Evolution of ideas about fatigue of metals due to volume loading and friction
Published from "Sosnovskiy L.A., Makhutov N.A., Troshchenko V.T. Evolution of ideas on fatigue of metals by volume loading and friction / Тр. VIго Международного симпозиума по трибофатике (ISTF 2010), 25 октября  1 ноября 2010 г., Минск (Беларусь) / Редкол.: М.А. Журавков (пред.) [и др]. Минск: БГУ, 2010. Т. 1. С. 7784."
L. A. Sosnovskiy, N. A. Makhutov, V. T. Troshchenko
One hundred and fifty years have passed in 2010 since А. Wohler started publishing his outstanding series of papers devoted to the fatigue of metals (1860–1870). Twenty years before Ponsele was first who used the common word fatigue to explain the impossible failure of axes of stagecoaches in France after a run of 40–60 thousands km. Though problems of the fatigue of materials and structures keep all engineering world in exertion, history of studies in this field has not yet been written. It is big with wonderful, tragic, and merry events – and this is a lot of many famous scientists and real history of evolving engineering. As the power of single units grows and the velocity ant output of machines increase, fatigue failure remains one of the most often and harmful threats for health and life of staff. Material loss caused by fatigue fails of, as a rule, the most chief machine units, is also gross. Nobody, including us, knows who will be the author of this history. However, this topic seems not without interest to us. But we only touch upon the subject since this paper is just brief notes on how the notion the fatigue of materials got firmly established and widened and how it was reflected by its integral characteristic, i.e. the fatigue curve or the SNcurve, or, better, the Wohler’s curve, as it is called in many countries, though, unfortunately, informally. 
Authors of modern books treat mechanical fatigue as the process of the gradual accumulation of damages of a material under the effect of alternating stresses or deformations that causes changes in its structure and properties, the nucleation and propagation of cracks and fracture. “So, the term fatigue of metals means the behavior of metals that undergo repeated stresses”, – that is the general meaning of the notion under consideration, which is “far from ideal and too indefinite and unclear”, though “generally accepted” and “widely used”. It was given by H. J. Gough in 1926 [1]. Here the material behavior may mean whatever one likes – damage, cracks, or fracture etc., regardless of physical mechanisms that are discovered sometimes under certain conditions of loading of one or another material, including different scale levels, viz. atomic, submicro, micro, meso, and macrolevels. The Gough’s generalized definition remained applicable even when I. V. Kragelskii established the fatigue wear mechanism at sliding (1939). During the next decades volume or mechanical fatigue and surface or frictional fatigue were distinguished. The new adjectives used with the former term changed nothing in its meaning; they only concretized conditions of origination and the zone where fatigue processes occur. And, finally, when mechanics of complex or wearfatigue damage (briefly called tribofatigue) emerged at the end of the last century, the former term fatigue arrived again on the scene.
Fig. 1 presents the summary classification of basic notions that involve the term fatigue [2, 3]; these papers contain their substantiation and explanation. Here we will not repeat them; suffice to look at Fig. 1 to conclude that fatigue is a fundamental and, maybe, unlimitedly wide notion. To corroborate the conclusion, we note fatigue curves for a human being that were proposed by HeleShaw in 1911 [4] and fatigue curves for a community of people [5] that were plotted by the L.A. Sosnovskiy of the present paper in 1999. The analysis of the basic integral characteristic of the fatigue resistance, i.e. the fatigue curve, presented below will be based on the following three units of terms shown in Fig. 1: 1) volume fracture; 2) surface damage; 3) complex wearfatigue damage and fracture.
Fig. 1. Tribofatigue as complex discipline
1. Volume Fracture.As the life of the term fatigue and its service for science started from the study of volume deformation and fracture, here we present the description of results of the earliest fatigue tests given by Gough [1].
“…the tests were carried out with cast iron beams that were stricken by a rocking weight. When the striking force caused the beam inflection equal to half of the inflection sufficient for fracture during single loading, none of the beams withstood 4,000 strikes. When the striking force was reduced so that it caused the inflection equal to just one third of the maximal single inflection, each beam withstood 4,000 strikes. Other beams were bent using a rotating cam; they withstood 100,000 inflections equal to one third of the maximal single inflection. However, when the inflection reached half of the maximal single inflection, the beams failed after approx. 900 bends. Thus, truth was established, which meant that fracture may result from the repeated stress lower than the static strength.
During 1860–1861, William Fairburn carried out several tests for the Trade Council, which became outstanding. A riveted beam made of welding iron was tested. The applied load was increased or decreased dynamically using a mechanism that was driven by a waterwheel. The beam was 22 feet long and 16 inches high; it consisted of sheets 3/16 и 5/16 inches thick. The calculated static failure load for this beam was 12 tons. Fairburn found that a load of 3 tons repeated 3 million times was insufficient to fail the beam. However, if a greater load was applied, fracture occurred at a lesser number of repetitions. He concluded that there was a safe load for this structure, which could be applied infinitely long without fracture. Otherwise, if fracture finally occurred, it required so great number of loadings that this number exceeded the possible number of loadings during the normal life of the bridge. Fairburn pointed out that 12 million loadings corresponded to 328 year of the bridge’s life, provided that 100 loadings a day occurred.”
This passage describes many things, which are contained in Table 1, viz. mechanical fatigue, impact fatigue, and the notion that we currently call lowcycle and highcycle fatigue. Here we also find approaches to quasistatic fatigue as well as to longlife fatigue or gigafatigue. Unfortunately, we omit here attempts of search for the shape of fatigue curves based on the number of cycles until fracture which were made formerly by various researchers. We do not also consider here the first curves of mechanical, impact, impactmechanical, thermal, and thermomechanical fatigue because it would lead us too far and would require a book rather than an article to discuss the subject. We shall consider only the generalized characteristic of resistance to mechanical fatigue [2, 3, 6], which is called the total fatigue curve since it is plotted within the whole possible range of the governing parameters, viz.: 0 ≤ σ ≤ σ_{b} МPа and 1 ≤ N_{σ} ≤ 10^{10} циклов (рис.2)
Fig. 2. Schematic of total mechanical fatigue curve
The total fatigue curve consists of four characteristic portions (I, II, III, and IV) and has three inflection points (L, K and G). Coordinates of the inflection points correspond to the critical values of the limiting stresses and lifes, viz. the quasistatic fatigue limit σ_{L}, N_{Lσ} the lowcycle fatigue limit σ_{K}, N_{Kσ}; and the highcycle fatigue limit σ_{G}, N_{Gσ}); The slope of portions I, II, III, and IV relatively to the abscissa axis is characterized by the corresponding angle α, its cotangent is called the parameter of slope of the fatigue curve. A specific feature and the main advantage of the schematization of the total fatigue curve presented in Fig. 2 is that every its portion corresponds to the basic type of its phaseplane portrait or the hysteresis loop. The hysteresis loop is plotted in the coordinates stress–deformation and contains the following portions: the open plastic hysteresis loop (I), the nonclosed elastoplastic hysteresis loop (II), the closed mechanical hysteresis loop (III), and the degenerate mechanical hysteresis loop (IV). Thus, we indicate the governing parameter of volume fatigue fracture, i.e. the severe (quasistatic) plastic deformation (I), elastoplastic deformation (II), microplasticity (III), and nanoplasticity (IV).
Here we again especially emphasize the surprising fact that the notion and the term fatigue turned out to be applicable essentially to all conditions of reciprocal loading irrespective of a physical mechanism of damage and failure. Therefore, it is no wonder that the term the fatigue of materials got firmly established and widened leading to the terms random fatigue, fatigue at severe loading etc.
2. Surface Damage.Volume fatigue fracture occurs during the reciprocal loading of the simplest object, viz. a material specimen or a single structure member. Surface fatigue fracture occurs during contact between two solids pressed by the contact load and moving relatively to each other; the solids make up a specific object, viz. a friction pair.
It is strange, yet it is fact that during several decades tribologists have developed various wear theories (see, e.g., [7, 8]) but the fatigue theory has emerged only in the beginning of the forties of the last century. By the way, it is quite obvious (Fig. 3) [9] that the motion of any indenter over a surface produces a wave of the alternating deformation ahead of the indenter; the repetition of the wave during friction causes necessarily surface fatigue with all the ensuing consequences. In this case the following three specific features appear compared to volume or mechanical fatigue: a) under these conditions volume fracture does not occur, only the damage and fracture (wear) of a thin surface layer are observed; b) surface fatigue is commonly accompanied by various physicalchemical phenomena since an active third body, i.e. a lubricant and wear debris, is formed in the gap between the rubbing solids; c) repeated contact occurs during impact of hard particles and fluid drops and flows that carry hard particles [10] as well as under the effect of various radiations [11]. In case c) we deal, essentially, with microfatigue, since the effect of a huge number of particles is required for the fatigue process to begin. Corrosion can be apparently considered as a fatigue phenomenon since electrochemical effects repeat during a long period.
Fig. 3. Oscillogram of displacements of surface of polymer material when metallic indenter slides over it
Here we again omit great history including who plotted the surface fatigue curve and when it has been done. Fig. 4 shows the first almost total curve of friction fatigue resulted from the experimental data [12, 13]. It contains similar typical portions as the volume fatigue curves (compare to Fig. 2), but it is naturally that the interpretation of portions related to surface damage should be specific. The interpretation is given in Fig. 5 where every portion (I, II, III, and IV) corresponds to a predominant or typical wear friction and wear mechanism. Their classification was introduced by I.V. Kragelskii (see, e.g. [7]).
Fig. 4. Friction fatigue curve for steel 45 / polymer Ф4ВМ pair
Fig. 5. Schematic of total friction fatigue curve
Basic modes of damage during friction illustrate schematically the motion of a single asperity embedded into a plane at the velocity ϑ). Microcutting (I) is quasistatic fracture through shear in friction, whose slight mode is called abrasive wear. It occurs at stresses τ_{W}>>τ_{L}. In this case commonly h/r > 0,1 at dry friction and h/r>0,3 at lubricated friction, where h is the indentation depth of a single asperity of a radius r. The shear strength τ_{b} corresponds to the number of cycles N_{τ} = 1. Plastic deformation (II) occurs at stresses τ_{L} > τ_{W}> τ_{K} (τ_{K} is close to the shear stress). In this case lowcycle or elastoplastic wear occurs, when h/r < 0,1 at dry friction and h/r<0,3 at lubricated friction. Cohesion tearoff of the material appears in the intermediate zone (I–II), which is typical for adhesive wear. It may run provided that the gradient of the shear resistance τ_{0} in the surface layer is negative, i.e. dτ_{0}/dh < 0. Elastic deformation (III) occurs at stresses τ_{K} > τ_{W}> >τ_{G}. In this case highcycle or quasibrittle wear occurs when h/r < 0,01 at dry friction and h/r<0,001 at lubricated friction. In region IV at stresses τ_{W}< τ_{G} the fracture of films or the third body occurs that causes oxidation wear (the highlife region). It may occur if the gradient of the shear resistance in the surface layer is negative: dτ_{0}/dh > 0. Damage processes in region IV are substantially governed by two surface effects, viz. the Rehbinder effect and the Roscoe effect. Transition from one predominant wear mode (mode of friction failure) to another is controlled by the critical values of the limiting stresses τ_{L}, τ_{K}, τ_{G}, that correspond to the wear lifes N_{L}_{τ}, N_{K}_{τ}, N_{G}_{τ}.
The total friction fatigue curve (Fig. 5) contains the characteristics of resistance to friction fatigue similar to those presented in Fig. 2. In particular, Fig. 5 shows the critical transition points L(τ_{L}, N_{Lτ}), K(τ_{K}, L_{Kτ}) and G(τ_{G}, N_{Gτ}), as well as the parameters of the slope α of some curve portions. We note that in Fig. 4 the loading parameter is either the contact load F_{N} (see Fig. 4), or the specific friction force (the friction stress) τ_{w}=fF_{N} (see Fig. 5), where f – is the friction coefficient. Additionally, the fatigue life N_{σ} on the mechanical fatigue curve (see Fig. 2) corresponds to the number of cycles required to disintegrate a specimen, while the fatigue life N_{τ} on the friction fatigue curve (see Figs. 4 and 5) corresponds to a preset critical value of the wear (i_{lim} = 1000 µm in Fig. 4). A similar approach is used to plot the mechanical fatigue curve, when the limiting state is governed by a critical value of the length of a main fatigue crack, e.g. 1 mm.
The analysis of tribological publications will show that no total curves of contact, erosion and other modes of fatigue exist and partial curves plotted within restricted regions, e.g. in the lowcycle region, are very rare even in papers published in the beginning of the 21^{st} century. However, one can suppose that the situation will change soon; the hope is based on the fact that the section “Fatigue and Wear” was first organized during the work of the World Tribology Congress (WTС III, Washington, 2005). Tribologists “increase” the share of fatigue wear up to 30–60% compared to other wear modes.
If the inflection point K is absent on the total curve of friction fatigue (see Fig. 5) and region I is of no interest, the friction fatigue curve is plotted in the highcycle region and is simply called the friction fatigue curve (Fig. 6, а). It serves to determine the friction fatigue limit τ_{f} just as the mechanical fatigue curve within the highcycle region serves to determine the mechanical endurance limit σ_{–1} (Fig. 6, b).
Thus, here the term friction fatigue has a common meaning of surface damage and fracture during sliding friction irrespective of their mechanisms (see Fig. 5) just as the term mechanical fatigue has a common meaning of volume damage and fracture during repeated deformation irrespective of their mechanisms. Similarly, the term contact fatigue has a common meaning of surface damage and fracture during rolling friction irrespective of their mechanisms.
The equation of each of the portions (I, II, III, and IV) of the friction fatigue curve (see Figs. 4 and 5) is as follows:
, (1)
where the slope index (see Fig. 6, а) is
m_{τ} = ctg α; (2)
C_{τ} is a constant. Equation (1) yields the following formula for the wear life in, e.g. highcycle region (III):
, , (3)
According to it, the number of cycles required to reach the critical or limiting state of a friction pair is inversely proportional to the specific friction force in a power of m_{τ}. The structure of equations (1) и (3) is similar to that of the equations for mechanical fatigue; they are known and used in tribology [7].
Fig. 6. Schematics of curves of friction (а) and mechanical (b) fatigue
3. WearFatigue Damage and Fracture. If at least one of the members of a friction pair undergoes volume cyclic deformation so that contact stresses and stresses induced by the offcontact load act in the friction zone simultaneously, we deal with a specific object called tribofatigue system. Basic types of complex or wearfatigue damage and fracture of such systems are established in tribofatigue [14–16] and are included in Intergovernmental Standard [2]; they are shown in Fig. 1 and Table 1 contains their definitions and examples of their appearance. To obtain and analyze characteristics of complex damage at least four fatigue curves should be plotted (Fig. 7), viz. one mechanical fatigue curve, one contact fatigue curve and at least two curves of mechanorolling fatigue (for the direct effect related to the influence of the friction and wear processes on changes in characteristics of fatigue resistance and for the back effect related to the influence of cyclic stresses on changes in characteristics of wear resistance. For this reason it is necessary to introduce the coefficients of the direct (K_{D}=σ_{1}/σ_{1p}) and back (K_{B}=p_{f}/p_{fσ}) effects in addition to the common characteristics. Table 2 contains the summary of all the parameters determined from the experimental data presented in Fig. 7. We note that, since mechanorolling fatigue curves are plotted using two loading parameters, under these conditions the obtained number of values of the limiting stresses is as great as a researcher wants. As a result, the multicriterial diagram of limiting states of tribofatigue systems is plotted (Fig. 8). Its abscissa axis is the tribological scale and its ordinate axis is the strength scale. The points σ_{1} and τ_{f} are the most important characteristics of volume and surface fatigue and the curves AB and CD characterize complex wearfatigue damage and fracture (p_{f}_{σ} and σ_{1p}). In case of the direct effect the limiting state is reached by the mechanical fatigue criterion and the friction and wear processes are accompanying. In case of the back effect the limiting state is reached by the wear resistance criterion and the mechanical fatigue processes are accompanying.
Fig. 7. Determination of basic characteristics of wearfatigue damage (point number indicates sequence of tests)
Fig. 8. Multicriterial diagram of limiting states of steel 45 (shaft) / steel 25ХГТ (roller) tribofatigue system during mechanorolling fatigue
Table 1. Basic Types of WearFatigue Damage
Typical tribofatigue system 
Complex damage and fracture 
Definition 
Crankpin / connectingrod end with sliding bearing 
Mechanosliding fatigue 
Wearfatigue damage due to the effect of kinetic interaction between the phenomena of mechanical fatigue and sliding friction 
Wheel / rail 
Mechanorolling fatigue 
Wearfatigue damage due to the effect of kinetic interaction between the phenomena of mechanical fatigue and rolling friction (rolling friction with slippage) 
Spline shaft / bushing 
Fretting fatigue 
Wearfatigue damage due to the effect of kinetic interaction between the phenomena of mechanical fatigue and fretting 
Propeller shaft / sea water 
Mechanocorrosion fatigue 
Fatigue of the material under the simultaneous effect of alternating stresses and corrosive environment 
Turbine blades / fluid or gas flow carrying solid particles 
Mechanoerosion fatigue 
Wearfatigue damage due to the effect of kinetic interaction between the phenomena of mechanical fatigue and erosion 
Pipes of oil pipeline 
Corrosionerosion fatigue 
Wearfatigue damage due to the effect of kinetic interaction between the phenomena of corrosion and erosion 
Table 2. Notations and Numerical Values of Basic Characteristics
Characteristics of properties 
Mechanical fatigue curve 
Contact fatigue curve 
Mechanorolling fatigue curves 

N(σ_{a}) 
N(p_{0}) 
N(σ_{a}, p_{0} = const) 
N(p_{0}, σ_{a} = const) 

Endurance limit, МPа 
σ_{–1} = 165 
p_{f} = 1760 
σ_{–1p} = 256 
p_{fσ} = 2200 
Abscissa of inflection point of fatigue curve, cycles 
N_{Gσ} = 9x10^{6} 
N_{Gp} = 2,6x10^{7} 
N_{Gσp} = 5x10^{6} 
N_{Gp}_{σ} = 2x10^{7} 
Slope index of fatigue curve 
m_{σ} = 7,5 
m_{p} = 14,5 
m_{σp} = 11,6 
m_{pσ} = 24,6 
Direct effect coefficient 
– 
– 
K_{D} = 1,62 
– 
Back effect coefficient 
– 
– 
– 
K_{B} = 1,25 
By now, a very small number of curves of mechanorolling and mechanosliding fatigue are obtained. A huge number of curves of frettingfatigue for the direct effect are known while no such curves are available for the back effect. In brief, experimental studies in tribofatigue should be intensified. A wide front of captivating and useful studies is to come. Here we add that no total curve of wearfatigue damage and fracture is known by now.
In closing, we again emphasize that nowadays the term fatigue is not related to mechanisms of highcycle fatigue only such as the dislocation, vacancy, thermofluctuational and other mechanisms. Today we speak of fatigue whenever damage and fracture involving any mechanism are caused by alternating stresses or deformations that vary in time according any law and are applied to any object under study. In case of mechanical fatigue it is a single member of a structure or a specimen. In case of wear it is a friction pair such as solid/solid, particles/solid, fluid/solid and so on. In case of wearfatigue damage it is a tibofatigue system.
Here we should note that a number of methods and testing machines (machines of SI series) have been developed to determine experimentally characteristics of fatigue damage and fracture [see, e.g. 15, 16]. All tests are carried out with unified objects having the same dimensions; this allows one to compare correctly diverse test results.
Of course, the above analysis is not doubtless and the list and classification of the terms are far from completeness. Therefore, this paper can be considered as debatable. But discussions are exactly the way to a received opinion.
Nomenclature
σ_{K}, σ_{L}, σ_{G} are limiting stresses at mechanical fatigue; σ_{1}, N_{σ} are endurance limit and life at symmetrical bending; m_{σ} is parameter of slope of mechanical fatigue curve; N_{Kσ}, N_{Lσ}, N_{Gσ} are characteristic life corresponding to abscissas of inflection points on mechanical fatigue curve; τ_{K}, τ_{L}, τ_{G} are limiting stresses at friction fatigue under conditions of sliding friction; τ_{f}, N_{τ} are friction fatigue limit and wear life; m_{τ} is parameter of slope of friction fatigue curve; K_{D}, K_{B} are coefficients of direct and back effects; σ_{1p} is endurance limit at mechanorolling fatigue by parameter of contact stresses р_{0} in case of direct effect; p_{fσ} is endurance limit at mechanorolling fatigue by parameter of cyclic stresses σ in case of back effect; р_{0} is the maximal contact stress in center of contact area during rolling; σ_{a} is amplitude of cyclic stresses during bending.
References
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