80
V. L. Kotov et al.
ε
3
+
ε
s M
−
1
s
= 0,
(5.35)
where M = V /C 0 .
The coefficients in Eq. (5.35) depend only on the parameters of the shock adiabat
of the medium C 0 , s and cavity expansion velocity V .
Consider a linear approximation of ε following from Eq. (5.35), using Taylor’s
series expansion ε =
1
3
√
s
1 +
−
ε
M
1/3 ≈
1
3
√
s
1 −
ε
3M
ε ≈ 3M
1 + 3M
3
√
s
.
(5.36)
Designating ε = V /c, the following expression is obtained:
c ≈
3
√
sV + C 0 /3.
(5.37)
A solution for a rigid-plastic medium was earlier obtained for the cylindrical
cavity expansion problem (Forrestal and Longcope 1982).
It is also to be noted that the dimensionless velocity of a medium is determined
only by Hugoniot’s shock adiabat and does not depend on the plasticity condition of
the medium.
5.7 Determining Stresses in a Medium
with the Mohr-Coulomb Yield Condition
It is assumed, in what follows, that the plasticity criterion of the medium is described
by Mohr-Coulomb law
f 2 () ≡ τ 0 + κ p = τ 0 + μσ r ,
(5.38)
where τ 0 is cohesion, κ is internal friction coefficient, p =
σ r + 2σ φ
3 is pressure,
μ = κ/1 + 2κ/3.
With the account of solution (5.34) and yield criterion (5.38), ODE (5.32) and
boundary condition (5.33) can be written as:
S
+ 2
T 0 + μS
ξ
= −2
ξ −
ε
3
ξ 2
ε
3
ξ 3 = −2
ε
3
ξ 2 + 2
ε
6
ξ 5 , ε < ξ < 1
(5.39)
S(ξ = 1) = ε
3
− ε
6
(5.40)
where T 0 = τ 0
ρ s c
2 .
With the account of boundary conditions (5.40), the dimensionless stress will take
the form