The classical work of Biot (1957, 1965) and of Ramberg (1963) has had a profound influence on the way we think about the mechanics of folding. Even though these treatments are small amplitude, two dimensional, linear theories, they highlight the essential characteristics of all subsequent, more elaborate theories. The essential characteristics here are, given adequate contrasts in constitutive parameters between a layer and its embedding medium, a wavelength selection process begins operating early in the deformation process; eventually a critical wavelength is amplified at an exponential rate to form strictly periodic waveforms independent of initial geometrical irregularities (unless, of course, these initial geometrical irregularities happen to have a wavelength coincident with that of the critical wavelength which is preferentially amplified).

As more complicated constitutive relationships have been investigated in recent years and as more general (large amplitude) and three dimensional theoretical treatments have been developed, it appears that modifications to the Biot/Ramberg linear theories arise:

1. For viscous, non-linear (specifically, power law) materials, the critical wavelength which is amplified is less than that predicted for a linear-viscous material; for instance, if the power law exponent in the stress-strain rate relation is 0.5 then the wavelength amplified is 0.8 of that expected by the Biot theory with the viscosity defined by (6).
2. For purely viscous materials deforming in three dimensions, two wavelengths are always amplified no matter if the deformation history is biaxial shortening, plane shortening or shortening plus extension in the plane of the layer. These two wavelengths combine to give the impression of superimposed fold systems; in a particular section plane through the fold system this may result in fold system profiles that are a-periodic.
3. Combining linear elastic behaviour with viscous behaviour immediately introduces the possibility that parasitic folds will develop. Hence, in two dimensions, we have the possibility of fold systems characterised by two dominant wavelengths and not just one as in the classical theories.
4. Elastoviscous materials undergo a spectrum of responses ranging from the domination of homogeneous shortening and weak buckling at relatively low strain rates, to little layer parallel shortening and predominantly buckling deformation at relatively high strain rates (for a given set of physical parameters).

The position in this spectrum of behaviour is governed by the Deborah Number which expresses the balance between the time scale for amplification of the fold system and the time scale for viscous relaxation of the system. For Deborah Numbers less than one, homogeneous shortening dominates with the result that initial geometrical irregularities are mainly passively amplified and have a dominating influence on the final shapes of deformation features. In some situations (eg. a favourable wavelength) the dynamic fold growth rate may become significant in later deformation stages. However, the wavelength selection mechanism which generates the Biot type wavelength is still not possible here because the limb dips of initial perturbations can reach quite high angles into finite amplitude fields during early homogeneous shortening and passive growth stages. Then initial geometrical irregularities again control final fold shapes. For Deborah Numbers greater than one, a Biot type wavelength selection process dominates over layer parallel shortening and deformation characterised by dynamic buckling dominates. In such a process initial geometrical irregularities are "de-amplified" so that initial geometrical irregularities play a minor role in governing final fold shapes. Since there is a strong relationship between viscosity, elastic moduli, competency contrast and the stress induced by a given imposed strain rate, the generalisations mentioned above depend intrinsically upon where one is in viscosity, elastic moduli, competency contrast, strain rate space.

5. The consideration of higher forms of non-linearity in the constitutive relations leads to even more complicated geometrical response. Specifically, the introduction of elastic softening into the simple Maxwell elastoviscous material behaviour results in the development of localised packages of folds within a layer rather than the infinitely extended periodic waveform predicted by the classical Biot/Ramberg theories. These localised packages of deformation have a distinctly fractal spatial distribution, the position and precise geometry of the localised fold package being critically dependent upon the initial kinematic framework. This kind of behaviour has all the hallmarks of a chaotic dynamic system.

6. Particle-in-cell finite element simulations of grain growth evolution during folding demonstrate the possibility that we may be able to map more accurately the deformation history by incorporating detailed information about relative grain growth rates and deformation rates. Grain size variation under applied stress leads to an additional softening mechanism which, in the limit of fast recrystallization, produces a simple power-law rheology, but which, in the case of recrystallization at rates comparable to deformation, would produce a more general behaviour where history dependent rheology influences the deformation itself.