A viscoelastic strain energy principle expressed in fold–thrust belts and other compressional regimes

A mathematical folding theory for stratified viscoelastic media in layer parallel compression is presented. The second order fluid, in slow flow, is used to model rock rheological behavior because it is the simplest nonlinear constitutive equation exhibiting viscoelastic effects. Scaling and non-dimensionalization of the model system reveals the presence of Weissenberg number (Wi), defined as a ratio of time scales τ*/(H*/V*). V*/H* is the strain rate (s−1) imposed by an assumed far field velocity V* acting on a layer of thickness H*, while τ* (s) is related to the relaxation of normal stresses. Our most significant finding is a transitional behavior as Wi→½, which is independent of the viscosity contrast. A change of variables shows that lengths associated with this transition are scaled by a parameter α=[(1−2Wi)/(1+2Wi)]1/2, which is inversely proportional to local strain energy. On this basis a scaling law representing a distribution of non-dimensional wavelengths (wavelength/layer thickness) is derived. Geologically this is consistent with a transition from folding to faulting, as observed in fold–thrust belts. Folding, a distributed deformation scaling as Wi−1, is found to be energetically favored at non-dimensional wavelengths ranging from about three to seven. Furthermore, the transition from folding to faulting, a localized deformation scaling as (αWi)−1, is predicted at a non-dimensional wavelength of about seven. These findings are consistent with measurements of thrust sheets in the Sawtooth Mountains of western Montana, USA and other fold–thrust belts. A review of the literature reveals a similar distribution of non-dimensional wavelengths spanning a wide range of observational scales in compressional deformation. Specific examples include lithospheric scale folding in the central Indian Basin and microscopic scale failure of ice columns between splay microcracks in laboratory studies.