5 Analyzing the Problem of a Spherical Cavity Expansion …
71
5.3 Formulating a Boundary-Value Problem for a System
of Two First-Order Ordinary Differential Equations
in the Plastic Region
Consider a self-similar solution of the system for variable ξ = r/ct, introducing
dimensionless variables U =
υ
c
, S =
σ r
ρ 0 c 2 and designating T =
Y
ρ 0 c 2 , ˜
f 1 () =
f 1()
ρ 0 c 2 , ˜
K 1 =
∂ ˜
f 1( θ)
∂θ
As a result of the substitution, partial differential equation system (5.5) is
transformed into a system of ordinary differential equations:
U
+ 2
U
ξ
=
(ξ − U )
(1 − θ ) ˜
K 1
S
, S
+ 2
T + μS
ξ
=
(ξ − U )
1 − θ
U
,
where 1 − θ = 1 − ˜
f
−1
1 (S), the stroke shows differentiation for
Finally, the boundary-value problem for a system of two ordinary differential
equations (ODE), written in the normal form, will be as follows:
U
=
2
U ˜
K 1 + ˜
f 2 φ
ξ
φ 2 − ˜
K 1
, S
=
2 ˜
K 1
˜
f 2 + U φ
ξ
φ 2 − ˜
K 1
, ε < ξ < 1,
(5.7)
U (ξ = ε) = ε, U (ξ = 1) = U
e
, S(ξ = 1) = S
e
(5.8)
where φ = (ξ − U )
(1 − θ ), ε = V /c.
Equation systems (5.7) and (5.8) include unknown parameter c—propagation
velocity of the interface between the elastic and plastic regions, or the plastic shockwave front. The unknown velocity is found by iterations, until boundary condition |U (ξ = ε) − ε| < δ is satisfied to assigned accuracy δ. At each iteration step,
the fourth-order accuracy Runge–Cutta numerical method is used when self-similar
variable ξ varies from the elastic–plastic interface boundary (ξ = 1) to the cavity
boundary (ξ = ε).
The values of U
e , S
e in boundary conditions (5.8) are determined from the condition of continuity of the velocity and stress along the boundary with the region of
elastic behavior of the material for ξ = 1.
5.4 Formulation and Solution of the Boundary-Value
Problem for Second-Order ODE’s in the Elastic
Deformation Region
Consider the equation of motion in equation system (5.1), neglecting the convective
components in the time derivative of velocity:
71
5.3 Formulating a Boundary-Value Problem for a System
of Two First-Order Ordinary Differential Equations
in the Plastic Region
Consider a self-similar solution of the system for variable ξ = r/ct, introducing
dimensionless variables U =
υ
c
, S =
σ r
ρ 0 c 2 and designating T =
Y
ρ 0 c 2 , ˜
f 1 () =
f 1()
ρ 0 c 2 , ˜
K 1 =
∂ ˜
f 1( θ)
∂θ
As a result of the substitution, partial differential equation system (5.5) is
transformed into a system of ordinary differential equations:
U
+ 2
U
ξ
=
(ξ − U )
(1 − θ ) ˜
K 1
S
, S
+ 2
T + μS
ξ
=
(ξ − U )
1 − θ
U
,
where 1 − θ = 1 − ˜
f
−1
1 (S), the stroke shows differentiation for
Finally, the boundary-value problem for a system of two ordinary differential
equations (ODE), written in the normal form, will be as follows:
U
=
2
U ˜
K 1 + ˜
f 2 φ
ξ
φ 2 − ˜
K 1
, S
=
2 ˜
K 1
˜
f 2 + U φ
ξ
φ 2 − ˜
K 1
, ε < ξ < 1,
(5.7)
U (ξ = ε) = ε, U (ξ = 1) = U
e
, S(ξ = 1) = S
e
(5.8)
where φ = (ξ − U )
(1 − θ ), ε = V /c.
Equation systems (5.7) and (5.8) include unknown parameter c—propagation
velocity of the interface between the elastic and plastic regions, or the plastic shockwave front. The unknown velocity is found by iterations, until boundary condition |U (ξ = ε) − ε| < δ is satisfied to assigned accuracy δ. At each iteration step,
the fourth-order accuracy Runge–Cutta numerical method is used when self-similar
variable ξ varies from the elastic–plastic interface boundary (ξ = 1) to the cavity
boundary (ξ = ε).
The values of U
e , S
e in boundary conditions (5.8) are determined from the condition of continuity of the velocity and stress along the boundary with the region of
elastic behavior of the material for ξ = 1.
5.4 Formulation and Solution of the Boundary-Value
Problem for Second-Order ODE’s in the Elastic
Deformation Region
Consider the equation of motion in equation system (5.1), neglecting the convective
components in the time derivative of velocity: