358
27 Non-elastic Uniaxial Elongation–Compression
ε z (t) =
R
ϕ(α 0 , ω 0 , t) cos ω 0 cos α 0 sin
2 α 0 dω 0 dα 0 dβ 0 .
(27.27)
To calculate the integral in the right part of formula (27.27), let us successively
replace variables assuming at first that
α 0 =
π
4
+ v,
(27.28)
and then
cos 2v = F ((), F (() =
λ 0 /λ 1
cos
1 +
b
a
(2 − sin 2)
.
(27.29)
By using formula (27.22), we will obtain
ε z (t) =
2
3
·
λ 1 (t)
a
J (u),
(27.30)
where
J (u) =
3π
4
u
0
F (() ·
F (()(( + 1
2 sin 2) − 2(λ 0 /λ 1 ) sin
1 + b
a (2 + sin 2)
×
×
F (()dd
√
1 − F (()
,
(27.31)
whereas u is the positive square of Eq. (27.24) for sin 2α = 1, e.g.
cos u
1 + b
a (2u − sin 2u)
=
λ 0
λ 1
.
(27.32)
The latter formula and the ratio (27.24) show that the parameter u equals the
opening of the slip fan in the plane where the maximum tangential stress acts. The
function J (u) can be easily tabulated for various values of the ratio b/a.
By substituting formula (27.30) into the definition λ 1 according to (27.16), we
will find
λ 1 (t) =
1
1 + c
a J (u)
·
σ z (t) + 2AA
2[t]
.
(27.33)
From formula (27.16) and condition (27.32), we can obtain
1
2 σ z (t) + AA
ψ[t] − BB
=
1
δ(u)
1 +
c
a
J (u)
,
(27.34)