Viscoelasticidad2007

Diapositiva 1

Viscoelasticidad lineal

Contenido

• Tensor de tensiones• Ensayo de fluencia. Componentes de la respuesta del material.

Procesos de relajación. Componente viscosa, flujo irreversible.• Comportamiento lineal. Principio de superposición. • Ensayo de relajación de tensiones• Modelos mecánicos: Kelvin-Voigt , Maxwell, Zener.• Tiempos de relajación• Distribución de tiempos de relajación.• Efecto de la temperatura. Principio de superposición tiempo-

temperatura.• Espectroscopía dinámico-mecánica

Bibliografía

A. Horta. Macromoléculas. UNED. Capítulos 33 y 34

J.M.G. Cowie. Polymers: Chemistry & Physics of Modern Materials. Blackie Academic & Professional. 1991

R.J. Crawford, Plastics Engineering, 3rd ed Butterworths 1998

N.G. McCrum, C.P. Buckley, C.B. Bucknall, Principles of Polymer Engineering, Oxford Sci. Pub, 1988

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Diapositiva 2

Respuesta elástica, sin rotaciones en los enlaces

Movimientos locales de las cadenas laterales o incluso de pequeñas partes de la cadena principal. Procesos de relajación secundaria

Movimientos conformacionales.Relajación principal o relajación α

Proceso de compresión de un conjunto de cadenas de polímero

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Diapositiva 3

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

∂∂

∂∂

+∂∂

∂∂

+∂∂

∂∂

+∂∂

∂∂

∂∂

+∂∂

∂∂

+∂∂

∂∂

+∂∂

∂∂

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=

zw

yw

zv

xw

zu

yw

zv

yv

xv

yu

xw

zu

xv

yu

xu

zzyzxz

yzyyxy

xzxyxx

ij

γγγγγγγγγ

γ

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=

zzyzxz

yzyyxy

xzxyxx

ij

σσσσσσσσσ

σTensor de tensiones

Tensor de deformación

Tensión y deformación en un sólido elástico

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Diapositiva 4

Ley de Hooke generalizada

Tension σxx

σxx

xxzz

xxyy

xxxx

E

E

E

σνγ

σνγ

σγ

−=

−=

=

ν: Coeficiente de PoissonE: Módulo de Young

Cizalla pura

G

G

G

yzyz

xyxy

xzxz

σγ

σγ

σγ

=

=

=

( )ν+=

12EGMódulo de cizalla


σxy

y

x

z

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Diapositiva 5

Comportamiento puramente elástico. Ley de Hooke

Comportamiento puramente viscosoLey de Newton

( ) xyExy Gγσ =

( )dt

d xyVxy

γησ =

y

x

σxy


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Diapositiva 6

0)()( σγ tJt =

Log J(t)

log t

vidrio

Relajación principal

goma Flujoirreversible

10-9 Pa-1

σ

γ

time

time

γ1

γ2γ3

σ0

FluenciaViscoelasticidad lineal

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Diapositiva 7

Log J(t)

log t

10-9 Pa-1

T0 > T1 > T2 > T3 > T4

Influencia de la temperatura


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Diapositiva 8

0)()( γσ tGt =

log G(t)

log t

vidrioRelajaciónprincipal

gomaFlujoIrreversible

109 Pa

γ

σ

time

time

γ0

Relajación de Tensiones Viscoelasticidad lineal

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Diapositiva 9


El modelo de Maxwell

γ1

γ2

Em

ηm

σ

σ

21 γγγ +=

tm ∂∂

= 2γησ

mEσγ =1

mm tEttt ησσγγγ +

∂∂=

∂∂+

∂∂=

∂∂ 121

ttE mm

m

∂∂

=∂∂

+γησησ

Ensayo de relajación de tensiones0

0

=∂∂

=

tγ

γγ

ση

σ

m

mEt

−=∂∂

τσσ dt−=∂

m

m

Eητ =

Tiempo de relajación

τσσ t

−=0

ln

⎟⎠⎞

⎜⎝⎛ −=

τσσ texp0

0 0 σσ =→t

??? 0 =∞→ σt

Ensayo de fluencia

0

0

=∂∂

=

tσ

σσ

tm ∂∂

=γησ puramente viscoso

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Diapositiva 10

El modelo de Zener

γ1

γ2

EmEa

ηm

σ1 σ2

σ

σ

221 σγσσσ +=+= aE

mE2

1σγ =

mt ησγ 22 =

∂∂

mm tEttt ησσγγγ 2221 1 +

∂∂=

∂∂+

∂∂=

∂∂

ttE

t a ∂∂+

∂∂=

∂∂ 2σγσ

( )γση

γγσa

m

mma EE

tE

tE

t−−

∂∂+

∂∂=

∂∂

( ) γη

γσση m

amma

m

m EEt

EEt

E−

∂∂+=

∂∂+

( ) γγησησ am

mam

m

m EtE

EEtE

+∂∂+

=∂∂+


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Diapositiva 11

Ensayo de relajación de tensiones ( ) γγησησ am

mam

m

m EtE

EEtE

+∂∂+

=∂∂

+0

0

=∂∂

=

tγ

γγ

σ

log t

GUγ0

GRγ0

0γσησ am

m EtE

=∂∂

+

0γστσ aEt

=∂∂

+

( ) ⎟⎠⎞

⎜⎝⎛−−−=

τσγγσ tEE aa exp000

00 0 γσσ UGt ==→

00 γγσ Ra GEt ==∞→

( ) ( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −−−−=⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −−−=

τγ

τγσ tGGGtGGG RUUURR exp1exp 00

( ) ⎟⎠⎞

⎜⎝⎛ −−+=

τtGGGtG RUR exp)( ( ) ⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−−−−=

τtGGGtG RUU exp1)(

m

m

Eητ =



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Diapositiva 12

Fluencia

( ) γγησησ am

mam

m

m EtE

EEtE

+∂∂+

=∂∂

+ 0

0

=∂∂

=

tσ

σσ

( ) γγησ am

mam E

tEEE

+∂∂+

=0

γγτσ+

∂∂

=tEa

'0

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+=

+=

mam

ma

mam EEEE

EE 11' ηητ

Tiempo de retardo

'0 τγσγ dt

E

d

a

=−

⎟⎠⎞

⎜⎝⎛ −

⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

'exp0

00

τγσσγ t

EE aa

00 0 σγγ UJt ==→

00 σσγ Ra

JE

t ==∞→

( ) ⎟⎠⎞

⎜⎝⎛ −−−=

'exp00 τ

σσγ tJJJ URR

( ) ⎟⎠⎞

⎜⎝⎛ −−−=

'exp)(

τtJJJtJ URR

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

−−+='

exp1)(τ

tJJJtJ URU

γ

log t

JUσ0

JRσ0


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Diapositiva 13

maU

m

mam EEJEE

ηητ =⎟⎟⎠

⎞⎜⎜⎝

⎛+=

11'

maR

m

m

m

EEJEηητ ==

Tiempo de retardo


U

R

JJ

='τ

τ

Relación entre tiempo de relajación y tiempo de retardo


Módulo antes de la relajación: maU

U EEJ

G +==1

Módulo después de la relajación: aR

R EJ

G ==1

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Diapositiva 14

Distribución de tiempos de relajación y de retardo ∫

∞

∞− ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

−+= ττ

τ ln'

exp1)()( dtLJtJ U

ττ

τ lnexp)()( dtHGtG R ∫∞

∞−⎟⎠⎞

⎜⎝⎛ −+=

Modelo de un único tiempo de relajaciónUn tiempo de relajación

0

0.2

0.4

0.6

0.8

1

1.2

0 100 200 300 400 500 600 700 800 900 1000

t (s)

rela

xatio

n fu

nctio

n

0

0.2

0.4

0.6

0.8

1

1.2

-4 -2 0 2 4

log(t / s)

rela

xatio

n fu

nctio

n

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

d(t)

/ dl

ogt

0

0.2

0.4

0.6

0.8

1

1.2

-4 -2 0 2 4 6

log (t / s)

Rel

axat

ion

func

tion

Single relaxation timemodel

KWW b=0,2

KWW b=0,4

KWW b=0,6

KWW b=0,8

Ecuación de Kohlrausch-Williams-Watts⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛−−=

β

τψ tt exp1)(

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−−=

−−

='

exp1)()(τ

ψ tJJJtJtUR

U


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Diapositiva 15

El principio de superposición de Boltzmann

∆σ1

∆σ2

∆σ3

t1 t2 t3

Stress

Strain

Respuesta a un programa de multiples escalones de carga

...)()()()( 332211 +−∆+−∆+−∆= ttJttJttJt σσσγ

∫ ∞−−=

ttdttJt )'()'()( σγ

∆σ1

∆σ2

t1 t2

σ(t)

γ(t)

Respuesta a dos escalones de carga

∆σ1 ∆σ2= - ∆σ1

t1 t2

σ(t)

γ(t)


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Diapositiva 16

Una fibra larga se carga con una masa de 0.1 kg unida a su extermo inferior. El alargamiento (%) se mide en varios instantes de tiempo después de la aplicación de la carga

Alargamiento (%) t(min)0.3 00.328 100.350 200.390 400.428 600.462 800.490 1000.514 1200.535 1400.555 1600.572 1800.585 2000.593 2200.6 240

La lectura en t=0 da la respuesta elástica.Suponiendo que el comporamitnto del material es viscoelástico lineal, calcular el alargamiento bajo las siguientes condiciones:Se carga el hilo con 0.1kg en t=0, se quita la carga en t=40min, se vuelve a cargar a t=80min, se quita la carga en t=120min. Calcular el alargamiento residual en t=240min.


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Diapositiva 17

Espectroscopía dinámico-mecánica

( )δωσσ += tt cos)( 0

( ) ( ) ( ) ( ) )(2)(1coscos)( 00 tEtEsintsintt +=−= δωσδωσσ

- 3

- 2

- 1

0

1

2

3

0 1 2 3 4 5 6

ω t / π

σ

E 1E 2γ

( )δγσ cos'

0

0=E

( )δγσ sinE

0

0'' =

'''

EEtan =δ

Modulo complejo E*

Deformación ≡ Parte real de γ* = γ0 eiωt

Tensión ≡ Parte real de σ* = E*γ∗

tt ωγγ cos)( 0=Muestra sometida a una deformación sinusoidal

Medida de la respuesta en tensión

Deformación y tensión alternativas con ω=1s-1 and δ=0.5 rad


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Diapositiva 18


IPNs PMA-i-PMMA IPN y las redes puras de PMA y PMMA entrecruzadas con 5% de EGDMA

PMA-i-PMMA con diferente densidad de entrecruzamiento

-50 0 50 100 150 200

Temperature / ºC

-2.0

-1.5

-1.0

-0.5

0.0

log

tan δ

%EGDMA

0.1

0.5

1

5

10

-50.00 0.00 50.00 100.00 150.00 200.00

Temperature / ºC

0.00

0.20

0.40

0.60

0.80

1.00

log

tan

6

7

8

9

10

log

E'

IPN PMA-i-PMMA 5%EGDMA

PMA 5% EGDMA

PMMA 5% EGDMA

-50.00 0.00 50.00 100.00 150.00 200.00

Temperature / ºC

-6.00

-4.00

-2.00

0.00

log

tan

6

7

8

9

10

log

E'

Ensayos dinámico-mecánicos

Compatibilización forzada

Polymer Communications 42, 10071 (2001)

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Diapositiva 19

0.01

0.1

1

-100 -50 0 50 100 150 200

Temperature (ºC)

tan

δ

Tangente de pérdidas de copolímeros al azar poly(butyl acrylate – co –methyl methacrylate) PBA/PMMA

20/80

40/60

60/4080/20

Ensayos dinámico-mecánicos PMMA

PBA


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Diapositiva 20


Una barra de polímero con una longitud de 60 mm tiene sección rectangular, con dimensiones 10mm x 4 mm. La barra está sometida a una fuerza en sentido longitudinal, oscilando sinusoidalmente entre -150 y +150 N a una frecuencia de 50Hz. La componente imaginaria del módulo de fluencia complejo es D’’ = 3.7 10-9 Pa-1. Calcular la potencia necesaria para mantener la oscilación.

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Diapositiva 21

PM M A EG DM A1%

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

-6 -4 -2 0 2 4 6

log(ta T)

log(

Da v

)

153.6149.0144.3139.7135.2130.6126.3121.8KW W

Principio de superposición tiempo-temperatura

PM M A EGDMA1%

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

0 1 2 3 4

log(t / s)

log(

D

)

153.6149.0144.3139.7135.2130.6126.3121.8

Curva maestra a 144.3ºC

Curva de KWW con β=0.6 desplazada hacia bajas frecuencias buscando la claridad en el diagrama

PMMA entrecruzado con 1% EGDMA

-2

-1

0

1

2

3

4

5

6

7

2.3 2.35 2.4 2.45 2.5 2.55

1000/T

log

τ

Diagrama de Ahrrenius.

Dependencia con la temperatura de los tiempos de relajación

τ(144.3)=4s


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Diapositiva 22

0

0.5

1

1.5

2

2.5

3

0.01 0.1 1 10 100 1000

111.646

109.849

108.16

106.408

104.666

102.879

100.939

99.1933

97.1199

95.6455

0

0.5

1

1.5

2

2.5

3

0.01 0.1 1 10 100

F(Hz)

tan(

δ)

114.759

113.341

111.646

109.849

108.16

106.408

104.666

102.879

100.939

99.1933

99.4766

97.1199

95.6455

93.9324

92.2005

90.439

88.5715

86.7315

85.0308

83.2035

81 4147

Curva maestra superpuesta sobre la isoterma de 108ºC.

Tanδ de una red de poliestireno entrecruzada con un 0.1% en peso de EGDMA.

Ensayos dinámico-mecánicos en tensión.

Principio de superposición tiempo-temperatura


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Diapositiva 23

El módulo elástico de un polímero amorfo, cuya temperatura de transición vítrea es 70ºC, medido en un ensayo de relajación de tensiones en el rango lineal a la temperatura de 90ºC viene dado por la siguiente ecuación:

con t en minutos. El factor de deslizamiento que permitiría construir una curva maestra para E(t) puede calcularse de acuerdo con la ecuación de Williams-Landel y Ferry.

(a) Calcula la deformación, un mes después de la aplicación de la carga, de una barra de ese polímero con sección transversal de 3 x 5 mm y longitud inicial 100mm de la que se cuelga un peso de 200g a la temperatura de 90ºC. (b) Repite el cálculo si la carga se aplica a 70ºC. (c) Repite el cálculo si la carga se aplica 130ºC. (d) Explica los resultados ayudándote de un diagrama esquemático que represente E(t) a las tres temperaturas utilizando las escalas adecuadas.


MPattE ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−+=

5.0

5000exp12002)(

)(6.51)(4.17

logg

gT TT

TTa

−+

−=

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Viscoelasticidad2007

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Transcript of Viscoelasticidad2007