Natural Erosion of Sandstone as Shape Optimisation

Igor Ostanin,Alexander Safonov,Ivan Oseledets

doi:10.1038/s41598-017-17777-1

Igor Ostanin, Alexander Safonov + Show 1 more

Open Access

https://doi.org/10.1038/s41598-017-17777-1

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Abstract

Natural arches, pillars and other exotic sandstone formations have always been attracting attention for their unusual shapes and amazing mechanical balance that leave a strong impression of intelligent design rather than the result of a stochastic process. It has been recently demonstrated that these shapes could have been the result of the negative feedback between stress and erosion that originates in fundamental laws of friction between the rock’s constituent particles. Here we present a deeper analysis of this idea and bridge it with the approaches utilized in shape and topology optimisation. It appears that the processes of natural erosion, driven by stochastic surface forces and Mohr-Coulomb law of dry friction, can be viewed within the framework of local optimisation for minimum elastic strain energy. Our hypothesis is confirmed by numerical simulations of the erosion using the topological-shape optimisation model. Our work contributes to a better understanding of stochastic erosion and feasible landscape formations that could be found on Earth and beyond.

Highlights

Natural arches, pillars and other exotic sandstone formations have always been attracting attention for their unusual shapes and amazing mechanical balance that leave a strong impression of intelligent design rather than the result of a stochastic process
In order to bridge the natural erosion with shape optimisation, we suggest the following simplistic yet predictive model of erosion
Under certain conditions, differential laws of natural erosion, that originate in the fundamental law of dry friction, lead to optimal shapes that locally minimise the elastic strain energy stored in the rock bulk

Summary

Introduction

Pillars and other exotic sandstone formations have always been attracting attention for their unusual shapes and amazing mechanical balance that leave a strong impression of intelligent design rather than the result of a stochastic process. Weathering, abrasion and wind deflation removes the granular material in the regions subjected to relatively small compressive stresses, leaving more stressed and consolidated material intact, which, after many loops of erosion, leads to a distinctive natural arch structure This idea, undoubtedly fascinating by itself, looks even more interesting in a light of recently emerged engineering techniques of shape and topology optimisation. We demonstrate that the negative feedback between stress and erosion, discovered in[2], leads to shape optimisation of the rock, locally minimising its elastic strain energy We illustrate this idea with a simple analytical model of erosion and the numerical modelling, based on the approaches, developed in the field of shape and topology optimisation. Assume that the volume of the stone is composed of regularly arranged cubic particles (voxels) (Fig. 2(a))

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