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:
  1. 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.
  2. 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.
  3. 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.
  4. 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.