A model of thermal contact conductance at high real contact area fractions

Abstract

Thermal contact conductance (TCC) is studied in the whole range of real contact area fractions between zero and unity. For this purpose, a two-scale model is developed in which the effective (macroscopic) TCC coefficient is obtained from the solution of the heat conduction problem at the scale of asperities. Additional thermal resistance at the real contact spots is included in the model. The model is applied for several real 3D roughness topographies for which the effective TCC coefficient is determined as a function of the real contact area fraction and the local TCC coefficient at real contact spots. An analytical function is found which approximates this relationship in the whole range of parameters, and a characteristic length-scale parameter is introduced which characterizes the effective TCC properties of a rough surface.

Introduction

As all engineering surfaces are rough, contact phenomena are by their nature multiscale. At the microscale, contact occurs through the interaction of surface asperities, and these interactions govern the effective (macroscopic) properties of the surfaces at the macroscale. Thus consideration of the lower scales is necessary for successful predictive modelling of the complex contact phenomena, including those related to heat transfer at the contact interface.

The contact interface constitutes a barrier to heat flow, and the related thermal resistance is mainly due to constriction of the heat flux to the spots of real contact of surface asperities [1]. Additional resistance may be caused by thin layers of oxides, contaminants, etc., which are usually present on engineering surfaces. Other mechanisms of heat transfer, such as radiation and convection in the micro-gaps at the asperity scale, may be important in some contact conditions, but these effects are not addressed in this work.

The models available in the literature typically relate the coefficient of thermal contact conductance (TCC coefficient) to the contact pressure and the properties of the surfaces such as hardness, roughness parameters, etc.; see for instance the pioneering works [2], [3] and a recent review paper [4]. In these models much effort is put into accurate description of the relation between the contact pressure and real contact area. This is a contact mechanics problem which introduces additional parameters related to mechanical behaviour of the contacting bodies. Here, a relatively simple approach is available for the case of predominantly plastic deformations of asperities, which combines purely geometrical considerations with the assumption that the contact pressure at real contacts is equal to the hardness of the softer material [2]. This approach has next been generalized to account for elastic and elastoplastic deformations, e.g. [3]. Alternatively, the classical Greenwood–Williamson approach [5] can be used, in which the response of a typical asperity is averaged with respect to the statistical distribution of asperity heights, radii, etc., see also [6], [7], [8]. Direct approaches are also available which are based on direct solution of the asperity interaction problem for a representative sample of rough surface, e.g. [9], [10], [11].

The aim of the contact mechanics modelling is thus to predict distribution of real contact spots (size, number, fraction) for a given macroscopic contact pressure. Modelling of heat conduction provides then the macroscopic TCC coefficient for a given distribution of real contact spots. If material parameters and surface geometry do not vary with temperature, then the mechanical contact problem and the heat conduction problem can be solved separately, which is in fact a common approach, cf. [2], [3], [12]. In this work, we concentrate on the problems of the second type.

The majority of models of TCC are developed for the typical tribological conditions in which the fraction of real contact area is relatively small, so that the real contact spots are assumed to be circular and separated. Clearly, both assumptions are not valid in the conditions characterized by large fractions of real contact area, e.g. approaching unity. Furthermore, usually the constriction resistance is only accounted for, i.e. it is assumed that the temperatures of the two bodies are equal one to the other at the points of real contact. This assumption may be inadequate if, for instance, contaminant or oxide layers are present on the contact surfaces.

The aim of this work is thus to address two issues. Firstly, the thermal contact conductance is studied in the whole range of the real contact area fraction between zero and unity. The case of large real contact area fractions is of particular interest as this range is not covered by the majority of available models. At the same time, such conditions are typically met in metal forming processes, due to high contact pressures [13] and macroscopic plastic deformation [14], [15], but also in other situations such as contact of soft materials, e.g. elastomeric materials.

Secondly, we study the effect of additional thermal resistance at real contact spots. This local resistance may result from the thin layers of contaminants, oxides, boundary lubricants, etc., present at engineering surfaces, and also from the constriction resistance due to lower scale roughness. The local thermal resistance at real contact spots is thus introduced to include phenomena related to scales smaller than those considered in the actual analysis of contact of rough surfaces.

The paper is organized as follows. The problem of heat conduction at the scale of surface asperities is presented in Section 2.1 along with the averaging scheme which yields the effective TCC coefficient, Section 2.2. Results of finite element computations carried out for sample rough surfaces are provided in Section 2.3. An analytical function approximating these results is introduced in Section 3.1 and a characteristic length-scale parameter is introduced in Section 3.2. Scaling of the constriction resistance and the correlation of the characteristic length with roughness parameters are studied in Sections 3.3 Constriction resistance, 3.4 Correlation with roughness parameters, respectively. Finally, an empirical relationship for the effective TCC coefficient is proposed in Section 3.5.