Page 53 - Rock Mechanics For Underground Mining

STRESS-STRAIN RELATIONS

are defined by

⎡ ⎤ ⎡ ⎤
xx ε xx
yy ε yy
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
zz ε zz
⎢ ⎥ ⎢ ⎥
[ ] = ⎢ ⎥ and [ ] = ⎢ ⎥
xy xy
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎣ yz ⎦ ⎣ yz ⎦
zx zx
The most general statement of linear elastic constitutive behaviour is a generalised
form of Hooke’s Law, in which any strain component is a linear function of all the
stress components, i.e.
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
ε xx S 11 S 12 S 13 S 14 S 15 S 16 xx
ε yy S 21 S 22 S 23 S 24 S 25 S 26 yy
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
ε zz S 31 S 32 S 33 S 34 S 35 S 36 zz
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ (2.37a)
xy S 41 S 42 S 43 S 44 S 45 S 46 xy
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ yz ⎦ ⎣ S 51 S 52 S 53 S 54 S 55 S 56 ⎦ ⎣ yz ⎦
zx S 61 S 62 S 63 S 64 S 65 S 66 zx
or
[ ] = [S][ ] (2.37b)
Each of the elements S ij of the matrix [S] is called a compliance or an elastic
modulus. Although equation 2.37a suggests that there are 36 independent compli-
ances, a reciprocal theorem, such as that due to Maxwell (1864), may be used to
demonstrate that the compliance matrix is symmetric. The matrix therefore contains
only 21 independent constants.
In some cases it is more convenient to apply equation 2.37 in inverse form, i.e.

[ ] = [D][ ] (2.38)

The matrix [D] is called the elasticity matrix or the matrix of elastic stiffnesses.For
general anisotropic elasticity there are 21 independent stiffnesses.
Equation 2.37a indicates complete coupling between all stress and strain compo-
nents. The existence of axes of elastic symmetry in a body de-couples some of the
stress–strain relations, and reduces the number of independent constants required to
define the material elasticity. In the case of isotropic elasticity, any arbitrarily ori-
ented axis in the medium is an axis of elastic symmetry. Equation 2.37a, for isotropic
elastic materials, reduces to

⎡ ⎤ ⎡ ⎤ ⎡ ⎤
ε xx 1 − − 0 0 0 xx
− −
ε yy yy
⎢ ⎥ ⎢ 1 0 0 0 ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
ε zz zz
⎢ ⎥ 1 ⎢ − − 1 0 0 0 ⎥ ⎢ ⎥
= (2.39)
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ E ⎢ 0 0 0 2(1 + ) 0 0 ⎥ ⎢ ⎥
xy xy
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ 0
⎣ yz ⎦ 0 0 0 2(1 + ) 0 ⎦ ⎣ yz ⎦
zx 0 0 0 0 0 2(1 + ) zx
35