GIÁO TRÌNH KHAI THÁC PHẦN mềm TRONG GIA CÔNG KHUÔN mẫu chapter x sheet forming processes

1 Chapter X: Sheet Forming (Membrane Theory) 1 Content of Membrane Analysis • Basic Assumptions • Static Equilibrium Equations • Strain State • Application 1: Hole Expansion • Application 2: Drawing • Application 3: Flaring Chapter X: Sheet Forming (Membrane Theory) 2 P C z Surface Normal O r r q r f f Basic Assumption 1 The sheet metal part is a surface of revolution, so that it is symmetric about a central axis. Also the thickness, the loads and stresses are axisymmetric during forming. q P C z Principal radii of curvature: , r r θ φ meridian curve C center of curvatures sin r r θ φ = hoop plane 2 Chapter X: Sheet Forming (Membrane Theory) 3 Examples Examples: q z C r 0 0 ,r r r θ φ = = ∞ 0 0 , sin R r r r r θ φ φ = + = , sin r r r θ φ φ = = ∞ Surface BC: Surface AB: Chapter X: Sheet Forming (Membrane Theory) 4 q z dq s f s q Basic Assumption 2 For thin plastically deforming shells, the bending moments are negligible and because of axial symmetry the hoop (σ θ ) and tangential (σ φ ) stresses are principal stresses. The stress normal to the surface can be neglected, so that the resulting stress state is the one of plane stress. 1 2 3 0 θ φ σ σ σ σ σ = = = Normal pressure is assumed small enough as compared to the flow stress of the material. 3 Chapter X: Sheet Forming (Membrane Theory) 5 P C Surface Normal f dr P’ 90 o f P C Surface Normal r r f f df r f df f dr Surface Normal P’ A Useful Relation A useful geometric relationship: cos dr r d φ φ φ =  PP r d φ φ ′ =  cos dr PP φ ′ = Chapter X: Sheet Forming (Membrane Theory) 6 Basic Assumption 3 Friction forces are neglected. Only uniform pressure loads normal to the surface (although small enough wrt the flow stress) and uniform edge tensions tangential to the surface are allowed: Pressure Load Edge Tension Load ( )s f 0 z 4 Chapter X: Sheet Forming (Membrane Theory) 7 Basic Assumptions 4 & 5 Assumption 4: (Not used to derive equilibrium equations) Work hardening is compensated by thinning of the sheet, so that the product of flow stress times current thickness is constant: = constant f f t T σ ⋅ = Assumption 5: The Tresca flow condition is assumed to be applicable: s f s q s f0 work- hardening s f T t f f = s T t f f0 0 = =s s f t T t q q = s Chapter X: Sheet Forming (Membrane Theory) 8 T t f f = s T t f f0 0 = =s s f t T t q q = s I VI V IV III II The Flow Criterion Region Stress State Flow Condition I 0 II 0 III 0 IV 0 V 0 VI 0 f f f f f f T T T T T T T T T T T T T T T T T T T T T T T T T T θ φ θ φ θ φ φ θ φ θ φ θ θ θ φ φ θ φ θ φ > > = > > = > > − = > > = − > > = − > > − = : Force per meridian-width in hoop direc tion : Force per hoop-width in meridian direc tion T T θ φ The entity T is also called a force-resultant. max min f T T T = − Recall: 5 Chapter X: Sheet Forming (Membrane Theory) 9 Static Equilibrium Equations (1) A typical infinitesimal membrane element as a free-body dq r T r q f df r+rd ( +d )( +d )T r f r dqT f T r q f df T r f dq z p Chapter X: Sheet Forming (Membrane Theory) 10 r dq hoop-plane T r q f df T r q f df dq/2 meridian plane dq/2 Static Equilibrium Equations (2) ( ) ( ) 2 2 d d T r d T r d θ φ θ φ θ θ φ φ + T r d d θ φ φ θ = Normal resultant of hoop-forces: Radial resultant in hoop-plane of hoop-forces (in normal direction) meridian-plane T r q f d df q df surface normal z f T r q f d d sinf q f r f 6 Chapter X: Sheet Forming (Membrane Theory) 11 Static Equilibrium Equations (3) ( ) ( ) ( ) 2 2 d d T dT r dr d T r d φ φ θ φ φ θ θ + + + = Normal resultant of meridian-forces: T rd d θ φ θ = ( ) ( ) 2 2 d d T rd dT rd φ φ φ φ θ θ + ( ) 2 d T drd φ φ θ + ( ) 2 d dT drd φ φ θ + ( ) 2 d T rd φ φ θ + = Chapter X: Sheet Forming (Membrane Theory) 12 Static Equilibrium Equations (4) Equilibrium in normal direction: ( ) ( ) ( ) ( ) sin 0 p r d r d T r d d T r d d φ θ φ φ φ θ φ θ φ θ φ ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ = Cancelling out the term d φ d θ and pulling p out yields: sin T T p r r φ θ φ φ = + T T p r r φ θ θ φ = + Or, using the geometric relation given in Slide 02.002: 7 Chapter X: Sheet Forming (Membrane Theory) 13 Static Equilibrium Equations (5) Equilibrium of forces in the tangential plane to the shell surface yields: ( ) ( ) ( ) cos 0 T dT r dr d T r d T r d d φ φ φ θ φ θ θ φ θ φ + + − ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅ = T r d φ θ ⋅ ⋅ dT r d T dr d dT dr d T r d φ φ φ φ θ θ θ θ + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ ( ) cos 0T r d d θ φ φ θ φ − ⋅ ⋅ ⋅ ⋅ = ( ) cos 0 dT r T dr dT dr T r d φ φ φ θ φ φ φ ⋅ + ⋅ + ⋅ − ⋅ ⋅ ⋅ = Expanding: Deleting d θ and cancelling terms: cos dr r d φ φ φ = Using from Slide 02.005 Chapter X: Sheet Forming (Membrane Theory) 14 Static Equilibrium Equations (6) 0 dT r T dr dT dr T dr φ φ φ θ ⋅ + ⋅ + ⋅ − ⋅ = Yields: ( ) 0 T T dT dr r θ φ φ − − = Hence: REMARKS: 1) As seen from the last equilibrium equation the stress distribution is a function of radius r only and is independent of the shape (r φ ) of the shell 2) The external axial force F delivers the boundary stress as: ( ) 0 0 0 2 sin F T r φ π φ = 8 Chapter X: Sheet Forming (Membrane Theory) 15 Review of Membrane Model The generalized stress distribution can be obtained from the Tresca flow condition and the static equilibrium equation in the tangential direction: ( ) 0 T T dT dr r θ φ φ − − = The tool pressure can be found from the static equilibrium equation in normal direction: T T p r r φ θ θ φ = + Remark: No dependency on shape of shell! Remark: Dependency on shape of shell! 0 0 0 0 0 0 f f f f f f T T T T T T T T T T T T T T T T T T T T T T T T T T θ φ θ φ θ φ φ θ φ θ φ θ θ θ φ φ θ φ θ φ > > = > > = > > − = > > = − > > = − > > − = Chapter X: Sheet Forming (Membrane Theory) 16 Strain State (1) 2 1 σ α σ = Defining the plane stress by the three principal stress components: 1 2 3 ; and 0 σ σ σ = We can introduce a stress ratio α : The hydrostatic stress is found by: 1 2 3 3 h σ σ σ σ + + = ⇒ ( ) 1 1 1 3 h σ α σ = + The deviatoric stress components are: ( ) 1 1 1 1 2 3 h σ σ σ α σ ′ = − = − ( ) 2 2 1 1 2 1 3 h σ σ σ α σ ′ = − = − ( ) 3 3 1 1 1 3 h σ σ σ α σ ′ = − = − + 9 Chapter X: Sheet Forming (Membrane Theory) 17 Strain State (2) The strain increments are given as: ( ) 1 2 3 1 2 ; and d d d d d ε ε ε ε ε = − + Introducing the strain ratio β : 2 1 d d ε β ε = ( ) 3 1 1 d d ε β ε = − + From the flow rule: 1 2 3 1 2 3 d d d d ε ε ε λ σ σ σ = = = ′ ′ ′ 1 2 1 2 d d ε ε σ σ = ′ ′ or: ( ) ( ) 1 1 1 1 1 1 2 2 1 3 3 d d ε β ε α σ α σ ⋅ = − − Hence: 2 1 2 α β α − = − and 2 1 2 β α β + = + by volume constancy Chapter X: Sheet Forming (Membrane Theory) 18 Strain State (3) Hence having found T θ and T φ , the stress-ratio α can be determined as: T T θ φ α = and using this stress-ratio, the strain-ratio β can be determined. So, knowing one of the strain components, the other components can be derived. Also the equivalent strain and equivalent stress (flow stress) can be determined: ( ) ( ) ( ) 2 2 2 . . 1 2 2 3 3 1 1 2 f v M σ σ σ σ σ σ σ σ   = = − + − + −   ( ) ( ) ( ) 2 2 1 1 1 3 1 2 f σ σ α α σ β β β = − + = + + + Similarly, the equivalent plastic strain increment can be determined as: 2 2 2 1 2 3 2 3 d d d d ε ε ε ε = + + ( ) ( ) ( ) 2 2 1 1 4 3 1 2 1 2d d d ε ε β β ε α α α   = + + = − + −   10 Chapter X: Sheet Forming (Membrane Theory) 19 Application 1: Hole Expansion (1) Note: 1) Both T θ and T φ are tensile. 2) At the hole rim we have T φ = 0 and T θ >0, from which we can conclude that T θ > Τ φ >0. 3) Hence: T θ = T f . 4) From tangential equilibrium we find: ( ) 0 T T dT dr r θ φ φ − − = ( ) 0 f T T dT dr r φ φ − − = f dT dr T T r φ φ = − ( ) ln ln f T T r C φ − − = + with T φ = 0 at r = r i : ( ) 1 f i T T r r φ =  −    f T T θ = Chapter X: Sheet Forming (Membrane Theory) 20 Application 1: Hole Expansion (2) Remark 1: The stress state varies from uniaxial tension at the edge of the hole towards equal biaxial tension at the periphery for large radii Remark 2: As the hole radius approaches zero r i  0, almost the entire shell is in a state of uniform biaxial tension in which f T T T θ φ = ≈ [...]...Application 1: Hole Expansion (3) Remark 3: The case of ri = 0 provides an approximate solution for hydraulic bulging: Since α = 1 we obtain β = 1 Hence: σ f = σθ = σφ εθ = ε φ = − 1 ε t 2 ε = 2εθ = −ε t But by definition: ε t = ln ( t t0 ) So: t = t0 eε t = t0 e −ε T f = σ f ⋅ t = Cε n ⋅ t0 e −ε Or: T f = C ⋅ t0ε n e −ε Chapter X: Sheet Forming (Membrane Theory) 21 Application 1: Hole Expansion (4) Checking... Chapter X: Sheet Forming (Membrane Theory) 23 Application 2: Drawing (2) Remark 1: The given relations are valid if and only if Tφ> 0 > Tθ Tθ = T f ln ( r0 r ) − 1 ≤ 0   ln ( r0 r ) ≤ 1 ( r0 e ) ≤ r ≤ r0 ri ≥ ( r0 e ) Remark 2: The stress resultant at the inner boundary is: (T ) φ i = T f ln ( r0 ri ) Remark 3: Note again that these results are independent of the shape of the die! Chapter X: Sheet. .. eε t = t0 e −ε T f = σ f ⋅ t = Cε n ⋅ t0 e −ε Or: T f = C ⋅ t0ε n e −ε Chapter X: Sheet Forming (Membrane Theory) 21 Application 1: Hole Expansion (4) Checking the assumption of constant Tf: ε Chapter X: Sheet Forming (Membrane Theory) 22 11 Application 2: Drawing (1) Note: 1) At the outer rim we have Tφ = 0 and Tθ < 0, from which we can conclude that Tφ > Tθ 2) Hence: Tφ – Tθ = Tf 3) Tθ < 0 always... always! 4) From tangential equilibrium we find: dTφ dr − (T θ r dTφ Tf with Tφ = 0 − Tφ ) at r = r0: = =0 dr r Tφ = T f ln ( r r0 ) dTφ dr − Tf r =0 Tφ = T f ln r + C Tθ = T f ln ( r r 0 ) + 1   Chapter X: Sheet Forming (Membrane Theory) 25 13 ... r0 ri ≥ ( r0 e ) Remark 2: The stress resultant at the inner boundary is: (T ) φ i = T f ln ( r0 ri ) Remark 3: Note again that these results are independent of the shape of the die! Chapter X: Sheet Forming (Membrane Theory) 24 12 Application 3: Flaring Note the following: 1) At the bottom outer rim we have Tφ = 0 and Tθ > 0, from which we can conclude that Tθ > 0 > Tφ 2) Hence: Tθ – Tφ = Tf 3) Tθ . ⋅ Or: 0 n f T C t e ε ε − = ⋅ Chapter X: Sheet Forming (Membrane Theory) 22 Application 1: Hole Expansion (4) Checking the assumption of constant T f : ε 12 Chapter X: Sheet Forming (Membrane Theory) 23 Application. T θ = Chapter X: Sheet Forming (Membrane Theory) 20 Application 1: Hole Expansion (2) Remark 1: The stress state varies from uniaxial tension at the edge of the hole towards equal biaxial tension. uniform biaxial tension in which f T T T θ φ = ≈ 11 Chapter X: Sheet Forming (Membrane Theory) 21 Application 1: Hole Expansion (3) Remark 3: The case of r i = 0 provides an approximate solution