MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes ...inimif/teched/Relax2010/Lecture05_fotherin… · MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and

MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5

Special Topics in Relaxation in Glass and Polymers

Lecture 5: Viscoelasticity IShear

Dr. Ulrich Fotheringham

Research and Technology Development

SCHOTT AG


2.1 Maxwell Modell

Mechanical model consisting of a spring and a dash-pot in a series.

Continuous strain:

Constant elongation:

derived from:

The time constant τ=(A×η)/(D×d) is called relaxation time.

z

x y

F

z

t

dA

FDF

z zz ⋅⋅

+=η

xDFx ⋅=dtdy

dAFy ⋅⋅= η

zyx =+

zyx FFF ==

0=+dtdy

dtdx

01=⋅+⋅

ηzz F

Ad

dtdF

D

)A/()dD(tz econst)t(F ⋅⋅⋅−⋅= η

Basic viscoelastic models I: Maxwell Model


Viscoelastic behaviouraccording to the Maxwell-model means that the material considered responds like an elastic body on short time scales and like a viscous fluid on long time scales.

Continuous strain:


derived from:

The time constant τ=(A×η)/(D×d) is called relaxation time (force response to elongation).


2.1 Maxwell Modell


Continuous strain:


derived from:

The time constant τ=(A×η)/(D×d) is called relaxation time.

Basic viscoelastic models II: Kelvin-Voigt Model

Mechanical model consisting of a spring and a dash-pot in parallel.

Viscoelastic behaviour according to the Kelvin-Voigt-model means that the material responds like a rigid body on short time-scales and like an elastic body on long time-scales.

Both the Maxwell and the Kelvin-Voigt model are necessary to describe glass.

The case of an instantaneous, fixed elongation may not be realised with a Voigt-model.

Continuous strain:

derived from:

The time constant τ=(A×η)/(D×d) is called retardation time (elongation response to force).

xDFx ⋅=

dtdy

dAFy ⋅⋅= η

zx

y

F

z

zyx ==

yxz FFF +=

zFdtdz

dAzD =⋅⋅+⋅ η

( ))/()(1 η⋅⋅⋅−−⋅= ADdtz eDF

z


Some exercises:

1. Consider “Basic viscoelastic models II: Kelvin-Voigt Model”. A Kelvin-Voigt Model acting together with the tyre as additional spring is what you find as damper in cars.

Introduce numbers:

Spring:

Calculate τ of the Kelvin-Voigt Model.

2. Consider again “Basic viscoelastic models I: Maxwell Model”. At the bottom, it is said that the formula

giving Fz(T) for constant elongation follows from the homogeneous linear differential equation

Find the solution via Laplace-Transform. Write down the subsidiary equation.

01=⋅+⋅

ηzz F

Ad

dtdF

D

)A/()dD(tz econst)t(F ⋅⋅⋅−⋅= η

mm/N20D =

m/sN2000dA

⋅=⋅η


Shear Viscoelasticity and Bulk Viscoelasticity

Combining Maxwell- and Kelvin-Voigt-elements, one may describe the viscoelasticbehaviour of inorganic glasses and polymers. However, one has to distinguish between shear on one side and dilatation/compression on the other.

Shear

Mostly not the angle ε’ describing the change of γ’ is referred to but the angle εdescribing the change of γ. One has to introduce a factor of 2 for compensation.

Next step: drop the distinction between “x” and “y”. It is all in one.

A

x,y,z

d

F

ε

' γ

'

γ

ε

xxx GxDF 'εσ ⋅=→⋅=

dtd

dtdy

dAF y

yy

'εηση ⋅=→⋅⋅=

xxx 2GxDF εσ ⋅⋅=→⋅=dt

ddtdy

dAF y

yy

εηση ⋅⋅=→⋅⋅= 2


Relations for Maxwell- and Kelvin-Voigt-model in case of shear

For the simple Maxwell-model and constant shear stress this leads to:

J(t) is called compliance (describes strain response to stress).

For the simple Maxwell-model and constant shear strain this leads to:

G(t) is the thus defined time-dependent shear modulus (stress response to strain).

For the simple Kelvin-Voigt-model and constant shear stress this leads to:

The combination of simple Kelvin-Voigt-model and constant shear strain does not exist.

0000 )(:

221

22σσ

ηησσ

ε ⋅=⋅

+=⋅+= tJt

Gt

G

00)/()/(

0 )(2:)2(2)( εεεσ ηη ⋅⋅=⋅⋅⋅=⋅⋅⋅= ⋅−⋅− tGeGeGt GtGt

( ) ( ) 00)/()/(0 )(:1

211

2σσ

σε ηη ⋅=⋅

−⋅=−⋅= ⋅−⋅− tJe

Ge

GGtGt


Visco-Elastic Characterization of Glass

Pyrex® and BK7

Hemanth Chaitanya KadaliMechanical [email protected]

AdvisorDr. Vincent Blouin

Materials Science and Engineering

Committee Members

Dr. Paul Joseph Dr. Lonny Thompson

Mechanical Engineering

Clemson University, SC-29634

1

Shear relaxation experiments from Clemson University I

Kadali et al. have built a spring relaxometer as introduced by S. Rekhson et al.


II Creep Recovery Setup

6

Shear relaxation experiments from Clemson University II

To impose pure shear on a sample, torsion is an appropriate way. The elongation of a helical spring implies torsion of every cross-section => spring elongation means shear.

III Creep Testing on Helical Spring sample

7


II Creep testing

8

An Example of an LVDT Result

9

Shear relaxation experiments from Clemson University III


1st Realistic Picture of Shear Relaxation: Burger-Model

As we have seen, the behaviour of inorganic glasses is more complicated and cannot be described with either the Maxwell- or the Voigt-model alone.

The simplest representation is by a Burger-model:

G 2η2

G 1

η1

t

εcreep

delayed elasticity

Instantaneous elasticity


Burger-Model (continued)

In case of constant stress one gets:

The case of constant strain is more complicated and requires a careful derivation:

λ+, λ- are the roots of the equation G1×G2+(G2×η2+G1×η2+G2×η1)×λ+η1×η2×λ2=0.

( ) 00)/(

122

)(:12

122

111 σσ

ηε η ⋅=⋅

−⋅+

+= ⋅ tJe

Gt

G BurgerGt

⋅

+−⋅

+⋅

−⋅⋅

= ⋅−

⋅+

−+

−+ t

1

1t

1

102 eGeG)(

G2 λλη

λη

λλλε

σ

21

212112212222122122

2)()22()(

ηηηηηηηηηηη

λGGGGGGGGG −+++±++−

=±


Kohlrausch-Kinetics for Shear Relaxation

An even better coincidence of theoretical and experimental data than with the Burger model is obtained, if the single exponential function from above is replaced with a stretched exponential or Kohlrausch(-Williams-Watts)-function.

For constant stress one gets:

J(t) is the time-dependent compliance; τd and bd are retardation parameters.

For constant strain one gets

G(t) is the time-dependent shear modulus; τx and bx are relaxation parameters.

1 2 3

σ/σ(0)1

0,5Kohlrausch

single exponential function

t /τ

( )00

/t

00)t(J:e1

G21

G21

2t

G21 db

d σση

ε τ ⋅=⋅

−⋅

−++= −

∞

( )0

/t00 )t(G:eG2)t(

xbx εεσ τ ⋅=⋅⋅⋅= −

( ) 1b0,eb/t <<− τ


Boltzmann´s superposition principle, relaxation retardationIn general (not only for shear) it is assumed that the stress effects arising from strain contributions imposed at different times overlay without interfering and vice versa:

One may write introducing relaxation function Ψ

and retardation function Φ

Laplace-Transform yields:

and

This allows the mutual conversion of relaxation and retardation parameters.

( )( )( ) ( )( )

tdtd

)t(deG2eG2)t(t

00

/ttttimetheat0

xbx/ttxbx ′⋅

′′

⋅⋅⋅≈⋅⋅⋅= ∫∑′−−′−−

′ε

ε∆σττ

tdtd

)t(dttG2)t(t

0 x0 ′⋅

′′

⋅

′−⋅⋅= ∫

ετ

Ψσ

tdtd

)t(dtt1G21

G21

2tt

G21)t(

t

0 d00′⋅

′′

⋅

′−−⋅

−+

′−+= ∫

∞

στ

Φη

ε

( ) ( ) ( )εΨσ LsLG2L 0 ⋅⋅⋅⋅= ( ) ( ) ( )σΦη

ε LLsG21

G21

s21

G21L

0⋅

⋅⋅

−−+=

∞∞


Relaxation and Viscosity

Consider the special case of constant stress in the long-time limit. According to the nature of the Laplace-Transform

only the L-values belonging to small s-values are sensitive to things happening on a broad time scale. This means for the retardation formula:

Replacing the low s, high t – limit of L(σ)/L(ε) with the expression that follows from the relaxation formula gives:

( ) ( ) ´dtet́ffL ´stt

0⋅⋅= −∫

( ) ( ) ( ) ( ) ( )ση

εσΦη

ε Llimk2

1LlimLLsG21

G21

s21

G21L

t,0st,0s0 ∞→→∞→→∞∞⋅=⇒⋅

⋅⋅

−−+=

( )

( ) ( ) ( ) ´dtt́´dtet́limLlimG

s2kLlimG2

0

´stt

0t,0st,0s0

t,0s0

⋅=⋅⋅==⇒

⋅⋅=⋅⋅⋅

∫∫∞

−

∞→→∞→→

∞→→

ΨΨΨη

ηΨ


Addenda

Prony Series

For computational purposes, the Kohlrausch-function may be represented by a number of single exponentials (Prony-series):

In order to allow for the comparatively fast relaxation for t < τ and the comparatively slow relaxation for t > τ, the Prony-series must comprehend both τi < τ and τi > τ.

Temperature dependence of the relaxation times

As known, the temperature dependence of shear viscosity is well described by the Arrhenius law for a limited temperature range such as the one of glass transition:

Consequently, it is also used for shear relaxation and retardation times:

TRH

e ⋅= 0ηη

TRH

0e ⋅=ττ

( ) 1v,i0v,evei

iii

/ti

/t ib

=∀>⋅= ∑∑ −− ττ


Some exercises:

1. Consider the correspondence of the Maxwell model with spring and dashpot and the behaviourof a viscoelastic material. It has been said that F=D·x translates into σ=G·ε’=2·G·ε. Check dimensions in both cases (Newtons, Pascals etc.). Compare the dimensions of the right sides of both equations.

2. Consider the Burger model. What is ε(0) in case of constant stress and σ(0) in case of constant strain?

3. Compare single-exponential and stretched-exponential (Kohlrausch) behaviour. If b=0.5, what is the amount of Exp(-t/τ) and Exp(-(t/τ)b) after t=0.1·τ, t=τ, t=10·τ?

4. Which are the three time domains of a constant stress experiment?

5. How is η/G0 related to the Kohlrausch-function Exp(-(t/τx)b)?


Shear relaxation experiments from Clemson University IV

preliminary results

I. Obtain Strain-time curve through creep recovery experiment

III Characterization of glass in transition region

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10-3

Time(Sec)

Stra

in

0 50 100 150 200 250 300 350 4000

0.2

0.4

0.6

0.8

1

1.2

1.4

Time(Sec)

Disp

lacem

ent(m

m)

Area of interest

(t)*d*k*4+

d*D*k*8= (t) 23 δε

ΠΠ20

Retardation Spectrum on log time scale at different temperatures

10-2

10-1

100

101

102

103

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time(Sec) on Log Scale

Dis

plac

emen

t(mm

)

a) 522•C

b) 536•C

c) 547•C

22


Shear relaxation experiments from Clemson University V

preliminary resultsIII. Carryout curve fitting for the resultant retardation curve to determine

retardation parameters i.e. retardation times and retardation weights.

• Retardation function (Prony Series) (m1 = 5 is the best fit)

• Determine Retardation weights •1j (j=1 to 5) & Retardation weights •1j (j=1 to 5)

such that the resultant curve overlaps experimental curve

• A total of 10 parameters are obtained in this process

Retardation Parameters

−∑=Φ= ij

ij

m

j

ttλ

υ exp)(1

1

23

Calculated and Experimental Spectrum (Curve Fitting)

o : Experimental curve

- - - :Simulated curve

10-2 10-1 100 101 102 1030

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (sec)

Ret

arda

tion

func

tion

25


Shear Retardation Parameters

•1j •1j0.0011 0.00150.0124 1.99440.2187 2.48100.2532 8.62570.4939 45.9273

−∑=Φ= ij

ij

m

j

ttλ

υ exp)(1

1

26

Shear relaxation experiments from Clemson University VI

preliminary results

Shear Relaxation Parameters for Pyrex®

w1j •1j0.01133 0.03766383145

0.052928 1.76269908520.148225 2.1680468630.186358 6.9306352740.191795 32.469224890.39312 220.6745301

30

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MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes ...inimif/teched/Relax2010/Lecture05_fotherin… · MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and