New aspects of fatigue and fracture mechanics

En&wing

Fractun

Mechmicr,

1974, Vol. 6, pp. 773-793.

Peqjamon

Press.

Printed

in Gnat

Britain

NEW ASPECTS OF FATIGUE AND FOCI MECH~CS~ ALFRED M. FREUDENTHAL Department of Civil, Mechanical and Environmental Engineering, School of Engineering and Applied Science, Washington, D.C. 20006, U.S.A. Abstract-Recent Scanning Electron Microscope observations of the development of microstructural changes under reversed cyclic straining in soft, medium-scow and high strength metals have established the existence of two dominant fatigue mechanisms associated with, respectively, the development of glide fissures by reversed plastic slip in soft metals and of slip-less ‘shear cracking in strong metals. as well as of

combinations of various complexity of both in medium-strengthmetals. The stage of micro-crackmultiplication and growth preceding the stage of propagationof the final macro-crackand following a very short stage of formation of point-defects, the significance of which has essentially been lost between the attempts to reduce fatigue either to a lattice defect phenomenon in physics or to a single crack-propagation phenomenon in mechanics represents the principal engineering aspect of fatigue except when, as the result of the pre-existence of a dominant local defect or stress-concentration, the fatigue life is completely determined by the propagating crack. Analytical treatment of this stage for consideration in fatigue life prediction and reliability as~sment must await the development of suitable methods of statistical fracture mechanics. Some aspects of this approach are discussed.

1. INTRODUCTION

ITCAN be said of fatigue research that it belongs to those research areas in which, as in economics or sociology, large expenditures of effort have produced a surprisingly small amount of useful knowledge. There is an abundance of “theories” freely constructed on the basis of the models that reflect their author’s exclusive belief in and preoccupation with either lattice defect theories, or continuum theories of various types or thermodynamic theories or simple empirical materials testing concepts. The only test these theories are usually expected to meet is to reproduce the basic trend of the experimental or observational evidence of decreasing fatigue life with increasing amplitude of the cyclic loading, subject to considerable scatter. Since every one of the theories is specifically constructed to reproduce this trend and all of them easily do within the characteristically wide range.of scatter, they can be neither confirmed nor disproved by comparison with this insu~ciently disc~~nating aspect of reality. Their life-span is therefore subject solely to Plan&s law, which states that theories are not abandoned because they have been disproved but only because those who believe in them are dying out. It is therefore hardly surprising that, for instance, the belief in the “theory” of linear fatigue damage accumulation shows no sign of weakening, given the fact that such belief satisfies not only the inner need of engineers for simplicity, but also that of mathematicians for uniqueness. To recognize the complexity of a multifaceted physical phenomenon, such as fatigue, is obviously not only incompatible with those needs but also with the universalist expectation of most physicists, as welI as of many research engineers, of the existence of a single explanation or “model” of any phenomenon, which would justify the focusing of attention exclusively on that one specific aspect tInviM Lecureat the Tenth Anniversary Meeting of the Society of Engineering Science, Raleigh, North Carolina, 5-7 November 1973. 775

776

ALFRED M. FWXJDENTHAL

with which the particular research worker is most familiar, hoping that this aspect will provide the master key to the understanding of the phenomenon in its totality. While in research such hopes are rarely fulfilled, it requires a certain level of sophistication in the development of research techniques to discourage the expectation of universality of fatigue theories that concentrate on a single, not always demonstrable, feature, such as “debonding,” “ dislocation pile-ups, ” “vacancy condensation” or “plastic strain accumulation,” by the ability to actually confront the model with the physical reality of the lack of uniqueness in the origin and development of fatigue damage. 2. NEW ASPECTS OF FATIGUE RESEARCH This level has only recently been attained in fatigue research as a result of several specific developments that make it possible to face the different aspects of fatigue on a realistic basis: (i) The scanning electron microscope provides exactly the range of magnification which permits the direct identification of those microstructural changes produced by reversed cyclic straining that clearly demonstrate that during the stage of fatigue initiation “damage” proceeds by microcrack-generation that seems to originate at pre-existing strain-incompatibility centers [ 11, impurities and microstructural defects intensified by reversal of the applied strain, while they are smoothed or obliterated by unidirectional strain. Such observation makes it possible to confront “universal” theoretical models involving lattice defects or plastic strain accumulation with a sufficiently high resolution of the reality of wide-spread micro-structural crack generation clearly related to reversed cyclic straining[2], and thus to demonstrate the fallacy of associating fatigue with a single “cause,” such as reversed slip or concentration of plastic strain or the motion of lattice defects. (ii) The introduction into fatigue research of the fracture mechanics concepts of analytical classification of the severity of cracks and other defects with respect to their stability, resulting in the multiplication of systematic observations of crack-propagation rates during the terminal stage of fatigue, which is characterized by the steady propagation, under cyclic loading of specific intensity, of individual cracks or defects of well defined shape and severity in sufficiently large metal specimens or structural parts[3]. These observations provide the basis for the increasingly popular rational approach to the assessment of the expected duration of the terminal stage of fatigue. (iii) The introduction of concepts of order statistics and, in particular, of the statistics of extremes, in dealing with the characteristic scatter of the results of fatigue experiments and observations in the form of either number of load-cycles required to produce variously defined conditions of failure (“fatigue life”), or of crack-propagation rates under specific combinations of load-intensity and specimen or structural geometry [4]. The immediate result of these developments has been the recognition of the necessity not only to deal separately and on a different basis with the stages of initial micro-crack generation (“fatigue initiation”) and terminal macro-crack propagation, .but also to consider the relative significance of the intermediate transition from the initial to the terminal stage (Fig. 1) that is formed by the period of gradual shift of the growth of distributed micro-cracks from the smaller to the larger crack-sizes as a result of the differential in the micro-crack propagation rates implied by the empirical fracture mechanics relation (dc/dN) - (ofi)-. Under the assumption of constant stress and form F of the crack therefore (dc/dN) = MC”” where A4 = F. u-.

771

New aspects of fatigue and fracture mechanics

Micro to macro crack (2)

/

N, cycles

Fig. 1. Schematicdiagramrelatingcrack-lengthtofatiguelife.

If, for purposes of illustration, it is assumed that the distribution of the largest micro-crack sizes c follows the (Frechet) ~st~bution of extreme values[5]

F.(c)=exp[-(g-] the distribution form

of the associated

largest crack propagation

(K > 1) rates (dc/dN)

F,(y) = exp [- (zr”‘“‘]

(1) = y is of the

(2)

where the characteristic value of the crack-propagation rate y = tr is related to the characteristic value of the crack-size c = u by the equation u = Mu”‘~. Since (2K /a) < K the initial distribution of crack-growth rates has a wider spread [6] than that of the initial micro-crack size at the time of their emergence into the order of magnitude which just%es the application of isotropic fracture-mechanics crack-growth relation, particulariy since in this range of growth-rates the values of LYare quite largef7j; this stage might be considered as the end of the crack-generation process and the start of the transition because with increasing number of load cycles the distribution of crack propagation rates must gradually shift towards thelargest values which tend to outrun the smaller values. Integration of the crack-propagation equation at constant stress between the initial crack-size co at NT = 0 and the momentary size c at ffr leads to the relation

0

52 c

(r-m2

= = 1 - ~(y./c,,).Nr

= z-’

where NT denotes the length (in cycles) of the transition stage, y. = (dcIdN),, the initial crack-propagation rate and z the (statistical) variable (c/G,)(~-~“~. The distributionfunction of z cannot be easily determined, since it is a function of the distribution of the ratio (yolcO) which, being the ratio between two Fr6chet distributions, cannot be

ALmZED M. FREUDENTHAL

7%

expressed

in closed form. However,

in first approximation

(for co = const)

(4) and therefore

for values c > co or z 9 1 and (l/z) 4 1 FI(z)=exp[-A(l-$T”a)]-exp[-A(l+$z-’)l = exp(-A)

exp

_ %Az-1

= ,,,e-“z-’ >

(5)

where A = [yaO(%)3””

= -Inm

and

n =%A.

The form of the distribution F,(z) and the related distribution of the ratio (c/c,,) are rapidly shifting towards large crack-sizes not only with increasing number of load-applications but also with increasing initial values y, = u. that increase with increasing stress level. The trend of the shift of the central tendency is directly illustrated by equation (3) written in the form c = co 1 _ E$J [

yOLZ]z’~‘-‘~

which suggests that the order of magnitude of the duration NT of the transition can be estimated on the basis of the assumption c &co or cu-2 = -M(~~-* 2

I (NT)_’

(6) stage

(7)

as an inverse function of the expected initial rate of growth of the largest micro-crack. In estimating values of M and of a it should, however, be considered that the effective stress intensity (Famof an isolated crack is, in general, significantly higher than that of the same crack in a group of collinear or/and of parallel cracks of the same size when the distances between cracks are of the order of magnitude of the crack size@]. The growth rates in such fields of cracks are therefore probably smaller than those of isolated cracks of the same stress-intensity until the decreasing distance between the tips of collinear cracks reverses this trend, leading to crack-coalescence and discontinuous jumps in the crack-growth rate. The form of the distribution of NT follows from the forms of equations (7) and (2): in first approximation (co = const) the distribution of NT compatible with these equations is the third asymptotic distribution of smallest values[9]. In this oversimplified approach the effect on the length of the transition stage of micro-crack coalescence has been disregarded. Such coalescence, which is the more likely the higher the volume density of propagating cracks, obviously shortens the transition stage, the more the smaller the specimen or part. In the case of highly localized, intense stress-fields resulting from the existence of notches, cracks, macroscopic defects and other features producing sharp stress-gradients, the effect, on the

New aspects of fatigue and fracture mechanics

779

rapid emergence of an outrunning crack, of the high density of micro-cracks generated by the cyclic loading in accelerating growth and coalescence of large microcracks, may sharply reduce or practically eliminate the transition stage in the total fatigue process. Therefore the results of fatigue tests on small specimens cannot be directly used in the design of large specimens or structual parts. This applies in particular to damage accumulation rules, since the relative significance of each of the three stages in the total fatigue process, which varies with load intensity and specimen geometry, determines the possibility of domination of the fatigue damage accumulation process by a single one of the three stages. Thus, the uncertain degree of domination of the transition stage in the fatigue process of large specimens and structural parts without pre-existing severe defects or sharp stress-concentrations precludes the formulation of a general and reasonably reliable damage accumulation and life prediction rule for such parts or specimens, while the existence of a severe defect ensures the domination of the terminal crack-propagation stage with its reasonably well-defined crack-growth rules. In small plain material specimens in which not only the transition but also the terminal stages are necessarily short, so that the fatigue life is roughly the number of load-cycles required to produce a sufficiently sharp localization of the micro-crack generation process at the potential point of emergence of the terminal crack, the formulation of a dependable damage accumulation rule on the basis of the “fatigue initiation” stage alone faces the difficulty of the existence of an uncertain threshold of cyclic load amplitude below which growing microcracks are not generated, and would thus require a detailed knowledge of the nature of the crack-generation process in the particular metal. However, the introduction into these small specimens of a sharp notch or of a specific defect of even moderate severity removes the relation of the fatigue life to the crack generation stage and introduces the notch (or defect) as the single nucleus of the terminal crack propagation stage with respect to which both the micro-structure of the metal and the nature of microcrack generation process are of secondary significance. Since small specimen fatigue test results of plain specimen thus reflect mainly the fatigue initiation stage, of notched specimen the terminal crack propagation stage, they serve two distinctly different purposes in the materials evaluation for fatigue performance. The outlined recent developments in the approach to fatigue research and analysis make it possible to clearly delineate the central material mechanics, structural design and structural reliability problems of fatigue, in order to face each of them separately, and to establish the relations between them on the basis of the recognition of the nature of their interdependence. The key to the study of this interdependence is the multi-faceted micro-mechanism of fatigue damage generation disclosed by recent observations of the fatigue process initiation in different metals with the aid of the SEM, and its relation to and implications for an understanding of those problems. The central material mechanics problem of fatigue is the correlation of the microstructural damage produced by repeated cyclic straining with a mechanically definable and measurable parameter, and the utilization of this correlation in establishing the criteria for the production of fatigue resi%ant metal alloys, as well as the procedures for the evaluation of their fatigue performance. The central structural analysis and design problem is the utilization of the solution of the material mechanics problem in establishing the relation between materials evaluation procedures and procedures of analysis and design for fatigue of structual elements, and deriving therefrom such modifications of the procedures of design and analysis of fatigue sensitive structures

EFMVOL.6NO.4-K

780

ALFRED M. FWXJDENTHAL

which would include a rational procedure of prediction of their “safe” service life. The central structural reliability problem is the rational assessment of the risk associated with such prediction in the light of different mission spectra and selected procedures of surveillance and maintenance of structural systems. It appears that of the three recent developments referred to above each is the response to the needs of one particular problem area, and provides the principal approach to research in this area. The interrelation between these research areas arises from the fact that each approach considers a different aspect of the same global engineering problem of fatigue, as distinct from partial problems formulated in terms of specilic disciplines, such as metal physics, mechanics, materials testing, structural analysis, or statistical theory, and considered exclusively in the framework of the particular discipline, by which crucial aspects of the problem of fatigue disappear from sight, taking with them important keys to the understanding of some of the technically most significant features of the fatigue phenomenon. 3. hfRCIIANICS AND MICROMRCHANICS OF FATIGUE GENERATION Perhaps the principal example of recent progress in the utilization of the results of research in one area of fatigue in providing improved understanding of the new approaches to critical problems in a different area, is the effect of the clear identification of the different processes of micro-crack generation in different types of metals under cyclic straining on the formulation of “fatigue” as a phenomenon in terms of concepts of mechanics of quasi-isotropic solid media, such as stress, strain, elasticity, plasticity, damage, damage-accumulation and others. The importance of this formulation arises from the fact that since the methodology of structural analysis and design and the related materials testing is based on solid mechanics, useful engineering procedures of fatigue analysis and materials evaluation must necessarily be formulated in those terms. Observation by optical microscope at high magnification and, more recently, by scanning electron microscope of changes in the microstructure of polycrystalline metals resulting from reversed repeated straining has conclusively demonstrated that fatigue damage is initiated because the imposed, mechanically defined and therefore continuous, continuously differentiable reversed cyclic strain field is not compatible with and therefore cannot be accomodated by the real inhomogeneous, locally anisotropic and frequently discontinuous polycrystalline microstructure without severe local distortions, either inside the cyclically strained body or on its surface. The severity of these “strain incompatiblity centers” in terms of the local intensity of the built-up elastic micro-textural stress-fields associated with these distortions is, in general, intensified by strain-reversal and strain repetition, while the irreversible deformation accompanying unidirectional straining tends to reduce it or to obliterate its source. The possibility and extent of the build-up or the resolution of these “incompatibility centers” by micro-slip, micro-creep, or micro-cracking determines the character, the intensity and the duratio duration of the process of “fatigue initiation” by micro-crack generation. The build-up, under cyclic straining, of “strain-incompatibility centers” of a level of severity that can only be resolved by micro-cracking is not a “unique” process that could be attributed to the presence or absence of a specific type of deformation. Thus, for instance, micro-slip can produce high elastic textural stresses initiating crackgeneration when stopped by a grain boundary [Fig. 2(a)], or it can relieve existing local elastic stresses and thus reduce the tendency to crack-generation; however, such relief

Fig. 2. (a) Point ruptures at intersection of slip zones and grain boundary. (reversed cyclic torsion, CU); tbb) Slip-band fissures (reversed cyclic torsion, F-stage, Cu): (c) Microcrack formation by intersecting slip processes (reversed cyclic torsion, Cu).

Fig. 3. (a) GliQe bands, fissures and grain bmuxiaries (reversed cyclic torsion, 70/3O brass); (b) Surface fissures at 2 X 10” cycles of &@45% reversed cyclic torsion strain (70/30 brass, F-stage); 63 Surface cracks due to reversed cyclic torsion (70/30 brass, F-stage).

Fig. 4. Magnificationof surface crack due to reversed cyclic torsion (70/30 brass, F-stage).

Fig. 5. (a) Local surface eruption due to reversed cyclic torsion (Cu, very early F-stage); @) Sucfaceridgewithextrusionduetoreversedcyclic torsion(Cu,very earlyF-stage).

Fig. 6. (a) Point ruptures in surface slip bands due to reversed cyclic torsion (70/30 brass, F-stage); (b) Point ruptures coalescing into fissures (reversed cyclic torsion, Cu, F-stage).

Fig. 7. Diffuse slip characterizing S-stage (reversed cyclic torsion, Cu). Fig. 8. Concentrated

slip with extrusions due to reversed cyclic torsion (early F-stage, Ti).

Fig. 9. Localized point defects and fissures due to reversed cyclic torsion (“elastic” fatigue, Ti). Fig. 10. Slipless shear crack corssing local glide bands (reversed cyclic “elastic” torsion, Ti).

Fig. 11. Sliple:ss shear cracks due to reversed cyclic torsion breaking through goxin (F-stage, Ti). Fig. 12. Slipless

shear crack due to reversed cyclic torsion encountering (F-stage, Ti).

lndary

grain-boundary

Fig. 13. (a) Slip-less distortion of micro-structure characteristic of H-stage (reverse:d cyclii: torsi 0% Ti); (b) Combination of slipless shear and tension cracking (reversed cyclic tc H-stage, high-strength steel alloy).

Fig. 16. Point defects due to reversed cyclic torsion in medium strength low-alloy si:d. Fig. 17. Microcrack development

due to reversed cyclic torsion in medium-strength steel (16 X l@ cycles).

low-alloy

Fig. 18. Slipless cracking in high-strength, high alioy steel due to reversed cyclic tc3rsiq cycles). Fig. 19. Slipgenerated micro-cracks in alloy steel due to reversed cyclic torsion.

Fig. 21. (a) Fatigue crack and surrounding micro-fissure (reversed cyclic tension in mc:dium strength aluminum alloy); (b) Magnification of fatigue crack of Fig. 21(a).

Fig. 22. (a) Fatigue crack and surrounding micro-fissures under reversed cyclic torsion in titanium; (b) Micro structure in front of tip of fatigue crack of Fig. 22(a).

Fig. 23. Fiel d of slipless shear micro-fissures surrounding propagating shear faltigue ( rack in high strength high-alloy steel.

New aspects of fatigue and fracture mechanics

781

by unrestrained slip intersecting the surface may lead to a surface distortion from which, upon repetition, a surface fissure is developing [Fig. 2(b)]. Intersection of slip-processes may directly produce micro-cracks [Fig. 2(c)] or it may relieve an existing tendency of micro-crack-generation at a slip-restraining boundary. Similarily, micro-creep by relieving built-up elastic (textural) stresses at grain boundaries of a certain direction are likely to magnify the textural stress intensity at different locations to micro-crack generating intensity [ lo]. When microslip is extensive and diffuse as in metals that are-easily and permanently deformable (“plastic”) at low stress levels, the surface distortions produced by reversed slip enforced by reversed straining are the primary nuclei of micro-crack generation; when in (“pseudo-elastic”) strong metals micro-slip is blocked or sharply localized and surface distortion restrained by the overall elastic deformational response up to levels of stress at which the metal fails by wide-spread yielding or by instable crack-propagation, the nuclei of micro-crack generation under reversed straining are the locations at which the growing incompatibility between the continuous isotropic elastic strain field and the reversed localized discontinuous, anisotropic micro-slip or, if micro-slip is effectively blocked, between the continuous elastic strain field and the locally discontinuous and inhomogeneous microstructure that produces the high strength, cannot be resolved without the formation of micro-cracks. The larger the cyclic strain-amplitude that pseudo-elastic metals can sustain before wide-spread macroscopic yielding, the more severe the textural stresses around the strain-incompatibilities that are being built up either because of the extent of the micro-slip or of the inhomogeneity of the enforced joint deformation of disparate elements of the microstructure. The larger the cyclic strain-amplitude that plastic metals can sustain (below the range of extensive cracking in the very first few cycles), the more severe the surface distortions that are the sources of the developing fissures and microcracks. Thus the crack-generation process in either type of material is strain-amplitude controlled, plastic metals by the plastic strain amplitude, pseudoelastic metals by the apparently elastic strain-amplitude. The intention and the strain amplitude is to avoid the introduction of the concept of “pseudo-elastic” conventional implication that “elastic” strain is associated with no material damage of any kind. With respect to the fatigue initiation process in high-strength metals such implication would be in error, although, within the range of apparent elasticity (pseudoelasticity) of such metals, the linear relation between stress and strain remains sufficiently accurate to associate strain-amplitudes with stress amplitudes: the distributed micro-cracks do not measurably affect either the shear or the bulk modulus, The fact of strain-amplitude control of the fatigue initiation process can therefore be dissimulated in strong metals and presented as stress-amplitude control which, from the structural designer’s point of view, has the merit of bringing fatigue design into the framework of conventional unidirectional load- or stress-controlled design procedures, while fatigue design involving plastic metals must clearly reflect the non-convertible plastic strain-control of the fatigue process which precludes the use of load-controlled design-procedures. The well-known effect on the character of the plastic deformation of the magnitude of the plastic strain must obviously be reflected in an intensified form in the character of the fatigue damage generation by reversed slip, as, first demonstrated by Gough and Wood and systematically investigated by Wood [ 111, who used the observed difference in the character of the changes in the microstructure of plastic metals, such as pure

782

ALFREDM.FREUDENTHAL

copper, produced by cyclic strain reversal of different amplitudes to introduce the classification of the “plastic” fatigue process into a high-amplitude (H) range of reversed straining’ associated with wide-spread crystal-fragmentation by extensive cross-slip and domain-formation of decreasing thermal stability, and reflected on the surface by extensive surface-rumpling and significant second-order strain accumulation (“Ronay effect”)[l2]; an intermediate (F) amplitude range of reversed straining associated either with sharp concentrations of reversed slip into rather widely spaced glide bands [Fig. 3(a)] that are or are developing into fissures [Fig. 3(b)], and, on close inspection, are revealed as microcracks [Fig. 3(c)] along which the metal is completely disrupted (Figs. 4); it appears that during the early part of this process concentrated distortions in the form of local eruptions [Fig. 5(a)] or ridges and plate-like extrusions [Fig. 5(b)] penetrate the surface, leaving, upon removal by etching, point defects [Fig. 6(a)] that develop into fissured glide bands or from which micro-cracks start [Fig. 6(b)]; a low-amplitude safe (S) range of reversed straining associated with widely diffuse non-concentrating micro-slip that produces no propagating micro-cracks (Fig. 7). Since at strain amplitudes within this range the deformational response is nearly enough elastic to justify the conversion of the limiting strain-amplitude of the S-range into a stress-amplitude, this “endurance limit” under constant amplitude of cyclic stress acquires its physical significance. However, the fact that in “plastic’* metals this limit is not only very low but also quite difficult to determine experimentally, obliterates its technical significance in such metals. A similar classification of the fatigue initiation process at cyclic strain amplitudes of different magnitude in high-strength pseudo-elastic metals would have to be based on the observation of amplitude dependent differences in the character of the development of microcracks, in the absence of any justification for the assumption that the fatigue initiation process in such metals can be correlated with reversed slip. Even when, as in titanium, the formation by reversed cyclic slip of concentrated glide bands with extrusions (Fig. 8) that leave point-defects and fissures behind (Fig. 9) is not blocked, the slipless shear-crack that characterizes the fatigue initiation stage in pseudo-elastic metals proceeds clearly across the glide bands (Fig. 10). There is, however, evidence that the character of the usually trans-granular slipless shear crack propagation under intermediate (F) cyclic strain amplitudes that may either break through a grain-boundary (Fig. 11) or be blocked by it, producing an apparent trace of highly concentrated unresolved elastic strain in the interfering crystal (Fig. 12) changes significantly in the range of high (H) cyclic strain amplitudes: below the yield point slipless accommodation of the polycrystalline microstructure to the enforced large strain-amplitudes proceeds either by a combination of transgranular shear cracking, grain boundary cracking and crystal fragmentation [Fig. 13(a)] which obviously involves a combination of shear and tension cracking [Fig. 13(b)], while at the yield point sudden extrusive slip provides a more effective micro-mechanism of accommodation of microstrain to macro-strain which, however, only reduces but does not completely obliterate the textural stress relieving effect of shear- and grain-boundary cracking. Thus the range of high cyclic strain is not associated with an unique micromechanism of fatigue initiation, but rather with a pre-yield pseudo-elastic and post-yield predominantly plastic mechanism, with a likely transition of simultaneous operation of both mechanisms. While, on the strength of the definite change in the operating micromechanism of fatigue crack generation, the classification of the fatigue initiation process into H and F stages appears therefore equally justified for pseudo-elastic met-

783

New aspects of fatigue and fracture mechanics

though on a different basis than for plastic metals, such classification should provide for the dual character of the H-stage in pseudo-elastic metals with clearly defined yield limit by a distinction between pre-yield (HE) and post-yield (HP) stages. A safe (S) fatigue-less stage undoubtedly exists, although the exact determination of its upper (“endurance”) limit, in the absence of any other distinction but the absence of propagating micro-cracks, is probably even more difficult than in plastic metals. als,

Table 1 Plastic fatigue Free Cross slip, cell struct. formation

Restrained CIptd

fragmentation

Mixed-mode macro-crack

F

Surface fissures

Incompatibility cracking

Coalescence of glide fissures

S I

Diffuse slip

“Elasticplastic” fatigue

Isolated point defects

!i

I5 i! $ ,pjj E j, A? *$E

.5 $ “0 E

Elastic fatigue Free

Restrained

Shear and tension cracking

Strain incompatib~ity cracking

Mixed-mode macro-crack Slipless shear cracking

Mixed-mode cracks Isolated defects

Shear and grain boundary cracking

Line-up of shear cracks Arrays of point defects

No macro-crack development

Isolated shear cracks

I

Table 1 summarizes the classification of the different ~cromechanisms of fatigue initiation in plastic metals (“plastic fatigue”) and in pseudo-elastic metals (“elastic fatigue”) at the three stages (H, F and S) of strain-amplitude cycling. This classification is based on a theoretical, sharp distinction between essentially plastic, predominantly face-centered cubic metals, such as copper or ahuninum, the elastic limit of which is very low and the elastic range of which therefore practically vanishes under reversed cyclic straining (Fig. 14), and “pseudo-elastic” metals, such as titanium (Fig. 15), with a very high, well-defined yield limit at which yielding becomes extensive, but below which reversed cyclic straining causes substantial microstructural damage by slipless .c & 20 I-

Fig. 14. Cyclic torque-strain diagrams of plastic metal (Or).

ALFRED M. FREUDENTHAL

784

0

I 4

I

I

I

a

12

16

Degrees

I 20

twst

Fig. 15. Torque-twist(indegrees)diagramof titanium. shear cracking without significantly reducing the apparent elastic range of cyclic strain. It appears therefore that the justification for the classification of the fatigue process in terms of character of the deformation and the range of cyclic strain amplitudes rests on the simple, almost obvious fact that when reversed plastic strain cycles of one type or another produce micro-cracks, their amplitude will control the rate of such generation, while the rate of generation of microcracks resulting from relief of severe elastic incompatibility (textural) stresses, in the absence of operating micromechanisms of relief by slip, creep or other energy-dissipating permanent deformation processes, will necessarily be controlled by the elastic ‘strains amplitude. The limitations of this classification are therefore inherent in this basis for its success in the cases of truly or at least nearly enough plastic and pseudo-elastic metals and metal alloys: when the cyclic straining of a metal involves both elastic and plastic deformation of the same order of magnitude, the alternative roles of the plastic deformation in either micro-crack generation or in micro-textural stress relief are no longer clearly assignable, with the result that the classification presented in Table 1, although remaining instructive, is not directly applicable with relation to the large group of the technically important alloys of medium and upper medium strength because of the coexistence and strong interaction of elastic and plastic deformation in the process of micro-crack generation, except when either the plastic or the elastic component of the reversed cyclic strain clearly dominates this process. Thus, for instance, in slow-hardening ferrous alloys such as low-carbon and stainless steels the plastic strain dominates the cyclic deformation process in the yield range because of the easy glide on the multitude of slip systems in body-centered metals and produces microstructural configurations that resemble those of the H-stage of plastic metals with vanishing elastic range; on the other hand, the rather high stability under cyclic strain of the yield limit of these metals produces a high level of the (endurance) limit of the S-range, reducing or obliterating the F-range, as in mild carbon steel. In the intermediate (F) range of medium strength alloys the interaction between point-defects (Fig. 16), elastic shear-cracking, and diffuse plastic

New aspects of fatigue and fracture mechanics

785

deformation in creating or relieving micro-textural stresses, and in generating or blocking microcracks produces rather complex ~~ros~~~ co~a~ons containing slip-generated as well as slip-less microcracks (Fig. 17), domination of the latter increasing with increasing alloy strength (Fig. IS), while in medium strong alloys, both ferrous and non-ferrous, slip-generated micro-cracks prevail (Fig. 19). Thus the F-range may either practically vanish or, when slip is severely restrained by precipitates or by a highly non-homogeneous micro-structure, the post-yield H-range might disappear while the pre-yield H-range becomes undistinguishable from the F-range and the level of the S-ranges is sharply reduced because of the severe textural stresses that are charactristic of the non-homogeneous microstructure: the F-range thus appears as the only remaining stage, but its complex character precludes its un~bi~ous classificatibn with respect to the type of strain-control. Wood[4] has attempted to overcome the difficulty inherent in the simple classification into plastic and pseudo-elastic fatigue by suggesting two additional classes of “elastic-plastic” and “alloy” fatigue. This ’1 rocedure, however, obliterates the principal merit of the simple classification according to the two principal crack-generation processes of focussing the attention on the essential duality of this process. It appears therefore more desirable to retain the simplicity of the dual class~~a~on of the fatigue i~tiation process, with the implicit recognition of the existence of the large intermediate class of the technicaky most important medium and medium-high strength metals in which both processes coexist and interact, without attempting to force the complex processes arising from such interaction into further “classes”, the delimination of which would necessarily be diffuse. While the classification according to the nature of the controlling type of strain applies to the process of micro-crack generation alone, and not to the subsequent stages of transitional and terminal crack growth, the effect of these stages on the fatigue life of small, plain fatigue specimens is hardly significant enough to produce a substantial change in the resulting trend of the relation between fatigue life and the controlling strain amplitude. This would explain the existence of an almost material-independent well-reproducible relation between the fatigue life and the reversed plastic strainamplitude of truly plastic metals (OHFC copper, pure aluminum) as well as of medium. strength metals with a distinct plastic range (titanium, stainless steel) of the form f2e,,r)eN” = const. known as the Corn-Hanson relation; using the rule of thumb, according to which the fatigue life in this range N - 10’ when eP1= + 0*01 while for. &O-l the life reduces to N - 10 cycles, this relation takes the simple form EPl = +ePr - (lON)-“* or O*la$ - l/N, where l/N can be considered as an average rate of fatigue damage per cycle, and seems to fit the observed trend of test results fairly well, though with considerable scatter, up to an order of magnitude of 0.005 < enI < 0.01 or 10’~ N < lo’, depending on the metal. Conversion of this relation into a stress-dependent expression for the fatigue life or the damage rate would require the introduction of a relation representing the form of the loading part of the stabilized cyclic stress-strain diagram, such as the frequently used expression lPl = Co”, where 5
786

ALFRED M. FREUDENTHAL

not only implied by the existing vast record of S-N diagrams for medium and high strength metals, but has more recently also been reported as the result obtained by applying rules and data of fracture mechanics to the prediction of fatigue life [ 141. Since this relation can also be converted into the form --CG . N” = const. where a is considerable larger than l/~, it is obvious that the (negative) slope, in double logarithmic representation, of the 4, vs N relation is much larger in the plastic range than that of the lel vs N relation in the elastic range, while the reverse is true for the g vs N relations. The only argument that might justify the introduction of a deterministic fracture mechanics model into the study of the fatigue crack generation process is that of pre-existence or of very early formation of that single microcrack or defect which subsequently grows into the terminal macrocrack, so that in comparison to the stage of crack propagation the stages of crack generation and transition can be disregarded. Except for the case of pre-existence of a defect which is severe enough to localize the crack generation process already in the first few cycles, no evidence exists to support this argument. The observation, early in the cyclic straining, of point defects and micro-fissures on the surface of plastic metals does not imply that one of these is the nucleus of the terminal crack, and that therefore a realistic model of fatigue of such metals could be based on a mechanism of macro-crack propagation in a plastic medium, even if such a mechanism could be devised without recourse to the unsubstantiated or vague assumptions that usually form the basis of such attempts; for instance, the combination of the assumptions [ 151 that ‘tincreased plastic strain eP is accommodated by the production of new crack surface” so that “crack-growth in the plastic zone is proportional to the size D of a plastic ‘region’ (that itself is a function of a’) and to the plastic strain tb ” that cannot be substantiated in spite of their analytic disguise, leads to the “crack-growth equation” dc/dN

= CepD = C,cp,cr2= C,k2e,‘28+”

if a relation of the form c = kgP is introduced to represent the “cyclic” stress strain relation, and thus reproduces the Coffin-Manson relation with K = (2p + I)-‘. This fact has been used 1161 to interpret the apparent material independence of this relation (O-4 < K < 0.6) as the result of the uniqueness of the postulated crack-propagation mechanism and its independence of the micro-structure. However, converting the Coffin-Manson relation with K = (2p + l)-’ into a relation of the form aN” = const., the exponent w = p/(2/3 + 1) varies between O-15 < o C O-25 or 4 < o-’ C 6-7 which, since w-l- (Y,suggests that this relation reproduces observed crack propagation processes in pseudo-elastic rather than in plastic metals, a contradiction that invalidates the introduced “plastic” crack-growth relation from which it has been derived. Thus, the possibility of conversion of E-N relations into a-N relations and vice versa, which is always given provided the associated stable cyclic stress strain relation is known, does not imply equivalence of the basic relation that involves the controlling mechanical parameter with its converted form unless the dependence between the alternative parameters is unique and invariable. Thus the basic relations in Fig. 18 are between the plastic strain range and the length of the crack-generation stage for plastic metals and between the elastic range and the length of this stage in pseudo-elastic metals. In the intermediate range between the extreme classifications, it might be assumed that the former governs the H-range while the latter reflects the conditions in the F-range.

New aspects of fatigue and fracture mechanics

787

The large difference between the range of values of (n/~) and of (Y is of considerable significance in the formulation of rules of fatigue damage accumulation for fatigue life prediction, since for technical metals such rules are usually formulated to extend over stress amplitudes within both ranges. GENERATION AND PROPAGATION The crack propagation rate of the “out~n~ng” terminal crack depends on the microstructure in two ways: (i) in a “global” macromechanical sense through the value of the parameters of the crack-velocity-stress-amplitude intensity relation; (ii) in a micromechanical sense through the character and extent of the crack-generation process preceding the appearance of the terminal crack. The “global” dependence is determined by the crack-growth tests under cyclic loading on differently shaped metal specimens with pre-existing or carefully prepared cracks or typical defects. On the basis of these test results the length of the terminal phase of fatigue crack propagation can be predicted by integration of the crack-growth equation, provided the critical unstable length of the crack in a representative specimen of the material has been determined by a suitable test. Crack growth experiments have in general been performed under pseudo-elastic conditions of medium and high strength metals, and their results are applicable as long as such conditions prevail. The sharp localization of plastic deformation in the vicinity of the crack tip and the control of its magnitude by the overall elastic cyclic strain- and associated load (or overall stress) amplitude justifies the assumption of control of the crack growth process by the elastic stress amplitude intensity factor, expressed in the usual growth rate equation, in spite of the frequently not insignificant plastic deformation at and around the advancing crack-tip. The significant difference in both the purpose, the results and their implications of plain and notched small specimen fatigue tests is due to this fact which also explains the observation that the endurance limit of plain specimens of different alloys of the same metal shows some slight correlation with the unidirectional yield limit, while the notch endurance limit shows none (Fig. 20). The early localization of the out~nning crack reduces the significance of the initial severity of notch or other pre-existing defect, since as soon as a crack is formed, its propagation is governed primarily by its own stress-intensity, which very rapidly becomes higher than that of most any pre-existing stress concentration. The principal rule in fatigue design is therefore to avoid stress-intensity factors that, for the particular metal, could produce early iocalization of the crack-generation process into one single outrunning crack. Therefore the existence of a very sharp notch effectively obliterates the influence of the m~cros~ctu~ of a metal on such loc~i2ation, since its stress intensity is usually too high to be affected by changes in the inelastic deformation mechanism that can be produced through changes in the microstructure. The result is the well-known insensitivity of the fatigue endurance limit in structures of most structural metals to microstructural changes that produce improvements in unidirectional properties, such as the yield-limit or fracture-toughness, since in structures stress-concentrations of high intensity might, to some extent, be controled, but cannot be avoided. The disproportionately damaging effect of even a very small number of high ampIitude Ioad cycles in a Ioad spectrum, that has been widely observed, particularly in aircraft structures of higher strength aluminum alloys 117, may therefore be due to the combination of two effects: the initiation and rapid localization of the crack-generation processes in locations of sufficiently high stress-intensity the number 4. CRACK

788

ALFRED M. FREUDENTHAL l00oao r

Wrwpht olumiiwmalloys

R

aluminum

alloys

l%ri+nan@nt -mold cast aluminumalloys

-

I

Tensile strength Yield strength Endurance limit Notch endumnce limit I

5x106 cycles

Sand -cast aluminum allays

Fig. 20. Mech~ic~

properties of some aIuminum alloys.

of which is itself a function of the increasing overall stress-amplitude, and the increase of fatigue crack growth rates in microstructures that are also responsible for increase of the yield limit. The fact that the crack-growth rate in fatigue is governed by its own stress-intensity rather than by the intensity factor of the pre-existing defect reduces the significance of the latter to its role in the initiation of the crack-generation process, which is quite different from its general role in “fracture mechanics” implying fracture under unidirectional loading. It appears therefore that attempts to transfer specific fracture mechanics concepts, the validity of which depends on specific conditions of unidirectional loading, assumed to exist in an elastically constrained ideal elastic plastic medium, being satisfied, such as the f-integral, are rather futile, since they suggest a complete disregard of the reality of the fatigue process. The only fracture mechanics concept that retains its theoretical validity in fatigue is the critical fracture toughness with respect to the start of f~s~u~fe propagation of the outing terminal crack, provided that credible theoretical and experimental parameters, the comparison of which determines the critical condition, can be estimated or derived from tests with an acceptable degree of approximation. In the fatigue process itself the fracture mechanics concept of the stress-intensity factor serves only as a very useful parameter for the empirical correlation of the observed crack-propagation rates and the applied cyclic load amplitudes. Even this limited use of fracture mechanics concepts, however, loses some of its credibility in the case of fatigue generation processes which, in the absence of pre-existing stress-concentrations, are not rapidly concentrated in the terminal crack,

New aspects of fatigue and fracture mechanics

789

with the result that the much more slowly developing “outrunning” crack propagates through a field of microcracks rather than through a continuous solid. The existence of this micromechanical dependence of the crack-propagation process on the preceding crack-generation and transition processes is well illustrated for different metals by Figs. 21-23. Figure 21(a) and (b) shows the terminal crack in a medium strength aluminum alloy that has propagated through a dense field of fissures, Figure 22(a) and (b) shows the terminal crack in titanium and the broken down microstructure in front of the advancing crack tip and Fig. 23 shows the field of slip-less shear cracks surrounding the propagating terminal crack in a high strength Ni-Co-Cr-Mo alloy steel specimen; at advanced stages of propagation of much larger cracks in much larger specimens the conditions will much more closely approach the assumption of a single crack propagating in a quasi-continuous medium. However, from the point of view of the designer as well as from that of the metals producer such conditions represent a highly undesirable state, the avoidance of which should be made the principal goal of both fatigue design and material development of fatigue resistant materials. 5. FATIGUE

MECHANISMS

AND RELIABILITY

The most important contribution of the study of the micro-mechanisms of fatigue to the problems of fatigue reliability is the possibility it provides to adjust the basis of the reliability approach to the physical reality by (i) distinguishing between reliability with respect to either the end of the crack generation and transition stage (start of crack propagation stage) or the end of the crack propagation stage (terminal fatigue life); (ii) using this distinction for the selection of the appropriate, physically germane form of the distribution function of the “fatigue life” in accordance with the selected basis for its definition. The fact that the rates of progress of all three stages of fatigue are governed by the largest values of the respective controlling parameter, such as extent of reversed slip or intensity of local textural stresses in the crack generation stage, micro-crack-size or growth rate in the transition stage and outrunning crack size or growth rate in the terminal stage, in conjunction with the assumption that no more unfavorable unimodal distribution of these (largest) values can be introduced than the second asymptotic (Frechet) distribution, leads easily to the conclusion that the duration of any of these stages which is necessarily an inverse function of the controlling parameter is distributed in accordance with the third asymptotic (Weibull) distribution. This argument provides further justification for the use of this form of distribution in the structural reliability analysis with respect to fatigue [ 181. However, the distinction between two definitions of “fatigue life,” based either on the start or on the end of the terminal crack propagation process raises the important problem of the admissibility of the two-parametric forms of this distribution that for reasons of analytic simplicity and convenience has so far been generally used in the reliability analysis on the basis of the assumption that an absolute lower limit of the fatigue life (minimum life)[191, even when it exists, is small enough to be neglected. Considering this assumption in the light of the alternative definitions of the “fatigue life” it appears that it can be justified only with respect to the definition of “fatigue life” as the end of the transition stage and the start, at various fatigue critical locations of the structure, of the process of growth of the outrunning largest crack. Depending on the existence and severity of stress concentrations in a structure and the resulting early

790

ALFRED M. FREUDENTHAL

localization of the crack propagation process, this process may start already in the course of the first few or few hundred load cycles, so that the assumption of a negligeably small “minimum life” appears justified. With respect to the end of the stage of crack propagation by “catastrophic” failure, whether local or global, the “minimum life” might be as short as the shortest period of crack-generation and transition required to produce an “outrunning” crack that becomes unstable as soon as it has developed, or it might be much longer but it cannot nearly be zero. In fact, except for small notched specimens, the “minimum life” may be a significant fraction of the expected terminal fatigue life. Interpretation of the small number of test results, presumably all on small specimens, has led to the tentative conclusion that a minimum life of about one-tenth of the expected life may be a credible assumption[20]; for components and structures this ratio might be higher. Hence for such structures the representation of the distribution of the terminal fatigue life by a three-parameter distribution function for smallest extremes -In [l - P(N)1 = [++$Ivalid for N > N,, the minimum life, might be a more credible assumption than that of setting No = 0 and using the simpler two parameter form, which, however, as a basis for the structural reliability analysis may lead to unduly pessimistic fatigue reliability assessments. The principal, well known difficulties in the use of this distribution are the interrelation of the three parameters V, No and 01,which precludes their independent estimation, requiring instead tables of auxiliary parameters for consecutive estimation, and the reliability of the estimation, by backward extrapolation, of the “minimum life” No. It has been suggested that an estimate of the minimum life No can be based on a visual inspection of the “linearity” of the plot, on extreme-value probability paper, of the fatigue lives obtained in tests of a sufficiently large sample, as a result of the following transformation of the distribution function into the straight line form x=log(N-No)=log(V-No)+y/d’ where

(9) y =ln{-In

[l -P(N)]}

and

l/a’=O*434/(~

log standing for the common, In for the natural logarithm; the logarithms of (N - NJ plotted against the reduced variate y with plotting position P(N) = m/(n + 1) should be scattered about this straight line. Hence the estimate of N, that transforms the curved plot N = No + (V - No)e y/Uinto the straight line x = a + y/a’ represents a satisfactory estimate. However, the mutual influence of the parameters No and a! contradict the assumption that the existence of N, is always reflected by the curved appearance of the relation between N and y. This is illustrated in Fig. 24 which presents these relations in the normalized form 12= E + (1 - e) e’l” (10) where n = N/V and E = No/V, which, in this form, dimensionless parameters E and l/cu that are both enclosed large values of CYthe normalized life n is practically a linear not zero. Thus, when E = 0, the function is linear; but when

depend only on the two between zero and one. For function of y, although E is the function appears linear,

791

New aspects of fatigue and fracture mechanics 0.999 O~Q@E

0.998 0.990 0.990

0.950

;:g 0.400 0.300 0.200 ~ 0~100 0.00 O-010

j:g$f i 0.10

-A

0.1s

-

020

0.2% 0.30

Relative

life

16 a/v, /Vs

Fig. 24. Normalized form of third asymptotic extremal (minimum) distribution function of fatigue lives (equation 10) on extremal probably paper with e = O+l, O-3and O-5and different values of the shape parameter CX.

e and therefore NO are not necessarily zero. When both (Yand E are large, the scatter of n is small, when both Q!and E are small the scatter is wide; with respect to scatter, the effect of the two parameters has the same direction. They can therefore compensate each other, a fact which produces obvious problems in the assessment, with respect to the existence of a minimum life, of observations of fatigue life, particularly in small samples. The most significant aspect of the existence of a “minimum” life is its effect on the “scatter factor” S in a sample of the population of size Y,defined as the ratio between the expected life of the population and the “safe life” NR for a specified reliability level L(K) = 1 - P(&) = R, which is the time to the first failure at this level. Assu~ng a three-parameter extreme value distribution of fatigue lives in terms of the normalized variable n, the distribution function of the (normalized) smallest life n 1in a sample of size v is obtained from = [l - P(n)]’

1 -Pl(n,)

= exp [- v(e>“],

(11)

which shows that P,(n,) is of the same extremal form as P(n), characteristic value reduced by v-“~. Solving for n, = nR at IL(&) nR

while the expectation

= E + (1 -

of P(n) according E(n)

=

d +

l)v-“’

but with the

If0

In f [ ( )I

to theory of extreme (1 - E)l-Yl + l/a).

w values (13)

ALFRED M. FREUDENTHAL

792

The scatter factor at reliability level R is therefore s

R -E(n)_

(1 - E)r(l + l/cr) (1 - e)F”“[ln (l/R)]“”

E + E

nR

+

(14)

or (15)

where -11.3

Son = v”“r(l+

l/a)

In $ [ 01

(16)

represents the scatter factor for the distribution with zero minimum life (E = 0) which, for R = e-l, takes the simple form S, = Y”- r(l + l/a). The existence of l > 0 reduces this value by the factor (E

9

(y)

=

iw -

1+0*1e I+ ml + II 1+ E(V”= - 1) 1+ (Y”” - l)E

(17)

which illustrates the combined effect of l and a on the expected scatter factor, with sample size v as a parameter, since r(l + l/a) varies only slightly (0.89-0.93) within the relevant range of values 3 < a < 6. The approximate form of the function (E, a) is easily evaluated: thus, for instance for v = 200 a minimum (normalized) life of not more than E = 0.1 with a = 4 reduces the scatter factor from So = 3.4 to about SR = 2.7. However, the scatter factor So, = 6 for R = 0.9 is reduced to about SR = 3.9, while the scatter factor SOR= 10.8 for R = 0.99 is reduced to about SR = 5.2, indicating that the existence of even a moderate value of the minimum life produces a substantial reduction of the scatter factor, particularly at high reliability level. This effect is the more pronounced the smaller the value of a: thus for a = 3 for which S, increases to S, = 5.3, a minimum (normalized) life c = 0.1 reduces S, to SR = 3.6, S,, = 11.4 for R = 0.9 to S, = 5.4, while S,, = 24.6 for R = 0.99 is reduced to SR = 6.8. Even if the (normalized) minimum life were only E = O-05, the scatter factors at R = O-9 and R = 0.99 would be S, = 7.3 and S, = 10.6 respectively, which is still a significant reduction of SOR. The interaction between the range of scatter indicated by l/a and the (normalized) minimum life E illustrated by the above examples is therefore of considerable practical significance, particularly with respect to the reliability assessment of structures of high strength steels the fatigue test results of which demonstrate a significantly wider scatter than similar tests of medium strength structural aluminum. On the other hand the existence of an endurance limit in such steels suggests, although only by implication, the existence of an at least moderate minimum life, an implication that arises from the conditions of compatibility between the distributions P(A&, P(S),., and the (S - MP relations[l8]. Thus at the reliability level R = 0.9 for the same sample size v = 200, the scatter factor for a = 4 and E = 0 (So, = 6) is even slightly larger than the scatter factor for (Y= 3 and E = O-1 (SR = 5.4), while at the reliability level R = 0.99 a minimum life of not more than E = 0.05 removes the difference between the scatter factors SRO= 10.8 for a = 4 and E = 0 and SR = 10.6 for a = 3 and E = 0.05. The existence of a minimum life under random or program cyclic loading, and the

New aspects of fatigue and fracture mechanics

793

effect of the stress-amplitude interaction on the minimum lives at the different constant stress-amplitudes as well on the resulting minimum life must necessarily have a strong influence on fatigue design procedures and reliability analysis of structures of strong metals and decisively affect the development of rules of fatigue damage accumulation as well as of accelerated fatigue testing procedures for such metals. It appears therefore than an urgent study, both theoretical and experimental, of the use of dist~butions with non-zero ~nimum life, is of considerable practical importance. It is not unlikely that the extension of such study to structural composites, which are known to display a disturbingly wide scatter under cyclic loading, may provide the key to their effective and safe use in critical structural elements. Acknowledgements-The electronmicrographs used for illustration of the discussion of the micromechanisms of fatigue (Figs. 2-23) have been obtained in various detailed studies of these mechanisms by members of the Institute for the Study of Fatigue and Structural Reliability of George Washington University, in particular Professor W. A. Wood and Professor C. M. Gilmore, and lirst published in Technical Reports of the Institute of which the author is the Director.

REFRRENCES [l] M. Ronay, On strain incompatibility and grain boundary damage in fatigue, Acta Tech. Hung. 96199-218 (1966). W. A. Wood, Four basic types of metal fatigue, Techn. Rep. No. 10, ONR-AFML. Inst. for The Study of Fatigue, Fracture and Struct. Reliability, Geo. Wash. Univ. Washington, D.C. (1972). A.S.T.M., Fatigue crack propagation, Spec. Techn. Publ. No. 415 (1%7). A. M. Freudenthal, Reliability of reactor components subject to fatigue and creep, Proc. 2nd Int. Conf. Struct. Mech. in Reactor Techn., Berlin (1973). E. Gumbel., Statistics of Extremes, p. 260 Columbia Univ. Press, New York (1958). ibid. p. 266. A. M. Freudenthal, Fatigue and fracture mechanics, Engng. Fracture Mcch. 5 403-414 (1973). M. Ratwani, Wechselwirkung von Rissen, Rep. Inst. fi Festkoerpermechanik. Freiburg (1972). A. M. Freudenthal, Statistical aspects of brittle fracture, (Ed. H. Liebowitz), Fracture Vol. 2, p. 600, Academic Press, New York (1%8). C. Zener, Elasticity and Anelasticity of Metals, p. 126 ff., Univ. of Chicago Press, (1948). W. A. Wood, Study of&fetal Strucfures and ~eir~echanicaiP~pe~ies,Pe~onPress, Oxford (1971). M. Ronay, (a) On Second Order Strain Accumulation’~n Torsion Fatigue, Br, J. appl. Physics, 16,727 (l%S); J. Inst. Met. 94, 392. (b) M. Ronay and A. M. F~udenthal, Second order effects in dissipative media, Proc. R. Sot. A. 292, 14-50 (1966). J. D. Morrow, Cyclic plastic strain energy and fatigue of metals A.S.T.M. Special Techn. Publ. No. 378, pp. 61-75 (1%5). T. W. Crooker, Basic concepts for design against structural failure by fati@re crack propagation, Naval Res. Lab. Rep. 7347 (1972). B. Tomkins, Fatigue crack propagation, Phil. Mug. 18, 1041 (1968). A. S. Tetelman and A. J. McEvily, Fracture of Structural Materials, p. 374, Wiley, New York (1%7). E. Gassner and K. F. Horstmann, Einfluss des Start-, Lande-Lastwechsels auf die Lebensdauer der B~n~~sp~chten Flugel von Verkehrsflug~ugen, Adv. Aero. Sci. 314, 763-780 (l%l). A. M. F~udenth~ and E. J. Gumbel, Physical and statistical aspects of fat&e, Adu. Mach. 4,138111. A. M. Freudenthal and E. J. Gumbel, Minimum life in fatigue, J. Am. Statistical Assoc. 49 575-l ll(lP54). A. M. Freudenthal, Reliability analysis based on time to the first failure, Aircraft Fatigue, p. 33. Pergamon Press, Oxford (1972). (Received 29 January 1974)