东南大学：《弹性力学》课程教学课件（英文讲稿）11 Three-Dimensional Problems Dimensional Problems

Particular Case - Lamé Strain PotentialsConsider the further special case with=0,=0 =+x=GV×(V2p)+(a+2G)V(v2) =0The displacement is commonly written as2Gu= VΦ, 2Gu, = Φi=== 0=0,= 8, +2G, =Examples ofharmonicfunctions1, x,y,z, xy,yz,zx, x?- y?,y?-22,=? -x2, R?-3x?,R?-3y?,R? -32?, r"cosno,Vr?+(z+c)°-z -cr2 +(z -In(R+ 2InrLR16

Particular Case – Lamé Strain Potentials • Consider the further special case with 2    0, 0 φ    u   φ          2 2 G G        φ   2 0 • The displacement is commonly written as The displacement is commonly written as     , 2 ,2 111 G Gu u    i i      2 , , , , , 111 0 24 2 1 ij i j j i ij ji ij kk kk u u G G                    1 , 1 2 2 ij kk ij ij kk G G          , , 1 2 2 ij ij ij G G               • Examples of harmonic functions Examples of harmonic functions         2 2 2 22 2 2 2 2 2 2 2 2 2 2 2 1, , , , , , , , , , 3 , 3 , 3 , cos , n x y z xy yz zx x y y z z x R x R y R z r n                 2 2 2 2 2 1 ln , , , ln , ln r zc zc r zc zc r Rz R r         16

Particular Case - Lamé Strain PotentialsIncylindrical coordinates1 aadap2Gw2Gu,2GuOrra0Ozasdasa'd1 1 as1 0p1 0gdeaz2,Or2ag?0222n200?r Orr or1 a2sasa2s1 og1 Tre.To=n2a0r oro0Orozr a0oz· For axi-symmetric problemsadad2Gu27arazadoda"sa?1 ad1 adO00022022Or?r orOrozr or17

Particular Case – Lamé Strain Potentials • In cylindrical coordinates 1 2 , 2 , 2 , Gu Gu Gw r rr z             2 22 2 2 2 22 2 22 2 11 11 , , , r z r rr r z rr r z                               2 22 2 11 1 , , r z rz r r r r z rz                       • For axi-symmetric problems 2 ,2 , Gu Gw r r z         2 2 22 22 2 1 1 , , r z rz r r r z r r z rz                           r r r z r r z rz 17