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

2. Chapter X: Sheet Forming (Membrane Theory) 2 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. Principal radii of curvature: meridian curve C meridian plane center of curvatures hoop plane

3. Chapter X: Sheet Forming (Membrane Theory) 3 Examples Examples: Surface BC: Surface AB:

4. Chapter X: Sheet Forming (Membrane Theory) 4 Basic Assumption 2 For thin plastically deforming shells, the bending moments are negligible and because of axial symmetry the hoop (   ) and tangential (   ) 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. Normal pressure is assumed small enough as compared to the flow stress of the material.

5. Chapter X: Sheet Forming (Membrane Theory) 5 A Useful Relation A useful geometric relationship:

6. 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

7. 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: Assumption 5: The Tresca flow condition is assumed to be applicable:

8. Chapter X: Sheet Forming (Membrane Theory) 8 I VI V IV III II The Flow Criterion The entity T is also called a force-resultant. Recall:

9. Chapter X: Sheet Forming (Membrane Theory) 9 Static Equilibrium Equations (1) A typical infinitesimal membrane element as a free-body

10. Chapter X: Sheet Forming (Membrane Theory) 10 Static Equilibrium Equations (2) Normal resultant of hoop-forces: Radial resultant in hoop-plane of hoop-forces (in normal direction)

11. Chapter X: Sheet Forming (Membrane Theory) 11 Static Equilibrium Equations (3) Normal resultant of meridian-forces:

12. Chapter X: Sheet Forming (Membrane Theory) 12 Static Equilibrium Equations (4) Equilibrium in normal direction: Cancelling out the term d  d  and pulling p out yields: Or, using the geometric relation given in Slide 02.002:

13. Chapter X: Sheet Forming (Membrane Theory) 13 Static Equilibrium Equations (5) Equilibrium of forces in the tangential plane to the shell surface yields: Expanding: Deleting d  and cancelling terms: Using from Slide 02.005

14. Chapter X: Sheet Forming (Membrane Theory) 14 Static Equilibrium Equations (6) Yields: 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  ) of the shell 2)The external axial force F delivers the boundary stress as:

15. 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: The tool pressure can be found from the static equilibrium equation in normal direction: Remark: No dependency on shape of shell! Remark: Dependency on shape of shell!

16. Chapter X: Sheet Forming (Membrane Theory) 16 Strain State (1) Defining the plane stress by the three principal stress components: We can introduce a stress ratio  : The hydrostatic stress is found by: The deviatoric stress components are:

17. Chapter X: Sheet Forming (Membrane Theory) 17 Strain State (2) The strain increments are given as: Introducing the strain ratio  : From the flow rule: or: Hence:and by volume constancy

18. Chapter X: Sheet Forming (Membrane Theory) 18 Strain State (3) Hence having found T  and T , the stress-ratio  can be determined as: and using this stress-ratio, the strain-ratio  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: Similarly, the equivalent plastic strain increment can be determined as:

19. Chapter X: Sheet Forming (Membrane Theory) 19 Application 1: Hole Expansion (1) Note: 1)Both T  and T  are tensile. 2)At the hole rim we have T    and T  >0, from which we can conclude that T  >    3)Hence: T   T f  4)From tangential equilibrium we find: with T   at r = r i :

20. 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

21. Chapter X: Sheet Forming (Membrane Theory) 21 Application 1: Hole Expansion (3) Remark 3: The case of r i = 0 provides an approximate solution for hydraulic bulging: Since  = 1 we obtain  = 1. Hence: But by definition: So: Or:

22. Chapter X: Sheet Forming (Membrane Theory) 22 Application 1: Hole Expansion (4) Checking the assumption of constant T f :

23. Chapter X: Sheet Forming (Membrane Theory) 23 Application 2: Drawing (1) Note: 1)At the outer rim we have T    and T  T   2)Hence: T  – T  =  T f  3) T  < 0 always 4)From tangential equilibrium we find: with T   at r = r 0 :

24. Chapter X: Sheet Forming (Membrane Theory) 24 Application 2: Drawing (2) Remark 1: The given relations are valid if and only if T  > 0 > T . Remark 2: The stress resultant at the inner boundary is: Remark 3: Note again that these results are independent of the shape of the die!

25. Chapter X: Sheet Forming (Membrane Theory) 25 Application 3: Flaring Note the following: 1)At the bottom outer rim we have T   and T  > 0, from which we can conclude that T  >  > T   2)Hence: T  – T   T f  3) T  > 0 and T  ≤  always! 4)From tangential equilibrium we find: with T   at r = r 0 :