viscoelástico

8/6/2019 viscoelstico

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Um fluido que evidencie um comportamento newtoniano caracterizvel por uma

simples equao constitutiva que estabelece uma relao de proporcionalidade diretaentre tenso e taxa de deformao,

Sendo a constante de proporcionalidade a viscosidade do fluido. Em

condies estveis de temperatura e de presso, esta viscosidade uma propriedade do

material facilmente mensurvel.

O primeiro dos modelos de comportamento molecular propostos paracaracterizao de um fluido viscoelstico foi avanado por Maxwell (1867) e baseava-

se numa analogia com o sistema mecnico mola amortecedor (elemento elstico elemento viscoso). Neste sistema mecnico, Figura abaixo, uma solicitao sbita

originar uma resposta essencialmente elstica, observando-se, tambm neste caso,

semelhana de um fluido viscoelstico, a capacidade do sistema em relaxar as tensesinicialmente desenvolvidas no processo de deformao ao fim de algum tempo. Pelo

contrrio, se a velocidade de deformao imposta for baixa no se observar qualquer

interveno do componente elstico, ou seja, a resposta do sistema agora

essencialmente dissipativa (inelstica).

Segundo Di Benedetto et al. (2001), at um certo nmero de aplicaes de carga e um

certo nvel de deformao no material, possvel empregar um modelo viscoelsticolinear para o modelar o comportamento do mesmo. Os limites referidos podem ser

vistos na figura 1:

Figura 1: Limites de comportamento das misturas asflticas (Di Benedetto et al., 2001)O campo Viscoelstico Linear na figura 1 atingido com cargas significativamente

menores que as limites do material. Com cargas baixas, tambm coerente considerar o

material como isotrpico (Kim et al., 2004) e com dano inexistente ou desprezvel

(Gibson et al., 2003). Os ensaios citados at ento buscam impor ao material um

comportamento viscoelstico linear. Assim, o modelo constitutivo que descreve o

comportamento tenso-deformao dos materiais viscoelsticos lineares expresso

pelas equaes 1 e 2:


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onde:I(t : deformao em funo do tempo;

D(t X : curva de fluncia ou creep compli nce;

W(t :tenso em funo do tempo;

E(t X : curva de relaxao ou rel x tion modulus;

X:instante de incio de aplicao de carga;

X0:instante inicial de aplicao de carga.

H vrios modelos para descrever D(t X e E(t X em funo do tempo. A forma maiscomum em pesquisas sobre misturas asflticas so as sries de Prony, vistas nas

equaes 3 e 4:

onde: Eg: mdulo de equilbrio a longo tempo;

Ei: rigidez elstica de cada elemento Maxwell;

Vi:tempo de relaxao de cada elemento Maxwell;

D0: compli ncia inicial ou vtrea;

Di: compli ncia de cada elemento Kelvin;

Xi:tempo de retardao de cada elemento Kelvin;

n: nmero de elemento Maxwell/Kelvin na srie.

Maxwell model

Main article:Maxwell material

Maxwell model

The Maxwell model can be represented by a purely viscous damper and a purely elasticspring connected in series, as shown in the diagram. Themodel can be represented by

the following equation:

.


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Underthis model, ifthe materialis put under a constant strain, the stresses gradually

relax, When a materialis put under a constant stress, the strain has two components.

First, an elastic component occurs instantaneously, corresponding to the spring, and

relaxes immediately upon release ofthe stress. The second is a viscous componentthat

grows with time as long as the stress is applied. The Maxwell model predicts that stress

decays exponentially with time, which is accurate for most polymers.One limitation of

this model isthatit doesnot predictcreep acc rately. The Maxwell model forcreep or constant-stressconditions post latesthatstrain will increase linearly with

time. However, polymers for the most partshow the strain rate to be decreasing

withtime.[2]

Application to soft solids:thermoplastic polymers in the vicinity oftheir melting

temperature, fresh concrete ( neglecting its ageing), numerous metals at a temperature

close to their melting point.

[edit] KelvinVoigt model

Main article:KelvinVoigt material

Schematic representation of KelvinVoigt model.

The KelvinVoigt model, also known as the Voigt model, consists of a Newtonian

damper and Hookean elastic spring connected in parallel, as shown in the picture. Itis

used to explain the creep behaviour of polymers.

The constitutive relation is expressed as a linear first order differential equation:

This model represents a solid undergoing reversible, viscoelastic strain. Upon

application of a constant stress, the material deforms at a decreasing rate, asymptotically

approaching the steady-state strain. When the stress is released, the material gradually

relaxes to its undeformed state. At constant stress (creep), the Modelis quite realistic as

it predicts strain to tend to /E as time continues to infinity. Similar to the Maxwell

model, the KelvinVoigt model also has limitations. The model is extremely good

with modellingcreep in materials, but with regardsto relaxationthe model ismuch less accurate.

Applications: organic polymers, rubber, wood when the load is nottoo high.

[edit] Standard linear solid model

Main article:Standard linear solid model


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Schematic representation ofthe Standard inear Solid model.

The Standard inear Solid Model effectively combines the MaxwellModel and a

Hookean spring in parallel. A viscous materialis modeled as a spring and a dashpotin

series with each other, both of which are in parallel with a lone spring. Forthis model,

the governing constitutive relation is:

Under a constant stress, the modeled material willinstantaneously deform to some

strain, which is the elastic portion ofthe strain, and afterthatit will continue to deform

and asymptotically approach a steady-state strain. This last portion is the viscous part of

the strain. Althoughthe Standard Linear Solid Model is more accurate thanthe

Maxwell and Kelvin-Voigt modelsin predicting material responses,

mathematically it returnsinaccurate results for strain under specific loading

conditions and is rather difficultto calculate.

[edit] Generalized Maxwell Model

Main article:Generalized MaxwellModel

Schematic ofMaxwell-WiechertModel

The Generalized Maxwell model also known as the MaxwellWiechert model (afterJames ClerkMaxwell and E Wiechert[4][5]) is the most general form ofthe linear model

forviscoelasticity. Ittakes into accountthatthe relaxation does not occur at a singletime, but at a distribution oftimes. Due to molecular segments of differentlengths with


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shorter ones contributing less than longer ones, there is a varying time distribution. The

Wiechert model shows this by having as many springdashpotMaxwell elements as are

necessary to accurately representthe distribution. The figure on the right shows the

generalised Wiechert model[6] Applications : metals and alloys attemperatures lower

than one quarter oftheir absolute melting temperature (expressed in K).

[edit] Prony series

Main article:Prony series

In a one-dimensional relaxation test, the materialis subjected to a sudden strain thatiskept constant overthe duration ofthe test, and the stress is measured overtime. The

initial stress is due to the elastic response ofthe material. Then, the stress relaxes overtime due to the viscous effects in the material. Typically, either a tensile, compressive,

bulk compression, or shear strain is applied. The resulting stress vs. time data can befitted with a number of equations, called models. Only the notation changes depending

ofthe type of strain applied:tensile-compressive relaxation is denotedE, shearisdenoted G, bulkis denotedK. The Prony series forthe shear relaxation is

where is the long term modulus once the materialis totally relaxed, i are the

relaxation times; the highertheir values, the longerittakes forthe stress to relax. The

data is fitted with the equation by using a minimization algorithm that adjustthe

parameters ( ) to minimize the error between the predicted and data values

.[7]

An alternative form is obtained noting thatthe elastic modulus is related to the long

term modulus by

Therefore,

This form is convenient when the elastic shear modulusG0is obtained from data

independent from the relaxation data, and/or for computerimplementation, when itisdesired to specify the elastic properties separately from the viscous properties, as in .[8]

A creep experimentis usually easierto perform than a relaxation one, so most data isavailable as (creep) compliance vs. time.

[9]Unfortunately, there is no known closed

form forthe (creep) compliance in terms ofthe coefficient ofthe Prony series. So, if one

has creep data, itis not easy to getthe coefficients ofthe (relaxation) Prony series,

which are needed for example in.[8] An expedient way to obtain these coefficients is the

following. First, fitthe creep data with a modelthat has closed form solutions in both

compliance and relaxation; for example the Maxwell-Kelvin model (eq. 7.18-7.19) in[10]

orthe Standard Solid Model (eq. 7.20-7.21) in[10]

(section 7.1.3). Once the

parameters ofthe creep model are known, produce relaxation pseudo-data with the


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conjugate relaxation model for the same times of the original data. Finally, fit the

pseudo data with the Prony series.

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