Self-similar voiding solutions of a single layered model of folding rocks

TL;DR

This paper models the formation of voids in geological folding using a nonlinear differential equation derived from an obstacle problem, revealing a scaling law that links void size to pressure and stiffness.

Contribution

It introduces a novel single-layer model with a free boundary for void formation in folding rocks, deriving and analyzing a unique solution with a scaling law.

Findings

Existence and uniqueness of the solution to the differential equation.

Identification of a scaling law relating void size to pressure and stiffness.

Foundation for future multilayered models of voiding in geological folds.

Abstract

In this paper we derive an obstacle problem with a free boundary to describe the formation of voids at areas of intense geological folding. An elastic layer is forced by overburden pressure against a V-shaped rigid obstacle. Energy minimization leads to representation as a nonlinear fourth-order ordinary differential equation, for which we prove their exists a unique solution. Drawing parallels with the Kuhn-Tucker theory, virtual work, and ideas of duality, we highlight the physical significance of this differential equation. Finally we show this equation scales to a single parametric group, revealing a scaling law connecting the size of the void with the pressure/stiffness ratio. This paper is seen as the first step towards a full multilayered model with the possibility of voiding.