s s U s L e A = P A l l + dl dε = dl l l

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1 P (ε) A o B s= P A s B o Y l o s Y l e = l l 0.% o 0. s e s B 1 s (e) s Y

2 s s U s L e A = P A l l + dl dε = dl l l

3 ε = dε = l dl o + l lo l = log l o + l =log(1+ e) l o Β F Α E YA C Ο D ε YF

4 B YA A YA B C YD C ε YA YD D YD YA > YD 4

5 τ 1 = k k J = 1 6 {( ) + ( ) + ( ) } = = 0 Y J = 1 6 ( ) = 1 1 = Y J = 1 6 {( ) + ( ) + ( ) }= 1 Y 5

6 ( ) + ( ) + ( ) = Y = 1 [( ) +( ) +( ) ] 1/ = Y Y k Y = k 45 ε 1 ε e 1 ε p 1 ε 1 = ε 1 e +ε 1 p = E + ε 1 p ε = ε e +ε p = νε 1 e +ε p = ν E +ε p ε = ε e +ε p = νε 1 e +ε p = ν E +ε p 6

7 ε 1 p +ε p +ε p = 0 ε p =ε p = ε p 1 ε p 1 ε p ε p ε = max( ε 1 p, ε p, ε p ) ε = ε 1 p ε = α (ε 1 p ε p ) +(ε p ε p ) +(ε p ε 1 p ) α ε = α ε p 1 + ε 1 p = α ε 1 p =ε p 1 α = ε = (ε 1 p ε p ) +(ε p ε p ) +(ε p ε 1 p ) β ε = β (ε 1 p ) +(ε p ) +(ε p ) 7

8 ε = β ε 1p =ε 1 p β = ε = {(ε 1 p ) +(ε p ) +(ε p ) } P T P T x τ xy y = 0 = max = x + x 4 + τ xy = min = x x 4 + τ xy = x 4 + τ xy τ 1 = = x 4 + τ xy = k 8 x τ xy τ yx τ yx τ xy T x P

9 x 4 + τ xy = k ( ) + ( ) + ( ) = Y y = z =τ yz =τ zx = 0 ( ) = x + 4τ xy ( ) = x x 4 + τ xy ( ) = x + x 4 + τ xy ( ) + ( ) + ( ) = x +6τ xy = Y Y = k x 4 + τ xy 4 = k = x + τ xy x x 4 + τ xy = x + τ xy + x x 4 + τ xy x + 4 τ xy = 1 k k 9

10 x k + τ xy = 1 k x = 0 τ xy τ 1 τ 1 =±k τ 1 =± k 10

11 OA A A ε e OA W OA e = ε A 0 dε e = Eε A = A E = A ε A W e OA OAA A AO 0 ε A W e AO = dε e = A ε A = W OA OAO W OAO = W OA +W AO =0 A A A O C B ε ε A ε B A B C B ε B 11 A O B A ε ε A ε C ε B p ε C e ε B B

12 ε B = ε C +(ε B ε C ) ε C (ε B ε C ) B ε p B ε e B ε B = ε B p +ε B e ε B p =ε C ε B e = ε B ε C = B E AB ε = ε p + ε e = ε p + E dε = dε p + dε e = dε p + d E AB W AB = p ε B ε dε= B dε p e ε B + dε e ε A ε A p ε A e A ε A p = 0 ε A e = ε A ε p B =ε C ε e B = ε B ε C W AB = ε C dε p ε B ε + C dε e = W p e AB +W AB 0 W AB p = ε C 0 dε p ε A AB W p AB OABC W AB e = ε B ε C ε A dε e 1

13 AB W e ε B ε AB = E C ε e dε e = E(ε B ε C ) ε A Eε A = B E A E A BB OAA BC W e e ε BC = C dε e e ε = C dε e ε B ε C = dε e ε B e ε B ε C ε C e ε e C C C ε C e = 0 W e ε B ε BC = C dε e = E(ε B ε C ) 0 = B E OABC W e W p W =W e +W p W e =W e OA +W e AB +W e ε A BC = dε e ε B ε C + dε e ε B ε dε e 0 ε A C 0 ε B ε = C dε e ε B ε C dε e = OABC AB 1

14 W =W p = W AB p = 0 ε C dε p OABC C B B D B D B F A A B D E O D = B + d ε D =ε B + dε dε = dε p + dε e dε p = ε E ε C CBDE dw CBDE ds dw p dw = dw p = ds B (ε E ε C )+ 1 ( D B )(ε E ε C ) = B dε p + 1 ddε p dε p dw =dε p dw o p = * dε p 14 C εc E B D εe ε B ε D ε

15 dw p dw p o = ( * )dε p = ddε p 0 ddε p 0 1 {( ) + ( ) + ( ) } = Y V V = + + = V = = V = = V = 15

16 + + = 1 9 [{( )+( )} +{( )+ ( )} +{( )+( )} ] = 9 {( ) + ( ) + ( ) +( )( )+ ( )( )+ ( )( )} = 1 {( ) + ( ) + ( ) } + + = Y 1 S τ 1 n =, 1, 1 S S T 1 S 1 T T S n 1 1 S 1 S S S = S 1 = S = S S 1 T 1 T T T 1 S = S 1 T S = S T S = S 16

17 T 1 = T = T = T = T 1 + T + T = + + S T 1 T T n 1/ n = T 1 + T + T = + + = V V τ = T V = ( + + ) 9 = ( ) = = 1 ( ) + ( ) + ( ) τ 1 = τ = τ = τ 1 +τ +τ τ = S τ 17

18 1 ( ) + ( ) + ( ) = τ = Y τ = τ 1 +τ +τ = Y O π 1 n n = V ' O n = V ' ' 1 18

19 = 1 V = V = V + + = V = 1 {( ) + ( ) + ( ) } + + = Y Y /Y / π 1 π 19

20 π π Levy-Mises 0 = = 0 V = = V = = V = = V = = = 0

21 dε 1 = dε = dε dε 1 = dε = dε = dλ V = = V = 1 ( + ) dε 1 = dλ = dλ 1 ( + ) dε = dλ dε dλ = dε = (dε 1 + dε + dε ) = 1 ( ) + ( ) + ( ) 1

22 dε 1 = dε dε = dε dε = dε 1 ( + ) 1 ( + ) 1 ( 1+ ) cotθ = dε H = d dε dε dε = d 1 d / dε = d H θ ε

23 = 400ε 0.5 [MPa] = = 0 MPa =160 MPa 1 = = 0 = 1 ( ) + ( ) + ( ) = ( 1 ) = = = 0 MPa =160 MPa = = = 160 MPa 4 ε = 400 = 160 =

24 dε 1 = dε dε = dε dε = dε 1 ( + ) = 4 1 ( + ) = 0 1 ( 1+ ) = 4 dε dε dε 1 = dε dε = 0 dε = (dε 1 + dε + dε ) = dε 1 dε 1 = dε ε 1 = ε dε = dε = ε = 0 ε = ε 1 = 0.00 Reuss dε 1 = dε p e 1 + dε 1 4

25 dε 1 p = dλ dε 1 e = 1 E {d ν(d + d )} = 1 E {d 1 ν(d + d )}+1 ν E + + = 0 d V dε e 1 = 1+ν E d 1 ν + E d V = d G + d V K G K d ε 1 = 1 dλ + d G d ε = dλ + d G d ε = dλ + d G dε V = d V K 5

26 yz x x z ε x E Y z y Y E ε x y z dε x = x 1 ( y + z ) dλ + 1 E {d x ν(d y +d z )} dε y = y 1 ( z + x ) dλ + 1 E {d y ν(d z + d x )} dε z = z 1 ( x + y ) dλ+ 1 E {d z ν(d x + d y )} y y y = 0 d y = 0 Y x x = Y d x =0 z 6

27 dε z = z + Y dλ + 1 E d z = 0 dλ = d z E z + Y dε x = Y 1 z dλ ν E d z = 1 Y + z E z + Y ν d z = 1 Y E 1 z + Y + 1 ν d z ε xo = 1 E { x ν( y + z )} = 1 E (Y+ν zo) ε yo = 1 E { y ν( z + x )}= ν E (Y zo) ε zo = 1 E { z ν( x + y )}= 1 E ( zo +νy) = 0 zo = νy ε xo = 1 ν E Y ε x ε xo dε x = 1 E z zo Y 1 z + Y + 1 ν d z ε x ε xo = Y 4 log z + Y zo + Y + 1 ν ( z zo ) 7

28 ε x = Y 4 log z /Y ν ν 1 ν ( z + νy ) E z Y log z /Y +1 1 ν z νy Y/ Y Prandtl-Reuss dλ dλ = dε dε 1 = dε + d G + d V K dε = dε + d G + d V K dε = dε + d G + d V K λ ε 1 p =λ 8

29 ε 1 = ε 1 p + ε 1 e ε e 1 = 1 E { ν( + )}= 1 E { ν( 1+ ν = E ν E = 1 G + V K V + ν )}+1 E V ε p = {(ε 1 p ) +(ε p ) +(ε p ) } ε p 1 =λ ε p =λ ε p =λ ε p = λ + + = + + ε p = λ λ = ε p ε 1 = ε p + G + V K 9

30 ε = ε p ε = ε p + + G + V K G + V K 0

.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,

.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =, [ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b