Page 21 - Applied Petroleum Geomechanics

Stresses and strains 11

(A) σ 1 (B)

σ

β
L' L
σ

α α

σ 1
Figure 1.10 Illustration of normal and shear strains. s 1 , s are the applied stresses; L, L’
are the original and the deformed lengths, respectively; a, b are the angles after
deformations.

in terms of relative displacement of particles in the body. Strain is defined as
deformation of a solid due to stress. It is a relative change in shape or size of
an object due to externally applied stresses (forces), and it is dimensionless
and has no unit. There are two types of strains: normal and shear strains.
Normal strain describes the relative size change; it is elongation or
contraction of a line segment. Contractile normal strain is taken as positive
in rock mechanics sign convention. If a rock sample in a typical uniaxial
compression test is loaded in the axial direction, then displacements
(contraction and elongation) are formed in axial and lateral directions. The
strain in axial direction is equal to the relative displacement (the length
change) divided by the original length of the rock sample, as shown in
Fig. 1.10A, i.e.,
L L 0
ε ¼ (1.14)
L
Engineering shear strain is defined as the change in angle between two
line segments originally perpendicular, as illustrated in Fig. 1.10B, i.e.,

g ¼ a þ b (1.15)

In a three-dimensional domain, the normal strains can be written in the
following forms:

vu x vu y vu z
ε x ¼ ; ε y ¼ ; ε z ¼ (1.16)
vx vy vz