Page 192 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 165

For the case of plane strain, the stresses and strains highlighted in expressions
(4.57) and (4.58) are accompanied by a stress component σ zz given by

σ zz 5 νσ xx 1 σ yy 1 αET 2 T 0 Þ ð4:59Þ
ð
For the case of plane stress, the stresses and strains highlighted above are accom-
panied by a strain component ε zz given by

ε zz 52 ν σ xx 1 σ yy 2 α T 2 T 0 Þ ð4:60Þ
ð
E
3. Three strain displacement relations

ε xx 52 @u
@x
ε yy 52 @v
ð4:61Þ
@y
!
ε xy 52 1 @u 1 @v
2 @y @x

4. Boundary conditions
In the two-dimensional case, traction boundary conditions, considering the surface
of the body as traction-free, take the following form

σ xx n x 1 σ xy n y 5 0
ð4:62Þ
σ xy n x 1 σ yy n y 5 0
For displacement boundary conditions

u 5 F 1 ðHÞ
ð4:63Þ
v 5 F 2 ðHÞ
where H is a generic point of the bounding curve C 0 . Mixed boundary conditions
may of course arise. It may be shown that the solution of the two-dimensional prob-
lem formulated here is unique.
In addition to the previous two simplified approaches to model three-dimensional
thermoelastic problems, there is one rigorous approach allowing to model three-
dimensional problems with reference to two-dimensional conditions. This approach
assumes that three-dimensional problems are characterised by axisymmetric conditions.
Axisymmetric conditions refer to situations in which three-dimensional geometries, loads
and boundary conditions can be described by the revolution around an axis of symmetry
of a two-dimensional geometry, loads and boundary conditions. Mathematical