Two-Dimensional Formulation.

Transcription

1 Two-Dimensional Formulation

2 Outline Introduction( 概论 ) Two vs. Three-Dimensional Problems Plane Strain( 平面应变 ) Plane Stress( 平面应力 ) Boundar Conditions( 边界条件 ) Correspondence between Plane Strain and Plane Stress ( 平面应变和平面应力的对应关系 ) Combined Plane Formulations( 平面问题统一体系 ) Anti-Plane Strain( 反平面应变 ) Air Stress Function( 应力函数 ) Polar Coordinate Formulation( 极坐标下的公式体系 )

3 Introduction Three-dimensional elasticit problems are ver difficult to solve. Thus we will first solve a number of two-dimensional problems, and will eplore three different theories: - Plane Strain - Plane Stress - Anti-Plane Strain Since all real elastic structures are three-dimensional, theories set forth here will be approimate models. The nature and accurac of the approimation will depend on problem and loading geometr. The basic theories of plane strain and plane stress represent the fundamental plane problem in elasticit. While these two theories appl to significantl different tpes of two-dimensional bodies, their formulations ield ver similar field equations. 3

4 Two vs. Three Dimensional Problems Three-dimensional Two-dimensional z z Spherical cavit z 4

5 Plane Strain Consider an infinitel long clindrical (prismatic) bod as shown. If the bod 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 u u(, ), v v(, ), w 0. z R 5

6 Plane Strain Field Equations Displacement-strain relation: u v 1 u v w 1 u w 1 v w,,, z 0, z 0, z 0. z z z Constitutive relations: 1 0 z z z z, z z 0, z z 0 E E G G G G = G,, G 1 1 z, E E E 1 1, 1 E E E E z 6

7 Plane Strain Field Equations Equilibrium Equations z z z z z z F F z z Fz Strain Compatibilit 0, 0, 0. F F 0, 0. 7

8 Plane Strain Field Equations Beltrami-Michell Equation: D Constitutive Law: Add to both sides: 1 F F Using Equilibrium on the RHS: 1 1 F F 1 8

9 Plane Strain Field Equations Navier s Equations G u G u v w F 0, z u v w F 0, G v G z u v w z z G w G Fz 0. u v ( ) 0, G u G F u v ( ) 0. G v G F G G u u F 0. 9

10 Eamples of Plane Strain Problems P z z Long clinders under uniform loading Semi-infinite regions under uniform loadings 10

11 Plane Stress Consider the domain bounded two stress free planes z=h, where h is small in comparison to other dimensions in the problem. Since the region is thin in the z-direction, there can be little variation in the stress components σ z, τ z, τ z through the thickness, and thus the will be approimatel zero throughout the entire domain. Finall since the region is thin in the z- direction it can be argued that the other nonzero stresses will have little variation with z. Under these assumptions, the stress field can be simplified as (, ) (, ) (, ) z z z 0 11

12 Plane Stress Field Equations Displacement-strain relation: u 1 1 1, v, w z, u v, v w z, u w z z z z Constitutive relations: z z G z z 0 0, 0 z z z z G E E E E E E 1 1, E E, G z G G G E E, G E E z G G, G E G 1 1

13 Plane Stress Field Equations Equilibrium Equations z z z z z z F F z z Fz Strain Compatibilit 0, 0, 0. F F 0, 0. 13

14 Plane Stress Field Equations Beltrami-Michell Equation: D Constitutive Law: 1 Add 1 to both sides: 1 Using Equilibrium on the F RHS: 1 F 14

15 Plane Stress Field Equations Navier s Equations E u v E u, F 0, E u v E v, F 0. E u v. 1 G 1 u v 0, 1 G u F G G v G 1 u v 1 G 1 u u F 1 0. F 0. 15

16 Eamples of Plane Stress Problems Thin plate with central hole Circular plate under edge loadings 16

17 Plane Elasticit Boundar Value Problem Displacement Boundar Conditions u u (, ), v v (, ) on S b b u S i S o Stress/Traction Boundar Conditions T T (, ) n n T T (, ) n n n ( b) ( b) ( b) n ( b) ( b) ( b) Plane Strain Problem: Determine in-plane displacements, strains and stresses {u, v, ε, ε, ε,,, } in R. Out-of-plane stress z can be determined from in-plane stresses. on S t S = S i + S o Plane Stress Problem: Determine in-plane displacements, strains and stresses {u, v, ε, ε, ε,,, } in R. Out-ofplane strain ε z can be determined from in-plane strains. R 17

18 Correspondence Between Plane Formulations Plane strain and plane stress field equations had identical equilibrium equations and boundar conditions. Navier s equations and compatibilit relations were similar but not identical with differences occurring onl in particular coefficients involving just elastic constants. So perhaps a simple change in elastic moduli would bring one set of relations into an eact match with the corresponding result from the other plane theor. Therefore the solution to one plane problem also ields the solution to the other plane problem through a simple transformation scheme. 18

19 Correspondence Between Plane Formulations E 1, 1 1 E 1, G u F v E 1, E 1, ; E 1 E 1 F F ( ) ; 1 G Plane Strain G u v 0, 1 G u v F 0. 1 ; E 1 1 E 1 E E 1 1 Plane Stress 1 1,, E E 1 ; E E, 1 E, G ; 1 F F ( ) (1 ) ; G 1 u v 0, 1 G u F G 1 u v 0. 1 G v F 19

20 Combined Plane Formulations Define a new material constant κ that is related to ν 3 For plane strain: 3 4 or ; For plane stress: or. 1 1 Constitutive relations: G G ,, G 4 1 G 4 1 G G G 1 3, 1 3, G ; G G 1 0

21 Combined Plane Formulations Beltrami-Michell Equation: 4 F F ( ). 1 Navier s equations G u v 1 3 ; F 0, 1 G v u u v 1 3 ; ; F 0. G 1 G u v 0, 1 G u F G u v 0. 1 G v F G G u u F

22 Anti-Plane Strain An additional plane theor of elasticit called Anti-Plane Strain involves a formulation based on the eistence of onl out-of-plane deformation starting with an assumed displacement field: u v 0, w w(, ). Strains z z 0, 1 w 1 w, z. Equilibrium Equations z z Fz 0, F F 0. Stresses z 0, G, G. z z z z Navier s Equation G w F z 0.

23 Air Stress Function Method Numerous solutions to plane strain and plane stress problems can be determined using an Air Stress Function technique. The method will reduce the general formulation to a single governing equation in terms of a single unknown. The resulting equation is then solvable b several methods of applied mathematics, and thus man analtical solutions to problems of interest can be found. 3

24 Conservative Bod Forces If a force field is capable of being represented as the gradient of a scalar function, it is referred to as conservative: V, V F F, F V z z This terminolog arises from gravitational theor. Consider the work done when moving one particle in a gravitational field. The conservation of energ demands V V V dv d d d dz F d Fd Fz dz z F r C C C C The curl: V F F e e 0 ijk k, j i ijk i kj Conservative force fields are irrotational. The above relation serves as the constraint condition. 4

25 Particular Cases of Conservative Bod Forces Gravitational Loading F F g V g V 0, & 0 Inertial forces due to a constant angular velocit ω a r a, a r F a, F a 1 V V Inertial forces due to rigid-bod accelerations are conservative if and onl if angular velocit is constant. 5

26 Air Stress Function Method Equilibrium equations for plane problems F 0, F 0. In the case of a bod force derivable from a potential function, i.e. a conservative bod force Solution to the homogeneous equations F V, F V V V 0, 0. 6

27 Air Stress Function Method B the theor of differential equations, V A, A, V,, (, ) (, ) V B B, V A, B(, ) (, ) (, ) A(, ), B(, ) V,,, V where = (,) is an arbitrar form called Air s Stress Function. This stress form automaticall satisfies the equilibrium equation. 7

28 Air Stress Function Method Beltrami-Michell Equation F F 4 1 V Plane strain: 3 4, Plane stress:. 1 For harmonic bod force potentials, i.e. gravit This relation is called the biharmonic equation and its solutions are known as biharmonic functions. The governing Air stress function equation is identical for plane strain and plane stress, and is independent of elastic constants. If onl traction BCs are specified for a simpl connected region, the stress field for both cases is also identical. 8

29 Air Stress Function Formulation The plane problem of elasticit can be reduced to a single equation in terms of the Air stress function. Traction boundar conditions would involve the specification of second derivatives of the stress function; however, this condition can be reduced to specification of first order derivatives. T n n n n ( n) T n n n n ( n) The plane problem is then formulated in terms of an Air function with a single governing biharmonic equation. 9,.

30 Polar Coordinate Formulation Strain-Displacement relationship ur 1 u 1 1 ur u u r, ur, r. r r r r r Hooke s Law G G r r, r, r r. G 4 1 G 4 1 G G G 1 3, 1 3 r r r, r G r For plane strain: 3 4 ; For plane stress:. 1 30

31 Polar Coordinate Formulation Equilibrium equations 1 1 r r r F 0, r r r F 0. r r r r r r Beltrami-Michell equation 4 F 1 r Fr F r. 1 r r r Navier s equation G G 1 u u F 0. ur 1 ur 1 ur u u r G ur ur 1 u G F 0, r r r r r r r 1 r r r r u 1 u 1 u ur u G 1 ur ur 1 u G F 0. r r r r r r 1 r r r r 31

32 Air Stress Function in Polar Coordinates 11 V, V, 1. er cos sine sin cos e e Q Q Q Q Q Q Q Q Q Q ij ik jl kl i1 j1 11 i1 j 1 i j1 1 i j 11 Q11Q11 11 Q11Q1 1 Q1Q11 1 Q1Q1 cos sin cos sin Q1Q1 11 Q1Q 1 QQ1 1 QQ sin sin cos cos 1 Q11Q1 11 Q11Q 1 Q1Q1 1 Q1Q sin cos cos sin sin cos r V, V,. r r r r r r r 3

33 Air Stress Function in Polar Coordinates cos sin r r cos cos sin sin cos sin r r r r r r cos cos sincos r r r r sin cos sin sin sin cos r r r r r r sin cos sin cos cos r r r r sin sin cos sin cos sin r r r r r r sin cos sin cos sin r r r r cos sin cos sin cos cos r r r r r r 33

34 Air Stress Function in Polar Coordinates Beltrami-Michell equation 1 4 V V r r r r r r r r 1 r r r r Traction boundar conditions f ( r, ) T n n ( n) r r r r r n r n r r r r r ( n) f ( r, ) T r nr n 1 nr n r r r. The plane problem is then formulated in terms of an Air function with a single governing biharmonic equation. 34, r R S

35 Outline Introduction( 概论 ) Two vs. Three-Dimensional Problems Plane Strain( 平面应变 ) Plane Stress( 平面应力 ) Boundar Conditions( 边界条件 ) Correspondence between Plane Strain and Plane Stress ( 平面应变和平面应力的对应关系 ) Combined Plane Formulations( 平面问题统一体系 ) Anti-Plane Strain( 反平面应变 ) Air Stress Function( 应力函数 ) Polar Coordinate Formulation( 极坐标下的公式体系 ) 35