Page 174 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 147

Figure 4.6 Normal vector n i and stress vector t i acting on a surface in a two-dimensional Cartesian
coordinate system. Modified after Jaeger, J.C., Cook, N.G., Zimmerman, R., 2009. Fundamentals of
Rock Mechanics. John Wiley & Sons (Jaeger et al., 2009).

Figure 4.7 Stress components.

n
ð4:10Þ
i t 5 σ ji n i
where n i is the outward unit normal of the surface π i and σ ij is a symmetric second-
order tensor. The tensor σ ij is generally known as the stress tensor and describes the
stress state of any infinitesimal three-dimensional element of a considered material sub-
jected to loading (cf. Fig. 4.7). The stress tensor is characterised by nine stress compo-
nents that in rectangular Cartesian coordinates read
2 3 2 3
σ x
σ ij 5 4 σ xx σ xy σ xz 5 5 4 τ yx τ xy τ xz 5 ð4:11Þ
τ yz
σ yx
σ yy
σ y
σ yz
σ z
τ zy
σ zy
σ zx
τ zx
σ zz
The diagonal components, σ kk , of the stress tensor expressed in Eq. (4.11) act nor-
mal to the coordinate planes and are called normal stresses. The off-diagonal