586 Appendix G Te st result Out-of-plane viewIn-plane view Medium mesh prediction Out-of-plane viewIn-of-plane view 0.22 mm 0.96 mm 0.40 mm 0.18 mm 1.00 mm 0.33 mm Out-of-plane viewIn-plane view Coarse mesh prediction Fine mesh prediction Out-of-plane viewIn-plane view 0.97 mm 0.36 mm 0.12 mm 0.88 mm Te ar 0.27 mm 0.39 mm Figure G.34. Comparison of various mesh refinements to experimental damage. The failure of the fine mesh to better predict material failure indicates that advanced codes such as hydrocodes and Particle-In-Cell (PIC) methods require significant expertise in order to be used correctly. In this case, the poor predictions may be due to mesh size, or they may well be due to the shortcomings in the deformation/failure constitutive model. In fact, the failure to arrive at a converged solution indicates that there may be competing shortcomings in various features of the model that may be offsetting each other. Because of this, these tools require expert users in order to predict damage prior to the impact event. Even without expert users, hydrocodes and PIC methods can be Appendix G 587 15° 30° 45°0° VALUE –INFINITY +INFINITY –1.50E + 04 –5.00E + 04 –1.30E + 10 +2.50E + 04 +5.00E + 01 +0.50E + 04 –2.50E + 04 033 Figure G.35. Comparison of residual stress fields for different impact angles. used to parametrically evaluate different facets of blade design. In this role, they can be useful design tools that can supplement experimental data in order to eliminate designs that are FOD intolerant. An additional benefit of hydrocode modeling is that the output of these models can be used to determine the relative magnitudes of residual stresses due to impact. For example, if a mesh refinement is chosen based on its correlation with impact damage size, that mesh could be used to predict residual stresses. An example of a parametric residual stress study based on an experimentally correlated mesh and a varying impact angle is shown in Figure G.35. Once the residual stresses are calcu- lated, they can be superimposed over cyclic stresses in order to predict life with greater accuracy. Numerical modeling of FOD is similar to modeling of all other ballistic impact events. Accurate predictions require physically realistic deformation and failure models within the code used. These need to be validated against experimental data to give trust in the results. A recent study carried out in the UK by QinetiQ to investigate the effects of FOD on a titanium alloy fan blade illustrates the state-of-the-art capability in ballistic modeling [12]. Detailed numerical FOD simulation The study included determination of damage mechanisms and identification of the thresh- old perforation velocity of a hard steel sphere. The DYNA suite of Lagrangian hydrocodes [13] was used in the numerical simulations. The code features sophisticated contact algo- rithms and interface treatments. Deformation and failure models were constructed for steel and Ti-6Al-4V. These models were validated against both low rate mechanical tests and against relevant high rate tests such as plugging of a thin sheet by a projectile. In the numerical simulation of an FOD event, each of the elements needs to translate the input energies and velocities into stresses and strains. There also needs to be a physically based means of determining when the material represented in each element fails. This is generally carried out by using a constitutive model to determine stresses and strains and a separate failure model. The failure model is applied to each element at each time step to determine if the failure criterion has been exceeded for that volume of material. The 588 Appendix G accuracy of the overall prediction is dependent upon the accuracy of the constitutive and failure models. The material models used by QinetiQ for the blade and the impactor materials are based on the modified Armstrong–Zerilli model shown in Equation (G.1).  =  C 1 +C 5  n   T  293 +C 2 exp   C 3 +C 4 ln ˙  T  (G.1) where C 1 to C 5 and n are empirical constants and , , ˙ and T are respectively stress, strain, strain rate and temperature in  K;  293 is the shear modulus at 293  K and  T is the shear modulus at the current temperature. The constants for the deformation model are taken from mechanical tests carried out at different strain rates and at different temperatures. For each test the effect of stress state on the measured flow stress is understood and this allows an effective uniaxial von Mises flow stress to be calculated for each separate condition. The variation of von Mises stress with strain allows the constants for the material to be calculated. Once these constants are calculated from simple mechanical tests, there is no need for them to be further changed, if the deformation model is accurate. Inaccurate predictions indicate that the model must be modified. The fracture model used in the simulations was the Goldthorpe Path-Dependent Fracture Model [14, 15] where the accumulated damage is given in Equation (G.2) dS =067 ∗ exp  15 ∗  n −004 ∗  −15 n  ∗ d +A ∗  s (G.2) where  n is the stress triaxiality (pressure/flow stress or P/Y), d is the effective plastic strain, and  s is the maximum principle shear strain; A is a constant determined from torsion tests and S is the damage parameter. The damage is then incremented in each time step and fracture occurs when the damage reaches a critical value S c . The critical damage value is derived from tensile tests and has been shown to be a material parameter rather than a fitting constant. The damage essentially comprises a tensile component due to void growth and a shear failure component due to shear localization. For the current illustration, constants for the titanium alloy were obtained from inter- rupted tensile tests on a standard untreated batch of Ti-6Al-4V. The values for the spherical ball were not available and were taken as a standard UK rolled homogeneous armour (RHA) for simplicity. The Mie-Gruniesen equation of state parameters used were standard values for this steel. Initial modeling runs were used to confirm that the mesh used for the study produced a converged result in terms of hole size, stress state, etc. It should be noted that when an element reaches the critical failure condition, the element including its mass and energy is deleted from the calculation, thereby opening up a gap in the mesh. This is considered to be the best that the hydrocode can achieve in terms of crack formation. Appendix G 589 A key issue in the simulation of the impact process is the degree to which the projectile deforms, particularly if it is spherical. This is crucial since the contact area on the blade will change dynamically during the impact and therefore influence the damage and fracture within the blade. To investigate this effect, the spherical projectile was simulated assuming a rigid body and also assuming a standard modified Armstrong-Zerilli model for RHA since these represent two extremes of behavior. It was found that the deformation of the spherical projectile was quite pronounced when the softer (deformable) material model was used. In reality, the ball is harder than RHA but will exhibit a small deformation. For the case used in this illustration, the numerical modeling with a deformable pro- jectile gave a predicted threshold speed for perforation of about 300 m/s which is in good agreement with the experimental data. As was expected, the majority of the damage is shear and a shear crack was observed running ahead of the projectile. It is worth noting that this crack took several time steps in the microsecond range to cross the thickness of the blade. In carrying out the study, the importance of the constitutive model was illustrated because the predicted threshold speed for the spherical projectile described using the RHA model was significantly higher than for the rigid material model. In addition, changing the damage parameters also affects the threshold perforation speed. This emphasizes the crucial need to characterize the actual materials being used since the results are sensitive to both the deformation and the failure models. It is also interesting to note that in all the impact cases, the von Mises stress field, which can be related to the residual stress, becomes constant very quickly (about 100s) after the impact event. This is important when attempting to link the impact information to the much longer time cycling response of the blade. An advantage of these simulations, and the trust that can be placed in them, is that a great deal of the information obtained can be used for quantitative analysis. Therefore, it is useful to compare bulge dimensions, plug velocities, spallation, and masses with experimental data. Although this may demand additional experiments, the techniques are available to accurately measure these phenomena. This is important since the general plug sizes and shape will influence the damage around the hole and may even indicate initial cracking around the hole. This may influence the subsequent life cycle of the blade due to fatigue cracking. The capability of quantitatively observing the damage progression due to stress triax- iality is a major advance in understanding the blade response. The models used in this study are considered superior to previous failure models, which were largely based on effective plastic strain, since the latter cannot differentiate between tensile, shear, and compressive failures. In particular, understanding the time-scale of these mechanisms might allow better design of the blade to withstand the impact process by better material processing or geometrical changes. 590 Appendix G A remaining issue is that the information generated from these simulations needs to be used as input into the simulation tool used to predict the fatigue life reduction incurred as the result of service incurred FOD. The simulation of a real impact event, as opposed to a laboratory simulation using a spherical ball, was carried out under the Air Force HCF program [16]. There, a sphere impactor of equal mass to a cube were compared in a simulated leading edge impact event. The cube has been found to produce damage in laboratory experiments that looks much like that found in blades that have suffered FOD in the field. A sharp-edge blade specimen with a 0.127-mm leading edge radius (Figure G.32) was used for this study. The impact angle was fixed at 45  , and the simulation used a 1.33-mm steel ball as the reference mass. Figure G.36 compares the two local model mesh geometries. The cube is oriented such that a sharp cube edge impacts the blade specimen leading edge. This produces the characteristic V-notch FOD which is considered to be a worst case impactor orientation. Here, the impact velocity was 1000 ft/sec. It was found that the sphere impact produces more local bending distortion around the FOD site, while the cube impact creates a deeper notch effect. The local bending distortion is produced by an interaction of blade edge radius, impactor radius, and impactor velocity. Slower and larger impactors produce more local bending distortion around the FOD site. As velocity increases, less local bending distortion is observed. It was also found that the exit side (top) compressive stress distributions are different for the two impactors. The sphere impact produced compressive stress all around the surface of the exit side while the cube impact created a very slight tensile stress at the exit side surface notch location with compressive stresses produced deeper into the blade specimen. The calculations also compared the two models loaded to a nominal 20 ksi stress level. Both models showed tensile stresses on the projectile entrance side. The cube model produced localized tensile stress at the exit-side notch location. The results showed that Figure G.36. Local sphere and cube impact models. Appendix G 591 the 40 ksi nominal stress significantly reduces the compressive stress field on the exit-side of the cube impact, and a very localized tensile stress at the exit side notch root was clearly observed. The sphere impact still showed a large zone of compressive stress at the exit side. Fatigue life to failure was investigated comparing the sphere impact to the cube impact. Velocity cases of 600, 800, 1000, and 1200ft/sec were investigated for the 45  impact angle. Figure G.37 shows the effect of the cube and the sphere on fatigue life to failure. Figure G.37 plots calculated fatigue life for the 0–20-ksi cycle and the 0–40 ksi-cycle. The effect of “with and without residual stress” is also shown. Figure G.37 clearly shows that the cube impact is more damaging to fatigue life than the sphere impact. Figure G.37 shows that fatigue life decreases for the increasing velocity, but the curves also appear to be converging as velocity increases. This suggests a possible upper limit on velocity for a minimum fatigue life point. For all cases, including residual stress reduced the predicted fatigue life to failure. The difference in fatigue life between “with and without residual stress” decreases as the cyclic stress range increases. This is observed by comparing the cube 0–40 ksi “with and without residual stress” curves. This cube versus sphere study illustrated that FOD geometry has a significant effect on fatigue life, local residual stress distribution, and FOD site geometry. This points out the important fact that in simulating real FOD in the laboratory, the geometry and properties of the impactor play important roles in determining the type (geometry) of impact damage, the residual stress distribution, and the subsequent fatigue life. Equal mass sphere to cube impact comparison 45° impact angle Sharp edge blade specimen results 1.0E + 03 1.0E + 04 1.0E + 05 1.0E + 06 1.0E + 07 1.0E + 08 1.0E + 09 1.0E + 10 600 700 800 900 1000 1100 1200 Impact velocity (ft/sec) Fatigue life to failure Ball 0 – 20-ksi cycle with residual stress Ball 0 – 20-ksi cycle without residual stress Ball 0 – 40-ksi cycle with residual stress Ball 0 – 40-ksi cycle without residual stress Cube 0 – 20-ksi cycle with residual stress Cube 0 – 20-ksi cycle without residual stress Cube 0 – 40-ksi cycle with residual stress Cube 0 – 40-ksi cycle without residual stress Figure G.37. Fatigue life comparison of cube impact to sphere impact. 592 Appendix G Post-impact life prediction The study of fatigue life prediction from damage sites has been examined for the past century using fracture mechanics concepts to describe crack growth. Unfortunately, a best life prediction method has not been agreed upon. This section attempts to describe several recently developed life prediction methods for FOD’d airfoils. Advantages and disadvan- tages of each methodology will be briefly discussed. In general, the methodologies fall into two different categories: total life and fracture mechanics. Crack initiation In the case of FOD, for HCF, total life prediction can be thought of as a crack initiation prediction. This type of model, described in Chapter 2, assumes that once a crack is initiated, the airfoil in question has effectively failed. The crack initiation method used to evaluate FOD in this section uses an equivalent stress life prediction parameter ( equiv  that is defined in Equation (G.3)  equiv =05  E  w   max  1−w (G.3) where  equiv is the alternating Walker equivalent stress, E is the elastic modulus,  is the total strain range, and  max is the maximum stress. The Walker equivalent stress exponent w is a material and temperature-dependent parameter that collapses variable mean stress data into a single life curve. Given the focus of FOD life prediction for aircraft engine components is in the intermediate and long life (HCF) regime, elastic cycling conditions typically dominate so that E = psu ∼. Parameter  psu is the elastic equivalent stress or pseudostress used for the Walker model. This quantity basically assumes the plastic stress and strain are small and approximates the total stress range as the elastic stress range. This can be used to establish Equation (G.4) as:  equiv =05   psu  w   max  1−w (G.4) This approach is essentially identical to the equivalent strain parameter that has been shown in the literature to collapse R  data for a number of different materials. This approach requires an elastic–plastic stress analysis, but does not require a plastic strain range term that is typically extremely small in the HCF life regime of interest to aircraft engine components. This approach also predicts a decrease in the importance of R  on life in the short life regime. An additional advantage of this approach is that a single curve collapses test data over the entire life regime (Figure G.38). In order to arrive at an accurate life prediction, the stress analysis includes the effects of local plasticity at the notch root. This can be done using any number of available Appendix G 593 1.E + 03 10 100 R = –1.0 S eq = 7611 * N f –.6471 + 65.39 * N f –.03582 R = 0.1 R = 0.5 R = 0.8 Avg Regression Results –3s Regression Results Alternating equivalent stress 1.E + 04 1.E + 05 1.E + 06 Cycles 1.E + 07 1.E + 08 1.E + 09 Figure G.38. Application of equivalent stress parameter to data with different stress ratios. numerical stress analysis programs. Application of the elastic–plastic stress analysis, in conjunction with the life prediction parameter described above, results in good correla- tion with experimental data. An example of the correlation for the winged specimens described above is shown in Figure G.39. Once the equivalent stress is determined for the appropriate notch geometry and residual stress, the life of the notched speci- men can be predicted by correlating the stress to life using a curve like that shown in Figure G.38. FOD Fod + stress relief Average seq + 3 σ seq – 3 σ seq 60.00 70.00 50.00 40.00 30.00 20.00 Maximum notch depth (inch) Weibull modified equivalent stress (Ksi) 0 0.01 0.02 0.03 0.04 Figure G.39. Equivalent stress for a given notch depth on ballistically impacted winged specimens. 594 Appendix G Crack growth Unlike crack initiation, crack growth methodologies do not assume that failure occurs when cracks initiate. Crack growth methodologies allow the cracks to grow until they reach a critical size or arrest. The first question that must be answered is how can methodologies developed for smooth notches, often without residual stresses, be applied to FOD damage that does not conform to those assumptions. The USAF conducted a study to determine whether or not fracture mechanics methods could be applied to FOD [17]. The results from this study clearly indicate that fracture mechanics can be applied to FOD and that an increase in the crack tip stress intensity factor (K) is necessary to account for increased FOD depths. This conclusion was reached even in the presence of significant residual stresses. The analysis of fatigue cracks emanating from notches is discussed in Chapter 5 in the notch fatigue section. The introduction and use of small crack theories is a vital part of the analysis since many cracks at FOD-induced notches are truly in the small crack regime. Faster crack growth rates than predicted by long crack LEFM are found under these conditions when stress levels are too high and small-scale yielding conditions are exceeded or the crack is so small that it is affected by microstructural conditions [18]. The Kitagawa diagram becomes an important tool under such conditions. In order to apply fracture mechanics to FOD, there are essentially six steps to determine an endurance limit stress. These steps are (1) calculate normalized elastic K (initial crack dimensions must be assumed) for airfoil/notch geometry, (2) calculate elastic K max and K min , (3) calculate residual K for airfoil/notch geometry, (4) calculate K and R-ratio with residual K, (5) compare K and R-ratio to K th material capability, and (6) iterate on stress to converge on solution. The following paragraphs describe each of these steps in more detail. Calculate normalized elastic K for LE (Leading Edge)/Notch geometry Stress intensity analysis must be conducted on LE and notch geometries which cover the configurations of interest. Interpolation/extrapolation on notch depth is used to determine the normalized elastic K for the given airfoil/notch geometry. Obviously, this approach will result in some error in K predictions, but should be adequate to capture the behavior of FOD in the non-catastrophic regime, especially considering the amount of scatter in experimental data. Calculate elastic K max and K min Calculation of the elastic K max and K min is simply the normalized elastic K times the applied stress. To predict the endurance limit stress, an initial value of stress must be selected and iterated to a solution through the remaining steps. Calculate residual K for airfoil/notch geometry Residual K can be assumed to be solely a function of notch depth. This is a very simplistic approach to capturing the effects of residual stress local to the FOD notch. This approach Appendix G 595 does not distinguish between the probable causes of the residual stress, FOD projectile impact, and local notch yielding. A more accurate representation would be to calculate the residual K trends with detailed impact and notch yielding FEA analysis. This will not only increase confidence in the methods, but also make them more robust by defining K residual as a function of airfoil, and notch parameters. Since notch residual stresses have a significant affect on crack growth from FOD damage, it is suggested to use the most accurate representation of residual stresses possible. Calculate K and stress-ratio with residual K The final calculations for the modeling are to determine the local K and stress-ratio, including the effects of the residual K. This is simply determined by subtracting the residual K from the maximum and minimum K, and recalculating R-ratio. Since residual K will reduce the elastic K  , the local R-ratio will always be less than the applied R-ratio. Compare K and R-ratio to K th material capability Finally, the predicted K and R-ratio are compared to the K th material capability. If the predicted K is above the material capability, cracks would be predicted to initiate. Iterate on Stress to Converge on Solution To determine the stress at which the onset of crack initiation will occur, the applied stress must be iterated until the predicted K and R-ratio match the material capability. Like crack initiation models, many crack growth models assume material failure once the nucleated crack has begun to grow (initiate). However, in the case of FOD damage, residual stress plays a significant role in crack initiation. It is possible in certain cases of FOD to grow a crack through the residual stress region or the vibratory stress field to a point where the crack will arrest. In order to capture this behavior, the following model has been proposed. Worst case notch (WCN) The WCN model is described in Chapter 5. It assumes that the lowest threshold stress for onset of HCF is controlled by whether or not microcracks can continue to grow after having been initiated early in life by FOD and/or LCF. It makes use of a crack- size–dependent threshold stress intensity. The WCN model predicts the conditions for onset of crack initiation, as well as regimes of crack growth and arrest, or crack growth to failure. The WCN model can use simple parametric notch-stress equations or finite element analysis to predict S th as a function of FOD-notch depth and sharpness, including residual stress effects. Since it explicitly treats the growth and arrest of microcracks, the WCN model is also applicable to cases where beneficial surface treatments are employed to enhance component life. An example of predictions of S th made using the WCN . cycling response of the blade. An advantage of these simulations, and the trust that can be placed in them, is that a great deal of the information obtained can be used for quantitative analysis. Therefore, it. the spherical ball were not available and were taken as a standard UK rolled homogeneous armour (RHA) for simplicity. The Mie-Gruniesen equation of state parameters used were standard values for. the Walker model. This quantity basically assumes the plastic stress and strain are small and approximates the total stress range as the elastic stress range. This can be used to establish Equation