Page 184 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 157

in such problems are found in both isothermal and nonisothermal conditions (Boley
and Weiner, 1997).

4.8 Generalities about stress strain relations

The definitions of the strain and stress tensors, the equations of compatibility and equi-
librium, as well as the boundary conditions alone are not sufficient to characterise the
actual mechanical behaviour of materials. The characterisation of the mechanical
behaviour of any material is supplied by so-called stress strain relations (often termed
constitutive equations), that is mathematical expressions that relate stresses to strains.
These relations complement the description of the mechanical behaviour of materials
with the elements of continuum mechanics proposed in the previous sections.
The general mathematical expression that relates stresses and strains reads

ð4:47Þ
dσ ij 5 M ijkl dε kl
where σ ij is the relevant stress tensor, M ijkl is the general constitutive tensor of the mate-
rial and ε kl is the total strain tensor. Eq. (4.47) is written in incremental form to
describe both linear and nonlinear relations. Linear stress strain relations can be
expressed without resorting to incremental formulations. A multitude of stress strain
relations can be formulated.
Modelling stress strain relations of continuous materials may be made by means of
the total stress tensor in Eq. (4.47). Modelling stress strain relations of porous materials
should be made by means of the effective stress tensor in Eq. (4.47). When variations in
pore fluid pressures are zero, that is when (fully or partially) drained conditions are
ensured during loading, the effective stress coincide with the total stress and analyses
disregarding the influence of the pore fluid on the mechanical response of the mod-
elled material(s) may be carried out. When pore fluid pressures vary, that is when
undrained conditions occur during loading, the above does not hold.

4.9 Thermoelasticity
4.9.1 Perfect thermoelasticity

The concept of perfect thermoelasticity is associated with a mechanical behaviour of
materials that is governed by a linear relation between stresses and strains under noni-
sothermal conditions. This concept derives from the one of perfect elasticity for iso-
thermal conditions (cf. Fig. 4.11). Materials characterised by a linear stress strain
relation under nonisothermal conditions are said to follow a linear thermoelastic
behaviour.