Figure "Material changes during crack development" (Ref. 12.6.3.2.3-7): Dynamic fatigue in the crack initiation phase of metals can be plausibly explained with the aid of the following model (Ref. 12.6.3.2-3). Purely elastic strain of the metal lattice should be tolerated indefinitely. This means that the fatigue strength should be found in this range. However, the actual fatigue strength is considerably lower. Crystals have different elastic properties, depending on their orientation (E moduli, also see Fig. "Single crystal material for mistuning"). A material consists of many crystals that are arranged with different lattice orientations relative to one another. If an external load acts on this structure, neighboring crystals (especially grain pores) hinder one another`s elastic strain. This restriction leads to high stress levels, corresponding to the grain orientation (micro-stress). The result is local, plastically deformed zones that are strain-controlled by the surrounding grains (top left diagram). This effect appears even at relatively low macro-stresses in a hysteresis of the cyclical stress-strain curve and, therefore, at increased damping. Depending on the material, the deformation resistance increases during plastic deformations, i.e. the material becomes harder. This is not only true for tensile tests with large plastic strain and LCF loads, but also for HCF loads, in this case at grain pores in the micro-range and at small flaws in the notch effect range. Under vibrating loads, the hardening process occurs over a long time and a correspondingly high number of deformation cycles. Hardening and micro-stress reinforce one another in a self-energizing process called reciprocal hardening (also see Ill. 12.6.1-10). This process ends only when the entire affected zone has hardened to the point that only elastic strain occurs (stationary state). If the dynamic loads are sufficiently high to exceed the strength of the most highly stressed hardened zone, electron microscope-detectable cracks form and/or the structure is damaged accordingly (second diagram from left). Crack initiation leads to local resiliency and, since this process is strain-controlled, stress reduction in these areas. The notch effect of the crack tip acts against this self-repair process.
If the macro-stresses are below the fatigue strength, the relaxing effect will be dominant and there will be no growth in micro-cracks above the perceptibility limit (third diagram from left).
If, however, the macro-stress is greater than the fatigue strength, the notch effect of the macro-cracks will be dominant and result in crack growth and macroscopic crack initiation in the fatigue fracture (right diagram). The limiting curves of the damage/crack sizes also reveal the limits within which possible non-destructive testing methods can be used.
At the fatigue strength limit, reciprocal hardening and reciprocal disruption cancel each other out.
The top graph shows the ranges of the various damage stages in a way similar to the Woehler diagram, but with standardized dynamic loads (Ref. 12.6.3.2-4).

The depicted fatigue model permits the interpretation of further phenomena that can be of considerable practical importance:
“Strength training” (conditioning) for materials (Fig. "Characterizing blade fatigue strength with 'a x f' value", Ref. 12.6.3.2-3): In an incremental test, the specimen is first subjected to low dynamic stress and, after running through 10⁷ load changes, the stress levels are incrementally increased several times until a dynamic fatigue fracture occurs. Specimens in this type of test, which have a sufficiently low initial stress stage, have considerably higher dynamic strength than samples that are immediately stressed above the fatigue strength. This can be explained by the fact that plastic strain and micro-cracking cause relaxation at structural weak points, mitigating their effect. Therefore, design data that are determined in this type of test can be higher than the characteristic values that are actually attainable in the engine parts. This is a dangerous situation because it increases the risk of part failure due to HCF.

Damage due to dynamic loads: This effect occurs when the dynamic loads are temporarily high enough to cause growth-capable micro-cracks to form below the fatigue strength. This load limit can be plotted in the Woehler diagram as a damage line.
These damages can occur, for example, due to large temporary dynamic loads on parts. Typical examples are compressor blade rubbing (Volume 2, Ill. 7.1.3-4) and loads on the blading during surges (Figs. "Estimating engine loads during surges" and "Vibrations by shaft caused flow disturbance"). Therefore, it must also be possible to use sufficient statistical numbers of damaged parts to verify this type of damage through dynamic tests below the fatigue strength. Experience has shown that this type of verification has often yielded usable results.

Notes: Naturally, the described model does not satisfactorily explain the dynamic fatigue processes for cast alloys with large grains (in the centimeter range), directionally-solidified alloys, or single-crystals. In these cases, cracks usually initiate at surface weak points (e.g. damaged grain boundaries) and internal flaws (e.g. pores, Fig. "Influences on thermal fatigue").
The bottom diagram shows the phases of macro-crack initiation, fatigue cracking to crack instability, and the eventual fracture of the part (see Chapter 12.2). The crack size is plotted over the life span, i.e. the number of load cycles.

Figure "Scatter of dynamic strength" (Ref. 12.6.3.2-9): The example of a high-strength Al alloy is used to show the influence of the size of dynamic loads on the life span statistical spread. The steeper the lines for the various loads, the smaller the scatter becomes (top diagram). One can easily recognize that the scattering is minimal at high dynamic loads (in the LCF range, bottom diagram). Loads in the range of the fatigue strength cause the scattering to increase considerably. The same is true of the notch effect, which decreases with increasing loads, and is therefore smaller in the LCF range than in the HCF range.
The following important conclusion must be drawn from this behavior: in order to investigate the influence of small flaws and weak points on dynamic strength, tests must be conducted in the HCF range. Accelerated tests with high loads are less sensitive to notches, and are therefore not on the safe side.

Figure "Dynamic strength at temperature and notches" (Ref. 12.6.3.2-4): In this dynamic strength diagram according to Haigh, the tolerable stress amplitude is plotted over the mean stress for a life span of 100 hours (2.16×10⁷ load cycles) for a Co-based alloy (for an explanation of the terms see Fig. "Terms and examples of dynamic loads"). The time limit is necessary in order to take into account the creep of the material during the dynamic loading. The dynamic strength of the material is naturally temperature-dependent. However, it is not always directly dependent on the thermal strength. At high temperatures the tolerable notch stress (multiply “A1” by 3.4) is considerably higher than the respective nominal stress in smooth samples (“A2”). This means that the notch effect weakens at high temperatures. This is an important realization for the acceptance of larger flaw sizes at higher temperatures than at lower temperatures, at least as far as the dynamic loads are concerned. One example is pores and small cracks in turbine blade leaves. The mean stress-dependency of the notched specimen (“B1”) is lower (horizontal curve progress) than that of the smooth specimen (“B2”). These effects should be related to creep deformations in the notch and the increased bracing effect these create. Less highly-stressed neighboring areas de-stress a notch through plastic deformation in the notch base.
It is also interesting that smooth specimens have load zones (e.g. grey zone, “C”), in which the tolerable strain amplitude increases as the mean stress increases. This effect should also be attributed to creep deformations.

Figure "Stage 1 at dynamic fatigue cracks": Austenitic alloys exhibit typical transcrystalline (bottom right diagram) cleavage cracks in the case of dynamic overstress, especially in the HCF range (Fig. "LCF as lifespan determining"). The selected example is an HCF fracture in an integral cast turbine disk (right diagram). These fracture structures are especially pronounced in large grains of the type common in Ni-based casts (especially in directionally-solidified or single-crystal materials). With minimal oxidation (fresh fracture surfaces), the crack facets are obvious due to their mirror-smooth surfaces (top right diagram). These fracture surfaces rarely reveal microscopically-analyzable crack growth lines, and must be distinguished from similar fracture characteristics in forced fractures that were created by impact stress at very high temperatures (Fig. "Indications of turbine blade overheating").

Figure "Background of material data for design": In the case of damage or problems in engine parts, one should always consider the possible damage-promoting influence of material design data that are not sufficiently relevant to the part. For example, in the case of turbine disks, the LCF-stressed part volume in the hub region is considerably larger than that of typical LCF specimens. In cast parts, larger cross-sections mean coarser grains and more missing residual melt, due to the slower cooling rate. This increases the likelihood of larger cavities being created during solidification. This tends to lower dynamic strength relative to thinner cross-sections of separately cast specimens.
In forged materials, thick cross-sections lead to poorly formed zones and problems with structural development in inner areas due to slower temperature changes during the heat treatment. For example, large material-specific weak points (segregations) with unfavorable orientations relative to the main stress can remain in the part. As a rule, the probability of strength-reducing flaws and, therefore, poor dynamic strength, increases with larger stressed volumes and/or surface areas.
Even if the specimens are separately cast (forged), they should be from the same batch as the part being analyzed. For example, the specimens and turbine blades should be from the same cluster. Ideally, specimens are manufactured together with the part (e.g. in the case of coatings or shot peening).
The problem of non-representative structures and flaws can be reduced by taking the specimen from the design-relevant part zone (integral specimen).
Specimen treatment is a question of design philosophy (Fig. "Dynamic strength differing at samples and parts"). Either the specimen surface sufficiently represents the critical part zone, or one assumes an “ideal” specimen surface (e.g. polished, without significant hardening or residual stresses). The resulting values, i.e. the determined life span must then be corrected to account for the differences between the specimen and the actual part surface. Minimum values from “ideal” specimens must contain corrections that correspond to the design.
The selection of the specified specimen form also influences the result. Differences in the length/diameter ratio in tensile tests lead to different relative fracture strains (A = D l/l₀), since the large plastic component of the fracture strain “Dl” only changes insignificantly with the test length “l₀”.
Changing the load arrangement and, therefore, also the specimen shape, leads to even greater uncertainties regarding the relevance of the specimen values to part behavior. Large deviations in testing volume and/or testing surface in bending specimens relative to tensile specimens are typical (Fig. "Material data of brittle materials (ceramics)"). However, uncertainties can be minimized through proper specimen selection, such as specimens with stressed volumes similar to those of the engine part (e.g. suitably shaped testing disks for powder-metallurgical materials).

Figure "Dynamic strength differing at samples and parts": The part-relevant determination of dynamic strength values and/or the transfer of specimen data to the part (see Ills. 12.6.2-23 and 12.6.3.4-19) can be difficult. For example, with compressor rotor blades, the question is which dynamic strength to determine under which influences. It is also an issue of the design philosophy, whether the specimens used for determining material data are selected with as realistic surfaces as possible with regard to the specifications for part design, or whether the selected specimens have comparable standard surfaces with the least possible manufacturing influences (polished surface).

Examples of influences on the dynamic strength of parts:

• part-typical machining surface (e.g. grooves, hardening, residual stress)
• operating influences (e.g. fretting, oxidation)
• structural characteristics of critical zones (e.g. forging, casting, heat-treatment)
• thickness of cross-sections.

If one assumes that the most universally applicable values should be determined, then one must understand possible deviations in the parts.

A special role is played by machining-dependent differences between the specimen surface and the part surface. It is often not possible to use comparable machining procedures on the typical thin round specimens as on the engine part. Typical differences include:

• relatively large elastic deflection of the thin specimen under the machining forces. This has effects such as vibration of the specimen during machining. This can lead to damaging chattermarks, for example.
• no comparably high machining speed in the sample
• rapid heating of the small specimen volume
• considerable bending of the machining surface of the cylindrical specimen shaft.

In the case of flexural modes, the critical zone in compressor blades is in the transition radius to the root platform (top right diagram). In the exceptionally thin edges, especially on the blades of the rear stages, the grain size can even be as large as the cross-section thickness. The vibration-stressed volume and surface are small in comparison to the specimens (top left diagram). The opposite is true of the grain size.
Stress gradients vary depending on the cross-section, even at the same stress amplitudes at the surface (bottom left diagram). This means that surface characteristics such as weak points and residual stresses have a cross-section specific influence on the dynamic strength.
In the bottom right table, the influences are evaluated on the basis of the differences between the specimen and engine part.

Figure "Usability of dynamic strength by surface quality" (Ref. 12.6.3.2-11): This diagram, which is based on Nall and Lipson, is intended to explain the influence of machining on the expected dynamic strengths (flexing cycles). Depending on the tensile strength (low-alloy steel), the influence of machining processes is both absolutely and relatively serious and varying. If, in this example, the polished surface has the highest dynamic strength (a hardened surface was evidently not tested here), then ground surfaces are still relatively good. The grinding process was apparently conducted without inducing tensile stresses or unfavorable structures (loss of hardness, etc.). Machined specimens are considerably worse (evidently without hardening and/or induction of compressive stress). It must be assumed that hot rolled or forged surfaces have larger flaws (e.g. loss of hardness due to decarbonization, scratches/nicks), which considerably reduce dynamic strength. For the part-relevance of the specimens, the comparability of surface conditions of forged or cast surfaces is especially important.

Figure "Dynamic strength and surface treatment " (Ref.12.6.3.2-10): In order for dynamic fatigue fractures to occur, a dynamic load greater than the dynamic strength is necessary. The lower the dynamic strength, the lower the levels of dynamic loads that are dangerous.
In the case of high-frequency vibrations, these are primarily flexural modes. For this reason, the strength in the more highly stressed part surface is especially important. The table shows typical machining influences on flex cycle strength with the aid of two representative forged materials, one each from titanium alloys and nickel alloys.
In both materials, especially at room temperature, one can recognize the serious damaging influence of grinding. This is especially true for titanium alloys and lengthwise machining. The dynamic strength of an Ni alloy is considerably higher at the testing temperature of 540°C than it is at room temperature. This effect could indicate a breakdown of damaging tension residual stresses (creep) in the ground surface.
This shows the degree of special care that must be given to smoothing out FOD damage with a grinding process on an installed part (e.g. fading out compressor blades). A reworking process that increases hardness is definitely recommended in this case. Because this is not possible with normal blasting processes, this may be the opportunity to use laser peening. Unlike shot peening, laser peening does not use particles.
With titanium alloys, electrochemical treatments have advantages even in comparison with mechanically milled surfaces. This can be explained by the low tendency of titanium alloys to peen harden. This means that the beneficial influences of machining (hardening, compressive stresses) are of only limited use.
Almost independent of the intensity of the machining process, non-reworked, electrical discharge machined (EDM) surfaces of the nickel alloy have the expected low dynamic strength levels (brittle recast layer with high tension residual stress), but these are still greater than those of the normal, roughly worked ground surfaces. A surprising result is the very high dynamic strength at room temperature after the EDM surfaces were shot peened. These properties decrease considerably at 540 °C, which is most likely related to an increase in the shot peening effect due to creep (relaxation).

Figure "Characterizing blade fatigue strength with 'a x f' value" (Ref. 12.6.3.2-13): It can be shown that, for a unilaterally fastened prismatic rod, in the case of fundamental flexural modes, the highest stress s_bmax at the point of fastening will be proportional to the product of the amplitude “a” and the eigenfrequency “f”. In practice, dynamic strength is generally given in the form of an “axf value” (top diagram). The fatigue strength is reached at the amplitude “a”. Unilaterally fastened blades without supporting shrouds are sufficiently similar to the bending rod. Therefore, under these conditions, the axf value is a material characteristic value for the dynamic strength of blades. Experience in practical operation has demonstrated this.
The fatigue strength of blades can deviate considerably from that of typical round specimens of the same material. This behavior can be explained by production-dependent characteristics in the surface and structure. For this reason, the fatigue strength must be determined in original parts for reasons of quality assurance. For blade design, OEMs must provide experiential data for the necessary axf values of blades in order to ensure sufficient operating safety. Quality assurance is also done with dynamic tests on blades that have been statistically chosen from production batches, and a minimum axf value must be attained. The top right diagram shows axf values for material families. One can see that the value for titanium alloy blades is very high. This underlines the advantages of using titanium alloys in compressor blades. Determining the axf values often happens in incremental tests. At first, the stress amplitudes are considerably below the normally expected fatigue strength, and a run-through specimen (>10⁷ load cycles without perceivable damage signs) is subjected to increasing strain amplitudes. This process is not without problems, as the bottom diagrams show:

In the Woehler diagram, the conditioning effect (“strength training”) and the damage effect are both shown. See Fig. "Scatter of dynamic strength" for an explanation of the mechanisms at the structural level (Ref. 12.6.3.2-3).
Conditioning (middle diagram): In multiple-stage tests that begin with loads below the fatigue strength and run through several million load cycles, it is observed that the specimens also tolerate dynamic loads above the fatigue strength for more than 10⁷ load cycles. This effect is especially pronounced in medium-strength steels, and can increase the fatigue limit by up to 30%.
Damage (bottom diagram): This effect occurs if the first dynamic stress stages in a multiple-stage test are set above the dynamic strength, and the test is stopped before crack initiation, i.e. after only a few load cycles. In a subsequent test with loads below the fatigue strength, the specimen may fail. Therefore, the specimen was damaged by the higher load stages. The damage depends on the size and number of the previous load cycles (see damage accumulation, Fig. "Dynamic fatigue life span estimations (Miner rule)"). This effect occurs above the damage line, which means that loads between the damage line and Woehler curve are damaging. Below the damage line, the disruption in the structure comes to a standstill and has almost no effect (latent damage). Above the damage line, the disruption continues. The grey area marks the disruption zone.

Figure "Dynamic fatigue life span estimations (Miner rule)" (Ref. 12.6.3.2-12): This diagram shows the simplest form of estimating the damage (life span reduction) due to dynamic fatigue. This procedure is based on the hypothesis of linear damage accumulation put forth by Palmgren and Miner. This Miner Rule is widely used in the industry. It is assumed that a load cycle causes damage that can be expressed as 1/N_i . “N_i” is the fracture load cycle in a single-stage test for the given levels of stress amplitude. The stress amplitudes of a dynamic operating load (complex of loads acting on a component) can be combined into load stages (top left diagram). A fracture load cycle N_i for the given level of stress amplitude of a class can then be determined in the Woehler diagram of the pertinent material (top right diagram). Here, an example is given for i=4. The total damage or life span reduction for a complex of loads adds up to s =S n_i /N_i for each load class with “n_i” load applications per 10⁶ load cycles (load horizon). According to the definition, the dynamic crack initiation occurs at a damage total of “s” = 1.0. The tolerable load cycle number “N” in the depicted case is N=10⁶/s. The stress amplitudes below the fatigue strength can be incorporated through a fictional lengthening of the Woehler line before it becomes horizontal.
The bottom diagram uses the example of a high-strength Al alloy to show the limits of the Miner Rule in an impressive manner. Although all depicted load sequences had the same complex of loads, the dynamic fatigue damage, i.e. the life span reduction, of the operating loads was the greatest. Even apparently minor changes to systematic load sequences have very different damaging effects (also see the examples in Figs. "Unpleasant surprises by 'standard' materials" and "LCF fracture of a fan disk")