386
29 Plane-Plastic Strain
Hence, using identical transformations, we can obtain the following equations:
γ xy (t) + ε ˙
γ xy (t) − 4εε x (t) ˙
δ 0 (t) =
=
(t)
−(t)
{r[ζ, t] + ε ˙
r[ζ, t]} sin 2[ζ + δ 0 (t) + χ 0 ]dζ,
(29.34)
ε x (t) + ε ˙
ε x (t) + εγ xy (t) ˙
δ 0 (t) =
=
(t)
−(t)
{r[ζ, t] + ε ˙
r[ζ, t]} cos 2[ζ + δ 0 (t) + χ 0 ]dζ.
Let us calculate integrals in the right parts of formulas 29.35). To do it, we use the
expressions (r + ε ˙
r) for the sum (29.33). We obtain:
−
{r[ζ, t] + ε ˙
r[ζ, t]} sin 2[ζ + δ 0 (t) + χ 0 ]dζ =
= λ 0 (t)
g N(() sin 2[δ 0 (t) + χ 0 ],
(t)
−(t)
{r[ζ, t] + ε ˙
r[ζ, t]} cos 2[ζ + δ 0 (t) + χ 0 ]dζ =
= λ 0 (t)
g N(() cos 2[δ 0 (t) + χ 0 ],
where
N(() =
1
cos 2
−
1
4
sin 4
.
Taking into account these expressions, formulas (29.35) give as follows:
γ
2
xy (t) + ε ˙
γ xy (t)γ xy (t) + [ε x (t) − ε y (t)]
2
+
+ε[˙ ε x (t) − ˙
ε y (t)][ε x (t) − ε y (t)] =
=
λ 0 (t)
g
N(()
γ xy (t) sin 2[δ 0 (t) + χ 0 ] +
+[ε x (t) − ε y (t)] cos 2[δ 0 (t) + χ 0 ]} .
Taking into account equations (29.32) and (29.21), we can make sure that the latter
equation is identical to the following: