The Skin Effect of the Anelastic Earth

Assuming that the rheology of the earthʹs crust is defined by stress-strain relations containing a memory mechanism, it is seen that the stress field and the maximum shear stress (mss), caused by elongated mountain ranges, are limited mostly to the crust. In a geological, relatively short time, where relaxation is reached, the mss assumes the depth distribution (2~x)exp(-2~x) where x is the depth h measured in units of wavelength w of the mountain range; the mss reaches its maximum value, of about 1/3 the surface load, at the depth x = 1/2~ (x = h/w). Contrary to the case of the perfectly elastic earth model, where the mss at the surface is about 0.25 the maximum surface load, in the case of the anelastic earth model the mss is asymptotically nil at the surface and for depths larger than w, the mss is less than 1.2% the maximum surface load. It is seen that the relaxation of the medium causes a rotation of the planes of the mss in a layer near the surface and that, in the case when the surface load is applied periodically, there is a phase shift between the surface load and the stress at depth which is less than four days when the period is one year; it is smaller for shorter periods. It is also seen that the creation of reservoirs lasting a finite time causes a small variation of the mss in the crust which, when the reservoir is removed, leaves a residual mss with maximum at the surface and decreasing exponentially with depth; for a load lasting one year it takes 250 days to reduce to 40% the stress at depth. As a consequence of the residual mss, caused by the load of the topography, after earthquakes the local mss will not drop to zero; this decreases the time required by tectonic forces to release the next earthquake and increases the rate of seismicity.