Finite Element Simulation on Thermoforming ... - Semantic Scholar

Finite Element Simulation on Thermoforming Acrylic Sheets Using Dynamic Explicit Method

Finite Element Simulation on Thermoforming Acrylic Sheets Using Dynamic Explicit Method Y. Dong, R.J.T. Lin* and D. Bhattacharyya Centre for Advanced Composite Materials (CACM), Department of Mechanical Engineering, University of Auckland, Private Bag 92019, Auckland, New Zealand Received: 1 July 2005 Accepted: 23 September 2005

SUMMARY After optimising the critical material parameters obtained from hot tensile tests, a dynamic explicit software package, PAM-FORM™, is used to simulate the thermoforming process of polymeric sheet. A hyperelastic constitutive law based on the Mooney-Rivlin model has been successfully adopted to carry out the initial simulation on bubble inflation and to identify the material parameters. It has shown a good agreement of the deformation profile with the experimental results. In this paper, further investigations are concentrated on the thickness distribution analysis and the strain states of the bubble inflation along with a comparison to the results from kinematic Grid Strain Analysis (GSA). The numerical simulation of pressure forming of a cup, whose forming mechanisms have been explained reasonably well with the available Williams' analytical solutions, is also presented. For a more academic case, the adaptive mesh refinement scheme has been employed in the simulation of thermoforming a complex-shaped rectangular container to well predict the wall thickness distribution. The final simulation results of the deformation at different stages of forming process and the analyses of final part geometry are also presented.

1. INTRODUCTION

•

Thermoforming, as a common and simple way of forming polymeric parts, is a deformation process of a polymer in the rubbery state above its glass transition temperature1. The contact of the softened polymeric sheets with the mould surface at different stages of thermoforming can affect the final geometry of the component. In this paper, studies are carried out with the help of a dynamic explicit finite element software package, PAM-FORM™, to understand the thermoforming mechanisms of a polymeric sheet and their effects on the final geometry.

•

In general, there are five major steps in a thermoforming process: •

Clamping (to fix the sheet blank);

*Author to whom all correspondence should be addressed. Tel: 64 9 3737599 ext.84543/89780, Fax: +64 9 373 7479. E-mail: [email protected] ©

Rapra Technology Limited, 2006

Polymers & Polymer Composites, Vol. 14, No. 3, 2006

• •

Heating (to heat the sheet up to the forming temperature); Shaping (to form the sheet into a designed cavity); Cooling (to make the products more rigid and retain their final shape); Trimming (to remove the unwanted excessive material).

Amongst the various thermoforming processes2, vacuum forming and pressure forming are the most commonly used manufacturing methods as shown in Figure 1. In the present finite element simulation process, ‘shaping’ is the main step to be investigated with the assumption that the majority of the deformation and thickness change in the sheet occur during this step3.

2. FINITE ELEMENT ANALYSIS 2.1 Hyperelastic Model In the simulation of thermoforming acrylic sheet, it is assumed that the sheet undergoes a nonlinear, large elastic deformation, which merits the

307

Y. Dong, R.J.T. Lin and D. Bhattacharyya

Figure 1. Two basic thermoforming methods: (a) vacuum forming and (b) pressure forming2

B

A Plastic sheet

(a) Seal

Mould Clamp

Vacuum C Formed part Thick areas

A

Thin corners

B

Blow air plate

(b)

Mould

O-ring seal C Air pressure

Vent or vacuum

application of the hyperelastic models4-10 with the path-independent characteristics11. A variety of hyperelastic models and related constitutive relationships, as well as their respective material types and deformation characteristics, are listed in Table 1 which includes both the Mooney-Rivlin and Ogden models adopted in this work. Oden and Sato12 first successfully implemented the hyperelastic Mooney-Rivlin and Neo-Hookean

308

models to numerically analyse the stretching and inflation of rubber-like membranes. Feng and Huang13 investigated the square membrane inflation by applying the minimum potential energy principle, which showed that the results of the numerical analysis was in a good agreement with an earlier experimental work by Yang and Lu14. Furthermore, Allard et al.15 studied both experimentally and analytically, the thermoforming of axisymmetric circular and elliptical membranes



W=C10 (I1-3) +C01 (I2-3)

W=C10(I1-3) +C01 (I2-3)2

MooneyRivlin6

Schidmt7-8

Bidman10

W=C10(I1-3) +C20(I1-3)2 +C30(I1-3)3 +C01(I2-3)

W=C10 (I1-3) +f(I2-3)

RivlinSaunders5

Ogden9

W=C10 (I1-3)

Form

Neo-Hookean4

Name

Uniaxial tension, uniaxial compression and plane strain of thin sheet (pure shear)

Uniaxial and biaxial tensions over the large strain range

ABS PPO and PC-PBT alloy HDPE

8% Sulphur Rubber

Large elastic deformation for biaxial tension with the areal elongation up to 1000 percent

Large elastic strain deformation for uniaxial tension, biaxial tension and simple shear

ABS PMMA PMMA/ABS

HIP polystyrene cellulose acetate butyrate

General formation of Mooney-Rivlin model with the specification of the elastic response of different high polymeric materials subject to large deformation

Small strain deformation for uniaxial and equibiaxial tension of sheets

Deformation Characteristics

Vulcanised Rubber

Natural Rubber

Experimental Materials

Table 1 Constitutive Relationships for hyperelastic materials

Uniaxial tension (λ1=λ, λ2=λ3=λ-1/2) Equibaxial Tension (λ1=λ2=λ, λ3=λ-2)

Constitutive Relationships


309


under both frictionless and sticky conditions using the Neo-Hookean model. The calculated results coincided reasonably well with experimental data on latex rubber membranes, which also suggested that the surface contact of the materials should be intermediate between frictionless and perfectly sticky cases. In addition, Charrier et al.16 analysed the free and constrained inflation of non-axisymmetric membranes with a similar model assuming a nonslip contact condition. A reasonably good agreement was again achieved when comparing with the experimental results. Delorenzi and Nied17 as well as Vlachopoulos et al.18 explicitly examined the simulations of both thermoforming and blow moulding processes with the quasi-static treatment and hypothesis of modelling the polymeric sheet as a rubber-like hyperelastic membrane using Mooney-Rivlin or Ogden model. Bourgin et al.19, on the other hand, have applied the dynamic equations and explicit time integration scheme in sheet stamping process to simulate membrane inflation behaviour with Mooney-Rivlin model, which has shown promising preliminary results. The initial sheet sagging situation is well modelled with the virtual work principle and the assumption of sticky contact between the mould and the membrane. Furthermore, the important application of Marckmann et al.20 by sophisticated dynamic explicit simulation procedure has been successfully undertaken to model the complex box with an insert using Schimdt model7,8 in which the contact algorithm of the non-convex moulds with the initial coarse sheet mesh is resolved under adaptive mesh refinement scheme.

For uniaxial tension (i.e. constitutive relation becomes

), the

(3) where i is the stretch ratio in i-th direction (i = 1,2,3), is the principal stretch ratio (=L/Lo, L0 and L denote the gauge length before and after the test), and 11 are true stress of uniaxial tension and UT Cauchy Stress tensor in the first principal direction respectively. Similarly, the constitutive relation for equibiaxial tension (i.e. 1 = 2 = ) can be expressed as

(4) where BT and 22 are true stress of equibiaxial tension and Cauchy Stress tensor in the second principal direction respectively.

2.1.2 Ogden Model As an alternative approach, Ogden9 developed his strain energy function directly expressed by the principal stretch ratios

(5)

2.1.1 Mooney-Rivlin Model

where i and i are experimentally determined material parameters. These two parameters can be non-integer and/or negative with the only requirement of the positive strain energy function W in Equation 5.

Based on the investigation of vulcanised rubber, Rivlin and Saunders5 constructed a new strain energy function given by

For uniaxial tension, the constitutive relation of Equation (5) may be shown as

W = C10 (I1 - 3)+f (I2 - 3)

(1)

where the function f(I 2 – 3) was determined by experiments. f(I2 – 3) can be assumed to be proportional to (I2 – 3) as the simplest form leading to Mooney function6 W = C10(I1 - 3)+C01 (I2 - 3)

(6) The constitutive relation for equibiaxial tension may be indicated as

(2) (7)

Therefore, Equation 2 is also called Mooney-Rivlin strain energy function or Mooney-Rivlin model.

310



2.2 Governing Equation

2.4 Explicit Time Integration Scheme

In developing the governing equation in finite element formulation for thermoforming simulation, it is generally assumed that during the inflation of membrane, the polymeric sheet can be represented by a two-dimensional continuum. Its thickness is considered as a function of the position along the mid-surface (curvilinear distance). Irrespective of the body force due to gravity, the equilibrium condition is provided by the inertial effects and internal work on the undeformed configuration and the inflating pressure acting on the deformed configuration. Using the Principle of Virtual Work, the condition may be expressed in a hybrid Lagrangian-Eulerian form20-21:

The classical explicit second-order central difference method21 is used to solve equation (9) in which the velocity of the degree of freedom i, , and the correspondent acceleration, , are expressed as

(8) where V0 and δV represent the volume of the undeformed membrane and the boundary surface of the deformed membrane respectively, ρ0 is the mass density and is the acceleration vector. Tensors E and S signify the Green-Lagrange strain tensor and the 2nd Piola-Kirchhoff stress tensor respectively; δu is the virtual displacement vector compatible with the displacement boundary conditions. P is the inflating pressure and n is an external normal vector to the deformed membrane.

2.3 Spatial Discretisation Generally the governing Equation (8) can be discretised using 3-node triangular or 4-node quadrilateral thin shell elements since most thermoformed parts have thin-walled structure and hot polymeric materials behave more like a membrane. In this study, 4-node quadrilateral thin shell elements are exclusively employed, which only deform in their plane and remain flat, helping to save the processing time in analytical integration21. After discretisation and integration on each element surface, the problem can be reduced to a system of ordinary second order differential equations with the inertial effects: (9) in which M is the mass matrix, X(t) is the nodal displacement vector and Fext(t) and Fint(t) are the external and internal nodal force vectors respectively.


(10)

(11) where Δt is a time step. By using the special lump mass method22, the mass matrix M in equation (9) can be approached by a diagonal mass matrix denoted by with the mass conservation. Consequently, equation (9) can be simplified as the following system:

(12) The stability of the central difference method (see Appendix) can be achieved by satisfying the equations (13) and (14) under the undamped and damped motion conditions respectively.

(undamped) (damped)

(13) (14)

where max is the largest natural frequency of the structure, which is determined by the highest frequency of individual element in the finite element mesh and bounds the time step size. ΔtC(Δt ≤ ΔtC) is the critical time step. Computation of the new time step at each cycle is based upon the previous smallest time step among all the elements of the model, namely (15) where is the time step scale factor for the stability of the model (set to 0.9 as default or small value), N is the number of elements.

311


For the i-th shell element time step

(16) where Δti, Li, i and Ei are the time step, the length, the density and the Young’s modulus of the i-th element respectively. For more details of finite element formulations, the readers may refer to the literature23, 24.

2.5 Contact Interface The solution of contact interface problems involves the identification of interactive points on the boundary and the insertion of appropriate conditions to prevent penetration. Initially the polymeric sheet is assumed to be free except the clamping areas, which are subjected to the full constraints (i.e. all 6 degrees of freedom are fixed). When the polymeric sheet comes in contact with the mould surface, the sheet rapidly cools down and becomes stiffer. As a result, the inflation pressure is not sufficient to deform it further. So each point in contact with the mould surface becomes fixed and its displacement is easy to calculate. Assuming tc to be the contact time of a given point K on the sheet and tf to be the final pressure inflation time, the displacement vector VK(t) of point K at current time t can be expressed as20 (17) Therefore position XK(t) of point K is given by (18) where XK' is the corresponding point K′ on the mould surface. The friction between the membrane and the mould surface is assumed to follow the Coulomb friction law22 which gives (19) where is a positive frictional parameter, Tn is the magnitude of the normal traction, and Ts is the tangential traction. Once the magnitude of Ts is at the limit condition, slip occurs with an imposed tangential traction on each surface opposite to the direction of slip and equal to |Tn|. Therefore,

312

when an assumption of “perfect stick” at the contact interface is made, penetration is not allowed between the plastic membrane and the mould as the mould surface in the finite element model is a rigid predefined boundary.

2.6 Adaptive Mesh Refinement The commercial application software PAM-FORMTM is a simulation tool for sheet forming process, which is based on the aforementioned dynamic explicit theories. Using PAM-FORMTM, either or both of the two mesh refinement methods, automatic selective adaptive meshing and uniform remeshing24, the material models can be produced, Figure 2 (a)-(c). Automatic selective mesh refinement is a mesh generating method based on the angle changes between the facets of two neighbouring quadrilateral or triangular thin shell elements. When the angles formed between a shell element and its neighbours exceed the maximum angle criterion (i.e. a default angle of 7°), an element is automatically refined and split into 4 smaller triangular or quadrilateral elements depending on the shape of the element, Figure 2 (d). The change of angles between the elements is regularly monitored during the forming simulation for a timely activation of the automatically selective mesh refinement process. This ensures that the number of elements be maintained at a sufficient level for more accurate simulating results, especially where there is a complicated geometry or high strain gradient. The maximum level of automatic selective mesh refinement (LEVMAX) is denoted to indicate the allowable extent of the remeshing scheme whereas the uniform re-meshing process is a simultaneous refinement of all the elements within the adaptive meshing domain without considering the angle change criterion (both shown in Figure 2 (b) and (c)). However, even though the accuracy of the simulation can be improved by increasing the number of elements, the maximum level of mesh refinement should be user-defined taking into account the required computation time.

3. GRID STRAIN ANALYSIS Grid Strain Analysis (GSA) uses real grids, printed on sheets before forming, and digitised afterwards to generate the actual strain maps and thickness distributions closely fitting the real geometry of formed 3-D parts25. For large strain analysis of continuous fibre reinforced thermoplastic materials, Christie26 employed bicubic Hermite elements27 in GSA that are different from linear and bilinear quadrilateral elements but can handle



Figure 2. Adaptive meshing for shell elements: (a) initial mesh; (b) automatic selective mesh refinement; (c) uniform mesh refinement; (d) maximum angle criterion

(a) Initial mesh

(b) Automatic selective mesh refinement

(c) Uniform mesh refinement

(d) Maximum angle criterion

better the curvature and continuity requirements while remaining computationally efficient. GSA is normally implemented according to the data point mapping technique between the undeformed sheet and the final deformed component. Principal surface strains are presented with pairs of arrows representing the magnitudes and orientations of the principal strains in the plane of the sheets. Thickness strains are calculated from in-plane strains with the assumption of volume constancy and can be displayed as contour plots of percentage strains.


4. SIMULATION OF THERMOFORMING ACRYLIC SHEETS 4.1 Bubble Inflation Comparisons between the simulation and experimental results of bubble deformation profiles validated the acquired material parameters11. The simulation of transparent acrylic sheet was carried out to give a clear idea on thickness distribution by finite element analysis and GSA approaches.

313


Material parameters describing the hyperelastic model and the geometry for bubble inflation simulation have been given in a previous paper11. In the simulation work using PAM-FORMTM, four deformation states with the thickness contour are demonstrated in Figures 3 and 4 (a-d) with two differential pressures 20 kPa and 40 kPa respectively. Arrow diagrams in GSA and numerical simulation work are also illustrated in Figures 5 and 6 (a-c) to analyse the principal surface strains in the hoop and longitudinal directions. Apparently there are two major strain states along the outer surface of the inflated bubbles, which are designated using two different dash lined zones A and B in both cases. Zone A displays the wall thinning based on the plane strain deformation due to the significant hoop strain ε1, which becomes greater with the increase of the inflation pressure from 20kPa to 40 kPa. In particular, the relationship of ε1≥10ε2 in GSA is well

maintained in the central portion of the bubble where ε2 is the longitudinal strain. In zone B where it is the interim area between the inflated bubble and the sheet edge, unbalanced biaxial stretching is quite predominant. However, strain states are quite obscure near the sheet edge where less material flow occurs. Both numerical simulation and GSA results have given the qualitative agreement on the deformation states of the two aforementioned critical zones. Further investigation of the thickness distribution in the central portion of the bubble, as shown in Figures 7 and 8, was carried out to validate the numerical results by comparing them with the experimental measurements using a digital micrometer. Regardless of the measurement errors, uniform thickness distribution occurs in

Figure 3. Four stages of bubble inflation simulation process using transparent acrylic sheet at the forming temperature of 160 °C with the differential pressure 20 kPa

Thickness Thickness 2.51 2.54 2.57 2.59 2.62 2.65 2.68 2.7 2.73

0.276 0.897 1.52 2.14 2.76 3.38 4 4.62 5.24

Z (b)

(a) Thickness Thickness

X

2.47 2.5 2.54 2.57 2.6 2.63 2.66 2.7 2.73

(c)

314

Y

2.42 2.46 2.5 2.53 2.57 2.61 2.65 2.68 2.72

(d)



Figure 4. Four stages of bubble inflation simulation process using transparent acrylic sheet at the forming temperature of 160 °C with the differential pressure 40 kPa

Thickness

z Thickness

0.276 0.897 1.52 2.14 2.76 3.38 4 4.62 5.24

z

2.17 2.23 2.3 2.36 2.43 2.5 2.56 2.63 2.69

Z y

(a)

y

(b) Y z

X

z

Thickness

Thickness

2.12 2.19 2.26 2.34 2.41 2.48 2.55 2.62 2.69

2.07 2.15 2.22 2.3 2.38 2.45 2.53 2.61 2.68

y

(c)

the hoop direction except near the clamped edge. Good agreement of relative thickness in regards to the nodal points is observed in bubble inflation simulation with the differential pressure 20 kPa. Nevertheless, the relative thickness is overestimated with the pressure increasing to 40 kPa, which might be further improved by considering the thermal effects due to the process temperature distribution from the centre sheet to the clamped edge. Both Williams28and Lai et al.29 confirm our present bubble inflation results analytically and experimentally.

4.2 Cup-Forming The blank sheet size of 100 mm × 100 mm was chosen in this cup-forming simulation for both opaque and transparent sheets with nominal thickness of 3 mm and 2.76 mm respectively. Figure 9 shows the symmetric half geometry of the cup mould and its matching meshed model. For the boundary condition, the four edges of the square blank were


y

(d)

assumed to be clamped with the corresponding boundary nodes fully fixed. The material parameters for cup-forming simulation are given in Table 2. Considering the stability of the simulated solution and the optimisation of the final part geometry, the square sheet was subjected to initial uniform fine mesh (66 x 66 elements) to alleviate the problems of large deformation and distortion of the elements. Four different stages of the cup-forming simulation are displayed with their corresponding thickness contours in Figures 10 and 11. Due to the difference of the material characteristics, the two cups showed the different extents of material thinning phenomenon from top to flange. The thickness of the opaque cup ranges from 0.75 mm on the cup ceiling to 2.94 mm at the bottom flange, which represent 75% and 1.7% of material thinning respectively. The thickness of the transparent cup varies from 0.73 mm (73.6%) to 2.6 mm (5.8%). The thicknesses of the clamped areas remain nearly unchanged from their original

315


Figure 5. Schematic of strain deformation states in the two critical zones A and B (forming temperature 160 °C and differential pressure 20 kPa): (a) principal membrane strain states in the overall bubble model; (b) strain states in the enlarged one quarter bubble model; (c) strain states in GSA

Table 2 Material parameters describing Ogden model1 Parameter Opaque Type 3

Transparent Type

1.2 x 10

1.2 x 10-6

Number of integration point

3

3

Thickness (mm)

3

2.76

Membrane hourglass coefficient

0.9

0.9

Out-of-plane hourglass coefficient

0.9

0.9

Rotational hourglass coefficient

0.9

0.9

0.866

0.866

2 -2 –1.14 x 10-5 –4.9 x 10-4

2 -2 –5.698 x 10-5 –8.13 x 10-4

Density (kg/mm )

Transverse shear correction factor Ogden parameters: α1 α2 β1 β2

-6

Note: Maximum inflation pressure =1.4 MPa for the cup-forming simulation (opaque and transparent), maximum differential pressure = 0.2 MPa for the complex shape thermoforming (opaque). Both of the simulations are subject to the isothermal condition with the uniform forming temperature 160 °C 1 Ogden model in PAM-FORMTM establishes a constitutive relationship with a two-term Mooney-Rivlin model: n = 2, β1 = 2C10, α1 = 2, β2 = -2C01 and α2 = -2

316



Figure 6. Schematic of strain deformation states in the two critical zones A and B (forming temperature 160 °C and differential pressure 40 kPa): (a) principal membrane strain states in the overall bubble model; (b) strain states in the enlarged one quarter bubble model; (c) strain states in GSA

thicknesses. In general, the symmetry with respect to the XZ- and YZ-plane is consistently maintained during the simulation. To compare these simulated results with Williams′ analytical solution28, Figures 12 and 13 present the normalised relative wall thickness (t/to) results along the pathways denoted as XY and YZ. the arc length calculated from the bottom rim of the cup is represented by S to define the position, and t, to give the measured wall thickness and original thickness respectively. Identical thickness distribution along the XY and YZ paths further validates the isotropic property of the blank material. Thickness calculations using PAM-FORM™ agree reasonably well with Williams′ analytical solution especially at the bottom portion of the cup for both opaque and transparent acrylic sheets, Figure 9(a).


Nevertheless, at the vertical wall of the cup, when the arc length S is greater than 22.2 mm for both materials, numerical thickness values are overestimated. The difference might have resulted from the assumption of equibiaxial stretching at all times in Williams′analytical solution which could not be entirely achieved in this simulation work despite the same “perfect stick” assumption. The numerical results show a lower relative thickness in the acute corners while slightly higher values are obtained in the ceiling and the vertical wall regions. This is because when the sheet is in contact with the ceiling and the vertical wall areas of the cup mould, the sheet movement against the mould is prohibited and the thickness of sheet membrane at the contact regions can no longer change due to the “perfect stick” assumption. On the other hand,

317


Figure 7. Thickness distribution analysis in the hoop direction of the middle portion for the bubble inflation using transparent acrylic sheet (forming temperature 160 °C, differential pressure 20 kPa and 40 kPa respectively)

Figure 8. Thickness distribution analysis in the longitudinal direction of the middle portion for the bubble inflation using transparent acrylic sheet (forming temperature 160 °C, differential pressure 20 kPa and 40 kPa respectively)

318



Figure 9. Cup-forming female mould mesh (a) and mould geometry (b)

(a)

(b)

Z

X y

Y

Figure 10. Four stages of cup-forming simulation process using opaque acrylic sheet

Thickness 0.3 0.975 1.65 2.332 3 3.67 4.35 5.02 5.7

(a) Thickness 0.9531 1.2 1.45 1.69 1.94 2.18 2.43 2.68 2.92

(c)


Thickness 1.43 1.62 1.81 2.01 2.2 2.39 2.59 2.78 2.97

(b) Thickness 0.7476 1.02 1.3 1.57 1.84 2.12 2.39 2.66 2.94

Z

X Y

(d)

319


Figure 11. Four stages of cup-forming simulation process using transparent acrylic sheet

Thickness

Thickness 1.78 1.91 2.04 2.17 2.3 2.43 2.56 2.69 2.82

0.276 0.897 1.52 2.14 2.76 3.38 4 4.62 5.24

(a)

(b)

Z

X Thickness

Thickness

1.07 1.26 1.45 1.64 1.83 2.02 2.22 2.41 2.6

Y

0.7319 0.9649 1.2 1.43 1.66 1.9 2.13 2.36 2.6

(c)

(d)

Figure 12. Comparison between cup-forming simulation result and Williams′ analytical solution28 for opaque acrylic sheet

1 0.9

Line XZ

Line YZ

Relative thickness t/t0

0.8

Z

X

0.7

Y

0.6 0.5 Cut from line XY 0.4

Cut from line YZ

0.3

Williams' analytical solution28

0.2 0.1 0

0

10

20

30

40

50

60

70

80

Arc length S (mm)

320



Figure 13. Comparison between cup-forming simulation result and Williams′ analytical solution28 for transparent acrylic sheet 1

Relative thickness t/t0

0.9 0.8

Line XY

Line YZ

Z

X 0.7

Y 0.6 0.5

Cut from line XY Cut from line YZ

0.4

Williams' analytical solution28

0.3 0.2 0.1 0

0

10

20

30

40

50

60

70

80

Arc length S (mm)

Figure 14. (a) 3-D computer model of the complex rectangular container; (b) meshed model


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the non-contact part of sheet membrane can still be subject to more stretching in the corner regions of the mould, resulting in further thinning in these areas. The numerical results clearly show that, as expected, the thinnest region of the final deformed cup occurs in the acute corners.

4.3 Thermoforming a Complex Rectangular Container Due to the limitation of the element numbers and mesh size, it has been shown to be very challenging to simulate thermoforming of a component with complex geometry using the quasi-static treatment17,18 or dynamic explicit procedure19-21. However, in order to gain more insight for defining the contact problems in the large deformation and remeshing procedure, a more academic case involving the forming of a relatively complex rectangular container model has been examined. As previously mentioned, the performance of a simulation can be greatly improved once the adaptive mesh refinement scheme is applied. In this part of study, a typical example of vacuum forming a rectangular container, as shown in Figure 14, has been simulated with the scheme of adaptive mesh refinement. This operation was achieved by seeking the geometrical conformity between the final formed part and the mould, as well as understanding the variation of thickness distribution through the numerical analysis. Due to the complexity of the mould geometry, especially around the corners and edges between the flat surfaces, high level (LEVMAX=2) of automatic selective mesh refinement was implemented. The area of the mould requiring special attention was the two-step container configuration, Figure 14 (a). The meshed mould model, Figure 14 (b), shows that the flat surfaces have been assigned the coarser mesh while the finer mesh has been applied to the complicated geometric areas (i.e. corners and edges). The undeformed membrane dimension was 250 mm × 110 mm, which was approximately the size of the open area of the rectangular container with its four edges fully clamped during forming. Opaque acrylic sheet with the thickness of 3mm was used to apply the material properties in this part of work, Table 2. The maximum differential pressure was controlled at 0.2 MPa for 400 ms. Due to the path-independent behaviour of hyperelastic models, a very short mathematical time span at the shaping stage of vacuum forming was selected even though

322

the real thermoforming time from clamping the sheet to demoulding can take several seconds. Four designated deformation states of the process are demonstrated in Figure 15 with the different thickness distribution represented by shaded area. The initial meshed model had 1000 elements, Figure 15(a), and the final one contained 13656 elements, Figure 15(d). During forming, the contours of the deformed sheet along the symmetry planes of XZ and YZ were calculated at various stages to show the conforming process of the acrylic sheet to the container mould, Figure 16. As the sheet membrane inflated from its initial shape, it underwent a period of free inflation between 0 to 88 ms, when the membrane did not come in contact with the cavity wall. It can be observed that due to the complex shape of the mould, the contact of the material with the mould occurred first (t = 88 ms) at the waist part (zone A) of the cavity, where the mould opening area was reduced, rather than the central area of the upper ceiling (zone B). It is also evident that in the two symmetry planes (XZ and YZ), the contacts at both sides of wall took place simultaneously at this instantaneous state, signifying the initiation of contact procedure with the mould surface even though the main part of the membrane was still subject to the free inflation during the time between 88 ms and 100 ms. At t = 100 ms, the membrane began to touch the mould upper ceiling along the symmetric axes XZ and YZ and subsequently the contact area between the membrane and the mould did spread towards the mould corners as the pressure forced the conformation of the material at the ceiling and the vertical walls. During the simulation, it has been observed that the time taken to fill in the cavity (Δt2 = 300 ms) greatly exceeds that of the free inflation period (Δt1 = 100 ms), which underlines the importance of the “fill–in” process during real thermoforming situation as it dictates the final dimensional conformance, even though the initial free inflation can provide the basic shape of the final part. In finite element simulation using dynamic explicit method, the difference of time between the two stages can be further explained due to the contact nature in the process. During the initial inflation process, the sheet underwent the non-contact stage with the mould and the time step calculated by PAM-FORM™ solver was relatively large with the large mesh size. Nevertheless, the non-linear contact problem inevitably triggered the reduction of the time step to avoid the instability of the numerical



Figure 15. Four stages of complex rectangular container forming simulation z z Thickness

Thickness

1.84 1.98 2.11 2.24 2.37 2.5 2.64 2.77 2.9

0.3 0.975 1.65 2.32 3 3.68 4.35 5.03 5.7

y

(a)

(b) z

Thickness

Thickness

y

z

1.35 1.56 1.77 1.97 2.18 2.38 2.59 2.8 3

1.63 1.79 1.96 2.12 2.28 2.44 2.6 2.77 2.93

y

y

(d)

(c)

solution when the sheet began to touch the mould surface. On the other hand, the implementation of the adaptive mesh refinement also increased the CPU calculating time because of the continuous mesh refinement at some critical areas, such as around the vertical wall and corner regions, to guarantee the accuracy of the final solution. Consequently, when running PAM-FORM™, the time step at each cycle was automatically selected based on the stability of the minimum size element. The time for calculation of the deformation for much larger size elements especially those non-refined elements was therefore greatly prolonged by choosing the small time step. Figure 17 presents the final contour of the model and the thickness distribution along the two symmetry planes XZ and YZ. The final shape of the formed container can be consistently produced in the numerical simulation even though the corner regions were not completely filled in. The thickness


distribution curve shows dramatic thinning around the vertical wall areas. This results from the perfect stick assumption; any early contact with the mould surface stops the displacement of the specific sheet portion whereas the non-contacting sheet areas can still experience additional stretching. This phenomenon can be further verified by the fact that the thinnest regions are around the small corners and the bending areas, where the conformance of the material with the mould is not fully complete. On the other hand, the clamped rims are relatively thick, which agrees with the real vacuum forming process. In a commercial manufacturing, serious attention can always be given to support the sharp corners or bending regions of the product to keep them from cracking using a plastic or wooden jig. The information gained from the numerical simulation can facilitate the mould design to take care of those crack-prone regions to avoid any possible forming problems and improve the final product quality.

323


Figure 16. Deformed sheet profiles and material paths in the shaping step along XZ and YZ symmetry planes at the forming temperature 160 °C

(a)

(b)

5. CONCLUSIONS From the analysis carried out in this study, the following conclusions can be drawn: •

324

The thickness distribution analysis in the bubble inflation simulation, has reconfirmed the uniform thickness in the hoop direction of the middle portion of the bubble, which is in agreement with the Williams′ theoretical

•

analysis and Lai′s 2-D free forming experimental work. This finding is explicitly interpreted by the plane strain deformation shown in postdeformation Grid Strain Analysis and well predicted in the numerical simulation with PAM-FORMTM. It has been demonstrated that the finite element simulation of thermoforming process can be successfully performed by using the hyperelastic models not only for simple



Figure 17. Computer calculated thickness distribution along XZ and YZ symmetry planes correspondent to the deformed regions at the bottom surface of the acrylic sheet

(a)

(b)

•

geometry cup forming, but also for the forming of a container with a more complex shape. During the simulation, the “perfect stick” condition can be assumed between the mould and polymeric sheet to obtain satisfying simulation results. Automatic selective mesh refinement has proved to be quite helpful during the simulation of a complex shape and it is also


observed that without the adaptive mesh refinement, the instability of the numerical solution will take place due to the deficiency of element subdivisions. The “large deformation” phenomenon of sheet portion close to vertical wall areas in the finite element simulation is very evident, which can be easily overcome by sufficient refined linear triangular or quadrilateral elements.

325


6. ACKNOWLEDGEMENTS The authors would like to gratefully acknowledge the assistance received from Mr. Miro Duhovic of the Department of Mechanical Engineering, at the University of Auckland and Mr. Allen Chhor, Senior Engineer in Pacific ESI (Engineering Systems International), Australia in dealing with the technical problems and model set-up in the implementation in PAM-FORM™ software. They are also thankful to the Foundation for Research Science and Technology New Zealand for partially supporting the project.

(A7) or (A8) in which .

7. APPENDIX Stability of the Central Difference Method The motion equation (9) can be simplified as

Consider the eigenvalues equation of motion

of A for undamped

(A1) (A9) by replacing (A2)

(A10)

where P(t) is the external nodal force due to pressure and body force load is eliminated. K and C are stiffness and damping matrices.

the requirement | | ≤ 1 leads to (undamped)

The modal matrix of eigenvectors in the modal equation are normalized with the respect to the diagonal mass, stiffness and damping matrices , K, C as below:

(13)

(damped) (14)

8. NOMENCLATURE

(A3) (A4) (A5)

A C

Damping matrix

C

Constant of curve-fitting exponential function for two-term Mooney-Rivlin model (MPa)

Cij

Empirically determined constants of strain energy function (i =0,1,2…M and j =0,1,2..N)

C10

Two-term Mooney-Rivlin model material parameter (MPa)

C01

Two-term Mooney-Rivlin model material parameter (MPa)

E

Green-Lagrange strain tensor

Ei

Young’s modulus of the i-th element

where I, , ξ are the unit matrix, vibration frequency and damping ratio respectively. Hence equation (A1) is rearranged as (A6) Combining the equations (10), (11) and (A6) leads to (A7) in matrix form

326



F

External surface force due to the pressure

Velocity of the degree of freedom i

ext

F (t) External nodal force vector int

F (t)

Internal nodal force vector

I

Unit matrix

Ii

Strain tensor invariants in the i-th directions ( i=1, 2, 3)

K

Stiffness matrix

L Li

Length of the i-th element

M

Mass matrix

Acceleration of the degree of freedom i

Time step scale factor α

Single term Ogden model material parameter

αi

Ogden model material parameter (i=1, 2, 3…)

β

Single term Ogden model material parameter (MPa)

βi

Ogden model material parameter (i=1,2,3…) (MPa)

Diagonal mass matrix

Modal matrix of eigenvectors

N

Number of elements

n

External normal vector

P

Inflating pressure

λ

Principal stretch ratio

P(t)

External nodal force due to pressure

λi

Stretch ratio in the i-th directions (i=1, 2, 3)

nd

Eigenvalues of undamped equation of motion

S

The 2 Piola-Kirchhoff stress tensor

μ

positive frictional parameter

S

Arc length calculated from the bottom rim of the cup

ρ0

Mass density

t

Time (ms)

tc

Contact time of a given point K on the sheet (ms)

tf

Final pressure inflation time (ms)

Tn

Normal traction

Ts

Tangential traction

ξ

t/t0

Relative thickness of the cup

LEVMAX

Δt

Time step (ms)

ΔtC

Critical time step (ms)

Δti

Time step of the i-th element (ms)

W

Strain energy function

V0

Volume of the undeformed membrane

VK(t)

Displacement vector of point K at current time t

dV0

Boundary surface of the deformed membrane

X(t)

Nodal displacement vector

XK(t)

Position of point K

XK

Correspondent point K′ on the mould surface

'


i

Density of the i-th element

σBT

True stress of equibiaxial tension (MPa)

σUT

True stress of uniaxial tension (MPa) Vibration frequency

max

Largest natural frequency of the structure Damping ratio Maximum level of automatic selective refinement

9. REFERENCES 1.

Throne, J.L., Technology of Thermoforming, Hanser Publisher, Munich, Germany, (1996).

2.

GE (General Electric) Engineering Thermoplastics Processing Guide, Plastics Division, General Electric Company. [Online] available: (http://www.geplastics. com/servlet/xslt/DefaultApplyXSL?URL=/ resins/techresearch/en/techniguides/ thermoProcessing/thermoprocessing. xml&xslURL=/resins/techresearch/en/ techniguides/xsl/techniguides.xsl)

327


3.

Warby, M.K., Whiteman, J.R., Jiang, W.G., Warwick, P. and Wright, T., Mathematics and Computers in Simulation, 61, (2003), 3-6, 209218.

4.

Treloar, L.R.G., Trans.Faraday Soc. Faraday Soc., 40, (1944), 59-70.

5.

Rivlin, R.S. and Saunders, D.W., Phil. Trans. R. Soc., A243, (1951), 251-288.

6.

Mooney, M., J. Appl. Phys., 11, (1940), 582592.

7.

Schmidt, L.R. and Carley, J.F., Polym. Eng. Sci., 15, (1975), 51-62.

8.

Schmidt, L.R. and Carley, J.F., Int. J. Eng. Sci., 13, (1975), 563-578.

9.

Ogden, R.W., Proc. R. Soc. Lond., A.326, (1972), 565-584.

10.

Bidman, V.L., Calculations of Rubber Parts (in Russian), Raschetina Prochnost, Moscow, (1958).

11.

Dong, Y., Lin, R.J.T. and Bhattacharyya, D., J. of Mat. Sci., 40, (2005), 399-410.

12.

Oden, J.T. and Sato, T., Int. J. Solids Structures, 3, (1967), 471-488.

13.

Feng, W.W. and Huang, P., Trans. ASME, J. of Appl. Mech., 41, (1974), 767-771.

14.

Yang, W.H. and Lu, C.H., Trans. ASME, 95, Series E, J. of Appl. Mech., 40, (1973), 7-12.

15.

16.

Bourgin, P., Cormeau, I. and Saint-Matin, T., J. Materials Proc. Tech., 54, (1995), 1-11.

20.

Marckmann, G., Verron, E. and Peseux, B., Polym. Eng.Sci., 41, (2001), 426-439.

21. Chevaugeon, N., Verron, E. and Peseux, B., Proceedings of European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS), Barcelona, (2000), 1-16. 22. Zienkiewicz, O.C. and Taylor, R.L., The Finite Element Method, Volume Ι, Fourth Ed., McGraw-Hill, London, (1994). 23. LS-DYNA Theoretical Manual, Livermore software technology corporation, U.S., (1998). 24. PAM-STAMP™ Solver Notes Manual, Pam System International (PSI), S.A., (the Software Company of ESI Group), France, (2000). 25. Bowis, M.E., Thermoforming WoodfiberPolypropylene Composite Sheets, PhD Thesis, The University of Auckland, New Zealand, (1997). 26.

Christie, G.R., Numerical Modelling of FibreReinforced Thermoplastic Sheet, PhD Thesis, The University of Auckland, New Zealand, (1997).

27.

Allard, R., Charrier, J.M., Ghosh, A., Marangou, M., Ryan, M.E., Shrivastava, S. and Wu, R., J. Polym. Eng., 6, (1986), 363-393.

Hunter, P.J. and Pullan, A.J., Finite Element Method/Boundary Element Method Notes, Department of Engineering Science, The University of Auckland, New Zealand, (2004).

28.

Charrier, J.M., Shrivastava, S. and Wu, R., J. Strain Analysis, 24, (1989), 55-74.

Williams, J.G., J. Strain Analysis, 5, (1970), 49-57.

29.

Lai, M.O. and Holt, D.L., J. Appl. Polym. Sci., 19, (1975), 1805-1814.

17.

Delorenzi, H.G. and Nied, H.F., Progress in Polymer Processing, Isayev, A.I. (ed.), Hanser Publisher, Munich, (1991).

18.

Vlachopoulos, J., Mirza, F.A., Koziey, B.L. and Ghafur, M.O., Composite Sheet Forming, Bhattacharyya, D. (ed.), Elsevier Science B. V., Amsterdam, (1997).

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19.


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