3.12 Principal Strains and Strain Invariants
69
where
J 1 = ε x + ε y + ε z
J 2 =
ε x ε xy
ε yx ε y
+
ε y ε yz
ε zy ε z
+
ε z ε zx
ε xz ε x
J 3 =
ε x ε xy ε xz
ε yx ε y ε yz
ε zx ε zy ε z
We can also write as
J 1 = ε x + ε y + ε z
J 2 = ε x ε y + ε y ε z + ε z ε x −
1
4
γ
2
xy + γ
2
yz + γ
2
zx
J 3 = ε x ε y ε z +
1
4
γ xy γ yz γ zx − ε x γ
2
yz − εγ
2
zx − ε z γ
2
xy
Hence, the three roots ε 1 , ε 2 and ε 3 of the cubic Eq. (3.25e) are known as the
principal strains and J 1 , J 2 and J 3 are termed as first invariant, second invariant and
third invariant of strains, respectively.
3.13 Octahedral Strains
The strains acting on a plane which is equally inclined to the three co-ordinate axes
are known as octahedral strains. The direction cosines of the normal to the octahedral
plane are,
1
√
3
,
1
√
3
,
1
√
3
.
The normal octahedral strain is:
(ε n ) oct = ε 1 l
2
+ ε 2 m
2
+ ε 3 n
2
∴ (ε n ) oct =
1
3
(ε 1 + ε 2 + ε 3 )
(3.26)
Resultant octahedral strain = (e R ) oct =
(ε 1 l)
2
+ (ε 2 m)
2
+ (ε 3 n)
2
=
1
3
ε
2
1 + ε
2
2 + ε
2
3
(3.27)
Octahedral shear strain = γ oct =
2
3
(ε 1 − ε 2 ) 2 + (ε 2 − ε 3 ) 2 + (ε 3 − ε 1 ) 2 (3.28)