Fractal characteristics of loess formation: evidence from laboratory experiments P. Lu, I.F. Jefferson * , M.S. Rosenbaum, I.J. Smalley Geohazards Group, Civil Engineering Division, Nottingham Trent University, Nottingham NG1 4BU, UK Received 11 October 2001; received in revised form 26 November 2002; accepted 11 December 2002 Abstract The fractal is presented as a method for describing the geometry of particles, with particular reference to the breakdown of granular soils and the formation of loess. The preliminary results are reported: (a) for the extent to which silt due to comminution exhibits a fractal distribution; (b) the tendency of fractal dimension to change with the comminution process; and (c) the relationship between fractal dimension describing particle size distribution and the grinding time. Laboratory simulation confirms the general tendency of fractal characteristics to reflect the size reduction process. D 2003 Elsevier Science B.V. All rights reserved. Keywords: Loess formation; Silt; Comminution; Particle size distribution; Fractal; Laboratory experiment 1. Introduction Approximately 10% of the world’s land area is covered with loessic materials (Pesci, 1990). The mechanisms causing the formation of such an exten- sive sedimentary deposit have been a major focus of interest for loess studies (Smalley, 1966a, 1978, 1990; Smalley et al., 2000). Field studies have been supple- mented by laboratory experiments to investigate these mechanisms (Whalley et al., 1982; Drewry, 1986; Wright, 1995; Jefferson et al., 1997). A major component of loess is silt size quartz. Whereas quartz silt can be produced in the laboratory by crushing sand, a number of internal controls are known to operate within the grains that affect and limit the size distribution; in many cases, these are related by a power law (Turcotte, 1986, 1992; Perfect, 1997; Lu et al., 2000). One example of such a dis- tribution is given by the Schuhmann equation (Eq. (2)). This has been observed in the fragmentation of rock masses and represents a scale invariance of the fragmentation mechanism. It may be noted that scale invariance is considered to be a basic property pos- sessed by a fractal distribution. The term ‘‘fractal’’ is perhaps best described as a general concept, which is relevant to describing the geometry of irregular objects. The fractal concept was originally introduced by Mandelbrot (1967) and has since been generalised for describing the geo- metrical properties of irregular settings or fragments (Mandelbrot, 1983). The fractal geometry and related 0013-7952/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0013-7952(02)00287-9 * Corresponding author. Tel.: +44-115-848-2130; fax: +44-115- 848-6450. E-mail address: ian.jefferson@ntu.ac.uk (I.F. Jefferson). www.elsevier.com/locate/enggeo Engineering Geology 69 (2003) 287 – 293