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