Why is the stress on the rocks at the bottom of a mountain due to the weight of the mountain shear stress and not compressional or longitudinal stress?
If we try to understand why a mountain cannot grow taller than 10km on Earth, the explanation is that the shear stress on the rocks at the base of the mountain for a mountain that is taller than 10km is greater than the elastic limit of rocks. But why is this shear stress and not longitudinal stress?
Longitudinal (vertical) stresses try to compress the rock while shear (horizontal) stresses act horizontally and tend to spread the base of the mountain out sideways and indirectly reduce the height of the mountain. It is well known that the compression strength of concrete is far greater than the shear strength of concrete and that is why we use rebars when laying concrete. I think we can safely assume mountain rock behaves similarly to concrete, but is less homogenous.
Shear stresses try to pull the rock apart. Consider a boulder with a big crack running through most of it, it is easy to imagine it is much easier to pull the boulder apart than to compress it. If mountain rock has imperfections and flaws, then the same applies.
When testing a concrete mix, a 'slump test' is often done where the uncured concrete is piled into a conical shape. If it fails, the failure mode is for the sides to shear off sideways and the end result is a spread out lump of concrete mix with grains of sand in it that are not individually deformed or compressed. The failure is materialised as a spreading out rather than a compression of the individual grains of sand in the mix which would be much more difficult.
To a large extent the core near the base of a mountain behaves a bit like an incompressible liquid and the pressure on the base core, due the weight of the rock above it, acts in all directs, vertically and horizontally. It is much easier to move rock horizontally near the base sides, than to compress its volume vertically near the centre.
Another factor to consider is that the outer crust of the Earth essentially 'floats' on the more fluid mantle below. For a huge mountain, the crust below it will partially sink to displace more volume as per Archimedes buoyancy law.
It is always possible to resolve the state of stress into the principal components, in which there is no shear stress normal to the principal directions. However, in practice, the criterion for rock failure nearly always is expressed in terms of the maximum allowable shear stress on all possible orientations of area elements (e.g., von Misces). It is also possible to express the failure criterion in terms of the principal stresses, but this is rarely used.
