For ν = 0.25 the horizontal deviatoric stress is 2/9 of the lithostatic stress. With ρ = 3000 kg m−3 and h = 2 km the horizontal deviatoric stress is −13.3 MPa. This stress is of the same order as measured surface stresses.
We next consider erosion. If the initial state of stress before erosion is that given above, erosion will result in the state of stress that existed before sedimentation occurred. The processes of sedimentation and erosion are reversible. However, in many cases the initial state of stress prior to erosion is lithostatic. Therefore at a depth h the principal stresses are σ1 = σ2 = σ3 = ρgh. (3.28)
After the erosion of h km of overburden the vertical stress at the surface is σ̄1 = 0 (an overbar denotes a stress after erosion). The change in vertical stress ∆σ1 = σ̄1 − σ1 is −ρgh. If only ε1 is nonzero, Equation (3–21) gives
∆σ2 = ∆σ3 =
1 − ν
The horizontal surface stresses after erosion σ̄2 and σ̄3 are consequently given by σ̄2 = σ̄3 = σ2 + ∆σ2 = ρgh − ν
(1 − ν) ρgh = ( 1 − 2ν 1 − ν ) ρgh. (3.30)
If h = 5 km, ν = 0.25, and ρ = 3000 kg m−3, we find from Equation (3–30) that σ̄2 = σ̄3 = 100 MPa. Erosion can result in large surface compressive stresses due simply to the elastic behavior of the rock. This mechanism is one explanation for the widespread occurrence of near-surface compressive stresses in the continents.
Determine the surface stress after the erosion of 10 km of granite. Assume that the initial state of stress is lithostatic and that ρ = 2700 kg m−3 and ν = 0.25.