5 Analyzing the Problem of a Spherical Cavity Expansion …
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To evaluate a critical pressure in a medium with the Mohr-Coulomb plasticity
criterion, the second equation in system (5.7) will be transformed into an equation
for a dimensional stress and velocity. Taking into account the fact that elastic–plastic
interface velocity c = O(V ), for V → 0 one has a Cauchy problem
dσ r
dξ
+ 2
Y + μσ r
ξ
= 0, ε ≤ ξ ≤ 1,
σ r (ξ = 1) = 2Y/3,
(5.22)
where the boundary conditions are determined from Eq. (5.18) for α = 0.
The solution of Cauchy problem (5.22) is function σ r (ξ ) =
2
3
Y ξ
−2μ
+Y
ξ
−2μ −1
μ
,
which, for ξ = ε, determines the critical pressure
P c =
2
3
Y ε
−2μ
+ Y
ε
−2μ
− 1
μ
.
(5.23)
The value of the critical pressure for μ = 0 can be found by passing to the limit
μ → 0 in (5.23)
P c =
2
3
Y
1 + ln ε
−3
.
(5.24)
In relations (5.23) and (5.24), the value of ε = V
c remains undetermined. To
determine it, an equation for the dimensionless velocity in equations system (5.7)
will be considered. Neglecting the convective summand and assuming 1 − θ ≈ 1,
yields
U
+ 2
U
ξ
= ξ
σ
r
K 1
,
where σ
r = −
2Y
3 (3 + 2μ)ξ
(−2μ−1)
, K 1 =
1 +
2
3
k
K = K
1 −
2
3
μ
−1 .
Finally, the problem is obtained with the initial conditions of the form
U
+ 2
U
ξ
= −2
Y
K
1 −
4
9
μ
2
ξ
−2μ
, ε ≤ ξ ≤ 1,
(5.25)
U (ξ = 1) =
Y
2G
The solution of Cauchy problem (5.25) is the function
U (ξ ) =
Y
18
(12G + 9K + 8Gμ)
K Gξ 2
−
2Y
9
(3 + 2μ)
K ξ 2 ξ
3−2μ
.