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

Two-Dimensional Formulation

Two-Dimensional Formulation

OutlineIntroductionPlane StrainPlane StressBoundary ConditionsCorrespondence between Plane Strain and Plane StressCombined Plane Formulations Anti-Plane StrainAiry Stress Function· Polar Coordinate Formulation2

Outline • Introduction • Plane Strain • Plane Stress • Boundary Conditions • Correspondence between Plane Strain and Plane Stress • Combined Plane Formulations • Anti-Plane Strain • Airy Stress Function • Polar Coordinate Formulation 2

Introduction: Three-dimensional elasticity problems are very difficult to solveThus we will first solve a number of two-dimensional problems.and will explore three different theories:-Plane StrainPlaneStress-Anti-Plane Strain Since all real elastic structures are three-dimensional, theories setforth here will be approximate models. The nature and accuracyof the approximation will depend on problem and loadinggeometry.. The basic theories of plane strain and plane stress represent thefundamental plane problem in elasticity. While these two theoriesapply to significantly different types of two-dimensional bodies.their formulations yield very similar field equations3

Introduction • Three-dimensional elasticity problems are very difficult to solve. Thus we will first solve a number of two-dimensional problems, and will explore three different theories: - Plane Strain - Plane Stress - Anti-Plane Strain • Since all real elastic structures are three-dimensional, theories set forth here will be approximate models. The nature and accuracy of the approximation will depend on problem and loading geometry. • The basic theories of plane strain and plane stress represent the fundamental plane problem in elasticity. While these two theories apply to significantly different types of two-dimensional bodies, their formulations yield very similar field equations. 3

Plane Strain: Consider an infinitely long cylindrical (prismatic) body as shownIf the body forces and tractions on lateral boundaries areindependent of the z-coordinate and have no z-component, thenthe deformation field can be taken in the reduced formu=u(x,y),V=v(x,y),0.=wRA

Plane Strain • Consider an infinitely long cylindrical (prismatic) body as shown. If the body forces and tractions on lateral boundaries are independent of the z-coordinate and have no z-component, then the deformation field can be taken in the reduced form x y z R ( , ) , ( , ) , 0 . u u x y v v x y w    4

Plane Strain Field Eguations1. Displacement-strain relation:12ayawawauavauawau1avazax=azaxayaaxnau1+vLIsotropic Hooke's Law: g. = as,2GEE2G86=元2GE0.=11 ++1v(ar+a))EEE1+VEEF5

Plane Strain Field Equations 1 1 1 , , , 0 , 0 , 0 . 2 2 2 x y x y z x z y z u v u v w u w v w x y y x z z x z y                                                       • Displacement-strain relation:                     2 , 2 , 2 0 1 1 , 1 1 1 , 0 x x y x y x y y z x y x y x y z x z y x x x y z x x y y y x x y z y x y x y x y z x z y z y G G G E E E E E E E                                                                                        ， 5  , ,  1 2 i j i j j i    u u 1 2 ; . i j k k i j i j i j i j k k i j G E E             • Isotropic Hooke’s Law:    

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