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

Simple Linear Elastic BVPsmi@se.edu.cn

Simple Linear Elastic BVPs

Outline·Reviewoffieldequations（线弹性力学控制方程回顾）·Thermoelasticity（热弹性力学本构关系）·Small strain theory in cylindrical coordinates （柱坐标）·Axial symmetry（轴对称）·Pressurized cylindrical shell（压力圆筒）·Spinningdisk（圆筒转动）·Interferencefitbetween two cylinders（圆筒过盈装配）· Small strain theory in spherical coordinates （球坐标系）·Spherical symmetry（球对称）·Pressurized spherical shell（压力球腔）·Gravitating planet（重力球）·Steady-state heat flow in spherical shell （球腔稳态热流）2

Outline • Review of field equations（线弹性力学控制方程回顾） • Thermoelasticity（热弹性力学本构关系） • Small strain theory in cylindrical coordinates（柱坐标） • Axial symmetry（轴对称） • Pressurized cylindrical shell（压力圆筒） • Spinning disk（圆筒转动） • Interference fit between two cylinders（圆筒过盈装配） • Small strain theory in spherical coordinates（球坐标系） • Spherical symmetry（球对称） • Pressurized spherical shell（压力球腔） • Gravitating planet（重力球） • Steady-state heat flow in spherical shell（球腔稳态热流） 2

Review of Field EquationsStrain-displacement relations: +u+ Strain compatibility: Sy,k + Su,j -Sik,jl -Sjlik = 0 Equilibrium: Oj,; + F, = Oj, + pb, = 0.Isotropic Hooke's Law:E1+vVVQi1EE(1+v) (1-2v Traction BCs on SRoDisplacement BCsesRon SueiDeformedOriginalConfigurationConfiguration3

 , ,  1 2  ij i j j i   u u     ij kl kl ij ik jl jl ik , , , ,     0 , , 0.    ij i j ij i j     F b   1 ; . 1 1 2 ij kk ij ij ij ij kk ij E E E                           Review of Field Equations • Strain-displacement relations: • Strain compatibility: • Equilibrium: • Isotropic Hooke’s Law: 3 • Traction BCs on St • Displacement BCs on Su

Thermoelastic Constitutive RelationsA temperature change in an elastic solid produces deformation The total strain can be decomposed into the sum of mechanical andthermal components.It is extremely important to understand that, the elastic stiffnesstensor (C) correlates mechanical stress and mechanical strain1+vVMTotalS+α△TSOkkN1EETotalTotal -8,=eM0,+2GeM=a(-8h)+2GS&kk, = Ae toal8, +2Ge,foal (3 + 2G) αAT8, = NeTol , +2Ge,Tol-3Kα△TS,EEαTVTotalTotalSo0kki111+v[1-2v(1-2v)4

Total M T 1 ij ij ij kk ij ij ij E E T                  Thermoelastic Constitutive Relations • A temperature change in an elastic solid produces deformation. • The total strain can be decomposed into the sum of mechanical and thermal components. 4 • It is extremely important to understand that, the elastic stiffness tensor (C) correlates mechanical stress and mechanical strain.           Total Total Total Total Total Total Total Total 2 2 2 3 2 2 1 1 2 1 3 2 T T kk ij ij i M M ij kk ij ij kk ij ij ij kk ij ij kk ij ij ij kk ij ij ij G T j G G G G E E T K T                                                            

Cylindrical Strain and Rotation&=(u+Vu); Q=u-Vu); u=u,e, +uge +ue,,ouOuduoueougPueere.u.e.egP6a0Ozaraaroueouou1 0u-eee4.e+OzOzr 0Oroue1(1OuUe0.00=0=0-2a0arrr11Ouououe1 OuQesQr22OzarOzra01ououeou1 (1ouaueWeCUr88+rear00ara0Oz2rroue1u1OuOu.602S22OzOzar00r5

Cylindrical Strain and Rotation     c 1 1 ; ; ; 2 2 1 1 1 1 1 0, 2 r z r r r r r r r z r r z z z z z r z z z r r z r u u u u u u u u u u r r z r r u u u u z r r z u u u r r                                                                                           r θ z ε u u ω u u u e e e e e e e e e e e e e u e e e e e e e e , 1 1 1 , ; 2 2 1 1 1 , , , , 2 1 1 1 , . 2 2 z z r z zr r z r r r z r z z r z zr r u u u u z r r z u u u u u u u r r z r r r u u u u z r r z                                                                                                           5

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

Axial Symmetry: Stresses in terms of displacements (generalized plane strainEEαTduurr+Veo(1+v)(1-2v(1-2v)drAEEαATdu+VEOe(1+v)(1-2v)(1-2v)drEEαTdu,(1-2v)(1+v)(1-2v)drStresses in terms of displacements (plane stress)EEEαTEαATdu,u,du,urO.O01-v21-v2dr(1-v)"dr(1-v)rr Equilibrium equations in terms of displacementsdo,+,-O=-F,-po"r(generalized plane strain)drr(1+v)(1-2v)1 du,α(1+) d△Turdudr2drdrr(1-v)E(1-v)dr10

• Stresses in terms of displacements (generalized plane strain) Axial Symmetry                      d 1 , 1 1 2 d 1 2 d 1 , 1 1 2 d 1 2 d 1 . 1 1 2 d 1 2 r r r r r r r z z z z E E T u u r r E E T u u r r E E T u u r r                                                                          • Stresses in terms of displacements (plane stress)     2 2 d d , . 1 d 1 1 d 1 r r r r r E E T E E T u u u u r r r r                                    • Equilibrium equations in terms of displacements (generalized plane strain)              2 2 2 2 2 d d d d 1 d 1 d d 1 1 1 2 . d d d d 1 d 1 r r r r r r r r F r r r u u u T ru F r r r r r r r r r E                                    10