82
V. L. Kotov et al.
5.8 Determining Stress in a Medium
with Mohr-Coulomb-Tresca’s Yield Condition
The function of plasticity for the Mohr-Coulomb condition, with the account of a
constraint on the maximal value of Tresca’s yield strength, in a dimensional and
dimensionless form has the following form:
f 2 ≡
τ 0 + μσ r , 0 < σ r ≤ σ M ,
τ M ,
σ r > σ M
, ˜
f 2 ≡
T 0 + μS, 0 < S ≤ S M ,
T M ,
S > S M ,
where σ M = (τ M − τ 0 )
μ and dimensionless quantities T M = τ M
ρ s c
2 , S M =
(T M − T 0 )
μ are introduced.
Stress S monotonely decreases with dimensionless coordinate ξ changing from ε
to 1, i.e., it has its minimal value for ξ = 1. The value of cavity expansion velocity
V M , at which S(ξ = 1) = S M , will be now determined.
From the relations on the shockwave for ξ = 1, it follows that σ r = ρ 0 cυ = ρ 0 c
2
θ .
The values of the shockwave velocity, volumetric strain, and ε, corresponding to
V = V M , will be designated as c M , θ M and ε M , respectively. To define c M , formula
(5.37) will be used. Then, to
c M =
3
√
sV M + C 0 /3, θ M = ε
3
M =
V
3
M
c
3
M
, σ M = ρ 0 c
2
M θ M = ρ 0
V
3
M
c M
determine V M , the following cubic equation is obtained:
τ M − τ 0
μ
3
√
sV M + C 0 /3
= ρ 0 V
3
M
(5.44)
In a similar way, the stress has its maximal value for ξ = ε. Symbol V 0 will
designate the value of cavity expansion velocity, for which equalities S(ξ = ε) = S M
or σ r (ξ = ε) = σ M hold. Using value V = V 0 in Eq. (5.43a): σ M = σ r (ξ = ε)
V =V 0 ,
the following nonlinear equation for determining V 0 is obtained
τ M − τ 0
μ
=
τ 0
μ
1 − ε
−2μ
0
+
ρ 0 V
2
0
1 − ε
3
0
3
(μ − 2)(2μ − 1)
+
2μ + 1
2μ − 1
· ε
1−2μ
0
−
μ − 1
μ − 2
· ε
4−2μ
0
,
(5.45)
where the following designations are introduced ε 0 = V 0 /c 0 , c 0 =
3
√
sV 0 + C 0 /3.
For the cavity expansion velocity varying in the range of V 0 < V < V M , it is
necessary to determine the value of dimensionless coordinate ξ = ξ M , for which the
dimensionless stress has the value S(ξ = ξ M ) = S M .
V. L. Kotov et al.
5.8 Determining Stress in a Medium
with Mohr-Coulomb-Tresca’s Yield Condition
The function of plasticity for the Mohr-Coulomb condition, with the account of a
constraint on the maximal value of Tresca’s yield strength, in a dimensional and
dimensionless form has the following form:
f 2 ≡
τ 0 + μσ r , 0 < σ r ≤ σ M ,
τ M ,
σ r > σ M
, ˜
f 2 ≡
T 0 + μS, 0 < S ≤ S M ,
T M ,
S > S M ,
where σ M = (τ M − τ 0 )
μ and dimensionless quantities T M = τ M
ρ s c
2 , S M =
(T M − T 0 )
μ are introduced.
Stress S monotonely decreases with dimensionless coordinate ξ changing from ε
to 1, i.e., it has its minimal value for ξ = 1. The value of cavity expansion velocity
V M , at which S(ξ = 1) = S M , will be now determined.
From the relations on the shockwave for ξ = 1, it follows that σ r = ρ 0 cυ = ρ 0 c
2
θ .
The values of the shockwave velocity, volumetric strain, and ε, corresponding to
V = V M , will be designated as c M , θ M and ε M , respectively. To define c M , formula
(5.37) will be used. Then, to
c M =
3
√
sV M + C 0 /3, θ M = ε
3
M =
V
3
M
c
3
M
, σ M = ρ 0 c
2
M θ M = ρ 0
V
3
M
c M
determine V M , the following cubic equation is obtained:
τ M − τ 0
μ
3
√
sV M + C 0 /3
= ρ 0 V
3
M
(5.44)
In a similar way, the stress has its maximal value for ξ = ε. Symbol V 0 will
designate the value of cavity expansion velocity, for which equalities S(ξ = ε) = S M
or σ r (ξ = ε) = σ M hold. Using value V = V 0 in Eq. (5.43a): σ M = σ r (ξ = ε)
V =V 0 ,
the following nonlinear equation for determining V 0 is obtained
τ M − τ 0
μ
=
τ 0
μ
1 − ε
−2μ
0
+
ρ 0 V
2
0
1 − ε
3
0
3
(μ − 2)(2μ − 1)
+
2μ + 1
2μ − 1
· ε
1−2μ
0
−
μ − 1
μ − 2
· ε
4−2μ
0
,
(5.45)
where the following designations are introduced ε 0 = V 0 /c 0 , c 0 =
3
√
sV 0 + C 0 /3.
For the cavity expansion velocity varying in the range of V 0 < V < V M , it is
necessary to determine the value of dimensionless coordinate ξ = ξ M , for which the
dimensionless stress has the value S(ξ = ξ M ) = S M .