Page 169 - Analysis and Design of Energy Geostructures

142 Analysis and Design of Energy Geostructures

2 3
ε ij 5 4 ε xx ε xy ε xz 5 ð4:1Þ
ε yz
ε yy
ε yx
ε zz
ε zx
ε zy
The diagonal components, ε kk , of the infinitesimal strain tensor expressed in
Eq. (4.1) are called normal strains and represent stretching of an element (i.e. in one
dimension, this stretching expresses the fractional shortening or lengthening of a line
element with respect to its initial length). The off-diagonal components, ε kl , are called
infinitesimal shear strains and measure angular distortion (i.e. in one dimension, this
angular distortion expresses the half variation in angle between two segments that are
parallel to the coordinate axes in the reference state). The infinitesimal shear strains are
one half of the engineering shear strains, γ , that is ε kl 5 1=2γ .
kl kl
The components of the strain tensor are as follows in relevant coordinate systems.
• Cartesian coordinates x, y, z:

ε xx 52 @u ε yy 52 @v ε zz 52 @w
@x @y @z
!
1
ε xy 5 γ 52 1 @u 1 @v
2 xy 2 @y @x
ð4:2Þ
!
1
ε yz 5 γ 52 1 @v 1 @w
2 yz 2 @z @y
!
1
ε xz 5 γ 52 1 @w 1 @u
2 xz 2 @x @z

where u; v and w are the components of the displacement vector, u i , in the x; y
and z directions, respectively.
• Cylindrical coordinates r, θ, z:

!
ε rr 52 @u ε zz 52 @w ε θθ 52 u 1 1 @v
r @θ
@r @z r
!
ε rθ 52 1 1 @u 1 @v 2 v
2 r @θ @r r
ð4:3Þ
!
ε rz 52 1 @u 1 @w
2 @z @r
!
ε zθ 52 1 @v 1 @w
@θ
2 @z