Page 256 - Numerical Analysis and Modelling in Geomechanics

F.PERGALANI, V.PETRINI, A.PUGLIESE AND T.SANÒ 237
direction j at point r′ and φ (r′) is the force density on the boundary in direction
j
j. The closed-form solution of the Green function has been derived for two-
dimensional elastodynamic problems of the whole space (Kummer et al., 1987).
Equation (8.5) shows that the displacement at any internal point can be
determined as a sum of the effects of forces applied at the boundaries. Such an
equation stems from the Somigliana identity, which is the base of the direct
approach of the BEM. Kupradze (1963) showed that the displacement field is
continuous across the boundary, S, if ф is continuous along S.
ij
Stresses and tractions can be calculated by direct application of Hooke’s law
except at the boundary singularities, i.e. when r→r′ on the boundary. In this
situation the Green function, G has a logarithm-type integrable singularity, but
ij,
its derivative can be computed only if the singularity is extracted (Kupradze,
1963). The integral can be calculated by a limiting process based on equilibrium
considerations around the singularity, in the following form:

(8.6)

where t is the i-th component of traction at the boundary and C is equal to zero
i
outside the boundary and equal to ± 0.5 for a smooth boundary. The signs + and
− are valid respectively for the interior and exterior domain. T ij (r, r′) is the
traction Green function and represents the traction in direction i at point r on the
boundary due to the application of a unit force in the direction j applied at r′.
The equations (8.5) and (8.6) are the basic formulations for solving the
problem of wave propagation. Consider the space in Figure 8.10, in the
homogeneous infinite domain external to the valleys on the right, under
incidence of elastic waves. It is usual to distinguish the resultant motion as
composed of the free field motion, i.e. the incident waves, and the “scattered”
waves, i.e. those reflected, diffracted and refracted from the boundary. In fact the
ground motion in this irregular configuration physically comes from the
interference of incoming waves with those generated by the boundaries. On the
contrary, in the limited areas of the valleys, on the right-hand side of Figure 8.10
and also called inclusions, the motion is due only to diffracted waves. The
resultant motion is expressed by:

(8.7)

s
where u° is the incoming and u the scattered motion. It is assumed that both the
waves also exist for z>0, that is in the air, fulfilling the same analytical
expression valid for z≥ 0.
Similarly for stresses and tractions the resultant value is:

(8.8)