Stress-Strain Behavior of Jointed Rock - Rock Mechanics Based on an Anisotropic Jointed Rock Model - page 45

(3.9)

in which

ℓ 1 = sin α , m 1 = cos α , n 1 = 0,

ℓ 2 = cos β cos α , m 2 = -cos β sin α , n 2 = -sin β ,

ℓ 3 = -sin β cos α , m 3 = sin β sin α , n 3 = -cos β .

Insertion of (3.6) and (3.7) into (3.4) and multiplication by [T] -1 yields

The calculation of [T] -1 leads to the relationship

(3.10)

Thus, (3.4) can be replaced by

(3.11)

with

Equation (3.11) describes the relation between stresses and strains for a transversely
isotropic rock in the global coordinate system.
The inverse relation of (3.11)

(3.12)

is obtained by inverting the elasticity matrix [D], which can be accomplished in consid-
eration of (3.10) as follows:

From the thermodynamic constraint of positive defi nite elastic strain energy the
theory of elasticity provides lower and upper bounds of the elastic constants for