## Abstract

Much modern engineering design work uses *S*–*N* curves and empirical applications thereof. In industry, currently popular methods for predicting fatigue life under complex loading use ad hoc cycle counting algorithms along with Miner's rule, in spite of its known weaknesses. Many ad hoc alternatives to Miner's rule have been proposed, each with limited experimental justification. Of these, Manson's double linear damage rule (DLDR) is widely considered to be good. In this paper, we bring a new perspective to empirical, as opposed to mechanistic, fatigue damage evolution models. It is first assumed, with reasonable justification, that there is a scalar, abstract, damage variable *ϕ*, whose evolution under cyclic loading satisfies , where *a* and *m* are unknown functions of load parameters. One main contribution of the paper lies in deducing what the functions *a* and *m* must be in order to obtain consistency with fatigue data in handbooks. A small correction to this basic power law model is then developed. The final explicit model initially has 10 unknown fitted parameters, but these are brought down to three unknowns; the accompanying discussion is the other main contribution of the paper. Finally, comparison with Manson's and other data suggests that, with two fitted parameters, our model works as well as the DLDR and much better than Miner's rule. For other parameter choices, our model reduces to Miner's rule. We conclude with speculation about ways in which the model might be extended beyond the scope of the DLDR.

## 1. Introduction

Fatigue life prediction is one of the big remaining problems of mechanical engineering design. Attacks on this problem include continuum damage theories, detailed studies of microstructures and cracks and, at the empirical end of the spectrum, cycle counting followed by the use of *S*–*N* curves and Miner's rule. Continuum damage theories involve sophisticated ideas that industry has not yet widely adopted in the face of practical difficulties. Studies of microstructure contribute more to fundamental understanding than to simple design formulae for daily use. Fracture mechanics is perhaps the most useful tool for predicting the remaining life of expensive structures with visible cracks where the attendant cost of monitoring and analysis is justified. The cycle counting approach, essentially the industrial state of the art for routine component design, is a mix of ad hoc ideas that serve in the absence of more attractive alternatives.

Fatigue calculations typically involve one of two situations. In one situation, we might have a visible crack in an expensive aeroplane and wish to predict remaining safe flying time. We may then spend significant resources studying the mechanics of *that* crack in that aeroplane. In contrast, an engineer designing a new component might need an estimate of its service life. There is, as yet, no specific observable microstructure, plastic zone, crack or actual component. In such cases, simple empirical analysis such as cycle counting (see ASTM Standard E 1049-85 2005) and *S*–*N* curves must usually suffice; it is here, in empirical fatigue modelling, that we aim to contribute.

Prior attempts to construct simple formulae for cumulative fatigue damage were largely *post facto* descriptions of data. Given the data on the failure of a certain specimen under multiple blocks of cyclic loading at various load levels, empirical formulae were sought to capture the failure data. Unfortunately, several disquietingly dissimilar formulae fit their individual datasets reasonably well.

In contrast, our approach makes *a priori* assumptions about the overall form of a damage *evolution* law and not about outcomes. This form is guided by its simplicity and ubiquitousness and not by multi-level failure data. The detailed mathematical form of our model is then *deduced*. The final good match obtained with experiment is thus more reassuring than an equally good match from a *post facto*, multi-level data-motivated model. We view this as one of the main advantages of our model over other damage accumulation rules; another advantage of our approach, as discussed later, is its greater scope for generalization.

In the following, we will first survey prior attempts to describe cumulative fatigue damage under variable cyclic loading. We will then present our new theory and see how it performs for a well-known dataset from Manson *et al*. (1967). Some other data are discussed in the electronic supplementary material.

We mention that our present theory builds on the earlier work by Cusumano & Chatterjee (2000), but takes the analysis significantly forward. There are now two load parameters in place of one, allowing non-zero mean loads (a non-trivial advantage); there is a small correction introduced to their basic power law model, which considerably improves the fit with experimental data, and there is clearer understanding of parameter fitting issues. Though Cusumano & Chatterjee (2000) also included fatigue limits in an ad hoc fashion, however, here we have focused on materials without a fatigue limit for clarity, brevity and simplicity.

Although our eventual goal is broadband (irregular) loading, here we only consider blocks of cyclic loading, not necessarily fully reversed. How our results here might be used for complex aperiodic loading or extended to materials with a fatigue limit, while being important questions, are left for future work.

We conclude this section with an outline of the flow of the paper. It is first assumed that there is a scalar, abstract, damage variable *ϕ*, whose evolution under cyclic loading satisfieswhere *a* and *m* are unknown functions of load parameters. The above form is simple, widely used for many different problems and (it turns out) gives good results for the problem studied here. The primary contribution of the paper lies in deducing what the functions *a* and *m* must be in order to obtain consistency with fatigue data in handbooks. A small correction to this basic power law model is then developed. The final explicit model initially has 10 unknown fitted parameters, but these are brought down to three unknowns; the accompanying discussion is another contribution of this paper. Finally, comparison with published data shows that the model is good.

## 2. Literature review

Much work on cumulative fatigue damage has been carried out since the middle of the previous century. Many cumulative fatigue damage theories have been proposed, most of them being incomplete to various degrees. Here, we mention relevant aspects of these theories. A more general survey may be found in Fatemi & Yang (1998).

The first, and industrially still the most widely used, cumulative fatigue damage theory is Miner's rule (1945), which is a linear life-fraction-summation rule that fails to incorporate experimentally well-documented load sequence effects (high load following low gives relatively longer life than low load following high).

Several prior papers consider ‘damage curves’ (damage versus life-fraction) parameterized by load (see Richart & Newmark (1948), and also Marco & Starkey (1954) who proposed a power law for damage accumulation, with a load-dependent exponent). These damage curves are, in a sense, integrated versions of evolution equations. However, these approaches are more limited than ours in that they do not really consider damage as a state variable with an initial condition; in that they retain a load-dependent time-scaling implied in any use of life-fractions; and in that their fit to the exponent in their power law is directly experimental, while we start one level lower with simple assumptions about the form of the evolution law, and then use general consistency conditions to derive the mathematical form of the load dependence of various quantities. However, our present approach may be viewed as a more general (e.g. we allow non-zero mean stresses), more powerful (with scope for extension to complex waveforms) and more promising (we have fewer fitted parameters, yet do well on experimental data) extension of the lines of thought of Marco & Starkey (1954).

Fatemi & Yang (1998) also discuss several papers that propose other load-dependent damage modelling approaches with varying degrees of similarity with Marco & Starkey (1954), but with various added observable data considerations such as reduction of endurance limits or rotation of *S*–*N* curves about this or that specified point. These papers are not discussed here in detail because the divergence from our approach is greater. We also mention that many papers have attempted to develop fatigue damage formulae related to other mechanical variables, such as crack lengths (from Shanley (1952) to Vasek & Polak (1991)); crack populations (Ma & Laird 1989); combinations of endurance limits and crack growth models (Bui-Quoc *et al*. 1971; Bui-Quoc 1981, 1982); energy or work associated with deformation (Kujawski & Ellyin 1984; Golos & Ellyin 1988); increases in surface hardness (Pavlou 2002); and a variety of other mechanical considerations, a broad sample of which may be found in Fatemi & Yang (1998).

Yet other empirical fatigue damage theories were proposed based on the observation that fatigue fracture involves a crack initiation stage and a crack propagation stage. Grover (1960) divided the damage growth into two such phases, and then proposed *post facto* (after examining multi-level load data) damage formulae. The idea of two such phases was adopted and adapted by Manson *et al*. (1967) and Manson & Halford (1981, 1986) into the well-known double linear damage rule (DLDR). The DLDR has been viewed as a credible alternative to Miner's rule. In this paper, we will constructively develop an empirical fatigue damage evolution model that also, qualitatively, has two phases (not explicitly related to crack initiation and growth); that, unlike Manson's assumptions regarding simple geometrical aspects of failure data plots, is based on simple assumptions at the level of damage evolution; that matches Manson's own data as well as his own theory does; that, unlike Manson's, already incorporates non-zero mean stresses; and that we think has potential for extension to complex loading waveforms.

Before we end this section, we mention the approach of continuum damage mechanics (Chaboche 1974), well developed in a true continuum setting, but also simplified to a scalar damage case (or a lumped damage variable) by some authors. These scalar treatments (e.g. Dattoma *et al*. 2006) also use damage evolution equations as we do. However, their mathematical form (see eqn (4) of Dattoma *et al*. 2006), as well as some modelling aspects, differs from ours, as will be clear below.

## 3. Initial assumptions

In industrial fatigue life estimation, when the load is biaxial or triaxial, it is commonly reduced to an effective scalar load (e.g. the largest principal stress). Often, fatigue failure starts with a surface crack, where the stress is biaxial at worst and may be almost uniaxial in many cases. Finally, most available fatigue data in handbooks are for uniaxially loaded specimens. With these thoughts, we assume uniaxial loading for simplicity.

What is damage? In continuum damage theories, damage might be something that affects effective stiffness moduli. In linear elastic analysis of a cracked specimen, however, damage might be a crack length. In elastoplastic analysis of the same specimen, damage might involve some combination (perhaps not explicitly laid down) of crack length and plastic zone size. Since there is no useful unique clarity on what damage really is, we deliberately view damage as an abstract variable. Damage might be, say, a crack length if we wish, but it might also be some unspecified function of crack length and the plastic zone ahead of the crack or, even, something quite different. Abstraction here will help us below.

We assume there is a single scalar damage variable, for several reasons. First, cycle counting and Miner's rule also, in effect, use a scalar damage variable. Second, near failure, components often have a single dominant crack; so late-stage damage can be modelled with a scalar damage variable (the crack length). Third, in our abstract approach, the damage ‘variable’ could, in principle, start with (say) a distribution of microscopic flaws and merge mathematically, as damage progresses, into a crack length, i.e. many complex phenomena could be described using a scalar damage variable. Finally, just scalar damage will, in fact, give a model that matches available data well.

## 4. Further assumptions

We assume that damage accumulates slowly (e.g. a crack grows over thousands of load cycles), and that it is possible, in principle, to write differential equations governing its evolution rate. We can then imagine a situation where the rapid temporal variation of loads is averaged out of these equations to give a slower evolution equation for the damage variable itself (see Cusumano & Chatterjee 2000), which we write as(4.1)where ‘load parameters’ include only slowly varying things such as the maximum and minimum stresses of a load cycle. We aim to deduce a useful form for *g*. Note that our assumed *g* does *not* include explicit dependence on *t*, implying that we are considering processes in which, under the same loading, the fatigue damage evolution in a specimen is the same whether we apply the load right now or some time later.

We assume that there is a well-defined state of zero damage; that for any specimen at any stage, *ϕ*≥0; that the initial damage for the unloaded specimen, *ϕ*_{i}, is small; that *g* is small when *ϕ* is small (damage accumulates slowly in the beginning); that failure occurs when some function *h*(*ϕ*)=0; and that the fatigue life depends sensitively on *ϕ*_{i}. These assumptions reflect practical observations and applications. Some people use *S*–*N* curves to predict the point of fracture, and others only to predict the first appearance of a visible crack. Our theory is indifferent to the distinction, and our definition of *h*(*ϕ*) is applicable to either case.

### (a) Power law model

We assume that, under some change of variable,1 the damage evolution is governed by a power law for all load parameters of interest. That is,(4.2)where *m* is greater than 1, and where *a* and *m* are arbitrary functions of load parameters. Note that power laws are used in many areas, including the popular Paris's law (Paris & Erdogan 1963; see also the concluding discussion in the paper). We will eventually allow ‘small’ corrections to the power law model; these will be easy to incorporate and will significantly improve performance.

The power law model with *m* greater than 1 offers an automatic simplification. Let failure occur at *ϕ*_{f}, i.e. . The time to failure is(4.3)For *ϕ*_{i}≪*ϕ*_{f}, *T*_{f} is independent of *ϕ*_{f} to leading order, and we can take *ϕ*_{f}=∞. Moreover, the time to failure is sensitive to *ϕ*_{i}, which suits us as mentioned above.

Deviations from the power law for extremely small *ϕ* are irrelevant if we do not encounter such near-perfect materials in engineering practice, and deviations for very large *ϕ* are irrelevant because failure is then imminent. Approximate validity of the power law model is only required over some suitable range of *ϕ*. Furthermore, as mentioned above, we will later include small variations from the power law.

### (b) Experimental data

Reliable and standardized experimental data in the form of *S*–*N* curves for a variety of materials and specimen geometries, with and without fully reversed loading, but always for cyclic data only, are available in Military handbook (1998) MIL-HDBK-5H. Every single *S*–*N* curve in MIL-HDBK-5H is of the formwhere *A*, *B*, *w* and *S*_{0} are fitted constants. Here, *S*_{0} is called the fatigue limit. Our theory is developed on the basis of these fits, for the special case when *S*_{0}=0,(4.4)Note that there are, in these handbooks, limits provided for the range of the stress ratios over which the fits are reliable .

We assume that all fatigue data for all materials of interest will always have the above form. Our aim is to capture, in a damage evolution model, the curve fits adopted by industry and summarized in MIL-HDBK-5H.

Imagine that a given specimen of a given material obeys equation (4.4). Now, we take the same specimen, preload it a little to change its initial damage state and then experimentally determine its *S*–*N* curve afresh. By assumption, the *S*–*N* curve will have the same form, but the fitted coefficients will be different. Thus, these three coefficients (*A*, *B* and *w*) are unknown functions of the initial damage state, *ϕ*_{i}. We will find them.

## 5. The range of *R*

Although the *S*–*N* curves given above come with restrictions on the values of , in industrial applications with cycle counting, one might easily encounter a load cycle where *R* lies outside this range. In such cases, some further ad hoc procedures are followed. Among these, it is common to simply discard all load cycles in which *S*_{max}<0, i.e. the load cycle is fully compressive.

Now, imagine a near-static tensile loading with a small superimposed fluctuation. For this loading, *R* is slightly below 1, i.e. outside the stated range. Let *R*=1−*ϵ*, with 0<*ϵ*≪1. For such loading, outside the fitting range though it might be, the *S*–*N* curve predictswhence *N* is very large. Thus, although the damage increment calculated using the formula might be incorrect, it is very small, and such errors become relatively insignificant if there are many other load cycles that cause more significant damage. With this thought, within our formulation, we will take the upper limit of *R* as 1.

Next, imagine a situation where *R* is negative and large, as can happen when *S*_{min} is significant and negative, but *S*_{max} is positive and small. In such cases, we note that typical values of *w* are between 0.5 and 0.8, and the *S*–*N* curve predicts (approx.)whence again the damage is small (because *S*_{max} is small), and such a situation may be allowed for similar reasons as above. With this thought, within our formulation, we will take the lower limit of *R* to be −∞. Of course, users unwilling to stray outside the handbook-recommended range can still use our theory, which conservatively ensures reasonable physical predictions over a larger range of *R*.

## 6. A functional equation

We write equation (4.4) as(6.1)where *N* is the life in cycles, *S*=*S*_{max} is the maximum stress, is the stress ratio and *w* is the Walker exponent (Walker 1970).

Our power law damage accumulation model iswhere the r.h.s. has no dependence on *N* (the specimen ‘knows’ its damage state, but not how many cycles it took to get there). Assuming an initial damage *ϕ*_{i}, the life of the component is given by equation (4.3) and the subsequent discussion as

Consistency between model and experiment (equation (6.1)) requires (emphasizing functional dependencies)which is a functional equation with three independent variables (*R*, *S* and *ϕ*_{i}) and five unknown functions (*A*, *B*, *w*, *m* and *a*). We assume that these unknown functions are as smooth as necessary for the subsequent development.

Functional equations can, in general, be hard to solve (Aczél 1966), but we have been fortunate here. For analytical convenience, we write and . We also write as some function , and write as some function . The functional equation can then be written asFinally, writing some new function , we obtain(6.2)The solution to the above functional equation may be found as follows. The r.h.s. is linear in ln *S*, regardless of *z* and *ψ*. Therefore, the l.h.s. must also be linear in ln *S*, in fact, *L* and *H* must individually be linear in ln *S*, and the coefficients of ln *S* therein must be independent of *z*. By similar arguments, *L* and *H* must also individually be linear in *z*, and the coefficients of *z* therein must be independent of *S*. Finally, the l.h.s. is linear in *ψ* and so *B*, *wB* and *A* must each be linear in *ψ*; the coefficients therein, by definition, must be independent of both *S* and *z*. The general solution to the functional equation can then be used to obtain the power law for damage evolution. That power law turns out to have six arbitrary constants, which we view as fitted parameters. Some of these constants can be read off directly from the *S*–*N* curve, and some can be chosen arbitrarily, as we will explain later. Instead of discussing this model in detail, however, we move on directly to a small modification to the damage evolution model, introduced indirectly using a change of variables.

## 7. Modified damage evolution model

### (a) Change of variables

Motivated by §6, we take as our abstract damage variable, where now , to get an exactly equivalent evolution lawwhere and are the functions of load parameters, as defined in §6, and the denominator has been retained for analytical convenience.

### (b) Quadratic correction term

Using the above exponential evolution law, fatigue life with initial damage *ψ*_{i} is(7.1)Note that the exponent is a linear function of *ψ*_{i}.

As seen in §6, our approach in this work is to obtain (and, with luck, solve) a functional equation that matches observed life with predicted life. In seeking a modification to the damage evolution law, therefore, it is analytically convenient to modify the expression for predicted life rather than the expression for the damage evolution itself. We anticipate (and will find, later, on comparison with experimental data) that the modification or correction to the underlying evolution law is small, in a sense that we will make precise and exploit below.

Thus, to slightly modify our model, we introduce a quadratic term in the expression for fatigue life (compare with equation (7.1))(7.2)where is a new unknown function of *S* and *z*.

The corresponding evolution law is easily obtained from equation (7.2) as follows. Letwhere *f* is a function to be determined. Starting from *ψ*_{i}, the fatigue life isHolding *S* and *z* as fixed, we differentiate the above with respect to *ψ*_{i} to get(7.3)Thus, from equations (7.2) and (7.3), we haveThus, the modified evolution law is(7.4)where functions , and are yet to be determined. Arbitrarily setting would give back our power law, disguised by a change of variable.

### (c) Functional equation

As before, we equate model and experiment to obtainwhich is to be solved for the six unknown functions , , , , and .

The general solution can be identified by inspection as follows (the discussion parallels that in §6). The r.h.s. is linear in ln *S* and in *z*. The l.h.s. is a quadratic polynomial in *ψ*_{i}. Thus, , and are each linear in ln *S* and in *z*. Similarly, , and are quadratic polynomials in *ψ*_{i}. Inserting these functional forms for the respective functions, we equate coefficients of polynomials on the two sides to obtain(7.5)

The above solution has nine arbitrary constants (fitted coefficients). In addition, the abstract and unmeasured *ψ*_{i} has to be fitted as well. In total, there are 10 parameters to be fitted in the modified damage evolution model (equation (7.4)). However, we will simplify the situation below.

## 8. Parameter reduction

We begin with 10 parameters to fit.

First, we observe that *ψ*_{i} is a parameter to the extent that some set of specimens, manufactured to some standard specifications, is expected to have approximately the same *ψ*_{i} (whatever it might be). In equation (7.4), we note that changing *ψ* to, say, *ψ*+*c*, for any constant *c*, does not affect the theory. Since *ψ* is abstract anyway, we can (for some suitably standardized specimen) take with no loss of generality. Note that , where for our initial damage variable *ϕ*, it was assumed that *ϕ*=0 was a special point (no initial damage; perfect specimen; infinite life). Here, however, *ψ*=0 implies *ϕ*=1, a more ordinary point. Setting has another advantage. The quadratic term used in the damage evolution model is to be viewed as a small correction to the original model. Specifically, this means that is small. In this context, setting (or any other reasonable, fixed number such as −1 or 1) restricts the quadratic term from growing unduly large for *negative* values of *ψ* because we never consider such values. Based on this reasoning, we set . There remain nine parameters to fit.

Recall that in the *S*–*N* curves for which numbers *A*, *B* and *w* are provided in handbooks, i.e.we have in our theory treated *A*, *B* and *w* as functions of *ψ*_{i}. Having set , we write the *S*–*N* curve asNow, from equation (7.5), the parameters *a*_{1}, *a*_{2} and *a*_{3} can be directly read off the *S*–*N* curve: , and , a convenient result of setting . There remain six parameters to fit.

Next, in equation (7.4), we note that changing *ψ* to, say, *cψ*, for any constant *c* greater than 0, does not change the form of the evolution law, does not shift *ψ*_{i} from zero, and therefore does not affect the theory. It is therefore possible to scale *ψ* such that, say, the parameter . This choice is somewhat arbitrary, but has no consequence as long as we do not attach any specific physical significance to the damage variable *ψ*. As mentioned earlier, the abstract nature of *ψ* helps us clean up the model at this stage by eliminating mathematically derived parameters that play no real predictive role. Thus, with no loss of generality, we set . There remain five parameters to fit.

At this point, we offer a simplification. Our approach in this paper is based on a small modification to a power law model (although we change variables so that the power law is disguised). We propose the basic power law model not because we can derive it from first principles, but rather because we cannot, and because similarly unjustified power law models have already proven useful in a wide variety of physical problems. In order to stay close to the power law model, we now impose the restriction that the function should have no components that grow very large, regardless of the loading condition. However, if *S* is small, then ln *S* is large. With this observation, and in the absence of experimental evidence that might force us to conclude otherwise, we now suggest choosing in equation (7.5). This leaves four parameters to fit (see equation (7.5)). They are *α*_{1}, *β*_{1}, *β*_{2} and *γ*_{2}. The damage evolution model now stands atwith . We recall that failure occurs when *ψ*=∞, and that *A*(0), *B*(0) and *w*(0) are read off the *S*–*N* curve.

From §5, we recall that *R* may approach 1, in which case approaches −∞. Also, *R* may approach −∞, in which case *z* approaches ∞. In the damage evolution model above, since the numerator of the r.h.s. is guaranteed as positive, the denominator must always be positive. As *ψ* starts from zero, it follows that we must have:an inequality that we will address later. Moreover, as *ψ* increases without bound with the progression of damage, we must also have for the full range of *z*, implying and .

Thus, we finally have three parameters to fit. They are *α*_{1}, *β*_{1} and *γ*_{2}. The final damage evolution model stands at(8.1)

We point out that most sizeable sets of published fatigue failure data under multi-level block loading are at zero mean load (where , and ), and so the parameter *β*_{1}, associated with variation in *z*, cannot be identified and may as well be dropped, giving a restricted model with only two fitted parameters (beyond the *S*–*N* curve). However, for a complete model, our theory admits three parameters as above, and multi-level fatigue testing data for multiple values of *z* will be needed (in this sense, our new theory points to new data needs).

## 9. Further remarks

### (a) Miner's rule as a special case

As stated by Lemaitre (1996), expressed in our notation, separable evolution equations of the formassociated with constant initial and final conditions, show ‘linear accumulation’, i.e. they match Miner's rule (1945). Accordingly, setting both *α*_{1} and *β*_{1} equal to zero, we match Miner's rule. It follows that our model should do at least as well as Miner's rule and can be expected to do better.

### (b) Sign of *α*_{1}

Incorporating the simplifications of §8 in equation (7.2) and taking logarithms,(9.1)*S*–*N* curves in MIL-HDBK-5H are plotted on log–log axes with ln *N* along the horizontal axis. The slope of the *S*–*N* curve (for given *ψ*_{i}) is therefore(9.2)Since the above slope is known to be negative, we must have(9.3)which is trivially satisfied here because we have chosen . If we were interested in a non-zero value for *ψ*_{i}, the inequality would imply a restriction on *α*_{1}.

Further, it is known from typical experimental data for materials without fatigue limits that, other things being equal, a pre-damaged specimen has a more steeply falling *S*–*N* curve than a virgin specimen. It may also be easily seen that if the *S*–*N* curve remains a straight line and if a pre-damaged specimen has a *less* steeply falling *S*–*N* curve than a virgin specimen, then, for sufficiently small loads the pre-damaged specimen has a longer life than the virgin specimen, a contradictory situation we disallow here. In contrast, a line with a *steeper* slope may intersect the old line to the left of *N*=1, and resulting discrepancies involving fractions of a cycle may be ignored for fatigue modelling purposes. Accordingly, we must havei.e.(9.4)

More interestingly, for a given *α*_{1}<0, if we pre-damage a specimen so that *its* *ψ*_{i} is sufficiently large, then inequality (9.3) might be violated. Among the possible interpretations of this observation, the simple and practical one2 is that this aspect of the theory is not applicable if *ψ*_{i} is too large (imminent failure). The consequence of allowing *ψ* to be unbounded even though the fitted *α*_{1} is strictly negative is merely that, preceding final failure, damage in a more lightly loaded specimen *briefly* proceeds faster than in a heavily loaded one; this will have negligible consequences on overall life predictions.

The above interpretation is consistent with our already adopted simplification of setting in equation (4.3). For the experimental data we will examine later in the paper, this restriction on *ψ*_{i} is reasonable indeed (compromising approx. the last 0.25 ‘cycles’ of life).

### (c) Further restrictions

Higher initial damage implies lower remaining life and so we must have

In particular, considering and , we have (see also §8)The first of the inequalities above may be used as follows.

We note that we have, in our theory, not non-dimensionalized *S*. This is because *S* would need to be non-dimensionalized against either some extra material parameter (such as ultimate strength, which is not a part of the *S*–*N* curve) or against some standard nominal stress irrelevant to the material, such as, 1 MPa. Neither non-dimensionalization would gain us anything; here, for clearer discussion, we assume the second case; alternatively, we simply take *S* to be in MPa.

The largest stress encountered in the fitting data, or the *S*–*N* curve's extrapolated load corresponding to a life of exactly one cycle, or alternatively the ultimate strength of the material, is here called *S*_{u}. All the three choices seem permissible, but here we adopt the last one (hence the subscript u). Note that, in MPa, for typical metals, *S*_{u}>1.

The inequality , on setting *z*=0 (a permissible value, corresponding to *S*_{max}>0 and *S*_{min}=0), givesWe observe that if, for some material or some choice of units, , then and the inequality above is reversed, leading to no new information (because already). Assuming here that , we have (including inequality (9.4))(9.5)

Now, imagine that is some reasonable non-zero positive number, and that *S*_{min} approaches *S*_{max}, so that *R* approaches 1 from below, and approaches −∞. Then implies (as *α*_{1} is bounded)

Next, imagine that is some reasonable non-zero negative number, and that *S*_{max} approaches 0, so that *R* approaches −∞ and is approximately equal to . Now becomesFor the above quantity to remain positive even as , we must haveCombining the above inequalities, we have(9.6)

Using inequalities (9.5) and (9.6), we reconsider our basic inequalityWe write and , and rearrange to getFor any given *S*_{max}, and for any suitable *α*_{1} and *β*_{1}, the inequality is most severely tested if we choose *S*_{min} to be as large and negative as possible, because we know . Accordingly, we choose , to obtainNow, because we also know that , the above inequality is most severely tested if we choose . Upon substitution and simplification, we obtain

The different inequalities derived above are not all, in the end, necessary. All restrictions considered above are met if *S*_{u}>1 and the following inequalities are obeyed by the parameters:(9.7)

## 10. Limited comparison with experiment

A usefully large set of experimental data for managing 300 CVM steel under two-level block cyclic fully reversed loading is given in Manson *et al*. (1967). Unfortunately, comparable data with variable mean stresses as well as multi-level block loading seem unavailable. So, for a limited comparison, we use Manson's data. Manson also proposed a DLDR that significantly outperformed Miner's rule in matching the data and we will compare the predictions of our model against Manson's DLDR. Comparison with some other data is provided in the electronic supplementary material.

In using Manson's data, we must regrettably ignore our theory's capacity to include variable mean stresses. Furthermore, the test material used was steel, which has a fatigue limit; but the loading was above the fatigue limit and so we use our theory anyway. Nevertheless, the good match we will obtain with Manson's data augurs well for our theory because we made a commitment to its mathematical form *before* looking at multi-level failure data. For fully reversed loading (*R*=−1), we have (a constant). Thus, *β*_{1} cannot be identified and we arbitrarily set it equal to zero here. Our model becomesAn *S*–*N* curve fitted to Manson's data iswhere *S* is in MPa (see electronic supplementary material for details). The constants *A*(0) and *w*(0) cannot be determined separately and we merely haveAlso, we have . We still have *α*_{1} and *γ*_{2} to fit. Our damage model now is(10.1)with and at failure .

We fit *α*_{1} and *γ*_{2} by minimizing an error measure defined as follows. Given a pair , for each two-level test, we use the first load level and the number of cycles *n*_{1} applied to find the damage state *ψ* at the end of that load block (using the damage model above). Then, using that *ψ* as an initial condition, for the second load level, we find the predicted number of cycles *n*_{2p} to failure. Manson's data provide the actual number of cycles to failure, *n*_{2}, observed experimentally. We take our error measure for that data point to be (the logarithm is used because there is a wide variation in the magnitude of *n*_{2} from data point to data point). Adding the error measures over all data points, we define(10.2)where superscript (*i*) stands for the *i*th data point, and there are *k* such data points. It now remains to minimize *F* subject to the constraints of the inequalities (9.7) (with *β*_{1}=0 and *S*_{u}=2033.966 MPa as per Manson *et al*. 1967).

A numerically obtained contour plot of *F* (for *k*=122 data points, we dropped one data point where the second-stage life was just 33 cycles) is shown in figure 1. Away from the relatively small region shown in the figure, *F* is complicated. However, having identified a meaningful region (which is easy here, because we have constraints on *α*_{1} and expect *γ*_{2} to be small), figure 1 is easy to draw. Note that the figure has been drawn based on an analytical ordinary differential equation that is valid only to the right of the vertical dashed line. In the figure, there is an unconstrained minimum where *F*=37.8. However, this lies to the left of the vertical dashed line and hence outside the region permitted by our inequalities (9.7). The optimal constrained point has and *F*=39.4, which is only marginally poorer; also, *γ*_{2} is, in fact, small there, supporting our initial hypothesis that the quadratic correction incorporated in our model is in some sense a small one. Note, however, that without the quadratic correction, i.e. with *γ*_{2}=0, the optimal point shifts further and *F*=53.3, which is significantly worse than 39.4.

Finally, we note that the same error measure, using the DLDR, is 40.5, which is really indistinguishable from 39.4; for Miner's rule it is 123.9, or distinctly worse. In order to get a better sense of how much the error really is, we note that for *F*≈40, the r.m.s. value of is approximately 0.57, which means that a typical second-stage life prediction is off by a factor of approximately . On the other hand, for *F*=123.9, a typical second-stage life prediction is off by a factor of approximately 2.7. Thus, both our model and the equally good DLDR do significantly outperform Miner's rule. We mention that all errors here are amplified owing to the two-level block loading, as follows. Suppose the expected life of a specimen at a certain load is 100±20 cycles. If we preload for 60 cycles, then the remaining life at the same load is 40±20 cycles, where the relative error is now greater.

## 11. Two stages of damage evolution

It is widely accepted that fatigue damage evolution usually has two distinct phases: crack initiation and crack propagation. In our abstract theory, direct connection with cracks is not possible. However, depending on whether the quadratic correction term in our model is significant or insignificant, we may divide the evolution process into two stages as well. The distinction between the two stages is not sharp. If we assume that the first phase (very loosely corresponding to initiation) ends when the quadratic term in the evolution model is 5% (or, say, 10%) of the linear term, then we can identify the corresponding point on the ln *S* versus ln *N* plane. Using the parameter values fitted to the data of Manson *et al*. (1967), the resulting transition is depicted in figure 2. To pursue the loose connection with initiation versus propagation, we note that the second phase occupies a relatively smaller proportion of total life for high cycle fatigue, which is consistent with many experimental observations, including those discussed by Manson *et al*. (1967).

## 12. Discussion and conclusions

If a theory is developed *without* an eye on a specific dataset, and still matches that data well, then one hopes that it will also match other relevant data. In contrast, a theory developed with an eye on a specific dataset may be less widely useful. From this viewpoint, our empirical theory may eventually find broad use.

To give the DLDR its due, multiple datasets acquired by other authors have borne out the useful predictive ability of this model. In this paper, in terms of fitting available data, we have so far done no more than to match the DLDR. Moreover, in the DLDR (see electronic supplementary material), there are two numerical parameters that have the aesthetically simple values of 0.35 and 0.25, chosen simultaneously for three different materials. In contrast, our model (for the zero mean loading case, where we have data) has two fitted parameters for one material only. Be that as it may, we do have a simple, constructively developed model that both performs well and offers hope for broad use.

The DLDR was motivated by the now accepted idea that there are two stages in the fatigue evolution process: crack initiation and crack propagation. Our approach, as discussed above, also allows a similar interpretation.

Our approach does have some advantages over the DLDR. A possible minor advantage is that we have easily incorporated non-zero mean stresses in the theory. The DLDR, stated in terms of cycles to failure, can, in principle, be tried for non-zero mean loading as well, though its performance for such data is not well known.

A greater advantage, though speculative, is that our approach is new, and is open to many possible generalizations. For example, models with *two* damage variables might be similarly developed. The DLDR is limited to one variable, pending inspired guesses for suitable functional forms for higher-dimensional damage curves.

A more speculative, and greater, advantage lies in that for an eventual practical application, one must seek truly incremental damage evolution models under broadband loading. We believe that there is potential for taking our approach one level lower, i.e. to move from per-cycle damage to actual evolution equations given present damage state, load, load rate and possibly some internal variables. We may need to look, for example, at the behaviour of fatigue cracks under occasional overloads, to gain insights into what the damage evolution should look like under sudden and large deviations from periodic loading. These questions are left for future work.

We end with a discussion of our basic choice of a power law model. Why a power law? Our real assumptions are that damage and its per-cycle evolution rate are non-negative. Non-negative quantities are often examined on logarithmic axes (e.g. *S*–*N* curves, the distributions of raindrop diameters; Marshall & Palmer 1948), or even the distributions of the first digits of large numbers of physical constants (see Benford 1938; Burke & Kincanon 1991). On logarithmic axes, power laws are just straight lines. Here, we note that the *S*–*N* curves used by us are power laws. Paris's law (Paris & Erdogan 1963) is a power law. So, often, is the relationship between stress and plastic strain assumed in nonlinear fracture mechanics (see Hutchinson 1968; Rice & Rosengren 1968). The power law is ubiquitous. It seems to us that, for our problem, something *other* than a power law would be a questionable choice.

## Acknowledgments

We thank the Defence Research Development Laboratories (DRDL), India, for financial support. David Chelidze and Jim Papadopoulos read and commented on drafts. A referee made useful suggestions for improvement.

## Footnotes

Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2007.0109 or via http://journals.royalsociety.org.

↵Suppose we have one damage variable

*ξ*in mind, such that . We introduce*ϕ*(*ξ*), such that , giving . Abstraction helps here.↵Alternatively, we might choose to interpret that largest-allowable value of

*ψ*_{i}to be the point of failure, instead of*ψ*=∞. This slight theoretical advantage would come at the cost of complications during data fitting. Alternatively, again, we might choose to*not*set*α*_{2}=0 in §8, so that we would now need to ensure , which could be ensured with suitably large*α*_{2}>0. However, with increasing*ψ*_{i},*α*_{2}would cause the*S*–*N*curve to become*shallower*with increasing damage, thereby pointing, perhaps, to a still smaller cubic term in the damage model. With these thoughts, we simply accept this minor inconsistency.- Received June 27, 2007.
- Accepted January 10, 2008.

- © 2008 The Royal Society