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-Seminar
Advanced Topics in Thermoforming
Part 2: 9:00-9:50
Constitutive Equations Applied
to Sheet Stretching
Let’s
begin!
Mini-Seminar
Advanced 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-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
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 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
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 e
• 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 s=f(e, e ;T)
• Linear viscoelasticity - The simple sum of
elastic and viscous responses to applied shear.
Usually shown as

s = f1(e) + f2( e )
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
s = E e
• Power-law behavior
s = E en
• Simple elongational Newtonian viscous behavior he is elongational Newtonian viscosity

s = he e
• Elongational power-law behavior- me is
elongational non-Newtonian viscosity

s = me ( e ) n
Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
• Stresses in terms of the strain energy function
W
si =
li
• Strain energy function in terms of the principal
invariants of the Cauchy strain tensor
W = W ( I , II , III)
Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
• The principal invariants of the Cauchy strain
tensor
I = l12  l22  l23
II = l  l  l
-2
1
-2
2
III = l12  l22  l23
-2
3
Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
• Stress-strain relationship in terms of Cauchy
invariants
 W  I   W  II   W  III 
si = 







 I  li   II  li   III  li 
• For an incompressible solid,
l1  l2  l3 = 1
or III = 1
Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
• For uniaxial stretching, l1=l, l2=1, l3=l-1/2
 2 1   W  2  W 
sl =  l - 2
 

l   II  l  I 

• For equal biaxial stretching, l1=l2=l, l3=l-2
 2 1   W 
2  W 
sl =  l - 4 2
  2l 

l   II 

 I 
Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
• The power-law form for the strain energy
function
W ( I , II ) =  Cij ( I - 3)  ( II - 3)
i
j
i, j
• The neo-Hookean solid form
W ( I ) = C10 ( I - 3)
• C10 is a constant related to the elastic modulus
Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
• The Rivlin form (developed for rubber elasticity)
W ( I , II ) = C01 ( I - 3)  f ( II - 3)
The Mooney form (also for rubber elasticity)
W ( I , II ) = C01 ( I - 3)  C10 ( II - 3)
• C01 and C10 are shape constants, described later
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 PlugAssist Thermoforming,” presented at the 2006
SPE ANTEC, they propose the following
constitutive equation:
W = (G(T)/4)[(I-3)+(II-3)]
Where G(T) is the temperature-dependent tensile
modulus
Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
• The Mooney stress-strain equation - uniaxial
2
 2 1 

sl =  l - 2C01  C10 
l 
l 

• The Mooney stress-strain equation - equal
biaxial

 2 1
sl =  l - 4  2C01  2l2C10
l 


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 l  0 the constants are
determined from the elastic modulus
E W W
=

6
I
II
Part 2: Constitutive Equations Applied to Sheet Stretching
Elastic Constitutive Equations
• For the Mooney model
E
= C01  C10
6
• Typically for many polymers
W
W

I
II
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
mn a
W =  l1  la2  la3
n =1 a n
m
n
n
n

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
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
1/ 2
 Fh0 
2
-2


2
C

2
C
=
l
1
1
l

 01
10
1
1
2

r



 

• 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
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
s = s 0 f (e ) g ( ) = s 0e 
m
n
• The current way of including fading memory

s   =  m  -  'h( I , II ) B( , ' )d '
0
Part 2: Constitutive Equations Applied to Sheet Stretching
Viscoelastic Constitutive Equations

s   =  m  -  'h( I , II ) B( , ' )d '
0
•
m  -  ' is the memory function
M
m  -  ' = 
i =1
Gi
li
e- - ' / li 
• where Gi and li are material parameters
• Typically only the first term of the series is
used
• B(,’) is the Finger strain tensor
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

h( I , II ) = 1  a ( I - 3)( II - 3)

-1/ 2
• for simple equal biaxial stretching

h(e ( )) = e
-2e 0 2e
e  (1 - e
-2e 0
)e

me -1
• where e=ln L(), e0 and m are called Wagner
constants, L() is the stretch ratio at  related
to time ‘.
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 Measurement
Hot tensile testing - Very difficult to get repeatable
data at elevated temperatures
Dynamic mechanical testing - Yields temperaturedependent 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 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
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
Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis
Two-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 D
[XD, YD, ZD]
Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis
Two-dimensional
• Force balance is made at each node between time 
and time D.
• If Fi,ext is the external force applied to node i,
being the pressure, p, times the normal to the
element, n, then Fi ,ext = p  ni
• If Fi,int is the internal force applied to node i, W is
the internal energy function and u is the
- W
displacement coordinate at node i, then
Fi ,int =
ui
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!)
 - W

Fi,int - Fi,ext  =  
 p  ni  = 0

N
N  ui

This equation set is then combined with the
appropriate material constitutive equation of state,
localized for each triangular element
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
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
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
Part 2: Constitutive Equations Applied to Sheet Stretching
Finite Element Analysis
Two-dimensional
• Keys to successful FEA include
– Selection of correct (small) D 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 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!
Part 2: Constitutive Equations Applied to Sheet Stretching
Sag
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
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  = ms
T cos  = To
Part 2: Constitutive Equations Applied to Sheet Stretching
Sag
The extent of deflection, y, below the horizontal is
 d 2 y  m  ds 
 2  =  
 dx  T0  dx 
Now (ds)2 = (dx)2 + (dy)2
Part 2: Constitutive Equations Applied to Sheet Stretching
Sag
The extent is then given as
d y m
 2  =
 dx  T0
2
And integrated to yield
  dy 
1   
  dx 
T0 
mx 
y =  cosh
- 1
m
T0

2 1/ 2



Part 2: Constitutive Equations Applied to Sheet Stretching
Sag
The position along the sheet is
 mx 
s = sinh  
m
 T0 
T0
And integrated to yield the total sheet length
S=
l/2
0
 mL 
T0 

sdx = cosh
m
 2T0 
-1
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
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.
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 twodimensional (catenary) sheet sag
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.
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!