2 notch sensitivity - puc-rio · chapter 2 – notch sensitivity . sif expressions. as in such...

2 Notch Sensitivity

Cracks may be considered short while their fatigue crack growth (FCG)

thresholds under fixed {∆K, Kmax} loading conditions are smaller than the long

crack threshold ∆Kth(R) = ∆KR, where R = Kmin/Kmax and ∆K, Kmax, and Kmin

are the range, the maximum, and the minimum of the applied stress intensity

factors (SIF). This behavior is natural, since otherwise the stress range ∆σ

required to propagate short cracks would be higher than the fatigue limit of the

material ∆SL(R) = ∆SR. Indeed, assuming as usual that the FCG process is driven

by the SIF range ∆K ∝ ∆σ√(πa), if very short cracks with size a → 0 had the

same ∆KR threshold the long cracks have, they would need ∆σ → ∞ to grow by

fatigue, a meaningless requirement7-9. Microstructurally short cracks, small

compared to the grain size gr, are much affected by microstructural barriers like

grain boundaries, thus cannot be well modeled using macroscopic stress analysis

and isotropic properties 10-15. Mechanically short cracks, on the other hand, with

sizes a > gr, may be modeled by Linear Elastic Fracture Mechanics (LEFM)

concepts if the stress field that surrounds them is predominantly LE, and if the

material can be treated as isotropic and homogeneous in such a scale. To check if

they may, the idea is to follow Irwin’s steps by first assuming such concepts are

valid, and then verifying if their predictions are validated by proper tests.

Therefore, in the sequence LEFM techniques are used to develop a model for the

FCG behavior of mechanically short cracks, in particular those that depart from

notches, and then its predictions are verified experimentally.

To reconcile the traditional fatigue (crack initiation) limit, ∆S0=2SL(R=0),

with the FCG threshold of long cracks under pulsating loads, ∆K0 = ∆Kth(R = 0),

Topper and his colleagues added to the physical crack size a hypothetical short

crack characteristic size a0, a wise stratagem that forces the SIF of all cracks, short

or long, to obey the correct FCG limits16-18:

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PUC-Rio - Certificação Digital Nº 1112084/CA

33 Chapter 2 – Notch Sensitivity

∆�� = ∆� ∙ �� ∙ (� + ��) , �ℎ�� = �� ∙ (

∆��∆��)� (1)

In this way, long cracks with a >> a0 (in Griffith’s plates under pulsating

loads) do not grow by fatigue if ∆KI = ∆σ√(πa) < ∆K0, while very small cracks

with a → 0 do not grow if ∆σ < ∆S0, since ∆KI = ∆σ√(πa0) < ∆S0√(πa0) = ∆K0 in

this case. Moreover, this clever idea reproduces the whole tendency of typical

∆σi×ai data points in Kitagawa-Takahashi diagrams19, where ∆σi is the stress

range needed to propagate a fatigue crack with size ai, see figure 3. This figure

also shows the fatigue limit ∆S0 and the stress range associated to the long cracks

threshold ∆σ(a) = ∆K0/√(πa), which limit the region that may contain non-

propagating cracks, as well as the El Hadad-Topper-Smith (ETS) curve, which

predicts that cracks of any size should stop when:

∆�� ≤ (∆��) �� ∙ (� + ��)⁄ (2)

Figure 3 - Stress ranges ∆σ(a) required to propagate cracks of size a under R = 0 in a

HT80 steel plate with ∆K0 = 11.2MPa√m and ∆S0 = 575MPa: long cracks, with a >> a0,

stop when ∆σ ≤ ∆K0/√πa, while very short cracks, with a → 0, stop when ∆σ ≤ ∆S0. The

ETS model predicts that any crack should stop when ∆σ ≤ ∆K0/√[π(a + a0)].

Steels typically have 6 < ∆K0 < 12MPa√m, ultimate tensile strengths

400 < SU < 2000 MPa, and fatigue limits 200 < SL < 1000MPa (the best high-

strength steels with very clean microstructures tend to maintain the trend

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SL ≅ SU/2). Consequently, the range of their fatigue limits under pulsating loads

estimated by Goodman is

∆�� ≅ �2 ∙ �� ∙ �� + ��⁄

260�� < ∆�� < 1300 �� (3)

Hence, the range of characteristic short crack sizes in steel components (in

large plates with a central crack subject to pulsating tensile loads) estimated

according to the ETS model is:

�� ∙ �∆��

∆��

≅ 7�� < �� < 700�� ≅ �� ∙ �∆��

∆�� (4)

Since such a0 values are small, the denomination “short crack characteristic

size” is justifiable. Indeed, they hardly reach the detection thresholds of traditional

non-destructive inspection methods20. For typical Al alloys (with

30 < SL < 230MPa, 70 < SU < 600MPa, 40 < ∆S0 < 330MPa, and

1.2 < ∆K0 < 5MPa√m) the range estimated for a0 is a little larger,

1µm < a0 < 5mm. So, it can be expected that short crack effects on materials with

high ∆K0 and low ∆S0 to be more pronounced in Al alloys than in steels.

As the generic SIF of cracked structural components is given by

KI = σ√(πa)⋅g(a/w), Yu, Duquesnay, and Topper used the geometry factor g(a/w)

to generalize equation (1), and redefined the short crack characteristic size by 18:

∆�� = ∆� ∙ �� ∙ (� + ��) ∙ �(�) ;

where �� = (��) ∙ (∆�� [∆�� ∙ �(

�� )])�

(5)

The largest stress range ∆σ that does not propagate micro-cracks in this case

is also the fatigue limit, as it should: if a << a0, ∆KI = ∆K0 ⇒ ∆σ → ∆S0.

However, when the crack starts from a notch, as usual, its driving force is the

stress range ∆σ at the notch tip, not the nominal stress range ∆σn normally used in

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SIF expressions. As in such cases the g(a/w) factor includes the stress

concentration effect of the notch, it is better to split it into two parts:

g(a/w)=η⋅ϕ(a), where ϕ(a) quantifies the effect of the stress gradient near the

notch root, which for micro-cracks tends towards Kt, ϕ(a → 0) → Kt, while the

constant η quantifies the effect of the other parameters that affect KI, like e.g. the

free surface. In this way, it is better to define a0 by:

∆�� = � ∙ �� ∙ ∆�� ∙ �� ∙ �� + �� ;

where �� = (��) ∙ (∆�� ∙ ∆��⁄ )�

(6)

The stress gradient effect quantified by ϕ(a) does not affect a0 since the

stress ranges at notch tips must be smaller than the fatigue limit to avoid cracking,

∆σ(a → 0) = Kt ∆σn = ϕ(0)∆σn < ∆S0. However, since SIF are crack driving

forces, they should be material-independent. Hence, the a0 effect on the short

crack behavior should be used to modify the FCG thresholds instead of the SIF,

making them a function of the crack size, a trick that is quite convenient for

operational reasons. In this way, the a0-dependent FCG threshold for pulsating

loads ∆Kth(a, R = 0) = ∆K0(a) becomes

∆��(�) ∆��⁄ = (∆�√�� ∙ �(� �))⁄ (∆��(� + ��) ∙ �(� �)) =⁄�

= �� + ��⁄ → ∆�� = ∆�� ∙ �1 + (�� )⁄ � � �⁄ (7)

It may also be convenient to assume that equation (7) is just one of the

models that obey the long crack and the microcrack limit behaviors, introducing in

the ∆K0(a) definition a data fitting parameter γ proposed by Bazant21 to obtain:

∆�� = ∆�� ∙ �1 + (�� )⁄ � �⁄ � � �⁄

(8)

Equation (8) reproduces the original ETS model when γ = 2, as well as the

bilinear limits showed in figure 3, ∆σ = ∆S0 and ∆σ = ∆K0/√(πa), when γ → ∞.

This additional parameter may allow a better fitting of experimental data, such as

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those collected by Tanaka et al.22 and by Livieri and Tovo23, see figure 4: most

data on short cracks are contained by the curves generated using γ = 1.5 and γ = 8.

The curves shown in figure 4 illustrate the influence of γ on the minima stress

ranges needed to propagate short or long cracks under pulsating loads, as a

function of the crack size a:

∆�� = (∆�� √� ∙ �)⁄ ∙ �1 + (�� )⁄ � �⁄ � � �⁄ (9)

Figure 4 - The additional parameter γ in ΔK0(a)/ΔK0 = [1 + (a0/a) γ/2]-1/γ may allow a better

fitting of the short crack FCG thresholds measured experimentally.

However, since fatigue damage depends on two driving forces, equation (8)

should be extended to consider the σmax (indirectly modeled by the R-ratio)

influence on the short crack behavior. Thus, if ∆KR = ∆Kth(a >> aR, R) is the

FCG threshold for long cracks and ∆SR = ∆SL(R) is the fatigue limit at the desired

R-ratio, then:

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∆�� = ∆�� ∙ �1 + (�� )⁄ � �⁄ � � �⁄ ;

where �� = (��) ∙ (∆�� ∙ ∆��⁄ )�

(10)

Figure 5 - Influence of γ in the fatigue limit curves ∆σ0(a) predicted by equation (9): the

larger the γ value is, the faster ∆σ0(a) tends to the bilinear limit defined by

∆σ0 = ∆K0/√(πa), the FCG threshold under pulsating loads for long cracks with size

a >> a0, and to ∆σ0 = ∆S0, the fatigue limit under pulsating stresses for very small cracks

with a << a0.

Albeit defect-free micro filaments (whiskers) can be made in lab conditions,

structural components always contain tiny defects like inclusions, voids,

scratches, etc., which behave as small cracks. The structural effects of such

(mechanically) short cracks can be evaluated using LEFM concepts, as detailed in

the following sections23-32.

2.1. Short cracks influence on the fatigue limit of structural components

Traditional SN and εN methods are used to analyze and design supposedly

crack-free components. But as it is impossible to guarantee that they are really

free of cracks smaller than the detection threshold of the non-destructive method

used to inspect them, SN or εN predictions may become unreliable when such tiny

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defects are introduced by any means during their manufacture or service. Hence,

structural components should be designed to tolerate undetectable short cracks.

Despite self-evident, this prudent requirement is still not included in most

fatigue design routines, which just intend to maintain the service stresses at

critical points below their fatigue limits, ∆σ < ∆SR/ϕF, where ϕF is a suitable

safety factor. Nevertheless, most long-life designs work just fine. This means that

they are somehow tolerant to undetectable or to functionally admissible short

cracks, but the question “how much tolerant” cannot be answered by SN or εN

procedures alone. Such problem can be avoided by adding a tolerance to short

crack requirement to their “infinite” life design criteria which, in its simplest

version, may be given by:

∆� < ��∆�� ∙ �1 + (�� )⁄ � �⁄ � � �⁄ � (��) ∙ √� ∙ � ∙ �(� �)⁄ !�

where �� = (��) ∙ (∆�� ∙ ∆��⁄ )�

(11)

Since the fatigue limit ∆SR = ∆SL(R) already reflects the effect of

micro-structural defects that are inherent to the material, equation (11)

complements it by describing the tolerance to small cracks that may pass

unnoticed in actual service conditions. The practical usefulness of this sensible

criterion is illustrated by an interesting case study32. Due to a rare manufacturing

problem, a component batch left the factory with unexpected tiny surface cracks

(only detectable by a microscope) which caused some non-negligible failures. The

effect of such small cracks on that component was evaluated knowing that its

rectangular cross-section is 2 by 3.4mm and that its steel had SU = 990MPa and a

fatigue limit measured under R = −1 = SL = 246MPa, assuming that Goodman

could be used to estimate SL(R) = SR = SLSU(1 – R)/[SU(1 – R) + SL(1 + R)], for

R > −1 . The FCG threshold ∆KR is also needed to model short crack effects, but

as it was not available, it was estimated by ∆KR(R ≤ 0.17) = ∆K0 = 6MPa√m and

∆KR(R > 0.17) = 7⋅(1 - 0.85R)33. This risky practice increases the predictions

uncertainty, but it was the only option available. Using the SIF of an edge cracked

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strip of width W loaded in mode I 34, the tolerable stress ranges under pulsating

loads can then estimated within a fatigue safety factor ϕF by:

∆�� ≤ ∆��∙��(

�� )� �⁄ � �⁄

�� (�∙��∙�)∙��∙��∙�∙�� (

�∙��∙�)∙��.��.��∙

��.��∙(� � ��∙��∙�) �∙(!�∙√�∙)

(12)

The R effect can be evaluated using the corresponding fatigue limit ∆SR,

threshold ∆KR, and characteristic short crack size aR. The maxima tolerable stress

ranges ∆σR(a) estimated in this way are shown in figure 6, using a logarithmic

scale to highlight the short crack effects. Note that the characteristic size aR varies

slightly from a0 = 59µm to a0.8 = 55µm in this figure.

Figure 6 - Larger stress ranges tolerable under several R–ratios in function of the edge

crack size a, for W = 3.4mm, η = 1.12, ∆K0 = 6MPa√m, a0 = 59µm, γ = 6, and φF = 1.6.

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This simple (but sensible) model indicates that the studied components

tolerate well small edge cracks up to a ≅ 30µm, which almost do not affect their

original fatigue limits. Such estimates can be very useful for designers and quality

control engineers. They can be used as a quantitative tool to evaluate the effect of

a production or operational accident that damages the surface of otherwise well-

behaved components, but they have some limitations. They assume that the short

crack grows unidimensionally (1D), thus can be characterized by its size a only.

However, as most short cracks are small regarding the structural component

dimensions, they are better described as 2D cracks that grow by fatigue in two

directions, maintaining their original plane under mode I loads but usually

changing their shape at every load cycle. Moreover, such estimates are valid for

mechanically short cracks only, with both a and a0 larger than the grain size gr.

The FCG behavior of micro-cracks with size a < gr is sensible to micro-structural

features and cannot be properly modeled using macroscopic material properties.

Such problems have academic interest10-15, but as the grains still cannot be

mapped in practice, they cannot be properly used for structural engineering

applications yet.

To model short 2D (mechanical) cracks that tend to grow both in depth and

width in the simplest possible way, it is assumed that: (i) the cracks are loaded in

pure mode I under quasi-constant ∆σ and R conditions, with no overloads or any

other event capable of inducing load sequence effects; (ii) material properties

measured testing 1D specimens may be used to simulate FCG or EAC behavior of

2D cracks; and (iii) 2D surface or corner cracks can be well modeled as having an

approximately elliptical front, thus their SIF can be described by the classical

Newman-Raju equations 32, 35-36. If such reasonable hypotheses hold as expected,

then the structural components tolerance to short or long fatigue cracks is given

by:

∆� < "∆�� ∙ #1 + ��# a⁄ �$ �⁄ $� $⁄ % √� ∙ � ∙ Φ(�, c, w, t)!�"∆�� ∙ #1 + ��# c⁄ �$ �⁄ $� $⁄ % √� ∙ & ∙ Φ%(�, c, w, t)!� (13)

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The SIF at the semi-axes a and c tips of semi-elliptical surface cracks in a

plate of thickness t are KI(a) = σ√(πa)⋅Φa = σ√(πa)⋅F⋅M/Q0.5 and

KI(c) = σ√(πa)⋅Φc = σ√(πa)⋅( F⋅M/Q0.5)⋅(a/c)⋅G , see equations (14). SIF for

quarter-elliptical corner cracks are even more complex35. Such complicated 2D

SIF functions enhance the operational advantage of treating the FCG threshold as

a function of the crack size, ∆Kth(a). Tolerance to EAC cracks can be treated

using these same principles, by properly changing the fatigue properties ∆KR and

∆SR by the corresponding material resistances to EAC in the desired environment,

KIEAC and SEAC, as explained later on.

�� = � ∙ √� ∙ � ∙ � ∙�

��

�� = � ∙ √� ∙ ∙ � ∙��

� � ,�� = �sec �� ∙

2 ∙ �� ∙ �� ∙ [1 − 0.025 ∙ �(�) ∙ ��

∙ 0.06 �(�) ∙ ��

=

1.13 − 0.09 ∙��

+ � .�.��(�

�)

− 0.54� ∙��

�+ ��

�−

�.��(�

�)

+ 14 ∙ �1 − �� ∙

��

�, � ≤

��

+ 0.04 ∙ �� + �

��.� ∙ �

�� ∙ �0.2 − 0.11 �

�� ,� >

� = 1 + 1.464 ∙ �

��.�� , � ≤

1 + 1.464 ∙ ��.�� , � >

� = 1.1 + 0.35 ∙ �

�� , � ≤

1.1 + 0.35 ∙ � �� ∙ �

�� , � >

(14)

2.2. Engineering estimates for notch effects on the FCG behavior of short cracks

The SIF of small mechanical cracks that start at the tips of notches with

depth b and tip radius ρ can be estimated by KI ≅ 1.12σn⋅f1(Kt, a)√(πa), where

f1 = σy(x)/σn is the stress concentration perpendicular to the crack plane at the

point (x = b + a, y = 0) ahead of such tips, induced by the ellipse with semi-axis b

and c and tip radius ρ = c2/b which is tangent to the notch tip. If the ellipse 2b axis

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is centered at the x-axis origin and is perpendicular to the nominal stress σn,

then37:

'� = &�'()*�,+)�,

&� = 1 +-*� �∙*∙%.∙-( √(� *��%�.∙-(� *��%�.�*∙%�∙(* %)∙(

(* %)�∙((� *��%�)∙√(� *��%� (15)

The high stress gradient ahead of elongated notch tips justifies the peculiar

behavior of short cracks that start from sharp notches: in the LE case, the

tangential stress ahead any elliptical hole with b ≥ c is σy(x/b = 1.2, 0)/σn ≅ 2,

independently of the elliptical notch Kt, see figure 7.

As the stress gradient around sharp notch tips is high, the SIF induced by

remotely applied loads on short cracks that start there first grows fast with their

growing sizes a, but after a small ∆a increment they may stabilize or even

decrease for a while before growing once again, since the notch effect on KI

diminishes sharply as the short crack grows. Indeed, the term √a that increases

KI = 1.12σ√(πa)⋅f1 can be overcompensated by the abrupt fall in f1 near the notch

tip, see figure 8. Such simple concepts can be used to evaluate the tolerance to

fatigue cracks that start from such notches using the KI(a) and ∆Kth(a) estimates.

For example, they can be used to evaluate if a large steel plate with SU = 600MPa,

SL = 200MPa and ∆K0 = 9MPa√m, can tolerate a circular central hole with

diameter d = 20mm or an elliptical hole with axes 2b = 20mm (perpendicular to

∆σn) and 2c = 2mm when it works under ∆σn = 100MPa at R = −132.

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Figure 7 - The ratio K1.2 = σy(x/b = 1.2, 0)/σn at the point that is just b/5 ahead the tip of

any elliptical hole is almost independent of its Kt (in the LE case).

Figure 8 - KI ≅ 1.12·σn√(πa)·f1(Kt, a) estimate for small cracks a ≤ b/5 that start from the

tip of an Inglis hole with b = 10mm.

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Assuming the buckling problem can be neglected in this case, the circular

hole has a safety factor φF = SL/Kfσn = 200/150 ≅ 1.33 against fatigue crack

initiation, since due to its large radius it has Kf ≅ Kt = 3. The elliptical hole, on the

other hand, would be not admissible according to traditional SN procedures, since

it has a too small tip radius ρ = c2/b = 0.1mm, thus Kt = 1 + 2b/c = 21 and notch

sensitivity q = (1 + α/ρ)−1 = [1 + 0.185⋅(700/600)/0.1]−1 ≅ 0.32 ⇒

Kf = 1 + q⋅(Kt − 1) = 7.33 according to Peterson 38. Then it would work under

σ = Kf ⋅σn ≅ 367MPa > SL. However, since this Kf is much larger than typical

experimental data 32-35, it should be reevaluated.

For cracks that start at the circular hole border, Kirsch can be used instead

of Inglis to estimate ∆KI ≅ 1.12⋅∆σn√(πa)⋅[1 + 0.5(d/2x) 2 + 1.5(d/2x)4], where a

is the crack size and x is the distance from the hole center. But

∆KI ≅ 1.12⋅∆σn√(πa)⋅f1, with f1 given by equation (15), must of course be used for

the elliptical hole (these two estimates are identical when b = c, as they should.)

The estimate ∆Kth(R < 0) ≅ ∆K0 can be used if it can be assumed (as usual) that

the crack does not propagate by fatigue when closed. Hence, it is assumed that

cracks induced at the notch tip by ∆σ only propagate by fatigue if

∆KI(a) ≥ ∆K0(a), with the range ∆K calculated considering only the tensile part

of ∆σ. Supposing that ∆Kth(a) = ∆K0/[1 + (a0/a)]−0.5 (by ETS),

L US 0.5S′ = , ∆S0=SU/1.5 (by Goodman), and a0 = (1/π)(∆K0/η∆S0)2 =

=(1/π)(1.5∆K0/1.12⋅SU)2 ≅ 0.13mm, the SIF ranges ∆KI(a) estimated for the two

holes are compared to the short crack threshold ∆Kth(a) in figure 9. So, estimating

the short cracks behavior, knowing that terminal failure includes the generation

and the propagation of a crack up to the fracture of the piece, this model predicts

that both the circular and the elliptical holes could support the load ∆σn without

failing by fatigue. However, the elliptical hole induces a tolerable crack, thus it is

much less robust than the circular one32.

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Figure 9 - A crack does not start at the circular hole (that tolerates cracks a < 1.5mm),

while the crack that begins at the elliptical hole stops at ast ≅ 0.33mm.

2.3. Analysis of notch sensitivity effects in fatigue

The notch sensitivity q still is quantified for structural design purposes by

empirical curves fitted to 7 experimental points compiled by Peterson a long time

ago38. It is used to estimate fatigue limits measured under fixed ∆σn and R in

notched TS with a SCF Kt ≥ Kf = 1 + q⋅(Kt − 1) by L L fS (R) S (R) K′∆ = ∆ , where

LS (R)′∆ is the unnotched fatigue limit under the same loads.

According to Frost, early data showing that small non-propagating fatigue

cracks are found at notch tips when ∆SL/Kt < ∆σn < ∆SL/Kf goes back as far as

194939. It is thus reasonable to expect that q is related to the fatigue behavior of

short cracks emanating from notch tips, or that such tiny cracks can be used to

quantitatively explain why Kf ≤ Kt. The notch sensitivity can in fact be calculated

in this way using relatively simple but sound mechanical principles that do not

require heuristic arguments, neither any arbitrary fitting parameter. To start with,

according to Tada 34, the SIF of a crack with size a that departs from a circular

hole of radius ρ on a semi-infinite plate under tensile stress is given within 1% by

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�� = 1.1215 ∙ � ∙ √� ∙ � ∙ ��, � ≡ � ⁄�� = �1 +

�.�

(��)+

�.�

(��)� ∙ �2 − 2.354 ∙ � �

�� + 1.206 ∙ � �

�� − 0.221 ∙ � �

�� (16)

Note that when a → 0 ⇒ x → 0, this equation tends to the expected limit,

lim→� ∆�� = 1.1215 ∙ 3 ∙ ∆� ∙ √� ∙ � (17)

Indeed, this equation combines the solution for an edge crack in a semi-

infinite plate with the stresses concentration factor of the Kirsch hole, Kt = 3.

Note also that when a → ∞, it reproduces the correct limit once again, the SIF for

an Irwin’s plate with a crack of length a (in fact, a + 2ρ ≅ a, as a → ∞), because

the crack tip is so distant from the hole that it is not affected by it:

lim→/ ∆�� = ∆� ∙ �(� ∙ � 2)⁄ (18)

Hence, for Kirsch holes such limits are ϕ(x = 0) = 3 and

ϕ(x → ∞) = 1/1.1215√2 ≅ 0.63. The FCG condition for the cracks that start at

such notch borders under pulsating loads is thus

∆�� = ∆� ∙ √� ∙ � ∙ � ∙ �(� �) > ∆��(�) = ∆�� ∙ 1 + �� ⁄ �� ⁄ �/�

⁄ (19)

Where ∆K0 = ∆S0√(πa0) ≡ ∆K0(a >> a0), and a0 = (1/π)[∆K0/(η⋅∆S0)]2. Note that

a0 cannot depend on the stress gradient factor ϕ(a/ρ), otherwise it would not be a

material property. This FCG criterion can be rewritten using two dimensionless

functions, one related to the notch stress gradient ϕ(a/ρ), and the other g(∆S0/∆σ,

a/ρ, ∆K0/∆S0√ρ, γ) which includes the effects of the applied stress range ∆σ, the

crack size a, the notch tip radius ρ, the fatigue resistances ∆S0 and ∆K0, and the

optional data fitting exponent γ (if it is used) 31:

�(� )� * >0∆��∆� 1∙2 ∆��

∆��∙��3

425∙��∙�� 3��2 ∆��∆��∙��

3�6 �⁄ ≡ � �∆��

∆& ,7 ,

∆��∆��∙87 , +� (20)

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Therefore, if x ≡ a/ρ and κ ≡ ∆K0/∆S0√ρ = η⋅√(πa0/ρ), a fatigue crack

departing from a Kirsch hole under pulsating loads grows whenever

ϕ(x)/g(∆S0/∆σ, x, κ, γ) > 1. Figure 10 plots some ϕ/g functions for several fatigue

strength to loading stress range ratios ∆S0/∆σ as a function of the normalized

crack length x for a small notch radius ρ ≅ 1.40⋅a0, comparable to the short crack

characteristic size, and for κ = 1.5 and γ = 6 32.

For high applied stress ranges ∆σ, the strength to load ratio ∆S0/∆σ is small,

and the corresponding ϕ/g curve is always higher than 1, so cracks will initiate

and propagate from this small Kirsch hole border without stopping during this

process. One example of such a case is the upper curve in figure 10, which shows

the function ϕ/g1.4 obtained for ∆S0/∆σ = 1.4. On the other hand, small stress

ranges with load ratios ∆S0/∆σ ≥ Kt = 3 have ϕ/g < 1, meaning that such loads

cannot initiate a fatigue crack from this hole, and that small enough cracks

introduced there by any other means will not propagate at such low loads. This is

illustrated by curves ϕ/g3, associated with the limit case where the local stress

range equals the material fatigue strength ∆S0/∆σ = 3, and ϕ/g4, associated with a

still smaller load, ∆S0/∆σ = 4.

However, three other curves must be analyzed in figure 10. The ϕ/g2.3 curve

crosses the ϕ/g = 1 line once, meaning that such an intermediate load can initiate

and propagate a fatigue crack from this hole border, until the decreasing ϕ/g2.3

ratio reaches 1, where the crack stops because the stress gradient ahead of its

small tip is sharp enough to eventually force ∆KI(a) < ∆Kth(a). Thus under this

∆σ = ∆S0/2.3 loading a non-propagating fatigue crack is generated at this small

hole border, with a size given by the corresponding a/ρ abscissa. The ϕ/g1.85 curve

intersects the ϕ/g = 1 line twice. This load level also generates a fatigue crack at

the hole border, which will propagate until reaching the maximum size obtained

from the abscissa of the first intersection point (on the left), where the crack stops

because it reaches ∆KI(a) < ∆Kth(a). Moreover, cracks longer than the second

intersection point will re-start propagating by fatigue under ∆σ = ∆S0/1.85, until

eventually fracturing this Kirsch plate. But the crack initiated by fatigue under

such an intermediate pulsating load range cannot propagate between these two

intersection points by fatigue alone, assuming the load {∆σ, σmax} remains

DBD



constant. Hence, it can only grow in this region if helped by a different damage

mechanism, such as EAC or creep. The FCG behavior of these two curves seems

different in figure 10, yet they are similar. Indeed, the ϕ/g2.3 curve crosses the

ϕ/g = 1 line twice if the graph is extended to include larger cracks, see figure 11.

This is so because a sufficiently long crack can always propagate by fatigue under

any given (even if small) ∆σ range if its SIF range ∆K = ∆σ√(πa)⋅g(a/w) grows

with the crack size a, as in this Kirsch plate. In fact, all ϕ/g curves become higher

than 1 for sufficiently large a/ρ values, even those that cannot initiate a crack by

fatigue, such as ϕ/g4.

Figure 10 - Cracks that can start from the border of a (small) Kirsch hole may propagate

by fatigue and then stop if their ϕ/g < 1 (ρ ≅ 1.40⋅a0, κ = 1.5, and γ = 6 in this figure).

DBD



Figure 11 - After leaving the region affected by the Kirsch hole, the cracks SIF steadily

grow as their size a increases (as usual for far field loaded cracks), thus even small

stress ranges ∆σ can propagate them by fatigue when they are sufficiently long.

Finally, the ϕ/g1.64 curve is tangent to the ϕ/g = 1 line in figure 10-11. This

means that this pulsating stress range ∆σ = ∆S0/1.64 is the smallest one that can

cause crack initiation and growth (without arrest) from the notch border by fatigue

alone. Hence, by definition, the fatigue SCF of this small Kirsch hole

(with ρ ≅ 1.40⋅a0, κ = η⋅√(πa0/ρ) = 1.5 and γ = 6) is thus Kf = 1.64 ⇒ q = 0.32.

Moreover, the abscissa of the tangency point between the ϕ/g1.64 curve and the

ϕ/g = 1 line gives the largest non-propagating crack size that can arise from it by

fatigue alone, amax. For any other ρ/a0, γ, and κ = η⋅√(πa0/ρ) combination, Kf and

amax can always be found by solving the system

� �� = 1

90! :; 19( = 0

⇒ �(,<() = �(,<( , �= , -, +)

9!'(��,9( =

9:-(��,��,>,�.9(

(21)

Kirsch holes induce relatively mild stress gradients. Larger holes compared

with the short crack characteristic size, with radii ρ >> a0, are associated to small

κ = η⋅√(πa0/ρ) values and do not induce short crack arrest. Figure 12 plots the

short crack fatigue behavior for κ = 0.3, which corresponds to a radius

DBD



ρ’ = 5⋅ρ = 7⋅a0 for γ = 6, to clarify this important point: practically all fixed stress

ranges ∆σ > ∆S0/Kt that can initiate a fatigue crack at this hole border under

pulsating loads can propagate them without arrest. Therefore, the notch sensitive

for this not so large hole is q ≅ 1. In other words, Kirsch holes with ρ > 7a0 in a

material with γ = 6 do not induce non-propagating fatigue cracks under fixed

pulsating loads, thus have q = 1. That is a nice way to mechanically interpret the

notch sensitivity concept

If for a given γ the system {ϕ/g = 1, ∂(ϕ/g)/∂x = 0} is solved for several

notch tip radii ρ using κ ≡ ∆K0/∆S0√ρ, then the notch sensitivity factor q is

obtained by:

.(-, +) ≡�=�-, +� − 1

�? − 1

(22)

Figure 12 - Fatigue behavior under fixed pulsating loads of short crack that depart from a

larger Kirsch hole border with κ = η⋅√(πa0/ρ) = 0.3 ⇒ ρ ≅ 7⋅a0, for γ = 6: almost all cracks

that start under ∆σ do not arrest their propagation, thus this hole has q ≅ 1.

DBD



This approach has four major assets: (i) it is an analytical procedure; (ii) it

considers the effect of the fatigue resistances to crack initiation and propagation

on q; (iii) it can use the exponent γ to flexible the original ETS model, but does

not need it neither any other data fitting parameter; and (iv) it can be easily

extended to other notch geometries. For example, the SIF of cracks that depart

from a semi-elliptical notch with semi-axes b and c, with b in the same direction

of the crack a, which is perpendicular to the (nominal) stress ∆σ, can be described

by:

∆�� ∙ �� ⁄ , ⁄ � ∙ ∆� ∙ √� ∙ � (23)

Where η = 1.1215 is the free surface correction factor and F(a/b, c/b) is the

geometrical factor associated to the notch stress concentration effect. Such

notches SCF Kt are given by34:

�� 1 � �∙�� ∙ �1 � �.�

��

� �.�� (24)

Using s = a/(a + b) two analytical expressions for F(a/b, c/b) were obtained

in31 by fitting results obtained by a series of finite elements (FE) analyses for

several types of semi-elliptical notches, made using the Quebra2D software,

which reproduce very well the results of Nishitani and Tada quoted by Bazant 21,

see figure 13:

� �� , � ≡ ��, �� ∙ �1 � ��∙��

�� ∙ � , �

� �� , � ≡ �´��, �� ∙ �1 � ��∙��

�� ∙ � ∙ 1 � ��

�∙��!�� ⁄ , " (25)

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Figure 13 - SIF for cracks that depart from semi-elliptical notches with c ≤ b.

Making g = ϕ, the minimum stress range needed to start and propagate a

fatigue crack from the edge of such notches can be calculated for several

combinations of κ and γ, leading to expressions for Kf and, consequently, for q,

see figure 14. Note that the notch sensitivity q(1/κ) estimated in this way is almost

linear for sensibilities q > 0, hence it can be approximately modeled by:

.�-, +� ≅.��+�- − .��+� =

.��+� ∙ ∆�� ∙ �)∆�� − .�(+)

(26)

Figure 14 - Notch sensitivity q(1/κ) estimated for a (circular) Kirsch hole.

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The parameters q0(γ) and q1(γ) that fit the quasi linear part of q(γ, κ) depend

only on γ, while the parameter 1/κ = ∆S0√ρ/∆K0 includes the material fatigue

limit and FCG threshold, as well as the notch tip radius ρ. Note that equation (26)

predicts q > 1 for high 1/κ values, if the notch has a large tip radius ρ compared to

the a0 value, larger than a radius ρsup given by:

∆�� ∙ �)�@A∆�� >

1 + .�(+)

.�(+) → )�@A > �1 + .�(+)

.�(+)∙

∆��∆��

�

(27)

It may seem strange to predict a notch sensitivity q > 1 (since in such cases

q = 1 must be used for fatigue design, since by definition Kf ≤ Kt), but these

values have a good physical interpretation: sensibilities q > 1 mean that the cracks

started by fatigue from the notch border do not stop under that fixed stress range,

thus never become non-propagating. This occurs when the stress gradient near the

notch tip is too gentle to affect the short crack behavior. In fact, in the absence of

compressive residual stresses, the only mechanical reason for cracks induced by

fatigue from a notch border to stop after growing for a while (under the same

constant load {∆σ, R = 0} that initiated them) inside an isotropic material is the

stress gradient near the notch tip. To stop a crack, the stress range decrease ahead

of the notch tip induced by the stress gradient around it must be able to surpass the

SIF increase induced by the crack size increment, in such a way that

∆K = η⋅ϕ(a)⋅∆σ√(πa) can fall as a grows until becoming smaller than the

propagation threshold ∆K0(a). The crack stop size ast (under pulsating loads) is

thus reached when:

∆�� = � ∙ ��?� ∙ ∆� ∙ �� ∙ ��? = ∆�� = ∆�� ∙ /1 + ��

� �⁄ 0 � �⁄ (28)

It is also possible to predict q < 0, i.e. negative notch sensibilities (up to

q ≅ −0.2 in the Kirsch hole case, see figure 14), which seems even more strange

than the q > 1 values. This occurs when the notch is too sharp, with a tip radius

ρinf given by:

DBD



∆�� ∙ �) �=∆�� <

.�(+)

.�(+) → ) �= < /.�(+)

.�(+)∙

∆��∆��0

�

(29)

However, values q < 0 also have a clear physical interpretation: in such

cases it is easier to start a fatigue crack from a notch free surface than from the

notch border. This occurs because the stress gradient ∂g(a)/∂a near the notch tip is

so large that the SIF of the crack quickly reaches the long crack condition limit,

which does not include any more the free surface η = 1.12, which affects the SIF

of the cracks that start from unnotched surfaces. In most materials, the value of

ρinf is on the order of a few micrometers41, meaning that small internal defects

with radii ρ < ρinf are not harmful, hence that the cracks will start, as usual, at the

free surface of the piece.

Traditional semi-empirical notch sensitivity estimates, like Peterson’s

q = (1 + α/ρ)−1, based on a length parameter α obtained by fitting only 7

experimental points, suppose that the notch sensitivity depends only on the notch

tip radius ρ and on the steel tensile strength, SU, and only on ρ for Al alloys. The

model proposed here, on the other hand, recognizes that q depends on ρ, ∆S0,

∆K0, and γ. There are reasonable relations between ∆S0 and SU, but none between

∆K0 and SU. This means that two steels of same SU, but very different ∆K0, should

behave identically according to Peterson-like q estimates, for example, whereas

they usually do not. To quantify such predictions, 450 steels and Al alloys with

reported SU, SL(R = −1), and ∆K0 values were gathered in ViDa’s

database 31-32, 42. The steels set was separated in 800, 1200, 1600, and 2000MPa

strength ranges, to use their mean fatigue limits and FCG thresholds in the notch

sensitivity analyses. All Al alloys were analyzed with respect to their mean

strength SU = 225MPa. Assuming R = −1, the mean fatigue limits were used to

calculate a0. The values so obtained were used to produce q×ρ curves for the

Kirsch hole, see figure 15, supposing γ = 6. Note how such curves reproduce quite

well the curves proposed by Peterson a long time ago38.

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Figure 15 - Notch sensitivity q for Kirsch holes, estimated using mean ΔK0 and ΔS0

values from 450 steels and Al alloys, supposing γ = 6. Note that q = 0 means that it is

easier to initiate a crack from a free surface than from the border of very small holes.

Notch sensitivities estimated for semi-elliptical notches of Al alloys,

estimated using the mean fatigue properties of the Al alloys mentioned above, are

shown in figure 16. They depend on ρ, ∆SL(R = −1) ≅ SL, ∆K0, γ, and also on

their c/b ratios. Thus they depend on the notch shape, not only on their tip radii ρ,

as assumed in traditional SN analyses. In fact, the c/b ratio effect is very

significant, and it cannot be ignored in practice.

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Figure 16 - Notch sensitivity q in function of the tip radius ρ of semi-elliptical notches in

Al alloys, estimated using a0 = (1/π)·(∆K0/1.12SL)2 = 264µm, SU = 225MPa, and γ = 6.

The notch sensitivity of steels can be calculated in the same way and

follows a similar pattern. Some typical values, estimated using the mean fatigue

properties of low-strength steels are shown in figure 17. The observations made

above for the parameter that controls q in the Al alloys are valid for the steels as

well. Further details on such calculations can be found on “Short crack threshold

estimates to predict notch sensitivity factors in fatigue” 31.

Figure 17 - Notch sensitivity q in function of the tip radii ρ of semi-elliptical notches

predicted for steels with SU = 800MPa, SL = 400MPa, ∆K0 = 8MPa√m, a0 ≅ 102µm, and

γ = 6.

DBD



A different approach to model the notch sensitivity problem, called the

theory of critical distances (TCD), is explored elsewhere43-46. Predictions made by

the TCD model are similar but not identical to the predictions obtained by the

stress gradient (SG) model proposed here. The TCD generalizes Peterson and

Neuber original ideas, but it seems that the SG model using sound mechanical

bases is easier to apply to notched components. The effect of micro-structural

defects on fatigue strength is deeply studied by Murakami41.

2.4. Experimental verification of the notch sensitivity predictions

Stop-holes are widely used as an emergency crack repair technique. The

hole is drilled at the crack front to remove its tip and to force it to re-initiate

before restarting its growth process. This simple trick may increase the cracked

component durability, but its efficiency depends on several variables, among them

the stop-hole radius ρ. To quantify how beneficial such holes can be, 23 pre-

cracked SEN(T) test specimen (TS) with w = 80mm and t = 8mm, see figure 18,

were repaired in this way and then fatigue tested under fixed force ranges at

R = 0.57 47. They were all cut from a plate of a 6082 T6 Al alloy (0.7-1.3Si, 0.6-

1.2Mg, 0.4-1.0Mn, 0.5Fe, 0.25Cr, 0.2Zn, 0.1Cu, 0.1Ti) with SY = 280MPa,

SU = 327MPa, E = 68GPa, AR = 12%, and HV50 = 95kg/mm2, in its LT

direction. This alloy is used e.g. in vehicles, railway components, and

shipbuilding.

The fatigue tests were all made at 30Hz on a 100kN computer controlled

servo-hydraulic machine, following ASTM E64748 procedures. The high R = 0.57

was chosen to avoid crack closure effects. After pre-cracking a specimen, a stop-

hole with radius ρ = 1, 2.5, or 3mm was carefully centered and drilled at its crack

tip. The cracked TS were removed from the testing machine, fixed on a milling

machine, drilled at low feedings with plenty refrigeration, and then finally reamed

to generate elongated notches with a tip diameter accuracy of 1.5µm, all with the

same size a = 27.5mm ⇒ a0/w = 0.344, to avoid interference of possible crack

length or residual ligament rl = w – a0 effects. Great care was taken to avoid

introducing residual stresses around the stop-hole by any means during its drilling

DBD



process. The repaired TS were then re-mounted on the test machine, and the

fatigue test was restarted under the previous loading conditions. Figure 18 also

shows typical data obtained from such repaired TS. Table 1 summarizes the

testing conditions after the introduction of the stop-holes, and the fatigue life

increments obtained from them. The pseudo-SIF of the repaired TS listed in this

table, ∆K* = 1.895∆P/t√w, were calculated using the SIF expression for the

SEN(T) with identical a/w, given by 49:

�� =�

1 ∙ √� ∙ /1.99 ∙ ��.�− 0.41 ∙ ��.�

+ 18.7 ∙ ��.�

− 38.85 ∙ ��.�+ 53.85 ∙ ��B.�0

(30)

Figure 18 - TS used to test the stop-hole size effect on their efficiency as a crack repair

method, and typical results obtained with them.

Table 1 - Crack re-initiation lives Nr after introducing the stop-hole at their tips.

DBD



The fatigue crack re-initiation lives at the stop-hole tips can be reliably

modeled by εN procedures. This modeling process requires the cyclic properties

of the 6082 T6 Al alloy, Hc = 443MPa, hc = 0.064, σc = 485MPa; b = −0.0695;

εc = 0.733; c = −0.82750; the nominal stress history (see Table 1); and the SCF of

the notches generated by the stop-holes. These can be estimated by Creager and

Paris51, for example Kt ≅ 12.38 for a stop-hole with radius ρ = 1, or else by

Inglis 52, Kt ≅ 1 + 2√(a/ρ) = 11.49 in this case; but the resulting notches SCF

were instead calculated by Finite Elements: Kt = 11.8, 8.1, and 7.6 for ρ = 1, 2.5,

and 3mm, respectively.

The life improvement induced by the stop-holes can be estimated by

calculating the stress and strain maxima and ranges at their borders by Neuber’s

rule, and then the crack re-initiation lives considering mean load effects. Such

effects cannot be neglected, since the R-ratio used in the tests was high. In fact

Coffin-Manson predictions are highly non-conservative, thus useless in this case.

Figures 19-22 show that the lives predicted by Morrow EL and by SWT are

similar in this case (but such a similarity cannot be assumed beforehand, since in

many other cases these rules can predict very different fatigue lives)32.

The lives predicted for the two larger stop-holes reproduced reasonably well

the tests results, see figure 19 for the ρ = 3mm results. However, the life

predictions for the ρ = 1mm stop-hole shown in figure 20 are too conservative in

comparison to the measured data. Such better-than-predicted fatigue lives of

course do not mean that the smaller hole is more efficient than the larger ones, as

the larger stop-holes are associated with longer fatigue crack re-initiation lives for

a given load. From a modeling point of view, the main result obtained from such

figures is that εN life predictions made using traditional procedures based on Kt,

Neuber, and Morrow or SWT were satisfactory for the larger stop-holes, but

severely underestimated the re-initiation lives for the smaller one.

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Figure 19 - Fatigue crack re-initiation lives for the stop-hole root with radius ρ = 3.0mm,

measured and predicted using the resulting semi-elliptical notch Kt.


measured and predicted using the resulting semi-elliptical notch Kt.

The better than expected fatigue lives obtained from the smaller stop-holes

could be due to compressive residual stresses. However, all stop-holes were

drilled and reamed following identical procedures, and their diameters were all

DBD



large enough to remove the previous crack tip plastic zones, leaving only virgin

material ahead of their tips. Hence, as the larger stop-hole lives were well

predicted supposing σres = 0, it is difficult to justify why high compressive

residual stresses would be present only around the smaller stop-hole tips. The

same can be said about the stop-holes surface finish. But the smaller stop-holes

generate larger Kt than the bigger ones, thus they induce a steeper stress gradient

ahead of their tips. This effect can significantly affect the growth of short cracks

and, consequently, the stop-hole fatigue notch sensitivity. Indeed, when using the

properly calculated fatigue SCF Kf instead of Kt with the traditional εN

procedures, considering the elongated notch sensitivity q by the method proposed

here, all estimated fatigue crack re-initiation lives reproduce quite well the

measured results, see figures 21-22. The Al 6082 T6 fatigue limit and fatigue

crack propagation threshold under pulsating loads needed to calculate Kf are

estimated as ∆K0 = 4.8MPa√m and ∆S0 = 110MPa, following traditional

structural design practices, and γ = 6.


measured and predicted using the resulting semi-elliptical notch Kf.

DBD



Figure 22 - Fatigue crack re-initiation lives for the stop-hole root with radius ρ =

1.0mm, measured and predicted using the resulting semi-elliptical notch Kf.

Note that figure 21 predictions are similar to figure 19, as the larger stop-

holes have q ≅ 1, whereas the life predictions for the smaller ρ = 1mm stop-holes

shown in figure 20 are much better than the Kt-based predictions shown in

figure 22. Note also that the word prediction can in fact be used here, since the

curves shown in such figures result from re-initiation life estimates made using

only mechanical principles and material data obtained from the literature, without

considering any of the measured data points. Thus they are really predicted, not

data-fitted curves. Moreover, an additional test made after these calculations

confirmed the prediction that the ρ = 1mm stop-hole could tolerate a

∆K* = 7MPa√m, as indeed it did, see figure 22.

2.5. Notch sensitivity effects on environmentally assisted cracking

EAC damage in aggressive media includes time-dependent nucleation and

propagation of cracks up to fracture under tensile stresses that may be well below

the material strength in benign environments. To enhance the stress role in such

problems, the stress crack corrosion SCC notation is sometimes preferred when

there is no need to separate EAC mechanisms. In fact, cracks only grow if driven

DBD



by tensile stresses, and the environment contribution is to decrease the material

resistance to the cracking process. Such problems are important for many

industries, because costs and particularly delivery times for special SCC-resistant

alloys are large and keep increasing. Major SCC problems occur e.g. in the oil

industry, since oil and gas fields can contain considerably amounts of H2S or

chlorides which may attack steel pipelines, and in the aeronautical industry, when

their light aluminum structures must operate in saline environments, like in

carriers, offshore platforms, or costal airports.

However, for structural analyses purposes most EAC problems have been

treated so far by a simplistic over-conservative policy on susceptible material-

environment pairs: if aggressive media are unavoidable during the service lives of

structural components, the standard design solution, e.g. ISO 151562 and NACE

TM001776, is to choose a material resistant to EAC in such media to build them.

A less expensive alternative solution may be to recover the structural component

surface with a suitable nobler coating, if such a coating is available. EAC-proof

coatings must be properly adherent, scratch resistant, and more reliable than

common corrosion-resistant coatings, because structural components can fail

without warning under such conditions. Such over-conservative design criteria

may be safe, but they also can be too expensive if an otherwise attractive material

is summarily disqualified in the design stage when it may suffer EAC in the

service environment, without considering any stress analysis issues. Indeed,

decisions based on this inflexible pass/fail approach may cause severe cost

penalties, since no crack can grow unless driven by a tensile stress caused by the

service loads and by the residual stresses induced by previous loads and

overloads.

In other words, the EAC behavior cannot be properly evaluated neglecting

the influence of the stress fields that drive them. Although EAC conditions may

be difficult to define in practice due to the number of metallurgical, chemical, and

mechanical variables that may affect them, sound structural integrity assessment

procedures must include proper stress analyses techniques for calculating maxima

tolerable flaw sizes. Such techniques are important in the design stage, but they

are even more useful to evaluate operating structural components not originally

designed for EAC service, when by any reason they must pass to work under such

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conditions due to some unavoidable operational change (a pipeline that must

transport originally unforeseen amounts of H2S due to changes in oil reservoir

conditions while a new one specifically designed for such service is built and

commissioned, e.g.). Economical pressures to take such a structural risk may be

inescapable, since loss of profits associated with the very long time required for

substituting the component can be too huge, especially in offshore applications.

Such risky decisions can in principle be controlled by the methodology proposed

following, which extends the analysis developed to mechanically quantify notch

sensitivity effects through the behavior of short fatigue cracks to EAC problems.

Indeed, if cracks behave well under EAC conditions, i.e. if Fracture Mechanics

concepts can be used to describe them, then a “short crack characteristic size

under EAC conditions” can be defined by 36,53:

��CDE = 1

� ∙ � ��CDE� ∙ �CDE��

(31)

In this way, all chemical effects related to EAC problems are assumed to be

properly described and quantified by the traditional material resistances to crack

initiation and propagation in the service medium under fixed stress conditions,

SEAC and KIEAC, supposing such pairs remain fixed. Such properties are well

defined and can be measured by standard procedures. Note that although EAC

problems are time-dependent, SEAC and KIEAC are not, as they quantify limit

stresses required for starting or growing cracks under EAC conditions. Hence,

supposing that the mechanical parameters that limit EAC damage behave

analogously to the equivalent parameters ∆Kth(R) and ∆SL(R) that limit fatigue

damage, a Kitagawa-like diagram can quantify the crack sizes a tolerable by any

given component that works in EAC conditions under a given tensile stress σ, see

figure 23.

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Figure 23 - A Kitagawa-Takahashi-like diagram proposed to describe the environmentally

assisted cracking behavior of short and deep flaws for structural design purposes.

This idea makes sense as well if KIEAC and SEAC are viewed as the limits for

∆Kth(R) and ∆SL(R) as R → 1, and can be further expanded. For example,

figure 24 presents a generalized Kitagawa diagram that shows four regions that

may contain non-propagating cracks. First, the region bounded by the material

resistances to crack initiation and large crack growth by fatigue in an aggressive

medium ∆SL(R) = 2SL/(1 – R) and ∆Kth(R)/√(πa), which limit the tolerance

region that may contain non-propagating fatigue cracks in that environment under

fixed amplitude loads at a given R-ratio; second, the region limited by SEAC and

KIEAC/√(πa) that may contain non-propagating cracks by EAC in that medium;

third, the crack tolerance region limited by ∆SLvac and ∆Kthvac, the R-independent

fatigue limit and FCG threshold of the given material in vacuum; and fourth, the

region limited by the intrinsic material properties SUvac and KICvac/√(πa), which

can only be measured in vacuum or in truly inert environments. The main

advantage of looking at this problem in such an integrated way is that it makes

natural the attempt to treat mechanical and chemical damage under a unified

analysis procedure, following e.g. Vasudevan and Sadananda’s unified approach

methodologies 54-55.

In other words, if cracks loaded under EAC conditions behave as they

should, i.e. if their mechanical driving force is indeed the SIF applied on them;

and if the chemical effects that influence their behavior are completely described

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by the material resistance to crack initiation from smooth surfaces quantified by

SEAC and by its resistance to crack propagation measured by KIEAC; then it can be

expected that cracks induced by EAC may depart from sharp notches and then

stop, due to the stress gradient ahead of the notch tips, eventually becoming

non-propagating cracks, exactly as in the fatigue case. In such cases, the size of

non-propagating short cracks can be calculated using the same procedures used

for fatigue, and the tolerance to such defects can be properly quantified using an

EAC notch sensitivity factor in structural integrity assessments. Hence, a criterion

for the maximum tolerable stress under EAC conditions can be proposed as:

�<( ≤ �(��CDE) ∙ �1 + (��CDE �)⁄ � �⁄ � � �⁄ � {(��) ∙ √� ∙ � ∙ �(� �)⁄� } ;

where ��CDE = (��) ∙ (��CDE � ∙ �CDE⁄ )�

(32)

In the same way, an equation analogous to (26) can be used to properly

define a “notch sensitivity under EAC conditions” by solving for a given γ (if it is

deemed necessary to better fit data) the system {ϕ/g = 1, ∂(ϕ/g)/∂x = 0} for

several notch tip radii ρ using κ ≡ KIEAC/(SEAC√ρ) to obtain

.CDE(-, +) ≡ 2�?� !�-, +� − 1

�? − 13 (33)

where qEAC and KtEAC = 1 + qEAC(Kt – 1) are the notch sensitivity and the

effective stress concentration factor under EAC conditions. In this way qEAC and

KtEAC can be seen as analogous to the q and Kf parameters used for stress

analyses under fatigue conditions.

DBD



Figure 24 - Modified Kitagawa diagram including fatigue and EAC limiting conditions for

crack growth, showing the contribution of mechanical residual stresses and of equivalent

chemical stresses involved in corrosion-fatigue problems.56

Such equations allow stress analyses under EAC conditions and can be used

for structural design purposes, thus they can possibly substitute the pass/non-pass

criterion used to “solve” most practical EAC problems nowadays. Indeed, they are

the bases for a mechanical criterion for EAC that can be applied even by structural

engineers, since it does not require expertise in chemistry to be useful. Moreover,

it can be properly tested, as follows. First, following expert advice57, the basic

EAC resistances were measured testing an Al 2024 – liquid gallium pair (Ga is

liquid above 30oC, but curiously it only boils at 2204oC). The main advantage of

this exotic material-environment pair is its very quick EAC (in fact, liquid metal

embrittlement) reactions, in the order of minutes. In comparison, sensible Al

alloys may take weeks to crack in NaCl water solutions. Moreover, contrary to

other liquid metals that may cause LME like Hg, Ga is a safe, non-toxic material.

This 2024 Al alloy was originally obtained in a T351 temper as a 12.7mm thick

plate (analyzed composition Al plus 4.44Cu, 1.35Mg, 0.54Mn, 0.18Zn, 0.16Fe,

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0.12Si, 0.02Cr, 0.01Zr, and less than 0.05 other elements), but it had to be

annealed to remove its residual stresses (in the original plate condition Ga induced

the test specimens to break during manipulation)56. All specimens were cut on the

TL direction of the plate identified by metallographic procedures. The basic

mechanical properties of the annealed material, E = 70GPa, SY = 113MPa,

SU = 240MPa, εengU = 16%, were measured following ASTM E8M58 standard

procedures at 35oC56.

EAC sensibility and reaction rates of the Al-Ga pair were qualitatively

evaluated also at 35oC in very slow 4.5⋅10−8/s strain rate tension tests made in

servo-controlled electromechanical testing machines, following ASTM G12959

and NACE TM001776 recommendations. The liquefied Ga was applied on the TS

surfaces with a brush, and light bulbs were used to maintain the warm temperature

during the tests. Then the basic EAC resistances were measured using incremental

load steps induced by calibrated force rings following ASTM E168160 and

standard procedures: SEAC tests started at 30MPa and used 2.5MPa steps and

KIEAC tests initiated at 7.5MPa√m and used 0.25MPa√m steps. The time between

successive load steps was at least one hour. The measured values were

SEAC = 43.6 ± 4.2 MPa (mean of 9 samples, with 95% reliability),

KIEAC = 8.8 ± 0.3 MPa√m (8 samples, 95% reliability)56. Some of the TS broken

during such standard EAC tests are shown in figures 25-26.

Figure 25 - TS used for measuring SEAC according to ASTM E1681 standard procedures.56

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Figure 26 - TS used for measuring KIEAC according to ISO 7579 standard procedures.56

Finally, using such standard EAC properties four pairs of C(T)-like notched

TS were designed to support a maximum local stress σ ≅ 90Mpa > 2·SEAC at their

notch tips. The dimensions chosen for such notches were

{b(mm), ρ(mm), b/w} = {20, 0.5, 0.33}, {12, 0.5, 0.2}, {20, 0.2, 0.33},

{40, 0.45, 0.67}. The idea was, of course, to play with their SCF/stress gradient

combination in order to assure tolerance to the short cracks that should start at the

tips of their notches, since they were loaded well above SEAC. The (different)

loads applied on each one of such notched TS were maintained for at least 48

hours. In spite of submitted to a much longer exposure than that required to

measure SEAC and KIEAC according to standard procedures, none of such notched

specimens failed during such tests, exactly as predicted beforehand. Figure 27

shows some of the unbroken notched TS after being loaded for a time 50 times

longer than that used for measuring SEAC under a maximum local stress at the

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notch tip higher than twice the material resistance to crack initiation under EAC

conditions, σmax > 2⋅SEAC56.

Figure 27 - Notched TS after being tested under σmax > 2⋅SEAC56.

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