Estudo termo-reológico e estrutural de peças obtidas por microinjeção

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2013

André Ri ardo

Correia dos Santos

Válega

Estudo termo-reológi o e estrutural de peças

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2013

André Ri ardo

Correia dos Santos

Válega

Estudo termo-reológi o e estrutural de peças

obtidas por mi roinjeção

DissertaçãoapresentadaàUniversidadedeAveiropara umprimentodos

req-uisitosne essários àobtenção do graude mestre emEngenharia Me âni a,

realizada sob orientação ientí a da Professor Doutor Móni a Sandra

Abrantes de Oliveira Correia, Professora Auxiliar do Departamento de

En-genharia Me âni a da Universidade de Aveiro e do Professor Doutor Rui

António da Silva Moreira, Professor Auxiliar do Departamento de

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Presidente/ President Professor Doutor JoséPaulo Oliveira Santos

ProfessorAuxiliardaUniversidadedeAveiro

Vogais /Committee Professor Doutor Rui Jorge Sousa Costa de Miranda Guedes

ProfessorAuxiliardaFa uldadedeEngenhariadaUniversidadedoPorto

Professor Doutor Móni aSandra Abrantes de OliveiraCorreia

Professora AuxiliardaUniversidadedeAveiro(orientador)

Professor Doutor Rui António da SilvaMoreira

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A knowledgements entes, amigos e olegas queestiveram onstatemente presentes durante o

meu per urso a adémi o. Porisso, quero deixar-lheso meu agrade imento

pessoal, eem espe iala:

Aos meuspais e avós,pelo in ansável einquestionável apoio durante toda

a minhavida e emespe ialnesta fase.

Àminha madrinhapelossábios onselhosdurante oper urso a adémi o.

A todos os ompanheirosde urso, e emparti ularao Bruno Barroqueiro,

Carlos Oliveira, Rui Bártolo, Rui Pais e Tiago Godinho por todo o apoio e

entre-ajuda durante o urso.

ÀProfessora Doutora Móni a Oliveira,pormeterdado aoportunidadede

trabalhar nesta área, por nun a ter duvidado de mim e tersido uma fonte

de motivação e in entivo onstante.

Ao ProfessorDoutorRuiMoreirapela sua disponibilidade, onselhos ertos

e ajuda na análisedosdados.

À EngªTatiana por toda aajuda naparteexperimentale nas simulações.

Aos olegasdelaboratórioquemeintegraramnumgrupodinâmi oea tivo.

A todosos que ontribuiram deforma dire taou indire taneste trabalho.

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prin ípio de sobreposiçãotempo temperatura;TTSP

Resumo Nas últimasdé adas, aminiaturizaçãodosequipamentoseletróni ose

apar-elhos me âni ostemsidouma tendên iaem onstante desenvolvimento. A

mi romoldaçãoporinjeçãoéumadasté ni asmaisutilizadasparaobtenção

de mi ro omponents plásti os para os mais variados propósitos. Através

desta té ni a é possível obter mi ro omponentes em grandes quantidades

e a baixo usto, usufruindo ainda do alto nível da automação inerente ao

pro esso. Contudo, durante o pro esso de moldação são originadas

ten-sões residuaisinternas nas peças moldadas, devido ao es oamento do

fun-dido e ao arrefe imento e solidi ação do plásti o. Neste trabalho

propõe-se uma metodologia que ontempla a simulação numéri a do pro esso de

obtenção de peças plásti as por injeção juntamente om uma análise

es-trutural. Esta última visa levar em onsideração o históri o do

pro es-samento bem omo o omportamento típi o de um material poliméri o

aquando em serviço. Desse modo, o trabalho in lui o estudo e denição

do omportamento vis oelásti o do material. Para tal, são abordadas

tam-bémmetodologiasdeanálisedo omportamentovis oelásti odosmateriais.

O trabalho divide-se em duas partes prin ipais: a primeira onde são feitos

ensaiosDMTApara ara terizaçãodo omportamentome âni odomaterial

e propostas metodologiasde análise do mesmo atravésdo usodo prin ípio

desobreposiçãotempo-temperatura;asegundapartein idenaobtençãodas

tensõesresiduaisatravésdasimulaçãonuméri adopro essodeinje çãoeda

simulaçãonuméri a deumensaioderelaxaçãodetensõesnoqual omodelo

vis oelásti opreviamente riadoé inserido. Osresultadosobtidospermitem

on luiraadequabilidade dametodologiadesenvolvida,permitindo postular

asuaviabilidadenasimulaçãodo omportamentodemi ropeçasquandoem

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time temperaturesuperpositionprin iple;TTSP

Abstra t In re ent de ades, the miniaturization of ele troni and me hani al

equip-ment is a tenden y in onstant development. Mi ro-inje tionis one of the

most used te hniques for obtaining plasti s mi ro- omponents for various

purposes. Throughthiste hniqueitispossibletoobtainmi ro- omponents

inlargequantitiesandatlow osts,yettakingadvantageofthehighlevelof

automationinherentto thepro ess. However, duringthe mouldingpro ess,

residualstressesstarttodeveloponthepart,duetothemeltowanddueto

the oolingandsolidi ationoftheplasti . Inthisstudyamethodologythat

addresses the numeri al simulation of the inje tion moulding oupled with

a stru tural analysis is proposed. The latter takes into a ount the ee t

of residual stresses on the part and the fa t that polymers are vis oelasti

materials. Forthispurpose,methodologiestoanalysevis oelasti behaviour

of polymers are also proposed. This workis omposed by two main parts:

the rst, where DMTAtests are performed to hara terize the me hani al

behaviour of the material and where analysis methods to handle resultant

data are proposed, namely by using the prin iple of time-temperature

su-perposition; the se ond part relates to obtaining the part residual stresses

throughthe numeri alsimulationoftheinje tion pro essandthe numeri al

simulation ofa stressrelaxation test,inwhi hthe previously developed

vis- oelasti modelisa ounted for. Theobtainedresultsprovidean insighton

the adequa y ofthe methodologyhere developed toassess thedurability of

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1 Introdu tion 1

2 Literature review 3

2.1 Vis oelasti ity. . . 3

2.2 Linear onstitutive equations (simple shear) . . . 4

2.2.1 Equations of hange . . . 4

2.2.2 Innitesimal straintensor . . . 5

2.2.3 Stresstensor . . . 5

2.2.4 Constitutive equation for linearvis oelasti ity(simple shear) . . . 5

2.3 Me hani al models for vis oelasti behaviour . . . 7

2.3.1 Maxwell Model . . . 7

2.3.2 Voigt-Kelvin Model. . . 8

2.3.3 ZenerModel . . . 9

2.3.4 Vis oelasti spe tra. . . 11

2.4 Polymeri materials. . . 12

2.4.1 Temperature inuen e onregionsof vis oelasti behaviour . . . 13

2.4.2 Thermoplasti s and thermosets . . . 15

2.4.3 Amorphousand rystallinethermoplasti s . . . 15

2.5 Linear time-dependent experiments (shear) . . . 17

2.5.1 Stressrelaxation . . . 17

2.5.2 Creep . . . 17

2.5.3 Sinusoidal soli itationsand dynami experiments . . . 18

2.6 Dynami and transient experiments. . . 20

2.7 Boltzmann Superpositionprin iple . . . 22

2.8 Time temperature superposition prin iple (TTSP) . . . 24

2.8.1 Mole ular interpretations oftime-temperature equivalen e . . . 25

2.8.2 Time temperature supeposition prin iplefailure . . . 27

3 Polymer vis oelasti behaviourassessment 31

4 Polymer mi ropart - numeri al analysis 51

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2.1 Testing modesandspe imengeometriesfor DMTA[17℄ . . . 22

3.1 Mi ropartpro essing onditions . . . 33

3.2 Log

α

T

(T )

obtained through rstand se ondmethods . . . 46

4.1 Inuen eof initial stresseson thestressed zone . . . 55

4.2 Realand Imaginary partvaluesfor alibrating vis oelasti model . . . 56

4.3 Vis oelasti modelinuen e on stressde rease, withinitial stresses . . . . 60

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2.1 Componentsof thestresstensor [3 ℄ . . . 6

2.2 Maxwell model- adaptedfrom [23℄ . . . 7

2.3 Maxwell's modelbehaviour . . . 8

2.4 Voigt-Kelvinmodel- adaptedfrom[23 ℄. . . 8

2.5 Voigt-Kelvin'smodelbehaviour . . . 9

2.6 Zenermodel- adaptedfrom [6 ℄. . . 9

2.7 Zener'smodel behaviour . . . 10

2.8 ImprovedZener's models . . . 11

2.9 Transitions inpolymer-adapted from[24 ℄ . . . 14

2.10 Regions of vis oelasti behaviour- adaptedfrom [21℄ . . . 14

2.11 Symboli representation oflongandshortrangerelationships ina exible polymermole ule- adaptedfrom [10℄ . . . 15

2.12 Typi al polymer stru ture - adaptedfrom[4℄ . . . 16

2.13 Phaselag between stress and strain- adaptedfrom[10 ℄ . . . 18

2.14 Ve torial representation of omplexmodulus- adaptedfrom [10℄ . . . 19

2.15 Range of frequen iesfor dierent te hniques [24℄ . . . 21

2.16 Boltzmann superposition prin iple- adapted from[6℄ . . . 23

2.17 Two-sitemodel - adaptedfrom[24℄ . . . 25

2.18 Spe i volume versus temperature for a typi al amorphous polymer-adaptedfrom [24℄ . . . 27 3.1 Moldow ® ll timepredi tion. . . 32 3.2 Moldow ® temperature at owfront predi tion . . . 32

3.3 Moldow ® mi ropart warpagepredi tion . . . 33

3.4 Mi ropart . . . 34

3.5 Triton Te hnology TTDMADynami Me hani al Analyser. . . 35

3.6 Sample mounting . . . 35

3.7 Matlabappli ation ow hart . . . 36

3.8 Cole-Cole plot for200

µm

ABSmi roparts. . . 37

µm

3.10 vanGurp Palmen plotfor 200

µm

ABSmi roparts . . . 38

µm

3.12 Densityversus temperature plot for ABS. . . 39

3.13 Lossmodulussuperpositionfor200

µm

ABSmi ropartswithverti alshifting 39 3.14 Storage modulus superposition for 200

µm

ABSmi roparts withverti al shifting . . . 40

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3.16 Storage modulus superposition for 300

µm

ABSmi roparts withverti al

shifting . . . 41

3.17 Mi ropartgeometry inuen e on tests . . . 41

3.25 Flow hartfor TTSP se ond method . . . 47

3.26 Master urveforlossfa torwithiterativemethod,Tref=

30

◦

C

for600

µm

3.27 Master urve for storagemodulus for 600

µm

3.28 Master urve for lossmodulus for 600

µm

3.29 Linear regressionfor a tivationenergy . . . 49

3.30 Comparison of Arrhenius and WLF equations urve tting with experi-mental datafor ABSat severaltemperatures . . . 50

4.1 Mi ropartinanalysis . . . 52

4.2 Initialstate of stress,reading fromle . . . 53

4.3 Partdeformation andshrinkage, s ale fa torof 50 . . . 54

4.4 Field of stresses when a 20

µm

strain is imposed (true deformation s ale fa tor) . . . 55

4.5 Vis oelasti model alibration . . . 57

4.6 Element 63657stress evolution . . . 58

4.9

1%

strain withoutinitial stresses . . . 61

4.10

1%

strain withinitial stresses . . . 62 5.1 Methodoly for inje tionmoulded parts rheologi aland stru tural analysis 64

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Introdu tion

Nowadays one of the most ommon te hniques to produ e parts is inje tion moulding.

A ording toCefamol-NationalMoldIndustryAsso iationinPortugal,thiste hnique in

parti ular,enables theprodu tionofparts atahighrate, in reasingenterprises

ompet-itivity. Thisimportant industry an supply omponents to world leading ompanies in

time, thanksto the Portuguese engineersexperien eandknow-how [1 ℄.

Inthelastde ades omponentsminiaturizationhasbeen agrowingtrendasa

onse-quen e of industry development. Mi roinje tion allows for produ ing parts invery high

quantities with a very low ost, thanks to the pro ess automation level. Produ ts

ob-tained bymi roinje tion mayhave a very distin t purpose. Automotive industry is one

of thelargest onsumers of these mi ro omponents and uses it in many imper eptible

ways. Mi ro omponentsarealsowidelyusedinele tri al/ele troni produ ts,inmedi al

omponents or even insimplethings su hasgears for wristwat hes.

Parts produ ed withthe mi roinje tion pro ess should also ensure good me hani al

hara teristi s su h as durability, physi al stability at working temperatures and even

small strain, sin e any deformation in su h a small pie e an make the whole part to

ollapse.

Inordertofulllthese riteria,itisne essaryaspe ial arewhen hoosingthe

poly-mer whi h will be inje ted. Polymers thatfulllthe riteria areusually moreexpensive

when ompared to ma ro inje tion polymers but,on ethe quantity usedis smaller, the

total pri e of the part is not mu h ae ted. In mi roinje tion theused material is

usu-allynon-re y led, whi h impliesthatthemold shouldbevery well proje ted inorderto

minimize wastes.

Highpre isionandnishingdemands auses themi roinje tionmoldstobedierent

from ma ro inje tion[2℄.

Similarlyto what happens on the onventional inje tion molding pro edures, mi ro

inje tiongeneratesinternal residualstressesinthe omponentsduetonumerousreasons,

in luding: stressesindu edbythemeltoworevenstressesdueto thepressureimposed

during thepolymer's ooling andsolidifying. Thelatterareresponsiblefor thewarpage

and deformation of the part if not properly ontrolled and may lead to a more fragile

omponent after its extra tion. In mi ro inje tion molding this ee t assumes an even

larger importan e be ause of the high pressure and inje tion velo ity needed to avoid

premature polymer solidifying due to thepart high surfa e/volume ratios. Shearstress

resultant from these extreme pro essing onditions may lead to breakage of mole ular

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in-tegrityofthemi ropart. Itisthennoti eabletheimportan eofthepro essing onditions

on the omponent future me hani al behaviour. In fa t, dierent pro essing onditions

may indu e dierent part me hani al properties due to the material distin t

thermo-rheologi alhistories. Thelattermaybetaken intoa ount through ommer ialsoftware

pa kages su h as Autodesk Moldow ®

whi h allow the user to simulate and assess the

omplete partprodu tion,from llingto ooling.

Asso iatedto thethermo-rheologi alhistory,itsindu edresidualstressesduring

pro- essing, it isdetrimental to a knowledge the material me hani al behaviour inorder to

postulate its durability during servi e. It is ommon knowledge that polymersare

vis- oelasti , nevertheless its me hani al behaviour is a onsequen e of its pro essing and

thereforeits thermo-rheologi history andrequiresfurther assessment.

An adequate vis oelasti model needs, therefore, to be provided and a suitable

methodology needsto bedeveloped inorderto postulatethepart inservi ebehaviour.

The present work aims at the predi tion of a mi ro part behaviour during its duty

servi e. Toa omplishthe latteranhybrid methodology isproposed. Firstly,the

mi ro-part thermo-rheologi al histori will be obtained through pro ess simulation by using

Autodesk Moldow ®

tostudypartlling, pa king andwarpage. Thelatterinformation

willbeusedaskeyinputtoAbaqus ®

,wherethemi ropartdutyservi ewillbestudied.

In order to obtain an assertive predi tion of the mi ro part duty servi e the adequate

vis oelasti onstitutivemodelmustalsobepromptedtoAbaqus ®

. Toadequatelymodel

themi ro-partvis oelasti ity,Dynami alMe hani al TemperatureAnalysis(DMTA)will

be arriedout to establish the mi ro-part history at dierent pro essing onditions and

hen e provide insight in what on erns its me hani al behaviour under ertain duty

y les.

In this resear h work, it is intended to develop methodologies and pro edures for

me hani al tests at mi ro s ale, ombining a omplete stru tural analysis based on an

inje tion simulationwithdatafromme hani al tests.

Themain obje tivesare:

literature reviewon polymers onstitutive models,polymertesting, vis oelasti ity

and polymersme hani al behavior;

to runDMTAteststoobtaindata,establishing theme hani albehaviourof mi ro

parts at spe i pro essing onditions;

to develop a polymer onstitutive model from the me hani al tests data using

te hniques su h asthetime temperature superposition prin iple;

toassessme hani alperforman eofami ro omponentwhensubje tedtoa ertain

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Literature review

2.1 Vis oelasti ity

Classi altheoryofelasti itydealswithme hani alpropertiesofelasti solids,for whi h,

ina ordan e withHooke's law, stress is always dire tlyproportional to strain insmall

deformations but independent of the rate of strain. Classi altheory of hydrodynami s

deals withpropertiesofvis ous liquids,for whi h,ina ordan e withNewton'slaw, the

stress is always dire tly proportionalto therate ofstrain but independent of the strain

itself.

Although Hooke's law approa hes the behaviour of many solids for innitesimal

strains,andthatNewton'slawapproa hesmanyliquidsbehaviourforinnitesimalrates

of strains, some deviations may be observed under ertain situations. Thesedeviations

may be distinguishedintwo:

stress-strain relations are more ompli ated when nite strains are imposed on

solids that andeform substantiallywithoutbreaking -theso allednon-Hookean

deformation;

polymeri solutions and undilutedun rosslinked polymersmaydeviatefrom

New-ton's lawwhen insteadyowwithnite strainrates - Non-Newtonianow.

Polymeri materials may present both vis ous resistan e to deformation and

elas-ti ity. Polymers' linear vis oelasti domain is limited to relatively small stresses. This

kind ofmaterials has hara teristi s between thoseof idealelasti s andthose fromideal

Newtonianuids. Anot ompletelysolidmaterialdeformsthroughtimewhensubje ted

to a onstant stress,insteadof having a onstant deformation,in otherword it reeps.

When su ha bodyis onstrained at onstant deformation, thestress required toholdit

diminishes gradually, or relaxes. When an external onstant stress is applied to a not

ompletely liquid material it stores some of the input energy, instead of dissipating it

all as heatduring its owand it mayre overpart of its deformation when thestress is

removed (thisis alledelasti re oil) [8℄.

Idealelasti solidsstoreallsupplieddeformationenergywhilstidealNewtonianuids

dissipate all supplied energy (ex ept hydrostati state of stress) [6℄. When sinusoidally

os illating stresses are applied to su h bodies, the resulting strain is neither exa tly in

phasewiththestress(asitwouldbeforaperfe tlyelasti solid)or

90

◦

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would be for aperfe tly vis ous liquid). It isthenpossibleto build uniaxial vis oelasti

behaviourmodels ombining solidsbasi physi almodels-springs andNewtonianuids

models - dampers[6 ℄.

In ertain ases,su hasvul anized( ross-linked)rubber,owisnotpossibleandthe

material has a unique onguration assumed inthe absen e of deforming stresses. The

vis ousresistan etodeformationisnoti edduetotherubber'sdelayona hangeinstress

[8℄. Insu hmaterialssomeoftheenergyinputisstoredandre overedinea h y le,and

some isdissipated intheformof heat. Materials whose behaviourexhibits su h

hara -teristi sare alledvis oelasti . Ifbothstrainandrateofstrainareinnitesimalandthe

time-dependent stress-strain relations an be des ribed by linear dierential equations

with onstant oe ients,linear vis oelasti behaviouro urs. Inanexperiment with

these materials the ratio ofstress tostrain isa fun tionof time(orfrequen y).

A " onstitutive equation" or "rheologi al equation of state" relates stress, strain

and their time dependen e. Finite strain/strain rates produ e ompli ated onstitutive

equations. Onthe other hand,innitesimal strain/strain rates ( orresponding to linear

vis oelasti behaviour) have relatively simpler onstitutive equations. Vis oelasti ity's

onstitutive laws maybeexpressedthroughthe Boltzmann Superposition prin iple [6℄.

2.2 Linear onstitutive equations (simple shear)

Polymersvis oelasti ityisdependantofmole ularstru ture,modesofmole ularmotion,

mole ular weight and its distribution, temperature, on entration, hemi al stru ture,

among others.

2.2.1 Equations of hange

Experimental measurements of me hani al properties relate stress and strain applied

to a body. Assuming that the pro ess is isothermal (therefore despising the equation

of onservation of energy), this relation depends not only on stress-strain onstitutive

equation but also on equation of ontinuity and the equation of motion. The earlier

expresses onservationof mass(Equation2.1)

∂

∂t

ρ = −

3

X

i=1

∂

∂x

i

(ρυ

i

)

(2.1)

andthe latter expressesthe onservation ofmomentum (Equation2.2

ρ

∂

∂t

υ

j

+

3

X

i=1

υ

i

∂

∂x

i

υ

j

!

=

3

X

i=1

∂

∂x

i

σ

ij

+ ρg

j

(2.2)

Intheseequations

ρ

isdensity,

t

istime,

x

i

arethreeCartesian oordinatesand

υ

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arenegligible. Here,internalstatesofstressandstrain anbe al ulatedfromobservable

quantities bythe onstitutive equationalone [10℄.

2.2.2 Innitesimal strain tensor

Like in a perfe tly elasti body, vis oelasti bodies exhibit a state of deformation at a

given point whi h an be spe ied by a strain tensor, representing relative hanges in

dimensions and angles of a small ubi al element in that position. The state of stress

is spe ied by a stress tensor whi h represents the for es a ting on dierent fa es of a

ubi alelement. Therateofstraintensor des ribesthetimederivativesof theserelative

dimensions andangles.

For an innitesimal deformation, the omponents of the innitesimal strain tensor,

inre tangular oordinates withthethree artesian dire tions

(1, 2, 3)

are:

γ

ij

=







2

∂u

1 ∂x

1 ∂u

2 ∂x

1

+

∂u

1 ∂x

2 ∂u

3 ∂x

1

+

∂u

1 ∂x

3 ∂u

2 ∂x

1

+

∂u

1 ∂x

2

∂u

2 ∂x

2 ∂u

2 ∂x

3

+

∂u

3 ∂x

2 ∂u

3 ∂x

1

+

∂u

1 ∂x

3 ∂u

2 ∂x

3

+

∂u

3 ∂x

2

∂u

3 ∂x

3







(2.3)

Here,

x

i

and

u

i

arethe point oordinates and its displa ement in thestrained state (

u

i

= x

i

− x

0 i

), respe tively. The rateof strain tensor,

˙γ

issimilar to strain tensor with

u

i

repla edby

υ

i

,thevelo ityofdispla ement [10 ℄.

2.2.3 Stress tensor

Stress omponentsof

σ

ij

thatappearin2.2 an berepresentedby:

σ

ij

=





σ

11

σ

12

σ

13

σ

21

σ

22

σ

23

σ

31

σ

32

σ

33





(2.4)

For a stress omponent

σ

ij

the rst index

i

indi ates the plane that the stress is perpendi ulartoandthese ondindex

j

denotestheparalleldire tioninwhi hthestress a ts, as seen in Figure 2.1. Normal stresses

σ

ii

are usually positive for tension and negative for ompression [10 ℄.

2.2.4 Constitutive equation for linear vis oelasti ity (simple shear)

When deformation is homogeneous/uniform, stress and strain omponents do not vary

with position and are independent of

x

i

. In ase of simple shear, where two opposite fa es aredispla ed by sliding, both tensors assume more simple forms. Assuming that

theplane

13

slides indire tion 1,straintensor isgiven by:

γ

ij

=





0 γ

12

0 γ

21

0

0 



(2.5)

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Figure2.1: Componentsof thestresstensor [3℄

where

γ

12

= γ

21

=

∂u

1 ∂x

2

= tan(α) ≈ α

and stresstensor isgiven by:

σ

ij

=





−P

σ

12

0 σ

21

−P

0

0 −P





(2.6)

where Pisanisotropi pressure. Strain

γ

12

andstress

σ

12

arefun tionsoftimeandthey arerelated bya onstitutive equationfor linearvis oelasti ity:

σ

21

(t) =

Z

t

−∞

G(t − t

′

) ˙γ

21

(t

′

)dt

′

(2.7)

where

˙γ

21

= ∂γ

21

/∂t

isthe shear rate,

G(t)

isthe relaxation modulus.

Strain an beexpressedinterms ofthe history of thetimederivative of thestress:

γ

21

(t) =

Z

t

−∞

J(t − t

′

) ˙σ

21

(t

′

)dt

′

(2.8)

where

˙σ

21

= ∂σ

21

/∂t

and

J(t)

is the reep omplian e.

The equation that expresses stress in terms of the history of thestrain rather than

theits timederivative is:

σ

21

(t) = −

Z

t

−∞

m(t − t

′

)γ

21

(t, t

′

)dt

′

(2.9)

wherem(t)isthememoryfun tionanditisequalto

−dG(t)/dt

and

γ

21

(t, t

′

)

isareferen e state.

Knowingtheshearrelaxationmodulus,thememoryfun tion,orthe reep omplian e

fun tionofamaterial,enables topredi titsstress-strainrelations aslongasmotionsare

(27)

2.3 Me hani al models for vis oelasti behaviour

Toperform al ulationsitisusefultohavethedesiredvis oelasti propertyasanequation

instead of a graph or table of data. It is then onvenient to have a general form that

ontains su ient parameters to t experimental data for a wide range of polymers.

This form an emulate the time or frequen y dependen e of the vis oelasti properties

byusingame hani al modelwithasu ient numberofelasti elements(representedby

springs) andvis ous elements (represented bydashpots) [10 ℄.

2.3.1 Maxwell Model

Oneofthesimplestmodelsthat anemulateavis oelasti systemistheMaxwellmodel,

depi ted in Figure 2.2. In this model, whi h is omposed by a series asso iation of

one spring and one dashpot, the total deformation is equal to the sum of ea h of the

deformations, i.e.,

ε = ε

s

+ ε

d

.

Figure2.2: Maxwellmodel- adaptedfrom[23℄

Here, if

k

represents spring's stiness and

µ

uid's vis osity the following equation an bewritten:

dε

dt

=

dε

s

dt

+

dε

d

dt

=

1 k

dσ

dt

+

σ

µ

(2.10)

A new on ept, the relaxation time of the element that an be understood as a

measure at the time required for stress relaxation an be written as

τ

i

= µ

i

/k

i

[10℄. Assuming thata onstant stress

σ

0

a ts,integrating Equation 2.10 results in the reep behaviour,obtained as:

ε(t) =

σ

0

µ

t + C1

(2.11)

C1

an be obtained from the initial ondition

ε(0)

as

σ

0

/k

, and dividing strain by stress, reep omplian e an be obtained as:

J(t) =

t

µ

+

1 k

(2.12)

Equation2.12suggeststhat reep omplian ein reaseslinearlyby

t/µ

as anbeseen inFigure 2.3.

(28)

(a)Maxwell's reep omplian e urve

-adaptedfrom[6℄

(b) Maxwell's relaxation modulus

urve-adaptedfrom[6℄

Figure2.3: Maxwell's model behaviour

However, this is not noti ed experimentally and what happens is the opposite, a

de rease over time, so this simple model is not good enough to represent omplian e.

Still, making

ε = ε

0

and integrating 2.10this model an predi tstressas:

ln(σ) = −

k

µ

t + C2

(2.13)

Here, assumingthat

σ(0) = σ

0

one an writethat relaxationisdepi ted by:

G(t) = k exp(−

kt

µ

)

(2.14)

Thisis on omitant withthebehaviourexperimentally expe ted [6℄.

2.3.2 Voigt-Kelvin Model

Anothersimplemodeltorepresentavis oelasti materialistheVoigt-Kelvinmodel. The

latter is also omposed byone spring and one dashpot, but this timethe asso iationis

made inparallel.

(29)

Here, the ratio between thespring anddashpot isdened astheretardation time,a

measure of the time required for the extension of the spring to a hieve its equilibrium

length whileretarded bythe dashpot. Inthis model, reep omplian e isgiven by:

J(t) =

1 k

h

1 − e

−kt

µ

i

(2.15)

Still,relaxation an be written as:

G(t) = k

(2.16)

Similarlytothepreviousmodel,thisonefailsinpredi tingthe reep omplian esin e

it la ks the initial elasti response. Oppositelyto theprevious model, this one doesnot

predi t stress relaxation sin e

G(t)

isgiven by a onstant whi h is not time dependent and sodoesnot hange overtime, asit an be seeninFigure 2.5[6 ℄.

(a) Voigt-Kelvin's reep omplian e

urve-adapted from[6℄

(b)Voigt-Kelvin's relaxationmodulus

urve-adaptedfrom[6℄

Figure 2.5: Voigt-Kelvin's modelbehaviour

2.3.3 Zener Model

There are more omplex models that an emulate a vis oelasti material's behaviour.

One of them isthe Zener model. This model lls some aws of previous models but it

stillis omposedbythem. It anberepresentedasone simpleMaxwellmodelinparallel

withone spring.

(30)

Thismodelstatesthat reep omplian e and relaxationaregiven by:

J(t) =

1 k

0

1 −

k

1

k

0

+ k

1

· e

−t

ρ1

(2.17)

G(t) = k

0

+ k

1

e

−t

λ1

(2.18)

where

ρ

1

and

λ

1

aretime onstants, given by:

ρ

1

=

µ

1

k

0

k

1

(k

0

+ k

1

)

(2.19)

λ

1

=

µ

1

k

1

(2.20)

Although the shapes of the urves produ ed bythis model are similar to those

ob-tained experimentally (Figure 2.7), it doesn't allow for a rigorous adjustment of the

behaviour of polymers sin e time onstants

ρ

1

and

λ

1

are insu ient. On e polymers often have a mi rostru tural omplex stru ture they present a ontinuous spe trum of

time onstantsandthereis aneed ofusingmodels withenoughspringsand dashpotsto

a hieve agoodapproximation tothe polymer realbehaviour[6℄.

(a) Zener's reep omplian e urve

-adaptedfrom[6℄

(b)Zener'srelaxationmodulus urve

-adaptedfrom[6℄

Figure2.7: Zener's modelbehaviour

Improved models

To a hieve a good aproximation, more omplex models whi h in lude paired or series

groups ofsimpler models weredeveloped,su hasimproved Zenermodels or generalized

Maxwell models. The main dieren e between these is the addition of one element

omposedonly bya spring asit an be seeninZener'smodel[6℄.

Improved Zenermodels reep omplian e andrelaxation anbeobtained as:

J(t) =

1 k

0

+

n

X

i=1

1 k

i

h

1 − e

−t

ρi

i

where

ρ

i

=

µ

i

k

i

(2.21)

G(t) = k

0

+

n

X

i=1

k

i

· e

−t

λi

_where

_λ

_i

₌

µ

i

k

i

(2.22)

Due to this large amount of parameters, rigorous representation of vis oelasti

be-haviour demands an enormous experimental eort whi h may be ome time- onsuming

(31)

(a)Parallelasso iation-adaptedfrom [6℄

(b)Seriesasso iation-adapted from[6℄

Figure2.8: Improved Zener'smodels

2.3.4 Vis oelasti spe tra

Maxwell elements in series have the properties of a single element with

J =

P J

i

and

1/µ

0

=

P 1/µ

i

. Voigt elementsinparallel have thesame properties asa single element with

G =

P G

i

and

µ

0

=

P µ

i

. However, whenmore ompli ated modelssu hasthose presentedbeforeareused, theseproperties relations arenot sodire t.

A group of parallel Maxwell elements represents a dis rete spe trum of relaxation

times, ea h one

τ

i

being asso iated with a spe tral strength

k

i

. In a parallel arrange-ment thefor es areadditive and vis oelasti fun tionssu hasG(t)maybeobtained by

summing termsequal to those inEquation2.14, obtaining:

G(t) =

n

X

i=1

k

i

e

−t

τi

(2.23)

AgroupofVoigtelementsinseriesrepresentsadis retespe trumofretardationtimes,

ea h beingasso iated witha spe tral omplian e magnitude

J

. In a seriesarrangement thestrainsareadditive,the omplian efun tion

J(t)

maybeobtainedbysumming2.15 overall theserieselements:

J(t) =

n

X

i=1

µ

i

1 − e

−t

τi

(2.24)

Theseequations antrelaxationand reepdatawithanydesireddegreeofa ura y

if

n

islargeenough bydeterminingthedis retespe trum of urves, ea hwithalo ation

τ

i

andintensity

k

i

. Experimentallyitisdi ulttodeterminetheparameters

τ

i

or

k

i

and its arbitrary hoi e may be enough to predi t ma ros opi behavior but wouldn't have

enoughvaluefortheoreti alinterpretation. This anbeavoidedbyanalysingvis oelasti

(32)

Ifthenumber of elements ina generalized Maxwell modelin reases indenitely, the

resultis a ontinuousspe trum, where ea hinnitesimal ontribution to systemrigidity

is

F dτ

and is asso iated with relaxation times between

τ

and

τ + dτ

. This ontinuous relaxationspe trumisdenedas

H dln(τ )

,where

H = F τ

. Inthese onditions,Equation 2.23 be omes:

G(t) = G

e

+

Z

∞

−∞

He

−t

τ

dln(τ )

(2.25)

where

G

e

is a onstant added to allow for a dis rete ontribution of the spe trum with

τ = ∞

for vis oelasti solidsand for vis oelasti liquidsitassumes value0.

Thesame analogy anbe establishedfor theVoigt-Kelvin model. Here ifthemodel

is indenitely extent it will represent a ontinuous spe trum of retardation times

L

. Equation 2.24maybewritten as:

J(t) = J

g

+

Z

∞

−∞

L

1 − e

−t

τi

dln(τ ) +

t

µ

0

(2.26)

Here, an instantaneous omplian e

J

g

is added to exist the possibility of a dis rete ontribution when

τ = 0

. This parameter an not be determined experimentally but should be takenina ount [10 ℄.

2.4 Polymeri materials

Industrial polymers onsist of large mole ules whi h are primarily ovalently bonded.

These mole ules may arrange themselves in main hains and may present side hains,

ir ular mole ulesand inter onne tions amongthem bymany me hanisms.

Atoms inside mole ulesmayintera t withatoms froma dierent mole ule. Thereby

largermole ules attra tea h othermore strongly than shorterones. Thisis ree ted in

polymersmelting points, whi h generally in rease within reasing hain length. Hen e,

melting points may be a good indi ator of the bond strength among mole ules of the

ompound.

Thesepolymer long hainsare omposed bysmall basi units alledmers,whi h are

repeated to generate the hain. The bigger the number of repeating mers, the larger

the degree of polymerization of the polymer, whi h an be evaluated by an average

number sin e dierent mole ules may have dierent hain lengths. As the number of

intermole ular attra tions between mers in reases, the total for e of attra tion and the

me hani al strength of the polymer generally in reases too, sin e me hani al failure is

asso iatedwithbreakingtheattra tivefor es between theatoms inmole ules. However,

in reasingthedegreeofpolimerizationindenitelydoesnotensureanin reaseinstrength

be ause the hain may be ome so long thatsome parts may behave independently and

an even break from the main hain. The large dimensions of these mole ules are also

responsiblefor theunique propertiesof polymersrelatively toother materials.

Thermalenergymay ausevibration oftheatomsandin reaseatomi andmole ular

motions. Thesemotions an be:

atomi vibrations in the arbon ba kbone hain, whi h may in rease the average

(33)

vibrationsperpendi ulartothebond dire tionwhi h an ause smallvibrationsin

bondangles;

rotation ofthe arbon atoms around thebonds ausing twisting ofthe hain.

Thesevibrations an be understoodasameasureofthekineti energy whi h

ontin-uously hanges with time. Therefore, the average interatomi distan e between atoms

in reases withtemperature.

Be ausebond within the hain ba kbone are ovalent,they areof highstabilityand

strength, and their rupture requires large amounts of energy. For es involved in

inter-a tion with neighbouring mole ules are signi antly weaker and failure of a polymer is

morelikelytoo urbybreakingthese onne tionsratherthanbyrupturingthemole ule

itself [18 ℄.

There anbestill onne tionpointsbetweenlinearpolymers hainsalongtheirlength

makinga ross-linkedstru ture. Whenthe ross-linkingagentinthepolymerisa tivated

by temperatures, hemi al ross links are obtained and therefore a thermoset polymer

may be obtained. Physi al ross-links may alsoo ur temporarily when longmole ules

inlinear polymersget entangle. Anotherphenomena that an o uris hain-bran hing,

where ase ondary haininitiates from a point on themain hain. These bran h points

leadto onsiderabledieren esinme hani al behaviourwhen omparingthesame

poly-mer withand withoutbran hingpoints.

Polymers hains may be of various lengths and therefore their mole ular mass will

also vary. Mass distribution is of importan e in properties of the polymer sin e it will

inuen iate the polymernalstate.

The manner in whi h polymers mers arrange themselves in the ba kbone lassies

thepolymer,makingita opolymerorahomopolymer. Ahomopolymerisasequen eof

identi almers inthe hainba kbone,forminglongequal groups. A opolymerisformed

when hemi al ombinationexistsinthemain hainbetweentwoor moredierentmers.

When ombination is made by groups a blo k opolymer is obtained. When they are

disposedinanyordera random opolymer isobtained.

Thesete hniques ofmodifying homopolymersmay alsoform grafts,long side hains

of a se ond polymer hemi ally atta hed to the base polymer, or blends. By applying

them,du tilityortoughnessofbrittlehomopolymersareenhan edaswellasthestiness

of some rubbery polymers. A rylonitrile-Butadiene-Styrene opolymer (ABS) is an

ex-ample ofablend. InABStheintrodu tion ofthe butadiene,therubberphase,improves

thepolymer impa tresistan e.Beyond enhan ing polymersme hani al properties, these

additivesmayalso improve polymer pro essabilityand resistan eto degradation[24 ℄.

2.4.1 Temperature inuen e on regions of vis oelasti behaviour

Thermoplasti s and elastomersshow verylarge hangesinpropertieswith temperature.

Thermoplasti s temperaturedependen ehastwopointsofmajor interest. Usually

poly-mers hange fromaglassystate to arubberystateaseitherthetemperatureisraisedor

the time-s ale of the experiment is in reased. At lowtemperatures, in theglassy state,

polymersare hard and brittle. Itsstiness relateswith hanges instored elasti energy

ondeformationwhi hareasso iatedwithsmalldispla ementsofthemole ulesfrom

equi-librium positions. Here,thermal energyis insu ient to over omethepotential barriers

(34)

rises, theamplitude ofvibrationalmotion be omesgreater,andthermal energybe omes

omparable to the potential energy barriers. Thereby, mole ules an adopt

onforma-tions where they rea h a minimum of free energy. These rubber-like deformations are

asso iated with hanges inthe mole ular onformations. Thissuggestthatthere isonly

one vis oelasti transition, from theglassy state to rubber. In pra ti e, there are more

relaxation transitions,asit an be seeninFigure 2.9[24 ℄.

Figure2.9: Transitionsinpolymer-adaptedfrom [24℄

The primary transition where the greatest hange inmodulus o urs is denoted

T

g

, and orresponds to a relaxation alled

α

. Here the modulus de reases abruptly in the glass transition region. Glass transition should be seen as a region instead of a xed

temperature and itsextent an getupto

20

◦

C

[24 ℄.

Beyond these regions of major importan e there are other regions requiring further

analysis. As the temperature isfurther in reased themodulus rea hesagain a plateau.

Thisnew plateu

E

2

an beseenin Figure2.10.

Figure 2.10: Regions of vis oelasti behaviour- adapted from[21 ℄

Here short-range motions of thepolymers o ur very fast unlike thelong-range

mo-tions of hains that arestill restri ted by the presen eof strong lo al intera tions with

neighbouring hains,asFigure2.11exemplies.

In aseof ross-linkedpolymerstheseintera tionsareprimary hemi albonds,known

as entanglements in linear polymers. In the rubbery plateau segments of hains an

(35)

Figure 2.11: Symboli representation of longand short range relationships ina exible

polymer mole ule-adaptedfrom [10 ℄

o ur.

In the rubbery region, linear and ross-linked polymers behave approximately but

as the temperature rises this behaviour starts to dier. Be ause of the primary

hem-i al bondsthe hanges in modulus relatively to those of the glasstransition are small.

Modulusonlystartstode reaseagainsigni antlywhentemperatureapproa hestothose

where hemi aldegradationbegins. TheabruptdrophighlightedinFigure2.10isrea hed

due to mole ular motions ausedby thein rease of temperature until a state where all

themole ulesareabletobeginto translate. Theselarge temperaturesareresponsibleto

over ome the potential energy barriersor thebreakof lo al hainintera tions whi h are

no longer ableto prevent themole ular ow[24 ℄.

2.4.2 Thermoplasti s and thermosets

Polymers are usually divided inthree main ategories: thermoplasti s, thermosets and

elastomers.

When onsidering thermoplasti s, the main hain mole ules are bonded by primary

ovalentbondswithinthe hain. Se ondaryfor esbetweentheindu edorpresentdipoles

are responsible for the attra tion between hains . Therefore, mole ules are apable of

individualmotionandwhen asolidpolymerissubmitted toheatit an melt,beingable

to owand llamould. This apabilityofbeingpro essedbyinje tionmouldingmakes

thermoplasti s themost ommonlyusedtype ofpolymer.

Inthermosets, ovalent bonds mayexist between dierent hains. By existingthese

primary bondsbetween dierent mole ules, intermole ular for es are strengthened and

the identi ation of any long single hain is very di ult if not impossible, reating a

single large network instead of existing many single mole ules. Sin e ovalent bonds

annot be broken without degrading the properties of thepolymer, these materials are

not abletobemoltenbyheating,but they har andde ompose. Be auseofthereferred

ovalentbondsthesepolymersarestrengthenedbutthe apabilityofmouldllingislost

[8℄.

2.4.3 Amorphous and rystalline thermoplasti s

Likesolidmaterials, polymersmaybe lassieda ordingtotheirmole ular stru turein

(36)

Many polymers may form regions of ristalinity if ooled slowly from melt regions

so thatthe longmole ules an be ome ordered in thesolid state asseen in Figure2.12

a). Thedegree of rystalinitymayvarybe ause oolingfroma moltenstatemayindu e

imperfe tions ausedbytanglingandtwistingoflinear hains. Also,inorderto rystalize,

themole ulesmusthaveregularstru ture. Threedimensionaljuxtapositionofmole ular

hains may also lead to formation of rystallites, though for this to happen they have

to be oriented in the same way. This way solid polymers that present a periodi al

tridimensionalorganizationare alled rystallinepolymersandthosewhopresentregions

of rystallinity are alled semi- rystalline polymers [24 ℄. These rystallites may a t as

rosslinks, making the rubbery plateau more pronoun ed in polymers with a ertain

degree of rystallinity than in the amorphous ones. When a polymer has a ertain

degree of ristalinity,regionsthat an be easilyidentiedinamorphouspolymersareno

longer so well dened. Usually these kind of polymers manifest a signi antly less fall

in modulus over theglass transition temperature. It an rea h up to one or two orders

of magnitude when ompared to amorphouspolymers. Also, the hange in modulus or

loss fa tor with temperature is more gradual, what indi ates a larger relaxation time

spe trum. At higher temperatures rystalline regionsredu e mole ular mobility and so

the polymer stops behaving as a rubber-like material [24 ℄. Also, rystalline polymers

exhibitadierent maintransitionthantheamorphousones. Theymeltfromanordered

rystal dire tly to a liquid. This usually happens at a region of temperatures alled

temperature ofmelting

T

m

. Here, the modulusdrops sharplyto therubberyplateau.

(a)Crystallinepolymerstru ture

(b)Amorphouspolymerstru ture

Figure 2.12: Typi alpolymerstru ture - adaptedfrom [4℄

Amorphous polymers are materials that do not have a long range periodi al

orga-nization, as Figure 2.12 b) suggests. These materials may present an elasti modulus

around

10

9

Pa at higher frequen ies, where the vis ous omponent is not sonoti eable.

Whenthematerialhashighmole ular mass,thereisaplateauinmodulusandmole ular

ow o urs at frequen ies below

10

−

5

Hz. This plateau in the rubbery region an be explainedduetotheentanglementofmole ulesandthephenomenaofphysi al

rosslink-ing that tend to restri t mole ular ow through the formation of temporary networks.

At longtimes these entanglements tendto indu e irreversible owunlike what happens

with permanent hemi al ross-links (for example when a rubber is vul anized). The

modulus valueinthisplateau isdire tly proportionalto theamount of these ross-links

per unit of volume. In fa t, lower mass samples tend to not show a rubbery plateau,

(37)

mole ular weight this plateau is more noti eable sin e it will take longer time for the

hains to disentangle, in reasing its length along the timeaxes [9 ℄. In the glassy state,

at lower temperatures, the stiness of polymer is related to hanges in stored elasti

energy aused bydeformation, whi h areexplained bysmall displa ements ofmole ules

fromtheirequilibriumpositions. Athighertemperatures,intherubberystate,mole ular

hainshavegreaterexibilityand anassumeformswheretheminimumfreeenergystate

anberea hed. Therefore,elasti deformationsareasso iatedwith hangesinmole ular

onformations. Infa t, thereareseveraltransitions inanamorphouspolymer. Atlower

temperaturesthere are usuallymany se ondary transitions, alled

β

transitions, involv-ing small hanges inmodulus. These transitions an be related to side-group motions.

The largest hange in modulus in this kind of polymers o urs in the glass transition

range oftemperature [24℄.

2.5 Linear time-dependent experiments (shear)

Constitutive equations des ribe theresponseof a linearvis oelasti material to various

patternsof stress andstrainin simpleshear.

2.5.1 Stress relaxation

Ifashearstrain

γ

isimposedinabrieftimeperiod

ξ

bya onstantrateofstrain

˙γ = γ/ξ

, Equation 2.7be omes

σ(t) =

Z

t

0 t

0 −

ξ

G(t − t

′

)

γ

ξ

is assumed to be mu h larger than the intervalof appli ation ofthestrain

ξ

,Equation 2.27 an be written as

σ(t) = γG(t)

(2.28)

where

G(t)

istherelaxationmodulus. Generally aratio ofa stresstostrain is alled amodulusandforaperfe tlyelasti solidtheshearmodulus

G

is

σ/γ

. Intheparti ular ase G(t)isthe shear modulus time-dependent equivalent[10 ℄.

2.5.2 Creep

Whenashearstress

σ

isappliedwithinabriefperiodandisheld onstant,thedependen e ofthestrain

γ

ontime anbeobtained fromEquation2.8onthesamewayasEquation 2.28 wasobtained fromEquation2.27 andis givenby

(38)

Here,

J(t)

isthe reep omplian e,whi hfor aperfe tly elasti solidisequalto

1/G

. However, for a vis oelasti materialthis isnot trueand

J(t) 6=

1 G(t)

[10 ℄.

2.5.3 Sinusoidal soli itationsand dynami experiments

Stress may be varied periodi ally with sinusoidal soli itations at a frequen y

f

in y- les/se (Hz) or

w = 2πf

in radians/se . A periodi experiment at frequen y

w

is qualitatively equivalent to an experiment at time

t = w

−

₁

. Regarding the following

onstitutive equationfor a sinusoidal strain:

γ = γ

0

sin(wt)

(2.30)

where

γ

0

isthemaximumamplitudestrain. Cal ulatingthederivativeone anobtain:

˙γ = wγ

0

cos(wt)

(2.31)

Substituting in Equation 2.7 and making the variable hange

s = t − t

′

it an be obtained:

σ(t) = γ

0

w

Z

∞

0

G(s)sin(ws)ds

sin(wt) + γ

0

w

Z

∞

0

G(s)cos(ws)ds

cos(wt)

(2.32)

Figure2.13: Phaselagbetween stress and strain-adapted from[10 ℄

The two omponents of material's stress response may be distinguished as one in

phase (

sin(wt)

) an one other out of phase (

cos(wt)

) with the imposed strain. This emphasises the ideathat vis oelasti materials have an intermediatebehaviourbetween

ideal Newtonian uidsand ideal solids [6 ℄. Sin e the quantities inbra kets inEquation

2.32 are only fun tions of frequen y and not the elapsed, Equation 2.32 an be written

as:

σ = γ

0

(G

′

sin(wt) + G

′′

(39)

Here,two frequen y dependant fun tionsaredened:

G

′

(w)

,theshear storage mod-ulus and

G

′′

(w)

,the shear lossmodulus. Asitis asinusoidal soli itationitisofinterest to formulatethe stressinan alternativeform, displayingthestressamplitude

σ

0

_(w)

and

thephaseangle

δ(w)

(2.36)

anda newrelation an thenbe established, thelossfa tor:

tan(δ) =

G

′′

G

′

(2.37)

Whenthe stressand strain vary sinusoidally, they an also be des ribed as omplex

numbers (refered withexponent

∗

) and sothe modulus isalso a omplex number given

by

G

∗

=

σ

∗

γ

∗

= G

′

+ iG

′′

(2.38)

where the real part is the elasti /storage part and the imaginary partis the energy

dissipation partasilustratedingure2.14 [6℄:

Figure2.14: Ve torial representation of omplex modulus - adaptedfrom[10℄

Hereis noti eablethat

G

′

is the omponent of

G∗

inphasewiththe strainandthat

G

′′

(40)

|G

∗

| =

σ

0

γ

0

=

p

G

′

₂

_{+ G}

′′

₂

(2.39)

Datafromsinusoidal an also be expressedintermsof a omplex omplian e

J∗ =

γ∗

σ∗

=

1 G∗

= J

′

− iJ

′′

(2.40) Where

J

′

,the storage omplian e, is theinphase omponent of thestress to strain

ratio to stress,whilst

J

′′

,the loss omplian e isthe

90

out of phase omponent.

G

′

and

J

′

aredire tlyproportionalto theaverageenergy storedina y leofdeformation whilst

G

′′

and

J

′′

are dire tlyproportional to theaverage dissipation or lossof energy as heat

inthesame y le.

Even if

J∗ = 1/G∗

, individual omponents of moduleand omplian e arenot re ip-ro ally related,thus an berelatedby[10 ℄:

J

′

=

G

′

(G

′

2

_{+ G}

′′

2

₎

=

1 G

′

1 + tan

2

_(δ)

(2.41)

J

′′

=

G

′′

(G

′

₂

_{+ G}

′′

₂

₎

=

1 G

′′

1 + tan

2

_(δ)

−

₁

(2.42)

G

′

=

J

′

(J

′

2

_{+ J}

′′

2

₎

=

1 J

′

1 + tan

2

_(δ)

(2.43)

G

′′

=

J

′′

(J

′

₂

_{+ J}

′′

₂

₎

=

1 J

′′

1 + tan

2

_(δ)

−

₁

(2.44)

The omplexmodule anbedeterminedexperimentallyforawiderangeoffrequen ies

w

throughDynami Me hani alAnalysis(DMA)or,sin eitmayvarywithtemperature, throughDynami Me hani alThermalAnalysis(DMTA).Tipi ally,frequen ies anvary

from

10

−

5

to

10

8

_Hz

but below

10

−

2

theexperiment is time onsuming and above

10

2

resonan emay o ur depending onthetype of essayand thetest spe imen[6 ℄.

2.6 Dynami and transient experiments

For understanding thevis oelasti behaviourof polymers, data arerequired overa wide

rangeoftime(orfrequen y)andtemperature. Thisrangeoftimemayberedu edthrough

theappli ation of the time-temperature superpositionprin iple aswill be dis ussed

fur-ther. Also the number of experiments may be redu ed through the relations between

properties. In order to over a wide range of both timeand temperature there are ve

main lasses ofexperiments:

Transient measurements: reep and stressrelaxation

Low-frequen y vibrations: freeos illating methods

High-frequen y vibrations: resonan emethods

(41)

Wave propagationmethods

Ea h te hnique hasits own rangeoffrequen y appli ability as an beseen inFigure

2.15.

Figure2.15: Range offrequen ies for dierent te hniques[24℄

Measurements of polymers' vis oelasti behaviour are sensitive to small hanges in

temperatureandso,a ontrolledtemperatureenvironmentisessential. Insomeparti ular

ases, su h asnylon, humidity must also be ontrolled sin e they are very hygros opi .

These measurements must be done in values of strain that does not ex eed one per

ent sin e that is taken as the limit value for linear vis oelasti ity [24℄. Within this

per entage the hange in ross-se tion is small and hen e stress ause by strain is also

small. When preparing an experiment, stresses indu ed by the lamping for es at the

end of the test spe imenwill indu e a dierent state of stress from theother zones. To

avoid the inuen e of this stress the overall length of the spe imen must be 10 times

its diameter or when dealing with oriented samples

10pE/G

where

E

is the Young's modulus inthe bre dire tion and

G

the shearmodulus.

Many dynami te hniques areused to overwide ranges of frequen ies, su h asfree

os illating pendulum methods, for ed vibration te hniques or DMTA. Free os illating

pendulumaresimplebut anonlygofrom

10

−

1

to

10Hz

anditsresultsdoesnotpresent mu hrepeatability. Onthe otherhand,for edvibration methodsaremore omplexand

mayyieldhigherreprodu ibilityand anextentthe frequen yrangeinmoreonede ade.

Nevertheless,the frequen yofvibrationsdependsonthestinessofthespe imen,whi h

varieswithtemperature [24℄.

However, another widely used te hnique is DMTA (Dynami me hani al thermal

analysis). Here, a small deformation is applied to a sample in a y li manner. This

allowstostudythematerialsresponsetostress,temperature,frequen yandothervalues.

DMTAdiersfromThermome hani alAnalysis(TMA)sin ethelatterappliesa onstant

stati for etoamaterialandevaluatesthematerial hangewithtemperature orintime,

reporting dimentional hanges, the oe ient of thermalexpansion. Ontheohter hand,

inDMTAanos illatoryfor eisapplied andmaterialsresponseinstiness anddamping

is obtained. Re ent DMTA ma hines are also able to measure the same properties as

(42)

Theamountofdeformationthatthesampleexhibitsisrelatedtoitsstiness. Thanks

to high pre ision sensors in DMTA equipments, su hs as LVDT sensors, damping may

bemeasuredwithhighpre isionandapplyingthistothemeasuredmodulus,thestorage

modulus andlossmodulusmaybe obtained.

TestsinDMTAmaybedoneinvarious manners,su hasdual antilever, three-point

bending, tension, ompression or shear sandwi h. Ea h testing mode and the geometry

of the sample has its own appli ability. It is di tated by thesample's physi al state at

thebeginning of the experiment and its di ulty inapplyingtheload. Some guidelines

to preferen ialgeometries and typesoftestsaregiven inTable 2.1.

Table2.1: Testingmodesand spe imengeometries for DMTA [17 ℄

Sample modulus Preferred Geometry Sample Thi kness (mm) Free Length (mm) Ideal Heating Rate

(

◦

C/min)

2

DMTA latest developments allows tests to be made from temperature ranges from

−150

◦

C

to

600

◦

C

andfrequen y ranges from

10

−

6

to

200Hz

.

2.7 Boltzmann Superposition prin iple

Boltzmann SuperpositionPrin iple statesthattheresponseofamaterialto agiven load

is independent of the response of the material to any pre-imposed load so the strain

response due to omplex loading is the sum of the strains due to ea h step, as an be

seen inFigure2.16. Thisisonly validinlinearvis oelasti region.

Considering a bar subje ted to uniaxial stress variable in time, reep stress an be

approximated by a series of in rements

∆σ

i

applied in times t'

i

. The total resulting strainint>t'

n

is given by Equation2.45.

(43)

Figure 2.16: Boltzmann superposition prin iple- adapted from[6℄

ε(t) = ∆σ

1

J(t − t

′

1

) + ∆σ

2

J(t − t

′

2

) + ... + ∆σ

n

J(t − t

′

n

)

(2.45)

where

J(t)

isthe Complian e fun tion.

Whenthe number ofin rementstends to innity,Boltzmann SuperpositionIntegral

is obtained:

ε(t) =

Z

t

−∞

J(t − t

′

)

dσ(t

′

)

dt

′

dt

′

where

J(t − t

′

) = 0

if

t < t

′

(2.46)

σ

i

(t) =

Z

t

−∞

G(t − t

′

)

dε(t

′

)

dt

′

dt

′

where

G(t − t

′

) = 0

if

t < t

′

(2.47) At time

t

′

1

stress

σ

1

isapplied and thestrainindu ed an be givenby

ε

1

(t) = σ

1

J(t)

(2.48)

A ording to linearvis oelasti ity

J(t)

remainsequal for allstresses ina given time. If stressesin reases by

σ

2

− σ

1

at time

t

′

2

thestrainin rease isgiven by

ε

2

(t) = J(t − t

′

2

)(σ

2

− σ

1

)

(2.49)

Also,ifstressin rement

σ

3

− σ

2

is applied,the strainin rease is

ε

3

(t) = J(t − t

′

3

)(σ

3

− σ

2

)

(2.50)

Furtherstrains ausedbystressin rements,whi h an beeitherpositiveornegative,

(44)

2.8 Time temperature superposition prin iple (TTSP)

The temperature dependen e of polymer properties is ofmajor importan e sin e either

polymers and elastomers present hanges in their me hani al properties with hanging

of temperature. Polymeri parts should be proje ted so that reep and relaxation

phe-nomena are slow, preferentially taking several years, albeit they an not be negle ted.

Hen e,experimentalpro edures toobtain reep omplian eandrelaxationmodulusdata

should also be long enough, what is not a eptable sin e it would be time onsuming.

Short time testsare thenmade and the results are extrapolated to working onditions.

Using the time temperature superposition prin iple it ispossible to redu e testing time

by repla ing it with shorter experiments at higher temperatures, sin e an in rease in

temperaturea eleratesmole ular motions. Thisprin iplealsoallowstoobtaina

mole -ular interpretation of the vis oelasti behaviour sin e generally polymers hange from

glass-like torubber-like behaviourasthetemperaturerisesor thetimes ale isin reased

[8℄,[18 ℄.

TTSP prin iple isbased onthehypothesesthat vis oelasti materials have a simple

rheologi al behavior, i.e. thatevery propertyhasthesame dependen y oftime[6℄.

If

P

designates any vis oelasti property, time temperature superposition prin iple states that:

P (t, T ) = P (t

r

, T

r

)

(2.51)

where

T

r

is the referen e temperature at whi h the properties are desired and

t

r

is theredu ed time,given by:

t

r

=

t

α

T

(T )

(2.52)

inwhi hintervenestheshiftingfa tor

α

T

(T )

. Thesame anbewritteninfrequen y:

P (w, T ) = P (w

r

, T

r

)

where

w

r

= wα

T

(T )

(2.53)

The implementation of this prin iple requires hoosing a referen e temperature as

wellasobtaining urvesforthedesiredpropertyto bemeasuredat severaltemperatures.

It is then possible to obtain a master urve through shifting measured properties at

ea h temperature by their respe tive shifting fa tors, in the time s ale. This master

urve may then serve asthe basis to the predi tion of the polymer behaviour in other

situations, sin e by shifting the isothermals, the time s ale is expanded and one an

extrapolate the behaviour through longer times. Experimental al ulation of thefa tor

α

T

(T )

isessentialfortheappli ationoftheTTSPsin ethesearethevaluestobeapplied to shift theisothermals horizontally.

Whilemakingmaster urves,variableshavingunitsoftime(orre ipro al)aresubje t

to horizontal shifts

α

T

while units of stress (or re ipro al) are subje t to verti al shifts

b

T

,both temperature dependent.

Whenall retardation andrelaxation me hanisms have thesame temperature

depen-den easwellasstressmagnitudesatalltimes,thematerial anbesaidas

thermorheolog-i allysimple. This anbenotedwhenbuildingamaster urveif,onadoublelogarithmi

plot ofvis oelasti fun tions,these an besuperimposed by shiftingthem only

(45)

temperature is the same and if the material is onsidered thermorheologi ally simple

[16 ℄. Someauthorsassumethatstressmagnitudesareproportionaltotheprodu tofthe

densityand temperature. So theverti alshift fa tor maybe al ulated by[7 ℄

b

T

=

T

0

ρ

0

T ρ

(2.54)

Anotherwaytodeterminethene essitytoin ludeaverti alshift(independentoftime

or frequen y shift), isto nd thehorizontal shift ofloss anglene essaryto superimpose

on alossangle versus

log(|G

∗

|)

plot, alledavan Gurp-Palmen plot[14 ℄.

2.8.1 Mole ular interpretations of time-temperature equivalen e

Site model theory

This theory states that there are two regions, separated by an equilibrium free energy

dieren e

∆G

1

− ∆G

2

,whereea hofthetermsarethebarrierheightspermoleasFigure 2.17 shows. Also,this modelis based on thetransition probabilty for ajump from site

1 tosite 2whenastress isappliedand a hange ofpopulations inbothsiteso urs [24℄.

Figure2.17: Two-site model- adaptedfrom [24 ℄

Aftersome assumptionsand mathemati al dedu tions it an be stated that the

fre-quen yofmole ular onformational hangesdependson thebarrierheight

∆G

2

and not on thedieren e between sites,asit anbe seeninEquation2.55 . [24 ℄

ν =

A

′

2π

e

−∆G2

RT

(2.55) Here,

A

′

is a onstant. This equation an be rewritten as:

ν = ν

0

e

−∆H

RT

where

ν

₀

=

A”

2π

e

∆S

R

(2.56)

Here,

∆S

is the numberof spe iesperunit volume. Equation2.56 shows that tem-perature ae tsthefrequen y ofmole ular jumps mostlybythea tivation energy