Though various methods of describing or characterizing the geometrical characteristics of folds have been proposed (Fleuty, 1964; Ramsay, 1967; Wilson, 1967; Elliot, 1968; Stabler, 1968; Hudleston, 1973a; Fletcher, 1977, 1979; Ramsay & Huber, 1987; Twiss, 1988; Johnson & Pfaff, 1989; Bastida, 1993; Grujic et al., 2002; Srivastava and Lisle, 2004), the mechanisms of folding continue to be difficult to study in the field of structural geology, as the folding deformation is rheologically dependant. This is far more complex than simple elastic deformation, highly variable fold geometries are generated depending on the rheological properties of what are usually non-linear materials.

Single-layer buckling has attracted the attention of theoreticians and experimentalists for nearly half a century. It's understanding is considered to be a key step to interpreting the complexity of natural folding and deformation. Ramberg (1960, 1964) and Biot (1961, 1964) proposed the theoretical and physical models to explain the rheological mechanism of single-layer buckle folding. The initial perturbation affects the buckling of a single-layer (Mancktelow and Abbassi, 1992; Zhang et al., 1996, 2000). The stress and strain are considered to be important factors affecting the buckling (Treagus and Sokoutis, 1992; Jeng et al., 2002). Lan and Huddleston (1996) employed finite elements to study the sharpness and development of buckle folds by curvature and finite strain, and suggested that two basic material properties affecting fold shape include non-linearity and anisotropy. They also discussed the rheological properties in numerical models (Huddleston & Lan, 1993, 1994; Lan & Huddleston, 1991, 1995, 1996). These studies are only concerned with simple buckling and agree that viscosity and thickness are the two most important factors affecting the buckling of a single-layer, but the coupling of two factors in complex fractal folds has not yet been studied.

Many factors affect folding and the fold complexity; e.g. the thickness, viscosity, multiple layers, layer parallel shortening, strain softening, initial perturbation, other instabilities, preferred wavelength, non-linear material and layer anisotropy, even the bond between layer and matrix, and many other unknown factors (Chapple, 1969; Fletcher, 1974,1977; Huddleston, 1973b; Abbassi and Mancktelow, 1990; Huddleston and Lan, 1994a, 1994b; Ramberg, 1964; Ramsay and Huber, 1987; Treagus, 1973, 1981, 1992; Zhang et al., 1996; Mancktelow, 1999), in which the thickness and viscosity of any layer is important. The question to be asked is: How does the coupling of thickness and viscosity affect the complexity of folds?

In this paper, we combine fractal theory with rheology to investigate how the fractal dimension (D) of complex folds is affected by the coupling of viscosity and thickness in single-layer buckling.