310
20 Problem Setting
Fig. 20.3 To the definition of
principal strains
s z
s y
s x
3 (z )
y
2
x
O
t yx
t xy
1
p/4
ε 1 = ε
y
1 + ε p
1 ,
ε 2 = ε
y
2 + ε p
2 ,
ε 3 = ε
y
3 + ε p
3 ,
γ xy = γ
y
xy + γ p
xy .
(20.27)
Taking into account that plastic strain occurs without changes in the volume (ε
p
1 +
ε
p
2 + ε
p
3 = 0), we can represent as follows:
ε 1,2 =
1
2
(ε
y
1 + ε
y
2 ± γ
y
xy ± γ
p
xy − ε
p
z ),
ε 3 = ε
y
z + ε
p
z .
(20.28)
By defining strains under Hooke’s law, let us represent formulas (20.28) as
follows:
ε 1,2 =
1
2
1
E
3σ (1 − ν) − (1 + ν)(σ z ∓ 2τ xy )
+ γ xy − ε z
,
ε 3 =
1
E
σ z − ν(σ x + σ y )
+ ε z ,
(20.29)
where σ is the mean stress
σ =
1
3
(σ x + σ y + σ z ),
and E and ν are the Young modulus and the Poisson coefficient of a material in an
elastic state.
The dependencies (20.29) show that principal strains will be fully defined if the
components of plastic strain γ xy and ε z are known. To calculate these, let us use
the definition of the tensor slip rate. Let ν = x and λ = y , whereas x and y are