Wear 249 (2001) 345–353 Fractal or fiction D.J. Whitehouse ∗ University of Warwick, Coventry, UK Received 10 May 2000; received in revised form 21 December 2000; accepted 8 February 2001 Abstract Fractal properties of surfaces have been explored by many investigators. Most have concluded that fractal characterisation is useful. This note questions the philosophy of using fractals to describe and control engineering surfaces. It concludes that the benefits are more virtual than real. The functional significance of fractal parameters is also examined and the overall question arises as to whether scale invariant parameters are appropriate. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Fractal; Surface roughness; Surface properties; Markov processes 1. Introduction There are two reasons for measuring surfaces. First is to help control manufacture, second is to help to ensure that the product performs well [1]. This note is primarily concerned with the former and in particular whether the recent interest in fractal characterisation is beneficial to manufacture. In manufacture there are two important areas: one is the manufacturing process such as grinding and the other is the means of applying the process, e.g. the path of the tool and/or machine tool characteristics. Surface assessment is used to control the first and to monitor the second. Control of the process at a rudimentary level has been achieved by using simple surface parameters such as the R a (the roughness average) to detect changes in the process. This approach is acceptable for statistical process control (SPC) because it can indicate that a process change has taken place; it cannot say what produced the change. For closed loop control the important process parameters have to be identified and measured. Input parameters such as feed, depth of cut, cutting speed etc. are not usually in doubt. What is important is the sur- face finish produced, the subsurface damage, grinding and cutting efficiency, effective cuts per unit distance, presence of microfracture, etc. also, deviations in the tool path, chat- ter and vibration to name just a few. It is the use or attempted use of fractal analysis in the problems outlined above which is being queried here. The ∗ Present address: 171 Cromwell Lane, Burton Green, Coventry CV4 8AN, UK. Tel.: +44-2476-473558; fax: +44-2476-471457. E-mail address: djw@djwhitehouse.forserve.co.uk (D.J. Whitehouse). first step is to find out what can be done at present. How random process analysis can be used is taken as the starting point, because it can also be related to fractal analysis and Markov processes which are other contenders as will be seen. Whether fractal characteristics are relevant to function or performance is not insignificant, on the contrary it is prob- ably more important than the implications in manufacture. This will be touched upon briefly in Section 5.2. 2. Random process analysis and manufacture There are two random process functions which can be used to help manufacture. These are (i) autocorrelation which is most useful in characterizing abrasive processes and (ii) power spectral analysis used mainly in single and multiple point cutting. 2.1. Autocorrelation 2.1.1. Form in abrasive manufacture Abrasive processes such as grinding have the property that there is a uniform probability of an abrasive cut anywhere on the surface being machined. This means that the centre positions of the machining grains represent a Poisson point process as shown in Fig. 1(a). Let the density of points be λ p . There are two considerations for the autocorrelation: one is the autocorrelation of the impression left on the surface by the grain, the other is the distribution of such points lead- ing to the probability of grain hits per unit profile distance. 0043-1648/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII:S0043-1648(01)00535-X