70
V. L. Kotov et al.
where K is elastic modulus of volumetric compression.
It is assumed that, in the region limited by radii r 0 = V t and r = ct, the medium
deforms plastically, with a linear dependence of yield strength on Mohr-Coulomb’s
pressure
σ r − σ θ = Y + kp,
(5.3)
where Y and k are initial value of yield strength and internal friction coefficient,
respectively.
In the adjacent region of elastic deformation, which is limited by coordinate
r e = c e t, stress tensor components are related to strains through Hooke’s law with
elastic moduli K and G, where c e =
K + 4G
3
ρ 0 is propagation velocity of
a plane elastic wave, G is shear modulus.
Yield condition s r − s θ = Y + kp gives equalities s r = 2(Y + kp)
3, s θ =
−(Y + kp)
3, wherefrom
σ r − σ θ = Y +
k
1 +
2
3
k
σ r = Y + μσ r , μ =
k
1 +
2
3
k
(5.4)
A one-dimensional problem of spherical cavity expansion in the plastic deformation region is then formulated. Partial differential equation system (5.1) for σ r and
υ, accounting for (5.4), will take the following form:
1
K 1
∂σ r
∂t
+ υ
∂σ r
∂r
+ (1 − θ )
∂υ
∂r
+
2υ
r
= 0,
∂σ r
∂r
+
2(Y + μσ r )
r
= −
ρ 0
1 − θ
∂υ
∂t
+ υ
∂υ
∂r
,
(5.5)
where:
1 − θ = 1 − f
−1
1 (σ r ), σ r = f 1 (θ ) ≡
2
3
Y +
1 +
2
3
k
f (θ ),
K 1 =
∂ f 1 (θ )
∂θ
≡
1 +
2
3
k
∂ f (θ )
∂θ
.
On the boundary of the expanding cavity of radius r 0 = V t, velocity V is assigned;
the outer surface of spherical layer r ∞ is free of stresses, at an initial time the velocity
and stresses in the medium are equal to zero:
υ(r 0 , t) = V, σ r (r ∞ , t) = 0, υ(r, 0) = σ r (r, 0) = 0
(5.6)