352
27 Non-elastic Uniaxial Elongation–Compression
or
0.5σ z (t) sin 2α cos ω = ψ(t) + (t)
a[ϕ νλ (t) + ε ˙
ϕ νλ (t)]+
+ b[ νλ + ε ˙
νλ ]+
+ g(r νλ + ε ˙
r νλ ) + 1.5c[ε z (t) + ε ˙
ε z (t)] sin 2α cos ω
−
(27.3)
− AA sin 2α cos ω − BB,
where the point above the symbol designating the function means, as earlier, the
differentiation of this function in time.
Taking into account the designations (20.17), formula (20.16) defining the slip
vector intensity in the considered case of uniaxial elongation can be represented as
νλ = 2
(α,t)
−
ϕ(α, ω 0 , t) cos(ω − ω 0 )dω 0 ,
(27.4)
whereas (α, t) is the boundary value of the angle ω 0 , defining the opening of the
slip fan in the plane with the normal line ν(α, β). By substituting formulas (20.17)
into the definition of the tensor intensity of slip (20.21), we will find
r νλ = 2
W
−W
dw 0
(α 0 ,t)
−(α 0 ,t)
ϕ(α 0 , ω 0 , t)(η λ cos ω 0 + ξ λ sin ω 0 ) cos
2 w 0 dw 0 ,
(27.5)
where the interval [−W, W ] belongs to the arch (20.24).
27.2 Calculation of the Integral (27.5)
To calculate the curvilinear integral (27.5) along the section [−W, W ] of the
arch (20.24), let us use the coordinate α 0 as the integration variable. From
Eq. (20.24), let us find
cos(β − β 0 ) =
1 (α, α 0 , ω)
sin α 0
cos
2 α + tg
2 ω
,
sin(β − β 0 ) =
2 (α, α 0 , ω)
sin α 0
cos
2 α + tg
2 ω
,
(27.6)