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 there 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 that 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 toward a full multilayered model with the possibility of voiding.
Dodwell, T. J., Peletier, M. A., Budd, C. J., & Hunt, G. W. (2012). Self-similar voiding solutions for a single layered model of folding rocks. SIAM Journal on Applied Mathematics, 72(1), 444-463. https://doi.org/10.1137/110822499