CH.3. COMPATIBILITY EQUATIONS. Continuum Mechanics Course (MMC) - ETSECCPB - UPC

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1 CH.3. COMPATIBILITY EQUATIONS Connuum Mechancs Course (MMC) - ETSECCPB - UPC

2 Overvew Compably Condons Compably Equaons of a Poenal Vecor Feld Compably Condons for Infnesmal Srans Inegraon of he Infnesmal Sran Tensor Inegraon of he Deformaon Rae Tensor 4/0/04

3 3. Compably Condons Ch.3. Compably Equaons 3 4/0/04

4 Inroducon Gven a dsplacemen feld, he correspondng sran feld s found: U X, u, E j j U U j Uk U k, j{,, 3} X j X X X j u u j, j{,,3} j Is he nverse possble?, u, 4 4/0/04

5 Compably Condons Gven an (arbrary) symmerc second order ensor feld,,, s a dsplacemen feld, u,, fulfllng u (, ), canno always be obaned: u u j j, j{,,3} 6 PDEs j 3 unknowns For o mach a symmercal sran ensor:, I mus be negrable. There mus es a dsplacemen feld from whch comes from. OVERDETERMINED SYSTEM COMPATIBILITY CONDITIONS mus be sasfed REMARK Gven u,, here wll always es an assocaed sran ensor,,, obanable hrough dfferenaon, whch wll auomacally sasfy he compably condons. 5 4/0/04

6 Compably Condons The compably condons are he condons a symmercal nd order ensor mus sasfy n order o be a sran ensor and, hus, es a dsplacemen feld whch sasfes: j u u j, j{,, 3} j They guaranee he connuy of he connuous medum durng he deformaon process. E X, Incompable sran feld 6 4/0/04

7 3. Compably Equaons of a Poenal Vecor Feld Ch.3. Compably Equaons 7 4/0/04

8 Prelmnary eample: Poenal Vecor Feld A vecor feld v, wll be a poenal vecor feld f here ess a scalar funcon, (named poenal funcon) such ha:,,, v Gven a connuous scalar funcon, here wll always es a poenal vecor feld v,. Is he nverse rue? v,,,3 v,, such ha, v, 8 4/0/04

9 Poenal Feld v,, such ha, v, In componen form,,, v, v, 0,,3 3 eqns. unknown OVERDETERMINED Dfferenang once hese epressons wh respec o : SYSTEM v, j j, j,,3 9 eqns. 9 4/0/04

10 Schwarz Theorem The Schwarz Theorem or equaly of med paral dervaves guaranees ha, gven a connuous funcon wh,,..., n connuous dervaves, he followng holds rue: j j, j 0 4/0/04

11 Compably Equaons Consderng he Schwarz Theorem, v v v y y z z vy vy vy y y y z yz vz vz vz z y zy z z In hs sysem of 9 equaons, only 6 dfferen nd dervaves of he unknown, appear:,,,, and y z y z yz They can be elmnaed and he followng denes are obaned: v vy v v vy v y z z y z z 4/0/04

12 Compably Equaons, he vecor feld verfes: A scalar funcon whch sasfes, v, wll es f v, v def y v 0 Sz y eˆ eˆ eˆ3 S def v vz 0 S where y SS y v z y z S v v def z z y v vy vz 0 S y z INTEGRABILITY (COMPATIBILITY) EQUATIONS of a poenal vecor feld v0 v v j 0, j,,3 j REMARK A funconal relaon can be esablshed beween hese hree equaons. v 0 4/0/04

13 3.3 Compably Condons for Infnesmal Srans Ch.3. Compably Equaons 3 4/0/04

14 Infnesmal Srans The nfnesmal sran feld can be wren as: u u u y u u z y z y z uy uy u z y yy yz y z y z yz zz u z symmercal z 6 PDEs 3 unknowns 4 4/0/04

15 Infnesmal Srans The nfnesmal sran feld can be wren as: yy zz u u u y 0 y 0 y u y u u z 0 z 0 y z u u z y u z 0 yz 0 z z y 6 PDEs 3 unknowns The sysem wll have a soluon only f ceran compably condons are sasfed. 5 4/0/04

16 Compably Condons The compably condons for he nfnesmal sran feld are obaned hrough double dfferenaon (sngle dfferenaon s no enough). u,,,,, y z y z yz u y u z yz z y,,,,, y z y z yz 6 equaons 6 equaons 66=36 equaons 6 4/0/04

17 Compably Condons The compably condons for he nfnesmal sran feld are 3 obaned hrough: 3 3 u yz u y u z 3 z y 8 equaons for y, z, yz 8 equaons for, yy, zz u yz u y u z 3 y y y z y y u yz uy u z... 3 z z z z y z 3 3 u yz 3 uy u z y y y zy y u yz uy u z z z z z y z u yz uy u z yz yz yz zy yz 7 4/0/04

18 Compably Condons All he hrd dervaves of u, uy and uz appear n he equaons: 3 u 3 3 3, y, z, y, y, y z, z, z, z y, yz 0 dervaves u 3 y 3, y, z, 3 y, y, y z, 3 z, z, z y, yz 3 uz 3 3 3, y, z, y, y, y z, z, z, z y, yz 0 dervaves 0 dervaves whch consue 30 of he unknowns n he sysem of 36 equaons: f n 3 u j, n,,...,36 jkl k l /0/04

19 Compably Equaons 3 u Elmnang he 30 unknowns,, 6 equaons are obaned: j k l 9 def yy zz yz S 0 z y yz def zz z S yy 0 z z def yy y Szz 0 y y def zz yz z y Sy 0 y z y z def yy yz z y Sz 0 z y y z def yz z y S yz 0 yz y z COMPATIBILITY EQUATIONS for he nfnesmal sran ensor S ε 0 4/0/04

20 Compably Equaons The s equaons are no funconally ndependen. They sasfy he equaon, In ndcal noaon: S ε 0 S S y Sz 0 y z Sy Syy Syz 0 y z S S z yz Szz 0 y z 0 4/0/04

21 Compably Equaons The compably equaons can be epressed n erms of he permuaon operaor, e. e jk jk S e e ml mjq lr j, qr 0 ejk Or, alernavely: j, kl kl, j k, jl jl, k 0, j, k, l,,3 REMARK Any lnear sran ensor ( s order polynomal) wh respec o he spaal varables wll be compable and, hus, negrable. 4/0/04

22 3.4 Inegraon of he Infnesmal Sran Tensor Ch.3. Compably Equaons 4/0/04

23 Prelmnary Equaons Roaon ensor Ω, : Roaon vecor θ, : Ω skew( u) ( uu) u u j j, j{,, 3} j 3 yz 0 3 u ( ) 0 3 z 3 3 y 0 3 4/0/04

24 Prelmnary Equaons Dfferenang, wh respec o : u u j j u u j j j k k j k 4 Addng and subracng he erm k : u j u u j uk uk k k j j j u u k u j u k k j k k j j k j jk jk 4/0/04

25 Prelmnary Equaons Usng he prevous resuls, he dervave of θ, s obaned: yz z y z y z y z y y z y y y z yz zz zy z z z z z y z 3 y y y 3 y yy y 3 y y y 3 y yz z z z y yz yz yy z z yz z z zz 5 4/0/04

26 Prelmnary Equaons 3 6 Consderng he dsplacemen graden ensor J,, u, J j u u u j u u j Jj j j, j,,3 j j j j θ J, Inroducng he defnon of,, he componens of are rewren: 3 3 yz z y : : 3: j j j 3 u u u y 3 z y z uy uy uy y 3 yy yz y z uz uz uz z yz zz y z 4/0/04

27 Inegraon of he Sran Feld The negraon of he sran feld ε, s performed n wo seps:. Inegraon of dervave of θ, usng he s order PDE sysem derved for, and. The soluon wll be of he ype: 3 The negraon consans c can be obaned knowng he value of he roaon vecor n some pons of he medum (boundary condons).. Known, and θ,, u s negraed usng he s order PDE sysem derved for u REMARK. The soluon wll be: If he compably equaons u u, y, z, c,,3 The negraon consans c can be obaned knowng he value of are sasfed, he dsplacemens n some pon of space (boundary condons) hese equaons wll be negrable. 7 yz,,, c,,3 ε 4/0/04

28 Inegraon of he Sran Feld The negraon consans ha appear mply ha an negrable sran ensor ε, wll deermne he movemen n any nsan of no no me ecep for a roaon c() ˆ () and a ranslaon c() uˆ () : ˆ,,, u, u, uˆ A dsplacemen feld can be consruced from hs unform roaon and ranslaon: u (, ) ˆ ( ˆ ( )) uˆ ( ) u ˆ S T ˆ ˆ T ( u*) ( u ( u ) ) ( ) 0 Ths corresponds o a rgd sold movemen. 8 4/0/04

29 3.5 Inegraon of he Deformaon Rae Tensor Ch.3. Compably Equaons 9 4/0/04

30 Compably Equaons n a Deformaon Rae Feld 4/0/04 There s a correspondence beween The concep of compably condons can be eended o deformaon rae ensor. ( ) j j j j j j u u u u u u u ( ) v v v v w j j j j j j d v dv v d v 30

31 Eample Deduce he velocy feld correspondng o he deformaon rae ensor: y 0 e 0 y d, e 0 0 z 0 0 e In pon,, he followng holds rue: v, e ω,, e e 0, v 0,, e 3 4/0/04

32 Eample - Soluon y 0 e 0 y d, e 0 0 z 0 0 e Consder he correspondence: u ( u) u v dv ( ) v Take he epressons derved for, 3 subsue, wh and, wh d, : d d z y 0 0 y z d d 0 0 y y z d d zz zy 00 z y z yz yy and θ ω, C 3 4/0/04

33 Eample - Soluon y 0 e 0 y d, e 0 0 z 0 0 e d dz 00 z d d 0 0 y z d z d zz 00 z z y yz d 3 y d 00 y d d y y d 3 yz d z 00 z y 3 yy y y 3 0 e C y y y, e dy e C /0/04

34 Eample - Soluon C C y 3 3 e C So, For pon,, : 0 ω, v 0 e 0 C 0 C y 3 e e C3,, Therefore, for any pon, 0 ω, 0 e y C C C /0/04

35 Eample - Soluon 0 ω, 0 ; y e y 0 e 0 y d, e 0 0 z 0 0 e Takng he epressons : : The componens of he veloces can be obaned: v v y v z d 0 d e e e y d 00 z 3 j j j 3 v v v d d d y z y 3 z v v v d d d y z y y y y 3 yy yz v v v 3: d d d y z y y y z z z z yz zz y y v y, e dy e C 35 4/0/04

36 Eample - Soluon 0 ω, 0 ; y e y 0 e 0 y d, e 0 0 z 0 0 e The componens of he veloces can be obaned: v v v d e e 0 y y y y 3 y y y z d yy 0 d 00 yz v y C v v y z z v z z d 00 z d 00 d yz zz e z z z v, z z e dz e C3 36 4/0/04

37 Eample - Soluon v z z e C3 v y e C v y C For pon,, : v, e e e So, v v y e C v Therefore, for any pon, y e e C z z e e C3,,,, v, e e e z y C 3 0 C e C /0/04

38 Summary Ch.3. Compably Equaons 38 4/0/04

39 Summary ε Gven u always ess: ε u u j j, j{,, 3} j Gven u wll es only f he compably condons are sasfed. Compably condons: Condons ha a symmercal nd order ensor mus sasfy n order o be an nfnesmal sran ensor and, hus, o es a dsplacemen feld whch sasfes: u u j j, j{,, 3} j They guaranee he connuy of he connuous medum durng he deformaon process. 39 4/0/04

40 Summary Compably equaons for he nfnesmal sran ensor: def yy zz yz S 0 z y yz def zz z S yy 0 z z def yy y Szz 0 y y def zz yz z y Sy 0 y z y z def yy yz z y Sz 0 z y y z def yz z y S yz 0 yz y z S ε 0 S e e ml mjq lr j, qr 0 j, kl kl, j k, jl jl, k 0, j, k, l,,3 ejk ejk 40 4/0/04

41 Summary Roaon ensor: Roaon vecor: Ω ( uu ) 3 θ u 3 z 3 y yz Dervave of θ, : yz z y z y y y z z z y z yz yz yy yz zz zy z y y z y y z z z z z zz z z z 3 yz y y y y y y 3 y yy y 3 3 y yz z z z y 4 4/0/04

42 Summary The negraon of he sran feld ε, : θ,, and.. Inegraon of dervave of usng he epressons derved for 3 ε. Known, and θ,, u s negraed usng: : : 3: The soluon wll be: yz,,, c,,3 j j j 3 u u u y 3 z y z uy uy uy y 3 yy yz y z u u u y z z z z z yz zz,,,,,3 u u y z c 4 4/0/04

43 There s a correspondence beween: The concep of compably condons can be eended o deformaon rae ensor. Summary 4/0/04 ( ) j j j j j j u u u u u u u ( ) v v v v w j j j j j j d v dv v d v 43