Uncertainty on Fatigue Damage Accumulation
for Composite Materials
Authors:
Henrik Stensgaard Toft, Aalborg University, Denmark
John Dalsgaard Sørensen, Aalborg University / Risø-DTU, Denmark
Contents
•
Introduction
•
Composite Material used for Testing
•
Constant Amplitude Fatigue Tests
•
Variable Amplitude Fatigue Tests
•
Conclusion and Future Work
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Introduction
Fatigue design of composite material is normally performed using a
deterministic design approach.
The uncertainties related to fatigue design are in general large which
allows for adopting a probabilistic design approach where the
uncertainties can be treated in a rational manner.
The objective of the present work is establish probabilistic models for:
•
Uncertainty on the SN-curve
•
Uncertainty on damage accumulation
Primary focus on composite materials used in wind turbine blades.
Introduction
Deterministic fatigue design is normally based on a:
•
•
Linear SN-curve (log-log scale)
Miner Rule for linear damage accumulation
Other types of SN-curves and damage accumulation models have
been proposed, see e.g. [1,2].
[1] V.A. Passipoularidis, Fatigue Evaluation Algorithms: Review, Materials Research Division, RisøDTU, 2009.
[2] N.L. Post, S.W. Case, J.J. Lesko, Modelling the variable amplitude fatigue of composite materials: A
review and evaluation of the state of the art for spectrum loading, Int. J. Fatigue, 30 (2008), pp. 20642086.
Introduction
Uncertainties can in general be divided into three groups:
•
Physical uncertainty (Aleatory uncertainty)
•
Model uncertainty (Epistemic uncertainty)
•
Statistical uncertainty (Epistemic uncertainty)
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Composite Material used for Testing
Material Database:
•
OptiDAT database (www.kc-wmc.nl)
Composite material:
•
Epoxy Glass Fiber Reinforced Plastic (GFRP)
Geometry:
•
•
•
Geometry R04 MD
Lay-up: [[45, 0]4; 45]
Size: 25 x 150mm
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Constant Amplitude Fatigue Tests
For composite materials the fatigue strength is dependent on the mean
stress. Therefore several SN-curves are collected in a Constant Life
Diagram using:
•
Constant amplitude data for 7 different R-ratios.
300
400
N=10 3
N=10 4
N=10 5
N=10 5
N=10
300
6
N=10 7
N=10
R=10.0
R=-2.5
R=-0.4
6
N=10 7
N=10 8
200
R=0.1
9
200
N=10 8 R=10.0
N=10 9
amp [MPa]
N=10
 min
 max
R=-1.0
N=10 3
R=-1.0
4
N=10
amp [MPa]
R
R=0.1
R=2.0
R=0.5
100
100
0
-800
-600
-400
-200
0
200
mean [MPa]
400
600
800
1000
0
-600
-400
-200
0
mean [MPa]
200
400
600
Constant Amplitude Fatigue Tests
SN-curves are fitted using the Maximum-Likelihood Method in order to
model the physical and statistical uncertainties.
log N  log K  m log   
The Likelihood function is given by:
 1  log N   log K  m log   2 
i
i
L  log K ,     
exp   
 

2

2 
i 1


 

n  n0
 log N i   log K  m log  i  


 

i  n 1


n
1
The statistical uncertainty is calculated using the Hessian Matrix:
Clog K , 
2

 log
K
   H log K ,    
 log K ,   log K   
1
log K ,   log K    
 2


Constant Amplitude Fatigue Tests
Parameters in SN-curves:
m

log K
logK

R-ratio
Tests n
Runouts n0
0.5
15
0
10.541
27.768
0.358
0.092
0.065
0.1
45
2
9.508
27.191
0.259
0.039
0.027
-0.4
28
0
7.582
23.398
0.435
0.082
0.058
-1.0
84
3
6.719
21.359
0.878
0.094
0.068
-2.5
10
2
11.983
35.231
0.633
0.197
0.143
10.0
34
0
22.211
58.664
0.644
0.110
0.078
2.0
6
3
29.686
73.780
0.354
0.143
0.103
Static tension and compression strength:
Test-type
Tests n
Mean [MPa]
Std. [MPa]
STT
66
556.5
64.2
STC
55
-458.6
33.2
Constant Amplitude Fatigue Tests
R = 0.1
R = -1.0
400
800
700
600
500
stress range 
stress range 
300
200
100
2
10
3
4
10
5
6
10
10
cycles to failure N
10
400
300
200
100
2
10
7
10
3
4
10
5
6
10
10
cycles to failure N
10
7
10
R = -0.4
R = 2.0
500
250
400
stress range 
stress range 
225
200
175
150
3
10
4
10
5
10
cycles to failure N
6
10
7
10
300
200
100
2
10
4
6
10
10
cycles to failure N
8
10
Constant Amplitude Fatigue Tests
300
R=-1.0
N=10 3
N=10 4
N=10 5
N=10
R=-2.5
R=-0.4
6
N=10 7
200
N=10 8 R=10.0
amp [MPa]
N=10 9
R=0.1
R=2.0
R=0.5
100
0
-600
-400
-200
0
mean [MPa]
200
400
600
Variable Amplitude Fatigue Tests
The model uncertainty related to Miners rule for damage accumulation is
determined based on variable amplitude fatigue tests.
Load spectra's
• Wisper spectrum
• WisperX spectrum
n
D
i 1
1
N   i 
40
30
Level [-]
20
10
0
-10
WISPER
WISPERX
-20
-30
0
0.5
1
1.5
Peak [-]
2
2.5
x 10
5
Variable Amplitude Fatigue Tests
Mean and standard deviation for accumulated damage at failure.
Spectrum
Tests N
Mean
Std.
COVD
Wisper
10
0.9000
0.5355
0.595
Wisperx
13
0.2763
0.1982
0.717
Reverse Wisper
2
0.2020
-
-
Reverse Wisperx
10
0.3179
0.1576
0.496
Wisper, Wisperx
23
0.5474
0.4886
0.893
All
35
0.4621
0.4196
0.908
•
High uncertainties compared e.g. welded steel details.
•
Significant model uncertainty related to Miners rule for composite materials.
•
Lognormal distribution (negative damage avoided)
Conclusion & Future Work
Fatigue design contains the following uncertainties:
•
•
•
Physical uncertainty:
Statistical uncertainty:
Model uncertainty:
SN-curves
Parameters in the SN-curves
Miners rule
Model uncertainty on Miners rule can be modelled by a Lognormal
distribution with mean and standard deviation equal to approx. 0,50.
Partial safety factors for fatigue design of wind turbine blades can be
calibrated using the stochastic models given in the present paper.
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Uncertainty on Fatigue Damage Accumulation
for Composite Materials
Authors:
Henrik Stensgaard Toft, Aalborg University, Denmark
John Dalsgaard Sørensen, Aalborg University / Risø-DTU, Denmark