LAGRANGIAN EQUILIBRIUM EQUATIONS IN CYLINDRICAL AND SPHERICAL COORDINATES
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1 Compute, Mateial & Continua 3 (2006) LAGANGIAN EQUILIBIUM EQUAIONS IN CYLINDICAL AND SHEICAL COODINAES.Y. Voloh Depatment of Mehanial Engineeing, John Hopin Univeity, Baltimoe, MD Abtat Lagangian o efeential equilibium equation fo mateial undegoing lage defomation ae of inteet in the developing field of mehani of oft biomateial and nanomehani. he main featue of thee equation i the neeity to deal with the Fit iola-ihhoff, o nominal, te teno whih i a two-point teno efeing imultaneouly to the efeene and uent onfiguation. hi two-point natue of the Fit iola-ihhoff teno i not alway appeiated by the eeahe and the total ovaiant deivative neeay fo the fomulation of the equilibium equation in uvilinea oodinate i ometime inauately onfued with the egula ovaiant deivative. Supiingly, the taditional ontinuum mehani liteatue doe not diu thi iue popely, exept fo ome bief notion on the two-point natue of the iola-ihhoff teno. We aim at patially filling thi gap by giving a full yet imple deivation of the Lagangian equilibium equation in ylindial and pheial oodinate.. Intodution Lagangian ala equilibium equation in ylindial and pheial oodinate fo mateial undegoing lage defomation ae aely diued in the liteatue. he mot influential monogaph on nonlinea elatiity and ontinuum mehani, inluding Antman (995); Chadwi (976); Cialet (988); Eingen (962); Geen and Adin (970); Geen and ena (968); Gutin (98); Haupt (2000); Jaunemi (967); Liu (2002); Lu e (990); Malven (969); Maden and Hughe (983); Ogden (984); uedell and oupin (96); On leave of abene fom the ehnion; voloh@jhu.edu; voloh@tehnion.a.il
2 Compute, Mateial & Continua 3 (2006) uedell and Noll (965); Wang and uedell (973); Wilmani (998), do not adde thi iue. Howeve, the Lagangian equilibium equation in ylindial and pheial oodinate an be vey ueful in olving nonlinea poblem analytially o emi-analytially. Sometime, it i poible to aume inompeibility of the mateial what allow fo uing a imple Euleian deiption fo obtaining ome elementay analytial olution. hi i not the geneal ae, howeve, whee we need the Lagangian equilibium equation of the fom Div 0 () in ylindial and pheial oodinate. hee equation an be deived fom the total ovaiant deivative of the t iola-ihhoff te teno. hough thi way may be elegant we pefe a moe taightfowad pedetian way, whih, howeve, doe not equie any nowledge of the geneal teno alulu fom the eade. 2. Cylindial oodinate We intodue othonomal bai in ylindial oodinate (Malven, 969) fo the efeene onfiguation. (2) (o,in,0) ; ( in,o,0) ; (0,0,) By diet alulation we have ;. (3) All othe deivative of the bae veto ae equal to eo. Analogouly, we have fo the uent onfiguation: (o,in,0) ; ( in,o,0) ; (0,0,), (4) ;. (5) Now, we wite the divegene opeato in the fom (Malven, 969) Div. (6) he plan i to ompute the ight-hand ide of thi equation tem by tem. We tat with the fit tem on the ight hand ide of Eq. (6) 2
3 Compute, Mateial & Continua 3 (2006) 37-42, (7) whee. N m N m With aount of othonomality of the bae veto we have. (8) Diffeentiating the Euleian bai, we get,, (9) 0. Now, ubtituting Eq. (9) in Eq. (8) we have. (0) Analogouly to Eq. (7)-(0) we alulate the lat two tem on the ight-hand ide of Eq. (6) 3
4 Compute, Mateial & Continua 3 (2006) 37-42, () (2),, (3) 0,, (4) 4
5 Compute, Mateial & Continua 3 (2006) 37-42, (5), (6),, (7) 0,. (8) Finally, ubtituting Eq. (0), (4), and (8) in Eq. (6) we have 5
6 Compute, Mateial & Continua 3 (2006) Div (9) 3. Spheial oodinate We intodue othonomal bai in pheial oodinate (Malven, 969) fo the efeene onfiguation (in o,in in,o) (oo,oin, in ) ( in,o,0). (20) By diet alulation we have the following noneo deivative of the bae veto ; o ; ; in in ; o Analogouly, we have fo the uent onfiguation: (in o,in in,o ) (o o,o in, in ) ( in,o,0). ` (2), (22) ; ; in ;. (23) o ; in o We will ue the following abbeviation fo the ae of impliity S in ; C o; in; o. (24) Now, we wite the divegene opeato in the fom (Malven, 969) Div. (25) S he plan i again to ompute the ight-hand ide of thi equation tem by tem. 6
7 Compute, Mateial & Continua 3 (2006) We tat with. (26) With aount of othonomality of the bae veto we have. (27) Diffeentiating the Euleian bai, we get, (28). Now, ubtituting Eq. (28) in Eq. (27) we have 7
8 Compute, Mateial & Continua 3 (2006) (29) Analogouly to Eq. (26)-(29) we alulate the lat two tem on the ight-hand ide of Eq. (25), (30), (3),, (32) 8
9 Compute, Mateial & Continua 3 (2006) 37-42,, (33) S S S S S S S S S S, (34) C S C S C S S S, (35), 9
10 Compute, Mateial & Continua 3 (2006) Div S C S C S C S S S S S S S S, (36), S S Finally, ubtituting Eq. (29), (33), and (37) in Eq. (25) we have 2 C S S S S 2 C S S S S 2 C S S S S. (37). (38) 4. Conluion Lagangian equilibium equation in ylindial (Eq. 9) and pheial oodinate (Eq. 38) have been deived in the peent wo. 5. efeene Antman, S.S. (995): Nonlinea poblem of elatiity. Spinge-Velag. Chadwi,. (976): Continuum mehani. Wiley. 0
11 Compute, Mateial & Continua 3 (2006) Cialet,.G. (988): Mathematial elatiity, Volume : hee-dimenional elatiity. Noth Holland. Eingen, A.C. (962): Nonlinea theoy of ontinuou media. MGaw-Hill. Geen, A.E., Adin, J.E. (970): Lage elati defomation. Oxfod Univeity e. Geen, A.E., ena, W. (968): heoetial elatiity. Oxfod Univeity e. Gutin, M.E. (98): An intodution to ontinuum mehani. Aademi e. Haupt,. (2000): Continuum mehani and theoy of mateial. Spinge. Jaunemi, W. (967): Continuum mehani. MMillan Company. Liu, I.S. (2002): Continuum mehani. Spinge. Lu e, A.I. (990): Nonlinea theoy of elatiity. Noth Holland. Malven, L.E. (969): Intodution to the mehani of a ontinuou medium. entie-hall. Maden, J.E., Hughe,.J.. (983): Mathematial foundation of elatiity. entie-hall. Ogden,.W. (984): Nonlinea elati defomation. Elli Howood. uedell, C., oupin,.a. (960): Claial field theoie. In: Flugge, S. (Ed.): Enylopedia of hyi, Vol. III/. Spinge. uedell, C., Noll, W. (965): he nonlinea field theoie of mehani. In: Flugge, S. (Ed.): Enylopedia of hyi, Vol. III/3. Spinge. Wang, C.C., uedell, C. (973): Intodution to ational elatiity. Noodhoff. Wilmani,. (998): hemomehani of ontinua. Spinge.
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