Stress and Strain Tensors Deformation and Strain

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1 MCEN 503/ASEN 50 Chaptr 4 Strss and Stran Tnsors Dformaton and Stran Fall, 006

2 Dformaton and Stran Dsplacmnt & Dformaton Dsplacmnt: A vctor or th magntd of a vctor from th ntal poston to a sbsqnt poston assmd by a body. Dformaton: An altraton of shap, as by prssr or strss. Eampl: Cas Cas Tm 0 Cas 3 Tm t

3 Dformaton and Stran Dformaton and Stran Stran charactrzs a dformaton Eampl: D stran L0 L ε I L L L 0 0

4 Dformaton and Stran Knmatcs of Contnos Body Tm 0 Tm t a a 3 a 3 a Tm 0: Undformd confgraton Rfrnc (ntal) confgraton Matral confgraton Tm t: Dformd confgraton Crrnt confgraton Spatal confgraton

5 Dformaton and Stran Knmatcs of Contnos Body ( 3 a, a, a, t) OR, d to contnos body a a ( 3,,, t) Lagrangan Dscrpton: Elran Dscrpton: Th moton s dscrbd by th matral coordnat and tm t. Th moton s dscrbd by th spatal coordnat and tm t.

6 Dformaton and Stran Lagrangan vs. Elran Lagrangan ( a, a, a3, t) (Tackng a matral pont) a Elran a (,, 3, t) (Montorng a spatal pont) Th spatal coordnats of ths matral pont chang wth tm. a t0 tt tt Dffrnt matral ponts pass ths spatal pont a

7 Dformaton and Stran Lagrangan vs. Elran Lagrangan Trackng a matral pont. Matral pont s fd bt th spatal coordnats hav to b pdatd. Elran Trackng a spatal pont. Spatal coordnats ar fd bt Matral ponts kp changng. Good for constttv modl Not good for constttv modl. Sold Mchancs Fld Mchancs Sold Mchancs

8 Dformaton and Stran Knmatcs of Contnos Body Tm 0 a Tm t a 3 a 3 a Usng ndformd confgraton as rfrnc: ( a, a, a3) ( a, a, a3 ) a Usng dformd confgraton as rfrnc: (,, 3) a (,, 3)

9 Dformaton and Stran Masr th dformaton Tm 0 P Tm t a P0 Q Q0 a 3 a 3 a P { a, a a 3 } { a + da, a + da a 3 da 3 } {,, 3 } { + d, + d d } 0, Q + 0, P Q +, 3 3

10 Dformaton and Stran Masr th dformaton

11 Dformaton and Stran Masr th dformaton

12 Dformaton and Stran Stran Tnsor: k k a a E δ k k a a δ Grn Stran Almans Stran

13 Dformaton and Stran Stran Tnsor: + + k k a a a a E + k k Grn Stran Almans Stran Applcabl to both small and fnt (larg) dformaton.

14 Dformaton and Stran Physcal Eplanatons of Stran Tnsor Tm 0 P Tm t P0 da Q0 d Q a a 3 a 3

15 Dformaton and Stran Physcal Eplanatons of Stran Tnsor Tm 0 P Tm t P0 da Q0 d Q a 3 a a 3

16 Dformaton and Stran Physcal Eplanatons of Stran Tnsor a Tm 0 Tm t n n v v 3 a a 3

17 Dformaton and Stran If << << a small dformaton + a a E + Th qadratc trm n Grn stran and Almans stran can b nglctd. Also, n small dformaton, th dstncton btwn Lagrangan and Elran dsappars. + E Cachy s nfntsmal stran tnsor

18 Dformaton and Stran E + E Cachy s nfntsmal stran tnsor E E + E E E

19 Dformaton and Stran << << a If small dformaton Not: In most of th cass, a << << small dformaton Bt,

20 Dformaton and Stran Engnrng Strans Coordnats:, y, z Dsplacmnts:, v, w Normal strans: ε ε ε v y y w z 33 z

21 Dformaton and Stran Engnrng Strans Shar Strans: γ γ γ y v y + v z w y yz + z w z + 3 3

22 Dformaton and Stran Strtchs at small dformaton

23 Dformaton and Stran Cachy s Shar Stran and Engnrng Shar Strans A A d B C B d C

24 Dformaton and Stran Cachy s Shar Stran and Engnrng Shar Strans A θ A θ y θ γ y θ + θ v d B θ C ( θ +θ ) B d C

25 Dformaton and Stran Cachy s Shar Stran and Engnrng Shar Strans [] z yz z yz y y z y ε γ γ γ ε γ γ γ ε Tnsor Not a tnsor!!! z yz z yz y y z y ε γ γ γ ε γ γ γ ε Engnrng Stran

26 Dformaton and Stran Transformaton of Coordnat Systm In gnral β k β k

27 Dformaton and Stran Transformaton of Coordnat Systm D X X X θ X cos sn θ θ snθ cos + snθ cosθ snθ cosθ ( ) ( ) + cos θ sn θ + sn + cos θ θ

28 Dformaton and Stran Transformaton of Coordnat Systm D Mohr Crcl

29 Dformaton and Stran Stran Invarants

30 Dformaton and Stran Stran Dvatons Man Stran θ 3 Stran dvaton tnsor 0I 0 δ Octahdral Shar Stran γ 3 ( ) ( ) ( ) ( )

32 Dformaton and Stran Dtrmn Dsplacmnt Flds from Strans E + Qstons: Can th dsplacmnts b dtrmnd nqly?

33 Dformaton and Stran 3 + Th stran flds ar nconsstnt bcas 3 Dtrmn Dsplacmnt Flds from Strans

34 Dformaton and Stran Compatblty of Stran Flds B C A D B C Compatbl stran flds A D Undformd B C C B C C A D A D Incompatbl stran flds

35 Dformaton and Stran Intgrablty Condton In gnral f, ( ) g, ( ) Intgrablty condton ( Compatblty of stran flds ) f g Intgraton of stran flds ylds nq dsplacmnt componnts.

36 Dformaton and Stran Compatblty of Stran Flds +

37 Dformaton and Stran Compatblty of Stran Flds, kl + kl, k, l l, k 0 St. Vnant Eqatons of Compatblty Totally 8 qatons, bt only 6 ar ssntal., +,,,3 3, +,3 + 3,, , 3,3, , 3,3 +,3 3, +,3 3, + 33,,33 + 3,3 3,3

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