Rethinking fatigue
crack growth modeling
I started this project with the idea of simply catching up
on the progress made in fundamental concepts for interpreting fatigue crack
growth. I had not given this subject much serious thought since leaving the
Lockheed Physical Sciences Laboratories at the end of 1969 shortly before my 1970
paper appeared in ASTM STP 482. At the beginning of 1970 I left the
laboratories to assume responsibility for providing technical guidance to the
Lockheed/USAF review of fatigue cracking problems on the C5A aircraft. This led
to a long and rewarding career providing guidance on aircraft damage tolerance
and structural integrity for Lockheed Corporation and a parallel career
teaching and lecturing on structural integrity and fracture mechanics
applications. Among these activities was about 5 years, from about 1967 to
1972, as chairman of ASTM subcommittee E24-5 on fracture mechanics applications.
This was a period when many of the current concepts were evolving. In 2007 we
moved to the east slopes of the Cascade Mountains in Washington State where I
promise to actually “retire”. However, the winter months of snow and cold here
are conducive to reading and thinking and thus the opportunity to catch up was
hard to resist.
I admit that over the years I have developed an attitude
about fatigue and fracture theory that guided my browsing through countless
publications and more than that of abstracts.
·
No theory or model is correct although some are
useful
·
The more convolute and detailed the theory or
arguments in its favor, the more likely the theory or model is flawed.
While I could never complete a review of such a vast amount
of materials, I did progress to the point where I believed that a fresh look at
modeling of fatigue crack growth was needed.
I am placing the first two notes from my “rethinking fatigue
crack growth” in iMechanica as attachments to this blog as the format of the
blog is not equation friendly. The first note is also given below as text as
the equations should be familiar. Just remember that the 2 in Neuber’s rule and
its fracture mechanics adaptation are actually exponents and that (m) and (1-m)
are also exponents.
I welcome your comments and more importantly, your
participation in testing and extending the concepts developed in these notes.
If there is sufficient interest other notes from my rethinking will follow. I
have selected iMechanica as my venue as I have no compulsion to publish another
formal paper and the potentials of an open and free exchange of ideas, example
applications (including those that seem contradictory) and, related data are
appealing. If you find these notes useful, all I ask is that appropriate credit
be given and that you share these note with others.
I will try to answer
simple question by e-mail. For any seriously interested, I still consult
occasionally and enjoy the opportunity to lecture on this or related subjects.
I can be contacted at ekwengr [at] frontier.com (ekwengr[at]frontier[dot]com).
Ken Walker
Technical
Note 1 October 2011
Adapting
Neuber’s Rule to Fracture Mechanics and its implications
This note stems from a long-standing curiosity
about the similarities between Irwin’s strain energy term K2/E and
Neuber’s rule in the form of Kt 2/E = σ ε. In this note the
similarities are explored and the implications of the resulting analogy to
Neuber’s rule in terms of fracture mechanics relatable quantities are examined.
A brief summary of Neuber’s rule applied to
notch fatigue.
Neuber’s rule [1] can be expressed as Kt
= (Kσ Kε)1/2, where Kt is the
elastic stress concentration, Kσ the true notch stress concentration
and Kε the true notch strain concentration. Neuber’s rule was
derived for the case of pure shear and applies to any arbitrary stress strain
relationship. Subsequent studies have shown that it also applies to other
stress states [2, 3].
Topper, Wetzel and Morrow
[3] adapted Neuber’s rule to notch fatigue by assuming that Kσ = σ/S and Kε = ε E/S. Thus,
(Kt S)2 /E = σ ε Equation 1
Where Kt is the elastic stress
concentration factor, E is Young’s modulus, S is the stress applied remote from
the notch, σ is the
true notch stress and, ε is the true notch strain. The application of equation
1 is best explained using Figure 1
Figure 1
Application of Neuber’s rule to notch fatigue.
The uniaxial tension stress-strain curve for 2024
T351 is assumed to apply. The elastic slope is extended beyond yield. A curve
is drawn representing stress and strain combinations that satisfy equation 1.
For this example the Kt S for a
zero–to-tension applied cyclic loading is 600 MPa. For large radius notches the
intersection of the selected stress-strain curve and the curve representing
solutions to equation 1 determines the estimated maximum true stress and true
strain at the completion of the first ½ cycle of applied stress and, the origin
for applying Neuber’s rule for the next ½ cycle. Subsequent ½ cycles are
estimated by shifting the origin to the end point of the previous ½ cycle. The
stress strain curve for each ½ cycle must reflect the effects of prior strain
history.
For constant amplitude cyclic loading and, materials
and stresses of primary interest, after the first loading to maximum applied
stress, subsequent cycles (up to about 2 x σ
max) will be assumed elastic and therefore subsequent cycles are the full
elastic range. No further significant change in peak stress is assumed to
occur. The complexities of cyclic strain hardening or strain softening will not
be considered at this time. With these simplifying assumptions, the cycles
following the first ½ cycle are essentially elastic, the peaks and valleys of
subsequent cycles can be determined without actually drawing the Neuber curves.
When applying the Neuber’s rule to fatigue it is
necessary to relate the notch stress given by the Neuber model to fatigue-life
data. For relatively sharp notches, this relationship is established by
replacing Kt with a fatigue notch factor Kf that accounts
for the differences between the theoretical Kt and the actual
fatigue inducing conditions at the notch. These conditions include
stress-strain gradient, and other differences.
The above background and simplifying assumptions
are adequate as a preface to initially exploring an adaptation of Neuber’s rule
for estimating a new maximum stress intensity factor (kmax) that is reduced from the elastic value as a result of
plastic deformations near a crack tip.
Stress-strain Intensity model
The facts that Neuber’s rule is shown to apply to
stress states other than pure shear where it could be mathematically proven and
it applies to any stress-strain law, allow speculation that it will apply in
the domain of stress intensity. Recognizing that the fracture mechanics (πa) 1/2
is analogous to Kt, Equation 1can be transposed and written as πa/E =
CσCε. Where a is ½ the length of an imbedded crack or the
length of an edge crack and CσCε is the equivalent to KσKε.
Both Cσ and Cε have dimension square root of the selected
dimensional units: Following the analogy to fatigue, Cσ is defined
as kσ/S and Cε is defined as kεE/S where kσ
is the “true” stress intensity and kε is the “true” strain
intensity. Then it follows:
S2πa/E
= kσkε = K2/E = G Equation 2
Where K is the elastic stress intensity and G is a
measure of strain energy. At fracture G becomes the strain energy release rate.
Equation 2 is the fracture mechanics equivalent to equation 1 and figure 2 is
the equivalent to figure 1
Figure2
Adaptation of Neuber’s rule to stress intensity
In figure 2 the applied stress is cycling 0 to
tension. The elastic stress intensity factor at the peak load and elastic
stress intensity factor range ΔK are 15 MPa m1/2. The G curve is
developed, using equation 2, by selecting values of stress intensity and
computing the corresponding values of strain intensity. Figure 2 also shows a
curve analogous to the stress-strain curve that relates crack tip stress
intensity and strain intensity (in this example the stress-strain intensity
curve is assumed to be simply a scaled down stress strain curve). The primary
assumption for the model is that an appropriate stress-strain intensity curve
does exist. Technical Note 2 will show that this curve can sometimes be deduced
from da/dn data. However, it should be possible to develop this curve directly
from CTOD measurements.
The curve
is treated similar to a stress strain curve. Its intersection with the G curve
identifies the stress-strain intensities for the region near the crack tip. The
assumption is that kmax has a unique relationship to the actual
stress distribution near the crack tip and that it is the same unique
distribution related to Kmax.
The initial “elastic” slope of this stress-strain
intensity curve is defined by E as used in equation 2. The assumption that E
applies requires that, at least in the elastic range, that the crack tip
stress-strain intensity curve be proportional to the stress -strain curve. While
it would be convenient to assume that the post yield portion is also
proportional, this possibility needs further study. However, we do know that kmax
and kε are single
valued proxies for the distribution of stress-strain in the region near the
crack tip and thus a curve representing these quantities should exhibit the
general characteristic of the material’s stress-strain curve.
The stress-intensity rang ΔK shown on figure 2 is
shown as the full elastic range that would be true for an open notch (up to
about 2 x kmax). There are a wide variety of opinions as to how or
if this range is affected by closure. What we do know is that whatever these
closure and reversed plasticity affects are that result in Δk (the “true”
stress intensity range) differing from the applied ΔK, the reasons for the
difference are not the same as those resulting in the differences between kmax
and Kmax.
Contributions of maximum and stress range to
crack Growth
When the stress-strain
intensity curve is in the initial linear range, equation 1 will be used to
approximate the contributions of max stress and stress range to crack growth.
The basis for this equation is developed in reference 2
ΔK effective =Δkm Kmax1-m Equation 1
Once beyond this
range, the equation will be modified to
ΔK effective =Δkm kmax1-m Equation 2
This modification
becomes necessary once we recognize that the relationships between kmax
and Kmax and between Δk and ΔK are not the same beyond the initial
linear range.
Discussion
On figure 2, both Kmax
(through the G curve) and, kmax
are proxies for the same stress-strain state near the crack tip. This confirms
that for fracture toughness (a single parameter), relating toughness to K is a
valid approach. Considering that the stress-strain intensity curves for plane
strain and plane stress will differ, use of the Neuber’s rule adaptation might
add to our understanding of the observed differences between plain strain and
plane stress toughness. With some additional thought, this line of reasoning
might be extendable to address the R curve as well. My work to-date has not
addressed the later stages of crack growth that logically merges with stable
tear or fracture.
Crack growth involves two stress parameters, kmax and Δk that are single valued proxies having unique
relationships to the actual stress strain distribution near the crack tip. Each
of these parameters relates to its elastic counterpart by different ground
rules. Thus attempting to explain crack growth in terms of Kmax and ΔK in a two parameter equation or concept or, ΔK alone, can have only limited success. The
direct approach of estimating the stress and strain distribution near the crack
tip has promise but is cumbersome to apply as an engineering tool. It also has
the disadvantage of requiring a fatigue data base related to stress and/or
strain.
The adaptation of Neuber’s rule developed in this note and
the use of the resulting kmax
to determine an effective stress appears to provide a relatively simple
approach to crack growth that addresses recognizable faults in current fracture
mechanics based approaches. For future reference this approach will simply be
referred to as the “stress-strain intensity” approach (unless someone has a
better idea) Subsequent notes in this series will address its applications to
gain a better understanding of da/dn curves.
References
1. Neuber
H., Theory of Stress Concentration for Shear Strained Prismatic Bodies with
Arbitrary Nonlinear Stress-Strain Law, Transactions .of the ASME, December
1961, p 544.
2. Walker
K., The Effect of Stress Ratio during the Crack Propagation and Fatigue for
2024-T3 and 7075-T6 Aluminum, ASTM STP 462, January 1970, p1-14
3. Topper,
T.H., Wetzel, R. M., and Morrow, JoDean, Neuber’s Notch Rule Applied to Fatigue
of Notched Specimens. ASTM Journal of Materials, Vol. 4. No. 1, March1969, p.
200.
| Attachment | Size |
|---|---|
| Technical Note 2 November 2011 final.pdf | 723.26 KB |
| Technical Note 1 Oct 2011.pdf | 397.72 KB |