384
29 Plane-Plastic Strain
dγ ν = r(( 0 , t)dd 0 ,
(29.27)
where dγ ν is the shift on the plateau with the normal line ν, (dγ ν = dγ νλ at α =
ω = π/2). By summing the shears (29.27) under the tensor rule, we will obtain the
following formulas for the components of plane-plastic strain:
γ xy (t) =
2 (t)
− 1 (t)
r(( 0 , t) sin 2(( 0 + χ 0 )dd 0 ,
ε x (t) = −ε y (t) =
1
2
2 (t)
− 1 (t)
r(( 0 , t) cos 2(( 0 + χ 0 )dd 0 .
(29.28)
Let us replace the variable 0 with ζ in formulas (29.26) and (29.28) assuming
that
ζ = 0 − δ 0 , δ 0 =
1
2
(( 2 − 1 ),
=
1
2
(( 1 + 2 ), r[ζ, t] = r(ζ + δ 0 , t).
(29.29)
These formulas will be converted into
g {r[ζ, t] + ε ˙
r[ζ, t]} = λ 4 (t) cos 2[ζ + δ 0 − χ (t)]−
−λ 5 (t) sin 2[ζ + δ 0 − χ (t)] − λ 0 (t),
(29.30)
γ xy (t) =
(t)
−
r[ζ, t] sin 2[ζ + δ 0 + χ 0 ]dζ,
ε x (t) = −ε y (t) =
1
2
(t)
−
r[ζ, t] cos 2[ζ + δ 0 + χ 0 ]dζ.
(29.31)
29.3.3 Continuity Condition
From the condition (25.6) of slip rate continuity at the boundary of the slip area, it
follows that at this boundary (ζ = ±), the tensor intensity of slips and its rate turn
to zero:
r[± t] + ε ˙
r[±, t] = 0.