Mecanica Del Medio

7/23/2019 Mecanica Del Medio

http://slidepdf.com/reader/full/mecanica-del-medio 1/157

10−8

Kn = λ/S λ

λ = 10−7

λ = 10−6

Kn ≤ 1

Kn > 1



V S

B R

B

t B R P

R B P

B P

P R

B



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



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



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+



a33b31c13 + a33b32c23 + a33b33c33



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



(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



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



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



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



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



xiX j

Ê 1

Ê 2

Ê 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



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



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



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 =



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

Ê 2 ·

Ê 1 =

Ê 1 ·

Ê 2 = 0

Ê 2 ·

Ê 2 = 1

Ê 2 ·

Ê 3 =

Ê 3 ·

Ê 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)



· 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



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



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



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



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



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



B X

V

S

R

t



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



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



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



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



∂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



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



(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



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



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



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



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



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



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



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



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



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 = 2E ijdxidxj


(dX )2 dx

dX

2− 1 = 2Lijninj



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



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



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



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



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



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



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



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



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



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



∂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



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



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



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



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



∂ 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



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



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



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



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)



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)



∂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



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



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



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



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



xi = xi(τ )

f i

vi = αdxi (τ )

dτ

wi = αdxi (τ )

dτ

ti



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



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



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)



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



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



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



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



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



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



v1 =

−

αx1 + βx2

R2

, v2 = βx1 + αx2

R2

, v3 = 0

R2 = x21 + x2

2 = 0 π



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



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)



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



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



P (xi)

P dA1 dA2 dA3

x1x2

x1x3

x2x3 dA

ni

dA1 = dA cos(ni, x1) = dAn1



cos(ni, xi) ni xi

dV = 13

h dA

h P



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



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



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



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



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



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



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



|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



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



σij = 1 4 6

4 5 86 8 3

xi



∆m

∆V ∆m

ρ = lım∆V →0

∆m

∆V

t

m =

ˆ V

ρd V

ρo = ρo (X j , t)



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



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



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)



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



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



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)



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



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



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

−



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



−

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



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



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



σij = C ijpqe pq

C ijpq



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



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



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



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



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



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



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



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



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



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)



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 )



∂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



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


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



∂ 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



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



∇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



∂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


e11 = σ11

E , e22 = −νσ11

E , e33 = −νσ11

E , e12 = e23 = e13 = 0



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



σ11

e11 = e

∂u1∂x1

= e

u1 = ex1, u2 = 0, u3 = 0



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


e11 = 1E

(σ11 − νσ22) , e22 = 1E

(−νσ11 + σ33)

e33 = − ν

E (σ11 + σ22) , e12 =

1 + ν

E σ12, e13 = e23 = 0



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




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



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



(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



∂ 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



∂ 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



∂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



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



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



∂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



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]



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



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



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



t = 0

to t = t xi = xi(X j ; t)

xi X j t

xi



t(n)i = σijnj

= − poni

po

po = −1

3σii



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



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


∂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



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



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


∂xi+ ρh

vi q i σij

ρ p u

∂vi∂xi

= 0

∂σij∂xj

+ ρbi = ρ dvidt

ρodu

dt = − ∂q i

∂x1+ ph



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



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



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)



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



( p, vi)

X 1 = x1e−t − x3

1− e−t

, X 2 = x2 − x3

et − e−t

, X 3 = x3e−t

p = αρ

α

β/r2 r β

Mecanica Del Medio

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