Page 221 - Analysis and Design of Energy Geostructures

194 Analysis and Design of Energy Geostructures

d. The thermally induced strain of materials subjected to thermal loads is

governed by the material linear thermal expansion coefficient α με= C

and the applied temperature variation ΔT ½ CŠ.

e. The unit measure of the linear thermal expansion coefficient is:

i. C
ii. m/ C

iii. C/m

iv. 1/ C

f. For an infinitesimal element that is free to deform and solely subjected to
a thermal load, the induced strain is spherical (volumetric) and can be
expressed in tensor form as
2 3
2 αΔT 0 0
ε ij 5 ε th 5 0 2 αΔT 0
ij;f 4 5
0 0 2 αΔT

where α με= C is the material linear thermal expansion coefficient and

ΔT ½ CŠ is the applied temperature variation.
g. The compatibility equations imply that the deformation of a continuum
material occurs without creating any gaps or overlaps.
h. The stress tensor describes the stress state of any infinitesimal three-
dimensional element of a considered material subjected to loading. It is a
symmetric tensor characterised by nine stress components that in rectan-
gular Cartesian coordinates reads:

2 3 2 3
σ xx σ xy σ xz σ x τ xy τ xz
σ ij 5 4 σ yx σ yy σ yz 5 5 τ yx σ y τ yz 5
4
σ zx σ zy σ zz τ zx τ zy σ z
The diagonal components σ kk [Pa] of the stress tensor expressed in
relation act normal to the coordinate planes and are called normal stres-
ses. The off-diagonal components σ kl [Pa] act tangential to the coordi-
nate planes and are called shear stresses.
The stress tensor is symmetric and this property can be proven con-
sidering the principle of balance of angular momentum. As a result, the
complementary components of the shear stress are equal:

σ xy 5 σ yx