Page 134 - Rock Mechanics For Underground Mining

ROCK STRENGTH AND DEFORMABILITY

˙
The total strain increment {ε} is the sum of the elastic and plastic strain in-
crements

˙ ˙ e ˙ p
{ε}={ε }+{ε } (4.26)
A plastic potential function, Q ({ }), is defined such that
∂ Q
p
˙
{ε }= (4.27)
∂
where is a non-negative constant of proportionality which may vary throughout the
loading history. Thus, from the incremental form of equation 2.38 and equations 4.26
and 4.27

∂ Q
−1 ˙
˙
{ε}= [D] { }+ (4.28)
∂
where [D] is the elasticity matrix.
It is also necessary to be able to define the stress states at which yield will occur
and plastic deformation will be initiated. For this purpose, a yield function, F({ }),
is defined such that F = 0 at yield. If Q = F, the flow law is said to be associated.
p
˙
In this case, the vectors of { } and {ε } are orthogonal as illustrated in Figure 4.32.
This is known as the normality condition.
For isotropic hardening and associated flow, elastoplastic stress and strain incre-
ments may be related by the equation

ep ˙
˙
{ }= [D ][ε]
where

∂ Q ∂ F T
[D] [D]
ep ∂ ∂
[D ] = [D] −
∂F T ∂ Q

A + [D]
∂ ∂
Figure 4.32 The normality condi-
tion of the associated flow rule.

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