Page 167 - Analysis and Design of Energy Geostructures

140 Analysis and Design of Energy Geostructures

The theories of plasticity and thermoplasticity address an irreversible mechanical
behaviour of materials (and general structural systems). An irreversible mechanical
behaviour is associated with the property of plasticity, which is common to many
materials (and general structural systems) whose deformation is caused by loads that do
exceed a certain limit (typically termed yield limit). In practice, it is unlikely for the
mechanical behaviour to be directly plastic due to loading (in those particular cases,
reference is made to a rigid, plastic behaviour). In contrast, plasticity typically follows
(e.g. progressively) elasticity. This phenomenon is common to materials such as soils,
whose reversible and irreversible mechanical behaviour are mathematically distin-
guished through the yield limit represented by the preconsolidation pressure.
When plasticity is addressed in the context of nonisothermal conditions while
accounting for preliminary elasticity, two main approaches can be considered. The first
approach resorts to the theory of thermoelasticity to account for the influence of
temperature variations on the reversible mechanical behaviour of the material, and
to the theory of isothermal plasticity to model the irreversible mechanical behaviour
of the material. The second approach resorts to the theory of thermoelasticity to
model the influence of temperature variations on the reversible mechanical behaviour
of the material and to the theory of thermoplasticity to model the irreversible mechan-
ical behaviour of the material while considering a dependence of the yield limit on
temperature. In the former framework, the modelled material (or general structural
system) is characterised by a thermoelastic, plastic behaviour. In the latter framework, the
modelled material (or general structural system) is characterised by a thermoelastic,
thermoplastic behaviour. Situations in which the behaviour of materials is irreversible are
typically associated with significant deformation levels caused by loading. These
situations can characterise energy geostructures in some applications.
In the following, the sign convention adopted is the typical one of soil and rock
mechanics. Contractive strains, downward displacements, compressive stresses and
increases in angles in the anticlockwise direction are considered to be positive. The
opposite is true, that is expansive strains, upward displacements, tensile stresses and
angles in the clockwise direction are considered to be negative. The Einstein’s sum
convention (Einstein, 1916) is used when advantageous for the presented mathematical
developments.

4.3 Strain
4.3.1 Concepts of deformation and strain
Materials subjected to loading exhibit variations of configuration. These variations of
configuration can be appreciated considering the displacement of each material point
before and after loading. Typical variations of configurations include a variation in
size, a variation in shape or a rigid-body displacement (cf. Fig. 4.2). The former two