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# 3章 塑性構成方程式 3 1 - mech.kagoshima-u.ac.jp

P ( ). Ao B. P. s= sB. Ao Y. lo sY s . l (e). l e= lo .. s e sY.. sB .. 3 1.. s .. sU.. sL.. e .. A . P. =. A. l l + dl .. dl d =. l l .. 3 2. l + l . d = . lo+ l dl = = log o =log(1+ e). lo l lo .. F. E.. YA .. C. D .. YF .. 3 3.. B.. YA A. YA B . C. YD C.. YA YD D. YD . YA >| YD |.. 3 4.. 1 3. 1 = k 2. k .. 1. J 2 = {( 1 2 )2 + ( 2 3 )2 + ( 3 1 ) 2}. 6. 2 = 3 = 0 Y . 1 2 2 1 2 1. J 2 = ( 1 + 1 ) = 1 = Y 2. 6 3 3.. 1 1. J 2 = {( 1 2 )2 + ( 2 3 )2 + ( 3 1 ) 2}= Y 2. 6 3. 3 5.. ( 1 2 ) 2 + ( 2 3 ) 2 + ( 3 1 )2 = 2Y 2.. 1. = [( 1 2 ) +( 2 3 ) +( 3 1 ) ]. 2 2 2 1/2. 2.. =Y.. Y k . Y = 2k . 45 .. 1 1 .. 1e 1p . 1. 1 = 1 e + 1p = + 1 p E.. 1. 2 = 2 e + 2p = 1e + 2 p = + 2 p E. 1. 3 = 3 e + 3p = 1e + 3 p = + 3 p E.. 3 6.. 1p + 2 p + 3p = 0.. 1 p 2 = 3 = . p p 2.. 1p 2 p 3p .. = max(| 1p |,| 2 p |,| 3p |).. = 1 p .. = ( 1p 2 p )2 +( 2 p 3 p )2 +( 3 p 1 p ) 2.. 2. p p 3 p 2. = 2 1 + 1 = 1 = 1 =. p 2 2 3.. 2. = ( 1 p 2 p )2 +( 2 p 3 p) 2 +( 3 p 1p )2.

3—1 3章 塑性構成方程式 3—1．単純引張（圧縮） 3—1—1．真応力と真ひずみ 延性材料である金属や合金を引張試

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### Transcription of 3章 塑性構成方程式 3 1 - mech.kagoshima-u.ac.jp

1 P ( ). Ao B. P. s= sB. Ao Y. lo sY s . l (e). l e= lo .. s e sY.. sB .. 3 1.. s .. sU.. sL.. e .. A . P. =. A. l l + dl .. dl d =. l l .. 3 2. l + l . d = . lo+ l dl = = log o =log(1+ e). lo l lo .. F. E.. YA .. C. D .. YF .. 3 3.. B.. YA A. YA B . C. YD C.. YA YD D. YD . YA >| YD |.. 3 4.. 1 3. 1 = k 2. k .. 1. J 2 = {( 1 2 )2 + ( 2 3 )2 + ( 3 1 ) 2}. 6. 2 = 3 = 0 Y . 1 2 2 1 2 1. J 2 = ( 1 + 1 ) = 1 = Y 2. 6 3 3.. 1 1. J 2 = {( 1 2 )2 + ( 2 3 )2 + ( 3 1 ) 2}= Y 2. 6 3. 3 5.. ( 1 2 ) 2 + ( 2 3 ) 2 + ( 3 1 )2 = 2Y 2.. 1. = [( 1 2 ) +( 2 3 ) +( 3 1 ) ]. 2 2 2 1/2. 2.. =Y.. Y k . Y = 2k . 45 .. 1 1 .. 1e 1p . 1. 1 = 1 e + 1p = + 1 p E.. 1. 2 = 2 e + 2p = 1e + 2 p = + 2 p E. 1. 3 = 3 e + 3p = 1e + 3 p = + 3 p E.. 3 6.. 1p + 2 p + 3p = 0.. 1 p 2 = 3 = . p p 2.. 1p 2 p 3p .. = max(| 1p |,| 2 p |,| 3p |).. = 1 p .. = ( 1p 2 p )2 +( 2 p 3 p )2 +( 3 p 1 p ) 2.. 2. p p 3 p 2. = 2 1 + 1 = 1 = 1 =. p 2 2 3.. 2. = ( 1 p 2 p )2 +( 2 p 3 p) 2 +( 3 p 1p )2.

2 3.. = ( 1 p )2 +( 2 p ) 2 +( 3 p )2. 3 7.. 3 p 2. = 1 = 1p =. 2 3.. 2. = {( ) +( 2 ) +( 3 ) }. p 2 p 2 p 2. 3 1.. P T . T. P P. T. x xy . y = 0 . yx 2. x x 2. 1 = max = + + xy 2 4 xy x 2. x x 2 x 2 = min = + xy 2 4. xy . 2. x 2 yx 1 2 = 2 + xy 4.. 2. 1 2 x 2. 1 = = + xy = k 2 4. 3 8.. 2. x 2. + xy = k2. 4.. ( 1 2 ) 2 + ( 2 3 ) 2 + ( 3 1 )2 = 2Y 2. y = z = yz = zx = 0 . 2 2. ( 1 2 ) 2 = x + 4 xy 2. 2 2 2. x x ( 2 3) 2 = x + xy = x + xy x 2 2 2. + xy 2 4 2 4.. 2. 2 2 2. x x ( 3 1) 2 = x + + xy = x + xy + x 2 2 2. + xy 2 4 2 4.. ( 1 2 ) 2 + ( 2 3 ) 2 + ( 3 1 )2. 2 2. = 2 x +6 xy = 2Y2. Y = 2k . 2 2. x 3 xy + =k 2. 4 4.. 2. xy . 2. x . + 4 =1.. 2k 2k . 3 9. 2.. 2. xy . x + 3 =1. 2k 2k .. x = 0 . xy 1 .. 1 = k . 2. 1 = k 3.. 1. 0. 0 1. 3 10.. OA . A . A. A. e OA .. O. A' . A C. A2 .. A E A2. WOAe = d e = = = A A. 0 2 2E 2. e WOA OAA' . A AO .. A A.. 0. WAOe = d e = = WOA. A 2.. OAO. B. WOAO = WOA +WAO =0 B. A. A.. O.

3 A' C B' . A C B. A Bp Be B . C . B B . 3 11. B = C +( B C ). C . ( B C) B. Bp Be . B = B + B. p e . B. Bp = C Be = B C =. E. AB .. d . = p + e = p + d = d p + d e = d p +. E E. AB .. B B p B e WAB = d = p d p + e d e A A A. A . Ap = 0 Ae = A. Bp = C Be = B C .. C B C. WAB = d p + d e = WAB p +W ABe 0 A.. C. WAB p = d p 0. AB . WABp OABC .. B C. WABe = d e A. 3 12. AB .. E( B C ) 2 E A2 B 2 A2.. B C e WAB = E d = = . e e A 2 2 2 E 2E. A'BB' OAA' . BC .. C e C e B C. WBCe = e d e = d e = e d e B B C C. Ce C C . C e = 0.. E( B C )2 2.. B C. WBCe = d e = = B. 0 2 2E.. OABC . We Wp . W =W e +W p .. A B C B C. W e =WOAe +WAB e +WBC e = d e + d e d e 0 A 0. = B C. 0. d e B C. 0. d e = 0.. OABC AB .. 3 13.. C. W =W p = WAB p = d p 0. OABC .. C . B . B D B D. B F. A. A.. B D . E . O C E B' D' . D = B + d C E B D. D = B + d .. d = d p + d e . d p = E C. CBDE dW CBDE dS . dWp . 1. dW = dW = dS B ( E C )+ ( D B )( E C ). p 2.

4 P 1. = B d + d d p 2. d p .. dW = d p . dWo p = * d p 3 14.. dW dWo = ( )d = d d 0. p p * p p . d d 0. p .. 1. {( 1 2 ) 2 + ( 2 3 )2 + ( 3 1 ) 2} = Y. 2. V . 1 + 2 + 3. V =. 3.. 2 1 2 3. 1 = 1 V =. 3. 2 2 1 3. 2 = 2 V =. 3. 2 3 1 2. 3 = 3 V =. 3.. 3 15. 1 2 + 22 + 32. 1. = [{( 1 2 )+( 1 3)} 2 +{( 2 1 )+ ( 2 3 )} 2. 9. +{( 3 1)+( 3 2 )} ]. 2. 2. = {( 1 2 ) + ( 2 3) + ( 3 1 ). 2 2 2. 9. +( 1 2 )( 1 3 )+ ( 2 1 )( 2 3 )+ ( 3 1 )( 3 2 )}. 1. = {( 1 2) 2 + ( 2 3) 2 + ( 3 1 )2 }. 3.. 2. 1 2 + 22 + 32 = Y. 3.. 3.. 1 1. 2 S1. T3. S n T2. 2 S. 2. 1 1 1 S2 T1. n= , , S3. 3 3 3 . 1 2 3. 3. 1. S1 S2 S3 . S = 3S1 = 3S 2 = 3S 3. S 1 2 3 T1 T2 T3.. T1 S = 1 S1 T2 S = 2 S2 T3 S = 3 S3. 3 16.. 1 2 3. T1 = T2 = T3 =. 3 3 3.. 12 + 22 + 32. T = T1 + T2 + T3 =. 2 2 2. 3. S T1 T2 T3 n . 1/ 3 . T1 T2 T3 1 + 2 + 3. n = + + = = V. 3 3 3 3. V . 1 2 + 22 + 3 2 ( 1 + 2 + 3 )2. = T 2 V 2 = . 3 9. 1. = 3 12 +3 2 2 + 3 32 ( 12 + 22 + 3 2 + 2 1 2 +2 2 3 +2 3 1).

5 3. 1. = 2 12 +2 2 2 + 2 32 2 1 2 2 2 3 2 3 1. 3. 1. = ( 1 2 )2 + ( 2 3 )2 + ( 3 1 )2. 3.. 1 3 2 3. 1 = 2 = 1 3 = 2. 2 2 2.. 2. = 2 + 22 + 32. 3 1. S .. 3 17. 1 3. ( 1 2 ) 2 + ( 2 3 )2 + ( 3 1 )2 = =Y. 2 2.. 2 2. = + 2 + 3 =. 2 2 2. Y. 3 1 3.. O .. 1 '1 n n = V .. 3. 3. '3. n= V 2. 2. O. '2. '1 1. 1. 3 18. 1 = 1 V. 2 2 2.. 2 2 = 22 V 2 3 2 = 32 V 2.. 1 2 + 22 + 32 = 12 + 2 2 + 22 3 V 2. 1. = {( 1 2 ) 2 + ( 2 3 )2 + ( 3 1 )2 }. 3.. 2. 1 2 + 22 + 32 = Y. 3 3.. 2.. 3 3.. 2. Y. 3. 2 / 3Y . 2 / 3 . 1 2 2. 2. 2. 3 1 3.. 3 19.. 3. n = V. 2. 1.. Levy-Mises .. 1 0 2 = 3 = 0 . 1. V =. 3.. 2 1 . 1 = 1 V = 2 = 2 V = 1 3 = 3 V = 1. 3 3 3.. 1 = 2 2 = 2 3. 3 20.. d 1 = 2d 2 = 2d 3.. d 1 d 2 d 3. = = = d . 1 2 3 .. 1 + 2 + 3. V =. 3. '1 . 2 1 . 1 = 1 V = 1 ( 2 + 3 ) . 3 2 .. 2 1 . d 1 = d 1 = d ( + 3) . 3 1 2 2 .. 2 2 d . d = d d =. 3 3 .. 2. d = (d 2 + d 2 2 + d 32 ). 3 1.. 1. = ( 1 2 )2 + ( 2 3 )2 + ( 3 1 )2. 2.. 3 21. d 1 . d 1 = ( + 3 ).

6 1 2 2 . d 1 . d 2 = 2 ( 3 + 1) . 2 . d 1 . d 3 = 3 ( 1 + 2 ) . 2 .. cot =. d .. d . H = d . d .. d d 1 d . = =. d / d H .. 3 22.. 1.. = 400 . [MPa]. 2 2.. 1 2 1 = 2 2. 1. 1 = 320 MPa 2 =1 6 0 M P a 1 2 3.. 1 = 2 2 3 = 0 . 1. = ( 1 2 )2 + ( 2 3 )2 + ( 3 1 )2. 2.. 2 2. 1 . = 1 1 + 1 + ( 1 )2. 2 2 2 . 3. = 1 = 3 2. 2. 1 = 320 MPa 2 =1 6 0 M P a . 3. = = 3 2 = 160 3 MPa 2 1.. 4.. 4 160 3 . = = = 400 400 .. 3 23. d 1 3 d . d 1 = ( + 3 ) = 1. 1 2 2 4 . d 1 . d 2 = 2 ( 3 + 1) = 0. 2 . d 1 3 d . d 3 = 3 ( 1 + 2 ) = . 2 4 1. d 1 = d 3 d 2 = 0 . 2 2. d = (d 2 + d 2 2 + d 32 ) = d 1. 3 1 3.. 3. d 1 = d . 2.. 3 3 1 = d = d = 2 0 2 0.. 2 = 0.. 3 = 1 = . Reuss .. d 1 = d 1 p + d 1e . 3 24.. d 1p = 1 d .. 1. d 1 = {d 1 (d 2 + d 3 )}. e E. 1 1 2 . = {d 1 (d 2 + d 3 )}+ d V. E E.. 1 + 2 + 3 = 0.. 1 +2 1 2 d 1 d V. d 1e = d 1 + d V = +. E E 2G 3K. G K .. d 1 . d 1 = 1d +. 2G. d 2. d 2 = 2d +. 2G. d 3. d 3 = 3d +. 2G.. d V. d V =.

7 K.. 3 25. yz x x . z .. y x . E Y. z .. Y.. E.. x y z .. 2 1 1. d x = x ( y + z ) d + {d x (d y +d z )}. 3 2 E. 2 1 1. d y = y ( z + x ) d + {d y (d z + d x )}. 3 2 E. 2 1 1. d z = ( + y ) d + {d z (d x + d y )}. 3 z 2 x E. y y y = 0 d y = 0 . Y x . x = Y d x =0 z . 3 26.. 2 Y 1 3 d z d z = z + d + d z = 0 d = . 3 2 E E 2 z + Y.. 2 1 1 2Y + z . d x = Y z d d z = d z 3 2 E E 2 z + Y . 1 3Y 1 1 . = + d z E 2 2 z + Y 2 .. 1 1. xo = { ( y + z )} = (Y + zo ). E x E. 1 . yo = { ( z + x )}= (Y zo ). E y E. 1 1. zo = { z ( x + y )}= ( zo + Y ) = 0. E E.. zo = Y.. 1 2. xo = Y. E.. z 3Y .. x 1 1 1. d x = 2 2 + Y + 2 d z xo E zo . z . 3Y 2 z + Y 1 . x xo = log + ( zo ). 4 2 zo + Y 2 z . 3 27. 3Y 2 /Y +1 1 1 2. x = log z + ( z + Y ) Y. 4 1 2 2 E. Y 2 /Y +1. z log z . 2 1 2 . z Y Y/ 2 . Prandtl-Reuss . d . 3 d . d =. 2 .. 3 1 d d . d 1 = d + 1 + V. 2 2G 3K. 3 2 d 2 d V. d 2 = d + +. 2 2G 3K. 3 3 d 3 d V. d 3 = d + +. 2 2G 3K.

8 1p = 1 . 3 28.. 1 = 1 + 1. p e . 1 1 1 2 . 1 = { ( 2 + 3 )}= { 1 ( 2 + 3 )}+ V. e E 1 E E. 1+ 2 1 2 . = 1 + V. E E.. = 1+ V. 2G 3K.. 2. p = {( p) 2 +( 2 p ) 2 +( 3 p )2 }. 3 1. 1p = 1 2 p = 2 3p = 3 . 2. p = 2 + 22 + 32. 3 1.. 3. = + 2 + 3. 2 2 2. 2 1.. 2 3 p p = =. 3 2 .. 3 p . 1 = + 1 + V. 2 1 2G 3 K. 3 29. 3 p . 2 = 2 + 2 + V. 2 2G 3 K. 3 p . 3 = + 3 + V. 2 3 2G 3 K.. 3 30.