Engineering Deisgn With Polymers

7/28/2019 Engineering Deisgn With Polymers

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Engineering Design with

Polymers

Society of Plastic Engineers

5-6-09

Ron Rorrer, Ph.D., P.E.

Associate ProfessorUniversity of Colorado [email protected]

Ronald A. L. Rorrer, 2009


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Outline Introduction Engineering view of polymers.

Fluid Solid

Mechanical models of viscoelastic behavior. Fluid Solid Advanced

Why Model? Thermomechanical Spectrum Time-Temperature Superposition Boltzman Superposition Viscoelastic correspondence principle FEA Example Creep

Rupture Buckling

Damping and Isolation Discussion Summary Questions References


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Introduction

Designing with non-polymeric materialsYoungs modulus, E, and yield strength, Sy

Designing with polymeric materials Consider that modulus and strength (yield

and ultimate) are now time/rate andtemperature dependent

Environmental factors can have a muchgreater impact on polymers than non-polymers.


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Introduction (continued)

Engineers must understand the viscoelasticresponse of polymers in order to use polymers inengineering designs.

Mechanical properties will be related to thethermomechanical spectrum.

The thermomechanical spectrum is the middleground and point of commonality between

engineers and polymer chemists. Throughout, the thermomechanical spectrum is

used as a focal point to discuss a variety oftopics of the mechanical response of polymers


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Engineering View

Polymers are long chain molecules Consider subdividing conceptually into to two

groups Fluid (plate of intertwined spaghetti analogy)-

thermoplastic fluids Solid (3-D cargo net analogy)-thermosets with

covalent bonding between the polymer chains

Of course, somewhat the exception isthermoplastic elastomers, that while notcovalently cross-linked act as if the material iscross-linked. However, these pseudo-cross-linksonly appear this way for short term elasticity.


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Fluid

Fluid or fluid behavior

Has no long term elasticity

What does this mean? Under constant loading material will continually deflect.

Constant load will eventually fail material.

Caveat

Loads or duty cycle may be such that failure does not occur in

application use lifetime. Examples

Polycarbonate window panels

Polypropylene classroom seating

Every student knows one can not sit too long or one will

fail!


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Solid

Solid behavior Has long term elasticity

What does this mean? In theory, if loads are not large enough to completely rupture all of the

crosslinks, the solid will Asymptotically approach a limiting deflection. Carry a constant load indefinitely.

Caveat Even though initial loading is carried, crosslinks can rupture in a

cascade fashion over time. Every student knows one can not sit too long or one will fail!

Examples Epoxy (carbon fiber/epoxy composite) EPDM roofing material Polybutadiene in vehicle tires


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Elements for Modeling

Most elementary models are composed of acollection of springs and dashpots to model

the elastic and inelastic aspects of viscoelasticbehaivor.

Spring

Dashpot

E =

=

,


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Fluid Model

Simplest Fluid Model

Maxwell Model

Spring in series withdamper

Apply constant creepstress to model,monitor strain

,

0

t

0t

E

= +


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Solid Model

Simplest Solid Model

Kelvin Model

Spring in parallel withdamper

Apply constant strainto model, monitorstress

,

( )0( ) 1 Ett E e =

t

( /)(11/ )0 e 0


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Advanced Viscoelastic Models

The Maxwell and Kelvin models are toosimplistic for most real world behavior.

However, they do capture the essence offluid and solid behavior.

In order to fit the behavior over many

decades of time other models are used.


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N-parameter

Conceptual extension n-parametermodels

n-parameter fluid

n-parameter solid

,

...

,

...

8

0

1( ) it ti

i

E t E E e=

= +


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Spring Pot -Fractional Model

where:= memory parameter, not necessarily integerE= modulus= viscosityD= fractional differentiation with respect to

=0 and the spring-pot reduces to a simple spring =1 and the spring-pot reduces to a dashpot. 0


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Fit to Experimental Data

Solid data from Gottenberg and Christiansen1 fit with a series ofexponentials and a fractional spring-pot model2

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

-5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0

Log t [log s]

Modulus[MPa]

Transformed Data

Exponential Series Fit

Four Parameter Fractional Model


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Why Model?

Experimental data can be used tocalculate response at a single time or

frequency However, a model allows analytical or

numerical prediction of response over allfrequency and times

Prior transformed data and models extendfrom 10,000 Hz to 1000 s.


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Thermomechanical Spectrum Shear testing of Nylon 6,6 Notes:

Glass transition temperature of Nylon 6,6 routinely reported as50 C, here 65-80 C

Strain level for Dynamic Mechanical Analysis often below

application levels


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Thermomechanical Regions

Glassy Region Material responds rigidly, region of highest storage

modulus, low damping

Leathery Region Transtion region, highest damping

Rubbery Region Material has low modulus

Terminal Region Thermoplastic fluid melts Crosslinked thermoset solid degrades


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Importance of Regions

Importance of regionsillustrated by ductile to brittletransition, TT, of impactproperties.

For structural purposespolymers are typically used ineither the rubbery or glassyregion.

A transition between regions

during use can be disastrousfor performance of a structuralmaterial or adhesive.

Temperature

Tg

ImpactStren

gth

TT

DuctileBrittle


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Time-Temperature

Superposition (TTS) Time-temperature equivalence is based

upon the observation that temperature

and rate are inter-related with respect topolymer mechanical properties.

Macroscopically, we can observe that the

effect of cold temperatures are similar tothe effect of high rates on polymerresponse.


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TTS

While TTS is not always valid, there are two shift factorsthat have been used to take data generated over a 1-3decade range of rates and create a master curve over 5-

15 decades of rate Applicability (Tg


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TTS Example of Amorphous

Polymer3


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Application of Boltzman

Superposition Results from constant

load creep test can be

used to predictresponse to timevarying load.

( ) ( )0 1t J t for t t =

( ) ( ) ( )0 1 1 1t J t J t t fo rt t = + >

( ) ( ) ( )0 10

t

ii

i

dt J t J t t d t

dt

= +


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Viscoelastic Correspondence

Principle The time dependent deformation of a

linear situation (both material properties

and deformation) can be analyzed bysubstituting the E(t) or 1/J(t) for E if thestatic solution is known.

Consider a cantilever beam with a

transverse end load. Static deflection

Time dependent deflection

( )( )

6

f xy x P

EI=

( ) ( ) ( )( , )

6 ( ) 6

f x f x J ty x t P P

E t I I= =


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FEA Example

This concept can be extended toFEA.

Example Polyamide

100 mm OD, 30 mm ID

Constrained on bottom half ofexterior

Loaded vertically on inner half ofbore with 100 N distributed load

Assume that at t1 E(t1)=0.5 E(t=0). Since FEA is linear, (0)=0.5(t=t1)

t=0

t=t1


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FEA Example

The maximum stressdoes not change.

This occurs becausethis problem does notexperience non-lineardeformation.

t=0

t=t1


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Creep Rupture

Due to viscoelasticity, polymers will fail underconstant load


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Miners Rule

Originally used to approximate fatigue failure of metals.

In many cases Miners rule has been shown to apply toboth creep and fatigue failure of polymers.

1

1

1

n

N

n n

n c

t t

t

=


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Creep Buckling

ViscoelasticCorrespondence

Principle can beextended to buckling

Euler Buckling2

2

E

cr

EIP

L

=

( )

2

2

1

KelvinSolid

cr Et n

I EP

L e

=

2

2

1

Maxwell

cr

I EP

EtL

= +


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Damping and Isolation

Fundamental difference between dampingand isolation. Damping-energy of vibration is dissipated as

heat due to viscous losses in material beingcycled.

Isolation-due to dynamic response of system,when excited at frequencies away from

natural frequencies (resonances) of system,lack of system dynamic response isolatesdriven components


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Vibrations and the

Thermomechanical Spectrum Damping is one of the

few situations where

it is desirable to use apolymer near theglass transition.

The temperature axis

due to timesuperposition isequivalently

Temperature(f= 1 Hz)

E/E=tan

E

1 1 2 2( , 1 ) ( , )T f Hz T f =


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Vibration Models

Most Dynamic Mechanical Analyses assume thatthe material being tested is a Kelvin Solid with

the complex modulus being written in thevarious forms found in the literature as

In the model that follows'AE

kL

=''AE

cL

=

* ' '' '(1 ) '(1 ) '(1 tan )E E iE E i E i E = + = + = + = +

A= area

L=the height of polymer represented by the spring and damper

n

k

m =

2c

km =


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Damping and Isolation of a SDOF

System

x

k

c

mF sin t0

A single dof system has a frequency response function as shown.

For w/wn 1.4 the system responds as anisolator.

Between 0.5 and 1.4 the system is dominated by damping

0

1

2

3

4

5

0 1 2 3 4 5/wn

XK/F

= 0

= 0.1

= 0.25

= 0.5

= 1.0

2 2sin 2n n

c k Fx x x t x x x

m m m + + = = + +


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Polyurethane Blends4

Polyurethanes areoften used as

vibration dampers Slight changes in

processing yield largechanges in properties

Blends of polyurethanecan shift peakdamping relative tofrequency


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Anamolous Blend

Most blends, have aTg between the twomaterials or evidences

two Tgs Of interest is a blend

of polyisoprene (PIP)and polybutadiene(PBD) which has apeak in losses outsideof this range.5


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Discussion

Polymers have all of the sameconsiderations for design as non-

polymers. The time dependence of viscoelasticity

requires additional consideration in the

design process.


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Summary

Design engineers must be aware of thedifferences between polymers and non-

polymers to create successful designs.


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Questions?


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Acknowledgement

The material for this webnair largelyabstracted from Gerdeen, J., Lord, H.,

Rorrer, R. A.L. Engineering Design withPolymers and Composites, CRC Press,Boca Raton, Florida, 2005, 349 pages.6


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References

1 Gottenberg, W. G., and Christiansen, R. M., Int. J. Eng. Sci., vol. 2, pp. 45-56, 1964

2 Welch, S. W. J, Rorrer, R. A. L., and Duren Jr., R. G., Mechanics of Time-Dependent Materials, Vol. 3, No. 3, pp. 277-301, 1999

3 Ferry, J.D.,Viscoelastic Properties of Polymers, John Wiley & Sons, NewYork, 1980.

4 Corsaro, R.D. and Sperling, L. H., Eds., Sound and Vibration Damping withPolymers, ACS Symposium Series 424, J.V. Duffy et al., p. 292, 1990.

5 Corsaro, R.D. and Sperling, L. H., Eds., Sound and Vibration Damping with

Polymers, ACS Symposium Series 424, C. M. Roland and C.A. Trask, p. 310,1990.

6 Gerdeen, J., Lord, H., Rorrer, R. A.L. Engineering Design with Polymersand Composites, CRC Press, Boca Raton, Florida, 2006, 349 pages.

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