application of a phenomenological viscoplasticity model to the stress relaxation behavior of...
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Application of A Phenomenological Viscoplasticity Model to The Stress Relaxation Behavior of Unidirectional and
Angle-ply Laminates at High Temperature
Y. Masuko and M. Kawai
Institute of Engineering Mechanics and Systems,
University of Tsukuba, Tsukuba 305-8573 , Japan
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Background
Objectives
Experimental results
Predicted results
Summary
Outline
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Matrix-Dominated Behavior of PMCs
PMC Laminates:
Polymer Matrix:
Time dependent responses
Off-axis loadingShear loading
・ Creep
・ Stress relaxation
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Unidirectional laminate
STRESS RELAXATION BEHAVIOR OF CFRP
Experimental Observation:
Unidirectional Laminates
Angle-Ply Laminates
Applicability of Viscoplasticity Model:Angle-ply laminate
Objectives
Ply Model
Laminate Model
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T800H/Epoxy#3631 (Cure temperature: 180˚C, Tg = 215˚C)
Angle-ply specimens:
Fiber Orientation:
Off-axis specimens:
50 50100
1.70
201
Unit: mm
Material System
Specimens
[]12 = 0˚, 10˚, 30˚, 45˚, 60˚, 90˚ []3s = [0]3s, [45]3s, [0]3s
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Experimental Procedure
Stress Relaxation Test (100˚C)
Time
a
c b
5 h
R
1 3
2
xf
xf
R 123
1 < 2 = xf < 3 )
a-b : Loading ( 1.0 mm/min; Stroke control )
b-c : Relaxation Period ( 5 hours; Stroke control )
・ Constant total strains for stress relaxation tests
Stroke control
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Stress-Strain Curves for CFRP
0 0.5 1 1.5 2 2.5 30
100
200
300
400
500
600
= 0
= 30
= 10
= 45 = 90
Experimental (100°C)
T800H/3631
1 mm/min
Axi
al s
tres
s x,
MP
a
Axial strain x, %
Time
Dis
plac
emen
t
1.0 mm/min
Unidirectional laminate
xy
[±30]3s
0 2 4 6 8 10 120
100
200
300
400
500
600
Axi
al s
tres
s x,
MP
a
Axial strain x, %
Experimental (100°C)T800H/36311 mm/min
[±45]3s
[±60]3s
Angle-ply laminate
xy
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Off-Axis Stress Relaxation of UD-CFRP
xy
xy
R = Const
R
0
50
100
150
200
250
300
0 1 2 3 4 5 6
Axi
al S
tres
s x,
MP
a
R = 0.63 %
R = 0.44 %
R = 0.22 %
ExperimentalT800H/3631
= 10 100C□,○ ,△
Time tR, h
0
10
20
30
40
50
60
70
80
0 1 2 3 4 5 6
R = 1.08 %
R = 0.60 %
R = 0.34 %Axi
al S
tres
s x,
MP
a
ExperimentalT800H/3631
= 30 100C□,○ ,△
Time tR, h
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Stress Relaxation of Angle-ply CFRP
xy
xy
R = Const
R
0
50
100
150
200
250
300
350
400
0 1 2 3 4 5 6
[±30]3S
R = 1.73 %
R = 1.14 %
R = 0.65 %
ExperimentalT800H/3631
100C□,○ ,△
Axi
al S
tres
s x,
MP
a
Time tR, h
0
10
20
30
40
50
60
0 1 2 3 4 5 6
R = 1.17 %
R = 0.68 %
R = 0.41 %
[±60]3S
ExperimentalT800H/3631
100C□,○ ,△
Axi
al S
tres
s x,
MP
a
Time tR, h
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Modeling of Time-Dependent Behavior (1/3)
ASSUMING Time-Dependent Elasticity:
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Stress Relaxation Modulus
12
xy
0
2
4
6
8
10
12
10-2 10-1 100 101
Time tR, h
= 30
R = 1.08 %
R = 0.60 %R = 0.34 %
T800H/3631
Experimental (100C)
x/ R
×10
3 , M
Pa
○
△
□0
2
4
6
8
10
12
10-2 10-1 100 101
Experimental (100C) T800H/3631
[±30]3S
R = 1.73 %R = 1.14 %
R = 0.65 %
○
△
□
x/ R
×10
4 , M
Pa
Time tR, h
xy
C x
R
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Modeling of Time-Dependent Behavior (1/3)
ASSUMING Time-Dependent Elasticity:
Nonlinear viscoelasticity (VE) modeling
Schapery model
Heredity integral form
Favored in polymer research
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Modeling of Time-Dependent Behavior (2/3)
ASSUMING Time-Dependent Plasticity:
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Loading-Unloading Behavior of UD-CFRP
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60
100
200
300
400
500 = 10
Axi
al s
tres
s x,
MP
a
Axial Strain x, %
Experimental (100C)1 mm/min
T800H/3631
Monotonic○
0
30
60
90
120
150
0 0.5 1 1.5 2 2.5
= 30
Experimental (100C)1 mm/min
T800H/3631
Axi
al s
tres
s x,
MP
a
Axial Strain x, %
Monotonic○
1212
xy
xy
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Modeling of Time-Dependent Behavior (2/3)
ASSUMING Time-Dependent Plasticity:
Nonlinear viscoplasticity (VP) modeling
Gates-Sun model
Nonlinear differential form
Technically, more profitable
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Modeling of Time-Dependent Behavior (3/3)
Nonlinear VE + VP modeling
Ha-Springer modelTuttle et al. model
A difficulty in distinguishing between VE and VP components
ASSUMING Time-Dependent ElastoPlasticity:
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Unidirectional Lamina
Angle-ply Laminate
Modified Gates-Sun model
Modified Gates-Sun model
+
Classical Lamination Theory
(CLT)
Viscoplasticity Modeling of Time-Dependent Behavior
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Off-axis stress-plastic strain curves
0 0.5 1 1.5 2 2.5 30
50
100
150
200
250
300
350
400
Axi
al s
tres
s x,
MP
a
Axial Plastic Strain %
xp,
Experimental (100C)1 mm/min
T800H/3631
= 90
■ = 10
▲ = 45 = 30◆
●
Sun-Chen Model (1989)
x
h 3
2sin4 2a66sin2 cos2
xp x
x
Ex
h xEffectiveStress
p xp h() Effective
Plastic-strain
a66 = 1.3
Effective stress - effective plastic strain curves
0
20
40
60
80
100
120
140
0 0.5 1 1.5 2 2.5 3
a66 = 1.3Experimental (100C)
Eff
ecti
ve S
tres
s
, M
Pa
Effective Plastic Strain
p , %
= 90
■ = 10
▲ = 45 = 30◆
●
1 mm/minT800H/3631
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Modified Gates-Sun Model
h x
Effective Stress:
0
20
40
60
80
100
120
140
0 0.005 0.01 0.015 0.02 0.025 0.03
Eff
ecti
ve S
tres
s
, M
Pa
Effective Plastic Strain
p, mm/mm
1 mm/min
Effective stress - effective internal strain curves
Effective Overstress:
H
r
Ý p K
1
mH
0
20
40
60
80
100
120
140
0 0.5 1 1.5 2 2.5 3
a66 = 1.3Experimental (100C)
Eff
ecti
ve S
tres
s
, M
Pa
Effective Plastic Strain
p , %
= 90
■ = 10
▲ = 45 = 30◆
●
1 mm/minT800H/3631
Effective stress - effective plastic strain curves
r Qi 1 e b ip
i
Hardening Variable:
Effective Plastic Strain Rate:
r
H
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Off-Axis Loading
where
h 1 1
m1K
1
m
h( ) 32
sin 4 2a66 sin2 cos2
Ý xp x r h()
1 m
Off-axis Specimen
Modified Gates-Sun Model
x
y
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Off-Axis Creep Curves for UD-CFRP
1212
xy
xy
C = Const
C
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6
C = 84 MPa
44 MPa
63 MPa
Time tc, h
Axi
al S
trai
n x
, % 100C
T800H/3631 □,○,△ = 30
Experimental
0
0.5
1
1.5
2
2.5
3
0 1 2 3 4 5 6
51 MPa
35 MPa
C = 68 MPa
Time tc, h
Axi
al S
trai
n x
, % 100C
T800H/3631 □,○,△ = 45
Experimental
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0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6
Time tc, h
Axi
al S
trai
n x
, %
Off-axis creep curve for = 10˚
Identification of Material Constants—1
p xp h()
r h()C
xp x 5h
C
Ex
x C
Effective stress - effective internal strain curves
0
20
40
60
80
100
120
140
0 0.005 0.01 0.015 0.02 0.025 0.03
Eff
ecti
ve S
tres
s
, M
Pa
Effective Plastic Strain
p, mm/mm
1 mm/min
r
Ý p H
K
1
m
r0
K
1
m
r
Q1 = 24 MPa
Q2 = 80 MPa
b1 = 750b2 = 45r0 = 17 MPa
C = 208 MPa = 10
■ Creep 5 h
= 90
■ = 10
▲ = 45 = 30◆
●
r
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Ý p H
K
1
m
Effective stress - effective internal strain curves
0
20
40
60
80
100
120
140
0 0.005 0.01 0.015 0.02 0.025 0.03
Eff
ecti
ve S
tres
s
, M
Pa
Effective Plastic Strain
p, mm/mm
1 mm/min
Identification of Material Constants—2
H
r
r0
Effective overstress
H r r0
Effectiveplastic strain rate
Ý p 1
h()Ý x
Ý xEx
K = 79 MPa ・ minm
m = 0.205
Effective stress - effective plastic strain rate curves
10-3
10-2
10-1
100
101
102
103
104
105
10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
Eff
ecti
ve O
ver
Str
ess
H,
MP
a
Effective Strain Rate , 1mm/mm/min
Ý p
T800H/3631Experimental (100°C)
○ Tension 1mm/min Base
= 90
□ = 10
△ = 45 = 30◇
○
Creep [5h] Base●
■◆
●▲
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Predicted Off-Axis Stress-Stain Curves(Modified Gates-Sun Model)
50
150
250
350
450
0 0.5 1 1.5 2 2.5
Axi
al S
tres
s x,
MP
a
Axial Strain x, %
T800H/3631
Experimental (100C)○
= 90 = 45
= 30
= 10
1 mm/min
Predicted
Time
Dis
plac
emen
t
1.0 mm/min
Material Constants
a66 = 1.3Q1 = 24 MPaQ2 = 80 MPab1 = 750b2 = 45r0 = 17 MPaK = 79 MPa ・ minm
m = 0.205
Unidirectional laminate
x
y
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0
10
20
30
40
50
60
0 1 2 3 4 5 6
R = 0.76 %
0.44 %
T800H/3631 = 45
Experimental○,△
100C
Axi
al S
tres
s x,
MP
a
Time tR, h
0
10
20
30
40
50
60
70
80
0 1 2 3 4 5 6
R = 1.11%
0.70%
0.56%
= 60
Experimental□,○,△T800H/3631
100C
Axi
al S
tres
s x,
MP
a
Time tR, h
Predicted Off-Axis Stress Relaxation Curves
Predicted Predicted
1212
xy
xy
R = Const
R
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0 0.5 1 1.5 20
50
100
150
R = 1.08 %
Axi
al S
tres
s x,
MP
a
Axial Strain x, %
DispExperimental (100C)T800H/3631
= 30
1 mm/minDisplacement
(100 mm)
Strain Gauge(2 mm)
GaugeR a tR b tR
2 c tR
3 d tR
4
Gauge
R = 0.87 %
StrokeControl
Disp R Constant x
y
Elastic Unloading due to Local Strain Recovery
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Predicted Off-Axis Stress Relaxation Curveswith Strain Recovery
1212
xy
xy
R = R (tR)
R
0
10
20
30
40
50
60
0 1 2 3 4 5 6
R = 0.76 %
0.44 %
T800H/3631 = 45
Experimental○,△
100C
Axi
al S
tres
s x,
MP
a
Time tR, h
0
10
20
30
40
50
60
70
80
0 1 2 3 4 5 6
R = 1.11%
0.70%
0.56%
= 60
Experimental□,○,△T800H/3631
100C
Axi
al S
tres
s x,
MP
a
Time tR, h
Predicted Predicted
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0 2 4 6 8 10 12
0
50
100
150
200
250
Axi
al s
tres
s x,
MP
a
Axial strain x, %
[±45]3s Experimental (100C)
T800H/36311 mm/min
○
Predicted
0 0.5 1 1.5 2 2.50
20
40
60
80
100
Axi
al s
tres
s x,
MP
a
[±60]3s Experimental (100C)
T800H/36311 mm/min
Axial strain x, %
○
Predicted
Predicted Stress-Strain Curves for Angle-Ply Laminates
3S3S
xy
xy
Time
Dis
plac
emen
t
1.0 mm/min
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0 2 4 6 8 10 120
50
100
150
200
250
Axi
al s
tres
s x,
MP
a
Axial strain x, %
[±45]3s Experimental (100C)
T800H/36311 mm/min
○
Predicted
ab
a’
b’
Fiber Rotation due to Deformation of Angle-Ply Laminate
x
y
tan b a
1 y
1 x
tan
(Sun, Herakovich, Wisnom)
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0 2 4 6 8 10 12
0
50
100
150
200
250
Axi
al s
tres
s x,
MP
a
Axial strain x, %
[±45]3s Experimental (100C)
T800H/36311 mm/min
○
Predicted
0 0.5 1 1.5 2 2.50
20
40
60
80
100
Axi
al s
tres
s x,
MP
a
[±60]3s Experimental (100C)
T800H/36311 mm/min
Axial strain x, %
○
PredictedWith rotaion With rotaion
Predicted Stress-Strain Curves for Angle-Ply Laminateswith Fiber Rotation
3S3S
xy
xy
Time
Dis
plac
emen
t
1.0 mm/min
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0
50
100
150
200
250
300
350
400
0 1 2 3 4 5 6
Axi
al S
tres
s x,
MP
a
Time tR, h
0.65%
R =1.73%
1.14%
[±30]3S
Experimental□,○,△T800H/3631
100C0
10
20
30
40
50
60
0 1 2 3 4 5 6
0.68%
0.41%
R =1.17%
[±60]3S
T800H/3631
100C
Experimental□,○,△
Axi
al S
tres
s x,
MP
a
Time tR, h
Predicted Predicted
Predicted Stress Relaxation of Angle-Ply Laminates
3S3S
xy
xy
R = Const
R
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0
50
100
150
200
250
300
350
400
0 1 2 3 4 5 6
Axi
al S
tres
s x,
MP
a
Time tR, h
0.65%
R =1.73%
1.14%
[±30]3S
Experimental□,○,△T800H/3631
100C0
10
20
30
40
50
60
0 1 2 3 4 5 6
0.68%
0.41%
R =1.17%
[±60]3S
T800H/3631
100C
Experimental□,○,△
Axi
al S
tres
s x,
MP
a
Time tR, h
PredictedWith recovery
PredictedWith recovery
Predicted Stress Relaxation of Angle-Ply Laminates with Strain Recovery
3S3S
xy
xy
R = R (tR)
R
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Stress relaxation effects at high temperature in unidirectional and angle-ply CFRP laminates were examined.
Simulation was also performed using a ply viscoplasticity model and CLT.
The stress relaxation effects are clearly observed in all specimens of unidirectional and angle-ply laminates.
The stress relaxation rate rapidly decreases to vanish in a short period, regardless of the ply orientations and the sustained strain levels.
Predictions using the ply viscoplasticity model and CLT together with a consideration of the local strain recovery agree well with the experimental results.
Conclusions
Good predictions of the stress relaxation behavior confirm that the stress relaxation behavior is consistent with the creep behavior.