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Two-scale models have become common practice to study the effect of roughness within tribology. For instance, in the context of lubrication, roughness has been treated with two-scale models since the early work of Patir and Cheng in the 1970's, In the context of contact mechanics, however, using two-scale models has proven much more challenging. This is because of the lack of scale separation. Indeed, in order to construct an efficient two-scale model including random features like roughness, one often demands the following conditions on length: L>>l>>d, where L is a reference macro-scale length, l is the size of the micro-scale cell and d is a reference length for surface roughness. The condition L>>l is required to see the micro-scale geometry as a point in the macro-scale problem and the condition l>>d is required to ensure that the results on a micro-scale cell are representative.
A common surface roughness is self-affine, with a power spectrum of the form
C(q) = C_0 q^{-2-2H}, for q_0>lambda_1), a direct simulation is very costly to perform, with millions of nodes needed to mesh a surface. Hence, two-scale models would be desired.
However, there is no clear point where a partition can be done. This is particularly true in non-conformal contacts, where the contact size is so small that it is usually of the order of lambda_0. As a result, the condition l>> d can rarely be satisfied.
In this work, the uncertainty in micro-scale contact mechanics results is discussed in situations where lis aproximatelly equal to d. Further a two-scale model is presented that incorporates the stochastic variability of the micro-scale results into the macro-scale model. In this manner, the need for the condition l>>d is eliminated. Exemplifying results on the rough ball vs. flat contact are provided.