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

Three-Dimensional Problems

Three-Dimensional Problems Dimensional Problems

OutlineDisplacement Formulation ReviewHalf-Space under Uniform Pressure and GravitySpherical ShellGeneral Solution - Helmholtz RepresentationParticular Case-Lame Strain PotentialGalerkin Vector PotentialLove Strain Potential -Axi-symmetryCompleteness of Displacement PotentialsHarmonic and Bi-harmonic FunctionsKelvin's ProblemBoussinesq's ProblemCerruti's ProblemDistributed Pressure onHalf-SpaceHertz Contact Problem2

Outline • Displacement Formulation Review • Half-Space under Uniform Pressure and Gravity Space under Uniform Pressure and Gravity • Spherical Shell • General Solution General Solution – Helmholtz Representation Helmholtz Representation • Particular Case – Lamé Strain Potential • Galerkin Vector Potential Galerkin Vector Potential • Love Strain Potential – Axi-symmetry • Comp p leteness of Displacement Potentials • Harmonic and Bi-harmonic Functions • Kelvin’s Problem • Boussinesq’s Problem • Cerruti’s Problem • Distributed Pressure on Half-Space • Hertz Contact Problem 2

Review of Displacement Formulation - RCCNavier's equationGv?u+(a+G)v(V·u)+F= 0Gui,kh +(a+G)uk,ki + F, = 0Displacement-strain relation:+Vu)8CiiHooke'slaw' = aTr()I +2G8,0,=180,+2GEEv2=G=(1+v)(1-2v)2(1+v)3

Review of Displacement Formulation – RCC     2 • Navier’s equation       2 0 0 G G G GF          u uF • Displacement-strain relation: , ,   0 G i kk k ki i u     G u F Displacement strain relation:     1 1 ε     u u  u u   ,  , ,  2 2 ij i j j i ε     u u  u u • Hooke s’ law: Tr 2   , 2 σ ε     G G     ij kk ij ij   I ε      , , ij kk ij ij E E G     3      , 1 1 2 21  G     

Review of Displacement Formulation - CylindricalNavier's equation210ugou.oueourduru,+F=0a+2.200ar00ozarrr2ou,oueaOu,ug1ur(α +F=0Oz0000raorraOu,ouOug1.W+F.=0ara0OzOzLrα?a2α21a1V2022Or.200?r2LarDisplacement-strain relation:augou1ou1oue(1ou,ugu88.Oz2a0a0Orarr1Ou.Ouroue1 ou.a80Oza02arOz2rHooke's law.4

Review of Displacement Formulation – Cylindrical • Navier’s equation        2 2 2 2 1 0 2 11 r rr z r r u u u uu u Gu G F r r r r rr z                                       2 2 2 2 11 0 1 r rr z u uu u u u Gu G F r r r r rr z uu u u                                        2 2 2 1 0 1 1 rr z z z uu u u Gu G F z r rr z                      2 2  2 2 1 1 r rr r       2 2  z     • Displacement-strain relation: 1 11 , , 2 r zr r r zr u u u uu u u r r z r rr                              Displacement strain relation: 2 11 1 , 2 2 z zr z zr r r z r rr u u uu zr r z                                • Hooke’s law. 4

Displacement Formulation - Axi-symmetricNavier's equationaOu.ou,u,v?u, -+G)+F=0T2arOrOzIaaurou.u,GVF=0OzOrOzra2a21a2Or2022arr Displacement-strain relationauOu.1Ou.ouu,6.C-2OzOrOzOrrHooke's lawOuauauauou,u2G0,=2(s, +8+8.)+2G6,=2(+6+6)+2G20azararOzarrrOuouOuauauu,2G0,=2(8,+8g+8:)+2G8,==2G6razazararOzI*EEv23(1+v)(1-2v)"2(1+v)5

Displacement Formulation – Axi-symmetric • Navier’s equation   2 2 0 r rr z r r u uu u Gu G F r rrr z                           2 2 2 0 rr z z z uu u Gu G F zrr z                2 2 2 2 2 1 r rr z          • Displacement-strain relation 1 , , 2 r r z zr r z zr u u u uu r r z rz                     Displacement strain relation    r r z rz 2   • Hooke’s law   22 22   rr z r rr z r uu u u uu u u   GG GG              2 2 , 2 2 2 2 ,2 rr z r rr z r rr z r r z rr z z z r z r z z rz rz GG GG rr z r rr z r uu u u u u G G GG rr z z r z                                                                          , 1 1 2 21 rr z z r z E E G                5

Review of Displacement Formulation - SphericalNavier'sequation1oue2cotpu.20u。22uROugOuR2uRcotpu,1Ouga+(a+G)V'UR+F.=0R?R3R?aRRRapapR'sinpaeaRRRsinp ae2auau.2cot@1ouOuROugaOuR2ucotpu+(a+G)+F.=0R2RRapR'sin'pR'sinpaeRapaRRapRsingae2aou1 QugOueOUR2cotp1QuR+）cotpu1Ue2uR+(a+G)1=0.R'singR'sin?RsingaeaRRRRopR'singaan0Rsingaoa2a20212.o1acotp72ROp?R'sin'00?OR?RaRR?apDisplacement-strain relation:1Oue1oue1 OURoug1 OURURURcotpu.UpERRe2RRRRaRRapaRRapRsin@ o111OuROueou1(10uaecotpuERO6002RRR2RsinpoaRa0Rsinoae.Hooke's law..6

Review of Displacement Formulation – Spherical • Navier’s equation   2 2 22 2 2 2 2cot cot 2 2 11 0 sin sin 2 2 cot 1 1 1 2 cot R R R R R u uu u u uu u u Gu G F R R R R RR R R R R u uu u uu u u                                                           2 2 22 2 2 2 cot 1 1 1 2 cot sin sin sin R R R u uu u uu u u Gu G R R R R RR R R R                                            2 2 2 22 0 2 2cot 1 1 1 2 cot 0 sin sin sin sin sin R R R F u uu u uu u u Gu G F R R R R RR R R R                                                          2 22 2 2 2 2 2 22 2 sin sin sin sin sin 2 cot 1 1 sin R R R R RR R R R R RR R R R                                 • Displacement-strain relation: 1 1 11   cot   , , sin 2 1 1 11 1 cot RR R R R R uu u u u u uu u R RR R R R RR R uu u u u                                     1 1 11 1       cot  , 2 sin 2 sin R R u uu u u u R RR R R R                                    • Hooke’s law. 6

Displacement Formulation - (Centro-symmetricNavier's equationd?dduR2ukd2uV2(a+G)V'u+ FR = 0,OR?RoRR2RdRdRd'u2 dur2UR2URdduR=0+FR=G(a+GaR?R?R oRPdRdR2URddur+FR=0dRRdRE(1-v)1ddR4a+2G=01+2G=HR?dRdR(1+v)(1-2v) Strain-displacement relation:OURUR60GR=8RaR.Hooke's law2UROUR2UROuROuR2GUR2GCR=/0aRRaRRRaREvE入二G2(1+v)(1+v)(1-2v)7

Dis placement Formulation – Centro-s ymmetric • Navier’s equation 2    2 2 d dd d 2   2 2 2 2 2 2 2 2 2 0, 2 2 2 R RR R R d u du u d u d dd du u Gu G F R dR dR R R R R d u u                              2 2 0 2 2 2 2 R R R R RR RR d u du u d G GF d du u d u u R R R R dR dR R                               2   2 2 0 1 1 2 02 R R R d du u G F dR dR R d d E G R F G                   • Strain-dis placement relation:          2 2 2 0 , 2 1 12 G R R R u F G dR R Rd                  p • Hooke ’s law , R R R u u R R         • Hooke s law . 2 2 2, 2 R R R RR R R uu u uu u G G RR R RR R                           , 1 1 2 21 E E G         7

Half-Space under Uniform Pressure and GravityqObservations and assumptionsxF,=0, F, =pg0IpgThu, = O,u, = u, (2)Navier's equationzououu.7+F=0L2arOzOrr52aauouaLou2pg=0OzOzOzOrerrr2u=(α+2G)Pg=0dz2By direct integrationd?udupgpgpg+ Bz+ A)=u.(z+d?2(几+2G)元+2Gdz1+2G8

Half-Sp y ace under Uniform Pressure and Gravity • Observations and assumptions q x   0, 0 F r z F g u u uz      x o h u u uz r zz   0,   • Navier’s equation z   2 2 0 r rr z r r u uu u Gu G F r rrr z                    2 2 1 z z u u G r rr        2 2 z rr u uu G z z rr                  0 z u g z               r rr   z z rr     2 2 2 0 z z d u G g dz         • By direct integration     2 d u du   gg g 2      2 2 2 2 22 z z z d u du gg g zA u zA B dz G dz G G                8

Half-Space under Uniform Pressure and GravityqStresses in terms of displacementsxOuOuu0OIpgI hOzerOrououOzOu.OuouOun+2GOozOzrOzapgdu+A), . =(+2G)=-pg(z+Ua0Adz1+2GdzThe traction BCs at z = Oa-q=G. (z=0)=-pgA山pgApgqq=Opg1+2Gpgpg9

Hal f-S pace under Uniform Pressure and Gravit y • Stresses in terms of displacements    p y q x r r r u u r r       2 z r u u G z r                 x o h r r u u r r        2 z r u u G z r                z r r z u u r r       2 , z z zr rz u u uu G G z z rz                                 , 2   2 z z r z du du g zA G g z A dz G dz                      • The traction BCs at z = 0 q q z gA A z  0  g             , 2 r z gq q z gz Gg g                     9

Half-Space under Uniform Pressure and GravityqLateral to in-depth stress ratiox元EvE(1-v)arCe0Ipg几+2GI h(1+v)(1-2v)/ (1+v)(1-2v)a.d.VZThe displacement BCs at z = hpgq+ Bu.pgq2(元+2G)=B=h+pg2(α+2G)pgO = u. (z = h)pg(h2 -z2)+2q(h-z)pgUu.2(元+2G)2(+2G)Dg(u.)max = u. (z= 0)= Pgh +2qhThe maximum displacement2(α +2G)occurs atthetop surface10

Half-Space under Uniform Pressure and Gravity • Lateral to in-depth stress ratio p y q x         1 2 1 12 1 12 r z z E E G                 x o h 1     • The displacement BCs at z = h z • The displacement BCs at z = h   2 2 2 2 z g q u zB g q G B h                          2 2 2 2 0z z g q G g B h G g uzh                               2 2 2 2 2 22 22 z gqq gh z qh z u hz Gg g G                                        2 22 22 2 0 Gg g G gh qh u uz                  The maximum displacement       max 0 2 2 z z u uz  G     p occurs at the top surface. 10