Multiscale thermomechanical modeling of frictional contact problems considering wear – Application to a pin-on-disc system
Introduction
In frictional braking systems, the disc-pad contact interface plays a key role in the operation of the system. Indeed, the contact interface is the location where kinetic energy is converted into frictional heat energy. Knowing that real surfaces are rough, the real contact occurs only on some randomly distributed small spots. Thus, severe high temperatures, resulting from the frictional heat, are reached within the contact surface [1]. This leads to wear and material transformation beneath the surface that affect braking performances. Moreover, wear is responsible of a continuous modification of the contact area during loading, which affects the interactions between the system components. Thereby, the contact surface and temperature vary according to the combination of heating and thermal expansion but also to wear which is itself influenced by temperature.
The aim of this paper is to propose a numerical strategy enabling to perform thermomechanical simulations considering the contact interface phenomena (wear, roughness, etc.). The main idea is to highlight, through numerical simulation, the impact of the contact interface on the thermomechanical behavior of the system and conversely.
From a numerical viewpoint, the main challenge is to develop a framework which considers both the scale of the global system components, and the scale of the contact interface phenomena which is much lower than the system's scale. Usually, only this scale is considered while the other is simplified or neglected. For example, [2], [3], [4], [5] have performed thermomechanical braking simulations considering a flat contact with a perfect interface, thus ignoring roughness and wear effects. Moreover, in these works, the frictional heat is distributed between the contacting solids using a heat partition coefficient which depends only on the thermal effusivity. However, the partition of heat depends on many other features such as speed, roughness and the contact area [6]. Hence, it is necessary to build an efficient strategy allowing to consider all these aspects at the large scale of the system.
As regards the contact interface modeling, from a mechanical viewpoint, most of the models use the concept of surface asperities which has been introduced in the classical statistical theory of Greenwood & Williamson [7]. Many improvements of this theory have been proposed, for instance by including interactions between asperities [8], [9], and considering solid layers beneath the surface [10]. Other multi-scale approaches have been developed based on the surface spectral density [11]. Numerical methods have been also used to solve the contact problem, using optimization techniques, either with the Finite Element Method (FEM) [12] or the discretized half space theory [13]. The latter approach is based on discrete convolution and quadrature programming [14]. It presents the advantage of a reduced computational time, especially if the Fast Fourier Transform (FFT) is used to accelerate calculations [14].
The contact interface models provide information about the contact area, stiffness and the pressure distribution. These data are relevant to perform a thermal contact analysis. For a static contact, the theoretical framework was proposed in the pioneering works [15], [16]. For a sliding contact, most of the works are based on the heat source theory developed in the works of [17], [18]. From this method, many solutions were proposed to study stationary and moving heat sources of different geometries [19]. Based on this theory, a numerical approach, that takes into account frictional heat generation, has been proposed recently in [6] using the FFT and optimization techniques. Thermal contact calculation gives information about surface temperature and the heat partition. The knowledge of these data is crucial for wear consideration.
A large variety of wear models has been proposed in [20] based on several theoretical and experimental works existing in the literature. From these models, the amount of wear volume can be expressed as a function of load, the sliding distance and a wear rate that depends on material properties and wear mechanisms. However, the difficulty in wear modeling is that there is no universal model that can be applied to all situations [21]. Thus, most of the existing models are empirical. Nevertheless, it appears that most of the wear mechanisms (e.g. abrasive and adhesive modes) can be described with Archard's law [22], [23]. In this case, wear volume is proportional to the frictional energy through the wear rate. Yet, this coefficient depends on the wear mechanism and has to be identified using appropriate experiments. For instance, the wear rate corresponding to adhesive wear of metals has been quantified by [22], depending on the operating conditions and material properties.
In this paper, we propose a multiscale strategy which considers the macro scale of the system and the micro scale of the contact interface. At the macro scale, the FEM is used to develop a numerical model considering all the system components. As regards the interface of concern, a flat surface is considered as in the classical FEM models. Then, the model is embedded with homogenized interface parameters which are computed using numerical micro contact interface models. The original idea behind this strategy is to avoid the explicit meshing of surface roughness at the macro scale, as it is computationally expensive. Thus, the studied contact surface remains flat at the macro scale and the interface parameters are used to simulate the micro contact phenomena. This idea has been originally proposed in [24], [25], [26], and has been improved in [9]. It consists of defining a local contact stiffness for each surface using a micro contact mechanics model developed in [9] and inspired from [8]. This methodology has already been used to study the effect of the disc-pad interface on the vibrational behavior of a braking system [27], and has shown very interesting results.
In this work, this methodology is extended to consider thermal and wear issues. Thus, in addition to contact stiffness, a heat partition coefficient will be integrated into the macro scale FEM model. This coefficient is calculated for each surface element using a thermal contact model developed in [6]. This model as well as the contact mechanics model are based on semi analytic methods which make use of discrete convolution, FFT and quadratic programming. Furthermore, to consider wear, roughness will be varied according to Archard's model [22]. The wear rate used depends on local surface temperature. The effect of plastic deformations on wear has not been introduced here. It would be very time consuming for the solving scheme, due to non-linearity of both contact and plasticity phenomena, especially in the multiscale framework of this study. As an approximation, the estimation of wear is done under elastic assumption and the wear coefficient is based on braking experiments and depends on temperature. Indeed, elastic calculations gives a good estimation of the contact area distribution, which is important for wear analysis.
Thus, the numerical simulation incorporates three mains steps. First, the micro contact simulations are performed to identify contact parameters that are integrated into the macro scale FEM model of the system. Thereafter, numerical simulation is run using the embedded FEM model. In the third step, the results of the FEM simulation are exploited in a second micro contact analysis simulating wear. At the end of this step, a second sequence of calculation can be performed using the new worn surface, and so on. This means that the simulation can be performed sequentially, where in every sequence, the three modeling steps are run. Thus, the multiscale numerical strategy enables to simulate the transient evolution of the contact interface considering wear effect. Its originality lies in its ability to give information about micro contact interface data considering the macro scale boundary conditions prescribed by the system. This is hardly feasible with experiments and classical numerical approaches. Moreover, a considerable gain of computational time is guaranteed using the micro contact accelerated models.
The different steps of the strategy are thoroughly explained in the first section. A special attention is devoted to the theoretical background of the micro contact models and to the transition between these models and the macro scale FEM model. As an application of the strategy, a thermomechanical analysis of a pin-on-disc system is performed. Results of simulation are discussed in the last section, and then compared to those of a classical numerical model considering a perfect contact interface. The goal of this comparison is to highlight the effect of the contact interface and its evolution on the system performances. Furthermore, a Complex Eigenvalue Analysis (CEA) is realized to study the dynamic behavior of the pin-on-disc system Indeed, recent works about friction-induced vibration for braking applications have shown that the contact interface plays a key role in the determination of unstable frequencies leading to squeal [27], [28], [29]. In particular, experimental works [30], [31] have shown that the topography of the pad/disc interface has a major influence on brake squeal generation. Hence, the CEA is realized considering roughness and its evolution with wear, with the aim of showing the influence of these aspects on the dynamic behavior of the system.
Section snippets
Global strategy
The aim of the multiscale strategy, proposed in this paper, is to integrate the micro contact interface phenomena in a macro scale FEM model without modeling them directly at this scale, which would be very time consuming. This is due to the large gap between the system's scale and the scale of micro contact phenomena. For this reason, the contact interface is flat in the FEM model, but instead will be embedded with contact interface parameters that traduce the real interface behavior (see Fig.
Numerical model of a pin-on disc system
A 3D finite element model is proposed for a pin-on disc system which has been developed from a bench test designed at the LAMCube laboratory to investigate friction material performances machined in brake pad [34], [35]. The model is made of several components. Essentially, there is a pin-housing, a disc and a friction pin maintained by a thin plate (see Figs. 1 and 7). At the remote extremities of the thin plate a normal load is applied which permits to enforce contact between the pin
Conclusions
A multi-scale thermomechanical strategy has been presented in this work to model complex systems including frictional contact. This strategy considers both the system's scale and the local contact interface behavior and evolution. The interface modeling is devoted to thermal, mechanical and wear issues.
At the macro-scale, a classical FEM approach considering a flat contact between the mating solids, thermal expansion and boundary conditions, is adopted. In order to integrate the micro-contact
Acknowledgments
The present research work has been supported by the International Campus on Safety and Intermodality in Transportation, the Hauts-de-France Region, the European Union, the Regional Delegation for Research and Technology, the Ministry of Higher Education and Research, the French National Research Agency (ANR COMATCO), the ELSAT project, and the National Center for Scientific Research. The authors gratefully acknowledge these institutions for their support.
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2022, Tribology InternationalCitation Excerpt :Early works on temperature evolution [24,31,70] and thermoelasticity [4,5] led to research on heat partition [8,72], multiple frictional contacts [20,71], influence of convective cooling [40], temperature dependent material parameters [46,76] and contributions to thermoelastic rough contacts [42,45]. Only a few investigations deal with thermomechanical contact simulations [9,17,23,39,41,53,67,73]. All these contributions have in common that they assume a stationary contact area during the whole sliding process or even stationary contact conditions.