Page 190 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 163

Therefore to solve this problem, the following fifteen equations need to be satisfied
throughout the material:
• six strain displacement relations;
• six stress strain relations;
• three equilibrium equations; and
• boundary conditions.
When the problem is formulated in this way and appropriate continuity restrictions
are set, the solution is unique [for the full development, see, e.g. Boley and Weiner
(1997)]. This means that there exists at most one set of 12 stress and strain compo-
nents, and one set of 3 displacement components [except possibly for rigid-body
motions, see, e.g. Boley and Weiner (1997)], which satisfies the above equations and
boundary conditions.
The three-dimensional modelling of thermoelastic problems usually resorts to numeri-
calmethods such as thefiniteelement anddifferencemethodstobecarried out.
Simplified approaches to model three-dimensional thermoelastic problems are presented
hereafter. These approaches may be considered for the analysis of energy geostructures.

4.9.5 Two-dimensional thermoelastic modelling
Two-dimensional analyses of thermoelastic problems may be of interest in some cases.
Such analysis approaches are proposed in the following by assuming that a temperature
distribution of the form T 5 Tðx; yÞ is known.
There exist two simplified approaches to model three-dimensional thermoelastic
problems as if they were two-dimensional. These approaches assume three-
dimensional problems to be characterised by plane strain and plane stress conditions.
A plane strain problem refers to circumstances in which all the strains and the displace-
ments associated to one coordinate direction can be considered to be zero. Plane strain
conditions refer to geometries of bodies characterised by one dimension much larger than
the other two. The geometry of the considered bodies may be associated to that of long
prisms subjected to a uniform and perpendicular distribution of loads along their principal
direction (cf. Fig. 4.13A C). Relatively small yet long cylindrical bodies loaded orthogo-
nally to their circular cross section may also be considered in plane strain conditions. The
mathematical formulation corresponding to plane strain conditions involves

ε zz 5 ε xz 5 ε yz 5 0 ð4:55Þ

A plane stress problem refers to circumstances in which all stresses associated to one
coordinate direction can be considered to be zero. Plane stress conditions refer to
geometries of bodies characterised by one dimension much smaller than the other
two. The geometry of the considered bodies may be associated to that of thin plates
subjected to a uniform and perpendicular distribution of loads along the cross section