Сягло И.С. Теоретическая механика
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Z
V (t)
ρ
d~v
dt
dV =
Z
V (t)
ρ
~
Φ dV +
I
S
~
f dS .
~
f dS
dS
~n
~
f
f
i
=
3
X
j=1
σ
ij
n
j
.
σ
ij
~
A
I
S
~
A~n dS =
Z
V
∇
~
A dV, =⇒
I
S
X
j
A
j
n
j
dS =
Z
V
X
j
∂A
j
∂x
j
dV.
I
S
f
j
dS =
I
S
X
j
σ
ij
n
j
dS =
Z
V
X
j
∂σ
ij
∂x
j
dV
Z
V (t)
(
ρ
dv
i
dt
− ρΦ
i
−
X
j
∂σ
ij
∂x
j
)
dV = 0.
ρ
dv
i
dt
= ρΦ
i
+
X
j
∂σ
ij
∂x
j
.
Z
V (t)
ρ[~r ×
d~v
dt
] dV =
Z
V (t)
ρ[~r ×
~
Φ] dV +
I
S
[~r ×
~
f] dS.
~v
ox
Z
V (t)
X
j
(y
∂σ
3j
∂x
j
− z
∂σ
2j
∂x
j
) dV =
I
S
X
j
(yσ
3j
n
j
− zσ
2j
n
j
) dS.
Z
V (t)
(σ
32
− σ
23
) dV = 0.
σ
32
= σ
23
σ
12
= σ
21
, σ
31
= σ
13
σ
ij
= −pδ
ij
,
δ
ij
f
i
= −pn
i
,
~
f = −p~n,
p
σ
ij
= −pδ
ij
+ 2µd
ij
+ λ(
X
k
d
kk
) δ
ij
,
µ λ
P
k
d
kk
= 0
ox
y
oy
v
x
= v(y), v
y
= 0, v
z
= 0,
d
12
= d
21
=
1
2
∂v
∂y
, n
x
= 0, n
y
= 1, n
z
= 0,
σ
11
= σ
22
= σ
33
= −p, σ
12
= σ
21
= µ
∂v
∂y
,
∆f
n
= ∆f
ny
= σ
11
∆S = −p ∆S,
∆f
τ
= ∆fτx = σ
12
∆S = µ
∂v
∂y
∆S.
∂v
∂y
> 0
µ
σ
ij
= λ(
X
k
ε
kk
)δ
ij
+ 2µε
ij
,
ε
ij
=
1 + ν
E
σ
ij
−
ν
E
(
X
k
σ
kk
) δ
ij
.
λ µ
E ν
i = j
E = µ
3λ + 2µ
λ + µ
, ν =
λ
2(λ + µ)
.
-
6
6
?
-
-
~v
x
y
~n
∆
~
f
τ
∆
~
f
n
∆S
-
6
-
-
y
x
∆
~
f
n
∆
~
f
τ
ox
σ
11
=
∆f
n
∆S
, ε
11
=
1
E
σ
11
=
1
E
∆f
n
∆S
, ε
22
= ε
33
= −νε
11
.
ε
11
ox
ox
σ
12
=
∆fτ
∆S
, ε
12
=
1 + ν
E
σ
12
=
1
2µ
σ
12
=
1
2µ
∆fτ
∆S
.
2µ
µ
d~v
dt
=
~
Φ −
1
ρ
∇p.