2.2 Fatigue life model for metallic surfaces that includes
strain hardening and shot peening
Applying a general expression for fatigue life for a stress level \(i\)to the case of drop impact results in Eq. 8a, introducing\(h_{\text{tot}}\) that corrects for the differences between the fatigue
test conditions and the actual conditions. The constants \(m\) and\(S_{f}\) are commonly used material parameters in fatigue tests:
\(S_{max,i}=h_{\text{tot}}\ S_{f}\ N_{i}^{-1/m}\)
The fatigue limit \(S_{D,i}\) for the actual erosion conditions is given
by:
\begin{equation}
S_{D,i}=h_{\text{tot}}\ S_{D}\nonumber \\
\end{equation}In this approach, the number of fatigue cycles to failure in a fatigue
test (\(N_{f}\)) equals the number of fatigue cycles of the incubation
period (\(N_{i}\)).
For metals showing strain hardening, the effects of additional surface
hardening and the related residual compressive stress state at the
surface due to the “water drop peening effect” should be included as
it will affect the fatigue life. The well-known Morrow’s equation with
mean stress correction24 can be used for this purpose.
In the present case the modified version as derived by Landgraf et
al.25 is used:
\begin{equation}
\frac{S_{a}}{\left({\frac{H}{H_{0}}S}_{f,0}-S_{m}-\sigma_{R}\right)}=\left(2N_{f}\right)^{-\left(\frac{1}{m}\right)}\nonumber \\
\end{equation}With:
\(S_{a}\) = stress amplitude,
\(S_{m}\) = mean stress,
\(\frac{H}{H_{0}}\) = ratio of increased surface hardness to initial
surface hardness,
\(\sigma_{R}\) = residual stress at the metal surface
\(N_{f}\) = fatigue life (number of cycles to failure)
\(S_{f,0}\) = fatigue strength coefficient of metal with initial
hardness \(H_{0}\), mean stress \(S_{m}\) = 0 (or \(R\) = -1), and
residual stress \(\sigma_{R}\) = 0,
\(R\) = stress ratio, ratio of minimum stress and maximum stress.
The fatigue strength coefficient (\(S_{f}\)) in Eq. 8a is to be
determined for a stress ratio of \(R\) = -0.5, given by the Rayleigh
wave amplitudes14. Using Eq. 9 and assuming that the
fatigue curves are available for a stress ratio R = -1 (the definition
used for \(S_{f,0}\)), the following expression for fatigue strength
coefficient \(S_{f}\) in Eq. 8a (with \(R\) = -0.5) was derived:
\begin{equation}
S_{f}=\left(1+\frac{1+R}{1-R}\right)\left(\frac{H}{H_{0}}.\frac{S_{f,0}}{2^{-\left(\frac{1}{m}\right)}}-\sigma_{R}\right)2^{-\left(\frac{1}{m}\right)}\nonumber \\
\end{equation}Eq. 10 now includes the (possible
beneficial) effect of surface hardening and residual stress. It is
assumed in this work that this beneficial effect occurs relative to the
‘water –hammer pressure’ (\(p_{\text{wh}}\)), see Eq. 1a, and the
maximum loaded surface area with a contact radius (\(r_{\text{wh}}\)),
see Eq. 5. By using a threshold water-hammer pressure (\(p_{wh,th}\)),
above which surface hardness and residual compressive stresses increase,
it was possible to construct the following model.
- if \(p_{\text{wh}}>p_{wh,th}\):
\begin{equation}
\frac{H}{H_{0}}=1+a\left(p_{\text{wh}}-p_{wh,th}\right)\nonumber \\
\end{equation}\begin{equation}
\sigma_{R}=b\left(p_{\text{wh}}-p_{wh,th}\right)\nonumber \\
\end{equation}- if \(p_{\text{wh}}\leq p_{wh,th}\):
\begin{equation}
\frac{H}{H_{0}}=1.0\nonumber \\
\end{equation}\begin{equation}
\sigma_{R}=0\nonumber \\
\end{equation}\(p_{wh,th}\) = threshold water-hammer pressure,
\(a,\ b\) = material dependent constants, see Table B-2 of Appendix B.
Similar to the work described in detail15 one can use
the Palmgren-Miner rule to account for the cumulative fatigue damage at
different stress levels. The basic form of this equation for variable
amplitude stress loading is:
\begin{equation}
D=\sum_{i=1}^{k}\frac{n_{i}}{N_{i}}\nonumber \\
\end{equation}In which \(n_{i}\) = number of cycles due to multiple drop impact at
stress level \(i\).
The droplet erosion incubation period is now defined as:
\begin{equation}
I_{p}=\frac{D_{f}}{D_{h}}\nonumber \\
\end{equation}\(I_{p}=\) Droplet erosion incubation period (h)
\(D_{f}=\) Cumulative fatigue damage at failure
\(D_{h}=\ \) Cumulative fatigue damage per hour
(h-1)
According to the general Palmgren-Miner approach14,15:\(D_{f}=1\). Using this approach a fatigue-based model for the droplet
impingement erosion incubation period of metallic surfaces can be
formulated. An expression for the incubation period can be
derived15:
\begin{equation}
I_{p}=\frac{d_{d}^{3}}{24\Phi_{v}}\frac{{\left(m-4\right)\left(h_{\text{tot}}S_{f}\right)}^{m}}{A^{4}}\ \frac{1}{\left[{S_{max(r0)}}^{\left(m-4\right)}-\left(h_{\text{tot}}S_{D}\right)^{\left(m-4\right)}\right]}\ \nonumber \\
\end{equation}\begin{equation}
\Phi_{v}={3.6\times 10^{6}\text{\ C}}_{v}v_{d}\nonumber \\
\end{equation}In which:
\(\Phi_{v}\) = volume of impacting water drops per unit area (mm/h)
With \(S_{max(r0)}\) according to Eq. 6 and using \(r_{0}\) according to
Eq. 4.
Furthermore \(S_{max(r0)}>h_{\text{tot}}S_{D}\) and for the
complementary condition \(S_{max(r0)}\leq h_{\text{tot}}S_{D}\) this
results in: \(I_{p}\rightarrow\infty\). Thus the condition\(S_{max(r0)}=h_{\text{tot}}S_{D}\) gives the threshold drop impact
velocity (\(v_{d,th}\)). For drop impact conditions\(\left(v_{d}\ ,\ d_{d}\right)\) resulting in\(S_{max(r0)}\leq h_{\text{tot}}S_{D}\), the fatigue damage will not
accumulate to fatigue failure and thus the life of the metal surface
will be infinite.
Results for stainless steel AISI 316 and aluminium
6061-T6
The droplet impingement incubation period was predicted for stainless
steel AISI 316 and aluminium 6061-T6 using the presented analytical
model and the fatigue life curves in Table B-1, and including the
additional surface hardening and residual compressive stress at the
surface due to the “water drop peening effect” with the data derived
in Appendix B that is summarised in Table B-2. For \(h_{\text{tot}}\), a
value of 1 was taken, this assuming that there are no corrections
necessary for the differences between the fatigue test conditions and
the actual droplet impingement erosion test conditions. Figures 2 and 3
show the predicted curves in comparison to the incubation periods and
confidence limits according to the multi-regression equation of
Heymann10, Eq. A-1 as given in Appendix A.
The Figures show that these
predicted incubation periods are nearly all within the 2.5 and 97.5 %
confidence limits of the multi-regression equation for the given metal.
These limits of the multi-regression equation are partly a result of
experimental uncertainty and partly due to differences between used test
set-ups10.
Discussion
The model predictions for stainless steel AISI 316 and aluminium 6061-T6
showed an excellent agreement with the multi-regression equation of
Heymann10 that is determined from an ASTM
interlaboratory test program. Nearly all incubation period predictions
were within the 95% confidence limits of the mentioned multi-regression
equation.
An essential aspect of the current model is the fatigue-based approach
in which now the surface hardening and residual compressive stress
effects caused by the impact of the droplets are taken into account as
well. The value for the threshold water-hammer pressure (\(p_{wh,th}\)),
which is used in the Eq. 11a and 11b, is based on the extensive research
by Thiruvengadam et al.43-45.
They performed water jet impact erosion tests with metals in a high
speed rotating disk facility. Observation of the specimens to determine
when denting or erosion occurred was implemented in the test sequence.
Depending on the impact velocity, the specimens were microscopically
inspected at intervals ranging from every few minutes to every hour. The
number of impacts taken for the initiation of permanent plastic
indentations on the surface was recorded at different test velocities.
Results for stainless steel AISI 316 (cold drawn), and aluminium 1100-O,
as a function of the impact velocity are shown in Figures 4a and b. The
number of impacts necessary for observing a small permanent plastic dent
is a clear function of the impact velocity.
This number of impacts varies for stainless steel AISI 316 (cold drawn),
in the velocity range 70 – 100 m/s, between 2 to 5 % of the incubation
period. For aluminium 1100-O, this number of impacts varies, in the
velocity range 40 – 60 m/s, also between 2 to 5 % of the incubation
period.
Thus it is concluded that permanent plastic indentations in the surface
of these metals are present during 95 – 98 % of the incubation period
(\(I_{p}\)). In this period a form of liquid impact peening of the metal
surface gives rise to an additional surface hardening and a residual
compressive stress state at the surface.
It is assumed in this work that this beneficial effect occurs relative
to the ‘water –hammer pressure’ (\(p_{\text{wh}}\)), see Eq. 1a, and
the maximum loaded surface area with a contact radius
(\(r_{\text{wh}}\)), see Eq. 5, are the parameters which govern the
beneficial effects of residual compressive stress and increase in
surface hardness, at a certain impact velocity.
The incorporated effect of surface hardening and residual compressive
stress for AISI 316 and aluminium 6061-T6 on fatigue strength can be
substantiated based on evidence from literature.
Soyama46 compares the improvements made to the fatigue
strength of stainless steel AISI 316L by cavitation peening, water jet
peening, shot peening and laser peening. For each peening method, the
optimum coverage was examined by measuring the fatigue life at constant
bending stress. The fatigue strength of the non-peened specimen was 280
MPa. For the treated samples the increase was: 25 % for cavitation
peening, 16 % for shot peening, 9 % for laser peening and 6 % for
water jet peening. Cho48 performed FEM simulations of
the repeated waterdrop impact, drop size of 0.2 mm, on 6061-T6
aluminium. In this computational study, residual effects of repeated
waterdrop impact onto an aluminium surface were investigated. The
results show that above a critical impact velocity (74 m/s), a residual
compressive stress zone is built up under the impact surface as a result
of local plastic deformation. The depth of the plastic deformation
increases with the impact velocity and number of impacts. At an impact
speed of 500 m/s, after 4 impacts, the maximum residual compressive
stress is -345 MPa (-1.06Rp0.2) and is obtained at 0.07
mm under the surface, and the depth of the compressive stress zone is
0.2 mm. Rajesh49,50 performed multi-droplet impact
FE-modelling to predict the residual stresses due to water jet peening
for three grades of aluminium. For this modelling approach, a transient
elastoplastic finite element analysis is used by considering the
impingement of a set of droplets in succession to one another over a
certain time period. The pressure is released following this sequence.
For aluminium 6063-T6 (Rp0.2 = 110 MPa), for drop impact
speeds between 532 and 604 m/s, and applying 1 to 4 “layers” of water
drops this was found to result in compressive stresses between
-0.36Rp0.2 and -0.61Rp0.2 at the
impacted surface. These results46-50 clearly confirm
that for appropriate water drop impact conditions, residual compressive
stresses and an increase in surface hardness due to strain hardening
occur.
In the current work, an analytical model for the prediction of the
droplet impingement erosion incubation period of metal surfaces is
presented. The model is based on the S-N curve of the metal, on the
effects of additional surface hardening and residual compressive stress
at the surface due to a “water drop peening effect”, such as
demonstrated for the case of Al 6061-T6 and AISI 316. Application of the
model to other metals requires the steps as shown in the flow diagram in
Figure 5.
The presented analytical model gives the interrelation of the physical
and mechanical properties of the metallic surfaces that determines the
droplet impingement erosion incubation period. As such, it becomes
possible to define guidelines for a longer droplet impingement erosion
incubation life based on optimised physical and mechanical properties.
Selected properties of the metals used in the presented analytical model
are summarised in Table 1. The required direction of the property (↑:
increase, and ↓: decrease) for an optimal long droplet impingement
erosion incubation life is indicated. From Table 1 it follows that a
higher fatigue strength affects the incubation period positively. Using
the corresponding equations shows that for instance an increase in the
fatigue strength coefficient (\(S_{f,0}\)) of AISI 316 with 10 %, and
using m = 7.8, the average value for AISI 316 in Table B-2, results,
with Eqs. 10 and 14, in an increase in the incubation period (\(I_{p}\))
with a factor of 2.1.
Conclusions
The following conclusions are drawn:
- In the current paper a fatigue
based analytical model for the prediction of the droplet impingement
erosion incubation period of metal surfaces loaded by impacting water
drops published by authors (Slot et al.15), was
tested against a wide range of liquid droplet erosion incubation
period tests. The model was extended for the use of S-N curves for
aluminium and stainless steel, by including the effects of additional
surface hardening and residual compressive stress at the surface due
to a water drop peening effect.
- The model predictions for stainless steel AISI 316 and aluminium
6061-T6, using S-N fatigue curves from different literature sources,
see Table B-1, and including the defined additional surface hardening
and a residual compressive stress state at the surface due to “water
drop peening effect”, showed for the droplet impact velocity range of
140 to 400 m/s an excellent agreement with the multi-regression
equation as determined from an ASTM interlaboratory test program.
Nearly all incubation period predictions were within the 95%
confidence limits of the mentioned multi-regression equation.
- The physical and metallurgical mechanisms resulting in the degradation
process of the metal surface during the incubation period (\(I_{p}\))
were identified, these consisted of: 1) surface plastic deformation
and, formation of dents, 2) surface hardening and residual compressive
stress as a result of these surface plastic deformations, 3) fatigue
crack initiation, 4) short fatigue crack growth.
- Selected properties of metals used in the presented analytical model
were identified with respect to the direction it should be adjusted
for enhanced droplet impingement erosion incubation life.