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

Spherical Shell under Uniform PressureCentro-symmetry and zero body forcesqb=(1+2GdRR2dR1d=3AR2=3ARuR? dRdRBR?uR=AR"+BUR = AR+UR?OuR2B2B2B2URou+2GA+2G22A-OR3R3R3RaRaRBB2BOUR2UR2G+2GAR3R3RRaRRE2E4GCR =(3+2G)AR3(1+v) R(1-2v)EE2G0。=。=(3元+2G)A+R3(1-2v)(1+v)11

S pherical Shell under Uniform Pressure • Centro-symmetry and zero body forces b q a     2 2 1 2 R R d d G Ru F dR R dR           0 a q   a     2 22 2 1 3 3 R R d d R u A R u AR R dR dR    2 3 b R R 2 B R u AR B u AR R      33 3 2 22 2 2 22 RR R R uu u BB B G A A GA RR R R R R                                 33 3 2 2 2 22 2 RR R uu u B B B G A A GA R RR R R R                                         3 3 4 21 3 2 12 1 R GE E GA B A B R R                    3 2 3 2 12 1 GE E GA B A R                3 1 B  R     11

Spherical Shell under Uniform PressureThe traction BCs at R = a and R = bE2E Bqb-qa=(oR)D1+va31-2vEEB-qb=(OR)1 +vb31-2v(1-2v)(a'qa-b3qt_ (1+v)(qa-qb)a3b32E(b3-a3)E(b3-ab3?(1+v)1-2v1-2vb'qbRUR2R32R3E(b3-a1+v1+b31a1b'qbaORR3b3-a3R3b3b311ab36qb2R3b3-a2RhsHere, stress is independent of Poisson's ratio.However, generally in 3-Dproblems with specified tractions, stress depends on Poisson's ratio.12

S pherical Shell under Uniform Pressure • The traction BCs at R = a and R = b  E EB 2 p b q a 3 2 ( ) 12 1 ( ) a R Ra E EB q A a E EB A              a q a         3 3 3 3 3 ( ) 12 1 1 2 1 b R Rb a b a b E EB q A b a q b q q q a b A B                 b             33 33 , 2 a b a b q q q q A B Eb a Eb a     3 3 3 3 (1 ) 1 2 1 2 b a b                 3 3 3 3 3 3 (1 ) 1 2 1 2 21 21 R ab b a u R aq b q Eb a R R                              3 3 3 3 33 3 3 3 3 1 1 1 1, R ab b a aq bq ba R b a R              3 3 3 3 33 3 33 3 1 1 1 1 2 2 a b b a aq bq ba R ba R                 • Here, stress is independent of Poisson’s ratio. However, generally in 3-D problems with specified tractions, stress depends on Poisson’s ratio. 12

General Solution - Displacement PotentialsHelmholtzrepresentation:V.@= 0.Vd+V×u=IrrotationalSolenoidalDilatation and rotation=V.u=.(V+×)=.+x =?=V×(VΦ+V×β)×+V××)=Vxu=22Navier's equationGv"u+(a+G)V(V·u)+F =0=GV?(VΦ+V×)+(α+G)(?)+F = 0GV×(V20)+(α +2G)V(V2)+ F = 013

General Solution – Dis placement Potentials • Helmholtz representation: u       Irrotational Solenoidal      , 0.              2   • Dilatation and rotation   kk    u    φ φ       2 11 1      ω         u φ   1 2      φ φ 22 2 ω         u   φ   2   φ   φ • Navier s’ equation equation     2 G G        u uF  0       2 2           G G   φ F 0       2 2          G G φ   2 0 F 13

General Solution - Displacement Potentials: If divergence and curl is taken of the previous equation[0 = V.[GV×(v?0)+(a +2G)(v?)+F|=(a + 2G)2V2$+V.F0 = V×[GV×(V2)+(α +2G)V(v2)+F= GV×V×(20)+V×F = -GVV?+V×Fv2V2=-V.F/(+2G)2?=V×F/G-With zero body forces, both the scalar andvector potential functions are biharmonicThese four harmonic functions are not independent.since they must satisfy the Navier's equationSummaryGV×(v)+(α+2G)V(v2Φ)+ F = 0V2?=-V.F/(a+2G), V=V×F/G14

General Solution – Dis placement Potentials • If divergence and curl is taken of the previous equation                 2 2 22 2 2 2 22 0 22 0 2 GG G G GG G                                                φ F F  φ      F  φ  F φ F   22 22 0 2 2 G GG G G G                                    φ F φ F φ F F φ F With zero body forces, both the scalar and vector potential functions are biharmonic. vector potential functions are biharmonic. • These four harmonic functions are not independent, since they must satisfy the Navier ’s equation • Summary since they must satisfy the Navier s equation. y       2 2 G G         φ   2 0 F   22 22           F 2 , G G φ F 14

Particular Case - Zero Body Forces Consider the special case?Φ= ?β= 0GV×()+(α +2G)V(2) = 0Both the scalar and vector potential functions areharmonic.This special case may lead to some useful solutionsHowever, there is no guarantee that every elastostaticsolution can be represented in terms of these fourharmonic functions15

Particular Case – Zero Bod y Forces • Consider the special case       2 2 2 2      φ 0 G G      φ  2 0    • Both the scalar and vector potential functions are harmonic. • This s p y ecial case ma y lead to some useful solutions. • However, there is no guarantee that every elastostatic sol i b di f h f lut ion can be represente d in terms o f t hese four harmonic functions. 15