Page 202 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 175

Eq. (4.89) expresses that elastic deformation arises as long as the stress state is inside
the yield surface. Eq. (4.90) expresses that plastic deformation arises as long as the stress
state lies or travels on the yield surface.
According to Prager (1949), when loading of a material characterised by a plastic
behaviour happens, the stress state must stay on the yield surface. This is the so-called
condition of consistency that should be satisfied to ensure an appropriate description of
the physical process involved in plastic deformation. The condition of consistency
requires that a yield criterion is satisfied as long as the material is in a plastic state.
The condition of consistency for the yield function of a material characterised by
perfect plasticity reads

df 5 @σ ij dσ ij 5 0 ð4:91Þ
@f

Classical examples of perfectly plastic yield criteria are those that have been pre-
sented, for example, by Coulomb (1773), Tresca (1864), von Mises (1928), Drucker
and Prager (1952), Lade and Duncan (1975), Matsuoka and Nakai (1974), Matsuoka
(1982) and Hoek and Brown (1980). Especially the Mohr Coulomb yield criterion,
which is based on the work of Coulomb (1773), may be used in a relatively straight-
forward way in the analysis and design of energy geostructures to address key features
of the mechanical behaviour of geomaterials such as soils and rocks.

4.10.5 Hardening plasticity
The concept of hardening plasticity is associated with the mechanical behaviour of
materials that while yielding present a further variation (e.g. increase) in stress for an
increase in strain (cf. Fig. 4.20). The concept of hardening plasticity has been applied

Figure 4.20 Concept of hardening (and softening) plasticity.