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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
Mini-SeminarAdvanced Topics in Thermoforming
Part 2: 9:00-9:50Constitutive Equations
Applied to Sheet Stretching
Let’s begin!
Mini-SeminarAdvanced Topics in Thermoforming
• All materials contained herein are the intellectual property of Sherwood Technologies, Inc., copyright 1999-2006
• 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
Mini-SeminarAdvanced 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
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
Part 2: Constitutive Equations Applied to Sheet Stretching
Fundamentals- Stress- Strain- Rate-of-Strain
Part 2: Constitutive Equations Applied to Sheet Stretching
Definitions• Stress - Applied load per unit area.
Usually given the symbol • Strain - Deformation resulting in applied
load per unit area. Usually given the symbol orwhere
• Stress and strain apply primarily to elastic materials
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
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 =f(, ;T)
• Linear viscoelasticity - The simple sum of elastic and viscous responses to applied shear. Usually shown as
= f1() + f2( )
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
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)
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
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
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
Part 2: Constitutive Equations Applied to Sheet Stretching
Constitutive Equations
Part 2: Constitutive Equations Applied to Sheet Stretching
Constitutive Equations
• Simple Hookean elastic behavior - E is elastic modulus
= E • Power-law behavior = E n
• Simple elongational Newtonian viscous behavior - e is elongational Newtonian viscosity
e
• Elongational power-law behavior- e is elongational non-Newtonian viscosity
e( ) n
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
ii
W
),,( IIIIIIWW
Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations• The principal invariants of the Cauchy
strain tensor23
22
21 I
23
22
21
II
23
22
21 III
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
iii
i
III
III
WII
II
WI
I
W
1321
Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations• For uniaxial stretching, 1=, 2=, 3=-1/2
• For equal biaxial stretching, 1=2=, 3=-
2
I
W
II
W
2
212
I
W
II
W 24
2 221
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
j
ji
iij IIICIIIW
,
)3()3(),(
)3()( 10 ICIW
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
)3()3(),( 1001 IICICIIIW
)3()3(),( 01 IIfICIIIW
Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive EquationsIn 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)]
Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations• The Mooney stress-strain equation -
uniaxial
• The Mooney stress-strain equation - equal biaxial
1001
2 22
1CC
102
0142 22
1CC
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
0
II
W
I
WE
6
Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations• For the Mooney model
• Typically for many polymers
10016CC
E
II
W
I
W
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
Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations• The Ogden Model
n and n are curve-fitting constants
• Usually m<3 yielding 2, 4, or 6 constants
• when m=2, 1=2 and 2=-2, the Mooney equation results
m
n n
n nnnW1
321
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
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
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
21
2/1211001
0 11222
CC
r
Fh
Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations -
Plug Stretching• Comparison of
plane strain model with FEA models that include viscoelasticity
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
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
nmgf 00 )()(
0
')',(),(' dBIIIh
Part 2: Constitutive Equations Applied to Sheet Stretching
Viscoelastic Constitutive Equations
• is the memory function
• where Gi and i are material parameters
• Typically only the first term of the series is used
• B(’) is the Finger strain tensor
0
')',(),(' dBIIIh
'
M
i i
i ieG
1
/''
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 ln L(), 0 and m are called Wagner constants, L() is the stretch ratio at related to time ‘.
2/1)3)(3(1),(
IIIaIIIh
1222 )1())(( 00 meeeeh
Homework assignment for TF Conference 2007
Analyze the four papers presented by Hosseini and Berdyshev at the 2006 SPE ANTEC, to wit:
1. “A Solution for Warpage in Polymeric Products by Plug-Assist Thermoforming”
2. “A Solution for Rupture of Polymeric Sheet in Plug-Assist Thermoforming”
3. “Modeling of Deformation Processes in Vacuum Thermoforming of Prestretched Sheet”
4. “Rheological Modeling of Warpage in Polymeric Products Under High Temperature”
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?]
Part 2: Constitutive Equations Applied to Sheet Stretching
Typical temperature-dependent stress-strain curves for an amorphous polymer
Part 2: Constitutive Equations Applied to Sheet Stretching
ABS temperature-dependent stress-strain curves
Part 2: Constitutive Equations Applied to Sheet Stretching
The forming window overlay on the stress-strain field
Part 2: Constitutive Equations Applied to Sheet Stretching
Forming Window MeasurementHot tensile testing - Very difficult to get repeatable
data at elevated temperaturesDynamic 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
Part 2: Constitutive Equations Applied to Sheet Stretching
Instrumented plug device for measuring thermoformability - Transmit Technology Group
Part 2: Constitutive Equations Applied to Sheet Stretching
Temperature-dependent elastic modulus as a determinant for the forming window
Part 2: Constitutive Equations Applied to Sheet Stretching
Temperature-dependent elastic modulus as a determinant for the
forming window
Part 2: Constitutive Equations Applied to Sheet Stretching
The maximum applied stress restricts the forming window to the cross-hatched
region
Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis
Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element AnalysisTwo-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
Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element AnalysisThree-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
Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element AnalysisTwo-dimensional
Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis Two-dimensional
• Location of each node of triangle at time [X, Y, Z]
• After differential deformation, location of each node of triangle at time
[X, Y, Z]
Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element AnalysisTwo-dimensional
• Force balance is made at each node between time and time
• 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
ii u
WF
int,
iexti npF ,
Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element AnalysisTwo-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
N N
ii
extii npu
WFF 0,int,
Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element AnalysisTwo-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
Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element AnalysisTwo-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
Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element AnalysisTwo-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
Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element AnalysisTwo-dimensional
• Keys to successful FEA include– Selection of correct (small) 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
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
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!
Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element AnalysisTwo-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!
Part 2: Constitutive Equations Applied to Sheet Stretching
Sag
Part 2: Constitutive Equations Applied to Sheet Stretching
SagAs sheet heats, it tends to drape or sag under
its own weightIf 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
Part 2: Constitutive Equations Applied to Sheet Stretching
Sag
The sheet weight, , is the sheet density times its local thickness, = h
The tension, T, in the sheet is factored into vertical and horizontal components, where s is the sheet length: T sin = s T cos = To
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
dx
ds
Tdx
yd
02
2
Part 2: Constitutive Equations Applied to Sheet Stretching
Sag
The extent is then given as
And integrated to yield
2/12
02
2
1
dx
dy
Tdx
yd
1cosh
0
0
T
xTy
Part 2: Constitutive Equations Applied to Sheet Stretching
Sag
The position along the sheet is
And integrated to yield the total sheet length
0
0 sinhT
xTs
1
0
2/
0
0
2cosh
T
LTsdxS
l
Part 2: Constitutive Equations Applied to Sheet Stretching
SagReview:To is temperature-dependent tensile strength
is sheet unit weightL 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
Part 2: Constitutive Equations Applied to Sheet Stretching
Sag MathematicsAdvantage: 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 , local unit sheet weight, must decrease. Can be rectified by trial-and-error or by making = (s) in equation and solving arithmetically.
Part 2: Constitutive Equations Applied to Sheet Stretching
Sag MathematicsOne 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
Part 2: Constitutive Equations Applied to Sheet Stretching
Sag MathematicsCaveat - 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.
Part 2: Constitutive Equations Applied to Sheet Stretching
End of
Part 2
Constitutive Equations Applied to Sheet Stretching
Part 2: Constitutive Equations Applied to Sheet Stretching
Part 3
Trimming as Mechanical Fracture
Begins promptly at 10:00!