Evolution of three–dimensional folds for a non–Newtonian plate in a viscous medium

Abstract
We derive analytical solutions for the three–dimensional time–dependent buckling of a non–Newtonian viscous plate in a less viscous medium. For the plate we assume a power–law rheology. The principal axes of the stretching Dij in the homogeneously deformed ground state are parallel and orthogonal to the bounding surfaces of the plate in the flat state. In the model formulation the action of the less viscous medium is replaced by equivalent reaction forces. The reaction forces are assumed to be parallel to the normal vector of the deformed plate surfaces. As a consequence, the buckling process is driven by the differences between the in–plane stresses and out of plane stress, and not by the in–plane stresses alone as assumed in previous models. The governing differential equation is essentially an orthotropic plate equation for rate dependent material, under biaxial pre–stress, supported by a viscous medium. The differential problem is solved by means of Fourier transformation and largest growth coefficients and corresponding wave numbers are evaluated. We discuss in detail fold evolutions for isotropic in–plane stretching (D11 = D22), uniaxial plane straining (D22= 0) and in–plane flattening (D11 = –2D22). Three–dimensional plots illustrate the stages of fold evolution for random initial perturbations or initial embryonic folds with axes non–parallel to the maximum compression axis. For all situations, one dominant set of folds develops normal to D11, although the dominant wavelength differs from the Biot dominant wavelength except when the plate has a purely Newtonian viscosity. However, in the direction parallel to D22 there exist infinitely many modes in the vicinity of the dominant wavelength which grow only marginally slower than the one corresponding to the dominant wavelength. This means that, except for very special initial conditions, the appearance of a three–dimensional fold will always be governed by at least two wavelengths. The wavelength in the direction parallel to D11 is the dominant wavelength and the wavelength(s) in the direction parallel to D22 is determined essentially by the statistics of the initial state. A comparable sensitivity to the initial geometry does not exist in the classic two–dimensional folding models. In conformity with tradition we have applied Kirchhoff's hypothesis to constrain the cross–sectional rotations of the plate. We investigate the validity of this hypothesis within the framework of Reissner's plate theory. We also include a discussion of the effects of adding elasticity into the constitutive relations and show that there exist critical ratios of the relaxation times of the plate and the embedding medium for which two dominant wavelengths develop, one at ca. 2.5 of the classical Biot dominant wavelength and the other at ca. 0.45 of this wavelength. We propose that herein lies the origin of parasitic folds well known in natural examples.

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