Page 226 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 199

r. For the cubic concrete sample:
The thermally induced volumetric stress is proportional to the volu-
metric thermal strain:

σ 5 3Kε th
th
As the sample is fully restrained, the prevented strain is equal to the
th
volumetric thermal strain ε 5 αΔT.
26
th
Thus: σ 5 3 KαΔT 5 3 20 10 10 10 5 6 MPa
For the cubic soil sample:
The same calculation is performed with a 30 MPa bulk modulus and
an equal coefficient of thermal expansion.
th
Thus: σ 5 9 kPa
s. i. The principal stresses are the normal stresses acting on the principal
planes, along which zero shear stresses are observed. For a stress
tensor in the form of a diagonal matrix, the principal stresses are
thus the diagonal components σ kk of the tensor, that is:

σ 1 5 10 kPa acting on the yz plane

σ 2 5 5 kPa acting on the xz plane

σ 3 5 2 kPa acting on the xz plane
ii. The three stress invariants are defined as

8
I 1 5 tr σ ij 5 σ ii 5 σ xx 1 σ yy 1 σ zz
>
>
>
< 1 2 2 2
I 2 5 σ ii σ jj 2 σ ij σ ij 5 σ xx σ yy 1 σ yy σ zz 1 σ zz σ xx 2 σ 2 σ 2 σ
2 xy yz zx
>
>
> 2 2 2
: I 3 5 detσ ij 5 σ xx σ yy σ zz 1 2σ xy σ yz σ zx 2 σ xx σ 2 σ yy σ 2 σ zz σ
yz zx xy
with
2 3 2 3
10 0 0
σ xx σ xy σ xz
σ ij 5 4 σ yx σ yy σ yz 5 5 4 0 5 0 5
0 0 2
σ zx σ zy σ zz
Thus
8
I 1 5 10 1 5 1 2 5 17
<
I 2 5 10 5 1 5 2 1 2 10 5 80
I 3 5 10 5 2 5 100
: