Page 224 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 197

2 3 2 3 2 3
σ xx σ xy σ xz p 0 0 s xx σ xy σ xz
4
4 σ yx σ yy σ yz 5 5 0 p 0 1 4 σ yx s yy σ yz 5
5
0 0 p
σ zx σ zy σ zz σ zx σ zy s zz
where σ kk [Pa] are the normal stresses, σ kl [Pa] are the shear stresses and
s kk [Pa] are the deviatoric stresses.
k. 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 tem-
perature change to the material that are required as well to maintain the
continuity of the material.
l. Stresses arise in a material because of the application of temperature varia-
tions as a result of the presence of a restraint (i.e. development of the ther-
mally induced strain prevented), of a nonuniform temperature distribution
(i.e. differential thermal deformation) or of a combination of these causes.
m. Considering the elastic superposition principle, the combined effect of sev-
eral loads acting simultaneously is equal to the algebraic sum of the effect
of each load acting individually that can be computed separately. For the
considered body subjected to the fields A and B:

u i;AB 5 u i;A 1 u i;B

ε ij;AB 5 ε ij;A 1 ε ij;B

σ ij;AB 5 σ ij;A 1 σ ij;B
n. Assuming a reversible material response upon loading and unloading, the
action of an equal but opposite force field than an initially applied field
cancels the overall effect of these fields.
o. A material free to deform can be subjected to stress due to the applica-
tion of a nonuniform temperature distribution causing differential ther-
mal deformation.
p. In the thermoelasticity framework, the total strain at each point of a
material subjected to a mechanical load and to a thermal load can be
mathematically expressed in compact form as:

ε ij 5 C ijkl σ kl 2 β ðT 2 T 0 Þ
kl
where σ kl [Pa] is the stress tensor, C ijkl [1/Pa] is the elastic compliance

matrix, β [1/ C] is the linear thermal expansion coefficient vector and
kl