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Statistical evaluation of the critical distance in the finite life fatigue regime
M. Benedetti – University of Trento, Italy
C. Santus – University of Pisa, Italy
1
• Introduce the optimized V-notched specimen for assessing the (fatigue) critical distance
• Introduce the procedure to find the statistical distribution of the critical distance
• Extend this procedure to the finite life regime
• Assess the fatigue strength of another specimen, with blunter radius, and its statistical distribution
• Application to the Aluminium alloy 7075-T6
Aims and objectives
2
The Theory of Critical Distances (TCD), Line Method – Point Method (Fatigue)
2
fl
0
fl
( )d (Line Method,LM)
( / 2) (Point Method,PM)
L
y
y
x x
L
=
=
x
( )y x
Notch root
y
Fatigue limit
(plain spec.)
2L/ 2L
Point Method
Line Method
Area Method
Volume Method
Other local methods, such as SED
Critical Distance orMaterial characteristic length
2
th
fl
1 KL
=
How to obtain L?- Literature (similar materials)- Experiment (Δσfl, ΔKth)
3
Critical distance determination with the SIF threshold
Plain specimen
Notched specimen blunt
Notched specimensharp
Notched specimenultra-sharp
Crack thresholdC(T) / M(T) specimen
No
tch
sev
erit
y
2
th
fl
1 KL
=
thK
fl
- Challenging though ASTM standard, the crack length measure is required
- Specimen C(T), or M(T), may not fit into the samples, especially for material supply in bars
- Precracking technique dependence 4
Critical distance determination by combining a notched specimen with the plain specimenN
otc
h s
ever
ity
L
thK
fl
N,fl
N,fl
C. Santus, D. Taylor, M. Benedetti. Determination of the fatigue critical distance according to the Line and the Point Methods with rounded V-notched specimen. International Journal of Fatigue, 2018
C. Santus, D. Taylor, M. Benedetti. Experimental determination and sensitivity analysis of the fatigue critical distance obtained with rounded V-notched specimens. International Journal of Fatigue, 2018
Plain specimen
Notched specimen blunt
Notched specimensharp
Notched specimenultra-sharp
Crack thresholdC(T) / M(T) specimen
The length which satisfies the TCD criterion, either Line or Point Method
5
Critical distance determination by combining two notched specimens N
otc
h s
ever
ity
L
thK
fl
N,fl
N,fl
L. Susmel, D. Taylor. The theory of critical distances as an alternative experimental strategy for the determination of KIc and DKth. Eng Fract Mech, 2010
S. Cicero, P. González, B. Arroyo, J.A. Álvarez. Analysis of environmentally assisted cracking processes in notched steels using the point method. Procedia Struct Integrity 2019
Plain specimen
Notched specimen blunt
Notched specimensharp
Notched specimenultra-sharp
Crack thresholdC(T) / M(T) specimen
Two notches with different severity can be used as well, and no referring to the plain specimen.This is the common approach for the brittle fracture (static) critical distance.
6
Proposed optimal specimen
• V-notch axisymmetric specimen:easy to manufacture, no boundary effects
• Relatively open angle: 90°, 60°
• Sharp root radius, at least well controlled: the nose radius of the tool
R
D
A
A
Max stress concentration:A = 0.3 D/2
Ultrasharp V-notch
Nominal R = 0.1 mm
Actual R = 0.12 mm
Sharp V-notch:
Nominal R = 0.2 mm
Actual R = 0.21 mm
7
Procedure for the critical distance calculation
Line Method Point Method
flf
N,fl
K
=
1/ 1/
N,UU N,UU
0 0
f f
550 min
min 0
1
12
2 (1 )
2 2
s s
i
i
i
K Kl l
s K K
ll l l l
D DL l L l
−
=
= =
−
− = + =
= =
2
fN
L
N
Singular term
( )y s
Kx
x =
x
/ 2D
A
N
y
L
Rounded notch
maximumstress
x
/ 2D
A
N
R
y
• Determination of the Fatigue stress concentration factor Kf
• Dimensionless and singularity based (Line Method) critical distance l0
• Conversion from l0 to the critical distance l with a FE-based analytical procedure (linearity)• Determination of the actual critical distance L
8
Critical distance determination, statistical assessment
fl
N,flL
N,av
fl
Experimental
uncertainties
High
uncertainty
N,fl
No
tch
sev
erit
y
Plain specimen
Notched specimen blunt
Notched specimensharp
Notched specimenultra-sharp
Crack thresholdC(T) / M(T) specimen Monte Carlo simulation: assuming
Normal distributions of the specimen strengths, the obtained distribution of the critical distance is Skew-Normal L 9
SmallKf
L
N,av
fl
Experimental
uncertainties
Lower
uncertainty on
the derived CD
LargerKf
Proposed procedure for the statistical assessment of the critical distance
Line Method, L
M. Benedetti, C. Santus. Statistical properties of threshold and notch derived estimations of the critical distance according to the line method of the theory of critical distances. International Journal of Fatigue, 2020
2
2
1 ( ) ( )( ) 1 erf exp
22 2
L LPDF L
− −= + −
fl N,fl N
2 2
fl N N,fl
Fatigue limits and standard deviations: , , ,
Equivalent coefficient of variation (CV) of the input data:
( / ) ( / )
2
Normalized coefficient of variation (NCV)
of the dimensionless critical
S S
S S
+ =
distance:
1
, :mean and standard deviation
of thecriticaldistance
=
Equations provided for:
,
and then for:
(skewness)
and finally:
, ,
sk
→
Probability
Density
Function
10
Non symmetrical
Right-skewed, 0sk
Aluminium alloy 7075-T6 experimental data
105
106
107
40
60
80
100
120
140
160
180200220240
plain
ultrasharp notch (R0.12)
7075-T6, 150 Hz, RT, R = −1
Str
ess
amp
litu
de,
a
N,a (
MP
a)
Number of cycles to failure, Nf
Δσfl, S
ΔσN,fl, SN
Kf, Σ
L & PDF(L)
Plain specimen
Notched specimen blunt
Notched specimensharp
Notched specimenultra-sharp
Crack thresholdC(T) / M(T) specimen
Specimens used for Lassessment and prob. distr.
10% CDF
90% CDF
50% CDF
R = 1 mm
R = 0.21 mm
R = 0.12 mm
R = 0
R = ∞
11
Aluminium alloy 7075-T6 experimental data
105
106
107
40
60
80
100
120
140
160
180200220240
plain
ultrasharp notch (R0.12)
7075-T6, 150 Hz, RT, R = −1
Str
ess
amp
litu
de,
a
N,a (
MP
a)
Number of cycles to failure, Nf
Δσfl, S
ΔσN,fl, SN
12
L & PDF(L)
Same procedure applied for any value of Nf
Aluminium alloy 7075-T6, critical distance and statistical distribution
105
106
107
40
60
80
100
120
140
160
180200220240
plain
ultrasharp notch (R0.12)
7075-T6, 150 Hz, RT, R = −1
Str
ess
amp
litu
de,
a
N,a (
MP
a)
Number of cycles to failure, Nf
Δσfl, S (Nf)
ΔσN,fl, SN (Nf)
13
Aluminium alloy 7075-T6, critical distance and statistical distribution
105
106
107
40
60
80
100
120
140
160
180200220240
plain
ultrasharp notch (R0.12)
7075-T6, 150 Hz, RT, R = 0.1
Str
ess
amp
litu
de,
a
N,a (
MP
a)
Number of cycles to failure, Nf
← previously load ratio R = -1now load ratio R = 0.1
14
Cumulative density function (CDF) of the critical distance
Low skewnessHigher st. dev. at lower Nf
High skewnessAlmost constant st. dev.
Load ratio R = -1 Load ratio R = 0.1
15
Limit notch radius for an accurate assessment of the critical distance
Plain specimen
Notched specimen blunt
Notched specimensharp
Notched specimenultra-sharp
Crack thresholdC(T) / M(T) specimen
Specimens used for Lassessment and prob. distr.
R = 1 mm
R = 0.21 mm
R = 0.12 mm
R = 0
R = ∞ Normalized coefficient of variation (NCV):
1 St.dev.to mean value ratioof thecriticaldistance
Equivalent coefficient of variation of theinput strenghts
= =
Rlim By imposing a certain level of NCV, the limit radius value can be obtained: Rlim
If the notched specimen radius is higher than Rlim the assessment of L has an excessive standard deviation
16
Limit notch radius for an accurate assessment of the critical distance
Load ratio R = -1 Load ratio R = 0.1
NCV = 5, or higher, can be obtained with the Ultrasharp specimen, and among the two load ratios, R = 0.1 requires a sharper radius at equal NCV 17
Fatigue strength prediction of blunter specimens
Plain specimen
Notched specimen blunt
Notched specimensharp
Notched specimenultra-sharp
Crack thresholdC(T) / M(T) specimen
Fatigue strength of these specimens assessed, along with their statistical distribution
Specimens used for Lassessment and prob. distr.
Plain specimenfl
Critical distance L
N,flNotched specimen
18
Fatigue strength prediction of blunter specimens
Load ratio R = -1 Load ratio R = 0.1
Accurate prediction on the entire finite life regime, both in terms of mean values and trends, and also statistical distributions
90% CDF
10% CDF
50% CDF
19
Conclusions
• The critical distance shows a Skew normal (not symmetric) distribution.
• This distribution is Right-skewed, i.e. the skewness is positive.
• This procedure is extended to the fatigue finite life regime, the distribution trend depends on the load ratio.
• A limit (maximum) radius of the notched specimen is found and its value was larger for the finite life than at the fatigue limit.
• Blunter specimens are assessed, and the probability distribution is in agreement with the experimental data scatter on the entire high cycle fatigue regime.
20
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