13.4 Other Invariants in Plasticity Theory
171
and second, that the deviator of plastic deformation coincides with the plastic
deformation tensor.
D ε p = T ε p =
⎛
⎝
e xx e xy e xz
e yx e yy e yz
e zx e zy e zz
⎞
⎠ ,
e xx = ε
p
x , e xy =
1
2
γ
p
xy , . . .
,
(13.20)
since
e 0 =
1
3
(e xx + e yy + e zz ) = 0;
ε 0 =
1
3
(ε x + ε y + ε z ) =
1
3
(ε
y
x + ε
y
y + ε
y
z ).
(13.21)
13.4 Other Invariants in Plasticity Theory
We have already talked (p. 166) that other invariants can be formed from the
principal tensor values. In the mechanics of elastic deformations, the following three
invariants of the stress tensor are frequently used, which are expressed through the
previously defined I σ , II σ , I I I σ , (13.14),
J 1 = σ ii = I σ ,
J 2 = σ ij σ ij = I
2
σ − 2I I σ ,
J 3 = I
3
σ − 3I σ I I σ + 3I I I σ = σ ij σ jk σ ki .
(13.22)
Similar invariants can be written for the strain tensor.
Since I σ = 0, invariants of type (13.22) for the stress deviator will be
J
1 = I σ = 0,
J
2 = σ
ij σ
ij = −2I I σ ,
J
3 = σ
ij σ
jk σ
ki .
(13.23)
In an expanded form, formula (13.23) can be written as follows:
J
2 = σ
2
x + σ
2
y + σ
2
z + 2(τ
2
xy + τ
2
yz + τ
2
zx )
= σ
2
1 + σ
2
2 + σ
2
3 =
1
3
(σ x − σ y )
2
+ (σ y − σ z )
2
+ (σ z − σ x )
2
+ 6(τ
2
xy ) + τ
2
yz + τ
2
zx )