# Unified Approach to Fatigue Damage Evaluation

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A.K. Vasudevan

atigue damage limits the service life of many structural components that are subjected to variable stress. Fatigue is irreversible and usually unavoidable. High-priority objectives for preventing fatigue include containment of the fatigue damage, diagnostic tools to detect the beginnings of fatigue damage, monitoring the damage evolution during service, prognostics for reliable prediction of remnant service life, and fatigue-resistant materials by design. The key to success for mitigating life-cycle costs is the development of fundamental understanding of fatigue damage evolution in engineered materials. NRL has been working in recent years to replace fundamentally unsound empiricism extant in fatigue life prediction with a new systematic interpretative framework derived from basic principles.

**INTRODUCTION**

Fatigue damage can result from fluctuating stress. At low stress, material behaves elastically and reversibly. However, if stress is greater than the yield point, then irreversible plastic deformation occurs. If a single peak stress is high enough, fracture or other monotonic failures occur. If the stress is variable, such that there is a peak stress and an amplitude, then damage can accumulate with each cycle. After a sufficient number of cycles, failure can occur even if the peak stress is too low to cause fracture.

Fatigue is the principal cause of premature failure of engineering components. Sometimes these failures can be quite catastrophic, leading to severe property damage and loss of life. Fatigue cracks can nucleate at preexisting flaws in materials and tend to occur at stress concentrators in components such as notches or holes. Generally, fatigue damage is unavoidable. Once formed, the propagation rate of a crack is strongly influenced by load history and environmental factors. Thus, the accurate prediction of fatigue life is a complex problem. It is very difficult and rare to predict fatigue life accurately to within a factor of two. Indeed it is not only necessary to improve fatigue life prognostics to avoid catastrophic failures, but methods for optimizing inspection, maintenance, and parts replacement intervals would have a tremendous impact on service life costs and systems readiness. Clearly, new approaches to the problem of fatigue prognostics are needed.

**The Fatigue Crack Growth Threshold**

It was recognized in the 1960s that the fracture mechanics stress intensity
*K* is an appropriate parameter for describing fatigue phenomena. The actual stress near a crack tip is equal to stress intensity divided by the
distance from the tip and multiplied by some angular factors. Peak stress intensity *K _{max}* is the parameter that determines whether a component fractures. In the case of cyclically varying

*K*, it was found that many fatigue phenomena correlated very well with the stress intensity amplitude Δ

*K = K*. In particular, Δ

_{max}- K_{min}*K*must exceed a threshold value Δ

*K*for a fatigue crack to grow. However, it also was recognized that a second parameter was necessary to adequately describe empirical fatigue data to account for the variation of Δ

_{threshold}*K*with peak stress intensity. The customary and experimentally expedient choice for this second parameter has been the load ratio

*R = K*, which is the ratio of minimum to maximum stress intensity.

_{min}/K_{max}**Crack Closure**

In the customary approach, Δ*K* is considered the only fatigue driving force, so that the *R* effects are considered the results of other phenomenology. In particular, the concept of "crack closure" was proposed to account
for observations that Δ*K _{th}* is low at high values of

*R*and increases with decreasing values of

*R*(Fig. 1). The concept, known as "plasticity-induced crack closure," was advanced and generally accepted in the 1970s. In this concept, it is assumed that the plastic deformation at and ahead of the tip of the growing crack leaves a residual stretching of the material. Upon unloading, this results in a buildup of residual compressive stresses at the crack tip. Upon further unloading, the crack faces come into contact (as illustrated in Fig. 2) — in effect, shielding the crack tip from further changes in applied load. Thus if the specimen is completely unloaded (

*R = 0*), a higher

*K*must be applied to open the crack and overcome the residual compressive stresses so that the measured Δ

_{max}*K*necessary for crack advance is increased. At high enough values of

_{th}*R*, however, the minimum applied load is high enough to hold the crack open and is higher than the residual compressive stress, thus no change in Δ

*K*is observed.

_{th}Another effect with similar manifestation in the fatigue behavior is if there is an obstruction in the crack. Then, in effect, the crack tip is wedged open. The stress intensity actually acting on the crack tip is then somewhat
higher than the applied stress intensity. As a result, the effective stress intensity amplitude Δ*K _{effective}* is less than the applied Δ

*K*. Consequently, a higher Δ

*K*will be measured than if the crack faces did not interfere. Crack face interference can occur, for example, if an oxide layer forms on the crack surfaces, if debris is in the crack, or if slight misalignments of the crack faces cause surface roughness to interfere on opposing crack faces. Although the true effect of these crack obstruction processes is to wedge a crack open, and this is unrelated to plasticity-induced crack closure, the terminology "crack closure" is generally used to describe all of these effects.

_{threshold}The crack closure concept has been successful in empirical parameterization of a wide range of fatigue data. It has become the default interpretation of load ratio effects and is used in some form or another in fatigue life prediction models. However, well-known ambiguities in its experimental determination make it virtually impossible to predict. In fact, in an ASTM-sponsored round-robin test involving 10 laboratories, each using 3 different methods for the determination of crack closure for the same material, 30 different values spread within a factor of 3 were reported. The practical value of this is that it is little more than an adjustable fitting parameter.

**Fundamental Problems with Plasticity-Induced Crack Closure**

Plasticity-induced crack closure relies on the assumption that as a crack opens, the material at the crack tip stretches. As the crack subsequently closes, compressive stresses occur at the tip and premature crack face contact occurs in the wake of the crack. However, based on fundamental concepts of dislocation theory, NRL has shown that plasticity either at the crack tip or in the wake does not cause the crack to close. This is shown schematically in Fig. 3. In crystalline materials, plastic deformation occurs by the creation and movement of dislocations, which are each characterized by a Burger's vector related to the orientation and size of the dislocation. A Burger's vector of a dislocation is a conserved quantity. Thus, dislocations are created in pairs with opposite Burger's vector. Every dislocation emitted from the crack to form the plastic zone ahead of a crack tip is accompanied by a dislocation of opposite Burger's vector being absorbed into the crack, creating a ledge. Such a ledge always opens in the wake of the crack. Any closure contribution induced by the dislocation in the plastic zone cannot be greater than the opening contribution induced by the ledge dislocation. This statement is true for each dislocation pair emitted from the crack; therefore, it is true for any aggregate of many dislocations pairs — whatever their distribution.

A more intuitive way of conceptualizing this is to consider the conservation of mass, which is equivalent to conservation of Burger's vector for a dislocation. In addition, plasticity occurs with constant volume. Conservative plastic flow of mass from the crack is equivalent to removal of material from the crack and redistributing it into the plastic zone associated with the crack tip. The closure contribution from this redistributed matter cannot be greater than the opening of the crack due to its removal. By using continuum dislocation theory, NRL also has shown that other forms of crack closure are also smaller than generally thought and, in some cases, have negligible effects on crack tip driving force.

The most important consequence of NRL's dislocation analysis of crack closure is that if the various crack closure mechanisms are either nonexistent or small, then some other formulation is needed for the various phenomena that have hitherto been attributed to crack closure. This has led to a complete reformulation of how fatigue crack growth is interpreted and represented.

**The Unified Approach to Fatigue**

To achieve the reformulation of fatigue crack growth behavior, it is only necessary to adopt six seemingly commonsense principles:

1. Fatigue crack growth requires two driving forces, both
Δ*K* and *K _{max}*. Δ

*K*creates fatigue damage through the irreversible plasticity, but

*K*is necessary to open and increment the crack. The customary approach recognizes only Δ

_{max}*K*as a driving force, and uses the load ratio

*R*to describe how "open" the crack is.

2. Each driving force has a threshold criterion. For a fatigue crack to propagate, two conditions must exist: Δ*K* must be large enough to create cumulative damage, and *K _{max}* must be large enough to increment the crack. The corresponding thresholds are termed Δ

*K*and

_{th}*K*. In terms of dislocations, the equivalent

_{max,th}*K*must be large enough to create new dislocations, and Δ

_{max}*K*must be large enough to overcome friction in the reverse dislocation flow. The customary approach assumes that only Δ

*K*is important.

3. The correct representation of fatigue is as a three-dimensional map of
*da/dn*, Δ*K*, and *K _{max}* (Fig. 4). The preferred two-dimensional representation of fatigue data should be constant

*da/dn*contours projected onto the Δ

*K*vs

*K*plane. Such a representation reveals clearly the roles of Δ

_{max}*K*and

_{th}*K*for threshold behavior as defining a roughly L-shaped curve. In fact, for

_{max,th}*da/dn*greater than zero, similar curves can be defined, and the characteristic parameters Δ

*K**and

*K**are defined for constant

_{max}*da/dn*contours. In addition, the shapes of the curves fall into a limited number of classifications associated with the type of fatigue mechanism. The mechanism also may change with

*da/dn*. Customary representation of fatigue data is plots of da/dn vs Δ

*K*at fixed

*R*and plots of Δ

*K*vs

_{th}*R*. Typically, however, only one or two values of

*R*are used so that the customary description necessarily is incomplete.

4. Fatigue crack growth is driven by the net local stress intensity acting at the crack tip. This idea is more often stated as the principle of similitude, i.e., that for a given stress intensity and given fatigue mechanism, the crack growth rate will be the same, regardless of specimen geometry. This also is equivalent to saying that, for a given fatigue mechanism, if one knows the stress intensity then one can deterministically predict the crack growth rate.

5. Internal stresses must be included in the fatigue driving forces. This is a consequence of the similitude concept. An immediate consequence of accounting for superimposed internal stresses is that if the internal stress is not a function of applied stress, then at first order, internal stress contributes to
*K _{max}* and

*K*equally. Thus, internal stress produces effects on

_{min}*K*but not on Δ

_{max}*K*. Customary interpretation of internal stress effects is that they can be treated as a form of crack closure, such that the effect is to reduce Δ

*K*.

_{effective}6. Environment and fatigue mechanism affect the characteristic parameters
Δ*K** and *K* _{max}* differently. The simplest way to think about this is to consider that environmental influences such as hydrogen, which can diffuse into the material, can directly affect Δ

*K**, but environmental effects acting primarily at the crack tip surfaces, such as oxidation, affect primarily

*K**. The preferred representation of environmental effects is parametric plots of Δ

_{max}*K**vs

*K**for the full range of

_{max}*da/dn*. Ideal behavior in the absence of any environmental effect is manifested as the line Δ

*K* = K**. The curves on these Δ

_{max}*K* - K**plots, which we usually refer to as "trajectories," reveal certain characteristic behaviors identifiable with the environmental mechanism when they deviate from ideal behavior. Customary treatment, on the other hand, is that environmental effects act as perturbations to

_{max}*da/dn*.

We refer to this as a Unified Approach^{1} because these six elements provide a unified framework for the representation of all fatigue crack growth phenomena. We examine three key topical areas to demonstrate how
the Unified Approach has enabled new understanding and interpretation of certain common fatigue phenomena.

**Short Crack Phenomena**

Understanding of short crack growth behavior is essential for fatigue life prognostics because most of the life of a fatigue crack, before detection or failure, is spent as a short crack. Fatigue cracks are considered "short" if they are small compared to the important characteristic dimensions in the region in which they are growing (for example, grain size, plastic zone size, or specimen dimensions). The distinction between "short" cracks and "long" cracks has been made because short cracks exhibit a wide range of complex behaviors. These complex behaviors include growth at stress intensities below the long crack growth thresholds, arrest at stress intensities above the threshold, and anomalous acceleration and deceleration of the crack growth rates. Figure 5(a) shows some examples.

Conventional thinking based on crack closure arguments is that nascent short cracks have not yet developed enough plasticity in their wake to support crack closure. Long cracks in this picture, on the other hand, have developed a crack wake plastic zone and exhibit steady-state crack closure. Thus, the underlying concept is that short cracks represent intrinsic behavior while long cracks are dominated by crack closure.

The Unified Approach interprets this completely differently.^{2} Long crack behavior is considered the fundamental behavior because the long crack thresholds are intrinsic to the material and are independent of crack
size. Short crack behavior is simply the result of internal stress variations on the short crack length scale since short cracks invariably form at notches, defects, and other stress concentrations. Internal stresses also arise
from thermomechanical stress, phase transformations, residual welding stresses, forging, rolling, machining, and surface treatments, and they can be induced by transient loads in the cyclic loading history. The magnitude of internal stresses may be tensile or compressive, and they decrease rapidly as a short crack propagates away from microscopic stress concentrations. Since crack growth is due to the superimposed effects of internal and applied stresses, deceleration and acceleration can occur if the net stress exhibits a minimum or maximum as a function of distance along the crack path. This is illustrated in Figure 5(b).

It is worth repeating that the primary effect of internal stresses is on the
*K _{max}*, not Δ

*K*. Hence, short-crack growth behavior considered anomalous in the Δ

*K*crack-closure interpretation is actually a manifestation of internal stress effects on

_{effective}*K*and the principle of similitude. Practical progress on fatigue prognostics now focuses on tractable problems of modeling, measuring, and controlling internal stress distributions.

_{max}**Overload Effects**

The importance of proper accounting of internal stress also is crucial to the understanding of transient and spectral load effects. Real load histories of engineering components are, of course, never perfect sine waves. This is most readily apparent for aircraft structures. These structures experience in-flight vibrations interspersed with occasional huge overloads or underloads that correspond to extreme maneuvering, turbulence, takeoffs, and landings. However, there are other examples. Ship hull structures undergo spectrum loading due to variable sea conditions. Motor, generator, turbine rotors, and gears experience cyclic stresses with occasional large transient stresses. Propulsion shafting experiences low-amplitude cyclic bending and torsional and longitudinal fatigue loading with occasional shocks as ships pass through variations in water density.

We have shown that overload effects are quite easily understood in terms of induced compressive residual stresses and the resultant effect on
*K _{max}*. We further have shown that, based on dislocation analysis, a crack tip shielding effect due to dislocations occurs some distance ahead of the crack tip. This accounts for delayed crack growth retardation effects that have heretofore been considered puzzling in the crack closure context. In
the Unified Approach, overload retardation effects can be most easily understood without crack closure.

**Environmental Effects**

The effects of environment on fatigue are unavoidable and almost always detrimental. This is particularly important for the Navy because corrosion due to seawater and humid salt-fog exposure is particularly damaging
to many structural steels and aluminum alloys. The environmental effects on fatigue are not only cyclically stress-dependent through the process known as corrosion-fatigue but are also time-dependent and directly
*K _{max}*-dependent via the stress-corrosion-cracking mechanism. Environmental effects on fatigue cannot be properly accounted for unless

*K*is explicitly recognized as a driving force, hence the Unified Approach provides a natural and superior interpretative framework.

_{max}As an interesting example, Fig. 6(a) shows constant
*da/dn* contours for cast iron in air. Except for the threshold curve, the constant *da/dn* contours are all composed of two L-shaped curves. One needs some very convoluted mechanisms to explain this in terms of crack closure. However, if we take the *K* _{max}* and ?

*K**values for each of the L-shaped curves and plot them on the ?

*K**vs

*K**trajectory map explained earlier and shown in Fig. 6(b), the true nature of this complex behavior is revealed. Two competing fatigue mechanisms are operating. One mechanism, corresponding to the trajectory that approaches the ideal fatigue line, is characteristic of pure environmentally assisted fatigue. The mechanism is that the environmental species (hydrogen) diffuses into the material at a certain rate, embrittling the material for a distance ahead of the crack tip and reducing the ?

_{max}*K**. As the crack growth rate increases, the crack tip can outrun the diffusing hydrogen and propagate through mostly unaffected material. The other competing mechanism is the trajectory that diverges from the ideal line with increasing

*K**. This is characteristic of stress-dependent monotonic cracking modes, in particular, stress-corrosion cracking. A more detailed analysis, which cannot be pursued here due to space limitations, shows that the competing mechanisms are the result of competing intergranular vs transgranular mechanisms.

_{max}**SUMMARY**

The Unified Approach to fatigue crack growth developed at NRL is a departure from the conventional fatigue interpretative framework in that the relative importance of crack closure is reduced and the role of internal stresses is more important. The examples described here illustrate the power of the Unified Approach to fatigue crack growth for providing clear physical insight into the fundamental mechanisms by using a correct and consistent interpretative framework.

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