Chapter 7a. Elements of Elasticity, Thermal Stresses

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1 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

2 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 ν z α T ν ε νε νε ( ν)( ν) [( ) ] Δ ν z α T ν ε νε νε ( ν)( ν) [( ) ] Δ 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

3 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

4 -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

5 -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

6 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

7 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)( ) ( )( ) ( )

8 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

9 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

10 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 (plus quadatic, cubic, and quatic tems) 6 5 F 1 a a 1 a a a a 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

11 , 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

12 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

13 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

14 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

15 -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

16 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

17 ample I: Cicula Hole ma b D t in clindical coodinates d d ( b, ) and ( b, 9 )

18 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

19 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

20 lasticit Solution b b b ( 1 ) ( 1 )cos ( 1 b ) ( 1 b )cos b b ( 1 )sin afte nomalization (b 1) ( ) ( )cos ( ) ( )cos 1 ( )sin equilibium equations: individual tems ( ) ( )cos ( )cos ( ) ( )cos ( )sin 5 1 ( )sin 5

21 6 ( )sin ( )cos 5 -diection ( )cos ( ) ( )cos ( ) ( )cos 5 1 ( )cos ( ) ( ) ( ) ( ) ( ) ( ) diection ( )sin 5 ( )sin 5 ( )sin compatibilit 1 1 ( ),

22 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 )

23 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

24 -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

25 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

26 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 ( )

27 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

28 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

29 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 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

30 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

31 ( 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

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