Specificity of Boundary Conditions for Laser-Stimulated …
299
Let us formulate the system (1)–(3) in terms of the solid phase using Hooke’s
laws. The continuity equation for the solid state of a substance, given the fact that
the density here is a constant, and the function of the source-sinks is zero (R m = 0),
may be written in the form:
v 0 = 0
(15)
The equations of motion (2) for a solid medium, taking into account the laws of
Hooke [13], will have the form:
ρ 0 ¨
u = ρ 0 f + (λ + μ)∇divu + μ · u,
(16)
where u is the displacement vector, f is the mass density of the bulk force, v 0 =
du/dt. The above equation characterizes the response of the elastic properties of a
solid to the action of a pulse, and condition (6) (taking Hooke’s laws into account)
is a boundary condition for it.
The third equation for energy in a condensed medium will have the form:
ρ 0
dU 0
dt
+ div Q 0 − P
(0)
i j v i j = 0.
(17)
In (17), ρ 0 is the density of a condensed medium, U 0 is the internal energy of a
unit mass of a solid, P
(0)
i j is the elastic tensor, Q 0 is the energy flux that has passed
into the volume of a solid medium.
Equation (17) in fact is a heat equation. This can be easily verified, taking into
account the definition of the generalized flux Q 0 = q 0 −λ 0 ∇T 0 and the determination
of the internal energy U 0 for temperatures lower than the phase transition temperature:
U 0 = −ϕ 0 + C
(0)
v T 0 .
In the latter formulas q 0 is the external energy flux in the bulk of the solid phase;
λ 0 is the coefficient of its thermal conductivity; T 0 is the temperature of the solid
phase, which is a function of time and coordinates; C
(0)
v is the mass specific heat
(specific heat of a unit mass) ϕ 0 is the interatomic bond energy of a unit mass of a
solid substance and this is in fact the specific energy of the phase transition.
Given all this, and also neglecting the terms P
(0)
i j v i j , (17) can be reduced to the
form:
ρ 0 C
(0)
v
∂ T 0
∂t
= div(λ 0 ∇T 0 ) − divq 0 .
(18)
It is taken into account that ϕ 0 = const and
dT 0
dt
=
∂ T 0
∂t
, since there is no convective
velocity in the solid phase. This equation must be considered together with the
boundary condition (12).
Introducing the coefficients L d and L allows us to separate the gas part of the
problem, considering it with the boundary condition (12), from the heat conduction
problem, which has the boundary condition:
299
Let us formulate the system (1)–(3) in terms of the solid phase using Hooke’s
laws. The continuity equation for the solid state of a substance, given the fact that
the density here is a constant, and the function of the source-sinks is zero (R m = 0),
may be written in the form:
v 0 = 0
(15)
The equations of motion (2) for a solid medium, taking into account the laws of
Hooke [13], will have the form:
ρ 0 ¨
u = ρ 0 f + (λ + μ)∇divu + μ · u,
(16)
where u is the displacement vector, f is the mass density of the bulk force, v 0 =
du/dt. The above equation characterizes the response of the elastic properties of a
solid to the action of a pulse, and condition (6) (taking Hooke’s laws into account)
is a boundary condition for it.
The third equation for energy in a condensed medium will have the form:
ρ 0
dU 0
dt
+ div Q 0 − P
(0)
i j v i j = 0.
(17)
In (17), ρ 0 is the density of a condensed medium, U 0 is the internal energy of a
unit mass of a solid, P
(0)
i j is the elastic tensor, Q 0 is the energy flux that has passed
into the volume of a solid medium.
Equation (17) in fact is a heat equation. This can be easily verified, taking into
account the definition of the generalized flux Q 0 = q 0 −λ 0 ∇T 0 and the determination
of the internal energy U 0 for temperatures lower than the phase transition temperature:
U 0 = −ϕ 0 + C
(0)
v T 0 .
In the latter formulas q 0 is the external energy flux in the bulk of the solid phase;
λ 0 is the coefficient of its thermal conductivity; T 0 is the temperature of the solid
phase, which is a function of time and coordinates; C
(0)
v is the mass specific heat
(specific heat of a unit mass) ϕ 0 is the interatomic bond energy of a unit mass of a
solid substance and this is in fact the specific energy of the phase transition.
Given all this, and also neglecting the terms P
(0)
i j v i j , (17) can be reduced to the
form:
ρ 0 C
(0)
v
∂ T 0
∂t
= div(λ 0 ∇T 0 ) − divq 0 .
(18)
It is taken into account that ϕ 0 = const and
dT 0
dt
=
∂ T 0
∂t
, since there is no convective
velocity in the solid phase. This equation must be considered together with the
boundary condition (12).
Introducing the coefficients L d and L allows us to separate the gas part of the
problem, considering it with the boundary condition (12), from the heat conduction
problem, which has the boundary condition: