Page 170 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 143

where u; v and w represent here the components of the displacement vector in the
r; θ and z directions, respectively.
• Spherical coordinates r, θ, φ:

! !
ε rr 52 @u ε θθ 52 u 1 1 @v ε φφ 52 u 1 cotθ 1 1 @w
v
r @θ rsinθ @φ
@r r r r
!
ε rθ 52 1 1 @u 1 @v 2 v
2 r @θ @r r
ð4:4Þ
!
1
ε rφ 52 1 @u 1 @w 2 w
2 rsinθ @φ @r r
!
cotθ
ε θφ 52 1 1 @w 2 w 1 1 @v
2 r @θ r rsinθ @φ
where u; v and w represent here the components of the displacement vector in the r; θ
and φ directions respectively.
Fig. 4.3 expresses in graphical form and with reference to a two-dimensional case
in Cartesian coordinates the geometrical meaning of infinitesimal strains.

4.3.3 Volumetric and deviatoric strains

In many cases, it is useful to decompose the strain tensor in a volumetric (i.e. spherical)
part and a deviatoric (i.e. distortional) part. The above can be mathematically expressed as

Figure 4.3 Two-dimensional representation of infinitesimal strains in a rectangular Cartesian coor-
dinate systems. Redrawn after Lancellotta, R., 2008. Geotechnical Engineering. CRC Press.