Page 185 - Analysis and Design of Energy Geostructures

158 Analysis and Design of Energy Geostructures

Figure 4.11 Concept of perfect elasticity.

In many situations, the reversible mechanical behaviour of materials is associated
with linear stress strain relations, while their irreversible behaviour (typically achieved
when the applied loads exceed the aforementioned limit) with nonlinear stress strain
relations. However, the reversibility of the mechanical behaviour of materials does not
imply linearity and various materials are characterised by a reversible behaviour follow-
ing nonlinear stress strain relations, especially at small strain levels. Elasticity can be
linear or nonlinear. In the following, reference is made to a linear thermoelastic
behaviour of materials.

4.9.2 Thermoelastic stress strain relations

The total strain at each point of a material characterised by a thermoelastic behaviour
is generally given by the sum of two contributions. The first contribution comprises
the strains induced by the application of a force (or displacement) field that is required
to maintain the continuity of the material by means of the generalised Hooke’s law.
The second contribution comprises the strains induced by the application of a temper-
ature change to the material.
The previous considerations can be mathematically expressed in compact form as

ε ij 5 C ijkl σ kl 2 β ðT 2 T 0 Þ ð4:48Þ
kl
where C ijkl is the elastic compliance matrix (i.e. inverse of the elastic stiffness tensor,
D ijkl ), β is a vector that comprises the linear thermal expansion coefficient of the
kl
material, α,and T 2 T 0 5 ΔT is the applied temperature variation. Strains induced
by thermal loads are proportional to the applied temperature variation, ΔT, and to
the linear thermal expansion coefficient of the material, α. Table 4.1 summarises the
values of linear thermal expansion coefficient for many materials of practical interest
in the context of energy geostructures. The temperature variations associated with
the geothermal operation of energy geostructures should range, at worst, between