Page 207 - Analysis and Design of Energy Geostructures

180 Analysis and Design of Energy Geostructures

In the context of the modelling of hardening and softening of materials under non-
isothermal conditions, it is noteworthy that soils exhibit a characteristic behaviour.
Soils can exhibit strain hardening at constant temperature, that is the size of the elastic
domain increases for increasing plastic strain. However, at constant void ratio, soils also
show thermal softening for increasing temperature levels, that is the size of the elastic
domain decreases with increasing temperature. In other words, the material undergoes
plasticity earlier at higher temperatures. An in-depth analysis of this phenomenon has
been presented, for example, by Hueckel and Baldi (1990).

4.10.6 Critical state plasticity
The concept of critical state plasticity is associated with the mechanical behaviour of
materials for which a continuous distortion caused by shearing eventually leads to a
well-defined state. The concept of critical state has been formulated independently
by Roscoe et al. (1958) and Parry (1958), and it has found major applications in the
analysis of geomaterials such as soils based on the works of Roscoe et al. (1958),
Parry (1958), Schofield and Wroth (1968) and Roscoe and Burland (1968),
for example.
Under critical state conditions, the material flows as a frictional fluid (Schofield and
Wroth, 1968), so that yielding occurs at constant volume and constant stresses. The
critical state is defined by the following two equations (Schofield and Wroth, 1968)
ð4:109Þ
q 5 M c p

~
Γ 5 v e 1 λlnp ð4:110Þ
~
where M c , Γ and λ represent material parameters, v e is the specific volume of the

material (v e 5 1 1 e, with e being the void ratio of the material) and p is the relevant
mean stress (in general, the mean effective stress). Eq. (4.109) and (4.110) identify the
so-called critical state line (CSL) in different planes (cf. Fig. 4.22). The CSL repre-
sented in the v e 2 lnp plane is parallel to another key setting for the mechanical analysis
of geomaterials, i.e. the Normal Compression (or Consolidation) Line (cf. Fig. 4.23).
~
The slope of both of these lines represents the compression index, λ. Unloading a
material from a stress state lying on the NCL (normal compression line) involves a
stress path along the so-called unloading reloading line (URL). The slope of the URL
represents the swelling index, ~ κ. The intersection between the NCL and URL coin-
cides with the preconsolidation pressure. Variations in void ratio of the material that
are associated with elastic and plastic strains can be determined through an analysis of
the stress paths along the NCL and URL.