Page 51 - Rock Mechanics For Underground Mining

PRINCIPAL STRAINS

gives for the normal strain components

∂u x ∂u y ∂u z
ε xx = , ε yy = , ε zz = (2.35)
∂x ∂y ∂z
and
1
∂u x
= xy − z
∂y 2
1
∂u y
= xy + z
∂x 2
Thus expressions for shear strain and rotation are given by

1
∂u x ∂u y ∂u y ∂u x
xy = + , z = −
∂y ∂x 2 ∂x ∂y
and, similarly,

∂u y ∂u z 1 ∂u z ∂u y
yz = + , x = −
∂z ∂y 2 ∂y ∂z
(2.36)
1
∂u z ∂u x ∂u x ∂u z
zx = + , y = −
∂x ∂z 2 ∂z ∂x
Equations 2.35 and 2.36 indicate that the state of strain at a point in a body is
completely defined by six independent components, and that these are related simply
to the displacement gradients at the point. The form of equation 2.34a indicates that
a state of strain is specified by a second-order tensor.

2.8 Principal strains, strain transformation, volumetric strain
and deviator strain

Since a state of strain is defined by a strain matrix or second-order tensor, determina-
tion of principal strains, and other manipulations of strain quantities, are completely
analogous to the processes employed in relation to stress. Thus principal strains and
principal strain directions are determined as the eigenvalues and associated eigen-
vectors of the strain matrix. Strain transformation under a rotation of axes is defined,
analogously to equation 2.13, by

∗
[ ] = [R][ ][R] T
∗
where [ ] and [ ] are the strain matrices expressed relative to the old and new sets
of co-ordinate axes.
The volumetric strain, , is defined by

= ε xx + ε yy + ε zz

The deviator strain matrix is defined in terms of the strain matrix and the volumetric
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