room temperature viscosity and delayed elasticity … · room temperature viscosity and delayed...
TRANSCRIPT
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