Chapte 7a lements of lasticit, Themal Stesses Mechanics of mateials method: 1. Defomation; guesswok, intuition, smmet, pio knowledge, epeiment, etc.. Stain; eact o appoimate solution fom defomation. Stess; constitutive elationship. Load; integation, equilibium equations 5. Displacement; invesion of defomation-load elationship, integation of stain, eneg method lasticit method: 1. quilibium equations; diffeential fom of Newton's law fo nomal stesses, smmet of shea stesses. Compatibilit; continuum estictions fo stains. Constitutive elationships; stess-stain elationships (isotopic o anisotopic mateial, linea o nonlinea mateial, etc.). Bounda conditions; loads and suppots 5. Solution; analtical o numeical
Isotopic mateial ε ν( ) z α Δ T γ G ε ν( ) z α Δ T γ z z G ε z ν( ) z α Δ T γ z z G 1 ( ν ) G Geneal (-D) state of stess α T ν ε νε νε ( ν)( ν) [( ) ] Δ 1 1 1 1 ν z α T ν ε νε νε ( ν)( ν) [( ) ] Δ 1 1 1 1 ν z α T ν ε νε νε ( ν)( ν) [( ) ] Δ 1 1 1 1 ν z z G γ, z G γ z, z G γ z Plane (-D) state of stess z z z γz γz G γ γ G α ε νε T 1 ν ( ) Δ 1 ν ε ν α Δ T α ε νε T 1 ν ( ) Δ 1 ν ε ν α Δ T ε ν z α Δ T
Plane (-D) state of stain εz γz γz z z γ G γ G (1 ν ) ν (1 ν) ε (1 ν) αδ T (1 ν ) ν (1 ν) ε (1 ν) αδ T ( 1 ν) ε νε ( 1 ν) α ΔT ( 1 ν)( 1 ν) ( 1 ν) ε νε ( 1 ν) α ΔT ( 1 ν)( 1 ν) z νε νε ( 1 ν) αδt ( 1 ν)( 1 ν) Displacements and Stains nomal stain shea stain u() u( d) u( d) u() d d v( d) v() d α α1 o 9 - γ d v() v( d) ε, ε v, ε z w z γ v, γ z w v, γ z z z w z v, w v, z w z
-D Compatibilit (in the - plane) plane stain plane stess (appoimate) ε z γ z γ z z z z v w, z z v ε z ν( ε ε ), z z ε, ε v, γ v ε ε γ quilibium equations z Β z d d dz taction foces bod foces z d d z d d z d d z d dz d dz z d dz d dz d dz d dz Β d d dz Β d d dz
-diection: [ ( d) ( )] d dz [ ( d) ( )] d dz [ z( z dz) z( z)] d d B d d dz ( d) ( ) ( d) ( ) z( z dz) z( z) d d dz B z B z z B z z z z Bz z Bounda conditions suface tactions Φ dr da, Φ dr da, Φ dr z z da dr z dr da z m, dr n z z z Φ z Φ z m n Φz z z n i mj n k ( m n 1) fee bounda conditions Φ Φ Φz igid bounda conditions
slip bounda conditions u u uz (u v w ) u n ( ui u j uzk) ( i mj nk) u um uzn hdostatic bounda conditions shea-flow Φ n o Φ Φ n Φ p p Φ m p n Φz z s S T s S T s S V suface nomal n i bounda nomal S S j Sz k bounda tangent s s j sz k S o s s S z z z z Stess field solution fo plane stess poblems 1. quilibium equations. Compatibilit. Constitutive elationships. Bounda conditions 5. Solution
Plane stess in isotopic mateials (without themal stains and bod foces) 1. quilibium equations z z z z z z z z z z z. Compatibilit ε ε γ. Constitutive elationships ε ν, ε 1 ( ν) G ν, γ 1 ( ν) G G Compatibilit elationship in tems of stess ν ν ( 1 ν)( ) ( )( ) ( )
Ai Stess Function F (, ) (without bod foces and themal epansion) F, F, F z z z z z F F z F F z z z z z i.e., the equilibium equations ae automaticall satisfied ε ε γ ε ν, ε 1 ( ν) G ν, γ 1 ( ν) G G ν ν ν 1 ( ) F ν ν F F F F 1 ( ν) F F F
hamonic diffeential (Laplacian) opeato f f bihamonic diffeential (Laplacian-squae) opeato ( ) compatibilit elationship fo plane stess in an isotopic medium F Polnomial solutions in Catesian coodinates quadatic stess function F a1 a a a, a 1, a (unifom stess distibution) cubic stess function F a1 a a a (plus quadatic tems) a 6a, 6a1 a, a a pue bending (linea stess distibutions) F a1 a
6 a, 6 a 1, quatic stess function F a1 a a a a5 (plus quadatic and cubic tems) F ( )F a1 a5 8a shea bending 1 a 6 a a 5 1 a 6 a a 1 a a a 6 a, 6 a, a a quintic stess function F a a a a a a 1 5 5 (plus quadatic, cubic, and quatic tems) 6 5 F 1 a a 1 a a a a 1 6 5 a a5, 5a1 a, 5a6 a bending unde unning load of constant intensit (see tetbook) Displacements calculated fom stesses integation of stains ample: pue bending
, v M, u c L b M M I c, b M ε c b v M ε ν c b u M c b f v ν M c b f γ v f M f c b f M a 1 c b M f a 1 a c b f a 1 f a1 a u M a a c b 1 v ν M M a1 a c b c b bounda conditions M L u L a 1 and a c b ν M ML v L a c b c b
v L, a ML c b v a L L M L c b 1 v M c b L [( ) ν ] accoding to Saint-Venant's pinciple, the discepanc at the ight end ma be ignoed
Plane Stess Poblems in Pola Coodinates, z,,, z cos, sin and, tan 1 z z z γ z γ z constitutive elationships G γ γ G α ε νε T 1 ν ( ) Δ 1 ν ε ν α Δ T α ε νε T 1 ν ( ) Δ 1 ν ε ν α Δ T ε ν z α Δ T
equilibium equations: ( d ) ( d) ( d) ( ) B B ( d) ( ) ( ) d ( ) d O -diection: ( d)( d) d dz ( ) d dz ( d ) ( d) d dz d O ( d) ( ) d B ( ) ( ) d ( d) ddz ( ) ddz B d d dz o 1 ( d)( d) ( ) d 1 ( d) ( ) d B 1 ( ) 1 B 1 B
-diection: ( d ) ( d) ( ) B ( d) ( ) ( d) d dz ( ) d dz ( d)( d) ddz ( ) ddz d dz d d ( ) d B dd dz o O 1 1 B 1 B Stain-displacement elationships ε, ε u 1 u, γ 1 u ε u ( d) ε u u ( d) u ( ) u ( ) u ( ) d d d d O O
u ( d, ) u (, ) u ( d) γ 1 ε 1 u ( ) u (, ) d d d d O O u (, d ) u (, ) u ( d ) u ( ) γ u (, ) u γ u ( ) d d d d O O aismmetic case ( / ) ε, ε u, γ u no shea case (u C, u u ) ε, ε u, γ
ample I: Cicula Hole ma b D t in clindical coodinates d d ( b, ) and ( b, 9 )
lasticit Method: quilibium equations Compatibilit Constitutive elationships Solution Bounda conditions Mechanics of Solids Method: "Solution" Constitutive elationships quilibium equations Compatibilit Bounda conditions equilibium: 1 1 compatibilit: ( ) 1 1 bounda conditions: ( b, ) and ( b, ) lim (, ) (1 cos ), lim (, ) (1 cos ), and lim (, ) sin
Ai Stess Function in Clinbdical Coodinates 1 F 1 F, F, 1 1 F 1 F 1 F z z z 1 1 F 1 F 1 F F F (, ) f() g ()cos 1 f 1 f 1 g 1 g g f () C 1 ln C Cln C C g () C 7 5 C6 C 8 C 6C7 C8 C1(1 ln ) C C 5 cos C 6C7 C1( ln ) C C 5 1C6 cos 6C7 C8 C5 6C6 sin out of 8 constants C is ielevant C 1 C6 (boundedness) 5 bounda conditions
lasticit Solution b b b ( 1 ) ( 1 )cos ( 1 b ) ( 1 b )cos b b ( 1 )sin afte nomalization (b 1) 1 1 1 ( ) ( )cos 1 1 1 ( ) ( )cos 1 ( )sin equilibium equations: individual tems 1 1 1 ( ) ( )cos 5 1 6 ( )cos 5 1 1 1 ( ) ( )cos 5 1 1 ( )sin 5 1 ( )sin 5
6 ( )sin 5 1 1 ( )cos 5 -diection 1 1 6 ( )cos 5 1 1 1 ( ) ( )cos 5 1 1 1 ( ) ( )cos 5 1 ( )cos 5 1 1 1 1 1 1 1 ( ) ( ) 6 1 1 1 1 1 ( ) ( ) ( ) ( ) 5 5 5 5 -diection 1 1 6 1 ( )sin 5 ( )sin 5 ( )sin 5 1 6 1 5 5 5 compatibilit 1 1 ( ),
cos 1 8 ( ) cos cos cos bounda conditions ( b, ) and ( b, ) lim (, ) (1 cos ), lim (, ) (1 cos ), and lim (, ) sin b b b ( 1 ) ( 1 )cos ( 1 b ) ( 1 b )cos b b ( 1 )sin stess concentation factos ( b, ) cos ( b, ) ( b, 9 )
ample II: Concentated (Line) Load P d d invaiance to scaling: a (, ) f() g() g() 1 ( ) 1 B cos a and equilibium equations cos a, a cos, 1 -diection 1
-diection 1 compatibilit 1 1 ( ), cos a cos cos cos a a a bounda conditions (, 9 ) and (, 9 ) π / π/ cos d P π/ / a π aπ a cos d (1 cos ) d P π/ π/ P cos π
Themal Stess ε ε γ without themal etension: ε ν, ε ν, γ G ( ) with themal etension (ΔT T ): ε ν αt, ε ν αt, γ G ( αt) uniaial stess fee ends, v T T ( ), u c b d d ( αt) αt a 1 a
bounda conditions on the top and the bottom (, ± c) and (, ± c) bounda conditions on the left and the ight ( ±, ) ( ) and ( ±, ) appoimate bounda conditions fo the ends (Saint-Venant's pinciple) F( ± ) and M( ± ) c c ( ) d and ( ) d c c c α T( ) d ca c and c α T( ) d a1 c c c α α α T ( ) T ( ) d T ( ) d c c c c c igidl held ends, v T T ( ), u c b ε α T d ( αt) d ( ) αt ( )
Disk with aismmetic tempeatue distibution ε, ε u 1 u, γ 1 u u u ( ) and u ε, ε u, γ ( ), ( ), equilibium equations 1 1 d d constitutive equations αt du ε νε u ( ) [ ν 1 ν 1 ν 1 ν d α( 1 ν) T] αt u du ( ε νε ) [ ν 1 ν 1 ν 1 ν d α( 1 ν) T] d du u 1 du u [ ( ) T] ν ν α 1 ν [ ] d d d d d du u 1 du u ( ) ν ν ( ) α( 1 ν) d d d u d u 1 du u ν du u dt ν ( ) ( ) α( 1 ν) d d d d d dt d
d u 1 du u dt α( 1 ν) d d d d 1 d ( u) (1 ) dt d α ν d d solution b integation 1 d ( 1 1 d u ) α( ν ) T ( ) C d d ( u ) α( 1 ν) T( ) C 1 u α (1 ν ) T( ) d C1 C b α (1 ν) u T() d C1 b C
ample: Hot spot solid disk with fee edge T T( ) a bounda conditions b C ( a) C1 du u ν α 1 ν T 1 ν [ d ( ) ] u du ν α 1 ν T 1 ν [ d ( ) ] u du (1 ) T() d C α ν 1 T() d (1 ) T() C α ν 1 d α (1 ν) α (1 ν) α (1 ν) [ T() d α (1 ν ) T() C ν T() d νc α (1 ν) T()] 1 ν 1 1 1 α T() d C1 α (1 ν) α (1 ν) αν (1 ν) [ T( ) d C T( ) d αν (1 ν ) T( ) νc α (1 ν) T( )] 1 ν 1 1
1 C () 1 α Td T () α(1 ν) C1 α(1 ν) 1 a T() d a 1 1 [ a 1 1 a α T( ) d T( ) d] α T() d T() d T() a a unifoml heated cicula spot a T c T T T T if c and T else 1 T T() d if c and T c 1 T() d if c< a a a T c () T d a 1 α T c ( 1 ) if c a αt c c α ( ) and a T c c ( ) if c < a a
( a) and c ( a) α T (unheated im) a in plane in plane if c α T c in plane ma{ in plane} α T if c < a at c