Numerical and Experimental Studies on Thermoforming Process. Sogang University

Transcription

1 Numerical and Experimental Studies on Thermoforming Process

2 Thermoforming Process Hot plate Atmosphere Seal Mold Air on Air on Vacuum or atmosphere

3 Introduction Thermoforming Process Advantage Low forming pressure Low cost of mold design Short cycle time Disadvantage Large deformation of material Highly non-linear behavior of material Hard finding of optimal processing conditions

4 Objective of this research Rheological Properties of Polymer Development of Simulation Algorithm Thermoforming Experiment Finite element formulation with Hyperelastic material model Simulation technique : membrane analysis and 3-Dim. analysis Processing variable : temperature, thickness, plug geometry, boundary condition Better understand of Thermoforming Process

5 Scope of this Research Finite Element Formulation Total Lagrangian formulation Hyperelastic model equation Axi-symmetric, 3-dimensional geometric analysis Simulation Results and Discussion Difference between membrane analysis and 3-Dim. analysis Effects of point and wall boundary conditions Effects of initial sheet thickness Effects of initial sheet temperature Stress analysis Effects of plug assist and find optimum plug geometry Non-isothermal analysis Rheological Properties of ABS resins Thermoforming Experiments Various sheet temperatures, plug depths Conclusions

6 Previous Research Throne, William(1970), Tadmor(1979) : vacuum forming analysis without material behavior Oden, Sato(1970) : First finite element solution with neo-hookian model Schmidt and Carley(1975) : axisymmetric bubble simulation and experiment DeLorenzi, et al(1987,1991) : Complex geometry thermoforming simulation Mooney-Rivlin and Ogden model Song, Vlachopoulos(1991,1992) : Simulation without membrane analysis for simple geometry, plug assisted forming analysis Shrivastava(1993), Laroche(1995), Debbaut(1997) : Simulation with viscoelastic material model

7 Modeling Material Model : Hyper-elastic model 2-term Mooney-Rivlin Model 5-term Mooney-Rivlin Model W A Ogden Model WA (I1 3) A01(I (I1 3) A01(I2 3) A11(I1-3)(I2-3) A20(I1-3) A30(I1-3) 3 μ W α λ Simulation Technique : Equilibrium Equation with Newton-Raphson Technique n1 n n [λ α 1 n α 2 n 3) λ α 3 n 3] Membrane Approximation Analysis : possible to relatively thin sheet Full 3-dimensional Analysis : extend to relatively thick sheet multi-layer sheet

8 Finite Element Formulation Stress and deformation Tensor t t t t t t t t t t X x x x x x x x x x x x x x x x x x x Displacement gradient tensor t t t t t T S X X 2nd Piola-Kirchhoff stress tensor t t C I ( ) Green-Lagrange strain tensor t t T t C X X Cauchy-Green deformation tensor

9 Small deformation Large deformation 1 2 Total Potential Energy ij V s edv f uds ij S i 1 2 t t t t t t ij eijd t t V t t s t t f u i d S t t S V Governing Equation ij edv ij V f S s u ds 0 i t t t t t ij ed t t ij V t t V t f s u d tt S t t i 0 S

10 Finite Element Equilibrium Equation Small deformation KU K R R 0 T B CBdV V S T s s H f ds KU tt tt R 0 t t t t t ( K K ) U R F L Large deformation NL F tt tt Linearization R tt t t t T t t t K L B L C B L d V t t V t t T t t t t K NL BNL BNL d V t t V t t T t t t F B L d V t t V R H f d S tt s tt s tt t t S T F 0 t F F t U U

11 Total Lagrangian Formulation t t t t t ( K K ) U R F 0 L 0 NL 0 t t T t 0 0KL 0 0BL 0C 0BL d V V t t T t t 0 0KNL 0 0BNL 0S 0BNL d V V t t T t 0 0F 0 0B L 0S d V V R H f d S tt s tt s tt t t S T

12 Flowchart of the Algorithm Set X = 0, p = 0 p = p + p Calculate g, G, S, C Form stiffness matrix and load vector t t tδt ( K K ) U R F 0 L 0 NL Solve for u i = i + 1 t 0 No Iteration convergence u 2 t+t u 2 < tol Yes No u n+1 = u n + u p > desired p stop Yes

13 Hyper-elastic Model Equation Strain energy function i W Aij ( I1 3) ( I2 3) i 0 j 0 j 3-D (incomp. cond. with penalty method) W A ( I 3) A ( I 3) W W W 1 2 penalty G ( I3) penalty W ( 2A 4A ) G( I ) correction 1 GI ( 3) ln( I3) correction 0 1 Membrane approximation W A ( I 3) A ( I 3) S ij W ij W 2 C ij 2 W W S11 2 ( 1 C 22C 33 ) ( C 22 ) I 1 I 2 2 W W S 22 2 ( 1 C11C 33 ) ( C 11 ) I 1 I 2 2 W 2 W S12 2C C [ C12 C33 C21( C11C22)] I I 1 2

14 Thermoforming Simulation for Axi-symmetric System

15 Infinitely long cylinder geometry z r Fig. Finite Element representation of an infinitely long cylinder.

16 Infinitely long cylinder geometry exact solution FEM results Internal pressure [MPa] exact solution FEM results Stress [MPa] t 33 t 22 t Displacement of interior node [cm] Undeformed radial distance [cm] Fig. Comparison of FEM results for inflation of a thick tubular sheet with exact solution. Fig. Three principal stresses at P=0.8837MPa internal pressure.

17 Internal pressure [KPa] Treloar's experiment Mooney-Rivlin (2 term) Mooney-Rivlin (5 term) Ogden model (6 term) Extension ratio Comparison of various material models [Treloar s, 1944]

18 1.5 p = 47 KPa membrane app. 1.0 p = 9.8 KPa Height [cm] p = 39.2 KPa p = 19.6 KPa Thickness (h/h o ) p = 19.6 KPa p = 39.2 KPa 0.3 p = 9.8 KPa 0.2 p = 47 KPa Radial distance [cm] w/o membrane approximation w/ membrane approximation Radial distance (r/r o ) Simple supported end boundary condition

19 1.5 membrane app. 1.0 p = 9.8 KPa 1.2 p = 47 KPa 0.8 Height [cm] p = 39.2 KPa p = 19.6 KPa Thickness (h/h o ) p = 19.6 KPa p = 39.2 KPa 0.3 p = 9.8 KPa 0.2 p = 47 KPa Radial distance [cm] w/o membrane approximation w/ membrane approximation Radial distance (r/r o ) Clamped end boundary condition

20 Loading pressure [KPa] w/o membrane app. (simple supported end) w/o membrane app. (clamped end) w/ membrane app. AR = 10 AR = 20 AR = 30.5 AR = Pole height, u/r o Fig. Pole height vs. loading pressure for various aspect ratio.

21 Pole thickness, 1-h p /h o AR = 100 AR = 30.5 AR = 20 AR = Pole height, u/r o Fig. Pole height vs. pole thickness for different aspect ratio.

22 Pole thickness, 1-h p /h o Lai & Holt's exp. w/ membrane app. w/o membrane app. (simple supported end) w/o membrane app. (clamped end) Pole height, u/r o Fig. Comparison with experimental results [ Lai and Holt s, 1975 ]

23 Thickness,h/h o soft sheet two layer sheet stiff sheet Stiff sheet A10=0.744, A01=0.271 Mpa Soft sheet A10=0.406, A01=0. Mpa 0.3 U/r o = Radial distance,r/r o Co-extruded composite sheet inflation

24 Mold boundary z z (a) x y (b) x y z z (c) x y (d) x y Mold height : 6cm Mold radius : 9cm Sheet thickness : cm Material : modified PPO

25 Non-isothermal thickness distribution stress(psi) modified PPO 300 o F 335 o F 380 o F Thickness (h/h o ) isothermal sheet non-isothermal sheet a b c d e a : 160 o C b : 155 o C c : 150 o C d : 160 o C e : 170 o C stretch ratio Radial distance (r/r o ) [delorenzi and Nied, 1991] E 6( A10 A01) k exp T

26 Commercial food package

27 Thickness distribution relative thickness a10, a01 1.2, 0 1.5, 0 1.8, 0 kg f /cm 2 relative thickness a10, a01 1.2, , , 0.8 kg f /cm (a) radius [cm] radius [cm]

28 Thickness distribution relative thickness a10, a01 1.2, 0 1.2, , , 0.8 kg f /cm 2 relative thickness a10, a01 1.8, 0 1.8, , , 0.8 kg f /cm radius [cm] radius [cm]

29 Multi-layer thickness distribution st layer a10=3, a01=0. 2nd layer a10=2, a01=0.5 3rd layer a10=1, a01=1 relative thickness radius [cm]

30 Axi-symmetric Plug Assisted Forming (a) (b) Sheet radius : 20cm Sheet thickness : 1cm Plug radius : 10cm Plug height : 20cm (c) (d) Fig. Consecutive feature of axi-symmetric plug assisted forming process. (a),(b) : stretch step with plug, (c),(d) : air inflation step after plug assisted.

31 Standard Deviation of Thickness Standard deviation Mean thickness Variation factor (standard deviation) = 1 n xi = sheet thickness of node i n = total node number x 1 n i 1 x i n n 1 i1 ( x i x) Plug height [cm] Plug length [cm] Optimum plug geometry Plug height : 10cm Plug radius : 15cm

32 Thermoforming Simulation for 3-dimensional Geometry System

33 Simulation Results and Discussion Confirmation of the Algorithm 20 FEM Simulation Analytic Solution 12 FEM Simulation Analytic Solution Nominal Stess f(kg/cm 2 ) f =2A 10 ( -1/ ) + 2A 01 (1-1/ Nominal Stress f(kgf/cm 2 ) f = 2A 10 ( -1/ ) + 2A 01 ( -1/ (a) Extension Ratio (b) Extension Ratio Fig. Comparison of analytic solution with FEM simulation. (a) Simple extension, (b) Equi-biaxial extension. A 10 =1.85, A 01 =0.05.

34 Free Inflation Behavior z z (a) x y (b) x y z z (c) x y (d) x y Fig. Consecutive features of free inflation and thickness distribution. Initial thickness = 0.2 cm. Membrane approximated algorithm. (a) P = 0.04, (b) P = 0.3, (c) P = 0.39, (d) P = 0.42kgf/cm 2.

35 Thickness Distribution Thickness / initial thickness Membrane approximation 3-Dim. simple supported end 3-Dim. clamped end center Thickness / initial thickness Membrane approximation 3-Dim. simple supported end 3-Dim. clamped end center (a) Node number (b) Node number Fig. Comparison of thickness distribution along the cross section of sheet (a) initial thickness 0.2 cm, P = kgf/cm 2. (b) initial thickness 1.0 cm, P=0.949 kgf/cm 2.

36 Stress Analysis Plane direction stress [kgf/cm 2 ] Thin gauge sheet Thick gauge sheet center Thickness direction stress [kgf/cm 2 ] Thin gauge sheet Thick gauge sheet center Node number (a) Node number Fig. Comparison of the distribution of stresses along the cross section of sheet for both thick( P = kgf/cm 2 ) and thin(p = kgf/cm 2 ) gauge sheet (a) plane direction stress (b) thickness direction stress. (b)

37 Mold Geometry Fig. Mold Geometry. 25cm x 35cm x 17cm (width x length x height).

38 Initial thickness Shaping on Mold z z (a) x y (b) x y z z (c) x y Fig. Consecutive features of no-slip mold shaping and thickness distribution. Initial thickness = 0.2 cm. 3-dimensional algorithm with simple supported end b.c. (a) P = , (b) P = , (c) P = , (d) P = kgf/cm 2 (d) x y

39 Thickness Distribution.9.8 Membrane Analysis 3-Dim. Analysis.9.8 Membrane Analysis 3-Dim. Analysis Thickness / initial thickness center Thickness / initial thickness center (a) Node number (b) Node number Fig. Features of slip and no-slip mold shaping and thickness distribution. (a) slip mold boundary condition, (b) no-slip condition.

40 [kgf/cm 2 ] Stress Analysis z z x y x y Fig. Features of slip and no-slip mold shaping and stress distribution. Initial thickness = 0.082cm. 3-dimensional algorithm with simple supproted end b.c. (a) Slip mold b.c. (b) No-slip mold b.c. at P = kgf/cm 2

41 Temperature dependancy Simulation Results and Discussion (kgf / cm 2 ) ABS o C (A 10 =276.8, A 01 =0.246) o C (A 10 =156.4, A 01 =44.11) 160 o C (A 10 =72.59, A 01 =77.8) o C (A 10 =19.32, A 01 =93.06) o C o C 160 o C ln[6(a 10 +A 01 )] (A 10 +A 01 ) = k exp(-t) k = M Pa = / o C o C stretch ratio Fig. Stress vs. stretch data for ABS resin. Curve fit with 2-term Mooney-Rivlin model and displacement rate is 1.05in/sec. [Goldsmith, J., 1987] Temperature( o C) Fig. Temperature dependancy of 2-term Mooney-Rivlin parameter of ABS resin for uniaxial-stretching results.

42 Temperature dependency Initial thickness.5.4 Temp = o C Temp = o C Temp = o C z Thickness [cm] x y Node number Fig. Figures of no-slip mold shaping and thickness distribution of ABS sheets for various temperatures. Initial thickness= 0.5 cm.

43 Temperature dependency.125 Standard deviation of thickness Mean thickness Variation factor (standard deviation) = 1 n 1 n i 1 n n 1 i1 xi = sheet thickness of node i n = total node number x x i ( x i x) Temperature [ o C] Fig. Standard deviation of sheet thickness for various processing temperatures.

44 3-D Plug Assisted Forming (a) (b) (c) (d) Fig. Consecutive feature of plug assisted forming process. (a),(b) : stretch step with plug, (c),(d) : air inflation step after plug assisted.

45 Thickness Distribution 35cm 1.0 plug case.3 plug case.2 plug case.1 direction of measuring thickness 25cm Thickness / initial thickness Node number no plug plug case.1 plug case.2 plug case.3 Fig. Thickness distribution of sheet along the center line for various plug cases.

46 Standard Deviation and Mean Thickness Standard deviation.016 Mean thickness [cm] Plug height [cm] Plug area [cm 2 ] Plug height [cm] Plug area [cm 2 ] Fig. Standard deviation and mean thickness of sheet vs. plug height and plug width.

47 Comparison of Experimental with Simulation Results

48 Materials ABS( Acrylonitrile-Butadiene-Styrene) Rubber contents Acrylonitrile contents Molecular weight ABS-a Low Low High ABS-b High High Low

49 Hot Tensile Test Wc Wo G L D Lo Type V (ASTM D638) Wc : width of narrow section(mm) L : Length of narrow section Lo : Length D : Distance btw. grips G : Gage length Fig. Test specimen for uni-axial extension test. Specimens are injection molded.

50 Hot Tensile Test 60 stress [kg f /cm 2 ] ABS-a ABS-b 130 o C 140 o C 150 o C 160 o C 170 o C stretch rate : 500mm/min stretch ratio

51 Hot Tensile Test 70 ABS-a 20 ABS-b stress [kg f /cm 2 ] o C 140 o C 150 o C 160 o C 170 o C A 10 A stress [kg f /cm 2 ] o C 140 o C 150 o C 160 o C 170 o C A 10 A stretch ratio stretch ratio Curve fitting : f=2a 10 (

52 Mooney-Rivlin Parameter (A 10 +A 01 ) = k exp(-t) k = 695 MPa = / o C 4 6(A 10 +A 01 ) = k exp(-t) k = 2.26 MPa = / o C ln[6(a 10 +A 01 )] 3 ln[6(a 10 +A 01 )] Temperature( o C) Temperature( o C) Fig. Curve fit results with 2-term Mooney-Rivlin model for ABS-a and ABS-b.

53 Oscillatory test 1e+6 Temp : 250 o C 1e+7 Ref. Temp : 210 o C G' [dyne/cm 2 ], * [poise] 1e+5 1e+4 G'(dyne/cm 2 ) 1e+6 1e+5 1e+4 1e+3 ABS-a ABS-b frequency 1e frequency ABS-a ABS-b

54 Elongational viscosity 1e+7 ABS-a ABS-b 1e+7 ABS-a ABS-b elongational viscosity [pa.s] 1e+6 1e+5 1e o C elongational rate = 0.01 elongational viscosity [pa.s] 1e+6 1e+5 1e o C elongational rate = e time [sec] 1e time [sec]

55 Thermoforming Equipment IBM PS/2 Machine control box Plug Heater Sheet Heater Temperature controller Vacuum pump Mold

56 240 Measured Temp. 200 Temperature [ o C] Heating time [sec] Fig. Temperature calibration.

57 Thermoformed ABS sheet 40 cm 32cm 52 cm Initial sheet Thermoformed sheet

58 Thickness distribution ABS-a heating time 30sec heating time 40sec heating time 50sec ABS-b heating time 30sec heating time 40sec heating time 50sec relative thickness relative thickness center distance [cm] center distance [cm] Fig. Thickness distribution of Thermoformed ABS sheet along the center line. Only vacuum applied.

59 Thickness distribution ABS-a ABS-b plug 14 cm ABS-a ABS-b plug 21 cm relative thickness relative thickness center distance [cm] center distance [cm] Fig. Thickness distribution of Thermoformed ABS sheet along the center line. Plug assisted.

60 Simulation Fig. Simulation results. (a) only vacuum applied, (b) plug assisted thermoformed sheet.

61 Thickness distribution experimental data 3-dim. analysis membrane analysis relative thickness center distance [cm] Fig. Comparison of simulated and experimental results. Only vacuum applied.

62 Comparison of 3-dim. and membrane analysis - Free inflation study Sheet thickness 15 mm ( W/T = 26.7 ) 10 mm ( W/T = 40 ) 5 mm ( W/T = 80 ) 2.5 mm ( W/T = 160 ) 1.0 mm ( W/T = 400 ) W L

63 Comparison of 3-dim. and membrane analysis Applied pressure [kg f /cm 2 ] W / T = 26.7 W / T = 40 W / T = 80 W / T = 160 W / T = 400 Applied pressure [kg f /cm 2 ] W / T = 26.7 W / T = 40 W / T = 80 W / T = 160 W / T = loading height [cm] relative thickness

64 Thickness distribution Experimental data Membrane Analysis Experimental data Membrane Analysis relative thickness relative thickness (a) center distance [cm] 0.0 (b) center distance [cm] Fig. Comparison of simulated and experimental results. Plug assisted. (a) plug 14cm, ( b) plug 21cm

65 Sheet temperature measurement Temperature [ o C] sheet length [cm] Temperature [ o C] sheet length [cm] sheet width [cm] sheet width [cm] After 30sec heating of sheet After plug assisted

66 Non-isothermal analysis Experimental data Isothermal analysis Non-isothermal analysis relative thickness center distance [cm] inner corner outer corner 0.00 Fig. Comparison of simulated and experimental results. Only vacuum applied.

67 Non-isothermal analysis Experimental data Isothermal analysis Non-isothermal analysis relative thickness center distance [cm] inner corner outer corner 0.00 Fig. Comparison of simulated and experimental results. Plug assisted with 14cm height.

68 Non-isothermal analysis Experimental data Isothermal analysis Non-isothermal analysis relative thickness center distance [cm] inner corner outer corner 0.00 Fig. Comparison of simulated and experimental results. Plug assisted with 21cm height.

69 Conclusions 1. This developed algorithm is confirmed by showing good agreement of simulation results with exact solutions. 2. The simulation with hyper-elastic material model can predict the experimental results quite well. 3. Point boundary conditions and wall boundary conditions have a considerable effect on the inflation behaviors and stress distributions of sheets. 4. Membrane analysis corresponds well with 3-dimensional analysis over W/T 100 from free inflation test. 5. As the sheet temperature increases, the thickness distribution becomes even.

70 Conclusions 6. Simulation with non-isothermal analysis can improve the accuracy compared with isothermal analysis. 7. Plug assisted forming technique is very useful to optimize the sheet thickness distribution. 8. More temperature sensitive ABS in hot tensile test shows more temperature sensitive thickness distribution in thermoforming experiment.

71 stress A10, A01 kg f /cm 2 1.2, 0 1.2, , , , 0 1.5, , , , 0 1.8, , , strain