent forces and elastic restoring forces are considered. With this model, states of sticking and the transition to sliding states can be described. Although developed for the atomic scale, the validity of the equations with regard to the wavelength of the potential is not restricted. However, the system parameters themselves are scale dependent. The equations derived from the model therefore also allow an energy-related interpretation of macroscopic friction processes.

In the model by Greenwood and Williamson [3], statistical information on the roughness of surfaces is provided by a probability distribution of spheres with respect to their sizes and positions. With these generated topographies, studies are possible, in particular to determine the relationship between the normal force and the real contact surface. For this purpose, the analytical solutions of the Hertzian contact theory [4] were used, which includes the non-linear relationship between normal force and elastic deformation of elementary bodies (in this case spheres). Extensions to include plastic deformations have also been implemented in this model.

Persson's models represent an extension of this statistical analysis [5]. In these models, contact properties are derived from power spectra for the surfaces and their roughness. Adhesion, plasticity and lubrication can also be considered. Special attention is paid to the fractal properties of surfaces and the associated Hurst exponent [6]. Under certain conditions, it has been possible to determine probability distributions of pressures and contact areas of measured surfaces.

An observation of adhesive forces in the contact model was carried out, e.g. in the JKR (Johnson, Kendall, Roberts) model [4]. The corresponding term considering the adhesion is especially important for the contact properties when interactions occur between bodies with small radii. The approaches according to half-space solutions and the Hertzian contact [4] were also utilized to specify the tangential force distribution. In this case it is decisive to estimate the size of local sticking and sliding zones. Nevertheless, a local friction law is required for the determination of the tangential force, which is usually not known a priori.

Friction laws are highly complex. In real systems the COF depends in particular on the normal force, the temperature and the relative speed. Corresponding stationary friction laws are therefore adopted to take these dependencies into account. The load history as well as the boundary layer dynamics can also have a significant influence on the current coefficient of friction. For this purpose, as one example, dynamic friction laws in the form of differential equations have been formulated for brake systems [7,8].

This present paper is primarily intended to project the local frictional laws onto the macroscopic friction and make a fundamental contribution to contact mechanics in terms of the relation between the mesoscopic and macroscopic scale. For the case of macroscopic sliding, it is particularly focused on the influence from the contact kinematics for asperities with large contact angles as well as the resulting dynamics of force direction and magnitude. The basic idea was outlined in Ref. [9] and is examined in detail in this work. The configuration in Fig. 1 intentionally represents relations between the heights of the asperities and their lateral extension, which results in the fact that the real contact angles are not negligibly small but shall also be larger than 10°. Such contacts with steeper flank angles are quite common, especially in systems where abrasion plays a major role or in tribologically virgin surfaces.

Fig. 2 illustrates that at a contact point i the local normal force Ni and the local friction force Ri are perpendicular and tangent to the contact surface, respectively. The COF at the contact point is mentioned in the following as the local COF μloc and can be calculated using μloci=Ri/Ni. The resultant of both forces is assigned as Fi, which is orientated with a certain angle αi to the horizontal (see Fig. 2(a)). The direction and the magnitude of this resultant Fi change within a certain limits (see Fig. 2(b)) following the contact point i during the sliding process. In the case of very small contact angles, Ni in this scheme would almost be vertical and Ri almost horizontal. The tangent of the angle αi would therefore correspond to the local COF. Under these conditions and a quasi-static view, in this case only the magnitude but not the angle of the resulting force would change over time.

However, what would be interpreted macroscopically as the COF in measurements is the global COF, which is defined as μglo=Fx/Fy (see Fig. 1). In principle, this value can deviate significantly from the local COF due to dynamic large contact angles, whose influence can become particularly significant if the local COF is very small (e.g. in the case of lubricated contacts), because in this case the influence of the contact angle on the direction of the resulting force can be greater than that of the COF. Experimental studies have shown that even very small amounts of lubricant drastically reduce the global COF. It is a long-term goal of the authors to apply this effect to minimally lubricated systems [10].

In the following studies, systems with high elasticity (such as rubber) will be investigated, where plasticity is still neglected. However, large deformations are to be allowed, which is why conventional solutions according to Hertz would not provide precise solutions here. For this reason the analytical and semi-analytical approaches are not appropriate. Against this background, continuum mechanical approaches are used instead in these studies. The computational contact mechanics undergoing large deformations regime focus on applying numerical algorithms and discretization techniques to solve the contact problems, where a variety of difficulties are involved, including extreme non-linearity and non-smoothness, potential ill-conditioning and enormous computational costs of contact detection [11].

The research on this topic has been distributed in several aspects. The first aspect is the choice of the contact discretization techniques to describe and enforce the contact constraints. One of those techniques is the well-known node-to-surface (NTS) algorithm [12], where the contact constraints are only enforced between the discrete points on one of the contact surfaces (denoted as “slave”) and the corresponding pieces (surface or edge) on the so-called “master” contact surfaces. Despite the computationally inexpensive implementation, De Lorenzis et al. [13] have shown that the NTS formulation cannot pass the contact patch test, since the concentration of forces at slave nodes violates the balance of moments at the element level. To overcome these drawbacks, the surface-to-surface (STS) algorithm was first proposed by Simo et al. [14], which enforces the contact constraints through integration over a new developed intermediate contact surface, where the contact discretization stems from the projection of the slave and master nodes. The STS formulations pass the contact patch test, but cannot fulfill the Ladyzhenskaya-Babuška-Brezzi (LBB) stability condition due to the overconstrained problems [15], so that a unique solution cannot be achieved upon mesh refinement. A further improvement was the application of mortar methods for the contact problems by Puso and Laursen [16]. The mortar method satisfies the contact patch test and is LBB stable, at the expense of a heavy computational cost.

Another aspect of the research is to integrate the above contact constraints formulation into the virtual works. The most popular options are the Lagrange multiplier method [12] and the penalty method [17] from the optimization theory. However, the Lagrange multiplier method can cause oscillations problems and difficult convergence, while unphysical solutions can be introduced due to penetrations in the penalty method. Thus Simo and Laursen [18] combined these two methods into the Uzawa algorithm, where the Lagrange multipliers are iteratively approximated through an outer loop. In addition, Alart and Curnier [19] have developed the augmented Lagrangian (AL) method, where the Lagrange multipliers are present as additional unknowns in the systems and can be solved using Newton's iterative method.

The third aspect is the development of smoothing techniques, where one of the most significant approaches is the NURBS-based isogeometric analysis (IGA) (more details see Ref. [20]). In comparison to the conventional FEM, IGA adopts the NURBS functions as basis functions instead of Lagrange interpolation functions. It builds the framework for the exact geometrical description and automatic meshing. Due to the local-support characteristic and the adaptive continuity of NURBS basis functions, it is possible to refine geometries as well as meshing and achieve the desired degree of continuity at element boundaries.

The IGA framework was first introduced into computational contact mechanics by Temizer et al. [21] and De Lorenzis et al. [22], where NURBS-based contact formulations using the penalty method and the mortar method were developed for 2D frictionless and frictional contact, respectively. De Lorenzis et al. [23] have also adopted the AL method instead of the penalty method into the contact formulations and modified them for the 3D frictionless contact problem. Additionally Temizer et al. [24] have also developed contact formulations based on the Uzawa algorithm and mortar methods under the framework of IGA for the 3D frictional contact problem.

The present work is an extension of the developments in De Lorenzis et al. [23] and Temizer et al. [24]. NURBS-based IGA and the mortar-based approach in the previous studies are directly adopted herein. Alternative to the Uzawa algorithm in Temizer et al. [24], the AL method in De Lorenzis et al. [23] is extended for the 2D frictional contact problem in this paper. The 2D contact formulations via NURBS basis functions and mortar-based AL method are illustrated in detail in Section 2. The full variation and linearization of the formulations are deferred to Appendix A and Appendix B. Furthermore, The model is applied for the case of Hertzian contact as a benchmark test, which is presented and discussed in Section 2. Section 3 describes the contact problems between rough surfaces with a single asperity and multiple asperities. Moreover, the influences of contact angles on the global COF are also examined in Section 3. The main conclusions are finally summarized in Section 4.

Section snippets

Frictional contact formulation

This section is dedicated to describing the frictional contact problem through the continuum formulation. For this purpose, the kinematics, statics and virtual work for the frictional contact between two elastic bodies is analyzed sequentially. This problem is then turned into a constrained optimization problem, which can be solved using the augmented Lagrangian method.

Frictional contact with large deformations and dynamic contact angles

As already mentioned in Section 1, the present paper is intended to conduct studies on the influence of the contact angles on the frictional resistance. In a first step, the contact between two asperities (Section 3.1) serves as an example. In a second step the relative movement of multiple connected asperities over multiple connected asperities (Section 3.2) is simulated. Both calculations are performed in 2D allowing large deformations. To ensure that the contact angle can vary over a wide

Conclusions

This paper focused on the basic contact mechanics question of what influence the contact kinematics can have in terms of contact angles for topographies with steep flanks. For this purpose, simulations on a mesoscopic scale were performed using a single asperity contact and multiple asperity contacts. These simulations are particularly addressing the question of contact dynamics and the difference between local COF in the contact and the global COF as the quotient of the resulting horizontal

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.

CRediT author statement

Georg-Peter Ostermeyer: Conceptualization, Methodology, Resources, Writing - Review & Editing, Supervision, Project administration. Yan Tong: Methodology, Software, Investigation, Writing - Original Draft, Visualization. Michael Müller: Methodology, Writing - Original Draft, Writing - Review & Editing, Project administration.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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