The Theory of
Critical Distances,
in metal fatigue
The Theory of Critical Distances,
in metal fatigue
Introduction
Short/Long cracks
Non-propagating
cracks
Ciro SANTUS
[email protected]
Notch effect
Size effect
Summary
D.I.M.N.P. Facoltà di Ingegneria. Università di Pisa.
July 2007
The Theory of
Critical Distances,
in metal fatigue
Introduction
Introduction
Short/Long cracks
Short/Long cracks
Non-propagating
cracks
Notch effect
Non-propagating cracks
Size effect
Summary
Notch effect
Size effect
Summary
The Theory of
Critical Distances,
in metal fatigue
Introduction
Introduction
Short/Long cracks
Short/Long cracks
Non-propagating
cracks
Notch effect
Non-propagating cracks
Size effect
Summary
Notch effect
Size effect
Summary
Only one and simple formula
The Theory of
Critical Distances,
in metal fatigue
subtitle
The material critical distance is:
1 ∆Kth 2
a0 =
π ∆σ0
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
where:
Size effect
Summary
I
∆Kth is the (long crack) threshold stress intensity factor,
at a certain load ratio R
I
∆σ0 is the fatigue limit, at the same load ratio R
Only one and simple formula
The Theory of
Critical Distances,
in metal fatigue
subtitle
The material critical distance is:
1 ∆Kth 2
a0 =
π ∆σ0
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
how to remember this formula:
Size effect
Summary
I
π is there just for cosmetics
I
to have a length, ∆Kth and ∆σ0 need to be put in ratio
and squared
Only one and simple formula
The Theory of
Critical Distances,
in metal fatigue
subtitle
The material critical distance is:
1 ∆Kth 2
a0 =
π ∆σ0
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Why is this formula so important?
Size effect
Summary
The Theory of
Critical Distances,
in metal fatigue
The Kitagawa–Takahashi diagram
subtitle
Introduction
Δσ (log)
Short/Long cracks
Δσ = Δσ 0
Δσ =
1
Unnotched
(plain)
specimen
1 ⎛ ΔK th
a0 = ⎜⎜
π ⎝ Δσ 0
⎞
⎟⎟
⎠
2
Non-propagating
cracks
ΔK th
πa
Notch effect
Size effect
Summary
2
a (log)
2a
Long crack
It is the link from the less severe notch (plain specimen) to
the most severe notch (long crack)
The Theory of
Critical Distances,
in metal fatigue
The Kitagawa–Takahashi diagram
subtitle
Introduction
Δσ (log)
Short/Long cracks
Δσ = Δσ 0
1 ⎛ ΔK th
a0 = ⎜⎜
π ⎝ Δσ 0
Non-propagating
cracks
⎞
⎟⎟
⎠
2
ΔK th
Δσ =
πa
a (log)
Short crack,
anomalous
behavior
Long crack,
fracture
mechanics
Transition from the short cracks to long cracks regimes
Notch effect
Size effect
Summary
The El Haddad model
The Theory of
Critical Distances,
in metal fatigue
subtitle
To describe the short to long crack transition the intrinsic
crack model is used
p
∆Kth = ∆σ π(a + a0 )
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
M. H. El Haddad, T. H. Topper, K. N. Smith
Prediction of non propagating cracks
Engineering Fracture Mechanics, vol. 11, p. 573–584
(1979)
Summary
The Theory of
Critical Distances,
in metal fatigue
The El Haddad model
subtitle
To describe the short to long crack transition the intrinsic
crack model is used
p
∆Kth = ∆σ π(a + a0 )
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
Δσ (log)
Summary
Δσ = Δσ 0
Δσ =
ΔK th
π (a + a0 )
Δσ =
ΔK th
πa
a (log)
The El Haddad model
The Theory of
Critical Distances,
in metal fatigue
subtitle
To describe the short to long crack transition the intrinsic
crack model is used
p
∆Kth = ∆σ π(a + a0 )
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
Summary
At least it works for asymptotes:
I
I
∆σ (a/a0 → 0) = ∆σ0
√
∆σ (a/a0 → ∞) = ∆Kth / πa
Results of the Theory of Critical Distances
The Theory of
Critical Distances,
in metal fatigue
subtitle
Introduction
Short crack /
Long crack
Short/Long cracks
Theory of
Critical Distance
Non-propagating
cracks
Non-propagating
cracks
Notch effect
Size effect
Notch effect
Size effect
Summary
The Theory of
Critical Distances,
in metal fatigue
Introduction
Introduction
Short/Long cracks
Short/Long cracks
Non-propagating
cracks
Notch effect
Non-propagating cracks
Size effect
Summary
Notch effect
Size effect
Summary
Linking Fatigue to Fracture Mechanics
The Theory of
Critical Distances,
in metal fatigue
subtitle
I
Fatigue is a main crack leading to final fracture
Introduction
I
According to LEFM a short crack should not grow
Short/Long cracks
Non-propagating
cracks
Notch effect
Δσ
Size effect
Summary
ai
So why fatigue ?
Assuming an existing crack ai
(~ grain size) it follows:
ΔK = βσ πai ΔK th
Linking Fatigue to Fracture Mechanics
The Theory of
Critical Distances,
in metal fatigue
subtitle
The LEFM does not work while the crack is short
Introduction
Short/Long cracks
da/dn
(log)
Non-propagating
cracks
Notch effect
Size effect
Short crack
behavior
ΔK th
Long crack
behavior
ΔK (log)
The short crack threshold is not the same as long crack
Summary
Linking Fatigue to Fracture Mechanics
The Theory of
Critical Distances,
in metal fatigue
subtitle
El–Haddad models the short crack threshold, but does not
explain why
Δσ
(log)
Short/Long cracks
Non-propagating
cracks
ΔK th
Δσ th =
πa
Short crack,
anomalous behavior
Δσ th, EH =
Introduction
Notch effect
Size effect
Summary
ΔK th
π(a +a0 )
a (log)
Linking Fatigue to Fracture Mechanics
The Theory of
Critical Distances,
in metal fatigue
subtitle
Short crack classification:
Introduction
I
Mechanically short crack
Short/Long cracks
I
Microstructurally short crack
Non-propagating
cracks
I
Physically short crack
Notch effect
Size effect
Summary
S. Suresh
Fatigue of Materials
Cambridge University Press, 2nd edition (2006)
The Theory of
Critical Distances,
in metal fatigue
Mechanically short crack
subtitle
The crack is inside the cyclic notch plastic region
Introduction
Short/Long cracks
Notch (cyclic)
plastic zone
Non-propagating
cracks
Notch effect
Size effect
Summary
Mechanically
short crack
I
Small scale yielding assumption violated (assumed for
definition of K)
I
Crack size can be few millimeters
Microstructurally short crack
The Theory of
Critical Distances,
in metal fatigue
subtitle
The crack is confined in few grains
Grain
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
Summary
Microstructural
short crack
I
I
I
I
Continuum mechanics not valid (grain anisotropy,
neighbor grains constraints)
Dislocation mechanisms
Shear growth
Weakest path
Crack closure
The Theory of
Critical Distances,
in metal fatigue
subtitle
The (plain strain) long crack closure is explained as plastic
wedge
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
Summary
R. Pippan, F. Riemelmoser
Visualization of plasticity–induced crack closure under
plane strain conditions
Engineering Fracture Mechanics, vol. 60, p. 315–322
(1998)
The Theory of
Critical Distances,
in metal fatigue
Crack closure
subtitle
The threshold stress intensity factor can be split into two
components:
I
I
Introduction
Short/Long cracks
intrinsic ∆Kth,i : the crack remains close for a portion of
the cycle
Non-propagating
cracks
extrinsic ∆Kth,e = ∆Kth,eff : the crack has its own
resistance against propagation
Size effect
K
ΔKth= ΔKth, i + ΔKth, e
ΔKth, i = ΔKth, eff
ΔKth, e = Kop
Notch effect
Summary
Crack closure
The Theory of
Critical Distances,
in metal fatigue
subtitle
Further effects than closure, in extrinsic threshold
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
Summary
A.J. Mc, R.O. Ritchie
Crack closure and the fatigue–crack propagation
threshold as a function of load ratio
Fatigue & Fracture of Engineering Materials &
Structures, vol. 21, p. 847–855 (1998)
The Theory of
Critical Distances,
in metal fatigue
Physically short cracks
subtitle
Introduction
Long crack > 1 mm
K
Kop
Kcl
ΔKeff
Physically short
crack < 1mm
K
time
Partial closure
Kcl ~ Kop > 0
then ΔKeff reduces
Short/Long cracks
Non-propagating
cracks
Notch effect
ΔKeff
Kcl
Kop
Kop
time
No closure
Kcl ~ Kop = 0
or even < 0 !!!
I
Crack size ≤ 1 mm, much longer than grain size
I
Crack closure is not saturated yet
Size effect
Summary
The Theory of
Critical Distances,
in metal fatigue
Introduction
Introduction
Short/Long cracks
Short/Long cracks
Non-propagating
cracks
Notch effect
Non-propagating cracks
Size effect
Summary
Notch effect
Size effect
Summary
Non–propagating cracks in notched components
The Theory of
Critical Distances,
in metal fatigue
subtitle
Introduction
Δσ
Δσ 0
Δσ =
kt
ρ
Short/Long cracks
Δσ =
1 ΔK th
πD
β
Non-propagating
cracks
Notch effect
Size effect
Summary
D
D = cost.; ρ → 0;
kt → ∞
1
kt*
non-prop.
cracks
kt
R. A. Smith, K. J. Miller
Prediction of Fatigue Regimes in Notched Components
International Journal of Mechanical Sciences, vol. 20, p.
201–206 (1978)
Non–propagating cracks in notched components
The Theory of
Critical Distances,
in metal fatigue
subtitle
Introduction
Δσ
Short/Long cracks
Δσ =
Δσ 0
kt
Non-propagating
cracks
Δσ =
1 ΔK th
β πD
Notch effect
Size effect
Summary
1
kt*
non-prop.
cracks
kt
The non–propagating crack can be observed only for
crack–like notches
Non–propagating cracks in notched components
The Theory of
Critical Distances,
in metal fatigue
subtitle
Introduction
Δσ
Δσ 0
Δσ =
kt
Short/Long cracks
1 ΔK th
Δσ =
β πD
Non-propagating
cracks
Notch effect
Size effect
Summary
1
kt*
non-prop.
cracks
kt
The non–propagating crack stress concentration factor kt∗ is
related to the material critical distance:
r
r
D
D
∗
kt = β
=
a0
aD
Non–propagating cracks in notched components
The Theory of
Critical Distances,
in metal fatigue
subtitle
Introduction
Short/Long cracks
ρ
Notch:
a0
2a
Non-propagating
cracks
Notch effect
kt ≈1 + 2
a
ρ
Crack:
β =1
The crack tip critical radius ρ ∗ can be obtained for the
reference crack configuration:
r
r
a
a
∗
kt = 1+2
=
→ ρ ∗ = 4a0
∗
ρ
a0
Size effect
Summary
Non–propagating cracks in notched components
The Theory of
Critical Distances,
in metal fatigue
subtitle
Introduction
Short/Long cracks
ρ
Notch:
a0
2a
Notch effect
kt ≈1 + 2
Crack:
β =1
Notch behavior (no non–propagating crack)
ρ > ρ ∗ = 4a0
Non-propagating
cracks
a
ρ
Size effect
Summary
Non–propagating cracks in notched components
The Theory of
Critical Distances,
in metal fatigue
subtitle
Introduction
Short/Long cracks
ρ
Notch:
a0
2a
Crack behavior (non–propagating crack)
ρ < ρ ∗ = 4a0
Non-propagating
cracks
Notch effect
kt ≈1 + 2
Crack:
β =1
a
ρ
Size effect
Summary
Non–propagating cracks, fracture mechanics
The Theory of
Critical Distances,
in metal fatigue
subtitle
In static fracture mechanics the resistance curve is common
σ3
πσ i a1
E
2
G=
a1
Short/Long cracks
σ4
G, R
σ2
Introduction
Non-propagating
cracks
Static fracture
resistance curve R
Size effect
Summary
σ1
Δa′ Δa′′
Notch effect
a
Increasing the stress σ :
I
at σ1 the crack a1 does not increase
I
at σ2 the crack a1 increases up to a1 + ∆a0
I
at σ3 critical stable growth point is reached
I
at σ4 unstable growth is obtained
Non–propagating cracks, fracture mechanics
The Theory of
Critical Distances,
in metal fatigue
subtitle
Small–Long crack resistance curve
Introduction
Short/Long cracks
ΔK (log)
′ a = ΔK th
ΔK th,
Non-prop.
crack
a
a0
Non-propagating
cracks
′′ a = ΔK th
ΔK th,
Notch effect
Size effect
El Haddad resistance curve
ΔK th,a = ΔK th
a
a + a0
a (log)
D. Taylor
A mechanistic approach to critical–distance methods in
notch fatigue
Fatigue & Fracture of Engineering Materials &
Structures, vol. 24, p. 215–224 (2001)
Summary
Non–propagating cracks, fracture mechanics
The Theory of
Critical Distances,
in metal fatigue
subtitle
Introduction
ΔK (log)
Short/Long cracks
Non-propagating
cracks
Notch effect
(C)
(B)
(A)
ΔK th,a = ΔK th
Size effect
a
a + a0
Summary
a (log)
I
(A) Non–propagating notch crack
I
(B) Fatigue limit non–propagating notch crack
I
(C) Propagating crack leading to failure
Non–propagating cracks, fracture mechanics
The Theory of
Critical Distances,
in metal fatigue
subtitle
Introduction
ΔK (log)
Short/Long cracks
Non-propagating
cracks
Crack-like notch,
loaded at Δσn
(B)
Notch effect
ΔK th,a = ΔK th
a
a + a0
Size effect
Summary
a (log)
Non − prop. crack size ≈ 2 a0
I
Non–propagating fatigue crack size ≈ 2a0 , emanating at
a crack–like notch
I
It allows the mechanistic approach to local methods to
asses notch fatigue strength
Non–propagating cracks, fracture mechanics
The Theory of
Critical Distances,
in metal fatigue
subtitle
Introduction
ΔK (log)
Crack-like
notch
Short/Long cracks
ΔK th,a = ΔK th
a
a + a0
Non-propagating
cracks
Notch effect
Resistance curve
Size effect
Summary
Blunt notch
a (log)
I
Crack–like notch, slope lower than resistance curve,
then non–propagating crack
I
Blunt notch, same initial slope than resistance curve, no
non–propagating crack
Nucleation mechanism
The Theory of
Critical Distances,
in metal fatigue
subtitle
Introduction
Stage I
Stage II
Stage I
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
Stage I
Nucleation is:
I Persistent Slip Bands (PSB) formation
I
Intrusion/Estrusion at the surface
I
Stage I shear dominated growth (crystallographic
propagation)
I
Stage II tensile dominated growth (non–crystallographic
propagation)
Summary
Non–propagating crack in plain specimen
The Theory of
Critical Distances,
in metal fatigue
subtitle
At the fatigue limit non–propagating fatigue cracks
(microstructurally short) can be detected in steels
Δσ
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
Plain
specimen
Important role of small defects and nonmetallic inclusions, at
the surface initiation point
Summary
Fatigue limit as non–propagating crack condition
The Theory of
Critical Distances,
in metal fatigue
subtitle
At the fatigue limit non–propagating fatigue cracks
(crystallographic size) can be detected in non ferrous alloys
Δσ
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
Plain
specimen
More precisely very slow propagation instead of
non-propagating, in almost all non ferrous alloys (no clear
fatigue limit)
Summary
Fatigue limit as non–propagating crack condition
The Theory of
Critical Distances,
in metal fatigue
subtitle
At the fatigue limit non–propagating fatigue cracks are
confined within a few number of grains
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Y. Murakami
Metal Fatigue: Effects of Small Defects and Nonmetallic
Inclusions
Elsevier (2003)
Size effect
Summary
Microstructurally vs. Physically
The Theory of
Critical Distances,
in metal fatigue
subtitle
I
I
Plain specimen fatigue limit is microstructurally short
crack resistance
Crack–like notch fatigue limit is physically short crack
resistance
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
Summary
K. J. Miller
The two thresholds of fatigue behaviour
Fatigue & Fracture of Engineering Materials &
Structures, vol. 16, p. 931–939 (1993)
The Theory of
Critical Distances,
in metal fatigue
Microstructurally vs. Physically
subtitle
I
I
Plain specimen fatigue limit is microstructurally short
crack resistance
Crack–like notch fatigue limit is physically short crack
resistance
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
Δσ
(log)
Notched sp.
Plain sp.
Summary
Δσ = Δσ 0
Δσ =
Major barrier
Microstructurally
short crack
d
ΔK th
πa
a (log)
Physically Long crack
short crack
Microstructurally vs. Physically
The Theory of
Critical Distances,
in metal fatigue
subtitle
I
I
Plain specimen fatigue limit is microstructurally short
crack resistance
Crack–like notch fatigue limit is physically short crack
resistance
Metallurgical evidence:
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
Summary
I
The smaller the grain size, the higher the plain
specimen fatigue limit
I
The larger the grain size, the higher the threshold stress
intensity factor
The Theory of
Critical Distances,
in metal fatigue
Microstructurally vs. Physically
subtitle
Metallurgical evidence:
I The smaller the grain size, the higher the plain
specimen fatigue limit
I The larger the grain size, the higher the threshold stress
intensity factor
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
Summary
Δσ
(log)
Crack-like
notched sp.
Plain sp.
d1
d2
a (log)
The Theory of
Critical Distances,
in metal fatigue
Introduction
Introduction
Short/Long cracks
Short/Long cracks
Non-propagating
cracks
Notch effect
Non-propagating cracks
Size effect
Summary
Notch effect
Size effect
Summary
The Theory of
Critical Distances,
in metal fatigue
Classic approaches for kf
subtitle
Introduction
σ (log)
Short/Long cracks
Plain
specimen
Non-propagating
cracks
Δσ0
Notched
specimen
100
103
LCF
Δσn (net area)
106
HCF
Nf (log)
Notch effect is defined by the fatigue factor kf , at the fatigue
limit:
kf =
∆σ0
∆σn
Notch effect
Size effect
Summary
The Theory of
Critical Distances,
in metal fatigue
Classic approaches for kf
subtitle
Introduction
σ (log)
Short/Long cracks
Plain
specimen
Non-propagating
cracks
Δσ0
Notched
specimen
100
103
LCF
Δσn (net area)
106
HCF
Notch sensitivity is usually defined as:
q=
kf − 1
,
kt − 1
q = [0, 1]
I
q = 0 → kf = 1, no notch sensitivity
I
q = 1 → kf = kt , full notch sensitivity
Nf (log)
Notch effect
Size effect
Summary
Classic approaches for kf
The Theory of
Critical Distances,
in metal fatigue
subtitle
Peterson equation for kf :
Introduction
Short/Long cracks
kf = 1 + q(kt − 1),
1
q=
αP
1+
r
Non-propagating
cracks
Notch effect
Size effect
R. E. Peterson
Notch sensitivity
In: Metal Fatigue, Edited by G. Sines and J. L.
Waisman, McGraw–Hill, New York p. 293–306 (1959)
Summary
The Theory of
Critical Distances,
in metal fatigue
Classic approaches for kf
subtitle
Neuber equation for kf :
Introduction
Short/Long cracks
kf = 1 + q(kt − 1),
1
r
q=
1+
αN
r
Non-propagating
cracks
Notch effect
Size effect
Summary
H. Neuber
Theory of Notch Stresses
Springer, Berlin (1955)
The Theory of
Critical Distances,
in metal fatigue
Classic approaches for kf
subtitle
Stress gradient as phenomenological parameter:
1 dσ χ=
σmax dx x=0
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
χ=
1 ⎛ dσ ( x ) ⎞
⎜
⎟
σ max ⎝ dx ⎠ x =0
σ (x)
x
E. Siebel, M. Stieler
Dissimilar stress distributions and cyclic loading
Z Ver Deutsch Ing, vol. 97, p. 121–131 (1955)
Summary
The Theory of
Critical Distances,
in metal fatigue
Classic approaches for kf
subtitle
Stress gradient as phenomenological parameter:
1 dσ χ=
σmax dx x=0
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
kt / kf
kt / kf
Summary
Mechanistic approaches to kf
The Theory of
Critical Distances,
in metal fatigue
subtitle
First fracture mechanics notch sensitivity model
Introduction
Short/Long cracks
kt
kf = r
a0
1 + 4.5
r
M. Klesnil, P. Lukáš
Fatigue of Metallic Materials
Elsevier Science, Amsterdam (1980)
Fracture mechanics related length parameter, instead of
Peterson and Neuber semi–empirical ones
Non-propagating
cracks
Notch effect
Size effect
Summary
The Theory of
Critical Distances,
in metal fatigue
Mechanistic approaches to kf
subtitle
First fracture mechanics related notch sensitivity model.
Introduction
Short/Long cracks
Non-propagating
cracks
kt
kf = r
a0
1 + 4.5
r
Δσ 0
Δσ n kt
kf =
Δ
σn
Δσ0 as uniform
Δσ n
r
a0
Fatigue limit:
ΔK = ΔKth
Notch effect
Size effect
Summary
I
elliptical surface notch
I
non–propagating
crack a0
I
fatigue limit as long
crack threshold at the
crack tip
Theory of Critical Distances for notch sensitivity
The Theory of
Critical Distances,
in metal fatigue
subtitle
Critical distance approaches to notch sensitivity are:
Introduction
I
Point method
Short/Long cracks
I
Line method
Non-propagating
cracks
I
Area method
Notch effect
Size effect
Summary
D. Taylor
Geometrical effects in fatigue: a unifying theoretical
model
International Journal of Fatigue, vol. 21, p. 413–420
(1999)
Theory of Critical Distances for notch sensitivity
The Theory of
Critical Distances,
in metal fatigue
subtitle
Point method (PM)
Introduction
Short/Long cracks
Δσ th
Non-propagating
cracks
Notch effect
Δσ (r ) = β Δσ th
ΔK th
a
=
2r
2π r
Δσ (a0 / 2) = Δσ 0
a0 / 2
r
At the fatigue limit (for a crack) the stress amplitude at
a0 /2 is σ0 (plain specimen fatigue limit)
Size effect
Summary
Theory of Critical Distances for notch sensitivity
The Theory of
Critical Distances,
in metal fatigue
subtitle
Line method (LM)
Introduction
Short/Long cracks
Δσ th
Non-propagating
cracks
Notch effect
Δσ (r ) = β Δσ th
ΔK th
a
=
2r
2π r
Δσ av (2a0 ) =
2a0
1
2a0
Size effect
Summary
2 a0
∫ Δσ (r ) dr = Δσ
0
r
At the fatigue limit (for a crack) the average stress
amplitude over r = [0, a0 /2] is σ0
0
Theory of Critical Distances for notch sensitivity
The Theory of
Critical Distances,
in metal fatigue
subtitle
Area method (AM)
Introduction
Short/Long cracks
Δσ th
Non-propagating
cracks
Notch effect
Δσ (r ) = β Δσ th
ΔK th
a
=
2r
2π r
Size effect
Summary
a0
r
Δσ av (Area, r = a0 ) =
4
π a0 2
π / 2 a0
∫ ∫ Δσ (r ,θ ) r dr dθ ≈ 1.1Δσ
0
0
0
At the fatigue limit (for a crack) the average stress
amplitude over r = [0, a0 /2], θ = [−π/2, π/2] is σ0
Theory of Critical Distances for notch sensitivity
The Theory of
Critical Distances,
in metal fatigue
subtitle
Introduction
Δσ th
Short/Long cracks
Non-propagating
cracks
Finite peak
stress (notch)
Notch effect
Size effect
Summary
Δσ (a0 / 2) = Δσ 0
a0 / 2
At the fatigue limit either cracks or notches feature similar
local stress distributions
Theory of Critical Distances for notch sensitivity
The Theory of
Critical Distances,
in metal fatigue
subtitle
I
Point method, most commonly used
Introduction
I
Line method, physical interpretation (explained later)
Short/Long cracks
I
Area method, rarely used
Non-propagating
cracks
Notch effect
Size effect
Summary
Theory of Critical Distances for notch sensitivity
The Theory of
Critical Distances,
in metal fatigue
subtitle
Advantages with respect to the use kf :
Introduction
I
Nominal stress not required
Short/Long cracks
I
Any geometry, notch depth and opening angle
Non-propagating
cracks
I
Defined even for zero notch radius
Notch effect
Size effect
Summary
Theory of Critical Distances for notch sensitivity
The Theory of
Critical Distances,
in metal fatigue
subtitle
Examples
Introduction
Short/Long cracks
Zero radius notch
Non-propagating
cracks
Notch effect
Size effect
Summary
Nominalstress = ??
kf = ??
k t = ∞, q = 0
kf = ??
Δσ(PM)
a0/2
a0/2
Fatigue limit:
Δσ(PM) = Δσ0
Fatigue limit:
Δσ(PM) = Δσ0
Δσ(PM)
Theory of Critical Distances for notch sensitivity
The Theory of
Critical Distances,
in metal fatigue
subtitle
Validation of the critical distance methods:
Introduction
Short/Long cracks
D. Taylor, G. Wang
The validation of some methods of notch fatigue
analysis
Fatigue & Fracture of Engineering Materials &
Structures, vol. 23, p. 387–394 (2000)
Non-propagating
cracks
Notch effect
Size effect
Summary
Theory of Critical Distances for notch sensitivity
The Theory of
Critical Distances,
in metal fatigue
subtitle
Requirements for the use of critical distance methods:
I
∆σ0 , a0 , ∆Kth (just two of them)
I
structure elastic F.E. post processing
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
Summary
Theory of Critical Distances for notch sensitivity
The Theory of
Critical Distances,
in metal fatigue
subtitle
Requirements for the use of critical distance methods:
I
∆σ0 is usually available in technical literature
I
∆Kth sometimes not available (especially at R < 0)
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
Summary
Theory of Critical Distances for notch sensitivity
The Theory of
Critical Distances,
in metal fatigue
subtitle
a0 can be deduced from fatigue limit with notched geometry
Introduction
Short/Long cracks
Δσ0
Non-propagating
cracks
Notch effect
Δσ0
PM
at a0/2
Plain specimen
at fatigue limit
Notched geometry
at fatigue limit
Size effect
Summary
Mechanistic explanation for the Line Method
The Theory of
Critical Distances,
in metal fatigue
subtitle
Resistance curve approach offers a mechanistic approach to
explain Line Method
Introduction
Short/Long cracks
Non-propagating
cracks
D. Taylor
A mechanistic approach to critical distance methods in
notch fatigue
Fatigue & Fracture of Engineering Materials &
Structures, vol. 24, p. 215–224 (2001)
Notch effect
Size effect
Summary
Mechanistic explanation for the Line Method
The Theory of
Critical Distances,
in metal fatigue
subtitle
Notch behavior regimes
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
Summary
Mechanistic explanation for the Line Method
The Theory of
Critical Distances,
in metal fatigue
subtitle
Crack–like notch
Introduction
Short/Long cracks
Stress distribution
(without the crack)
Δσ (r )
Non-propagating
cracks
Notch effect
Size effect
Summary
2 a0
r
Mechanistic explanation for the Line Method
The Theory of
Critical Distances,
in metal fatigue
subtitle
Crack–like notch
Introduction
ΔK
Non-prop. crack
(fatigue limit)
Short/Long cracks
Resistance
curve
Non-propagating
cracks
Notch effect
Size effect
Summary
2 a0
r
Mechanistic explanation for the Line Method
The Theory of
Critical Distances,
in metal fatigue
subtitle
Average stress (before crack) to calculate ∆K(2a0 )
Introduction
Short/Long cracks
1
∆σav =
2a0
Z 2a0
0
Notch effect
Z 2a0
∆K(2a0 ) =
Non-propagating
cracks
∆σ (r)dr
0
h(r, a0 )σ (r)dr = ∆σav
p
Size effect
2a0 π A
(h(r, a): weight function, A: correction factor)
Summary
Mechanistic explanation for the Line Method
The Theory of
Critical Distances,
in metal fatigue
subtitle
By imposing resistance curve tangency:
Introduction
Short/Long cracks
∆K(2a0 ) = ∆Kth,2a0
it follows:
Non-propagating
cracks
Notch effect
Size effect
∆σav ≈ σ0
Line Method explained for Crack–like notch
Summary
Generalization of Line Method, crack–like notch
The Theory of
Critical Distances,
in metal fatigue
subtitle
Line Method also works for
I
short cracks: ∆σn = ∆σ0
I
blunt notches: ∆σn = ∆σ0 kt
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Then it can be used for any notch geometry
Size effect
Summary
Generalization of Line Method, crack–like notch
The Theory of
Critical Distances,
in metal fatigue
subtitle
Point Method is an approximation of the Line Method
Introduction
I
Valid for any kind of notch (radius, depth, angle)
Short/Long cracks
I
Simple to use along with Finite Element
Non-propagating
cracks
Notch effect
Size effect
Summary
The Theory of
Critical Distances,
in metal fatigue
Exercise on TCD methods
subtitle
Find fatigue limit for notched components shown below.
Axial load. R = −1. Material AA 7075–T6.
Introduction
Short/Long cracks
Non-propagating
cracks
Δσ n
Δσ g
Notch effect
Size effect
0.5
1
Summary
α = 60°
10
12
1
0.1
2.5
(a)
Symm.
(b)
The Theory of
Critical Distances,
in metal fatigue
Exercise on TCD methods
subtitle
d a /dN[ m/cycle ], (log)
Introduction
Material properties
Fatigue life, R=−1
Short/Long cracks
350
σ0 =
300
150 −190
250
MPa
200
150
100
S−N curve
Boller, Seeger 1987
50
0 4
10
10
Non-propagating
cracks
−6
d a /dN[ m/cycle ], (log)
σ [MPa]
400
10
Notch effect
−8
10
Size effect
MPa m
5
6
10
Nf,
N ,(log)
(log)
f
10
7
10
0
1
Fit
Wu 1998
Bu 1986
10
Delta
[ MPa m1/2
], (log)
ΔKKI[ MPa
m ], (log)
2
3.8
= 32 × 10−6 m = 32 µ
2 × 190
2
4.2
1
= 62 × 10−6 m = 62 µ
a0 (max) =
π 2 × 150
a0 ≈ 50 µ
1
a0 (min) =
π
Summary
ΔK th =
3.8 − 4.2
−10
10
10
2
The Theory of
Critical Distances,
in metal fatigue
Exercise on TCD methods
subtitle
Blunt notch
Introduction
Short/Long cracks
Non-propagating
cracks
k t = 2.33
200
Notch effect
kf = 2.22
Size effect
Summary
1
stress [ MPa ]
150
100
50
0
0
S1 stress distribution
Point Method, a0 / 2
a0 min / 2
a0 max / 2
0.05
0.1
0.15
depth [ mm ]
0.2
0.25
The Theory of
Critical Distances,
in metal fatigue
Exercise on TCD methods
subtitle
Sharp notch
Introduction
Short/Long cracks
300
stress [ MPa ]
250
200
k t = 8.14
Non-propagating
cracks
S1 stress distribution
Point Method, a0 / 2
a0 min / 2
a0 max / 2
Notch effect
Size effect
kf (a0 min) = 6.12
kf = 5.51
kf (a0 max) = 5.01
150
100
50
0.1
0
0
0.1
0.2
0.3
depth [ mm ]
0.4
0.5
Summary
The Theory of
Critical Distances,
in metal fatigue
Exercise on TCD methods
subtitle
Sharp notch, TCD low sensitivity to actual radius
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
Summary
MX
MX
r = 0.1 mm
r = 0.01 mm
The Theory of
Critical Distances,
in metal fatigue
Exercise on TCD methods
subtitle
Sharp notch, low sensibility to local notch radius
Introduction
Short/Long cracks
800
Peak value
insensibility
Non-propagating
cracks
S1 str. distr. r = 0.1
S1 str. distr. r = 0.01
Point Method, a0 / 2
a0 min / 2
a0 max / 2
700
600
500
Notch effect
Size effect
Summary
400
300
200
100
0
0
0.05
0.01
0.1
0.1
0.15
0.2
The Theory of
Critical Distances,
in metal fatigue
Exercise on TCD methods
subtitle
PM ≡ LM, only for sharp notch
Introduction
Short/Long cracks
800
600
stress [ MPa ]
Non-propagating
cracks
S1 str. distr. r = 0.1
PM r = 0.1
LM r = 0.1
S1 str. distr. r = 0.01
PM r = 0.01
LM r = 0.01
700
500
Notch effect
Size effect
Summary
400
300
200
100
0
0
0.05
0.01
0.1
0.1
depth [ mm ]
0.15
0.2
The Theory of
Critical Distances,
in metal fatigue
Introduction
Introduction
Short/Long cracks
Short/Long cracks
Non-propagating
cracks
Notch effect
Non-propagating cracks
Size effect
Summary
Notch effect
Size effect
Summary
The Theory of
Critical Distances,
in metal fatigue
Size effect in fatigue
subtitle
Introduction
Δσ
Short/Long cracks
Δσ
Non-propagating
cracks
Notch effect
r ′′
r′
Size effect
Summary
D′′
D′
r ′′ = r ′ / 2
D′′ = D′ / 2
(1)
(2)
Continuum mechanics stress distribution is the same
The Theory of
Critical Distances,
in metal fatigue
Size effect in fatigue
subtitle
Introduction
Δσ
Short/Long cracks
Δσ
Non-propagating
cracks
Notch effect
r ′′
r′
Size effect
Summary
D′′
D′
r ′′ = r ′ / 2
D′′ = D′ / 2
(1)
(2)
Component (2), the smaller, has higher fatigue strength
Material size vs. component dimension
The Theory of
Critical Distances,
in metal fatigue
subtitle
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Large dimension /
material size ratio
(1)
Little dimension /
material size ratio
(2)
Continuum mechanics dimensions scalability violates
material scale (meso–scale) proportions
Size effect
Summary
The Theory of
Critical Distances,
in metal fatigue
Sharp notch size effect
subtitle
Introduction
Δσ
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
r
a
ξ=
r
<< 1
a
All other dimensions
much larger
ξ = r/a constant, a ∈ [0, ∞], offers the most informative
fatigue size effect example
Summary
The Theory of
Critical Distances,
in metal fatigue
Sharp notch size effect
subtitle
Introduction
Δσ (log)
Short
crack
regime
aD
Long
crack
regime
aN
Short/Long cracks
Non-propagating
cracks
Notch effect
Blunt
notch
regime
Size effect
Summary
a (log)
B. Atzori, P. Lazzarin, G. Meneghetti
Fracture mechanics and notch sensitivity
Fatigue & Fracture of Engineering Materials &
Structures, vol. 26, p. 257–267 (2003)
The Theory of
Critical Distances,
in metal fatigue
Sharp notch size effect
subtitle
Introduction
Δσ (log)
Short
crack
regime
aD
Long
crack
regime
aN
Short crack regime, fatigue limit
∆σ = ∆σ0
Short/Long cracks
Non-propagating
cracks
Notch effect
Blunt
notch
regime
Size effect
Summary
a (log)
The Theory of
Critical Distances,
in metal fatigue
Sharp notch size effect
subtitle
Introduction
Δσ (log)
Short
crack
regime
aD
Long
crack
regime
aN
Long crack regime, fatigue limit
∆σ =
∆Kth
√
β πa
Short/Long cracks
Non-propagating
cracks
Notch effect
Blunt
notch
regime
Size effect
Summary
a (log)
The Theory of
Critical Distances,
in metal fatigue
Sharp notch size effect
subtitle
Introduction
Δσ (log)
Short
crack
regime
aD
Long
crack
regime
aN
Blunt notch regime, fatigue limit
∆σ =
∆σ0
kt
Short/Long cracks
Non-propagating
cracks
Notch effect
Blunt
notch
regime
Size effect
Summary
a (log)
The Theory of
Critical Distances,
in metal fatigue
Sharp notch size effect
subtitle
Introduction
Δσ (log)
Short
crack
regime
aD
Long
crack
regime
Short/Long cracks
Non-propagating
cracks
Notch effect
Blunt
notch
regime
aN
Size effect
Summary
a (log)
Asymptotic transition Short–Long crack
aD =
a0
β2
where a0 = 1/π(∆Kth /∆σ0 )2 (shape factor β correction)
The Theory of
Critical Distances,
in metal fatigue
Sharp notch size effect
subtitle
Introduction
Δσ (log)
Short
crack
regime
aD
Long
crack
regime
aN
Short/Long cracks
Non-propagating
cracks
Notch effect
Blunt
notch
regime
Size effect
Summary
a (log)
Asymptotic transition Long crack – Blunt notch
aN = kt2 aD
The Theory of
Critical Distances,
in metal fatigue
Sharp notch size effect
subtitle
Introduction
a0 > a
r
a
Short/Long cracks
Non-propagating
cracks
Notch effect
a0
Size effect
Summary
Short crack
Crack size smaller than the material critical distance
The Theory of
Critical Distances,
in metal fatigue
Sharp notch size effect
subtitle
Introduction
a > a0 , r < a0
r
a
Short/Long cracks
Non-propagating
cracks
Notch effect
a0
Long crack
Crack size larger than the material critical distance, but
crack tip radius smaller than the critical distance
Size effect
Summary
The Theory of
Critical Distances,
in metal fatigue
Sharp notch size effect
subtitle
Introduction
Short/Long cracks
r > a0
Non-propagating
cracks
r
Notch effect
Size effect
a
a0
Blunt notch
crack tip radius larger than the material critical distance
Summary
The Theory of
Critical Distances,
in metal fatigue
Introduction
Introduction
Short/Long cracks
Short/Long cracks
Non-propagating
cracks
Notch effect
Non-propagating cracks
Size effect
Summary
Notch effect
Size effect
Summary
Summary
The Theory of
Critical Distances,
in metal fatigue
subtitle
D. Taylor
The Theory of Critical Distances: A New Perspective in
Fracture Mechanics
Elsevier (2007)
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
I
The Kitagawa–Takahashi diagram introduce the critical
distance a0
I
a0 is the size of Short/Long crack transition
I
Anomalous short crack regime is the very basis of
fatigue
I
Critical distance methods (as Point Method PM or Line
Method LM) assume a0 as length for averaging stress
distribution
I
Critical distance methods capture notch effect and size
effect
Summary
Summary
The Theory of
Critical Distances,
in metal fatigue
subtitle
D. Taylor
The Theory of Critical Distances: A New Perspective in
Fracture Mechanics
Elsevier (2007)
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
I
The Kitagawa–Takahashi diagram introduce the critical
distance a0
I
a0 is the size of Short/Long crack transition
I
Anomalous short crack regime is the very basis of
fatigue
I
Critical distance methods (as Point Method PM or Line
Method LM) assume a0 as length for averaging stress
distribution
I
Critical distance methods capture notch effect and size
effect
Summary
Summary
The Theory of
Critical Distances,
in metal fatigue
subtitle
D. Taylor
The Theory of Critical Distances: A New Perspective in
Fracture Mechanics
Elsevier (2007)
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
I
The Kitagawa–Takahashi diagram introduce the critical
distance a0
I
a0 is the size of Short/Long crack transition
I
Anomalous short crack regime is the very basis of
fatigue
I
Critical distance methods (as Point Method PM or Line
Method LM) assume a0 as length for averaging stress
distribution
I
Critical distance methods capture notch effect and size
effect
Summary
Summary
The Theory of
Critical Distances,
in metal fatigue
subtitle
D. Taylor
The Theory of Critical Distances: A New Perspective in
Fracture Mechanics
Elsevier (2007)
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
I
The Kitagawa–Takahashi diagram introduce the critical
distance a0
I
a0 is the size of Short/Long crack transition
I
Anomalous short crack regime is the very basis of
fatigue
I
Critical distance methods (as Point Method PM or Line
Method LM) assume a0 as length for averaging stress
distribution
I
Critical distance methods capture notch effect and size
effect
Summary
Summary
The Theory of
Critical Distances,
in metal fatigue
subtitle
D. Taylor
The Theory of Critical Distances: A New Perspective in
Fracture Mechanics
Elsevier (2007)
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
I
The Kitagawa–Takahashi diagram introduce the critical
distance a0
I
a0 is the size of Short/Long crack transition
I
Anomalous short crack regime is the very basis of
fatigue
I
Critical distance methods (as Point Method PM or Line
Method LM) assume a0 as length for averaging stress
distribution
I
Critical distance methods capture notch effect and size
effect
Summary
Summary
The Theory of
Critical Distances,
in metal fatigue
subtitle
D. Taylor
The Theory of Critical Distances: A New Perspective in
Fracture Mechanics
Elsevier (2007)
Introduction
Short/Long cracks
Non-propagating
cracks
Notch effect
Size effect
I
The Kitagawa–Takahashi diagram introduce the critical
distance a0
I
a0 is the size of Short/Long crack transition
I
Anomalous short crack regime is the very basis of
fatigue
I
Critical distance methods (as Point Method PM or Line
Method LM) assume a0 as length for averaging stress
distribution
I
Critical distance methods capture notch effect and size
effect
Summary