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A Model of Heat Transfer in Brake Pads by Mathematical Homogenisation

Car braking often functions by friction between brake pads and steel discs connected to wheels. Part of the heat produced during braking is dissipated through the pad and the rear pressure mechanism. Overheating can cause the system to be blocked and provoke an accident. It is therefore important to know the mode of heat transfer in the brake pads. As a result of the characterisation of the porosity of brake pad prototypes, we propose a model of heat transfer inside these composite materials. They contain about ten varieties of solid grains (steel, brass, polymers, organic fibres, rubber, etc.) which can be classified into two groups: good heat conductors and average heat conductors. A structural analysis of the brake pads shows that they are porous materials obtained by embedment of metallic grains in a polygranular organic matrix. A highly interconnected intergranular porous network is situated between the organic constituents (o.e.) and the metallic grains (m.g.). This network enfolds the granular constituents with many points of linkage (o.e. o.e. and o.e. m.g.). Thus, the brake pad structure can, in a first approximation, be represented by a set of adjoining identical cubic cells (a few mm). Given these conditions, we propose the heat transfer model described below. Consider a periodic medium obtained by the juxtaposition of cubic cells of ε sides. Each cell is composed of a metallic grain (a good heat conductor), in partial contact with the less conducting organic matrix, and an insulating gas in the intergranular porous network. We assume that there is no heat exchange by convection in this gas. A brake pad is a limited domain of the space of this theoretical medium in which the period ε is smaller than the characteristic length of the domain consisting of two connected components which are distributed periodically. They are made up of a component of variable heat conductivity depending on an ε-periodic law and a gaseous fluid of zero heat conductivity. The heat equation in this domain, given the appropriate boundary conditions, gives a boundaryvalue problem where ε is a parameter. The mathematical theory of homogenisation (ε -> 0) allows us to establish a tensor of conductivity of an equivalent homogeneous medium. The coefficients of this tensor depend on the volume ratios of the metallic grains, the organic matrix, the intergranular porous network as well as the respective average conductivities of each solid class. The aim of this mathematical and physical approach is to estimate the heat transfer in these composite materials and thus make it possible to determine an optimal structure correlated to their tribological qualities.