1. Mini-Seminar Dr. James Throne, Instructor
Advanced Topics in Thermoforming
Mini-Seminar Dr. James Throne, Instructor
8:00-8:50 - Technology of Sheet Heating
9:00-9:50 - Constitutive Equations Applied to Sheet Stretching
10:00-10:50 - Trimming as Mechanical Fracture

2. Mini-Seminar Advanced Topics in Thermoforming
Part 2: 9:00-9:50
Constitutive Equations Applied to Sheet Stretching

3. Let’s begin!

4. Mini-Seminar Advanced Topics in Thermoforming
All materials contained herein are the intellectual property of Sherwood Technologies, Inc., copyright
No material may be copied or referred to in any manner without express written consent of the copyright holder
All materials, written or oral, are the opinions of Sherwood Technologies, Inc., and James L. Throne, PhD
Neither Sherwood Technologies, Inc. nor James L. Throne, PhD are compensated in any way by companies cited in materials presented herein
Neither Sherwood Technologies, Inc., nor James L. Throne, PhD are to be held responsible for any misuse of these materials that result in injury or damage to persons or property

5. Mini-Seminar Advanced Topics in Thermoforming
This mini-seminar requires you to have a working engineering knowledge of heat transfer and stress-strain mechanics
Don’t attend if you can’t handle theory and equations
Each mini-seminar will last 50 minutes, followed by a 10-minute “bio” break
Please turn off cell phones
PowerPoint presentations are available at the end of the seminar for downloading to your memory stick

6. Part 2: Constitutive Equations Applied to Sheet Stretching
Outline
Fundamentals
Definitions
General Premise
General Premise for Thermoforming
Elastic Constitutive Equations
Viscoelastic Constitutive Equations
Forming Window Measurement
Finite Element Analysis
Sag

7. Part 2: Constitutive Equations Applied to Sheet Stretching
Fundamentals
- Stress
- Strain
- Rate-of-Strain

8. Part 2: Constitutive Equations Applied to Sheet Stretching
Definitions
Stress - Applied load per unit area. Usually given the symbol s
Strain - Deformation resulting in applied load per unit area. Usually given the symbol e or l where e = l-1
Stress and strain apply primarily to elastic materials

9. Part 2: Constitutive Equations Applied to Sheet Stretching
Definitions
Rate of strain (or strain rate) - The rate of deformation owing to applied stress. Usually given the symbol
Rate of strain is usually applied to materials that yield or flow under stress

10. Part 2: Constitutive Equations Applied to Sheet Stretching
Definitions
Viscoelasticity - The combination of elastic and viscous behavior. The general form for stress-strain-rate-of-strain is s=f(e, ;T)
Linear viscoelasticity - The simple sum of elastic and viscous responses to applied shear. Usually shown as
s = f1(e) + f2( )

11. Part 2: Constitutive Equations Applied to Sheet Stretching
Definitions
Elasto-Plastic Deformation - Material stretches elastically to a given extension, then rapidly deforms with little additional stress

12. Part 2: Constitutive Equations Applied to Sheet Stretching
General Premise
If a material responds elastically to applied load, it recovers fully and instantaneously once the load is removed (think rubber band)
If a material responds viscously to applied load, it remains completely deformed once the load is removed (think pudding)
If a material recovers a little but remains somewhat deformed once the load is removed, the material is considered viscoelastic (think silly putty)

13. Part 2: Constitutive Equations Applied to Sheet Stretching
General Premise For Thermoforming
An amorphous polymer is stretched primarily in its rubbery solid state
Polyethylene is typically stretched in its elastic melt state
Polypropylene is stretched either in its rubbery solid state (solid state forming) or, if it has good melt strength, in its elastic melt state

14. Part 2: Constitutive Equations Applied to Sheet Stretching
General Premise For Thermoforming
Ergo, for most polymers and most stretching regions on a given part, the elastic character of the polymer dominates
For certain polymers (PP, for example), and for certain regions on a given part for many other polymers, the viscous character of the polymer influences the local part wall thickness

15. Part 2: Constitutive Equations Applied to Sheet Stretching
General Premise For Thermoforming
There are four general stretching modes
Uniaxial stretching - Stretching only in one direction
Equal biaxial stretching - Stretching to the same elongation in two directions
Biaxial stretching - Stretching in two directions but not necessarily to the same elongation
Plane strain stretching - defined below

16. Part 2: Constitutive Equations Applied to Sheet Stretching

17. Part 2: Constitutive Equations Applied to Sheet Stretching
Simple Hookean elastic behavior - E is elastic modulus
s = E e
Power-law behavior
s = E e n
Simple elongational Newtonian viscous behavior - he is elongational Newtonian viscosity
s = he
Elongational power-law behavior- me is elongational non-Newtonian viscosity
s = me ( ) n

18. Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
Stresses in terms of the strain energy function
Strain energy function in terms of the principal invariants of the Cauchy strain tensor

19. Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
The principal invariants of the Cauchy strain tensor

20. Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
Stress-strain relationship in terms of Cauchy invariants
For an incompressible solid,
or III = 1

21. Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
For uniaxial stretching, l1=l, l2=1, l3=l-1/2
For equal biaxial stretching, l1=l2=l, l3=l-2

22. Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
The power-law form for the strain energy function
The neo-Hookean solid form
C10 is a constant related to the elastic modulus

23. Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
The Rivlin form (developed for rubber elasticity)
The Mooney form (also for rubber elasticity)
C01 and C10 are shape constants, described later

24. Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
In a recent paper by Hosseini and Berdyshev, “A Solution for Rupture of Polymeric Sheet in Plug-Assist Thermoforming,” presented at the 2006 SPE ANTEC, they propose the following constitutive equation:
Where G(T) is the temperature-dependent tensile modulus
W = (G(T)/4)[(I-3)+(II-3)]

25. Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
The Mooney stress-strain equation - uniaxial
The Mooney stress-strain equation - equal biaxial

26. Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
The coefficients C01 and C10 are curve-fit to stress-strain curves
They are also highly temperature-dependent
In the limit as the constants are determined from the elastic modulus

27. Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
For the Mooney model
Typically for many polymers

28. Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
If C01=0, the value for C10 is just the elastic modulus
This is usually valid for low levels of deformation
When C10=0, the model seems to correlate with PP creep data

29. Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
The Ogden Model
an and mn are curve-fitting constants
Usually m<3 yielding 2, 4, or 6 constants
when m=2, a1=2 and a2=-2, the Mooney equation results

30. Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations -
Plug Stretching
Plane strain - No relative effect of stretching is seen from the vertical

31. Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations -
Plug Stretching
Plane strain - No relative effect of stretching is seen from the vertical

32. Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations -
Plug Stretching
Mooney-Rivlin constitutive equation for plane strain
where F is the applied force, r is the instant location between the edge of the plug and the rim, and ho is the initial sheet thickness

33. Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations -
Plug Stretching
Comparison of plane strain model with FEA models that include viscoelasticity

34. Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
The Ogden model is the favorite for model builders today
The Mooney-Rivlin models are considered classical and are not usually used for model building

35. Part 2: Constitutive Equations Applied to Sheet Stretching
Viscoelastic Constitutive Equations
A simple way of including time-dependency in stress-stain equations
The current way of including fading memory

36. Part 2: Constitutive Equations Applied to Sheet Stretching
Viscoelastic Constitutive Equations
is the memory function
where Gi and li are material parameters
Typically only the first term of the series is used
B(q,q’) is the Finger strain tensor

37. Part 2: Constitutive Equations Applied to Sheet Stretching
Viscoelastic Constitutive Equations
h(I,II) is the damping function of the two strain invariants, in the Wagner form
for simple equal biaxial stretching
where e(q)=ln L(q), e0 and m are called Wagner constants, L(q) is the stretch ratio at q related to time q‘.

38. Homework assignment for TF Conference 2007
Analyze the four papers presented by Hosseini and Berdyshev at the 2006 SPE ANTEC, to wit:
“A Solution for Warpage in Polymeric Products by Plug-Assist Thermoforming”
“A Solution for Rupture of Polymeric Sheet in Plug-Assist Thermoforming”
“Modeling of Deformation Processes in Vacuum Thermoforming of Prestretched Sheet”
“Rheological Modeling of Warpage in Polymeric Products Under High Temperature”

39. Homework assignment for TF Conference 2007
Their first paper, “A Solution for Warpage in Polymeric Products by Plug-Assist Thermoforming,” was reprinted in TF Quarterly, 3rd Quarter 2006.
[Note: As Tech Editor of the Quarterly, I made the comment that the authors had tacitly assumed that warpage could be described as uniaxial deformation and recovery. In other words, the authors used the scalar forms for the Cauchy, Hencky, and the flow strain rate terms. Is this correct? Should they have used the tensor forms as they have in their other papers?]

40. Part 2: Constitutive Equations Applied to Sheet Stretching
Typical temperature-dependent stress-strain curves for an amorphous polymer

41. Part 2: Constitutive Equations Applied to Sheet Stretching
ABS temperature-dependent stress-strain curves

42. Part 2: Constitutive Equations Applied to Sheet Stretching
The forming window overlay on the stress-strain field

43. Part 2: Constitutive Equations Applied to Sheet Stretching
Forming Window Measurement
Hot tensile testing - Very difficult to get repeatable data at elevated temperatures
Dynamic mechanical testing - Yields temperature-dependent modulus through frequency of oscillation or more commonly, by direct measurement
Hot creep measurement - Instrumented device similar to HDT device, measuring viscosity
Instrumented plug - By changing the rate of plugging, the role of viscosity can be ascertained

44. Part 2: Constitutive Equations Applied to Sheet Stretching
Instrumented plug device for measuring thermoformability - Transmit Technology Group

45. Part 2: Constitutive Equations Applied to Sheet Stretching
Temperature-dependent elastic modulus as a determinant for the forming window

46. Part 2: Constitutive Equations Applied to Sheet Stretching
Temperature-dependent elastic modulus as a determinant for the forming window

47. Part 2: Constitutive Equations Applied to Sheet Stretching
The maximum applied stress restricts the forming window to the cross-hatched region

48. Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis

49. Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis
Two-dimensional
Replaces sheet surface with triangular grid connected through nodes
As grid stretches, triangles remain planar but increase in surface area
Material assumed to have constant volume; thus increase in local surface area means decrease in local thickness
Model works best for thin sheet

50. Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis
Three-dimensional
Replaces sheet surface with triangular grid connected through nodes
Sheet thickness accounted for by several initially parallel grids, connected through initially parallel node junctions
Often called brick model; volume in each thin brick constant
Model allows for localized compression, shear

51. Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis
Two-dimensional

52. Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis
Two-dimensional
Location of each node of triangle at time q
[Xq, Yq, Zq]
After differential deformation, location of each node of triangle at time q+Dq
[Xq+Dq, Yq+Dq, Zq+Dq]

53. Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis
Two-dimensional
Force balance is made at each node between time q and time q+Dq.
If Fi,ext is the external force applied to node i, being the pressure, p, times the normal to the element, n, then
If Fi,int is the internal force applied to node i, W is the internal energy function and u is the displacement coordinate at node i, then

54. Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis
Two-dimensional
An equilibrium force balance is applied over the entire surface of the sheet (no acceleration, please!)
This equation set is then combined with the appropriate material constitutive equation of state, localized for each triangular element

55. Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis
Two-dimensional
In addition to FEA, very accurate identification of mold surface is needed [X’j, Y’j, Z’j]
In certain FEA models, a coefficient of friction is needed between the sheet and the mold surface (this subject is under review and will be the topic at future Conferences, including this one!)
Triangles can rotate, translate, and grow in surface areas
Triangles cannot flex, bend, or fold

56. Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis
Two-dimensional
Consider a node fixed when its X,Y,Z coordinate approximates a mold X’,Y’,Z’ coordinate location
Keep in mind that although one or two nodes of a triangular element are affixed to the mold surface, the triangle can continue to stretch
Keep in mind that the local triangle nodes do not need to be affixed to the mold surface if all the nodes of adjacent triangles are affixed

57. Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis
Two-dimensional
When dealing with a plug, a tag is assigned to each node that is affixed to the plug
This allows the analysis to include these nodes when the mesh is mathematically stripped from the plug
The FEA is complete when no nodes are moveable or viable
Newer algorithms address only those nodes that are not immoveable or are tagged, in this way rapidly accelerating the analysis

58. Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis
Two-dimensional
Keys to successful FEA include
Selection of correct (small) Dq step
Predetermination of local mesh size (too small will generate excessive computer time, too large will generate strange surface bumps, instabilities)
Very accurate mold surface replication and mapping
Selection of a time-conservative iterative method such as
Newton-Raphson iteration
Galerkin weighted residuals

59. Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis
Two-dimensional
Recommended practice
If the part is symmetric (viz, five sided box), select only one portion rather than solving the entire structure
If the part is axisymmetric (viz, drink cup), select only a wedge portion rather than the entire structure
Select coarse mesh initially
Refine mesh in local areas
Repeat computation
Continue to refine mesh until (nearly) all nodes are at rest on mold surface

60. Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis
Two-dimensional
HOWEVER...
If your sheet is not uniformly heated OR…
If your sheet is sagging OR…
If your mold is not uniformly cooled OR…
Use the entire sheet and mold surface!

61. Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis
Two-dimensional
Keep in mind that the computer display of the stretching mechanism is NOT real time, ONLY COMPUTER TIME
If the model being used is elastic-only, keep in mind that the final computer prediction of wall thickness represents what happens in an instant! (There is no time parameter in Mooney-Rivlin or Ogden models)
If the model is viscoelastic and/or if the model includes a moving plug, there will be a time factor included… Make certain this matches real time!

62. Part 2: Constitutive Equations Applied to Sheet Stretching
Sag

63. Part 2: Constitutive Equations Applied to Sheet Stretching
Sag
As sheet heats, it tends to drape or sag under its own weight
If the sheet is of uniform temperature and is clamped on only two edges (think roll-fed), sag shape can be predicted by considering it to be a catenary or chain

64. Part 2: Constitutive Equations Applied to Sheet Stretching
Sag
The sheet weight, m, is the sheet density times its local thickness, m = rh
The tension, T, in the sheet is factored into vertical and horizontal components, where s is the sheet length: T sin q = ms T cos q = To

65. Part 2: Constitutive Equations Applied to Sheet Stretching
Sag
The extent of deflection, y, below the horizontal is
Now (ds)2 = (dx)2 + (dy)2

66. Part 2: Constitutive Equations Applied to Sheet Stretching
Sag
The extent is then given as
And integrated to yield

67. Part 2: Constitutive Equations Applied to Sheet Stretching
Sag
The position along the sheet is
And integrated to yield the total sheet length

68. Part 2: Constitutive Equations Applied to Sheet Stretching
Sag
Review:
To is temperature-dependent tensile strength
m is sheet unit weight
L is total initial span of sheet
As sheet heats, To decreases (linearly to perhaps exponentially, depending on the polymer)
Sheet sag increases as To decreases, as observed

69. Part 2: Constitutive Equations Applied to Sheet Stretching
Sag Mathematics
Advantage: Very compact, easy to obtain local slope as a function of horizontal position (important when calculating the view factor for radiant heat to sagging sheet)
Disadvantage: As sheet sags, S increases, but total sheet weight remains constant. Sheet must thin, meaning that m, local unit sheet weight, must decrease. Can be rectified by trial-and-error or by making m = m(s) in equation and solving arithmetically.

70. Part 2: Constitutive Equations Applied to Sheet Stretching
Sag Mathematics
One approach to the sag problem is given in Thermoforming Quarterly, 4Q06 where the view factor is determined as a function of two-dimensional (catenary) sheet sag

71. Part 2: Constitutive Equations Applied to Sheet Stretching
Sag Mathematics
Caveat - Problem can be solved using FEA
BUT to effectively use the FEA model, we need to know the temperature of every element. We can only get that by calculating the view factor - as we did in the first lecture – but now for a sagging sheet. This is NOT done in the FEA model.

72. Part 2: Constitutive Equations Applied to Sheet Stretching
End of
Part 2
Constitutive Equations Applied to
Sheet Stretching

73. Part 2: Constitutive Equations Applied to Sheet Stretching
Trimming as Mechanical Fracture
Begins promptly at 10:00!