东南大学：《固体力学基础》课程教学课件（英文讲稿）09 Simple Linear Elastic BVPs

Cylindrical Equilibrium EquationsX3V . = contraction on the first and third index of zd10terat.6,-0V.0LO0Ozrrrotre100gat-oTre+Ter二azOrr 00Treatr1 0Tgd0eoOzOra0rCeX2de-0V.6+F=0drdo,Otr10treaaTreT00rz三0arazr 001=0TozTreOtreate100e2tre +F。=0,0.TrTozOzOrr 00ot,10teagTr=o,e,+tre,+t,e.TrF=0T'=tre,+Ogee+To.e:OrOzr a0rT"=t,e,+Te-eo+o.e.6

6 Cylindrical Equilibrium Equations contraction on the first and third index of 1 1 1 r r zr r r r z r r rz z z rz z r r z r r r z r r r z r                                                                                    σ σ σ e e e r r rz r rz z z z                          σ 1 0, 1 0, 1 0. 2 r r r r r r rz z z z z z r rz F r r z r F r r z r F r r z r                                                                σ F 0 r r r rz z r z z r r rz z z z                           r θ z T e e e T e e e T e e e

Hooke's Law in Cylindrical CoordinatesX. Recall that. the elastic stiffnesstensor C is a fourth orderoisotropic tensor.Its components remain unchangedunder any orthogonal coordinateAsystems.0The isotropic Hooke's law stays thesame.EEαAT(1-2v)1+v1+vVEEα△TTotalM,+8+8.)+8gGijGi0CEE(1+v) (1-2(1-2v)EEαTEEαT2(1-2v1J(1-2v)1+v)11-EEET(1+v)(1+v)(1+v7

 x3 x1 x2 r  z dr z r r rz z d Hooke’s Law in Cylindrical Coordinates 7     Total 1 , . 1 1 2 1 2 M T ij ij ij ij kk ij ij ij kk ij ij ij T E E E E T                                                                 , 1 1 2 1 2 , 1 1 2 1 2 , 1 1 2 1 2 , , . 1 1 1 r r z r r z z r z z r r z z rz rz E E T E E T E E T E E E                                                                                                    • Recall that, the elastic stiffness tensor C is a fourth order isotropic tensor. • Its components remain unchanged under any orthogonal coordinate systems. • The isotropic Hooke’s law stays the same

Axial Symmetryea? Displacements and stressesu=u,[rle, +e,ze., o=o, [rle,e, +oo[rlese.+o.[rle.ee2? Strain-displacement relation:verdu,u,Ce2drr0e? Equations of motion:do, +,-0+F,=-po°rPlane strain, ordrAgeneralizedplane strain? Hooke's law in cylindrical coordinatesEEαT?Plane strain(1-v)6, + v8g +ve.)o(1-2v)(1+v)(1-2v)8, =0.EEαATve, +(1-v)g。 +ve.0e: Generalized plane strain(1-2v)(1+v)(1-2v)8 =const., F,=["2元ro,drEEoATve, +ve +(1-v)oa(1-2v)(1+v)(1-2v)8

Axial Symmetry • Displacements and stresses • Strain-displacement relation: • Hooke’s law in cylindrical coordinates d , d r r r u u r r                                1 , 1 1 2 1 2 1 , 1 1 2 1 2 1 . 1 1 2 1 2 r r z r z z r z E E T E E T E E T                                                           8 u e e      u r z r r r r r z z r r r z z z       , σ  e e e e e e        • Equations of motion: r r 2 r d F r dr r          0. z   • Plane strain • Generalized plane strain const., 2 . b z z z a      F r dr 

Axial Symmetry Plane stressEαT1+vEVTotalMYA+6=8.GiOiEE(1+v) [1-2v(1-2v)E(1+)α△TEαT0=0.08.(1+v)(1-2v)-2y1-ve3(1+v)α△T-(c, +6)8=8+8+81-v1-EEα△TO(1-v)EαATEIQe(1-v)e2er0er(α +o,)+α△T=0.Planestress. Boundary conditions: u,[a]=ua, u,[b]=u0,[a] =0a, ,[b]=09

Axial Symmetry • Plane stress         , , r a r b r a r b u a u u b u     a b     • Boundary conditions:           2 2 , 1 1 , 1 1 0, . r r r z z x y E E T E E T T E                                                           Total 1 , . 1 1 2 1 2 1 1 0 1 1 2 1 2 1 2 1 1 1 1 2 1 1 M T ij ij ij ij kk ij ij ij kk ij ij ij z r z z r kk r z r E E T T E E E E T T T                                                                                                                 9