Sheet metal forming with six components of strain

Transcription

1 Sheet metal forming with six components of strain Julian Allwood Micromechanics Seminar Friday 8 May 2009 Department of Engineering University of Cambridge jma42@cam.ac.uk 1

2 A question from the 1960 s In sheet forming processes, workpieces typically fail by fracture or ductile instability at a localised neck. Can the tendency to form this type of localised neck be approximated as a material property? If so, it would be possible to predict in advance whether a particular process can produce a given part - and to adjust the tooling, part-design or material selection accordingly. (An answer from the 2000 s ) Don t worry, we ve got equations for everything - we ll predict failure with our model. But. Our model will only work if you tell us the boundary conditions in a form that suits us. The perfect model will simulate the process perfectly and make the interior detail visible - but we can simulate the process anyway - just by operating it in real life. The industrial question is always how do I make this component? - so we need a way to capture the insights from the model as knowledge. 2

3 Characterising deformation by surface grids: Keeler and Backhofen (1963) proposed use of an array of circles marked on a sheet prior to processing, so measurement of their deformed (elliptical) shape can be used to predict strain experienced during the process. The measurements are shown on a strain diagram ε 1 Thins Thickens Excluded: ε 1 ~> ε 2 ε 2 ε 1 ε 2 Initial circle on surface Deformed circle 3

4 Characterising deformation by surface grids: Keeler and Backhofen (1963) proposed use of an array of circles marked on a sheet prior to processing, so measurement of their deformed (elliptical) shape can be used to predict strain experienced during the process. The measurements are shown on a strain diagram ε 1 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x A threshold curve is drawn to distinguish failed and unfailed circles, and called the Forming Limit Curve - it is hoped that this is a material property ε 2 4

5 Example - strains in deep drawing: Deep drawing is the most common sheet metal forming process, in which the edges of the workpiece are drawn inwards while a punch descends, with wrinkling avoided by a controlled blank holder: The measurements are shown on a strain diagram ε 1 Thins Thickens ε 2 5

6 Analytical prediction of Forming Limit Curves Marciniak and Kuczynski (1967) developed an analytical prediction of instability - local necking - to predict the forming limit curve. σ 1, ε 1 σ n, ε n σ t, ε t σ 2, ε 2 Within a region (A) of homogenous stress, there exists a groove (B) with slightly reduced thickness, at some angle θ. The forming limit is defined when the groove necks unstably. Key assumptions in M&K analysis: strain paths are linear (principal axes don t rotate) 2D state of plane stress, uniform in A and B Loading in A is proportional and aligned with (non rotating) principal axes:! A A 2 = "! 1 Strain increments in A proportional (Levy Mises): d! A 2 = "d! A 2# $1 1, " = 2 $ # Strain increment controlled by hardening law: d! A = d" A if " A = H! A H # Rotate A variables to B (angle θ):! A n =! A 1 cos 2 " +! A 2 sin 2 " Apply boundary conditions from A to B: f! A n =! B n, f" A nt = " B nt, d# A B t = d# t Remaining unknowns in B: Solve with flow rule in B. Update f and θ Instability defined when effective strain increment in B is 10 times that in A. Minimise with θ n.b. sensitivity to f ( )! A t =! A 1 sin 2 " +! A 2 cos 2 " # A nt =! A A ( 1 $! 2 )sin" cos" (with f = t B t A )! B t, d" B n, d# B nt, d" B 6

7 Analytical prediction of Forming Limit Curves Marciniak and Kuczynski (1967) developed an analytical prediction of instability - local necking - to predict the forming limit curve. ε 1 σ 1, ε 1 σ n, ε n σ 2, ε 2 σ t, ε t Within a region (A) of homogenous stress, there exists a groove (B) with slightly reduced thickness, at some angle θ. The forming limit is defined when the groove necks unstably. ε 2 Key assumptions in M&K analysis: strain paths are linear (principal axes don t rotate) 2D state of plane stress, uniform in A and B 7

8 Forming limits in Sheet Forming 8

9 Forming limit diagrams & Incremental Sheet Forming Iseki (1993) identified increased strains at failure with ISF Most strains are very close to plane strain (ie vertical axis) ε 1 Iseki (1993) ε 2 Do the assumptions of M&K analysis apply, or has something changed? 9

10 Paddle forming Modeling through thickness effects in ISF is arduous - can we find a simplified process with similar effects? If so, we can then model it with fine through-thickness detail. 10

11 Paddle forming Incremental forming with a tool making contact along a short line rather than at a point: Limit shape with axisymmetric punch Limit shape with paddle 11

12 Deformation of a pin initially vertical at r=2mm,θ=0: Tool travel Significant through-thickness shear strain arises in paddleforming, parallel to the direction of tool travel 12

13 Forming limit diagrams revisited What happens if we add a through-thickness shear strain to the analysis of Marciniak and Kuczynski? σ 1, ε 1 σ n, ε n σ t, ε t σ 2, ε 2 Key assumptions in original M&K analysis: strain paths are linear (principal axes don t rotate) so incremental strains can be integrated 2D state of stress, uniform in A and B through thickness strain insignificant Key assumptions in revised M&K analysis: Within a region (A) of homogenous stress, there exists a groove (B) with slightly reduced thickness, at some angle θ. The forming limit is defined when the groove necks unstably. strain paths are linear (principal axes don t rotate) so incremental strains can be integrated 3D state of stress, uniform in A and B Through thickness shear strain exists and is proportional Groove rotation about axis in plane of sheet does not affect relation between stresses in A and B Normal and through thickness stresses same in both regions 13

14 Forming limit diagrams revisited Loading in A is proportional and aligned with (non rotating) principal axes:! A A 2 = "! 1 Strain increments in A proportional (Levy Mises): d! A 2 = "d! A 2# $1 1, " = 2 $ # Strain increment controlled by hardening law: d! A = d" A if " A = H! A H # Rotate A variables to B (angle θ):! A n =! A 1 cos 2 " +! A 2 sin 2 " Apply boundary conditions from A to B: f! A n =! B n, f" A nt = " B nt, d# A B t = d# t Remaining unknowns in B: ( )! A t =! A 1 sin 2 " +! A 2 cos 2 " # A nt =! A A ( 1 $! 2 )sin" cos" (with f = t B t A )! B t, d" B n, d# B nt, d" B # %! A = % % $ % 1 " xy " zx " xy " yy " yz " zx " yz " zz & ( ( ( '( )! A xx ( ) ( )! = T " t #! # T "! = T " #! # T " t d!$ = T " t # d$ # T " d$ = T " # d!$ # T " t A = " )! xx! B zz =! A zz,! B z!x =! A z!x,! B z!y =! A z!y,!d" B!y!y f #! B!x!x =! A!x!x, f #! B!x!y A =!!x!y A = d"!y!y ( where f = t B t A ) Solve with flow rule in B. Update f and θ Instability defined when effective strain increment in B is 10 times that in A. Minimise with θ n.b. sensitivity to f 14

15 Generalised Forming Limit Diagrams The M-K model to predict the FLC can be extended to allow proportional loading with all six components of the symmetric stress tensor, and represented on a GFLD! tt =! 2 2 zx +! yz 15

16 Generalised Forming Limit Diagrams - implications Forming limits can be increased with nonplanar strain states: Add hydrostatic (normal) pressure Apply through-thickness shear 1. Better explanation of existing processes: Any sliding contact with tools may lead to through thickness shear Sensitivity to normal stress 2. New tests required to validate GFLD: Forming limit test near to shear in 2D Tests for various 3D states 3. Opportunity to design new processes: Seek to promote and exploit through thickness strains 16

17 1. From paddle forming to Incremental forming Through thickness strains exist in paddle forming and cause the MK forming limit prediction to rise: does through thickness shear strain occur in Incremental Sheet Forming? 17

18 1. From paddle forming to Incremental forming Through thickness strains exist in paddle forming and cause the MK forming limit prediction to rise: does through thickness shear strain occur in Incremental Sheet Forming? Yes! 18

19 2. Requirement for new forming tests: (a) Tests for formability near to line of pure shear (b) Tests for formability with three dimensional strain Bulge or cup tests? ε 1 ε 2 19

20 2a. Requirement for new forming tests: Test for formability in pure shear 20

21 2b. Linear paddle forming test process: First attempt to create a new test of formability including through thickness shear effects 21

22 2b. New forming tests: Test for formability with any 3D proportional loading 22

23 3. New processes: Male Paddle Forming Female Paddle Forming Rotary peen Forming 23

24 3. Female paddle forming: Female Paddle Forming 24

25 3. Joining with female paddle forming: 25

26 Conclusions: future use of forming limit diagrams If any through thickness shear strain occurs: The principal axes of strain will not be aligned with the sheet Measurement of scribed circles gives log principal surface stretches not principal strains The forming limit depends on the amount of through thickness shear strain - so the forming limit for every point in an incrementally formed product will vary - depending on the through thickness shear But also: If a particular geometry is required but cannot be formed - by ISF or any other process - the process/tool path can be re-designed to increase the through thickness shear - and hence the forming limit. Process design could promote shear strain - many sheet forming processes will not be plane stress We have a new class of paddle forming processes to explore 26

27 Opportunities to collaborate within CUED: We are aiming (1) to develop new processes to meet industrial needs (2) to develop tests and analysis to explain their behaviour. Several apparent overlaps for future collaboration: Continuum Damage Mechanics prediction of forming limits in real processes New industrially useable experimental approach to replace surface grids and track strain evolution during deformation Replacing the forming limit diagram - we know it has only a weak theoretical basis, but it is industrially ubiquitous. We need something that is experimentally useable - but with a better theoretical basis Pseudo-Eulerian modelling of incremental type processes - the stress (and strain rate) fields remain almost stationary relative to the tools, but existing codes do not exploit this. 27

28 References Allwood, J.M. and Shouler, D.R. (2009) Generalised forming limit diagrams showing increased forming limits with non-planar stress states, I. J. Plasticity, 25(7) Shouler, D.R. and Allwood, J.M.(2008) A novel sample design for formability testing in pure shear, ICTP, South Korea, September Jackson, K.P. and Allwood, J.M. (2008) The mechanics of incremental sheet forming, Journal of Materials Processing Technology, 209(3) Allwood, J.M. and Shouler, D.R. (2007) Paddle Forming: a Novel Class of Sheet Metal Forming Processes, Annals of CIRP 56(1) Allwood, J.M., Shouler, D.R. and Tekkaya, A.E. (2007) The increased forming limits of incremental sheet forming processes, Shemet 2007, Palermo 28