MCEN 503/ASEN 50 Chaptr 4 Strss and Stran Tnsors Dformaton and Stran Fall, 006
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
Dformaton and Stran Dformaton and Stran Stran charactrzs a dformaton Eampl: D stran L0 L ε I L L L 0 0
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
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.
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
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
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)
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
Dformaton and Stran Masr th dformaton
Dformaton and Stran Masr th dformaton
Dformaton and Stran Stran Tnsor: k k a a E δ k k a a δ Grn Stran Almans Stran
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.
Dformaton and Stran Physcal Eplanatons of Stran Tnsor Tm 0 P Tm t P0 da Q0 d Q a a 3 a 3
Dformaton and Stran Physcal Eplanatons of Stran Tnsor Tm 0 P Tm t P0 da Q0 d Q a 3 a a 3
Dformaton and Stran Physcal Eplanatons of Stran Tnsor a Tm 0 Tm t n n v v 3 a a 3
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
Dformaton and Stran E + E Cachy s nfntsmal stran tnsor E 3 3 33 33 E + E + 3 3 3 3 E + 3 3 3 3 E
Dformaton and Stran << << a If small dformaton Not: In most of th cass, a << << small dformaton Bt,
Dformaton and Stran Engnrng Strans Coordnats:, y, z Dsplacmnts:, v, w Normal strans: ε ε ε v y y w z 33 z
Dformaton and Stran Engnrng Strans Shar Strans: γ γ γ y v y + v z w y yz + z w z + 3 3
Dformaton and Stran Strtchs at small dformaton
Dformaton and Stran Cachy s Shar Stran and Engnrng Shar Strans A A d B C B d C
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
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 33 3 3 3 3 Tnsor Not a tnsor!!! z yz z yz y y z y ε γ γ γ ε γ γ γ ε Engnrng Stran
Dformaton and Stran Transformaton of Coordnat Systm In gnral β k β k
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
Dformaton and Stran Transformaton of Coordnat Systm D Mohr Crcl
Dformaton and Stran Stran Invarants
Dformaton and Stran Stran Dvatons Man Stran 0 + 3 + 33 θ 3 Stran dvaton tnsor 0I 0 δ Octahdral Shar Stran γ 3 ( ) ( ) ( ) ( + + + + + ) 0 33 33 6 3 3
Dformaton and Stran Dtrmn Dsplacmnt Flds from Strans E + Qstons: Can th dsplacmnts b dtrmnd nqly?
Dformaton and Stran 3 + Th stran flds ar nconsstnt bcas 3 Dtrmn Dsplacmnt Flds from Strans
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
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.
Dformaton and Stran Compatblty of Stran Flds +
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,,33 + 33, 3,3,33 + 33, 3,3 +,3 3, +,3 3, + 33,,33 + 3,3 3,3