mini-seminar dr. james throne, instructor 8:00-8:50 - technology of sheet heating 9:00-9:50 -...

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!


• 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


• 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


Fundamentals- Stress- Strain- Rate-of-Strain


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


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


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( )


Definitions • Elasto-Plastic Deformation - Material

stretches elastically to a given extension, then rapidly deforms with little additional stress


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)


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


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


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


Constitutive Equations


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


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


Elastic Constitutive Equations• The principal invariants of the Cauchy

strain tensor23

22

21 I

23

22

21

II

23

22

21 III


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


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


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


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


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)]


Elastic Constitutive Equations• The Mooney stress-strain equation -

uniaxial

• The Mooney stress-strain equation - equal biaxial

1001

2 22

1CC

102

0142 22

1CC


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


Elastic Constitutive Equations• For the Mooney model

• Typically for many polymers

10016CC

E

II

W

I

W


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


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


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


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


Elastic Constitutive Equations -

Plug Stretching• Comparison of

plane strain model with FEA models that include viscoelasticity


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


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


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

/''


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?]


Typical temperature-dependent stress-strain curves for an amorphous polymer


ABS temperature-dependent stress-strain curves


The forming window overlay on the stress-strain field


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


Instrumented plug device for measuring thermoformability - Transmit Technology Group


Temperature-dependent elastic modulus as a determinant for the forming window


Temperature-dependent elastic modulus as a determinant for the

forming window


The maximum applied stress restricts the forming window to the cross-hatched

region


Finite Element Analysis


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


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


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]



• 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 ,



• 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,



• 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



• 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



• 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



• 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



• 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



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!



• 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!


Sag


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


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


Sag

The extent of deflection, y, below the horizontal is

Now (ds)2 = (dx)2 + (dy)2

dx

ds

Tdx

yd

02

2


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


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


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


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.


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


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.


End of

Part 2

Constitutive Equations Applied to Sheet Stretching


Part 3

Trimming as Mechanical Fracture

Begins promptly at 10:00!

mini-seminar dr. james throne, instructor 8:00-8:50 - technology of sheet heating 9:00-9:50 -...

Documents

strain slide

stretching regions

property slide

stress strain

directions biaxial stretching

elastic melt state slide

strain rate

strain rate