29.3 Monotonous Plane-Plastic Strain
383
Fig. 29.1 Fan of slip
directions in monotonous
plane-plastic strain
or at least decreasing functions so that
˙
1,2 (t) > 0,
t t 0 , ˙
=
dd
dt
.
By equaling the shear resistance (29.23) to the tangential stress (29.20), let us
find as follows in the slip direction:
g [r(( 0 , t) + ε ˙
r(( 0 , t)] =
τ n (t) + AA
[t]
− cγ m (t) − cε ˙
γ m (t)
×
× cos 2[ 0 − χ (t)] − 2cεγ m (t) ˙
χ(t) sin 2[ 0 − χ (t)] −
ψ[t] − BB
[t]
.
(29.24)
Let us introduce designations:
λ 4 (t) =
τ m (t) + AA
[t]
− c [γ m (t) + ε ˙
γ m (t)] ,
λ 5 (t) = 2cεγ m (t) ˙
χ(t).
(29.25)
Then Eq. (29.24) will be written as follows:
g [r(( 0 , t) + ε ˙
r(( 0 , t)] = λ 4 (t) cos 2[ 0 − χ (t)]−
−λ 5 (t) sin 2[ 0 − χ (t)] − λ 0 (t), ( 0 ∈ [− 1 (t), , 2 (t)]) ,
(29.26)
whereas λ 0 (t) is again defined using formula (27.6).
In the considered case of plane-plastic strain, the formula (20.20) looks as
follows:
383
Fig. 29.1 Fan of slip
directions in monotonous
plane-plastic strain
or at least decreasing functions so that
˙
1,2 (t) > 0,
t t 0 , ˙
=
dd
dt
.
By equaling the shear resistance (29.23) to the tangential stress (29.20), let us
find as follows in the slip direction:
g [r(( 0 , t) + ε ˙
r(( 0 , t)] =
τ n (t) + AA
[t]
− cγ m (t) − cε ˙
γ m (t)
×
× cos 2[ 0 − χ (t)] − 2cεγ m (t) ˙
χ(t) sin 2[ 0 − χ (t)] −
ψ[t] − BB
[t]
.
(29.24)
Let us introduce designations:
λ 4 (t) =
τ m (t) + AA
[t]
− c [γ m (t) + ε ˙
γ m (t)] ,
λ 5 (t) = 2cεγ m (t) ˙
χ(t).
(29.25)
Then Eq. (29.24) will be written as follows:
g [r(( 0 , t) + ε ˙
r(( 0 , t)] = λ 4 (t) cos 2[ 0 − χ (t)]−
−λ 5 (t) sin 2[ 0 − χ (t)] − λ 0 (t), ( 0 ∈ [− 1 (t), , 2 (t)]) ,
(29.26)
whereas λ 0 (t) is again defined using formula (27.6).
In the considered case of plane-plastic strain, the formula (20.20) looks as
follows: