东南大学：《弹性力学》课程教学课件（英文讲稿）07 Two-Dimensional Formulation

Plane Strain Field Eguations· Equilibrium Equationsataga1xJaxayatagXYF0atagax0ayxy+ag0xayat0axayatatagy2A0azaxOy6

• Equilibrium Equations x y x x z x y z            0 , x x y y y z F x y z              0 , 0 . y y z x z z z F F x y z                0 , 0 . x y x x x y y y F x y F x y                    Plane Strain Field Equations 6

Plane Strain Field Equations.Navier's Equations(auavaaGvu+(a+G)ayoxiox( auaavOGV(A+GV+ayoylaxa(aurowGV2+α+GWayazaxaza(auav)GV(+G)4axoyax-a(auavGV0(+CaydyoxGVu+GVV1

• Navier’s Equations   2 u v w G u G x x y z                 2 0 , x F u v w G v G y x y z                         2 0 , 0 . y z F u v w G w G F z x y z                                   2 2 2 ( ) 0 , ( ) 0 . 0 . x y u v G u G F x x y u v G v G F y x y G G                                                 u u F Plane Strain Field Equations 7

Plane Strain Field Eguationsjl,ik = 0 (6 eqns) Strain CompatibilityCyu,k+ekkl.ijik,jlaaSCxy2axQxoy0. Beltrami-Michell Equation:2D Constitutive LawaxoyQaaaOOLAddto both sides:(1-v)(gaxaxayayaxoyaF(aFUsingEquilibriumontheRHS:(I-)(o,+oayaxaF.(aF1axay8

• Beltrami-Michell Equation: Plane Strain Field Equations                 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 D C o n s t i t u t i v e L a w : 2 A d d t o b o t h s i d e s : 1 2 U s i n g E q u i li b r i u m o n t h e R H S : 1 x y x x y y x y y y x y x x x y y x x y y x x y x y x y x y F F x y                                                                              2 1 1 y x x y F F x y                     8 • Strain Compatibility 2 2 2 2 2 2 y x y x y x x y             , , , , 0 ij k l k l ij ik jl jl ik         (6 eqns)

Examples of Plane Strain ProblemsPX.-X1Long cylindersSemi-infiniteregions underunder uniform loadinguniformloadings9

Examples of Plane Strain Problems x y z x y z P Long cylinders under uniform loading Semi-infinite regions under uniform loadings 9

Anti-Plane Strain: An additional plane strain theory of elasticity called Anti-Plane Strain involves a formulation based on theexistence of only out-of-plane deformation starting withan assumed displacement field: u = v = O, w = w(x, y)StrainsStresses0SC0.Oaa二T二xyy1 ow1aw2G82G520x2oyNavier's EquationEquilibriumEquationsatatF=0GVw+F=0.ayaxFF=010

Anti-Plane Strain • An additional plane strain theory of elasticity called Anti￾Plane Strain involves a formulation based on the existence of only out-of-plane deformation starting with an assumed displacement field: u v w w x y    0 , ( , ) . 0 , 1 1 , . 2 2 x y z x y x z y z w w x y                 0 , 2 , 2 . x y z x y x z x z y z y z G G               0 , 0 . y z x z z x y F x y F F            2 0 . z G w F    Strains Stresses Equilibrium Equations Navier’s Equation 10