17.9 On the Applicability Limits of the Strain Theory of Plasticity
263
Fig. 17.11 Case of a conic
point on the loading surface
β + δ
π
2
.
(17.31)
Let us introduce designations:
σ = τ o p, dσ = kdq, qe
p
= mdr,
(17.32)
1
2
1
G s
−
1
G
= A,
1
2
1
G t
−
1
G s
= B, N = 1 +
B
A
.
The singular vectors p, k, m are shown in Fig. 17.11. Taking into account these
designations, formula (17.26) is written as a vector equation:
de
p
= Adσ + B
σ
τ o
dτ o .
By using designations (17.32) and considering that τ o dτ o = σ dσ = τ o dq cos α, let
us write the previous equation as follows:
mdr = (Ak + pB cos α)dq.
(17.33)
Let us square both parts of Eq. (17.33). We obtain
dq
dr
=
A
2
+ (B
2
+ 2AB) cos
2 α
−1/2
.
Multiplying Eq. (17.33) by a single vector p gives
cos δdr = (A + B) cos αdq.
Let us find from two last equations:
cos δ =
N cos α
1 + (N 2 − 1) cos 2 α
1/2 .
(17.34)
The inequation (17.31) is equivalent to the following:
263
Fig. 17.11 Case of a conic
point on the loading surface
β + δ
π
2
.
(17.31)
Let us introduce designations:
σ = τ o p, dσ = kdq, qe
p
= mdr,
(17.32)
1
2
1
G s
−
1
G
= A,
1
2
1
G t
−
1
G s
= B, N = 1 +
B
A
.
The singular vectors p, k, m are shown in Fig. 17.11. Taking into account these
designations, formula (17.26) is written as a vector equation:
de
p
= Adσ + B
σ
τ o
dτ o .
By using designations (17.32) and considering that τ o dτ o = σ dσ = τ o dq cos α, let
us write the previous equation as follows:
mdr = (Ak + pB cos α)dq.
(17.33)
Let us square both parts of Eq. (17.33). We obtain
dq
dr
=
A
2
+ (B
2
+ 2AB) cos
2 α
−1/2
.
Multiplying Eq. (17.33) by a single vector p gives
cos δdr = (A + B) cos αdq.
Let us find from two last equations:
cos δ =
N cos α
1 + (N 2 − 1) cos 2 α
1/2 .
(17.34)
The inequation (17.31) is equivalent to the following: