Page 202 - Numerical Analysis and Modelling in Geomechanics

BACK ANALYSIS OF GEOTECHNICAL PROBLEMS 183

Figure 6.6 Elastic moduli of the sand layers obtained by the in situ dilatometer tests and
by the elastic back analyses of the displacement measurements.
The τ–δ relationship can be schematically subdivided into three main parts. In
the first, the average shear stress increases until its peak value is reached. This
corresponds, in the average normal stress-shear stress (σ–τ) plane, to the so-
called peak failure envelope characterised by peak cohesion and friction angle.
Relatively small additional increments of displacements produce a reduction of
the shear resistance, caused by an almost complete loss of the peak cohesion.
This leads to a second (fully softened) failure condition. Finally, further large
increments of the horizontal displacement bring the friction angle to its residual
value, which represents the only non-vanishing parameter of the residual failure
envelope. During this process also a reduction of the instantaneous elastic
modulus could take place.
In order to account for this material behaviour in a finite element stress
analysis it is necessary to define a law governing the reduction of cohesion,
friction angle and elastic moduli from peak to residual values. This was based on
the relationship graphically depicted in Figure 6.9.
It is assumed that shear strength and stiffness parameters are functions of a
measure of the (irreversible) plastic deformation, represented by the square root
of the second invariant J of the deviatoric plastic strains.
2
The friction angle keeps its peak value until J reaches a “peak” limit J . Then
2p
2
a reduction occurs until a second limit J is attained, which corresponds to the
2r
residual friction angle. Analogous relationships, with different limits on J , are
2
adopted also for the remaining mechanical parameters.
In order to account for the effects of dilatancy, the plastic flow rule is related
to the variation of the friction angle. In particular, an associated flow rule is