Rock Mechanics.pdf - Mining and Blasting

METHODS OF STRESS ANALYSIS
It is instructive to follow the procedure developed by Airy (1862) and described by
Timoshenko and Goodier (1970), in establishing a particular form of the field equation
for isotropic elasticity and plane strain. The differential equations of equilibrium in
two dimensions for zero body forces are
or
∂xx
∂x
∂xy
∂x
+ ∂xy
∂y
+ ∂yy
∂y
= 0 (6.1)
= 0
∂2xy ∂x∂y =−∂2 xx
∂x 2 =−∂2 yy
∂y 2
For plane strain conditions and isotropic elasticity, strains are defined by
where
εxx = 1
E ′ (xx − ′ yy)
(6.2)
εyy = 1
E ′ (yy − ′ xx) (6.3)
xy = 1
G xy
= 2(1 + ′ )
E ′
xy
E ′ = E
1 − 2
′ =
1 −
The strain compatibility equation in two dimensions is given by
∂2εyy ∂x 2 + ∂2εxx ∂y 2 = ∂2xy ∂x∂y
(6.4)
Substituting the expressions for the strain components, (equations 6.3) in equation
6.4, and then equations 6.2 in the resultant expression yields
1
E ′
2 ∂ yy
∂x 2 − ′ ∂2xx ∂x 2
+ 1
E ′
2 ∂ xx
∂y 2 − ′ ∂2yy ∂y 2
= 2 (1 + ′ )
E ′
∂2xy ∂x ∂y
=− (1 + ′ )
E ′
2 ∂ xx
∂x 2 + ∂2yy ∂y 2
which becomes, on simplification,
168
∂2xx ∂x 2 + ∂2xx ∂y 2 + ∂2yy ∂x 2 + ∂2yy ∂y
2 = 0