3 Failure Limits to Relief

Reaching the top of a high mountain is a great experience, yet there seem to be several limits. One is the height of the mountain itself. Material strength defines this limit to relief. Relief determines the driving stress, consisting of height, h, and density, ρ of the mass, accelerated by gravity, g and extenuated by slope, α. There are 3 failure modes in which the material strength can be surpassed: shear, compression, and tension. Failure criteria established for shear and compression have been demonstrated to be useful in certain settings, but often don't hold in steep bedrock landscapes (50-90°). E.g. threshold hillslopes limited by shear strength, often yield unrealistic low or infinite height. Compressive or crushing strength limitations predict realistic relief only in very weak materials. We propose another limit to relief based on tensile strength. Due to the Poisson effect of the normal stress, σ_n/compressive stress, σ_c, indirect tensile stresses (extensional strains), σ_t arise near free surfaces. The magnitude of these stresses is defined by the Poisson's ratio, ν and the relief. First order estimates are in good agreement with the heights of steep hard rocky slopes on Earth, and beyond!

In terms of the dynamics of steep rock walls the tensile strength limit criterion (TSLC) predicts critical yielding at the foot of the slope, causing surface parallel fractures that lead to further critical yielding and slope failure upward. This pattern of progressive failure has often been observed in steep rock walls, e.g. El Capitan, Half Dome. As each of the three limits to relief is best suited for a certain set of material properties and slope angles, dominant failure mechanisms can be associated with them, e.g. shear failure and sliding or toppling and fall due to tensile stresses.

We propose this criterion not contrary, but in addition to existing limit criteria. Further implications are that, (i) over-steepening or threshold slope angle doesn't necessarily exist, (ii) there is a transition point from one dominant limit to the other, which also implies a shift in the failure mechanism, and (iii) internal material property changes, e.g. due to chemical and/or mechanical weathering, can evoke a progressive reorganisation of yielding and potential failure without external forcing events.