616 Appendix H is provided, it shall be demonstrated once during each shut down. Components such as air-oil coolers with exposure to inlet sand and dust conditions shall be considered inlets for this test but a rig test may be performed to satisfy the requirements herein. Following the post-test performance check, the engine shall be disassembled to determine the extent of sand erosion, and the degree to which sand may have entered critical areas in the engine. The test will be considered satisfactorily completed when the criteria of 3.3.2.4 have been met and the teardown inspection reveals no failure or evidence of impending failure.” Background: The recommended text decreases the operational time in the extreme sand and dust environment from ten hours to two hours for turbofan and turbojet engines. Engine contractors have been unwilling in the past to guarantee their engines for ten hours (helicopter subjected to the severe Vietnam sand and dust environment typically used inlet filtration systems). The time requirement will have to be negotiated with each engine contractor in specific future specification negotiations based upon the intended usage in regions of the world where sand will be a concern. The sand concentration should be calculated with customer bleed air extraction. The anti-icing switch should be activated five times during each hour of sand ingestion at equally spaced intervals. The test should be conducted with a thrust bed and load cell measurement of thrust in lieu of calculating thrust by EPR. Disassembly and inspection between the coarse and fine sand tests should be conducted for 45.4 kg/s (100 lb./sec) airflow or smaller engines. VERIFICATION LESSONS LEARNED (A.4.3.2.4) The Engine V sand and dust test did not use the recommended sand and dust mixture due to commercial unavailability of the mixture. The specification for fine sand calls for a particle size distribution which cannot be obtained commercially. Specifically, calcite and gypsum could not be obtained with a particle size distribution to match the specified particle size distribution. Table XXXVIa and b shows the closest particle size distributions which the Engine V sand and dust test team could find along with the required size distribution. Appendix I ∗ Computation of High Cycle Fatigue Design Limits under Combined High and Low Cycle Fatigue Joseph R. Zuiker ABSTRACT Applications in rotating machinery often result in stress states that produce both low cycle fatigue (LCF) damage in addition to the damage produced from the high frequency or high cycle fatigue (HCF) vibratory loading. While the Haigh diagram takes into account the vibratory as well as the steady stress amplitudes for a fatigue limit corresponding to a (large) given number of cycles, it does not consider the combined effects of LCF and HCF. To account for the combined effects analytically, an initiation model for combined cyclic fatigue (CCF) is coupled with a threshold fracture mechanics crack propagation model to predict fatigue thresholds for CCF. The results are contrasted with the HCF allowable stresses represented in a constant-life Haigh diagram. Experimental data from the literature for a Ti-6Al-4V alloy are used to demonstrate the viability of the analysis and the limitations of the use of the Haigh diagram in design. Comments on the limitations on the use of a Haigh diagram for combined HCF–LCF loading are presented. NOMENCLATURE C Paris-Walker law constant CCF combined cycle fatigue d Paris-Walker law constant d damage parameter d i initiation phase damage parameter D diameter of rod HCF high cycle fatigue K t stress concentration factor LCF low cycle fatigue LEFM linear elastic fracture mechanics ∗ This document was contributed by Dr. Joseph Zuiker, a former employee of the Air Force Research Laboratory. It is based on unpublished work conducted by him while with the Air Force. Dr. Zuiker is currently with General Electric Company Power Systems Division. 617 618 Appendix I m Paris-Walker law constant n number of HCF cycles per LCF cycle N number of CCF cycles N iCCF number of cycles to crack initiation in CCF loading N iHCF number of cycles to crack initiation in HCF-only loading N iLCF number of cycles to crack initiation in LCF-only loading Q stress intensity range ratio =K HCF /K LCF r exponent for initiation life equation R stress ratio = min / max crack growth rate acceleration factor K HCF stress intensity factor range of HCF cycles K LCF stress intensity factor range of LCF cycles K th threshold stress intensity factor range K onset K LCF value at which HCF crack growth becomes active in CCF K tho K th at R =0 (in CTOD-based model) HCF strain range of HCF cycle stress range end endurance limit stress range below which no initiation damage is caused HCF stress range of HCF cycle LCF stress range of LCF cycle ∗ constant for initiation life equation a alternating stress aeq equivalent alternating stress at R =0 for a stress state at R =0. aHCF alternating stress of HCF cycle to be converted to equivalent R =0 cycle fs alternating stress causing failure in a specified number of cycles at R =−1 m mean stress mHCF mean stress of HCF cycle to be converted to equivalent R =0 cycle ult ultimate strength y yield strength INTRODUCTION Design of components for HCF must generally account for the detrimental effects of a superimposed mean stress. This accounting is often in the form of an alternating versus mean stress (Haigh) diagram that shows allowable vibratory stress amplitude as a function of applied mean stress for a specified life. In many cases little or no data are available for conditions other than fully reversed loading where the stress ratio R = min / max =−1, and tensile overload R = 1 or ultimate stress, and assumptions such as a straight line fit must be made in order to interpolate between these limiting cases. Appendix I 619 A more general Haigh diagram can be produced using data at various values of mean stress and a specified number of cycles to failure, e.g. 10 7 , as obtained from S–N curves and plotting the locus of points. For any of these plots, the number of cycles is typically taken to be those corresponding to a “runout” condition, perhaps 10 8 or even 10 9 , but there are few data available to demonstrate that a true runout condition ever exists for a material. This has been shown to be the case in several studies on titanium (cf. [1, 2]). For convenience and practicality, the number of cycles chosen is taken to correspond to the region where the S–N curve becomes nearly flat with increasing number of cycles, or is selected such that the number of cycles exceeds that which might be encountered in service. In some cases, neither condition may be satisfied. For design purposes, because of the statistical variability of fatigue data, particularly in the long-life regime where S–N curves tend to be close to horizontal, Haigh diagrams commonly represent a statistical minimum. For the purposes of the present discussion, only average material property data will be discussed. The straight line Goodman assumption and corresponding Haigh diagram are widely used in design for HCF. Henceforth, we shall consider only the Goodman assumption, but it is understood that any discussion of the Haigh diagram is equally valid for any other assumptions regarding the shape of the diagram. A critical issue in the use (or misuse) of the Haigh diagram in design is the degree of initial or service induced damage that may be present in a component, but may not be present in the material used for generation of the Haigh diagram. In the present study, we deal with damage induced by superimposed LCF. If such damage is present, the Haigh diagram is not valid for the material because it represents “good” or undamaged material. Therefore, a design methodology which considers the development of damage from sources other than the constant amplitude HCF loading must be used to account for the different state of the material. Turbine engine components, for example, which are subjected to HCF, are typically subjected to LCF in addition because the non-zero mean stress is achieved through the centrifugal loading typical of operation. Each startup and shutdown constitutes an LCF cycle. Thus, the component experiences combined HCF and LCF or CCF and, for design purposes, the effect of LCF loading on the HCF life should be considered. In this appendix, we present a simple model for the CCF of a typical turbine engine alloy and use data from the literature to predict the effect of superimposed LCF on the HCF capability of the material. Here, LCF refers to large amplitude, low frequency cycles whose total number is typically less than 10 3 –10 4 , while HCF refers to small amplitude, high frequency cycles at high mean stress, whose number generally exceeds 10 6 –10 7 . In the following sections, a prediction methodology is described including descriptions of the initiation life model, the propagation life model, the experimental data used to calibrate the model, and the assumptions concerning the interaction of the HCF and LCF cycles. Then, numerical predictions are presented to confirm the model accuracy and 620 Appendix I show its sensitivity to a variety of factors. Finally, we close with a discussion of the results, conclusions, and possible future efforts. It is important to note a principal difference between this work and the majority of the previous studies on CCF. While most of the literature has been concerned with the effect of superimposed HCF on the LCF life of materials and structures, this appendix deals with the effect of superimposed LCF on the HCF capability of the material and further and, further, addresses total life as a sum of initiation and propagation phases, the latter of which uses fracture mechanics analysis. LIFE PREDICTION METHODOLOGY In order to illustrate HCF–LCF interactions, analytical predictions are made of the total fatigue life and presented as a Haigh diagram for a material experiencing 10 7 HCF cycles divided equally over N LCF loading blocks. It is assumed that total life can be divided into two distinct phases: a crack initiation phase, and a crack propagation phase. Each CCF loading block consists of a low frequency cycle over which the material is loaded from zero stress to a given mean stress and held while n=10 7 /N high frequency cycles are superimposed about the mean stress as shown schematically in Figure I.1. The details of the analysis follow. Initiation life During initiation the material is assumed to be uncracked. Initiation damage, d i ,is accumulated over each HCF and LCF cycle until d i =1 at which point it is assumed that a crack of depth a i has initiated. The number of LCF cycles required to reach d i =1is σ m σ a 2 σ LCF 2 σ HCF Time n = 8 Stress (strain) ONE CCF LOAD BLOCK Figure I.1. Idealized combined cycle fatigue load block. Appendix I 621 defined as N iLCF . For LCF-only cycling applied at R =0 =2 a =2 m , a power law function of the applied stress range using a form similar to the Basquin equation is used such that N iLCF = ∗ 2 a r (I.1) where a is the alternating stress amplitude and ∗ and r are constants. In fitting the response of actual materials, multiple sets of constants are used over specific ranges of a such that Equation (I.1) forms a piece-wise linear approximation to the actual material response when plotted on a log-log scale. Equation (I.1), which is written for LCF-only loading R =0, can also be used for HCF cycles at R =0 by substituting an equivalent alternating stress amplitude, aeq . The equivalent alternating stress is obtained by moving along a line of constant life on a Haigh diagram from the point defining the HCF cycle m a at R = 0 to a point at R = 0. The form of the constant life line must be assumed. Here, we postulate that the straight-line Goodman assumption governs mean stress effects on initiation life in the same manner as it governs mean stress effects on total life. That is, straight lines passing through ult 0 exhibit constant initiation life. The fully reversed stress to initiation, fsi is defined as the y-axis intercept of a line passing through points defining the HCF load cycle at R = 0 mHCF , aHCF and ult 0. Fully reversed initiation stress, fsi can be defined in terms of aHCF , mHCF , and ult ; and substituted into the modified Goodman equation for fs , the fully reversed alternating stress amplitude. The equivalent alternating stress is then obtained by setting a = m = aeq and solving for aeq ,as aeq = 1 1 ult + 1 aHCF − mHCF aHCF ult (I.2) Thus, the initiation life due to HCF cycles, N iHCF , is obtained via Equation (I.1) by replacing a with aeq from Equation (I.2). To determine the initiation life under combined HCF–LCF loading, the linear damage summation model [3, 4] is used such that the initiation life, in CCF blocks, is N iCCF = 1 1 N iLCF + n N iHCF (I.3) where N iCCF is the initiation life under CCF in terms of CCF load blocks. The linear damage summation model has been criticized for its inability to account for load sequenc- ing affects. However, it is noted that when different cycles are mixed evenly over the life of a component, the Palmgren–Miner rule gives acceptable results (cf. [5, 6]). More advanced nonlinear damage summation models have been proposed. While many give 622 Appendix I better results than the linear damage summation model, they are often limited to specific materials or conditions and require experience to be used with confidence [7]. After N iCCF loading blocks, a crack, which is amenable to fracture mechanics techniques for predicting crack growth, is assumed to have formed in the component and grows according to LEFM to failure. The size, shape, and location of the crack must be assumed and, here, will be taken from experimental data in the literature. For cases in which N iCCF 1, it may be sufficiently accurate to round N iCCF to the nearest integer and begin crack propagation with the next load block. In other cases this may not be accurate and itis important to determine atwhat point in theload block the crackinitiates and crack propagation begins. As a first approximation, it is assumed thatall initiation damagein each cycle occurs duringthe loading portion of thecycle. Thus, if N iCCF is fractional, the first portion of the fractional cycle is attributed to the LCF cycle; the remainder of the fractional initiation damage is attributed to HCF cycles, and during the remaining portion of the load block the crack is assumed to have initiated and begins to grow in HCF. Initiation example Consider the case of a specified loading sequence consisting of n = 8000 HCF cycles per CCF load block. For a specified maximum stress and HCF stress range, the initiation lives are found as N iLCF =16×10 4 and N iHCF =3×10 7 . In this case, the initiation damage per CCF load block due to LCF is d iLCF =1/N iLCF =6250 ×10 −5 , the initiation damage per CCF load block due to HCF is d iHCF =n/N iHCF =2667×10 −4 , the total initiation damage per CCF load block is d iCCF =d iLCF +d iHCF =3292 ×10 −4 , and N iCCF =1/d iCCF = 3037975. Thus, after 3037 CCF load blocks, d i =0999679. During the loading portion of the LCF cycle in load block 3038, d i increases by 6250 ×10 −5 to 0999742 ×10 −n . Each HCF cycle then increases the damage by 3333 ×10 −8 until the crack initiates after 7740 HCF cycles in load block 3038. Thus, during HCF cycle 7741 in CCF load block 3038, the crack is considered to have initiated and begins to grow under the assumptions of fracture mechanics. Propagation life During the crack propagation phase, the crack grows under the assumptions of linear elastic fracture mechanics. Short crack behavior is neglected. During LCF and HCF cycles, the crack is assumed to grow in mode I following the Paris law as modified by Walker [8] to account for stress ratio effects as da dN ∗ =C K m 1–R d (I.4) Here, C and m are material constants describing the crack growth rate at R=0, and d is a material constant accounting for the higher crack growth rate at higher R for the same Appendix I 623 K, an effect attributed to K max or mean stress effects. For LCF cycles, N ∗ corresponds to a single LCF cycle, K LCF replaces K, and R=0. For HCF cycles, N ∗ corresponds to a single HCF cycle, K is replaced by K HCF , and in general R>0. Equation (I.4) holds for K>K th for individual LCF cycles as well as individual HCF cycles provided that the appropriate stress range and value of R are used in each case. In accordance with experimental observations, K th is assumed to be a decreasing function with increasing R. The values of K LCF and K HCF are calculated from LCF and HCF which are shown in Figure I.1. It can be deduced from the figure that K HCF is typically less than K LCF for a given crack length and, therefore, the threshold in LCF should be reached before that in HCF. However, when considering growth rate per block of cycles, the number of cycles per block, n, if large, could dominate the growth rate if both values of K for HCF and LCF are above threshold. In the case of tension–compression cycling R<0, the crack tip is assumed to be open, and the crack growing, only when the applied stress is positive. Thus, the minimum effective stress is always positive or zero, and R never drops below zero in Equation (I.4). This is, however, a minor point as we are most interested in loading typical of turbine engine components in which the mean stress is high, the vibratory stress is relatively low, and R HCF >0. The specimen is assumed to fail when K max surpasses K IC , or when the crack depth exceeds an appropriate length scale indicative of tensile overload in the specimen, whichever occurs first. Crack growth is calculated for each HCF and LCF cycle, and is assumed to occur during the loading portion of each cycle. Thus, growth increments are determined sequentially for an LCF cycle, n HCF cycles, another LCF cycle, and so on. Under these assumptions, several failure sequences are possible. The particular sequence encountered is a function of four characteristic crack depths that, in turn, are a function of the material properties and LCF and HCF stress ranges. They are • a i – the crack depth at initiation, which is defined by experimental data • a crit – the crack depth at which K IC is exceeded at the crack tip (or a depth appropriate to the specimen size if a crit exceeds characteristic specimen dimensions), which is a function of HCF , LCF , and K IC • a gLCF – the crack depth beyond which the crack grows during LCF cycles, which is a function of LCF and K th (at R=0 for LCF cycles) and • a gHCF – the crack depth beyond which the crack grows during HCF cycles, which is a function of HCF and K th (at R for HCF cycles). There are 24 possible permutations of these four crack depths, any of which will produce one of seven failure sequences which are shown in Figure I.2. Path 1 is not likely if reasonable initiation data are available. Path 2 is unlikely for load levels of interest. Paths 4 and 7 produce HCF-only crack propagation, which is a possible failure mode if K th in HCF (at high R) is sufficiently small in comparison with K th in LCF (at R=0), 624 Appendix I a crit ≤ a i ? 1) Fast fracture immediately after initiation a i < a g, HCF and a i < a g, LCF ? a i ≥ a g, HCF and a i ≥ a g, LCF ? a crit ≥ a g, LCF and a i ≥ a g, HCF ? a crit ≥ a g, HCF and a i ≥ a g, LCF ? 2) No propagation after initiation. Infinite life 3) Initiation followed by crack growth in CCF to failure 4) Initiation followed by crack growth in HCF only to failure 5) Initiation followed by crack growth in LCF only to failure a crit > a g, HCF and a g, HCF ≥ a i and a i ≥ a g, LCF ? 6) Initiation followed by crack growth in LCF only followed by crack growth in CCF to failure 7) Initiation followed by crack growth in HCF only followed by crack growth in CCF to failure Yes Yes Yes Yes Yes Yes No No No No No No Figure I.2. Flow chart of possible failure sequences under CCF. and HCF is sufficiently large to grow the crack. While this situation depends on the assumed relation of K th with R, neither of these HCF-only crack propagation modes has been observed in any of the numerical calculations reported here. Paths 3, 5, and 6, then, are of the most practical interest. Model Calibration In order to calibrate and exercise the model, crack initiation and propagation data on surface-cracked round bars [9] are used. In this study, electropotential drop techniques were used to determine the number of cycles required to produce 50m deep surface Appendix I 625 cracks in mildly notched K T =2 Ti-6Al-4V round bars with an / microstructure. Total life was measured in both mildly notched and smooth bars. Chesnutt et al. [10] and Grover [11] reported total life measurements on Ti-6Al-4V materials with a similar microstructure at lower stress levels (and longer lives) at various values of K T . Using these data, total life estimates for long life tests at K T =2 were interpolated and are shown, along with the short life data by Guedou and Rongvaux [9], in Figure I.3. A multi-part power law fit to the initiation life curve was generated by connecting the ultimate stress at N =1 to the LCF data from Guedou and Rongvaux [9]. A power law fit to the experimental data was extrapolated to lower stress values. Two scenarios were considered for low stresses. In the first, alternating stress ranges below 300MPa R=0 cause no damage. Thus the life is infinite for lower stresses and the final portion of the S–N curve is a horizontal line. This stress range was chosen to agree approximately with the observed runout behavior in the long life tests [10, 11]. The contrasting scenario assumes that no endurance limit exists. Any alternating stress causes a finite amount of damage. In this scenario, the S–N curve extends downward continuously. Both cases are shown in Figure I.3. The corresponding total life curve was generated by adding the analytical estimate of the propagation life to the initiation life measurement and correlated well with the experimentally measured total life values shown in Figure I.3. Crack propagation data at R =005 and 0.85 [9] were used to determine parameters C, m, and d for the Paris–Walker relation in Equation (I.3). The values used here are C =5376 ×10 −12 , m=3409, and d=13. The values of K th for Ti-6A1-4V are taken 10 3 10 4 10 5 10 6 10 7 200 300 400 500 600 700 800 900 1000 N I K t = 2 (Guedou and Rongvaux, 1988) N T K t = 2 (Guedou and Rongvaux, 1988) N T K t = 1 (Chessnutt et al., 1978) N T K t = 3.4 (Chessnutt et al., 1978) N T K t = 2 (Interpolated) N I K t = 2 (Predicted) N T K t = 2 (Predicted) N Stress range (MPa) ENDURANCE LIMIT: 2 σ a = 300 MPa NO ENDURANCE LIMIT Figure I.3. Predicted and measured values of N i and total life N T as a function of applied stress range at R=0. . presented. NOMENCLATURE C Paris-Walker law constant CCF combined cycle fatigue d Paris-Walker law constant d damage parameter d i initiation phase damage parameter D diameter of rod HCF high cycle fatigue K t stress. determine atwhat point in theload block the crackinitiates and crack propagation begins. As a first approximation, it is assumed thatall initiation damagein each cycle occurs duringthe loading portion. materials and structures, this appendix deals with the effect of superimposed LCF on the HCF capability of the material and further and, further, addresses total life as a sum of initiation and