Seismic Ground Motion - Dynamic Behavior of Concrete and Seismic Engineering - page 111

- a law associated with a relative constancy of the stress drop: D 0/ L c | c 2 , with c 2
| 10 -5 and 10 -4 (which gives 'V = C P D 0/ L c |P c 2 | 3. 10 5 to 10 6 , i.e. 0.3 at 3 MPa);
- a “dynamic” law: WV R/ L | c 3 , with c 2 | 0.2 (the rising time is proportional to
the total length of failure propagation).
By integrating these empirical scaling laws into simple models, we can show
that:
- the magnitude of the moment M W = 2/3 log M 0 - 6 varies like the logarithm of
the fault's surface;
- the seismic moment, and the very low frequency content vary as the cubed
fault's dimension and as 10 1.5 M W ;
- the corner frequency (which is linked to the emission length of the waves on
the failure area), varies as the inverse of the fault's dimension, and then as 10 -0.5 M W ;
- from then on, the level of the plateau of the acceleration Fourier spectrum is
roughly proportional to the fault's dimension, and then it varies as 10 0.5 M W .
3.4.2. Effects of propagation in the Earth's crust
Once emitted, seismic waves propagate inside the Earth's crust: these deep
propagation effects are characterized by three main phenomena, which are discussed
in the following sections.
3.4.2.1. Geometric expansion
This is simply the decrease in wave amplitude due to distance, in close contact
with energy preservation.
For “volume” waves, the wave front is spherical, and the conservation of energy
(which is proportional to the squared amplitude on this surface) involves using a
geometric expansion term varying in 1/R.
For surface waves in which the energy concentrates in the direct vicinity of the
surface, the wave front is cylindrical and the geometric expansion term varies in
1/R 0.5 .
These theoretical dependencies are of course, only valid for far fields, and for a
perfectly homogenous space (or semi-space). In practice, the existence of near-field
terms, the non-homogenity of the propagation medium and the multiplicity of the
waves bring about geometric expansions with terms which are markedly different
from the exponents - 1 and - 0.5.