288
19 Other Variants of Plasticity Theories
τ =
1
2
(σ 11 − σ 22 ) sin 2ω + σ 12 cos 2ω.
(19.4)
In elements where τ > τ s , plastic shift will occur
dγ = F (τ )dω,
causing plastic strain whose components in the axes x 1 x 2 will be
de
p
11 = −de
p
22 =
1
2
(de
p
11 − de
p
22 ) =
1
2
dγ sin 2ω =
1
2
F (τ ) sin 2ωdω,
de
p
12 =
1
2
dγ cos 2ω =
1
2
F (τ ) cos 2ωdω.
We will use the two-dimensional space of stresses assuming the following values
as coordinates
Q 1 =
1
2
(σ 11 − σ 212 ), Q 2 = σ 12 .
We will match these stresses to the following coordinates in a two-dimensional
space of strains with unit axes i 1 , i 2
q 1 =
1
2
(e
p
11 − e
p
22 ), q 2 = e
p
12 .
In the adopted designations, the initial loading surface is shown by the circumference Q 2
1 + Q 2
2 = const, and plastic strains are found using the formulas
q 1 =
1
2
F (τ ) sin 2ωdω,
q 2 =
1
2
F (τ ) cos 2ωdω,
(19.5)
whereas
τ = Q 1 sin 2ω + Q 2 cos 2ω.
(19.6)
Assume that additional loading is done at some stressed state. The components
of plastic strain will get gains
δq 1 =
1
2
F
(τ )δτ sin 2ωdω,
δq 2 =
1
2
F
(τ )δτ cos 2ωdω.
(19.7)