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7/23/2019 Mecanica Del Medio
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10−8
Kn = λ/S λ
λ = 10−7
λ = 10−6
Kn ≤ 1
Kn > 1
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V S
B R
B
t B R P
R B P
B P
P R
B
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S R t = 0
P t = t b
p
P p
B ∆B P ∆m
∆B ∆V ∆V > 0
ρ ∆V → 0
ρ
ρ = lım∆V →∞
∆m
∆V
m =
ˆ V
ρdV
m B
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a11x1 + a12x2 + a13x3 =b1
a21x1 + a22x2 + a23x3 =b2
a31x1 + a32x2 + a33x3 =b3
ai1x1 + ai2x2 + ai3x3 = bi; i = 1, 2, 3
3j=1
aijxj = bi, i = 1, 2, 3
j
aijxj = bi
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aijxj = bi
aijxj = bi
ambm = a pb p
aijxj = bi
airxr = bi
aitxt = bi
ak + bk? a1 + b1, a2 + b2, a3 + b3
aiixi
3i=1
3j=1
aijbij
3i=1
3j=1
3k=1
aijbjkcki
3i=1
3j=1
aibj
aijbi j
aijbij = a1bj + a2bj + a3bj
aijbij = a11b11 + a12b12 + a13b13 + a21b21 + a22b22 + a23b23 + a31b31 + a32b32 + a33b33
aijbjkcki
aijbjkckj = a1jbjkck1 + a2jbjkck2 + a3jbjkck3
aijbjkcki = a11b1kck1 + a12b2kck1 + a12b2kck1+a21b1kck2 + a22b2kck2 + a23b3kck3+a31b1kck3 + a32b2kck3 + a33b3kck3
aijbjkcki = a11b11c11 + a11b12c21 + a11b13c31 + a12b21c11 + a12b22c21 + a12b23c31+
a13b31c11 + a13b32c21 + a13b33c31 + a21b11c12 + a21b12c22 + a21b13c32+
a22b21c12 + a22b22c22 + a22b23c32 + a23b31c12 + a23b32c22 + a23b33c32+
a31b11c13 + a31b12c23 + a31b13c33 + a32b21c13 + a32b22c23 + a32b23c33+
a33b31c13 + a33b32c23 + a33b33c33
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aikxkxj = bij
ai1x1xj + ai2x2xj + ai3x3xj = bij
a11x1xj + a12x2xj + a13x3xj = b1j a21x1xj + a22x2xj + a23x3xj = b2j a31x1xj + a32x2xj + a33x3xj = b3j
j = 1;
a11x1x1 + a12x2x1 + a13x2x1 = b11a21x1x1 + a22x2x1 + a23x3x1 = b21a31x1x1 + a32x2x1 + a33x3x1 = b31
j = 2;
a11x1x2 + a12x2x2 + a13x3x2 = b12a21x1x2 + a22x2x2 + a23x3x2 = b22a31x1x2 + a32x2x2 + a33x3x2 = b32
j = 3;a11x1x3 + a12x2x3 + a13x3x3 = b13a21x1x3 + a22x2x3 + a23x3x2 = b23a31x1x3 + a32x2x3 + a33x3x3 = b33
3n
3n
ai
aij
aijk
ai = (a1, a2, a3) aij
aij =
a11 a12 a13
a21 a22 a23a31 a32 a33
aijk =
a111, a112, a113, a121, a122, a123, a131, a132, a133a211, a212, a213, a221, a222, a223, a231, a232, a233a311, a312, a313, a321, a322, a323, a331, a332, a333
ai = Aijbj bi = Bijcj ai cj
ai = Aijbj =Aikbk bk = Bkjcj ai = AikBkjcj
(aijk + ajki + akij) xixjxk = 3aijkxixjxk
aijkxixjxk
ajkixixjxk
aijkxkxixj
x
aijkxixjxk
akijxixjxk aijkxjxkxi x aijkxixjxk
(aijk + ajki + akij ) xixjxk = 3aijkxixjxk
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(X 1, X 2, X 3)
= (a1, a2, a3) a1 a2 a3
ˆ
i
ei · ej = δ ij =
1 i = j0 i = j
ˆ j
ˆ i
ei · E j = δ ij =
1
i = j0 i = j
E j · ei = δ ji =
1 i = j0 i = j
δ ij 1 2 3
ai
Dmnbn
= a1e1 + a2e2 + a3e3 = aiei
e1 =1e1 + 0e2 + 0e3,
e2 =0e1 + 1e2 + 0e3,
e3 =0e1 + 0e2 + 1e3
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α = αai
· = a1b1 + a2b2 + a3b3 = aibi
· = | |2 = a21 + a22 + a23 = arar
cos θ = aibi|ar| |bs|
δ ij
δ ij = ei · ej =
e1 · e1 =e2 · e2 = e3 · e3 = 1
e1 · e2 =e2 · e1 = e1 · e3 = e3 · e1 = e2 · e3 = e3 · e2 = 0
δ ij δ ij = δ ji δ ii = δ 11 + δ 22 + δ 33 = 3
a11 = α (b11 + b22 + b33) + βb11
a22 = α (b11 + b22 + b33) + βb22
a33 = α (b11 + b22 + b33) + βb33
a12 = βb12, a13 = βb13, a21 = βb21,
a23 = βb23, a31 = βb31, a32 = βb32
a11 = α (brr) + βb11
=αδ 11brr + βb11
a22 = αδ 22brr + βb22
a33 = αδ 33brr + βb33
a12 a12 = αδ 12brr + βb12
a21 = αδ 21brr + βb21, a31 = αδ 31brr + βb31,
a23 = αδ 23brr + βb23, a13 = αδ 13brr + βb13,
a32 = αδ 32brr + βb32
aij = αδ ijbrr + βbij δ ijaj = ai δ ijajk = aik δ ijδ jkbkm = bim
δ ijaj = ai
δ ijaj = δ i1a1 + δ i2a2 + δ i3a3
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δ 1jaj = δ 11a1 + δ 12a2 + δ 13a3 = a1
δ 2jaj = δ 21a1 + δ 22a2 + δ 23a3 = a2
δ 3jaj = δ 31a1 + δ 32a2 + δ 33a3 = a3
ai
δ ijaj = ai
δ ijajk = aik ajk δ ij aik
δ ijδ jkbkm = bim δ ik
bim
n1
n2
n3
α
β
γ
X 1
X 2
X 3
mi
ni
cos θ = mini
cos θ = 0 mi ni
n21 + n2
2 + n23 = 1
X j xi
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X j
xi
aij = ei · E j = cos (xi, X j)
xi X j
xiX j
X 1 X 2 X 3x1 a11 a12 a13x2 a21 a22 a23x3 a31 a32 a33
a23
x2
X 3
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a21j + a22j + a23j =1, j = 1, 2, 3
a2i1 + a2i2 + a2i3 =1, i = 1, 2, 3
ai1aj1 + ai2aj2 + ai3aj3 =0, i, j = 1, 2, 3; i = j
a1ja1i + a2ja2i + a3ja3i =0, i, j = 1, 2, 3; i = j
X j xi
X j
= X 1 E 1 + X 2 E 2 + X 3 E 3 = X p E p
xi
= x1e1 + x2e2 + x3e3 = xqeq
= X j E j = xiei · E j = ajixi =
= xiei = X j E j · eii = aijX j =
xi
X j
xi X j X 3 θ
X j
xi
X j
x1 = a11X 1 + a12X 2 + a13X 3 = cos θ + 2 sen θ
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xiX j
ˆE 1
ˆE 2
ˆE 3e1 cos θ sen θ 0
e2 − sen θ cos θ 0e3 0 0 1
x2 = a21X 1 = a22X 2 + a23X 2 + a23X 3 = 2 sen θ − cos θ x3 = a31X 1 + a32X 2 + a33X 3 = 3 X j x1 x2 x3 (cos θ+ sen θ, cos θ − sen θ, )
xiX j
X 1 X 2 X 3
x112
32
12√ 2
12√ 2
x212
32
12√ 2
12√ 2
x3 a31 a32 a33
a231 + a232 + a233 =1
a11a31 + a12a32 + a13a33 =0
a21a31 + a22a32 + a23a33 =0
a231 + a232 + a233 =1
1
2
3
2 a31 +
1
2√
2a32 +
1
2√
2a33 =0
1
2
3
2 a31 +
1
2√
2a32 − 1
2√
2a33 =0
a31 = ± 12
, a32 = ∓√ 32
, a33 = 0
a11 a12 a13a21 a22 a23a31 a32 a33
= 1
−1
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xi =aijX j
X j =ajixi
X 1 X 2 X 3 xi
x1 x2 x3
E j ei
X 1 X 2 X 3 x1 x2 x3
xixi =X mX m
=armxrasmxs
=armasmxrxs
=δ rsxrxs
=xrxr
n 3n
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e1 e2 e3
E 1
E 2
E 3
= αe1 + β e2 + γ e3
= A E 1 + B E 2 + Γ E 3
x1 x2 x3 α β γ
E 1
E 2
E 3
X 1 X 2 X 3 A B Γ
x1e1 + x2e2 + x3e3 = X 1 E 1 + X 2 E 2 + X 3 E 3
ei
E j
e1 = E 1 e2 = E 2 e3 = E 3
e1 · E 1 = E 1 · e1 = 1 e1 · E 2 = E 2 · e1 = 0 e1 · E 3 = E 3 · e1 = 0
e2 · E 1 = E 1 · e2 = 0 e2 · E 2 = E 2 · e2 = 1 e2 · E 3 = E 3 · e2 = 0
e3 · E 1 = E 1 · e3 = 0 e3 · E 2 = E 2 · e3 = 0 e3 · E 3 = E 3 · e3 = 1
ei · E j = E j · ei =
1 i = j0 i = j
ei
E j
ei · ˆE j =
ˆE j · ei = δ ij = δ ji
x1 x2 x3
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X 1
X 2
X 3
x1 = x1 (X 1, X 2, X 3) X 1 = X 1 (x1, x2, x3)x2 = x2 (X 1, X 2, X 3)
X 2 = X 2 (x1, x2, x3)
x3 = x3 (X 1, X 2, X 3) X 3 = X 3 (x1, x2, x3)
x1 x2 x3 X 1 X 2 X 3
e(1) = ∂
∂x1 e(2) =
∂
∂x2 e(3)
∂
∂x3
= x1e1 + x2e2 + x3e3
e(1) = ∇x1 e(2) = ∇x2 e(3) = ∇x3
∂xm
∂X j∂X j
∂xn =
∂xm
∂X 1∂X 1
∂xn +
∂ xm
∂X 2∂X 2
∂xn +
∂xm
∂X 3∂X 3
∂xn = δ mn
= v1e(1) + v2e(2) + v3e(3) = v1e(1) + v2e(2) + v3e(3)
·
· =
v1e(1) + v2e(2) + v3e(3)· v1e(1) + v2e(2) + v3e(3)
=v1v1 + v2v2 + v3v3 = | |2
· = vivi = vivi = | |2
e1 e2 e3
E 1
E 2
E 3
x1e1 + x2e2 + x3e3 =
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X 1 E 1 + X 2 E 2 + X 3 E 3
E 2
[x1e1 + x2e2 + x3e3] · E 2 =
X 1 E 1 + X 2 E 2 + X 3 E 3
· E 2
=x1
e1 · E 2
+ x2
e2 · E 2
+ x3
e3 · E 3
=X 1
E 1 · E 2
+ X 2
E 2 · E 2
+ X 3
E 3 · E 2
x2 =X 2
E 1 · E 1 =1
E 1 · E 2 = E 2 · E 1 = 0
E 1 · E 3 = E 3 · E 1 = 0
ˆE 2 ·
ˆE 1 =
ˆE 1 ·
ˆE 2 = 0
ˆE 2 ·
ˆE 2 = 1
ˆE 2 ·
ˆE 3 =
ˆE 3 ·
ˆE 2 = 0
E 3 · E 1 = E 1 · E 3 = 0
E 3 · E 2 = E 2 · E 3 = 0
E 3 · E 3 = 1
E i · E j = E j · E i = gij = δ ij
gij
δ ij
n
= x1e(1) + x2e(2) + x3e(3) = x1e(1) + x2e(2) + x3e(3)
e(i) · e(j) = e(j) · e(i) = δ ji
e(i) · e(j) = gij
e(i) · e(j) = gij
= xie(i) = xj e(j)
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· e(r) =xi
e(i) · e(r)
= xj
e(j) · e(r)
=xiδ ri = xjgjr
=xr
= xjgjr
xr = xjgjr
x1 =x1g11 + x2g21 + x3g31
x2 =x1g12 + x2g22 + x3g32
x3 =x1g13 + x2g23 + x3g33
xr = xj
gjr
a
xi
A
X j
a = A
xi
X j
=
gij δ ij
X j
xi
ai OP
ai X 1 X 2 X 3
a1 a2 a3 ai
bi = aijbj
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dj
cij
cij =
c11 c12 c13c21 c22 c23c31 c32 c33
bij = aipajqc pq
c pq = a piaqjbij
Dijk
Dijk = airajsaktbrst
brst = ariasjatkDijk
N N 3N
cijk...
X j brst... xi
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cijk... = airajsakt... brst...
brst... = ariasjatk... cijk...
m n m n
m
n α C cij
α C α cij A B
aij bij A B C C
aij
bij
n− 2
A aij aji A
aij − aji
H ijklm H ijklm
H ijklm H ijklm − H ijkml
bij = 1
2 (bij + bji) +
1
2 (bij − bji)
1
2 (bij + bji)
1
2 (bij − bji)
εijk =
1 i,j,k 1, 2, 3, 1, 2,...0 i ,j,k
−1 i,j,k 3, 2, 1, 3, 2,...
×
× = εijkajbk = ci
× · = εijkaibjck = λ
λ
xi aijX j
Aij aipajqB pq Aijk aipajqakrB pqr
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aikajnδ ij = aikδ in = δ kn
εijk = εrstδ irδ jsδ kt
= δ i1δ j2δ k3 − δ i1δ j3δ k2 + δ i2δ j3δ k1 − δ i3δ j2δ k1
εijk =
δ i1 δ j1 δ k1δ i2 δ j2 δ k2δ i3 δ i3 δ k3
f
x1
x2
x3
df = ∂f
∂x1dx1 +
∂f
∂x2dx2 +
∂f
∂x3d3 =
∂f
∂xidi
ai,j = ∂ai∂xj
, ϕ,i = ∂ϕ
∂xi, eij,k =
∂eij∂xk
ϕ
ϕ =
∇ϕ =
∂ϕ
∂xi
= ∇ = ∂yi∂xj
= ∇ · = ∂yi∂xi
= ∇× = εijk∂yk∂xj
∇2
= ∇ · ∇
=
∂
∂xi∂yj
∂xi
=
∂ 2yj
∂xi∂xi
A =
1 2 3
4 5 67 8 9
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bij = 12 (bij + bji) + 1
2 (bij − bji)
12 (1 + 1) 1
2 (2 + 4) 12 (3 + 7)
1
2 (4 + 2) 1
2 (5 + 5) 1
2 (6 + 8)12 (7 + 3) 1
2 (8 + 6) 12 (9 + 9)
= 1 3 5
3 5 75 7 9
1
2 (1 − 1) 12 (2 − 4) 1
2 (3 − 7)12 (4 − 2) 1
2 (5 − 5) 12 (6 − 8)
12 (7 − 3) 1
2 (8 − 6) 12 (9 − 9)
=
0 −1 −2
1 0 −12 1 0
3x1e1 + 5x32e2 + 7x1x4
3e3 vi,i vi,i3 vj,3j vi,i
vi,i = ∂vi∂xi
=∂v1∂x1
+ ∂v2∂x2
+ ∂v3∂x3
=3 + 15x22 + 28x1x33
vi,i3
vi,i3 = ∂vi∂xi∂x3
= ∂v1
∂x1∂x3+
∂v2∂x2∂x3
+ ∂v3∂x3∂x3
=0 + 0 + 84x1x23
=84x1x23
vj,3j
υj,3j = ∂ 2υj∂x3∂xj
= ∂υ2
1
∂x3∂x1+
∂ 2υ2∂x3∂x2
+ ∂υ3
∂x3∂x3
= 84x1x23
aki akj
δ ij
= viei = vj E j = a pqvq
σij
σ pq
X j
xi
σ pq = a piaqjσij
σij = amianjσmn
εijkT jk = 0 T 12 = T 21; T 13 = T 31; T 23 = T 32
bij
∂bijxj∂xk
bik
∂bijxixk∂xk
bik+bki
X j xi
e1 = 1225 E 1 − 925 E 2 + 45 E 3 e2 = 35 E 1 − 45 E 2 + 0 E 3 e3 = α1 E 1 − 1225 E 2 + α2 E 3 α1 α2 e1
aij
i = j = 2 i = 1, j = 3 0 1 −1
X j −29/25 4/5 −3/25 xi
ai = εijkbjk
a1, a2, a3
aik = εijkbj
aik = −ajk
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xi,j = δ ij ; xi,i = 3
∇2 (x pxq) = 2δ pq
Aij AiBj AijBj aijxixj aijkxjxj aijkxjxjxk Aijkll Aijklm Aij23 A1233 Aiijj Aii + Ajj
δ ij δ ijAkl δ ijAik δ ijAi j δ ijδ ik δ ijδ jk δ ijAjkδ kl
δ ijδ jkδ kl
δ ii
G
ui,kk +
1
1− 2ν uk,ki
+ X i = ρ
∂ 2ui∂ t2
drswrvs δ ijxixj σij = 2µeij +
λδ ijenn
s11 s12 s13
s21 s22 s23s31 s32 s33
=
1 1 0
1 2 20 2 3
sii
sijsij
sijsjk
xiX j
X 1 X 2 X 3x1
35√ 2
1√ 2
45√ 2
x245 0 −
35
x3 − 35√ 2
1√ 2
− 45√ 2
aijaik = δ jk xi = 2 E 1 + E 3
X 1 − X 2 + 3X 3 xi
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t = 0
t = to
t = 0 t = to
t = t
P
= X 1 E 1 + X 2 E 2 + X 3 E 3 = X k E k
X j
P p
= x1e1 + x2e2 + x3e3
xi
aij
E j · ei = ei · E j = aij = aji
aijajk = δ ik, ei = aij E j , E j = aij ei
= uiei = uj E j
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uj = bj + xi −X j
ui = bi + xi −X j
ui = xi −X j
uj = xi −X j
ui = xi − aijX j
uj = aijxi − X j
ui = xi − δ ijX j = xi −X i
ui = δ ijxj −X i = xi −X i
xi = xi (X j, t)
xi = xi (X j, t)
t
P (X 1, X 2, X 3)
(x1, x2, x3) (X 1, X 2, X 3)
B b
X j = X j (xi, x2, x3, t) = X j (xi, t)
X j xi t
J =
∂xi∂X j
= 0
J x1 x2 x3
ei = aij E j ; e1 = a1j E j , e2 = a2j
E j , e3 = a3j E j
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xiX j
X 1 X 2 X 3x1 a11 a12 a13x2 a21 a22 a23x3 a31 a32 a33
xi = aijX j
X j = ajixi
xi = aijajpx p
aijajp = δ ip
1 i = p0 i = p
T ij...
T ij . . . = aipajq · · ·A pq i j . . . Aij
Aj = aipajpB pq
ds2 = dxidxi
P
p
= uj E j
= uiei
aij
ei = aij E j
E j = aji ei
= uiei = uiaij E j =
X j
δ ij
δ ijX j = δ i1X 1 + δ i2X 2 + δ i3X 3
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i = 1
δ 1jX j =δ 11X 1 + δ 12X 2 + δ 13X 3
=X 1
i = 2
δ 2jX j =δ 21X 1 + δ 22X 2 + δ 23X 3
=X 2
i = 3
δ 3jX j =δ 31X 1 + δ 32X 2 + δ 33X 3
=X 3
δ ijX j = X i
xi = xi(X j, t)
xi X 1 X 2 X 3 t = 0
X j = X j(xi, t)
X j
x1
x2
x3
t = t
dxi = ∂xi∂X j
dX j
dX j = ∂X j∂xi
dxi
∂xi/∂X j F ij ∂X j/∂xi
H ij
F ikH kj = ∂xi∂X k
∂X k∂xj
= ∂X i∂xk
∂xk∂X j
= δ ij
F = H −1
H = F −1
ui
∂ui∂X j
∂ui∂xj
uj = xj −X j
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∂uj∂X i
= ∂xj∂X i
− δ ij
uj = xj −X j
∂ui∂xj
= δ ij − ∂ X i∂xj
∂xi∂X j
=
∂x1
∂X 1
∂x1
∂X 2
∂x1
∂X 3∂x2
∂X 1
∂x2
∂X 2
∂x2
∂X 3∂x3
∂X 1
∂x3
∂X 2
∂x3
∂X 3
= F ij
∂X j∂xi
=
∂X 1∂x1
∂X 1∂x2
∂X 1∂x3
∂X 2∂x1
∂X 2∂x2
∂X 2∂x3
∂X 3∂x1
∂X 3∂x2
∂X 3∂x3
= H ij
∂xi∂X j
∂X j∂xk
= ∂X i∂xj
∂xj∂X k
= δ ij
∂xi∂X j
= ∂ui∂X j
+ δ ij =
∂u1
∂X 1+ 1
∂u1
∂X 2
∂u1
∂X 3∂u2
∂X 1
∂u2
∂X 2+ 1
∂u2
∂X 3∂u3
∂X 1
∂u3
∂X 2
∂u3
∂X 3+ 1
∂ui∂xj
= δ ij − ∂ X i∂xj
∂X i∂xj
= δ ij − ∂ui∂xj
=
1− ∂ u1
∂x1−∂u1
∂x2−∂u1
∂x3
−∂u2
∂x11− ∂ u2
∂x2−∂u2
∂x3
−∂u3
∂x1−∂u3
∂x21− ∂ u3
∂x3
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P Q
(dX )2
= dX idX i
(dX )2
= δ ijdX idX j
X i
dX i = ∂X i∂xj
dxj
(dX )2
= ∂X k
∂xi
∂X k∂xj
dxidxj
C ij = ∂X k
∂xi
∂X k∂xj
pq
(dx)2
= dxidxi = δ ijdxidxj
dxi
dxi = ∂xi∂X j
dX j
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(dx)2
= dxkdxk = ∂xk
∂X i
∂xk
∂X jdX idX j
Gij = ∂xk∂X i
∂xk∂X j
P Q pq
(dx)2 − (dX )
2=
∂xk∂X i
∂xk∂X j
dX idX j − δ ijdX idX j
(dx)2 − (dX )
2=
∂xk∂X i
∂xk∂X j
− δ ij
dX idX j
Lij
Lij = 1
2
∂xk∂X i
∂xk∂X j
− δ ij
Lij = 1
2
∂uk∂X i
+ δ ik
∂uk∂X j
+ δ jk
− δ ij
= 1
2
∂uk∂X i
∂uk∂X j
+ δ ik∂uk∂X j
+ δ jk∂uk∂X i
+ δ ikδ jk − δij
Lij = 1
2 ∂ui
∂X j+
∂uj
∂X i+
∂uk
∂X i
∂uk
∂X j
E ij
E ij = 1
2
δ ij − ∂ X k
∂xi
∂X k∂xj
E ij = 1
2
∂ui∂xj
+ ∂ uj∂xi
− ∂ uk∂xi
∂uk∂xj
(dx)
2
− (dX )
2
= 2LijdX idX j
(dx)2 − (dX )
2= 2E ijdxidxj
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lij = 12
∂ui∂X j+ ∂uj∂X i
eij = 1
2
∂ui∂xj
+ ∂ uj∂xi
lij = eij
Lij E ij lij eij C ij Gij Lij E ij lij eij
u1 = X 2 + X 1 u2 = X 1 − X 2 u3 = −2X 3
∂u1
∂X 1= 1,
∂u1
∂X 2= 1,
∂u1
∂X 3= 0
∂u2
∂X 1= 1,
∂u2
∂X 2= −1,
∂u2
∂X 3= 0
∂u3
∂X 1= 0,
∂u3
∂X 2= 0,
∂u2
∂X 3= −2
J = 2 1 0
1 0 00 0 −2
= 2 > 0
u1 = X 2 + X 1, u2 = X 1 − X 2, u3 = −2X 3
Lij = 1
2
∂ui∂X j
+ ∂uj∂X i
+ ∂uk∂X i
∂uk∂uj
L11 = 1
2 ∂u1
∂X 1+
∂u1
∂X 1+
∂ u1
∂u1
∂u1
∂u1
+ ∂u2
∂X 1
∂u2
∂X 1+
∂u3
∂X 1
∂u3
∂X 1
= 1
2 (1 + 1 + 1 × 1 + 1 × 1 + 0 × 0) = 2
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L12 = 1
2
∂u1
∂X 2+
∂u2
∂X 1+
∂u1
∂X 1
∂u1
∂X 2+
∂u2
∂X 1
∂u2
∂X 2+
∂u3
∂X 1
∂u3
∂X 2
= 1
2 (1 + 1 + 1
×1 + 1
×(−
1) + 0×
0) = 1
L13 = 0 L21 = 1 L22 = −1 L23 = 0 L32 = 0 L33 = 0
Lij =
2 1 0
1 −1 00 0 0
u1 = x22 u2 =
3x2x3 u3 = 4x1 + 6x3
eij = 1
2 ∂ui∂xj
+ ∂ uj∂xi
e11 = ∂u1
∂x1= 0, e22 =
∂u2
∂x2= 2x3, e33 =
∂u3
∂x3= 6,
e12 = e21 = 1
2
∂u1
∂x2+
∂ u2
∂x1
= x2,
e13 = e31 = 1
2
∂u1
∂x3+
∂ u3
∂x1
= 2,
e23 = e32 = 1
2
∂u2
∂x3+
∂ u3
∂x2
=
3
2x2
eij = eji =
0 x2 2
x2 3x332x2
2 32
x2 6
x1 = X 1 + X 2, x2 = X 1−X 2, x3 = X 1 + X 2−X 3
x1 = X 1 + X 2 + 0X 3
x2 = X 1 −X 2 + 0X 3
x3 = X 1 + X 2 −X 3
∆ =
1 1 01 −1 01 1 −1
= 2
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X 1 =
x1 1 0x2 −1 0
x3 1 −1
2 = x1 + x2
2
X 2 =
1 x1 01 x2 01 x3 −1
2
= x1 − x2
2
X 3 =
1 1 x1
1 −1 x2
1 1 x3
2
= x1 − x3
X 1 = x1 + x2
2 ; X 2 = x1 − x2
2 ; X 3 = x1 − x3
F ij = ∂xi∂X j
=
∂x1
∂X 1
∂x1
∂X 2
∂x1
∂X 3∂x2
∂X 1
∂x2
∂X 2
∂x3
∂X 3∂x3
∂X 1
∂x3
∂X 2
∂x3
∂X 3
=
1 1 0
1 −1 01 1 −1
H ij = ∂X i∂xj
=
∂X 1∂x1
∂X 1∂x2
∂X 1∂x3
∂X 2∂x1
∂X 2∂x2
∂X 2∂x3
∂X 3∂x1
∂X 3∂x2
∂X 3∂x3
=
1
212 0
12 − 1
2 01 0 −1
1 1 01 −1 01 1 −1
12
12 0
12 −1
2 01 0 −1
=
1 0 00 1 00 0 1
F ikH kj = δ ij
J = |F ij | = 1 1 01 −1 0
1 1 −1
= 2
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P Q
Q
t = t
q
P
p
Q
P
P
dui = u(Q)i − u
(P )i
dui = ∂ui∂X j
P
dX j
p
dui = ∂ui∂xj
p
dxj
duidX
= duidX j
dX jdX
= duidxj
nj
dui =
1
2
∂ui∂X j
+ ∂uj∂X i
+
1
2
∂ui∂X j
− ∂uj∂X i
dX j = [lij + wij ] dX j
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lij = 1
2
∂ui
∂X j
+ ∂uj
∂X i
wij
wij = 1
2
∂ ui∂X j
− ∂uj∂X i
lij = 0
wi = 1
2εijkwkj
dui =
1
2
∂ui∂xj
+ ∂ uj∂xi
+
1
2
∂ui∂xj
− ∂ uj∂xi
dxj
= [eij + wij ] dxj
eij wij
W i = 1
2εijkW kj
wi = 1
2εijkwkj
dui = εijkW jdX k
dui = εijkwjdxk
= (3X 2 − 4X 3) E 1 + (2X 1 −X 3) E 2 + (4X 2 −X 1) E 3
A B
u1 = x1 − X 1 u2 = x2 −X 2 u3 = x3 −X 3
3X 2 − 4X 3 = x1 − X 1 2X 1 − X 3 = x2 −X 2 4X 2 − X 1 = x3 −X 3
x1 = X 1 + 3X 2 − 4X 3 x2 = 2X 1 + X 2 − X 3 x3 = −X 1 + 4X 2 + X 3
A x1 = −11 x2 = −1 x3 = 2 B x1 = −3
x2 = 6
x3 = 27
AB − − −
− −
C 2 6 3
o x1 = 0, x2 = 0, x3 = 0 c x1 = 2 + 3× 6− 4× 3 = 8
x2 = 2 × 2 + 6 − 3 = 6 x3 = −2 + 4 × 6 + 3 = 25 oc
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ui = AijX j
Aij
AijX j = xi −X i
(dx)2−(dX )2
(dx)2 − (dX )
2= 2lijdX idX j
(dx
−dX ) (dx + dX ) = 2lijdX idX j
dx ≈ dX
dx− dX = 2lij2dX
dX idX j
dx− dX
dX = lij
dX idX
dX jdX
ni = dX i
dX
X 1
n1 = dX 1
dX = 1; n2 =
dX 2dX
= 0; n3 = dX 3
dX = 0
dx− dX
dX = L11
X 1
L22 L33 X 2 X 3
X 2 X 3
θ
∂ui/∂xj = 1
n2 = ∂u1
∂X 2e1 + e2 +
∂u3
∂X 2e3
n3 = ∂u1
∂X 3e1 +
∂u2
∂X 3e2 + e3
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cos θ = n2 · n3 = ∂u1
∂X 2
∂u1
∂X 3+
∂u2
∂X 3+
∂u3
∂X 2
cos θ = ∂u2
∂X 3+
∂u3
∂X 2= 2l23
γ 23 = π
2 − θ
sen γ 23 = sen
π2 − θ
= cos θ = 2l23
γ 23
90
γ ij = 2lij ; i = j
X 2
x2
X 1
x1
L1 = dx1 − dX 1
dX 1
L1 = dx1
dX 1− 1
dx/dX
dx1 = (1 + L1) dX 1
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dx1 = (1 + L1) dX 1
(dx)2 − (dX )2 = 2LijdX idX j
[(1 + L1) dX ]2 − [dX ]
2= 2L11dX 1dX 1
(1 + L1)
2 − 1
(dX )2
= 2L11dX 1dX 1
(1 + L1)2 − 1 = 2L11
dX 1dX
dX 1dX
dX 1/dX
X 1
(1 + L1)2 − 1 = 2L11
L1 =
1 + 2L11 − 1, L2 =
1 + 2L22 − 1, L3 =
1 + L33 − 1
γ 23 = cos θ
cos θ = d 2
dx2
d 3
dx2
dx2dx3 cos θ = d 2d 3
dxi =
∂xk
∂X jdX j
dx2dx3 cos θ = ∂xk∂X 2
∂xk∂X 3
dX 2dX 3
dx2
dX 2
dx3
dX 3cos θ =
∂xk∂X 2
∂xk∂X 3
dx2
dX 2=
1 + 2L22dx3
dX 3=
1 + 2L33
2L23 = G23 = ∂xk∂X 2∂xk∂X 3
cos θ = 2L23√
1 + 2L22
√ 1 + 2L33
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cos θ = sen γ 23 = γ 23
γ 23 = 2L23
√ 1 + 2L22√ 1 + 2L33
γ 12 = 2L12√
1 + 2L11
√ 1 + 2L22
γ 13 = 2L13√
1 + 2L11
√ 1 + 2L33
P
p
Q
q
pq
dx
dx
dX = λ; (0 < λ < ∞)
P Q
pq
P Q
ε = dx− dX
dx
ε = dx
−dX
dX =
dx
dX −1 = λ
−1
−1 < < ∞
λ = 1
= 0
λ > 1
> 0
P
Q
λ < 1 < 0 P Q
(dx)2 − (dX )2 = 2LijdX idX j
(dx)2 − (dX )2 = 2E ijdxidxj
(dx)2 − (dX )2 = 2LijdX idX j
(dX )2 dx
dX
2− 1 = 2Lijninj
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dx
dX
2− 1 = 2Lijninj
λ2 − 1 = 2Lijninj
λ =
1 + 2Lijninj
(dx)2 − (dX )2 = 2E ijdxidxj
(dx)2
1−
dX
dx
2=2E ij
dxidx
dxjdx
1−
1
λ
2=2E ij ninj
1λ2
=1− 2E ij ninj
1
λ =
1− 2E ijninj
λ = 1
1− 2E ij ninj
= λ − 1
1
λ =
1
1 + ε1
1 + ε =
1− 2E ij ninj
ε = 1
1− 2E ij ninj− 1
E ij =
0 0 −tetx3
0 0 0−tetx3 0 t(2etx3 − et)
t = 0 t = 2
(1, 1, 1)
λ =
1 1− 2E ijninj
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n1
n2
n3
1√ 3
E ij ninj = E 11n1n1 + E 12n1n2 + E 13n1n3
+ E 21n2n1 + E 22n2n2 + E 23n2n3
+ E 31n3n1 + E 32n3n2 + E 33n3n3
E ij ninj = −2
3tetx3 +
2
3tetx3 − 1
3tet = −1
3tet
t = 2
λ = 1 1 + 2
tet
3
= 1 1 +
4e2
3
=
√ 3√
3 + 4e2
LOA =
ˆ AO
dX
λ = dx/dX dX = dx/λ
LOA = 1
λ
ˆ a0
dx
ˆ a0
dx =√
3
LOA =
1
λˆ a
0 dx
=
√ 3
λ
LOA = 1
λ
√ 3
=
3 + 4e2
P Q
dxi = ∂xi∂X j
dX j; dX i = ∂X i∂xj
dxj
F ij = ∂xi∂xj
; H ij = ∂xi∂X j
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P M
P Q
pm
pq
dX (1)i = H ijdx
(1)j , dX
(2)i = H ijdx
(2)j , dx
(1)i = F ijdX
(1)j , dx
(2)i = F ijdX
(2)j
ni = dX i
dX , nj =
dxjdx
λ = dx/dX
λ(1) = dx(1)
dX (1), λ(2) =
dx(2)
dX (2)
dx(1) = λ(1)dX (1) dx(2) = λ(2)dX (2) dX (1) =dx(1)/λ(1)
dX (2) = dx(2)/λ(2)
dx(1)i dx
(2)i = ∂xk
∂X i(1)
∂xk
∂X j(2)
dX (1)i dX
(2)j = (F ik)
(1)(F jk)
(2)dX
(1)i dX
(2)j
Lij = 1
2
∂xk∂X i
∂xk∂X j
− δ ij
dx(1)dx(2) = [2Lij + δ ij ] dX (1)i dX
(2)j
(dx)2
dx(1)
dx
dx(2)
dx = [2Lij + δ ij ]
dX (1)i
dx
dX (2)j
dx
dx(1)
dxdx(2)
dx = cos θ
cos θ = [2Lij + δ ij ] dX
(1)i
λ(1)dX (1)dX
(2)j
λ(2)dX (2) =
2Lij + δ ijλ(1)λ(2)
n(1)i n
(2)j
λ =
1 + 2Lij ninj
cos θ =[2Lij + δ ij ] n
(1)i n
(2)j
1 + 2Lij n(1)i n
(1)j 1 + 2Lij n
(2)i n
(2)j
cos Θ =[δ ij − 2E ij ] n
(1)i n
(2)j
1− 2E ijn(1)i n
(2)j
1− 2E ij n
(2)i n
(2)j
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X 1
X 3
cos θ13 = 2L13√
1 + 2L11
√ 1 + L33
P Q
P Q X 2 dX pq
λ =
1 + 2Lijninj
X 2
λ(2) =
1 + 2L22
X 1
X 3
λ(1) = 1 + 2L11, λ(3) = 1 + 2L33
Li j
= λ − 1
ε1 =
1 + 2L11, ε2 =
1 + 2L22, ε3 =
1 + 2L33
X 2 X 3 P Q M θ
cos θ =[2Lij + δ ij ] n
(i)i n
(j)j
1 + 2Lij n(1)i n
(1)j
1 + 2Lij n
(2)i n
(2)j
1 + 2Lij n
(1)1 n
(1)j =
1 + 2L22 1 + 2Lij n
(2)i n
(2)j =
1 + 2L33
cos θ = 2L23√
1 + 2L22
√ 1 + 2L33
∆23 = π/2−θ23
λ(1) = 1√ 1− 2E 11
; λ(2) = 1√ 1− 2E 22
; λ(3) = 1√ 1− 2E 33
E (1) = 1√ 1− 2E 11
− 1; E (2) = 1√ 1− 2E 22
− 1; E (3) = 1√ 1− 2E 33
− 1
x1 = X 1 − X 2 + X 3; x2 = X 2 −X 3 + X 1; x3 = X 3 − X 1 + X 2
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ni = 1
√ 2(e1 + e2)
n1 = 1√
2, n2 =
1√ 2
, n3 = 0
u1 = x1 −X 1; u2 = x2 −X 2; u3 = x3 −X 3
u1 = X 1 −X 2 + X 3 −X 1; u2 = X 2 −X 3 + X 1 −X 2; u3 = X 3 − X 1 + X 2 − X 3
u1 = − X 2 + X 3; u2 = X 3 − X 1 + X 2 − X 3; u3 = −X 1 + X 2
Lij = 1
2
∂ui∂X j
+ ∂uj∂X i
+ ∂uk∂X i
∂uk∂X j
L11 = 1
2
∂u1
∂X 1+
∂u1
∂X 1+
∂u1
∂X 1
∂u1
∂X 1+
∂u2
∂X 1
∂u2
∂X 1+
∂u3
∂X 1
∂u3
∂X 1
L11 = 1
2 [0 + 0 + (0)(0) + (1)(1) + (−1)(−1)] = 1
Lij =
1 − 1
2 − 12− 1
2 1 − 12
− 12 − 1
2 1
λ =
1 + 2
1
2 − 1
8 − 1
8 − 1
8 +
1
2 − 1
8
=√
2
θ12
n1 n2 e2
n(1)1 =
1√ 2
, n(1)2 =
1√ 2
, n(1)3 = 0, n
(2)1 = 0, n
(2)2 = 1, n
(2)3 = 0
2Lijn(1)i n
(2)j = 2
−1
2 × 1√
2× +1× 1√
2× 1
=
1√ 2
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2Lijn(1)i
n(1)j
=2 L1jn(1)1
n(1)j
+ L2jn(1)2
n(1)j
+ L3jn(1)3 n
(1)j
=2
L11n(1)1
n(1)1
+ L12n(1)1
n(1)2
+ L13n(1)1 n
(1)3 +
L21n(1)2
n(1)1
+ L22n(1)2
n(1)2
+ L23n(1)3 n
(1)1 +
L31n(1)3 n
(1)1 + L32n
(1)3 n
(1)2 + L33n
(1)3 n
(1)3
= 1
2Lijn(2)i
n(2)j
= 2
cos θ =
1√ 2√
1 + 1√
1 + 2=
1
2√
3
eij
e(n)i = eij nj
e(n)i n eij
e(n)i
n
e(n)i = eni
ni eij
eijnj = δ ijenj
(eij − δ ije) nj = 0
nj e
(e11 − e) n1 + e12n2 + e13n3 = 0
e21n1 + (e22 − e) n2 + e23n3 = 0
e31n1 + e32n2 + (e33 − e) n3 = 0
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e11 − e e12 e13
e21 e22 − e e23e31 e32 e33 − e = 0
e
−e3 + (e11 + e22 + e33) e2 + (e23e32 + e12e21 + e13e31 − e11e22 − e22e33 − e11e33) e +
e11 (e22e33 − e23e32) − e12 (e21e33 − e23e31) +
e13 (e21e32 − e22e31) = 0
I 1 = e11 + e22 + e33 = eii
I 2 = 1
2 (eijeji − eiiejj )
I 3 = det |eij| = e11
e22 e23e32 e33
− e12
e21 e23e31 e33
+ e13
e21 e22e31 e32
eij
e3 − I 1e2 + I 2e − I 3 = 0
eij
eij e1 e2 e3
e1 e2 e3
e1 e2 e3
e1
e2
e3
e(n)i =e(
)ni
=
e(1) 0 0
0 e(2) 00 0 e(3)
n1
n2
n3
e(n)1 = e1n1, e
(n)2 = e2n2, e
(n)3 = e3n3
n21 + n2
2 + n23 = 1
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I 1 = e1 + e2 + e3
I 2 = e1e2 + e2e3 + e1e3
I 3 = e1e2e3
V
= dX 1dX 2dX 3
V
= dX 1 (1 + e1) dX 2 (1 + e2) dX 3 (1 + e3)
∆V
V =
dX 1 (1 + e1) dX 2 (1 + e2) dX 3 (1 + e3) − dX 1dX 2dX 3dX 1dX 2dX 2
= e1 + e2 + e3
I 1
e
(n)
1 = e1n1, e
(n)
2 = e2n2, e
(n)
3 = e3n3
n1 = e(n)1
e1; n2 =
e(n)2
e2; n3 =
e(n)3
e3
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e(n)1
e12
+e
(n)2
e22
+e
(n)3
e3
= 1
lij eij
lij = δ ijlnn
3 + sij
eij = δ ijenn
3 + εij
δ ijlnn
3 =
lnn3
0 0
0 lnn
3 0
0 0 lnn
3
=
l11 + l22 + l333
0 0
0 l11 + l22 + l33
3 0
0 0 l11 + l22 + l33
3
lij
sij =
l11 − lnn
3 l12 l13
l21 l22
− lnn
3
l23
l31 l32 l33 − lnn3
lij
δ ijenn
3 =
enn3
0 0
0 enn
3 0
0 0 enn
3
=
e11 + e22 + e333
0 0
0 e11 + e22 + e33
3 0
0 0 e11 + e22 + e33
3
εij =
e11 − enn3
e12 e13
e21
e22 −
enn
3 e
23
e31 e32 e33 − enn3
eij
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e(n)i = eijnj
eN = e(n)i ni = eijnjni
|eS |2 =e(n)i
2
− |eN |2
n21 + n2
2 + n23 = 1
n3 =
1− n21 − n22
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∂n3
∂n1
= −2n1
2
1− n21 − n22
=;
−n1
n3
∂n3
∂n2= − n2
n3
∂n1
∂n2=
∂n2
∂n1= 0
eN = eN (n1, n2)
∂eN ∂n1
= 0, ∂eN
∂n2= 0
∂eN ∂n1
= ∂ (eijninj)
∂n1=eij
∂ni∂n1
nj + eijnj∂ni∂n1
=eij∂ni∂n1
nj + eji∂ni∂n1
nj ; eij = eji
∂eN ∂n1
=2eij∂ni∂n1
nj
2
e1j
∂n1
∂n1nj + e2j
∂n2
∂n1nj + e3j
∂n3
∂n1nj
= 0
e1j∂n1
∂n1nj + e2j
∂n2
∂n1nj + e3j
∂n3
∂n1nj = 0
n2
e1j∂n1
∂n2nj + e2j
∂n2
∂n2nj + e3j
∂n3
∂n2nj = 0
e1jnj − e3jn1
n3nj = 0
e(n)i = eijnj
e(n)
1 = e
1jnj
, e(n)
2 = e
2jnj
e(n)1
n1=
e(n)3
n3
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e(n)2
n2
= e(n)3
n3
e(n)1
n1=
e(n)2
n2=
e(n)3
n3= e
e
eN = eni
e(n)i = eni
e(n)
i
= eN
e
e1 > e2 > e3
e(n)i =
e1 0 00 e2 00 0 e3
n1
n2
n3
e(n)i · e
(n)i = e21n2
1 + e22n22 + e23n2
3
e(n)i 2 = |eN |2 + |eS |2
eN = e(n)i ni = e1n2
1 + e2n22 + e3n2
3
e21n21 + e32n2
2 + e23n23 = e2N + e2S
e1n21 + e2n2
2 + e3n23 = eN
n21 + n2
2 + n23 = 1
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n21
n22
n23
e21 e22 e23
e1 e2 e31 1 1 = e
2
1 (e2 − e3) − e2
2 (e1 − e3) + e2
3 (e1 − e2)
±e1e2e3
e21 (e2 − e3) − e22 (e1 − e3) + e23 (e1 − e2) = e21 (e2 − e3) − e22e1 + e22e3 + e23e1
− e23e2 + e1e2e3 − e1e2e3
= e21 (e2 − e3) − e1e2e3 + e23e1
− e22e1 + e1e2e3 + e22e3 − e23e2
= e21 (e2 − e3) − e1e3 (e2 − e3)
− e1e2 (e2 − e3) + e2e3 (e2 − e3)
=
e
2
1 − e2e3 − e1e2 + e2e3
(e2 − e3)= [e1 (e1 − e3) − e2 (e1 − e3)] (e2 − e3)
= (e1 − e2) (e1 − e3) (e2 − e3)
e21 e22 e23e1 e2 e31 1 1
= (e1 − e2) (e1 − e3) (e2 − e3)
n21
n21 =
e2N + e2S e22 e23eN e2 e31 1 1
(e2 − e3) (e1 − e2) (e1 − e3)
(e2 − e3)
e2N + e2S − eN (e2 + e3) + e2e3
n21 =
(eN − e3) (eN − e2) + e2S (e2 − e1) (e3 − e1)
n22 =
(eN −
e3) (eN −
e1) + e2S (e2 − e3) (e2 − e1)
n23 =
(eN − e1) (eN − e2) + e2S (e3 − e1) (e3 − e2)
e2N − eN (e2 + e3) + e2S + e2e3 > 0
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e2N − eN (e2 + e3) + e2S > −e2e3
eN − e2 + e3
2
2+ e2S >
e2 − e3
2
2
(e1 + e2)/2 (e1− e2)/2 eN , eS
eN − e1 + e32
2+ e2S <
e1 − e3
2
2
e1 e3 (e1 − e3)/2 eN , eS
eN −
e1
−e2
22
+ e
2
S >e1
−e2
22
e1 e2 (e1 − e2)/2 eN , eS
e(n)i
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eij = eji = 1
2
∂ui∂xj
+ ∂ uj∂xi
eij
u1 u2 u3 eij ui
e12 = 1
2
∂u1
∂x2+
∂ u2
∂x1
x1
2∂e12∂x1
= ∂ 2u1
∂x2∂x1+
∂ 2u2
∂x21
x2
2 ∂ 3e12∂x1∂x2
= ∂ 3u1
∂x1∂x22
+ ∂ 3u2
∂x2∂x21
e11 =
∂u1
∂x1
e11 x2
∂ 2e11∂x2
2
= ∂ 3u1
∂x1∂x22
e22 x1
∂ 2e22∂x2
1
= ∂ 3u2
∂x2∂x21
∂ 2
e11∂x2
2+ ∂
2
e22∂x2
1= ∂
3
u1∂x1∂x2
2+ ∂
3
u2∂x2
1∂x2
∂ 2e11∂x2
2
+ ∂ e22
∂x21
= 2 ∂ 2e12∂x1∂x2
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∂ 2
e11∂x2
3+ ∂
2
e33∂x2
1= 2 ∂e13
∂x1∂x3
∂ 2e22∂x2
3
+ ∂ 2e33
∂x22
= 2 ∂e23∂x2∂x3
e12 x1 x3 e13
x1 x2
2e12 = ∂u1
∂x2+
∂ u2
∂x1
2 ∂ 2e12∂x1∂x3
= ∂ 3u1
∂x1∂x2∂x3+
∂ 3u2
∂x21∂x3
2e13 =
∂u1
∂x3 +
∂ u3
∂x1
2 ∂ 2e13∂x1∂x2
= ∂ 3u1
∂x1∂x2∂x3+
∂ 3u3
∂x21∂x2
2
∂ 2e12∂x1∂x3
+ ∂ 2e13∂x1∂x2
= 2
∂ 3u1
∂x1∂x2∂x3+
∂ 3u2
∂x21∂x3
+ ∂ 3u2
∂x21∂x2
2
∂ 2e12∂x1∂x3
+ ∂ 2e13∂x1∂x2
= 2
∂ 2
∂x1∂x3
∂u1
∂x1
+
∂ 2
∂x1
∂u2
∂x3+
∂ u3
∂x2
∂u1
∂x1 = e11;
∂u2
∂x3 +
∂ u3
∂x2 = 2e23
∂ 2e12∂x1∂x3
+ ∂ 2e13∂x1∂x2
− ∂ 2e23∂x2
1
=
∂ 2e11∂x2∂x3
∂
∂x2
∂e12∂x3
− ∂ e13∂x2
+ ∂ e23
∂x1
=
∂ 2e22∂x1∂x3
∂
∂x3
−∂e12
∂x3+
∂ e13∂x2
+ ∂ e23
∂x1
=
∂ 2e33∂x1∂x2
eij
∂ 2eij∂xk∂xk
+ ∂ 2ekk∂xi∂xj
− ∂ 2eik∂xk∂xj
− ∂ 2ejk∂xk∂xi
= 0
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err = ∂ur
∂r
; eθθ = 1
r
∂uθ
∂θ
+ ur
r
; ezz = ∂uz
∂z
erθ = 1
2
1
r
∂ur∂θ
+ ∂ uθ
∂r − uθ
r
; erz =
1
2
∂ur∂z
+ ∂ uz
∂r
; eθz =
1
2
∂uθ∂r
+ 1
r
∂uz∂θ
∂ 2eθθ∂r2
+ 1
r2∂ 2err∂θ2
+ 2
r
∂eθθ∂r − 1
r
∂err∂r
= 2
1
r
∂ 2erθ∂r∂θ
+ 1
r2∂erθ
∂θ
∂ 2eθθ∂z2
+ 1
r2∂ 2ezz
∂θ2 +
1
r
∂ezz∂r
= 2
1
r
∂ 2eθz∂z∂θ
+ 1
r
∂ezr∂z
∂ 2ezz∂r2
+ ∂ 2err
∂z2 = 2
∂ 2erz∂z∂r
1
r
∂ 2ezz∂r∂θ
− 1
r2∂ezz
∂θ =
∂
∂z
1
r
∂ezr∂θ
+ ∂ eθz
∂r − ∂ erθ
∂z
− ∂
∂z
eθzr
1
r
∂ 2err∂θ∂z
= ∂
∂r
1
r
∂ezr∂θ − ∂ eθz
∂r +
∂ erθ∂z
− ∂
∂r
eθzr
+
2
r
∂erθ∂z
∂ 2eθθ∂r∂z
− 1
r
∂err∂z
+ 1
r
∂eθθ∂z
= 1
r
∂
∂θ
−1
r
∂ezr∂θ
+ ∂ eθz
∂r +
∂ erθ∂z
+
1
r
∂
∂θ
eθzr
u1 = 2x1 + x2 u2 = x3 u3 = x3 − x2
1/√
3 1/√
3 1/√
3
x3
1 0 0 1/√
3
1/√
3 1/√
3
u1 = x1 − x2
u2 = 3x1 + 2x2
u3 = 5x3
e11 = 0.003 e22 = −0.003
e12 = 0.004 e33 = e13 = e23 = 0
u1 = a1x1x2 u2 = a2(x12 + vx22 − vx32) u3 = a3vx2x3 a1 a2 a3 v v
xi
u1 = ax1 (x2 + x3)2 , u2 = ax2 (x3 + x2)
2 , u3 = ax3 (x1 + x2)2
a
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E 1
E 2
E 3
P
x1 = X 1 x2 = X 2 + aX 3 x3 = X 3 + aX 2 a
X 1 = 0 X 22 + X 32 = 1/1 − a2)
X 1 = 0 (1 + a2)x22 − 4ax2x3 + (1 + a2)x2
3 = 1− a2
x1 = X 1 −X 2X 3 x2 = X 2 + X 1X 3 x3 = X 3
X 212 + X 22 = a2
xi = aijX j+ci aij ci
r a2a pra ps = r2δ ij
x1 = 2X 1 + X 2, x2 = X 1 + 2X 2, x3 = X 3
X 1 X 2
n
C ijninj = 1/J 2
x1 = X 1t2 + 2X 2t + X 1, x2 = 2X 1t2 + X 2t + X 2, x3 = 1
2X 3t + X 3
1 −1
√ 2 1
−1
√ 2
−1 1
√ 2
eij =
1 −4
√ 3
−4 1 −√ 3√ 3 −√ 3 6
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vi = DX i
Dt =
dX idt
= X i
xi vi
D
Dt =
d
dt =
∂
∂t + vk
∂
∂xk
vk(∂ /∂xk)
∂ /∂t
vi = dX j
dt = X j
D
Dt =
d
dt
ui = xi−X i
vi = dxi
dt =
d (ui + X i)
dt
= dui
dt
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X i
t = 0
ui = ui(X j , t)
vi = ui = dui (X j , t)dt
= ∂ui (X j , t)
∂t
ui(xj , t)
vi (xj , t) = ui (xj , t) = dui (xjt)
dt =
∂ui (xj,) t
∂t + vk
∂ui (xj,t)
∂t
ai = vi = dvi (X j , t)
dt
ai (xj , t) = Dvi (xj,t)
Dt =
∂vi (xj , t)
∂t + vk (xj, t)
∂vi (xj , t)
∂t
f = f (X j , t)
f t X i
Df
Dt =
∂f (X i, t)
∂t
Xi
X i f
X i t
ϕ = ϕ(xi, t)
xi t
ϕ ϕ t xi
∂ϕ
∂t =
∂ϕ (xi, t)
∂t
xi
∂ϕ/∂t xi t
ϕ = ϕ(xi, t) X j
ϕ = ϕ (xi, t) = ϕ (xi (X j , t) , t)
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ϕ
dϕ
dtXj
= ∂ϕ
∂txi
+ ∂xi
∂tXj
∂ϕ
∂xit
∂ϕ
∂t =
∂ϕ
∂t
xi
, ∂xi
∂t
Xj
= vi
dϕ
dt =
∂ϕ
∂t + vi
∂ϕ
∂xi
∂ϕ/∂t
ϕ vi∂ϕ/∂xi
Dvi
Dt =
∂vi
∂t + vk
∂vi
∂xk
d
dt =
∂
∂t + vk
∂
∂xk
x1 = X 1e−t, x2 = X 2et, x3 = X 3 + X 2
e−t − 1
θ = e−t (x1 − 2x2 + 3x3)
θ = e−t
X 1e−t − 2X 2et + 3
X 3 + X 2
e−t − 1
= (X 1 + 3X 2) e−2t − 3 (X 3 −X 2) e−t − 2X 2
dθ
dt = −2 (X 1 + 3X 2) e−2t − 3 (X 3 − X 2) e−t
dθ
dt =
∂θ
∂t + vj
∂θ
∂xj
∂θ
∂t = −e−t (x1 − 2x2 + 3x2)
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∂θ
∂t
=
−e−t X 1e−t
−2 X 2e t+ 3 X 3 + X 2 e−t −
1=− (X 1 + X 3) e−2t − 3 (X 3 −X 2) e−t + X 2
v1 = ∂x1
∂t =
∂X 1e−t
∂t ;
= −X 1e−t
v2 = ∂x2
∂t =
∂X 2et
∂t ;
=X 2et
v3 = ∂x3
∂t =
∂ (X 3 + X 2 (e−t − 1))
∂t
=− X 2e−t
∂θ/∂xj∂θ
∂x1= e−t.
∂θ
∂x2= −2e−t,
∂θ
∂x3= 3e
−t
dθ
dt = ∂θ
∂t + vj∂θ
∂xj
dθ
dt =− (X 1 + 3X 3) e−t − 3 (X 3 −X 2) e−t + X 2
+−X 1e−t
e−t
+
X 2e−t −2e−t
+−X 2e−t
3e−t
dθ
dt = −2 (X 1 + X 3) e−2t − 3 (X 3 − X 2) e−t
dvi (X j)
dt = 0
∂vi∂xi
= 0
u1 = 0; u2 = x2 − 12 (x2 + x3) e−t − 1
2 (x2 − x3) et;
u3 = x3 − 12 (x2 + x3) e−t + 1
2 (x2 − x3) et
ui = xi
−X i
x1 − X 1 = 0
x2 − X 2 = x2 − 12 (x2 + x3) e−t − 1
2 (x2 − x3) et
x3 − X 3 = x3 − 12 (x2 + x3) e−t + 1
2 (x2 − x3) et
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X 1 = x1
X 2 = 12 (x2 + x3) e−t + 12 (x2 − x3) et
X 3 = 12 (x2 + x3) e−t − 1
2 (x2 − x3) et
x2 = 12X 2
e−t + et− 1
2X 3
e−t + et
x3 = − 12
X 2
e−t + et
+ 12
X 3
e−t + et
x2 = 12 (X 2 + X 3) et + 1
2 (X 2 −X 3) e−t
x3 =
1
2 (X 2 + X 3) e
t
− 1
2 (X 2 −X 3) e−t
v1 = Dx1
Dt = 0
v2 = Dx2
Dt = 1
2 (X 2 + X 3) et − 12 (X 2 −X 3) e−t
v3 = Dx3
Dt = 1
2 (X 2 + X 3) et + 12 (X 2 −X 3) e−t
a1 = Dv1
Dt = 0
a2 = Dv2
Dt = 1
2 (X 2 + X 3) et + 12 (X 2 − X 3) e−t
a3 = Dv3
Dt = 1
2 (X 2 + X 3) et − 12 (X 2 − X 3) e−t
a1 = Dv1
Dt = 0; a2 =
Dv2Dt
= 12 (X 2 + X 3) et + 1
2 (X 2 −X 3) e−t;
a3 = Dv3
Dt = 1
2 (X 2 + X 3) et − 12 (X 2 − X 3) e−t
v1 = 0,
v2 = 12 (x2 + x3) e−tet − 1
2 (x2 − x3) ete−t = x3,
v3 = 12 (x2 + x3) e−tet + 1
2 (x2 − x3) ete−t = x2
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a1 = 0; a2 = x2; a3 = x3
∂X i∂xj
=
∂X 1∂x1
∂X 1∂x2
∂X 1∂x3
∂X 2∂x1
∂X 2∂x2
∂X 2∂x3
∂X 3∂x1
∂X 3∂x2
∂X 3∂x3
d
dt
∂X i∂xj
=
∂
∂xj
dX idt
=
∂vi∂xj
= Y ij
Y
dij wij
Y ij =
∂v1∂x1
∂v1∂x2
∂v1∂x3
∂v2∂x1
∂v2∂x2
∂v2∂x3
∂v3∂x1
∂v3∂x2
∂v3∂x3
Y ij = ∂vi∂xj
= 12
∂vi∂xj
+ ∂ vj∂xi
+ 1
2
∂vi∂xj
− ∂ vj∂xi
= dij + wij
dij
wij
dij = dji = 1
2
∂vi∂xj
+ ∂ vj∂xi
wij = −wji = 1
2
∂vi∂xj
− ∂ vj∂xi
dij
dij = 0
d11
d22
d33
xi
d12 d13 d23
xi xj i = j
dij eij
dij
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dV = 0
∂vi/∂xi = 0 J = 1
dvi = ∂vi∂xj
dxj
= Y ijdxj
P Q dxi vi vi
dvi Q P P
vi + dvi − vi = Y ij |P dxj
dvi = (dij + wij) dxj
dij
P
dvi = ωijdxj
dvi
wi = εijk∂vk∂xj
xi
ωi = 1
2
wi = 1
2
εijk∂vk
∂xj
vi = ∇φ = ∂φ
∂xiei
ϕ ϕ
rot = ijk∂vk∂xj
= 0
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P
xi = xi (X j , t)
xi
vi = Dxi
Dt
xi
xi = X j t = 0
f i
xi
xi
f i xi = xi(τ ) τ dxi/dτ γ xi
f i xi γ
f i
f i = αdxidτ
α xi
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xi = xi(τ )
f i
vi = αdxi (τ )
dτ
wi = αdxi (τ )
dτ
ti
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vk(xi, t)
xi = xi(τ )
τ
xi
dxi (τ )dτ
= vi [xi (τ ) , t1]
xi(τ )
t1
f i C
f i C
S ˆ
S
f inidS
S 1 S 2 f i V
S 1
S 2
S
ˆ V
div f idV =
ˆ S 1
f inidS 1 +
ˆ S 2
f inidS 2 +
ˆ S
f inidS
ni S f ini
S 1
S 2
ˆ V div f idV = ± ˆ S 1 f inidS +
ˆ S 2
f inidS
f i
divwi = 0
ˆ S 1
winidS =
ˆ S 2
winidS
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divvi = 0
ˆ S 1
inidS =
ˆ S 2
inidS
∂ϕ(xi, t)/∂t = 0
ϕ(xi, t)
vk (xi, t) = vk (xi) = vk (xi (X j , t)) = vk (X j , t)
X 1 X 2
xi
p(xi) xi
v1 = ωx2, v2 = −ωx2, v3 = 0
ai
vixi = 0
v1 = ωx2
v2 = −ωx1
v3 = 0
x1 = x1(t)
x2 = x2(t)
dx1
dt = ωx2
dx2
dt = −ωx1
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d2x2
dt2 =
−ω
dx1
dtdx1
dt = − 1
ω
d2x2
dt2
− 1
ω
d2x2
dt2 = ωx2
d2x2
dt2 + ωx2 = 0
x2 = sen ωt; x2 = cos ωtx2 = C 1 sen ωt + C 2 cos ωt
dx2
dt = C 1ω cos ωt − Cω sen ωt
C 1ω cos ωt − Cω sen ωt = −ωx1
x1 = −C 1 cos ωt + C 2 sen ωt
t = 0
xi = X j
X 2 = C 1 sen θ + C 2 cos θ
C 2 = X 2
X 1 = −C 1 cos θ + C 2 sen θ
C 1 = −X 1
P
x1 = −X 1 cos ωt + X 2 sen ωt; x= − X 1 sen ωt + X 2 cos ωt; x3 = 0
x1 = sen (αX 1 + ωt) , x2 = 1− cos(αX 2 + ω7) , x3 = 0
X j = 0 X 1 = X 2 = X 3 = 0
x1 = sen (ωt)
x2 = 1− cos(ωt)
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x2 − 1 = − cos(ωt)
x21 = sen2 (ωt)
(x2 − 1)2
= cos2 (ωt)
x21 + (x2 − 1)
2= 1
x1x2
v1 = x1
1 + t (1); v2 = 2x2
1 + t (2); v3 = 3x3
1 + t (3)
dx1
dt =
x1
1 + t (4);
dx2
dt =
2x2
1 + t (5);
dx3
dt =
3x3
1 + t
dx1
dt =
dt
1 + t
ln x1 = ln (1 + t) + ln C 1
ln x1 = ln C 1 (1 + t)
eln x1 = elnC 1(1+t)
x1 = C 1 (1 + t)
t = 0 x1 = X 1 C 1 = X 1
x1 = (1 + t) X 1
x1 = (1 + t) X 1
ln x2 =2 ln(1 + t) + ln C 2
= ln(1 + t)2
+ ln C 2
= ln(1 + t)2 C 2
x2 = (1 + t)2 C 2
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t = 0
x2 = X 2
C 2 = X 2
x2 = (1 + t)2
X 2
x3 = (1 + t)3
X 3
P
x1 = (1 + t) X 1; x2 = (1 + t)2 X 2; x3 = (1 + t)
3 X 3
v1 = x1
1 + t
, v2 = x2
1 + 2t
, v3 = 0
t = 0
(a, b)
vi = dxi (τ )
dτ
dx1
dτ =
x1
1 + t
dx2
dτ
= x2
1 + 2t
dx1
x1=
1
1 + tdτ
ln x1 = 1
1 + tτ + ln C 1
ln x1 = ln e τ 1+t + ln C 1
ln x1 = ln
C 1e τ 1+t
e ln x1
= e
lnC 1e
τ 1+t
x1 = C 1e τ 1+t
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x2 = C 2e τ 1+2t
τ = 0 (a, b)
X 1 = a =C 1e0
C 1 =a
X 2 = b =C 2e0
C 2 =b
x1 = ae τ 1+t
x2 = be τ 1+2t
ln x1 = ln a + τ
1 + tτ
1 + t = ln x1 − ln a
τ = (1 + t) ln x1
a
τ = (1 + 2t) ln x2
b
(1 + t))ln x1
a = (1 + 2t) ln
x2
b
1 + t
1 + 2t ln
x1
a = ln
x2
b
lnx1
a
1+t1+2t
= ln x2
be ln(x1
a ) 1+t1+2t
= eln x2b
x1
a
1+t1+2t
= x2
b
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x2 = b
x1
a 1+t1+2t
t = 0
x2 = b
ax1
vi = dxi
dt
dx1
dt
= x1
1 + t
dx2
dt =
1
1 + 2t
dx1
x1=
dt
1 + t
ln x1 = ln (1 + t) + ln C 1
x1 = C 1 (1 + t)
dx2
x2=
dt
1 + 2t
ln x2 = 1
2 ln (1 + 2t) + ln C 2
x2 = C 2 (1 + 2t)12
t = 0 (a, b)
C 1 = a, C 2 = b
x1 = a (1 + t)
x2 = b (1 + 2t)12
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t = x1
a − 1
x2 = b
1 + 2x1
a − 1 1
2
x1 = C 1 (1 + t) , x2 = C 2 (1 + 2t)12
C 1 = x1
1 + t
C 2 = x2
1 + 2t
(a, b) r < t
C 1 = a
1 + r
C 2 = b
(1 + 2t)12
x1 = a1 + t
1 + r
x2 = b
1 + 2t
1 + 2r
12
r
r = a (1 + t)
x1 −1
r = 1
2
(1 + 2t)
b
x2
2− 1
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a (1 + t)
x1 − 1 =
1
2
(1 + 2t) b
x22 − 1
x2 =
b 11+2t
2a(1+t)x1
− 1
t = 0
x2 =
2a
x1− 1
α
x1 = X 1
1 + α2t2
, x2 = X 2, x3 = X 3 X 1 = x1 cos αt − x2 sen αt, X 2 = x1 sen αt + x2 cos αt, X 3 = x3
v1 = −2x1x2x3
R4 , v2 =
x21 − x2
2
R4
, v3 = x2
R2
R2 = x21 + x2
2 = 0
x1 = X 1 + atX 2, x2 = X 2, x3 = X 3
a a(t) a(0) = 0
dij dij
xi
vi = xi1 + t
v1 = α x2
1
R2x2, v2 = −α
x22
R2x1, v3 = 0
α R2 = x21 + x2
2 = 0
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v1 =
−
αx1 + βx2
R2
, v2 = βx1 + αx2
R2
, v3 = 0
R2 = x21 + x2
2 = 0 π
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ρbi
pi ρ =ρ(xj , t)
ρb1 = pi
f i
t(n)i P
V
S f i ρi
P
dV
dS
ni
dS P df i dS
df i
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df i dS
lımdS →∞
df idS
= t(n)i
t(n)i
M i t(n)i
P
−t(n)i = t
(−n)i
P
−ni
xi
t ni
t
t(n)i = t
(n)i (xj , t, n)
dS
ni
t(n)i
dS t
ni
ti (xj , t,−n) = −ti (xj , t, n)
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P i V t
P i =
ˆ V
ρbidV
f i t
f i =
ˆ S
t(n)i dS
Ri
Ri =
ˆ s
t(n)i dS +
ˆ V
ρbidV
ˆ s
t(n)i dS +
ˆ V
ρbidV = 0
t(n)i ni P
P
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t(e1) = t(e1)1 e1 + t
(e1)2 e2 + t
(e1)3 e3
t(e2) = t(e2)1 e1 + t
(e2)2 e2 + t
(e2)3 e3
t(e3) = t(e3)1 e1 + t
(e3)2 e2 + t
(e3)3 e3
t(ei) = t(ei)j ej
P
t(ei)j
σij
t(ei) = σij ej
t(e1) = σ11e1 + σ12e2 + σ13e3
t(e2) = σ21e1 + σ22e2 + σ23e3
t(e3) = σ31e1 + σ32e2 + σ33e3
σ11
σ22
σ33
σ12 σ13 σ23 σ21 σ31 σ32 σij
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P (xi)
P dA1 dA2 dA3
x1x2
x1x3
x2x3 dA
ni
dA1 = dA cos(ni, x1) = dAn1
dA2 = dA cos(ni, x2) = dAn2
dA3 = dA cos(ni, x3) = dAn3
cos(ni, xi) ni xi
dV = 13
h dA
h P
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ˆ V
ρbidV +
ˆ A
t(n)i dA −
ˆ A1
t(e1)i dA1 −
ˆ A2
t(e2)i dA2 −
ˆ A3
t(e3)i dA3 = 0
ˆ A
ρbi13hdA +
ˆ A
t(n)i dA −
ˆ A
t(e1)i n1dA −
ˆ A
t(e2)i n2dA −
ˆ A
t(e3)i n3dA = 0
lımh→0
dV = lımh→0
13
hdA = 0
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ˆ A
t(n)i − t
(e1)i n1 − t
(e2)i n2 − t
(e3)i n3
dA = 0
t(n)i = t
(e1)i n1 + t
(e2)i n2 + t
(e3)i n3 = t
(ej)i nj
t(ei) = t(ei)j ej
t(ei) = σij ej
t(n)1 = σ11n1 + σ12n2 + σ13n3
t(n)2 = σ21n1 + σ22n2 + σ23n3
t(n)3 = σ31n1 + σ32n2 + σ33n3
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V S
t(n)i
ρbi
ˆ S
t(n)i dS +
ˆ V
ρbidV = 0
ˆ S
σijnjdS +
ˆ V
ρbidV = 0
ˆ S
ainidS =
ˆ V
∂ai∂xi
dV
ai = ai(xj) ni S
ˆ V
∂σij∂xj
dV +
ˆ V
ρbidV = 0
ˆ V
∂σij
∂xj + ρbi
dV = 0
∂σij∂xj
+ ρbi = 0
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ˆ S
εijkxjt(n)k dS + ˆ
V
εijkxjρbkdV = 0
ˆ S
εijkxjσkrnrdS +
ˆ V
εijkxjρbkdV = 0
ˆ V
εijk∂xjσkr
∂xrdV +
ˆ εijkxiρbkdV = 0
ˆ V
εijk ∂xi
∂xr σkr + xj∂σkr∂xr
dV +ˆ
εijkxjρbkdV = 0
∂xj∂xr
= δ jr
ˆ V
εijk
δ jrσkr + xj
∂σkr∂xr
dV +
ˆ εijkxjρbkdV = 0
ˆ V
εijkσkjdV +
ˆ V
εijkxj
∂σkr∂xr
+ ρbk
dV = 0
ˆ V
εijkσkjdV = 0
εijkσkj = 0
εijkσkj = εi11σ11 + εi12σ12 + εi13σ13
+ εi21σ21 + εi22σ22 + εi23σ23
+ εi31σ31 + εi32σ32 + εi33σ33 = 0
i = 1
σ23 = σ32
i = 2
σ13 = σ31
i = 3
σ12 = σ21
σij = σji =
σ11 σ12 σ13
σ12 σ22 σ23
σ13 σ23 σ33
σij eij
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dij
t(n)1 = σ11n1 + σ12n2 + σ13n3
t(n)2 = σ12n1 + σ22n2 + σ23n3
t(n)3 = σ13n1 + σ23n2 + σ33n3
∂σ11
∂x1+
∂ σ12
∂x2+
∂ σ13
∂x3+ ρb1 = 0
∂σ12
∂x1+
∂ σ22
∂x2+
∂ σ23
∂x3+ ρb2 = 0
∂σ13
∂x1+
∂ σ23
∂x2+
∂ σ33
∂x3+ ρb3 = 0
σ
rs = ariasjσij
a pq
P
σij =
2 −1 5−1 4 05 0 1
P x1 + 2x2
−3x3 = 4
t(n)i = σijnj
(1, 2,−3)
1√ 14
, 2√ 14
,− 3√ 14
t(n)1 = 2× 1√
14− 1× 2√
14+ 5 × −3√
14
t(n)2 = − 1× 1√
14+ 4 × 2√
14+ 0 × −3√
14
t(n)3 = 5× 1√
14+ 0 × 2√
14+ 1 × −3√
14
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t(n)1 =σ11n1 + σ12n2 + σ13n3
=2× 1√ 14− 1× 2√
14+ 5 ×− 3√
14
= −4.010
t(n)2 =σ12n1 + σ22n2 + σ23n3
=− 1× 1√ 14
+ 4 × 2√ 14
+ 0 ×− 3√
14
= 1.871
t(n)3 =σ13n1 + σ23n2 + σ33n3
=5 × 1√ 14
+ 0 × 2√ 14
+ 1 ×− 3√
14
= 0.5345
t(n) =−
4.010e1 + 1.871 e2 + 0.535 e3
t(n)i =
(−15)2
14 +
72
14 +
22
14 =
278
14 = 4.456
cos θ = t(n)k nkt(n)i
cos θ =
1√ 14
− 15√
14
+
2√ 14
7√ 14
+− 3√
14
2√ 14
278
14
= −0.1122
θ = 96.44
|σN | = t(n)i ni
|σN | = − 15√ 14× 1√
14+
7√ 14× 2√
14+
2√ 14×− 3√
14
= −0.5
|σS |2 =t(n)i
2
− |σN |2
|σS | =
4.4562 − 0.52 = 4.428
2− σ −1 5−1 4− σ 05 0 1− σ
= 0
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σ3−7σ2−12σ +93 = 0
σ1 = 6.73; σ2 =3.59; σ3 = −3.59
σ1 = 6.73
(2− 6.73) n1 − n2 + 5n3 =0
−n1 + (4 − 6.73) n2 =0
n21 + n2
2 + n23 =1
n1 = −0.7020
n2 = 0.257
n3 = 0.664
x1 : 134.58
x2 : 75.11
x3 : 48.42
t(n)1
2σ21
+
t(n)2
2σ22
+
t(n)3
2σ23
= 1
t(n)1
2
(6.73)2 +
t(n)2
2
(3.59)2 +
t(n)3
(−3.59)
2 = 1
n1 = n2 = n3 = n
n21 + n2
2 + n23 =1
3n2 =1
n = ± 1√ 3
1√ 3
, 1√ 3
, 1√ 3
t(n)1 = 1√
3 (σ11 + σ12 + σ13)
t(n)2 = 1√
3 (σ12 + σ22 + σ23)
t(n)3 = 1√
3 (σ13 + σ23 + σ33)
σN
|σN |oct = t(n)i ni =
13 (σ11 + σ22 + σ33 + 2σ12 + 2σ13 + 2σ23)
σS
|σS | =
t(n)i 2 − |σN |2
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|σS |oct = 13
(σ11 − σ22)
2+ (σ22 − σ33)
2+ (σ33 − σ11)
2+ 6 (σ2
12 + σ223 + σ2
31)
σij =
σ1 0 00 σ2 00 0 σ3
|σN |oct = σ1 + σ2 + σ3
3 =
I 13
I 1
|σS |oct = 13
(σ1 − σ2)
2+ (σ2 − σ3)
2+ (σ3 − σ1)
2
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σ11 = Mx2
I , σ12 =
V
R2 − x22
3I
, σ22 = σ13 = σ23 = σ33 = 0; I = πR4
4
σij =
2 3 4x2
3 2 8x1
4x2 8x1 2
kPa
P (1, 2, 3) x1 + x2 + x3 = 10
σij =
1 0.5 −1
0.5 2 −1.5−1 −1.5 −1
kg/cm2
12 , 12 , 1√
2
P a b c
P
σij =
1 a ba 1 cb c 1
P ni
tn1 tn2 P
σij =
2 0 −4
0 −2 4−4 4 0
σij =
11 2 8
2 2 −108 −10 5
a > b > c
σij =
a 0 0
0 b 00 0 c
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σij = 1 4 6
4 5 86 8 3
xi
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∆m
∆V ∆m
ρ = lım∆V →0
∆m
∆V
t
m =
ˆ V
ρd V
ρo = ρo (X j , t)
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m = ˆ V o
ρ (X j, t)dV o
dm
dt = m =
d
dt
ˆ V
ρ (xi, t) dV = 0
ϕ
d
dt
ˆ V
ϕ dV =
ˆ V
dϕ
dt + ϕ
∂vi∂xi
dV
ddt
ˆ V
ρdV = 0
ˆ V
dρ
dt + ρ
∂vi∂xi
dV
dρ
dt + ρ
∂vi∂xi
= 0
ˆ V o
ρo (X j , 0)dV o =ˆ V
ρ (xi, t) dV
dV = J dV o
ˆ V o
ρo (X j , 0)dV o =
ˆ V
ρ (xi (X j , t) , t)dV
ρo = J ρ
Jρ
d (ρJ )
dt = 0
dρ
dt + ρ
∂vi∂xi
= 0
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dρ
dt
= ∂ρ
∂t
+ vi∂ρ
∂xi
∂ρ
∂t + vi
∂ρ
∂xi+ ρ
∂vi∂xi
= 0
∂ρ
∂t +
∂ (ρvi)
∂xi= 0
∂ρ
∂t +
∂ (ρv1)
∂x1+
∂ (ρv2)
∂x2+
∂ (ρv3)
∂x3
∂ρ
∂t + ρ
∂vi∂xi
= 0; d
dt
1
ρ
+
∂ vi∂t
= 0
dvidxi
= 0; dv1
dx1+
dv2dx2
+ dv3dx3
= 0
Ω
= rotΩ =∇×
Ω = εijkdΩk
dxj
m =
ˆ V 0
ρ0(xj , t) dV 0 =
ˆ V
ρ(xi, t) dV =
ˆ V o
ρ [xi (xj , t) , t] J dV 0
ρo =ρJ
dρodt
=d (ρJ )
dt
dρo/dt = 0 t = 0
d (ρJ )
dt
= 0
d (ρJ )
dt =
dρ
dtJ + ρ
dJ
dt
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dJ
dt = J
dvidt
d(ρJ )
dt =
dρ
dtJ + ρJ
dvidxi
= 0
d(ρJ )
dt = J
dρ
dt + ρ
dvidxi
= 0
∂vi∂xi
= 0
vi = εijk∂ Ωk∂xj
Ω vi θ = θ(xi, t)
ρθ
d
dt
ˆ V
ρθ dV =
ˆ V
dρ θ
dt + ρθ
∂vi∂xi
dV
d
dtˆ V ρθ dV =
ˆ V
ρ
dθ
dt + θ
dρ
dt + ρθ
dvi
dxi
dV
d
dt
ˆ V
ρθdV =
ˆ V
ρ
∂θ
∂t + θ
dρ
dt + ρ
∂vi∂xi
dV
d
dt
ˆ V
θρdV =
ˆ V
ρ∂θ
∂t dV
x1 = X 1 + αtX 3, x2 = X 2 + α tX 3, x3 = X 3 + αt (X 1 + X 2)
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J =
∂x1
∂X 1
∂x1
∂X 2
∂x1
∂X 3∂x2
∂X 1
∂x2
∂X 2
∂x2
∂X 3∂x3
∂X 1
∂x3
∂X 2
∂x3
∂X 3
J =
1 0 αt0 1 αt
αt αt 1
= 1 − 2 (αt)2
ρ = ρo
1 + 2 (αt)2
∂vi∂xi
= 0, dρ
dt = 0, ρ = ρ0, J = 1
V S
tni ρbi vi
P i (t) =
ˆ V
ρvidV
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F i = mai
d
dt
ˆ V
ρvidV =
ˆ S
t(n)i dS +
ˆ V
ρbidV
d
dt
ˆ V
ρvidV =
ˆ S
σijnj dS +
ˆ V
ρbi dV
ˆ V
∂σij∂xj dV +
ˆ V ρbi dV =
ˆ V ρ
dvidt dV
∂σij∂xj
+ ρbi = ρdvidt
∂σij∂xj
+ ρbi = 0
Jσij = ∂xi∂X A
∂xj∂X B
S AB
S AB
d
dtˆ V
εijkxjρvkdv =ˆ S
εijkxjt(n)k ds +
ˆ V
εijkxjρbk
ˆ V
εijk
xj
ρ
∂vk∂t − ∂ σqk
∂xk− ρbk
− σjk
dV = 0
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ˆ εijkσjkdV = 0
εijkσjk = 0
σij = σji
K
V S t
K = 12
ˆ V
ρvividV
vi
ˆ V
∂σij∂xj
vi + ρbivi
dV =
ˆ V
ρvidvidt
dV
ˆ V
∂σij∂xj
vi + ρbivi
dV =
d
dt
ˆ V
ρvividt
dV
ˆ V ∂σij
∂xj υi + ρbiυi
dV =
dK
dt
∂ (viσij)
∂xj=
∂σij∂xj
vi + ∂vi∂xj
σij
∂σij∂xj
vi = ∂ (viσij)
∂xj− ∂vi
∂xjσij
∂vi/∂xj
∂vi∂xj
= dij + ωij
dij = 12
∂vi∂xj
+ ∂ vj∂xi
, ωij =
12
∂vi∂xj
− ∂vj∂xj
vi∂σij∂xj
= ∂ (viσij)
∂xj− σij (dij + ωij)
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vi∂σij∂xj
= ∂ (viσij)
∂xj− σijdij
σijωij = 0
ˆ V
∂ (viσij)
∂xj− σijdij + ρbivi
dV =
dK
dt
ˆ V
∂ (viσij)
∂xj
dV +
ˆ V
ρvibidV = dK
dt +
ˆ V
σijdijdV
ˆ S
viσijnjdS +
ˆ V
viρbidV = dK
dt +
ˆ V
σijdijdV
t(n)i = σijnj
ˆ S
vit(n)i dS +
ˆ V
ρbividV = dK
dt +
ˆ V
σijdijdV
dU
dt =
ˆ V
σijdijdV
U
ˆ S
vit(n)i dS +
ˆ V
ρbividV = dK
dt +
dU
dt
ˆ S
vit(n)i dS +
ˆ V
ρvibidV .=
dW
dt
dU
dt
= d
dtˆ V
ρu dV = ˆ V
ρdu
dt
dV
u
S Q V
S
Q =
ˆ S
q inidS
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q i
ni
V
S
H =
ˆ V
ρh dV
h
H − Q
dK
dt +
dU
dt = P + (H − Q)
ddtˆ V
ρ vivi2 dV + ddt
ˆ V
ρudV =ˆ S
t(n)i vidS +ˆ V
ρbividV +ˆ V
ρhdV − ˆ S
q inidS
ˆ S
t(n)i vi − q ini
dS =
ˆ S
(σijnjvi − q ini) dS
ˆ S
(σijnjvi − q ini) dS =
ˆ V
∂ (σijvi)
∂xj− ∂q i
∂xi
dV
ˆ V ∂ (σijvi)
∂xj − ∂q i
∂xidV = ˆ
V vi
∂σij
∂xj+ σij
∂vi
∂xj − ∂q i
∂xidV
∂vi∂xj
= dij + ωij
ˆ V
vi
∂σij∂xj
+ σij∂vi∂xj
− ∂q i∂xi
dV =
ˆ V
vi
∂σij∂xj
+ σij (dij + ωij) − ∂q i∂xi
dV
ˆ V
vi
∂σij∂xj
+ σij∂vi∂xj
− ∂q i∂xi
dV =
ˆ V
vi
∂σij∂xj
+ σijdij − ∂q i∂xi
dV
d
dt
ˆ V
ρ
vivi2
+ ρu
dV =
ˆ V
ρvi
dvidt
+ ρdu
dt
dV
ˆ V
ρvi
dvidt
+ ρdu
dt
dV =
ˆ V
(ρbivi + ρh) dV +
ˆ V
vi
∂σij∂xj
+ σijdij − ∂q i∂xi
dV
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ˆ V ρvi
dvi
dt
+ ρdu
dtdV = ˆ
V
(ρbivi + ρh) dV + ˆ V vi
∂σij
∂xj
+ σijdij
−
∂q i
∂xidV
ρdu
dt − σijdij +
∂q i∂xi
− ρh = 0
du
dt =
1
ρσijdij − 1
ρ
∂q i∂xi
+ h
σij
dij
h
Q
dQ
dt = −
ˆ S
q inidS +
ˆ V
ρhdV
dK
dt +
du
dt =
dW
dt +
dQ
dt
d
dtˆ V
ρvivi
2 dV + ˆ
V
ρdu
dtdV = ˆ
S
vit(n)
i
dS + ˆ V
ρhdV − ˆ
S
q inidS
d
dt
vivi2
+ u
= 1
ρ
∂ (σijvi)
∂xj
+ biυi − 1
ρ
∂q i∂xi
+ h
−
S
V S
−
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d
dt
ˆ V
ρsdV
ˆ V
ρedV −ˆ S
q iniT
dS
e s T
ˆ V
ρds
dtdV
ˆ V
ρedV −ˆ V
∂
∂xi
q iT
dV
γ
γ = ds
dt − e − 1
ρ
∂
∂xi
q iT
∂ρ
∂t +
∂ (ρvk)
∂xk= 0
∂σij
∂xj+ ρb
i = ρ
∂υi
∂t
ρdu
dt − σijdij − ρh +
∂q i∂xi
= 0
ρo
ρbi
h
ρ ui vi σij q i
u
−
ds
dt = h − 1
ρ
∂
∂xi
q iT
0
s T
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−
V S
dW B dW S
dW B + dW S = d
ˆ V
udV
dW B =
ˆ V
ρbiduidV
dui
dW S =
ˆ S
t(n)i didS
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t(n)i = σijnj
dW S = ˆ s
σijnjduidS = ˆ S
σijduinjdS
ˆ S
σijduinjdS =
ˆ V
∂ (σijdui)
∂xjdV =
ˆ V
∂σij∂xj
dui + σij∂ dui∂xj
dV
dW S + dW B =
ˆ V
∂σij∂xj
dui + σijd∂ui∂xj
dV +
ˆ V
ρbidV
ˆ V
dW dV =ˆ V
∂σij∂xj
+ ρbi
dui + σijd
∂ui∂xj
dV
ˆ V
dW =
ˆ V
σijd (eij + ωij)dV
=
ˆ V
σijd (eij)dV
eij wij σijdwij = 0
dW = σijd (eij)
dW = σ11de11 + σ12de12 + σ13de13 +
σ21de21 + σ22de22 + σ23de23 +
σ31de31 + σ32de32 + σ33de33
W (eij)
σ11 = ∂W
∂e11; σ12 =
∂W
∂e12; σ13 =
∂W
∂e13;
σ12 =
∂W
∂e21 ; σ22 =
∂W
∂e22 ; σ23 =
∂W
∂e23 ;
σ31 = ∂W
∂e31; σ32 =
∂W
∂e32; σ33 =
∂W
∂e33
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σij = σji = 1
2 ∂W
∂eij+
∂W
∂eji
W
= x1
1 + te1 ; ρ =
ρo1 + t
= t(x1e1 + x3e3) ρ = ρ(t)
k
v1 = kx3 (x2 − 2)2 , v2 = −x1x2, v3 = kx1x3
x1 = (1 + a) X 1 + bX 2, x2 = bX 1 + (1 + a) X 2, x3 = X 3
a b a > b − 1
ρ = ρo
(1 + a)2 − b2
m
xi = 1
m
ˆ V
ρxidV
md2xi
dt2 = ρb
i
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σij = C ijpqe pq
C ijpq
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du
dt =
1
ρσijdij − 1
ρ
∂q i∂xi
+ h
du
dt =
1
ρσijdij
dij = deij
dt
du
dt =
1
ρσij
deijdt
du = 1
ρσijdeij
u u = u(eij)
du = ∂u
∂eijdeij
1ρσij = ∂u
∂eij
W = ρou, w = ρu
W = w
∂w
∂eij= ρ
∂u
∂eij
σij = ρ
∂u
∂eij
σij = ∂w
∂eij
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V
S
lij = eij = 1
2
∂ui∂xj
+ ∂ uj∂xi
=
1
2
∂ui∂X j
+ ∂uj∂X i
σ11 =C 1111e11 + C 1112e12 + C 1113e13 + C 1121e21 + C 1122e22 + C 1123e23 +
C 1131e31 + C 1132e32 + C 1133e33
σ11 = C 1111e11 + C 1122e22 + C 1133e33 + (C 1112 + C 1121) e12 +
(C 1113 + C 1131) e13 + (C 1123 + C 1132) e23
σij = C ijpqe pq
eij
σ pq
eij = S ijpqσ pq
σ12 = C 1211e11 + C 1222e22 + C 1233e33 + (C 1212 + C 1221) e12 +
(C 1213 + C 1231) e13 + (C 1223 + C 1232) e23
σij = σji
σij = C ijrsers = σji = C jirsers
C ijrs = C jirs
eij = eji
σij = C ijrsers = C ijsresr
C ijrs = C ijsr
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u
∂u
∂e11= σ11;
∂u
∂e22= σ22
σ11 = C 1111e11 + C 1112e12 + C 1113e13 + C 1121e21 + C 1122e22 +
C 1123e23 + C 1131e31 + C 1132e32 + C 1133e33
σ22 = C 2211e11 + C 2212e12 + C 2213e13 + C 2221e21 + C 2222e22 +
C 2223e23 + C 2231e31 + C 2232e32 + C 2233e33
∂ 2u
∂e11∂e22=
∂σ11
∂e22= C 1122
∂ 2u
∂e22∂e11=
∂σ22
∂e11= C 2211
∂ 2u
∂e11∂e22=
∂ 2u
∂e22∂e11
C 1122 = C 2211
∂ 2u
∂eij∂e pq= C ijpq = C pqij
σij = C ijpqe pq
σ11
σ22
σ33
σ12σ13
σ23
=
C 1111 C 1122 C 1133 C 1112 C 1113 C 1123C 2211 C 2222 C 2233 C 2212 C 2213 C 2223C 3311 C 3322 C 3333 C 3312 C 3313 C 3323
C 1211 C 1222 C 1233 C 1212 C 1213 C 1223C 1311 C 1322 C 1333 C 1312 C 1313 C 1323C 2311 C 2322 C 2333 C 2312 C 2313 C 2323
e11e22e33
2e122e132e23
C ijpq
σ pq
eij = S ijpqσ pq
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e11
e22e332e122e132e23
=
S 1111 S 1122 S 1133 S 1112 S 1113 S 1123
S 2211 S 2222 S 2233 S 2212 S 2213 S 2223S 3311 S 3322 S 3333 S 3312 S 3313 S 3323S 1211 S 1222 S 1233 S 1212 S 1213 S 1223S 1311 S 1322 S 1333 S 1312 S 1313 S 1323S 2311 S 2322 S 2333 S 2312 S 2313 S 2323
σ11
σ22σ33
σ12
σ13
σ23
S ijpq
σ = Ee
σij = C ijpqe pq
C
abcd = aaiabjackadmC ijkm
X j
xi
x1x2
aij = 1 0 0
0 1 00 0 −1
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a = 1
b = 1
c = 1
d = 1
C
1111 = a1ia1ja1ka1mC ijkm
C
1111 = a11a1ja1ka1mC 1jkm + a12a1ja1ka1mC 2jkm + a13a1ja1ka1mC 3jkm
C
1111 = a1ja1ka1mC 1jkm
C
1111 = C 1111
C
1123
C
1123 = −C 1123
C
1123 = −C 1123 = 0
C
abcd
C ijkm =
C 1111 C 1122 C 1133 C 1112 0 0C 1122 C 2222 C 2233 C 2212 0 0C 1133 C 2233 C 3333 C 3312 0 0C 1112 C 2212 C 3312 C 1212 0 0
0 0 0 0 C 1313 C 13230 0 0 0 C 1323 C 2323
x1x2
x1x3
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aij = −1 0 0
0 1 00 0 −1
C ijkm =
C 1111 C 1122 C 1133 0 0 0C 1122 C 2222 C 2233 0 0 0C 1133 C 2233 C 3333 0 0 0
0 0 0 C 1212 0 00 0 0 0 C 1313 00 0 0 0 0 C 2323
X 3
aij =
cos θ sen θ 0− sen θ cos θ 0
0 0 1
X j σkl = C klmnemn
xi
σ
pq = C pqrse
rs
e
rs = armasnemn
e
11 = cos2 θ e11 + 2 cos θ sen θ e12 + cos θ e22;
e
22 = sen2 θe11 − 2cos θ sen θe12 + cos θe22; e
33 = e33
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e
12 = (e22 − e11)cos θ sen θ + e12
cos2 θ − sen2 θ
;
e
13 = e13 cos θ + e23 sen θ; e
23 = −e13 sen θ + e23 cos θ sen θ
σ
ij = aipajqσ pq
σ
11 = σ11 cos2 θ + 2σ12 cos θ sen θ + σ22 sen2 θ
σ
22 = σ11 sen2 θ − 2σ12 cos θ sen θ + e22 cos2 θ; σ
33 = σ33
σ
12 = (σ22 − σ11)cos θ sen θ + σ12
cos2 θ − sen2 θ
;
σ
13 = σ13 cos θ + σ23 sen θ; σ
23 = −σ13 sen θ + σ23 cos θ
σ
33 = C 3311e
11 + C 3322e
22 + C 3333e
33 + 2
C 3312e
12 + C 3313e
13 + C 3323e
23
σ
33 =e11
C 3311 cos2 θ + C 3322 sen2 θ − 2C 3312 cos θ sen θ
+ e22
C 3311 sen2 θ + C 3322 cos2 θ + 2C 3312 cos θ sen θ
+ 2e12
(C 3311 −C 3322)cos θ sen θ + C 3312
cos2 θ − sen2 θ
+ 2e13 [C 3313 cos θ − C 3323 sen θ] + 2e23 [C 3313 sen θ + C 3323 cos θ]
σ33 = C 3311e11 + C 3322e22 + C 3333e33 + 2 (C 3312e12 + C 3313e13 + C 3323e23)
σ
33 = σ33
e11
C 3311 cos2 θ + C 3322 sen2 θ
−2C 3312 cos θ sen θ = C 3311
C 3311 −C 3311 cos2 θ −C 3322 sen2 θ + 2C 3312 cos θ sen θ = 0
C 3311
1− cos2 θ−C 3322 sen2 θ + 2C 3312 cos θ sen θ = 0
C 3311 sen2 θ − C 3322 sen2 θ + 2C 3312 cos θ sen θ = 0
(C 3322 −C 3311)sen2 θ − 2C 3312 cos θ sen θ = 0
C 3311 = C 3322; C 3312 = 0
σ
13 σ13 σ
23 σ23
C 1111 C 1122 C 1133 0 0 0C 1122 C 1111 C 1133 0 0 0C 1133 C 1133 C 3333 0 0 0
0 0 0 12 (C 1111 −C 1122) 0 0
0 0 0 0 C 1313 00 0 0 0 0 C 1313
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X 1
X 2
X 2 cos θ 0 − sen θ
0 1 0sen θ 0 cos θ
C 1111 = C 3333; C 1313 = 12 (C 1111 − C 1133)
X 1
1 0 00 cos θ sen θ0 − sen θ cos θ
C ijkm =
C 1111 C 1122 C 1122 0 0 0C 1122 C 1111 C 1122 0 0 0C 1122 C 1122 C 1111 0 0 0
0 0 0 12 (C 1111 −C 1122) 0 0
0 0 0 0 12 (C 1111 −C 1122) 0
0 0 0 0 0 12 (C 1111 − C 1122)
λ = C 1122, µ = 12 (C 1111 − C 1122)
C 1111 = λ + 2µ
λ µ µ
σ11
σ22
σ33
σ12
σ13
σ23
=
λ + 2µ λ λ 0 0 0λ λ + 2µ λ 0 0 0λ λ λ + 2µ 0 0 00 0 0 µ 0 00 0 0 0 µ 0
0 0 0 0 0 µ
e11e22e33
2e122e132e23
σij = 2µeij + λδ ijenn
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eij =
−
λδ ij
2µ (3λ + 2µ)
σnn + 1
2µ
σij
µ = 0, 3λ + 2µ = 0
e11e22e33
2e122e132e23
=
λ+µµ(3λ+2µ)
−λ2µ(3λ+2µ)
−λ2µ(3λ+2µ) 0 0 0
−λ2µ(3λ+2µ)
λ+µµ(3λ+2µ)
−λ2µ(3λ+2µ) 0 0 0
−λ2µ(3λ+2µ)
−λ2µ(3λ+2)
λ+µµ(3λ+2µ) 0 0 0
0 0 0 14µ 0 0
0 0 0 0 14µ 0
0 0 0 0 0 14µ
σ11
σ22
σ33
σ12
σ13
σ23
σm = σnn
3 =
σ11 + σ22 + σ33
3
e11 = −λ (σnn)
2µ (3λ + 2µ) +
σ11
2µ
e11 = 1
2µ (3λ + 2µ) [−λσnn + (3λ + 2µ) σ11]
e22 = 12µ (3λ + 2µ)
[−λσnn + (3λ + 2µ) σ22]
e33 = 1
2µ (3λ + 2µ) [−λσnn + (3λ + 2µ) σ33]
enn = σnn3λ + 2µ
eesf = enn
3 ; σesf =
σnn3
= σm
enn = 33λ + 2µ σm
eV = e11 + e22 + e33
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eV = 3σm
3λ + 2µ
= σm
K
K = 3λ + 2µ
3
K
σ11 + σ22 + σ33 = 0
e11 = −λ
2µ (3λ + 2µ) (0) +
1
2µσ11 =
σ11
2µ
e22 = σ22
2µ , e33 =
σ33
2µ
σ33 − (σ11 + σ22) e33 = −(σ11 + σ22)/2µ
σnn = 3Kenn
σ pq = 2µe pq
eij = 1
2µ
σij − 1
3δ ijσnn
+
1
9K δ ijσnn
E ν
σij =
σ11 0 0
0 0 00 0 0
e11 = λ + µ
µ (3λ + 2µ)σ11, e22 =
−λ
2µ (3λ + 2µ)σ11, e33 =
−λ
2µ (3λ + 2µ)σ11
E
ν
e11 = σ11
E
, e22 =
−νσ11
E
, e33 =
−νσ11
E
E = µ (3λ + 2µ)
λ + µ , ν =
λ
2 (λ + µ)
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λ = νE
(1 + ν ) (1− 2ν )
, µ = E
2 (1 + ν )
K = E
2 (1− 2ν )
σij = E
1 + ν eij +
νE
(1 + ν ) (1− 2ν )δ ijenn
eij = 1
E [(1 + ν ) σij − νδ ijσnn]
T
∆T
α
e11 = e22 = e33 = α (∆T ) , e12 = e13 = e23 = 0
eij = δ ijα (∆T )
eij = 1
E [(1 + ν ) σij − νδ ijσnn] + αδ ij (∆T )
σij = E
1 + ν eij +
νE
(1 + ν ) (1− 2ν )δ ijenn − E
1− 2ν δ ij (∆T )
σij = 2µeij + λδ ijenn − (3λ + 2µ) δ ijα (∆T )
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∂σij∂xj
+ ρbi = 0
σij = 2µeij + λδ ijenn
eij = 1
2 ∂ui∂xj
+ ∂ uj∂xi
−
σij = 2µ1
2
∂ui∂xj
+ ∂ uj∂xi
+ δ ij
∂un∂xn
σij = µ
∂ui∂xj
+ ∂ uj∂xi
+ δ ij
∂un∂xn
∂
µ
∂ui∂xj
+ ∂ uj∂xi
+ λδ ij ∂un∂xn
∂xj + ρbi = 0
µ ∂ 2ui∂xj∂xj
+ (λ + µ) ∂
∂xi
∂un∂xn
+ ρbi = 0
∇2 = ∂
∂xj∂xj=
∂ 2
∂x21
+ ∂ 2
∂x22
+ ∂ 2
∂x23
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µ∇2ui + (λ + µ) ∂
∂xi
∂un∂xn
+ ρbi = 0
∂ 2e11∂x2
2
+ ∂ 2e22
∂x21
= 2 ∂ 2e12∂x1∂x2
, ∂ 2e11
∂x23
+ ∂ 2e33
∂x21
= 2 ∂ 2e13∂x1∂x3
, ∂ 2e22
∂x23
+ ∂ 2e33
∂x22
= 2 ∂ 2
∂x2∂x3
∂
∂x1
−∂e23
∂x1+
∂ e13∂x2
+ ∂ e12
∂x3
=
∂ 2e11∂x2∂x3
, ∂
∂x2
∂e23∂x1
− ∂ e13∂x2
+ ∂ e12
∂x3
=
∂ 2e22∂x1∂x3
,
∂
∂x3
∂e23∂x1
+ ∂ e13
∂x2− ∂ e12
∂x3
=
∂ 2e33∂x1∂x2
∂ 2e11∂x2
2
+ ∂ 2e22
∂x21
= 2 ∂ 2e11∂x1∂x2
eij = 1
E [(1 + ν ) σij − νδ ijσnn]
e11 = 1
E (σ11 − νσ22 − νσ33)
e22 = 1
E (−νσ11 + σ22 − νσ33)
e33 = 1
E (−νσ11 − νσ22 + σ33)
e12 = 1 + ν
E
σ12
e13 = 1 + ν
E σ13
e23 = 1 + ν
E σ23
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∂ 2σ11
∂x22
+ ∂ 2σ22
∂x21 −
ν ∂ 2σ11
∂x21 −
ν ∂ 2σ22
∂x22 −
ν ∂ 2σ33
∂x21 −
ν ∂ 2σ33
∂x22
= 2 (1 + ν ) ∂σ12
∂x1∂x2
±v ∂ 2σ22∂x2
1
±v ∂ 2σ11∂x2
2
(1 + v)
∂ 2σ11
∂x22
+ ∂ 2σ22
∂x21
− ν
∂ 2 (σ11 + σ22 + σ33)
∂x21
+ ∂ 2 (σ11 + σ22 + σ33)
∂x22
= 2 (1 + ν ) ∂ 2σ12
∂x1∂x2
σ11 + σ22 + σ33 I 1
(1 + ν )∂ 2σ11
∂x22
+ ∂ 2σ22
∂x21 − ν
∂ 2I 1∂x2
1
+ ∂ 2I 1
∂x22 = 2 (1 + ν )
∂ 2σ12
∂x1∂x2
∂σ11
∂x1+
∂ σ12
∂x2+
∂ σ13
∂x3+ ρb1 = 0
∂σ12
∂x1+
∂ σ22
∂x2+
∂ σ23
∂x3+ ρb2 = 0
∂σ12
∂x2= − ∂ σ11
∂x1− ∂ σ13
∂x3− ρb1
∂σ12
∂x1
= − ∂ σ22
∂x2
− ∂ σ23
∂x3
− ρb2
x1 x2
∂ 2σ12
∂x1∂x2= − ∂ 2σ11
∂x21
− ∂ 2σ13
∂x1∂x3− ∂ ρb1
∂x1
∂ 2σ12
∂x1∂x2= − ∂ 2σ22
∂x22
− ∂ 2σ23
∂x1∂x3− ∂ ρb2
∂x2
[2 ∂ 2σ12
∂x1∂x2= −∂ 2σ11
∂x21
− ∂ 2σ22
∂x22
− ∂
∂x3
∂σ13
∂x1+
∂ σ23
∂x2
− ∂ ρb1
∂x1− ∂ ρb2
∂x2
∂σ13
∂x1+
∂ σ23
∂x2= −∂σ33
∂x3− ρb3
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2 ∂ 2σ12
∂x1∂x2
=
−∂ 2σ11
∂x21 −
∂ 2σ22
∂x22
+ ∂ 2σ33
∂x23 −
∂ ρb1
∂x1 − ∂ ρb2
∂x2
+ ∂ ρb3
∂x3
(1 + ν )
∂ 2σ11
∂x21
+ ∂ 2σ11
∂x22
+ ∂ 2σ22
∂x21
+ ∂ 2σ22
∂x22
− ∂ 2σ33
∂x23
−
ν
∂ 2I 1∂x2
1
+ ∂I 1∂ 2x3
2
= (1 + ν )
−∂ρb1
∂x1− ∂ ρb2
∂x2+
∂ ρb3∂x3
±∂ 2σ33
∂x21
, ± ∂ 2σ33
∂x22
, ± ∂ 2σ33
∂x23
, ± ∂ 2σ11
∂x23
, ∂ 2σ22
∂x32
(1 + ν )∇2I 1 −∇2σ33 − ∂
2
I 1∂x23
−ν
∂ 2I 1∂x2
2
+ ∂ 2I 1
∂x23
= (1 + ν )
−∂ρb1
∂x1− ∂ ρb2
∂x2+
∂ ρb3∂x3
±∂ 2I 1∂x2
3
(1 + ν )
∇2I 1 −∇2σ33 − ∂ 2I 1
∂x23
−
ν
∇2I 1 − ∂ 2I 1
∂x23
= (1 + ν )
−∂ρb1
∂x1− ∂ ρb2
∂x2+
∂ ρb3∂x3
(1 + ν )
∇2I 1 −∇2σ11 − ∂ 2I 1
∂x21
−
ν
∇2I 1 − ∂ 2I 1
∂x21
= (1 + ν )
∂ρb1∂x1
− ∂ ρb2∂x2
− ∂ ρb3∂x3
(1 + ν )
∇2I 1 −∇2σ22 − ∂ 2I 1
∂x22
−
ν
∇2I 1 − ∂ 2I 1
∂x23
= (1 + ν )
−∂ρb1
∂x1+
∂ ρb2∂x2
− ∂ ρb3∂x3
(1− ν )∇2I 1 = − (1 + ν ) ∂ρbi
∂xi
∇2I 1 = −
1 + ν
1− ν
∂ρbi
∂xi
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∇2σ11 +
1
1 + ν
∂ 2I 1
∂x21
+ 1
1− ν
∂ρbi
∂xi=
−∂ρb1
∂x1
+ ∂ ρb2
∂x2
+ ∂ ρb3
∂x3
±∂ρb1∂x1
∇2
σ11 + 1
1 + ν
∂ 2I 1∂x2
1
+ 1
1− ν
∂ρbi∂xi
= ∂ρbi
∂xi− 2
∂ρb1∂x1
∇2σ11 + 1
1 + ν
∂ 2I 1∂x2
1
= − ν
1− ν
∂ρbi∂xi
− 2∂ρb1∂x1
∇2
σ22 +
1
1 + ν
∂ 2I 1
∂x22 = − ν
1− ν
∂ρbi
∂xi − 2
∂ρb2
∂x2
∇2σ33 + 1
1 + ν
∂ 2I 1∂x2
3
= − ν
1− ν
∂ρbi∂xi
− 2∂ρb3∂x3
∇2σ12 + 1
1 + ν
∂ 2I 1∂x1∂x2
= −
∂ρb1∂x2
+ ∂ ρb2
∂x1
∇2σ13 + 1
1 + ν
∂ 2I 1∂x1∂x3
= −
∂ρb1∂x3
+ ∂ ρb3
∂x1
∇2σ23 + 11 + ν
∂ 2I 1∂x2∂x3
= −
∂ρb2∂x3
+ ∂ ρb3∂x2
∇2σij + 1
1 + ν
∂ 2I 1∂xi∂xj
= − ν
1− ν δ ij
∂ρbn∂xn
−
∂ρbi∂xj
+ ∂ ρbj
∂xi
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∂σ(1)ij
∂xi+ ρb
(1)i = 0
|
∂σ(2)ij
∂xi+ ρb
(2)i = 0
∂
σ(1)ij + σ
(2)ij
∂xi
+ ρ
b(1)i + b
(2)i
= 0
σij =
σ11 0 0
0 0 00 0 0
eij = 1
E [(1 + ν ) σij − νδ ijσnn]
e11 = σ11
E , e22 = −νσ11
E , e33 = −νσ11
E , e12 = e23 = e13 = 0
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e11 = ∂u1
∂x1
; e22 = ∂u2
∂x2
; e33 = ∂u3
∂x3
du1
dx2+
du2
dx1=
du1
dx3+
du3
dx1=
du2
dx3+
du3
dx2= 0
du1
dx1=
σ11
E ,
du2
dx2= − ν
E σ11,
du3
dx3= − ν
E σ11
u1 = σ11
E x1, u2 = −νσ11
E x2, u3 = −νσ11
E x3
u1 = u1 (x1) , u2 = 0, u3 = 0
e11 = e11 (x1) , e22 = e33 = e12 = e13 = e23 = 0
σij = E
1 + ν eij +
νE
(1 + ν ) (1− 2ν )δ ijenn
σ11 = E
1 + ν e11 +
νE
(1 + ν ) + (1 − 2ν )(e11 + e22 + e33)
= E
(1 + ν ) (1−
2ν ) [(1 − ν )e11 + ν (e22 + e33)]
σ22 = E
(1 + ν ) (1− 2ν ) [(1 − ν )e22 + ν (e11 + e33)]
σ33 = E
(1 + ν )(1− 2ν ) [(1 − ν )e22 + ν (e11 + e33)]
σ12 = E
1 + ν e12, σ13 =
E
1 + ν e13, σ23 =
E
1 + ν e23
σ11 = E (1− ν )
(1 + ν )(1− 2ν )e11, σ22 =
E
(1 + ν )(1− 2ν )e11, σ33 =
E
(1 + ν )(1− 2ν )e11
σ22 = σ33
∂σ11
∂x1+
∂ σ12
∂x2+
∂ σ13
∂x3+ ρb1 = 0
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σ11
e11 = e
∂u1∂x1
= e
u1 = ex1, u2 = 0, u3 = 0
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σ11 = σ11 (x1, x2) , σ22 = σ22 (x1, x2) , σ12 = σ12 (x1, x2) , σ13 = σ23 = σ33 = 0
σij = E
1 + ν eij +
νE
(1 + ν ) (1− 2ν )δ ijenn
σ11 = E
(1 + ν ) (1−
2ν ) [(1 − ν ) e11 + νe22 + νe33]
σ22 = E
(1 + ν ) (1− 2ν ) [νe11 + (1 − ν ) e22 + νe33]
σ12 = E
1 + ν e12, σ13 = σ23 = σ33 = 0
σ33 = 0
σ33 = 0 = E
(1 + ν ) (1− 2ν ) [νe11 + νe22 + (1 − ν ) e33]
e33 = − ν
1− ν (e11 + e22)
e33 σ11 σ22
σ11 = E
1− ν 2 (e11 + νe22) , σ22 =
E
1− ν 2 (νe11 + e22) , σ12 =
E
1 + ν e12
σij =
σ11 σ12 0
σ12 σ22 00 0 0
eij = 1
E [(1 + ν ) σij − νδ ijσnn]
e11 = 1E
(σ11 − νσ22) , e22 = 1E
(−νσ11 + σ33)
e33 = − ν
E (σ11 + σ22) , e12 =
1 + ν
E σ12, e13 = e23 = 0
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eij =
e11 e12 0e12 e22 0
0 0 e33
e33 x1 x2
∂σ11
∂x1+
∂ σ12
∂x2+ ρb1 = 0;
∂σ12
∂x1+
∂ σ22
∂x2+ ρb2 = 0; ρb3 = 0
x3
∂ 2e11∂x2
2
+ ∂ 2e22
∂x21
= 2 ∂ 2e12∂x1∂x2
, ∂ 2e33
∂x21
= 0, ∂ 2e33
∂x22
= 0, ∂ 2e33∂x1∂x2
= 0
∂ 2e33∂x2
1
= ∂ 2e33
∂x22
= ∂ 2e33∂x1∂x2
= 0
e33 = C 1x1 + C 2x2 + C 3 C 1 C 2 C 3
u1 = u1 (x1, x2) , u2 = u2 (x1,x2) , u3 = 0
e11 = ∂u1
∂x1, e22 =
∂u2
∂x2, e12 =
1
2
∂u1
∂x2+
∂ u2
∂x1
, e33 = e13 = e23 = 0
−
σij = E
1 + ν eij +
νE
(1 + ν ) (1− 2ν )δ ijenn
σ11 = E
(1 + ν ) (1− 2ν ) [(1 − ν ) e11 + νe22]
σ22 = E
(1 + ν ) (1− 2ν ) [νe11 + (1 − ν ) e22]
σ33 = νE
(1 + ν ) (1− 2ν ) [e11 + e22]
σ12 =
E
1 + ν e12, σ13 = σ23 = 0
eij = 1
E [(1 + ν ) σij − νδ ijσnn]
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e11 = 1
E (σ11 − νσ22 − νσ33)
e22 = 1E
(−νσ11 + σ22 − νσ33)
e33 = 1
E (−νσ11 − νσ22 + σ33) = 0
e12 = 1 + ν
E σ12, e13 = e23 = 0
σ33 = ν (σ11 + σ22)
e11 = 1
E
σ11 − νσ22 − ν 2 (σ11 + σ22)
e11 = 1 + ν
E [(1− ν ) σ11 − νσ22]
e22 = 1 + ν
E [−νσ11 + (1 − ν ) σ22]
e12 = 1 + ν
E σ12
σ33 = νE
(1 + ν ) (1− ν )
1 + ν
E (1− 2ν ) σ11 + (1 − 2ν ) σ22
σ33 = ν (σ11 + σ22)
σ33 x1 x2
∂σ11
∂x1+
∂ σ12
∂x2+ ρb1 = 0;
∂σ12
∂x1+
∂ σ22
∂x2+ ρb2 = 0; ρb3 = 0
x3
∂ 2e11∂x2
2
+ ∂ 2e11
∂x21
= 2 ∂ 2e12∂x1∂x2
µ∂ 2ui
∂xj+ (λ + µ)
∂ 2uj
∂xi∂xj+ ρbi = 0
µ∇2ui + (λ + µ) ∂ (enn)
∂xi+ ρbi = 0
µ∇2u1 + (λ + µ) ∂
∂x1(e11 + e22) + ρb1 = 0
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µ∇2u2 + (λ + µ) ∂
∂x2(e11 + e22) + ρb2 = 0
ρb3 = 0
∂σ12
∂x2= −∂σ11
∂x1− ρb1,
∂σ12
∂x1= −∂σ22
∂x2− ρb2
x1 x2
∂
∂x1
∂σ12
∂x2
=
∂
∂x1
−∂σ11
∂x1− ρb1
,
∂
∂x2
∂σ12
∂x1
=
∂
∂x2
−∂σ22
∂x2− ρb2
2 ∂ 2σ12
∂x1∂x2
=
−∂ 2σ11
∂x2
1
+ ∂ 2σ22
∂x2
2
+ ∂ ρb1
∂x1
+ ∂ ρb2
∂x2
∂ 2e11∂x2
2
= 1 + ν
E
(1− ν )
∂ 2σ11
∂x22
− ν ∂ 2σ22
∂x22
∂ 2e22∂x2
1
= 1 + ν
E
−v
∂ 2σ11
∂x21
+ (1 − ν ) ∂ 2σ22
∂x21
∂ 2e12∂x1∂x2
= 1 + ν
E
∂ 2σ12
∂x1∂x2
1 + ν E
(1− ν ) ∂
2
σ11
∂x22
− ν ∂ 2
σ22
∂x22
+
1 + ν
E
−ν
∂ 2σ11
∂x21
+ (1 − ν ) ∂ 2σ22
∂x21
=
1 + ν
E
∂ 2σ12
∂x1∂x2
1 + ν
E
(1− ν )
∂ 2σ11
∂x22
− ν ∂ 2σ22
∂x22
− ν ∂ 2σ11
∂x21
+ (1 − ν ) ∂ 2σ22
∂x21
=
−1 + ν
E
∂ 2σ11
∂x21
+ ∂ 2σ22
∂x21
+ ∂ ρb1
∂x1+
∂ ρb2∂x2
(1− ν ) ∂ 2σ11
∂x22
+ (1 − ν ) ∂ 2σ22
∂x21
− ν ∂ 2σ11
∂x21
−ν ∂ 2σ22
∂x22
+ ∂ 2σ11
∂x21
+ ∂ 2σ22
∂x2= −
∂ρb1∂x1
+ ∂ ρb2
∂x2
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(1− ν )
∂ 2σ11
∂x21
+ ∂ 2σ11
∂x22
+ ∂ 2σ22
∂x21
+ ∂ 2σ22
∂x22
= −
∂ρb1∂x1
+ ∂ ρb2
∂x2
∇2 (σ11 + σ22) = − 11− ν
∂ρb1∂x1
+ ∂ ρb2∂x2
∇2 (σ11 + σ22) = 0
σ11 + σ22
∂ 2
∂r2 +
1
r
∂
∂r +
1
r2∂ 2
∂θ2
(σrr + σθθ) = − 1
1− ν
∂ρbr
∂r +
1
r
∂ρbθ∂θ
+ ρbr
r
ϕ = f (r)cos nθ ϕ = f (r)sen nθ
ϕ
ϕ θ
d4φ
dr4 +
2
r
d3φ
dr3 − 1
r2d2φ
dr2 +
1
r3dφ
dr = 0
∂σ11
∂x1+
∂ σ12
∂x2+ ρb1 = 0,
∂σ12
∂x1+
∂ σ22
∂x2+ ρb2 = 0
∂ 2e11∂x2
2
+ ∂ 2e22
∂x21
= 2 ∂e12∂x1∂x2
K = K (x1, x2)
ρbi = −∂K ∂xi
ϕ = ϕ(x1, x2)
σ11 = ∂ 2ϕ
∂x22
+ K, σ22 = ∂ 2ϕ
∂x21
+ K, σ12 = − ∂ 2ϕ
∂x1∂x2
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∂ 3ϕ
∂x1∂x22
+ ∂K
∂x1 − ∂ 3ϕ
∂x1∂x22
+ pb1 = 0
− ∂ 3ϕ
∂x21∂x2
+ ∂ 3ϕ
∂x21∂x2
+ ∂K
∂x2+ ρb2 = 0
e11 = 1
E (σ11 − νσ22) , e22 =
1
E (σ22 − νσ11) , e12 =
1 + ν
E σ12
1
E
∂ 2σ11
∂x2
2 −ν
∂ 2σ22
∂x2
1+
1
E
∂ 2σ22
∂x2
1 −ν
∂ 2σ11
∂x2
2 =
2 (1 + ν )
E
∂ 2σ12
∂x1∂x2
∂ 2σ11
∂x22
+ ∂ 2σ22
∂x21
− ν
∂ 2σ22
∂x22
+ ∂ 2σ11
∂x21
= 2(1 + ν )
∂ 2σ12
∂x1∂x2
∂ 4ϕ
∂x42
+ ∂ 2K
∂x22
+ ∂ 4ϕ
∂x21
+ ∂ 2K
∂x21
−ν
∂ 4ϕ
∂x21∂x2
2
+ ∂ 2K
∂x21
+ ∂ 4ϕ
∂x21∂x2
2
+ ∂ 2K
∂x22
= −2 (1− ν )
∂ 4ϕ
∂x21∂x2
2
∂
4
ϕ∂x41
+ 2 ∂
4
ϕ∂x21∂x2
2+ ∂
4
ϕ∂x42
= − (1− ν )
∂
2
K ∂x21
+ ∂
2
K ∂x22
e11 =1 + ν
E [(1− ν )σ11 − vσ22] , e22 =
1 + ν
E [(1− ν )σ22 − vσ11] ,
e12 =1 + ν
E σ12
∂ 2e11∂x2
2
+ ∂ 2e22
∂x21
= 2 ∂ 2e12∂x1∂x2
(1− ν )∂ 2σ11
∂x22 +
∂ 2σ22
∂x21− ν ∂ 2σ11
∂x21 +
∂ 2σ22
∂x22
= 2
∂ 2σ12
∂x1∂x2
∂ 4ϕ
∂x41
+ 2 ∂ 4ϕ
∂x21∂x2
2
+ ∂ 4ϕ
∂x42
= −1− 2ν
1− ν
∂ 2K
∂x21
+ ∂ 2K
∂x22
7/23/2019 Mecanica Del Medio
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∂ 2K
∂x21
+ ∂ 2K
∂x22 = 0
∂ 4ϕ
∂x41
+ 2 ∂ 4ϕ
∂x21∂x2
2
+ ∂ 4ϕ
∂x22
= ∇4ϕ = 0
ϕ
P AB x2
x3
P
ABCD
Q r
ϕ = arθ sen θ
∇4ϕ = 0
∇4ϕ =
∂ 2
∂r2 +
1
r
∂
∂r +
1
r2∂ 2
∂θ2
∂ 2ϕ
∂r2 +
1
r
∂ϕ
∂r +
1
r2∂ 2ϕ
∂θ2
= 0
∂ϕ
∂r = aθ sen θ,
∂ 2ϕ
∂r2 = 0
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∂ϕ
∂θ = ar (sen θ + θ cos θ) ,
∂ 2φ
∂θ2 = ar (2 cos θ − θ senθ)
∂ 2ϕ
∂r2 +
1
r
∂ϕ
∂r +
1
r2∂ 2ϕ
∂θ2
=
2
ra cos θ
∂
∂r
2
ra cos θ
= − 2
r2a cos θ,
∂ 2
∂r2
2
ra cos θ
=
4
r3a cos θ
∂
∂θ
2
ra cos θ
= −2
ra sen θ,
∂ 2
∂θ2
2
ra cos θ
= −2
ra cos θ
∂ 2
∂r2 +
1
r
∂
∂r +
1
r2∂ 2
∂θ2
2
ra cos θ
=
4
r3a cos θ − 2
r3a cos θ − 2
r3a cos θ = 0
ϕ = arθ sen θ
σrr = 1
r
∂ϕ
∂r +
1
r2∂ 2ϕ
∂θ2 , σθθ =
∂ 2ϕ
∂r2 , σrθ = − ∂
∂r
1
r
∂ϕ
∂θ
σrr = 1r
aθ sen θ + 1r2
[ar (2 cos θ − θ sen θ)]
σrr = 2a
r cos θ
σθθ = 0
σrθ = − ∂
∂r
1
r (ar sen θ + arθ cos θ)
σrθ = − ∂
∂r [a sen θ + aθ cos θ]
σrθ = 0
r
P
σrrrdθ cos θ
P + 2
ˆ π2
0
σrrr cos θdθ = 0
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2
ˆ π2
0
2a
r cos θr cos θdθ = −P
4
ˆ π2
0
a cos2 θdθ = −P
ˆ cos2 axdx =
x
2 +
sen 2ax
4
4a
θ
2 +
sen 2θ
4
π2
0
= −P
πa = −P
a = −P π
ϕ = −P
π rθ sen θ
σrr = −2P
π
cos θ
r
d x1 x2 O
Q
cos θ = r
d
r = d cos θ
σrr = −2P
πd
d
O
e11 =1 + ν
E [(1− ν ) σ11 − vσ22] , e22 =
1 + ν
E [−νσ11 + (1 − ν ) σ22] ,
e12 =1 + ν
E σ12
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e33 = e13 = e23 = 0; σ33 = ν (σ11 + σ22)
σ11 = σrr = −2P
π
cos θ
r
σθθ = σrθ = σrz = σθz = 0 σzz = −ν 2P
π
cos θ
r
err = 1 + ν
E [(1− ν ) σrr ]
err = −2
1− ν 2
E
P
π
cos θ
r
eθθ = 1 + ν
E (−νσrr)
eθθ = 2ν (1 + ν )E P π cos θr
erθ = 0
err = ∂ur
∂r
eθθ = 1
r
∂uθ∂θ
+ ur
r
erθ = 1
2
1
r
∂ur∂θ
+ ∂ uθ
∂r − uθ
r
∂ur∂r
= −2
1− ν 2
E
P
π
cos θ
r
ur = −ˆ
2
1− ν 2
E
P
π
cos θ
r dr
ur = −2
1− ν 2
Eπ P cos θ ln r + f 1 (θ)
2ν (1 + ν )
πE
P cos θ
r =
1
r
∂uθ∂θ
+ ur
r
2ν (1 + ν )
E
P cos θ
π =
∂uθ∂θ
+ ur
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∂uθ∂θ
= 2ν (1 + ν )
πE P cos θ +
2
1− ν 2
Eπ P cos θ ln r − f 1 (θ)
uθ =
ˆ 2v (1 + ν )
πE P cos θ +
2
1− ν 2
Eπ P cos θ ln r − f 1 (θ)
dθ
uθ = 2ν (1 + ν )
πE P θ +
2
1− ν 2
Eπ P θ ln r −
ˆ f 1 (θ)dθ + f 2 (r)
1
r
∂ur∂θ
+ ∂ uθ
∂r − uθ
r
= 0
1
r
2
1− ν 2
πE P sen θ ln r +
df 1 (θ)
dθ
+
2
1− ν 2
Eπ P sen θ
1
r +
df 2 (r)
dr
− 1
r
2ν (1 + ν )
πE P sen θ +
2
1− ν 2
Eπ P sen θ ln r −
ˆ f 1 (θ)dθ + f 2 (r)
= 0
2P
πE
sen θ
r
1− ν 2− ν (1 + ν )
+
1
r
df 1 (θ)
dθ +
df 2 (r)
dr +
1
r
ˆ f 1 (θ) dθ − 1
rf 2 (r) = 0
1− ν 2− ν (1 + ν ) = 1− ν − 2ν 2 = (1 + ν ) (1− 2ν )
2 (1 + ν ) (1− 2ν )
πE
P sen θ + df 1 (θ)
dθ
+ rdf 2 (r)
dr
+ ˆ f 1 (θ) dθ
−f 2 (r) = 0
rdf 2 (r)
dr − f 2 (r) = M
2 (1 + ν ) (1− 2ν )
πE P sen θ +
df 1 (θ)
dθ +
ˆ f 1 (θ) dθ = −M
M = 0
f 2 (r) = C r
f 1 (θ) = A sen θ + B cos θ − (1 + ν ) (1− 2ν )
πE P θ sen θ
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df 1 (θ)
dθ
= A cos θ
−B sen θ
− (1 + ν ) (1− 2ν )
πE
P (sen θ + θ cos θ)
ˆ f 1 (θ)dθ = −A cos θ + B sen θ − (1 + ν ) (1− 2ν )
πE P
ˆ θ sen θdθ
ˆ xn sen axdx = −1
axn cos ax +
n
a
ˆ xn−1 cos axdx
ˆ θ sen θdθ = −θ cos θ +
ˆ cos θdθ = −θ cos θ + sen θ
A cos−B sen θ − (1 + ν ) (1− 2ν )
πE P [sen θ + θ cos θ] +
2 (1 + ν ) (1− 2ν )
πE P sen θ − A cos θ + B sen θ
− (1 + ν ) (1− 2ν )
πE P [−θ cos θ + sen θ] = 0
2 (1 + ν ) (1− 2ν )
πE P sen θ − 2 (1 + ν ) (1− 2ν )
πE P sen θ = 0
A B C
ur = −2
1− ν 2
Eπ P cos θ ln r + f 1 (θ)
f 1 (θ) = A sen θ + B cos θ − (1 + ν ) (1− 2ν )
πE P θ sen θ
ur = −2
1− ν 2
πE P cos θ ln r + A sen θ + B cos θ − (1 + ν ) (1− 2ν )
πE P θ sen θ
uθ = 2ν (1 + ν )πE P sen θ + 2
1− ν
2Eπ P sen θ ln r − ˆ
f 1 (θ) dθ + f 2 (r)
ˆ f 1 (θ)dθ = −A cos θ + B sen θ − (1 + ν ) (1− 2ν )
πE P [−θ cos θ + sen θ]
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f 2 (r) = C r
uθ =2ν (1 + ν )
πE
P sen θ + 2
1− ν 2
πE
P sen θ ln r + A cos θ
−B sen θ+
(1 + ν ) (1− 2ν )
πE P [−θ cos θ + sen θ] + Cr
uz = 0
x1
uθ = 0 θ = 0, ur = 0 θ = 0 r = d
ur = 0 = −2 1− ν 2
πE P ln d + B
B = 2
1− ν 2
πE P ln d
uθ = 0 = A + Cr
A = 0 C = 0
ur = −2 1− ν 2
πE P cos θ ln r +
2 1− ν 2πE P cos θ ln d−
(1 + ν ) (1
−2ν )
πE P θ
θ
ur = 2
1− ν 2
πE P cos θ ln
d
r − (1 + ν ) (1− 2ν )
πE P θ sen θ
uθ =2ν (1 + ν )
πE P sen θ +
2
1− ν 2
πE P sen θ ln r − 2
1− ν 2
πE P sen θ ln d+
(1 + ν ) (1− 2ν )
πE P (−θ cos θ + sen θ)
uz = 0
ur|θ=π2
= − (1 + ν ) (1− 2ν ) P
πE
uθ|θ=π2 = 2ν (1 + ν )
πE P − 2
1− ν 2
πE P ln
d
r +
(1 + ν )
πE P
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uz = 0
σ11 =a
x22 + b
x21 − x2
2
, σ22 = a
x21 + b
x22 − x2
1
, σ33 = ab
x21 + x2
2
σ12 =2abx1x2, σ13 = σ23 = 0
u1 = −ax2x3
u2 = ax1x3
u3 = 0
p
p = −Ke e
ν = 0.5
µ = 3/E
K = ∞
V e = 0
xi
µ∇2u1 + ∂I 13∂x1
+ ρb1 =0, µ∇2u2 + ∂I 13∂x2
+ ρb2 = 0
µ∇2u3 + ∂I 13∂x3
+ ρb3 =0, ∂u1
∂x1+
∂ u2
∂x2+
∂ u3
∂x3= 0
−L/2?x1?L/2 −h/2?x2?h/2
σ11 =Ax2 + Bx21 + Cx32, σ22 = Dx32 + Ex2 + F
σ12 =
G + Hy22
x1, σ13 = σ23 = σ33 = 0
A B C D E F G H
σ11 = ax22 + bx1, σ22 = −ax2
1 + bx2, σ12 = −b (x1 + x2)
u1(x1, x2) u2(x1, x2) a b
I E I D
(I E )1 = (3λ + 2µ) (I D)1
(I E )2 =λ (3λ + 4µ) (I D)21 + 4µ2 (I D)2
(I E )3 =λ2 (λ + 2µ) (I D)31 + 4λµ (I D)1 (I D)2 + 8µ3 (I D)3
λ µ
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u1 = kx2x3, u2 = kx3x1, u3 = k x21 − x2
2
ρb3 = ax1x2
a
u1 = Ax21x2x3 u1 = Bx1x3
2x3 u3 = Cx1x2x23 A B C
po p ci
ui = pci − 1
4 (1− ν )∇ [ po + (xncn) p]
σ11 = x22 + ν x2
1
−x22 , σ22 = x2
1 + ν x22
−x21 , σ33 = ν x2
1 + x22 , σ12 =
−2νx1x2, σ13 = σ23 = 0
σ11 = x22 + ν
x21 − x2
2
, σ22 = x2
1 + ν
x22 − x2
1
, σ33 = ν
x21 + x2
2
, σ12 = −2νx1x2, σ13 = σ23 = 0
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t = 0
to t = t xi = xi(X j ; t)
xi X j t
xi
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t(n)i = σijnj
= − poni
po
po = −1
3σii
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p
− p
σij = − pδ ij
σij = C ijpqe pq
p
d pdt + ρ ∂vi∂xi= 0
∂σij∂xj
+ ρbi = ρdvidt
ρdu
dt = σijdij − ∂q i
∂xi+ ρh
σij = − pδ ij
vi
q i σij ρ
p u bi
h
∂vi∂xi
= 0
dρ
dt = 0
∂σij∂xj
+ ρobi = ρodvidt
ρodu
dt = − ∂q i
∂xi+ ρh
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σij = − pδ ij
∂vi∂xi
= 0
ρ vi
σij
f ( p, ρ, T ) = 0
u
u = u (ρ, T )
d p
dt + ρ
∂vi∂xi
= 0
∂σij
∂xj + ρobi = ρo
dvi
dt
ρdu
dt = σijdij − ∂q i
∂xi+ ρh
σij = − pδ ij
∂q i∂xi
= −k∇T
f ( p, ρ, T ) = 0
f ( p, ρ, T ) = 0
ρ
p
T
u
q i
vi
σij
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τ ij
σij = − pδ ij + τ ij
τ ij = Aijpqd pq
Aijpq d pq
(F × t) /L2
τ ij = λ∗δ ijdkk + 2µ∗dij
σij = − pδ ij + λ∗δ ijdkk + 2µ∗dij
λ∗
µ∗
λ∗
µ∗
σii = − 3 p + λ∗3dnn + 2µ∗dii
= − 3 p + (3λ∗ + 2µ∗) dii
σij3
= − p + (3λ∗ + 2µ∗)
3 dii
K ∗ = 1
3(3λ∗ + 2µ∗)
K ∗ = 1
3(3λ∗ + 2µ∗) = 0
λ∗ = −2
3µ∗
S ij
σij = S ij + 1
3δ ijσnn
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dij = bij + 1
3
δ ijdnn
bij
S ij + 1
3δ ijσnn = − pδ ij + λ∗δ ijdkk + 2µ∗
bij +
1
3δ ijdnn
σnn = − 3 ( p −K ∗dnn)
S ij = 2µ∗bij
dρ
dt + ρ
∂vi∂xi
= 0
∂σij∂xj
+ ρbi = ρdvidt
σij = − pδ ij + λ∗δ ijdnn + 2µ∗dij
ρdu
dt = σijdij − ∂q i
∂xi+ ρh
vi q i σij
ρ p u
∂vi∂xi
= 0
∂σij∂xj
+ ρbi = ρ dvidt
ρodu
dt = − ∂q i
∂x1+ ph
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σij = − pδ ij + λ∗δ ijdnn + 2µ∗dij
p = p(ρ, T )
p = p(ρ, T )
q i = −k dT
dxi
dij = 1
2
∂vi∂xj
+ ∂ vj∂xi
−
σij = − pδ ij + λ∗δ ij∂vn∂xn
+ µ∗
∂vi∂xj
+ ∂ vj∂xi
∂
∂xj
− pδ ij + λ∗δ ij
∂vn∂xn
+ µ∗
∂vi∂xj
+ ∂ vj∂xi
+ ρbi = ρ
dvidt
ρbi − ∂p
∂xi+ (λ∗ + µ∗)
∂ 2vj∂xj∂xi
+ µ∗ ∂ 2vi∂xj∂xi
= ρdvidt
ρdvidt
= ρbi − ∂p
∂xi+ (λ∗ + µ∗)
∂
∂xi
∂vj∂xj
+ µ∗∇2vi
dvidt
= ∂vi
∂t + vj
∂vi∂xj
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ρdvidt
= ρbi − ∂p
∂xi+
1
3µ∗
∂ 2vj∂xi∂xj
+ 3∂ 2vi∂x2j
ρo = ρo (X j , 0)
u = C V T
C V p
p = ρRT
R
ρ = ρ( p) p = p(ρ?)
∂σij∂xj
= − ∂p
∂xjδ ij = − ∂ p
∂xi
− ∂ p
∂xi+ ρbi = ρ
dvidt
dvidt =
∂vi∂t + vk
∂vi∂vk
− ∂ p
∂xi+ ρbi = ρ
∂vi∂t
+ vk∂vi∂xk
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dv1dt
= a; dv2
dt = 0;
dv3dt
= 0
ρb1 = 0; ρb2 = −ρg; ρb3 = 0
a = d
dt =
− ∂ p
∂xi+ ρbi = ρ
dvidt
∂p
∂x1= ρa
∂p
∂x2= −ρg
∂p
∂x3
= 0
x3
p = ρax1 + f (x2)
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f (x2)
x2
f (x2) = −ρgx2 + c
c
p = ρax1 − ρgx2 + c= ρ (ax1 − gx2) + c
x2
p = pa
pa (0, h)
pa = −ρgh + c
c = pa + ρgh
p = pa + ρ (ax1 − gx2 + gh)
p = pa
gx2 = gh + ax1
x2 = h + ag
x1
a/g
x1
− ∂ p
∂xi+ (λ∗ + µ∗)
∂ 2vj∂xi∂xj
+ µ∗∂ 2vi∂x2j
+ ρbi = ρdvidt
− ∂ p
∂xi + µ∗∂ 2vi
∂x2j + ρbi = ρ
dvi
dt
− ∂ p
∂xi+ ρbi +
µ∗
3
∂ 2vj∂xi∂xj
+ µ∗∂ 2vi∂x2j
= ρdvidt
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