Page 189 - Analysis and Design of Energy Geostructures

162 Analysis and Design of Energy Geostructures

Figure 4.12 Configurations where (A) no stress arises and (B) stress arises.

because of the application of temperature variations as a result of the presence of a
restraint, of a nonuniform temperature distribution or of a combination of these
causes. In the context of energy geostructures, the combination of the previous causes
is more common.
Thermally induced stress arises in a material subjected to a uniform or nonuniform
temperature variation because, in the presence of a restraint, the development of the
strain induced by the applied temperature variation is prevented with respect to a case
in which free thermal deformation would occur. Thermally induced stress arises in a
material subjected to a nonuniform temperature variation [different than a temperature
field varying linearly with a set of rectangular Cartesian coordinates (Boley and
Weiner, 1997)] also in the case in which free deformation can occur because of the
differential thermal deformation of the infinitesimal elements that characterise the
considered material. This differential thermally induced deformation, together with
the fact that the material must remain continuous, indicates that each element restrains
the distortions of its neighbours and causes stress.

4.9.4 Three-dimensional thermoelastic modelling
The three-dimensional modelling of thermoelastic problems consists in the determina-
tion of 15 functions, assuming for simplicity the temperature distribution to be known.
In rectangular Cartesian coordinates, these functions are:
• six strain components ε xx ; ε yy ; ε zz ; ε xy ; ε yz ; ε zx ;
• three displacement components u; v; w; and
• six stress components σ xx ; σ yy ; σ zz ; σ xy ; σ yz ; σ zx :