Чеботарев А.Ю. Введение в механику сплошных сред
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F
Ω x ∈ R
3
, t ∈ (0, T )
F
t ∈ (0, T )
ξ = (ξ
1
, ξ
2
, ξ
3
) Ω
0
t = 0
F
0
(ξ, t)
F
0
(ξ, t) = F (γ(ξ, t), t)
(ξ, t) (x, t)
F
t
∂F
0
∂t
=
∂F
∂t
+
∂F
∂x
j
∂x
j
∂t
= F
t
+
~
∇F
F =
d
dt
= (x, t).
=
d
dt
=
t
+
k
∂
∂x
k
=
t
+
∂
∂x
h i.
d
dx
x = γ(ξ, t)
x = γ(ξ, t) = ξ +
t
Z
0
(ξ, t)dτ.
dx
dt
= (x, t), x(0) = ξ,
ξ → γ(ξ, t)
∈ C
1
ξ → x
J = det
³
∂x
∂ξ
´
dJ
dt
= J div , J(0) = 1.
J(t) > 0
I(t) =
Z
ω
t
F (x, t)dx =
Z
ω
0
F
0
(ξ, t)Jdξ.
dI
dt
=
Z
ω
t
µ
dF
dt
+ F div
¶
dx =
Z
ω
t
(F
t
+ div (F )) dx.
r
E(r) =
1
r
Z
|x|=r
x
2
1
dS.
x = 0
v
1
= ξ
1
e
t
, v
2
= ξ
2
+ ξ
1
+ t
2
, v
3
= 2t,
ξ = (ξ
1
, ξ
2
, ξ
3
) t = 0
t = 0 1
v
1
= x
2
− 2x
3
, v
2
= x
3
− x
1
, v
3
= 2x
1
− x
2
.
= d /dt
F = 1 div
ax
1
+ bx
2
+ cx
3
= C
1
x
2
1
+ x
2
2
+ x
3
3
= C
2
ρ, U, ,
n
, q
n
∈ C
1
(Q), ∈ C(Q),
Q = {(x, t) : x ∈ Ω
t
, t ∈ (0, T )} ⊂ R
4
n
, q
n
Z
ω
t
µ
dρ
dt
+ ρ div
¶
dx = 0 ∀ω
t
.
ω
t
dρ
dt
+ ρ div = 0
∂ρ
∂t
+ div(ρ ) = 0.
d
dt
Z
ω
t
ρF dx =
Z
ω
t
ρ
dF
dt
dx.
Z
∂ω
t
n
dS =
Z
ω
t
dx, = ρ
µ
d
dt
−
¶
,
7→
n
Q P : Q → L(R
3
)
n
= P h i.
P
ω
t
ρ
d
dt
= P + ρ .
P
∗
= P
P
P
P P h i
P P = −p(x, t)I
p(x, t)
P
Z
∂ω
t
q
n
dS =
Z
ω
t
Ψdx, Ψ = ρ
d
dt
µ
U +
2
2
¶
− ρ( · ) − div P h i.
7→ q
n
(x, t)
q
n
= − · .
ω
t
q
n
> 0
ω
t
ρ
dU
dt
= div P h i − · P − div .
D =
1
2
µ
∂
∂x
+
∂
∗
∂x
¶
P : D = div P h i − · P
ρ
dU
dt
= P : D − div .
P = P
∗
Ω
t
θ(x, t) ≥ 0
S(ω) =
R
ω
ρs dx ω s
dQ = θds Q
dQ = dU + dA,
dQ = θds dA
dU
dS(ω
t
)
dt
≥ −
Z
∂ω
t
( · )
θ
dS.
= − κ∇θ, κ
ρ
ds
dt
+ div
¡
θ
−1
¢
≥ 0.
ρ
dU
dt
= P : D + div(κ
~
∇θ).
v
1
= − g(r)x
2
, v
2
= g(r)x
1
, v
3
= h( r), r
2
= x
2
1
+ x
2
2
.
(P ) =
0 1 2
1 a 1
2 1 0
.
a
ω
ω
dρ
dt
= 0
div = 0
| | =
1,
n
= 0
= 0, P = −pI ω
∆θ > 0
θ
P
D
P = (−p + λ div )I + 2µD.
p λ, µ
λ µ
U = U(ρ, s)
dA = pd(ρ
−1
) = −
pdρ
ρ
2
.
P P : D
P = −
~
∇p + λ
~
∇(div ) + 2µ D,
P : D = −p div + Φ,
Φ = (λ +
2
3
µ)(div )
2
+ 2µD
0
: D
0
D
0
D
θ =
∂U
∂s
, p = ρ
2
∂U
∂ρ
,
ρ
dU
dt
= ρθ
ds
dt
− p div .
M0
dρ
dt
+ ρ div = 0, θ =
∂U
∂s
, p = ρ
2
∂U
∂ρ
,
ρ
d
dt
= −
~
∇p + (λ + µ)
~
∇(div ) + µ∆ + ρ ,
ρθ
ds
dt
= div (κ
~
∇θ) + Φ,
dρ/ dt = 0
div = 0
M0
M1
∂ρ
∂t
+
~
∇ρ = 0,
div = 0,
ρ
d
dt
= −
~
∇p + µ∆ + ρ .
ρ = Const M1
M2
div = 0,
d
dt
= −
1
ρ
~
∇p + ν∆ + , ν = ρ
−1
µ.
ν
M2
E(t) =
Z
ω
t
ρ
2
2
dx,
dE
dt
=
Z
∂ω
t
(
n
)dS −
Z
ω
t
Φdx, Φ = 2µD : D
Φ
d
dt
=
∂
∂t
+
∂
∂x
h i M0 M2
| |,
¯
¯
¯
¯
∂
∂x
k
¯
¯
¯
¯
,
1
ν
¿ 1,