Bulk Metal Forming II

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1 Bulk Metal Forming II Simulation Techniques in Manufacturing Technology Lecture 2 Laboratory for Machine Tools and Production Engineering Chair of Manufacturing Technology Prof. Dr.-Ing. Dr.-Ing. E.h. Dr. h.c. Dr. h.c. F. Klocke

2 Lecture objectives Overview of analytic and numeric methods in process modelling Introduction and fundamentals in Finite Element Modelling Examples of process simulations in bulk forming Overview of damage mechanisms and damage modelling in bulk forming Future developments in bulk forming simulation Seite 1

3 Outline 1 Fundamentals of FEM-Simulations 2 Simulation of bulk forming 3 Examples 4 Future Developments 5 Summary Seite 2

4 Stress conditions with corresponding Mohr's stress circles Uniaxial Biaxial Triaxial Seite 3

5 Start of yielding Seite 4

6 Yield stress and yield criterion F F F σ t σ z σ r σ r σ t Real process (multiaxial) Determination of flow curves (uniaxial) σ z Assumption for plastic flow (v. Mises) σ V = k f = ( σ σ ) + ( σ σ ) + ( σ ) σ 2 k f k f Yield stress σ V σ V Equivalent stress Yield criterion σ V = k f Seite 5

7 Comparison of the yield criterions according von Mises and Tresca τ σ von Mises V = k f = ( σ σ ) + ( σ σ ) + ( σ ) σ 2 2 k f σ 1 σ 2 =0 von Mises σ 3 σ 2 σ 1 Tresca Yield begin: w deviatoric reaches critical value τ τ max Tresca σ V = σ1 σ 3 = 2τ max = k f -σ 3 -k f k σ f 3 σ 2 σ 3 σ 1 Demonstration of the two yield criterions in a yield locus diagram for the state of plane stress. Yield begin: shear stress τ max reaches shear yield k -k f -σ 1 Seite 6

8 Levy-Mises flow rule Elastic Hooke s law: Mathematical dependence between stress and strain. 1 ε 1 = σ 1 E Levy-Mises flow rule: Plastic Mathematical dependence between yield stress and strain rate. The strain rate tensor and the deviatoric stress tensor are proportional to each other (λ = proportionality factor). Flow rule: d ϕ = dλ( σ σ ) dϕ2 = dλ( σ 2 σ m ) dϕ dλ( σ σ ) 1 1 m 3 = 3 m stress σ ε σ yield stress k f Alternative form (division by dt): & & λ( σ ) & ϕ & λ( σ ) & ϕ & λ( σ ) ϕ 1 = 1 & λ = f (, & ϕ) k f σ m 2 = 2 σ m 3 = 3 Proportionality factor λ (not constant): σ m Out of flow rule and v. Mises yield criterion follows: strain ε el plastic strain φ V (& ϕ + & ϕ & ϕ ) & 1 3 λ = k f 3 Seite 7

9 Calculation methods in plasticity theory elementary plasticity theory plasticity theory analytical methods graphical-, empirical-, analytical method numerical methods energy method stripe model disk model strict solution slip field calculation viscoplasticity upper and lower bound method error compensation method tube model finite element method based on simplifications based on theory closed solution approximate solution Seite 8

10 Analytical calculation methods with the stripe-, disk- and tube model stripe model strip tube model tube disk model disk Seite 9

11 Disk model Example: drawing of rods or wire α pµ p die Equilibrium condition in drawing direction: dσ z σ z + A + p α da ( 1 + µ cot ) = 0 A α 0 A 1 σ z + dσ z areas: A + da Assumptions: Source: Pawelski, O. (1976) σ z A v z (r) = const. v z (z) const. workpiece F Yield criterion according to Tresca with p (compressive stress) and σ z (tensile stress) 0: p + σ z = k f Elimination of p results in a differential equation of 1st order: dσ z µ cot α kf ( 1+ µ cot α ) σ z + = 0 da A A σ z + f x σ + g x = ( ) ( ) 0 z Seite 10

12 Viscoplasticity in bulk metal forming r die workpiece at starting point I II v Wz A A workpiece sr v = t z I II s 0 v r sr t = und v z = s t z Characterization of the velocity field by measurement of the temporary change of markers. No information about the stress distribution. Source: Lange, K. (1990) Seite 11

13 Principles of metal forming: Material Laws Low accuracy / low calculation time σ σ Elastic ϕ ϕ σ Elastoplastic with strain hardening σ ϕ Ideal-plastic ϕ Plastic with strain hardening Modelling of the material behavior based on mathematical material laws. Deduction of the modelling parameters from experimental data: Elastic material behavior: Young s Modulus, Poisson s ratio, elastic anisotropy Plastic material behavior: flow curve, strain hardening parameter, plastic anisotropy Usage of ideal-plastic material models are mostly sufficient for bulk forming. Usage of elastoplastic material models in sheet metal forming simulations. High accuracy / high calculation time Seite 12

14 Verification of the material law h 0 = 45 mm x 2 h 0 = 45 mm Yield stress k f / MPa x 2 x 1 x = ϕ = ϕ = ϕ 0,1sec 1sec 10 sec Force F St / kn Experimental Plastic, isotropic τ Elastic-plastic, isotropic Elastic-plastic, kinematic 0 0,5 1 1,5 2 2, Reib = f( σ n ) Plastic strain φ / - Stroke x St / mm Seite 13

15 Friction laws in metal forming τ R Coulomb friction Reality Shear friction Coulomb friction: τ Shear friction: τ R = µ R σ N = m k with k = k f 3 τ R Orowan σ N Wanheim and Bay developed a more general friction law with a continuous transition of the two friction laws mentioned above: τ σ N R = f α k Normal stress Reality Shaw / Wanheim und Bay τ R Shear stress µ,m Friction coefficients f Friction factor α Ratio of the contact areas σ N Seite 14

16 Outline 1 Fundamentals of FEM-Simulations 2 Simulation of bulk forming 3 Examples 4 Future Developments 5 Summary Seite 15

17 Idealization of the workpiece s geometry Geometry simplification Idealization of the workpiece s geometry: Simplification of the original geometry by neglecting unimportant details Assumption of constant wall thickness Making use of symmetry axes and planes Making use of rotational symmetry: Reducing 3D- Models to 2D-Models Advantages: Reduction of complexity Reduction of calculation time Disadvantages: Less simulation accuracy Greater deviation from reality 2D Symmetries Source: ABAQUS User s Manuals Seite 16

18 Type of elements Dimension Element type (load) Geometry 1D Rod (strain) 2D Beam (strain, bending) Membrane (2-dim. strain) Shell (strain, bending) 3D Volume element (strain) Tetrahedron Pentahedron Pyramid Hexahedron Seite 17

19 Continuum discretization in Finite Element Method F punch punch workpiece die Lagrange: Nodes and elements move with the material Remeshing is necessary when elements distort Suitable for simulations of instationary processes Lagrange Euler Eulerian: Nodes and elements stay fixed in space Material flows through a stationary mesh No remeshing necessary Suitable for simulation of stationary processes and fluid dynamics simulations ALE (Arbitrary Lagrangian Eulerian): Combination of Lagrange and Eulerian Models Reduction of mesh distortion by permitting material flow through the mesh but only within the object boundary shape Suitable for stationary processes only ALE Seite 18

20 Mesh optimization Methods of mesh optimization: Local und global mesh smoothing Local und global remeshing Without smoothing Advantages of mesh smoothing: Fast calculation time Small loss of accuracy Advantages of remeshing: No great element distortion Reduction of calculation time by increasing element size in uncritical areas With smoothing An ideal mesh is decisive for the quality level of results and fast convergence of FE-simulations Seite 19

21 Cup-extrusions process Semi-finished product Setup Cup Seite 20

22 Cup-extrusions process: effective strain, effective strain rate Effective strain φ V Effective strain rate dφ V /dt φ V 0.6 dφ V /dt Seite 21

23 Cup-extrusions process: mean stress, effective stress Mean stress σ m Effective stress σ V σ m, MPa 0.6 σ V, MPa Seite 22

24 Cup-extrusions process: axial and radial stress Axial stress σ z Radial stress σ r σ z, MPa 0.6 σ r, MPa Seite 23

25 Cup-extrusions process: damage and axial velocity Damage D 0, Axial velocity v z D 0.6 v z, mm sec Seite 24

26 Macromechanical damage criteria Cylinder specimen Collar specimen Crack D max = 1,17 Crack D max = 0,483 D max = 1,37 Crack Crack D max = 0,334 Seite 25 Norm. Cockroft & Latham Effective strain h0 = 45 mm h0 = 43 mm Damage criterion hcrack = 8,2 mm hcrack = 22,5 mm Forming process

27 Micromechanical damage criteria Damage criterion Forming process Cylinder specimen Crack Collar specimen Crack Mc Clintock Crack D max = 0,754 Crack D max = 0,73 Mod. Rice & Tracey Crack D max = 0,522 Crack D max = 0,424 Seite 26

28 Cause of material separation Void nucleation starts at inclusions Void nucleation Void growth Void coalescence Mechanisms of void nucleation: σ Z Fracture or decohesion of inclusions Nucleation due to decohesion at the grain border Initiation of cracks at existing voids σ Z Tensile test Crush test Decohesion Void Inclusions of MnS 3 µm 5 µm Seite 27

29 Forming history - experiment 4 Point Bending 4 Punkt Biegeversuch mit Umstülpung Experiment Versuchsergebnisse Die Matrize (1. Umformung) Phase 1 Step 1 C B A Phase 2 Step 2 (2. Umformung) B A GegenCounter stempel punch C A B σ1 A 0 B 0 C 0 Area Bereich σ1 A >0 B <0 C >0 Area Bereich σ1 A >0 B >0 C <0 Thickness 8 mm Blechdicke 8mm 200µm 200µm Übergang Transition C Sheet Blech Area Bereich 200µm Bereich Area BB Blank holder Niederhalter Bereich Area A A Thickness 1010mm mm Blechdicke Punch Stempel Seite 28

30 Forming history - FEM Forming history Area C zeitlicher Verlauf von σ und ϕ - Bereich C Spannung σ 1, MPa / MPa σ ,5 0,4 0,3 0,2 0, time t, s 6 0 Zeit t / σ 1 Forming history Area B Spannung σ 1, MPa / MPa σ 1 1 v ϕ v 1. Umformung Step 1 2. Umformung Step 2 0,6 0,4 0, Zeit time t t, / s σ 1 ϕ v 1. Umformung Step 1 2. Umformung Step 2 Umformgrad ϕ v / - zeitlicher Verlauf von σ und ϕ - Bereich B 1 v Strain, - Umformgrad ϕ v / - Strain, - Spannung / MPa 1. Umformung 2. Umformung Step 1 Step 2 Forming history Area B Stress, MPa σ 1 σ m C B C B Umformgeschichte - Bereich B 0 0,1 0,2 0,3 Umformgrad plastic strain, φ, -/ - 0,8 σ v 2. Umformung Step 2 1. Umformung Step 1 ϕ v Seite 29

31 Outline 1 Fundamentals of FEM-Simulations 2 Simulation of bulk forming 3 Examples 4 Future Developments 5 Summary Seite 30

32 Examples FEM-3D Geared shaft Fender Joint Seite 31

33 Examples FEM-3D DEFORM: rotary forging - mesh Seite 32

34 Examples FEM-3D DEFORM: rotary forging - nodal velocities Seite 33

35 Examples FEM-3D DEFORM: rotary forging - strain Seite 34

36 Examples FEM-3D DEFORM: rotary forging - strain rate Seite 35

37 Examples FEM-3D DEFORM: radial forging cross joint, mesh Seite 36

38 Examples FEM-3D DEFORM: radial forging cross joint, strain rate Seite 37

39 Examples FEM-3D ABAQUS: rolling of a slab temperature plastic strain Source: LSTC equivalent stress Seite 38

40 Outline 1 Fundamentals of FEM-Simulations 2 Simulation of bulk forming 3 Examples 4 Future Developments 5 Summary Seite 39

41 Future developments: improving prediction of ductile fracture Fundamental research Experiments with different geometries Visual identification of void nucleation Results: Location and time of first crack initiation Simulation of experiments Forming history Database with different forming histories Application in the industry Training the neural network Simulation of process Identification of critical locations in workpiece Forming history Using the trained neural network Determination of formability Seite 40

42 Ductile fracture prediction: implementation in DEFORM Serial transportation of DATA to USRUPD Nodes, Elements Saving data in memory DATAMANAGER n Received all data y Read point to be tracked from file and identify the element that contains this point Calculation of forming history Read stress rate, Calculate σ 1, σ 3, σ V and Θ and save values Calculate movement of tracking point and save coordinates for next step Continue simulation Seite 41

43 Artificial neural networks: test for formability Analogue experiments: Compression Other analogue experiments Indentation test Lateral extrusion Tensile test Fine blanking Bending Seite 42

44 Relevant mechanical parameters for the neural network Analogue experiment Simulation Interpretation stress Forming history plastic strain σ 1 : Consideration of void growth σ m : Consideration of multi axis character φ V : Consideration of equivalent strain α: Consideration of load direction σ V : Consideration of stress tensor Seite 43

45 Design of artificial neural networks Input Processing Output Real neuronal connections Dendrites Cell body Excitation Axon Musclefiber Connection Synapsis Artificial neural networks Source entries Target entries Input - Layer Hidden - Layer Output - Layer Seite 44

46 Artificial neural networks: damage prediction Damage < 100 % Damage > 100 % D ANN D ANN : Damage parameter ANN D C&L : Damage parameter Cockroft & Latham Seite 45

47 Future developments: Problem orientated metal forming simulation Simulation input Simulation output Problem Question to answer Geometry Material properties Interface conditions Simulation control Meshing parameters Material model Forces Velocity fields Strains Strain rates Stresses Temperatures Initial rod diameter Final rod diameter Workpiece and extrusion length Geometry of die Standard die Free geometry Punch velocity Does the die break? Does ductile fracture appear? Is annealing needed for recrystaization? Does the process work? Material How can I improve the process? Current status Simulation tools of the future Seite 46

48 Available programs on the market Seite 47

49 Outline 1 Fundamentals of FEM-Simulations 2 Simulation of bulk forming 3 Examples 4 Future Developments 5 Summary Seite 48

50 Summary -σ 3 k f σ 1 -k f k σ f 3 -k f σ 2 =0 von Mises Tresca A α 0 A 1 σ z + dσ z areas: A + da α pµ p σ z A workpiece die F Yield criteria and yield laws in metal forming Basic understanding of bulk forming simulation input variables, simulation and output parameters -σ 1 Assumptions: v z (r) = const. v z (z) const. Damage criteria and methods for damage prediction Seite 49

51 Questions Outline the Mohr`s stress circles with uni-, bi- and triaxial compressive stress conditions. Explain the differences between the yield criterion of von Mises and Tresca. Why are the two conventional friction models (shear and coulomb) not accurate enough for the description of forging processes? What are typical results of a bulk forming simulation? Explain the approach to damage and crack prediction? Seite 50

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