Room temperature viscosity and delayed elasticity in infrared TAS glass fibers

J-C. Sangleboeuf a, C. Bernard a, Y. Gueguen a, V. Keryvin a, T. Rouxel a,G. Delaizir bc, E. King bc, B. Bureau b, X. Zhang b, P. Lucas c.

a) Laboratory of Research in Applied Mechanics, LARMAUR FRE-CNRS 2717, Rennes, Franceb) Glasses and Ceramics Laboratory, UMR-CNRS 6226, Rennes, Francec) University of Arizona, Tucson, USA

FFAG IV

Shiga - Japan

- Material / mechanical test

- Stress relaxation/air quenched fibers

- Stress relaxation/heat treated fibers

Outline

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Material

TAS glassTellurium Arsenic Selenium

20% 30% 50% (mol%)

Cutting

Fiber drawing

Air Quenched

Low Tg (137°C) Viscoelastic behaviour at room temperature

Heat Treated

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Fiber bending test

Rolling

0 Time

RelaxationConstant strain imposed

Stress decreases

1

Instantaneous elasticity

Unrolling

tu

Ru

Anelasticity

RecoveryStrain decreases

2

R(t)

t∞t

R∞

Permanent strain

Delayed elasticity

R0

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Experimental results

0

5

10

15

20

25

30

35

40

0 20 40 60 80 100

Time (days)

Cur

vatu

re ra

dius

, R(t)

(cm

)

Change of the radius of curvature

Time (Days)tu2tu1

Mandrin radius R0

R∞1

radi

us o

f cur

vatu

re R

(t) (c

m)

R∞2

Ru1

Ru2

Relaxation

Recovery

Bernard et al, J.Europ.Ceram.Soc, 2007FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Stress and strain analysis

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Stress Relaxation

σ

σu

ttu

ε0

σ0

0

Instantaneous elastic strainεel

Unrolling

Instantaneous elasticity σu = E.εel

R : Radius of the fiber

R = 200 µm

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Model for Relaxation

Viscoelastic model : Burger’s cell

Ed

ηd

ε(t)

σ(t)

0

E η

Instantaneous elasticity

Delayed elasticity

Irreversible part

Eη=τ

Time constant:

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Identification for a 32 days test

E = 16.9 GPaη = 1.39 1017 Pa.s

Burger’s cell accurately models the system for the relaxation stage.

Ed = 25.3 GPaηd = 2.65 1015 Pa.s

0

10

20

30

40

50

60

70

80

0 5 10 15 20 25 30 35

Time spent on the mandrel, tu (Days)

Stress during relaxation stage

Stre

ss σ

u(t)

(MPa

)

Burger model

σu = E.εel

Best parameters:

Bernard et al, J.Europ.Ceram.Soc, 2007FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

0

10

20

30

40

50

60

70

0 50 100 150

Burger τ = 95 days τd = 1.2 days

Burger τ = 390 daysτd = 10 days

Identification for a 170 days test

Time spent on the mandrin, tu (days)

Problem to model for both short and long periods

Burger’s cell

Stre

ss σ

u(t)

(MPa

)

Stress during relaxation stage

Gueguen, Master Report, 2007FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Stretched exponential equation or Kohlraush-Williams-Watt equation (KWW):

t

K

Model for Relaxation

with 0<b<1

G(t) = E. ( / )− bKWWte τ

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Results

τKWW = 69 days

b = 0.2

0

10

20

30

40

50

60

70

0 50 100 150

KWW

Time (days)

Stre

ss (M

Pa)

σ(t) = E.ε0.0.2( / 69)− te

Stress during relaxation stage

Gueguen, Master Report, 2007FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Model for Relaxation

• Burger’s cell : only for short times (< 30 days)

• KWW equation : for both short and long times.

[ ] 0)()(0

≠⎥⎦⎤

⎢⎣⎡=+∞→

+∞→t

t

tLimtGLim εσ

but

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Modified KWW model

( ) ( ) ∞∞ +⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−×−= GtExpGEtG

b

τ

Equilibrium modulus G∞

Equilibrium stress σ∞

G(t) or σ(t)

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Results

τ=69 days b=0.2 τ=11.5 days b=0.26 and G∞ = 5.2 GPa

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

Time (days)

Stre

ss (M

Pa)

Gueguen, Master Report, 2007FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

0

10

20

30

40

50

60

70

80

0 20 40 60 80 100 120 140 160 180time (days)

stre

ss (M

Pa)

Influence of thermal history

Air quenched

Heat treated

Gueguen, Master Report, 2007FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

τ b (GPa) E (GPa)

Air Quenched 11,6 0,26 3,5

Heat Treated 16,0 0,36 6,916,9

Gueguen, Master Report, 2007

Results

G∞

• Residual stresses

• Structural relaxation

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Model for relaxation:

• Burger’s cell: only for short times (< 30 days)

• KWW equation : for both short and long times

• KWW modified : allows to represent the

equilibrium modulus G∞ or stress σ∞

Conclusion

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Fiber bending test

Rolling

0 Time

RelaxationConstant strain imposed

Stress decreases

1

Instantaneous elasticity

Unrolling

tu

Ru

Anelasticity

RecoveryStrain decreases

2

R(t)

t∞t

R∞

Permanent strain

Delayed elasticity

R0

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Strain Recovery

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

0 10 20 30 40 50 60 70 80 90 100

Time (days)

Stra

in ε

(t)

Recovery curves

Time (Days)

Stra

in ε

(t)

ε(t)R(t)

r=

tu2

Ru2

R∞2

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Model for Recovery

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

0 10 20 30 40 50 60 70 80 90 100

Time (days)

Stra

in ε

(t)

Burger’s model used for relaxation

Time (Days)

Stra

in ε

(t)

τd = 1.2 days

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Model for Recovery

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

0 10 20 30 40 50 60 70 80 90 100

Time (days)

Stra

in ε

(t)

Burger’s model, one set of parameters for all the curves

Time (Days)

Stra

in ε

(t)

τd = 9.51 days

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Model for Recovery

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

0 10 20 30 40 50 60 70 80 90 100

Time (days)

Stra

in ε

(t)

Burger’s model, one set of parameters for each curve

Time (Days)

Stra

in ε

(t)

τd1

τd2

τd3

τd4

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Model for Recovery

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

0 10 20 30 40 50 60 70 80 90 100

Time (days)

Stra

in ε

(t)

KWW model, one couple of parameters for all the curves

Time (Days)

Stra

in ε

(t)

τKWW = 6.63 days

b = 0.57

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Model for Recovery

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

0 10 20 30 40 50 60 70 80 90 100

Time (days)

Stra

in ε

(t)

KWW model, one couple of parameters for each curve

Time (Days)

Stra

in ε

(t)

τKWW1 , b1

τKWW2 , b2

τKWW3 , b3

τKWW4 , b4

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

05

10152025

3035

0 20 40 60 80Time (days)

τ (d

ays)

0

0,2

0,4

0,6

0,8

1

b

Strain recovery

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan Bernard et al, J.Europ.Ceram.Soc, 2007

Model for recovery:• Can’t use the same parameters for relaxation

and recovery

• The best fit is obtain with KWW equation with parameters depending on the time spent on the mandrin (tu).

Conclusion

FFAG IV – Nov. 4-7, 2007 – Shiga, Japan

Ongoing work

Heat capacity measurements

Air quenched

Heat treated

19 months old TAS fibers