Чеботарев А.Ю. Введение в механику сплошных сред
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R
n
R
n
r > 0 r
Φ R
n
Φ : R
n
× R
n
× . . . × R
n
→ R
1
,
Φ(a
1
, a
2
, . . . , a
r
) a
k
R
n
Φ
n
r
r
n
r
r = 1
R
n
Φ = a ·p, p ∈ R
n
R
n
R
n
r = 2
Φ(a, b) = a · L hbi
L(R
n
)
L : R
n
→ R
n
{e
i
}
n
1
T
ij
= e
i
T he
j
i
T
n = 3
det(T − λT ) = −λ
3
+ J
1
λ
2
+ J
2
λ + J
3
,
T J
1
= tr T =
P
T
ii
T
T
3
= det T
J
2
=
1
2
(tr T
2
− (tr T )
2
)
kT k = (max λ( T
∗
T ))
1/2
λ(A)
A
tr
A : B = tr(A
∗
B) = tr(B
∗
A) A
∗
A A
∗
haib = aA hbi
e
i
· e
j
= δ
ij
A : B = A
ij
B
ij
1 n
g(a, b) = a · b.
T
0
= T −
1
n
(tr T )I T
ε(a, b, c) = a ·(b × c)
R
3
ε
123
= ε(e
1
, e
2
, e
3
) > 0.
Ω ⊂ R
n
, n ≥ 1
R R
m
, m ≥ 2 L(R
m
)
{
i
}, i = 1, . . . , n
ϕ
grad ϕ =
~
∇ϕ =
∂ϕ
∂x
i
i
=
µ
∂ϕ
∂x
1
, . . . ,
∂ϕ
∂x
n
¶
.
div =
~
∇ · =
∂W
i
∂x
i
.
: R
3
→ R
3
rot =
~
∇ · = ε
ijk
∂W
k
∂x
j
i
,
ε
ijk
rot , ∇ϕ div
x T
x
=
∂W
∂x
(x + h) − (x) = T
x
hhi + o(|h|), |h| → 0
T
x
T
x
³³
∂W
i
∂x
j
´´
T
x
div T · = div (T
∗
h i) ∀ ∈ R
n
.
(div T )
i
=
∂T
ij
∂x
j
∆ϕ = div ∇ϕ, ∆ = div
µ
∂
∂x
¶
.
div
¡
∂
∂x
¢
∗
=
~
∇(div ) ∆ϕ =
n
P
1
∂
2
ϕ
∂x
2
i
, (∆ )
i
= ∆W
i
R
m
Z
Ω
f
∂ϕ
∂x
i
dx = −
Z
Ω
ϕ
∂f
∂x
i
dx +
Z
∂Ω
fϕn
i
dS,
= (n
1
, . . . , n
m
)
Ω
Z
Ω
div dx =
Z
∂Ω
( · )dS,
Z
Ω
div T dx =
Z
∂Ω
T h idS.
Z
S
(rot · )dS =
Z
C
d
S C
T ∈ L(R
2
) T h i =
(3a
1
+ a
2
; a
1
+ a
2
)
T T
∗
T
· T h i ≥ γ
¡
a
2
1
+ a
2
2
¢
, γ = 2 −
√
2.
λ = sup {T h i · , | | = 1}
| |
2
= a
2
1
+ a
2
2
B B
2
= T
Ω
T (x) h i = (ϕ (x) a
1
+ sin x
2
a
2
; x
2
a
2
+ x
3
a
3
; g (x) a
3
) ,
ϕ (x) = (3x
2
− x
1
) cos x
2
, g (x) =
3x
3
2
.
T (x)
=
R
∂Ω
T (x) h idS
Ω ⊂ R
3
Z
Ω
( · ∇ϕ + ψ · ∆ϕ)dx = 0
ϕ Ω
ψ = x
2
1
− x
2
2
T = T
∗
B = B
∗
R
Ω
f(x)ϕ(x)dx = 0 ϕ
Ω f ≡ 0
R
3
, Ω ⊂ R
3
t ∈ (0, T )
Ω
0
Ω
t
ξ → γ(ξ, t) ξ ∈ Ω
0
Ω
0
Ω
t
ξ ∈ Ω
0
t → γ(ξ, t)
(0, T )
ω ⊂ Ω
t
ρ ρ
=
Z
ω
ρ dx ;
∂ω
n
∂ω
F
s
=
Z
∂ω
n
dS ;
ω ∂ω
q
n
Q =
Z
∂ω
q
n
dS.
ω
t
dM
dt
=
d
dt
Z
ω
t
ρdx
= 0 M
d
dt
=
d
dt
Z
ω
t
ρ dx
= +
s
d
dt
=
d
dt
Z
ω
t
ρ( × )dx =
Z
ω
t
ρ( × )dx +
Z
∂ω
t
( ×
n
)dS
d
e
E
dt
=
d
dt
Z
ω
t
ρ(U +
1
2
2
)dx = Q +
Z
ω
t
ρ dx +
Z
∂ω
t
n
dS
e
E
Ω
∀ ω ⊂ Ω
ω → M(ω) ω
ω → E(ω) ω
ρ(x) =
dM
dω
= lim
x∈ω,|ω|→0
M(ω)
|ω|
U(x) = ρ
−1
dE/dω |ω| ω
M(ω) =
Z
ω
ρ(x)U(x)dx, E(ω) =
Z
ω
ρ(x)U(x)dx
F (ω)
F (ω
1
∪ ω
2
) = F (ω
1
) + F (ω
2
) ω
1
, ω
2
ω
1
∩ ω
2
= ∅
x = γ(ξ, t) ξ ∈ Ω
0
Γ
ξ
=
©
x ∈ R
3
: x = γ(ξ, t), t ∈ (0, T )
ª
ω
t
=
©
x ∈ R
3
: x = γ(ξ, t), ξ ∈ ω
0
ª
=
∂γ
∂t
dx/dt = (x, t)
t
t