Rheological stability with couple stresses and its application to geological folding

Abstract

The author’s theory of elasticity and viscoelasticity of initially stressed anisotropic solids is extended to include couple stresses and an additional refinement represented by a dependence of the deformation on the second gradient of the stress. The theory is intended to provide an approximate continuous model valid over a wide range for the mechanics of laminated media. The equations derived for the continuous model are rigorous for a medium initially at rest under the initial stress but are applicable as an approximation to stability problems of viscous and plastic media which exhibit initial strain rate with finite strain. The effect of gravity is included. A ‘couple-stress analogy’ renders immediately applicable to the medium with couple stresses a large class of solutions obtained earlier for plates and multilayers under initial stress. The problem of internal buckling is discussed. The theory is particularly suited to the analysis of folding and gravity instability of stratified sedimentary rock and remarkably simple results are shown to provide striking verification when applied to a wide variety of geological structures.

Folding and faulting of an elastic continuum

“…In this context, two independent characteristic lengths˜ and can be introduced in the constitutive relations through: η 1 = c 11˜ 2 and η 2 = c 22 2 . For a layered rock material where stiff strata of thickness h are alternated with compliant strata [7] (his eqn 7.11, where b = 2η 1 ; see also [8]) provides the following relation for the internal length:˜ = h/ √ 6. A value representative of the layers reported in figure 1a,b is h ≈ 25 cm, so that˜ ≈ 10 cm.…”

“…Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/ by/4.0/, which permits unrestricted use, provided the original author and source are credited. the linear theory of (Cauchy) elasticity, 1 while Cosserat elasticity has already been advocated as particularly suited to model layered rock and fibre-reinforced materials [7][8][9][10][11].…”