Page 178 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 151

It is often convenient to number the principal stresses (without mandatorily referring
to the position of the stress components in the stress tensor reported in Eq. 4.26)sothat

σ 1 $ σ 2 $ σ 3 ð4:26Þ
The principal directions are mutually orthogonal because the eigenvectors of a
symmetric tensor, such as the stress tensor, are mutually orthogonal. Eqs (4.17) (4.26)
written thus far for the stress tensor can also be written for the strain tensor.
In two dimensions, analogous results can be obtained with Eq. (4.21) that reduces to

2
λ 2 σ xx 1 σ yy λ 1 σ xx σ yy 2 σ 2 5 0 ð4:27Þ

xy
The two principal stresses in the plane may then be written explicitly as
r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2
λ 5 σ 1;2 5 σ xx 1 σ yy 6 σ xx 2σ yy 1 σ 2 ð4:28Þ
2 2 xy
with the principal directions that make an angle ϕ with the x-axis of
x
tanϕ 5 n y 5 σ xy ð4:29Þ
x
σ xx 2 λ
n x
In two dimensions, the coordinate points (σ 1 ; 0Þ and (σ 2 ; 0Þ represent peculiar
points of the so-called Mohr circle of stress [for further details, see, e.g. Timoshenko
(1953)], which is given by the following equation

2

σ 2 2 σ xx 1σ yy 5 σ 1 1 2 ð4:30Þ
2
2 4
x 1 y 1 1 σ x 1 x 1 xy σ xx 2σ yy
are the normal and shear stress components acting in any direc-
where σ x 1 x 1 and σ x 1 y 1
with the x axis (positive in the anti-
tion x 1 ; y 1 , where the x 1 axis makes an angle α x 1
clockwise direction).
The Mohr circle of stress represents the setting of all possible stress states acting on
a point along different planes (cf. Fig. 4.8). The formulas for the stress components
derived from the Mohr circle of stress are

8
> σ xx 1 σ yy σ xx 2 σ yy
> 5 1
σ x 1 x 1 cos2α x 1 1 σ xy sin2α x 1
2 2
>
>
>
>
>
>
>
<
5 σ xx 1 σ yy 2 σ xx 2 σ yy
σ y 1 y 1 cos2α x 1 2 σ xy sin2α x 1 ð4:31Þ
2 2
>
>
>

>
>
> σ xx 2 σ yy
> 52
>
σ x 1 y 1 sin2α x 1 1 σ xy cos2α x 1
>
: 2