substructural organization, dislocation plasticity and

10.1098/rspa.2003.1181

Substructural organization, dislocation plasticityand harmonic generation in cyclically stressed

wavy slip metalsBy John H. Cantrell†

Cavendish Laboratory, University of Cambridge, Madingley Road,Cambridge CB3 0HE, UK

Received 9 December 2002; accepted 28 April 2003; published online 12 January 2004

Organized substructural arrangements of dislocations formed in wavy slip, face-centred-cubic metals during cyclic stress-induced fatigue are shown analytically toengender a substantial nonlinearity in the microelastic-plastic deformation resultingfrom an impressed stress perturbation. The non-Hookean stress–strain relationshipis quantified by a material nonlinearity parameter β that for a given fatigue state ishighly sensitive to the volume fractions of veins and persistent slip bands (PSBs),PSB internal stresses, dislocation multipole configurations, dislocation loop lengths,dipole heights and the densities of secondary dislocations in the substructures. Theeffects on β of vacancy, microcrack and macrocrack formation are also addressed.The connection between β and acoustic harmonic generation is obtained. The modelis applied to calculations of β for fatigued polycrystalline nickel as a function of percent life to fracture. For cyclic stress-controlled loading at 241 MPa, the model pre-dicts a monotonic increase in β of ca. 360% over the fatigue life. For strain-controlledloading at a total strain of 1.75 × 10−3, a monotonic increase in β of ca. 375% overthe fatigue life is predicted.

Keywords: dislocation plasticity; metal fatigue; nonlinearity parameter;substructural organization

1. Introduction

Safe-life and damage-tolerant design philosophies of high-performance structureshave driven the development of various methods that attempt to evaluate non-destructively the accumulation of damage in such structures resulting from cyclicstraining. In most cases the effort focuses on the detection of a ‘flaw of subcriticalsize’ in the material and the monitoring of that flaw to ensure that the size remainssubcritical throughout the service life of the structure. Among the more commonmethods used for such evaluations are dye-penetrant inspections, eddy-current tech-niques, radiographic inspections, acoustic emission and linear ultrasonic techniques.Although these methods have proven useful, none have been able to provide an unam-biguous, quantitative assessment of damage accumulation from the earliest stages ofthe fatigue process or to provide a signature that is characteristic of a given stage offatigue.

† Present address: 245 East Queens Drive, Williamsburg, VA 23185, USA.

Proc. R. Soc. Lond. A (2004) 460, 757–780757

c© 2004 The Royal Society

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758 J. H. Cantrell

The process of cyclic stress-induced fatigue in metals may be roughly divided intofive stages: cyclic hardening/softening, strain localization and microcrack nucleation,propagation or coalescence of microcracks to form macrocracks, macrocrack propaga-tion, and fracture. Early nonlinear acoustical experiments (Buck 1976; Buck & Alers1979) indicate a strong dependence of a generated acoustic harmonic on the num-ber of fatigue cycles for both single-crystal and polycrystalline metals. More recentacoustic harmonic generation measurements (Cantrell & Yost 1994; Na et al . 1996;Nazarov & Sutin 1997; Frouin et al . 1999; Cantrell & Yost 2001) suggest that eachstate of the fatigue process may be characterized by a nonlinear relationship betweenan impressed stress perturbation and a microelastic-plastic straining of the materialthat is quantified by an experimentally determined material (acoustic) nonlinearityparameter. The nonlinearity parameter increases monotonically in the damage regionby hundreds of per cent over the fatigue life of those materials measured. In order togain a proper understanding of the relationship between the value of the nonlinearityparameter and the state of fatigue, it is necessary to understand the contributionto the nonlinearity parameter from each of the process stages. For definiteness, onlywavy slip, polycrystalline, pure metals are considered in the present study.

In wavy slip pure metals the hardening/softening stage is characterized by thegrowth of a vein structure formed by the generation and mutual trapping of dis-locations of opposite sign moving under the action of the impressed cyclic stresses(Basinski et al . 1969; Hancock & Grosskreutz 1969). Saturation of the vein struc-ture occurs at a critical value of dislocation density that signals the formation of apersistent slip band (PSB) structure and the end of the hardening/softening stage(Kuhlmann-Wilsdorf & Laird 1980). Crack nucleation occurs primarily at the inter-section of a PSB with a bounding surface (Brown 1981; Kim & Laird 1978; Mughrabiet al . 1983). The density of microcracks nucleated during the first 20–40% of fatiguelife is substantial (Lukas 1996). For high cycle fatigue the propagation or coalescenceof microcracks into macrocracks typically occurs at 80–90% of the fatigue life. Thefocus of the present research is on the early stages of the fatigue process leading tocrack initiation, although the effects of cracks on the material microelastic-plasticnonlinearity will be briefly addressed.

Model development begins with a consideration of the salient microstructural fea-tures of wavy slip metals developed in response to cyclic loading. The nonlinearrelationship between a perturbative stress imposed on the material and the uniquemicroelastic-plastic straining resulting from specific substructural (vein and PSB)arrangements is then derived. The connection to acoustic harmonic generation isobtained by substituting the derived relationships into Newton’s law. The effects ofvacancies and vacancy clusters resulting from dislocation glide in the PSBs as wellas that of microcrack nucleation on the nonlinearity parameter are considered. Theanalytical model is applied to a calculation of the nonlinearity parameter as a func-tion of the per cent full life for fatigued polycrystalline nickel using microstructuraldata obtained from the literature.

2. Substructural organization in metal fatigue

The present study is restricted to high cycle fatigue in face-centred-cubic (FCC) met-als with wavy dislocation slip (i.e. easy dislocation cross-slip), of which copper and

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Substructural organization, dislocation plasticity and harmonic generation 759

nickel are prototypical. A considerable understanding of the microstructural featuresformed during the fatigue process has been gained from the study of single crys-tals because of the ability to more precisely define the slip geometry and resolvedshear stresses in the material. Relevant reviews can be found in the literature (Christ1996; Neumann 1983; Suresh 1991; Laird 1978). Studies show that the initial cycles ofalternating strain generate dislocations that accumulate on the primary glide plane(Basinski et al . 1969; Hancock & Grosskreutz 1969). Continued cycling generatesmore dislocations, but the to-and-fro motion of dislocations during cyclic loadingpromotes the accumulation of dislocations in the form of mutually trapped primaryedge dislocation dipoles. The mutual trapping occurs as the result of a given dis-location moving into the force field of a second dislocation of opposite sign. Thenetwork of dislocation dipoles formed is often referred to as a vein structure (alsocalled bundles or loop patches). The process of mutual trapping continues until thevein structure is composed almost entirely of edge dislocation dipoles. Screw disloca-tion dipoles are generally annihilated due to cross-slip. The veins have a dislocationdensity of the order 1014–1016 m−2 and are separated by relatively dislocation-poorchannels (dislocation density of the order 1011–1013 m−2) having dimensions roughlyequal to that of the veins (Antonopoulos & Winter 1976). The veins promote an ini-tial hardening of annealed material by impeding the movement of dislocations on theprimary slip plane. The vein structure continues to grow both in dislocation densityand volume fraction (up to ca. 50%) with increasing number of cycles until a criticaldislocation density is attained.

The attainment of a critical dislocation density signals the formation of PSBs(Kuhlmann-Wilsdorf & Laird 1980). The PSBs in single crystals of pure metal arecharacterized by a periodic array of parallel walls forming a ladder structure. Therungs of the ladder (i.e. the parallel walls) are composed primarily of dislocationdipoles. The PSB walls have a dislocation density of the order 1014–1016 m−2 andoccupy a volume fraction of ca. 10% of the PSB. Lying between the PSB wall struc-tures are dislocation-poor channels having densities of the order 1011–1013 m−2. Sincethe vein structure is so hard, most of the plastic strain amplitude imposed duringcyclic loading is developed in the PSBs at this stage of the fatigue process. The PSBsare also characterized by internal stresses acting on secondary dislocations within thestructure (Brown 1981; Antonopolous et al . 1976; Essmann et al . 1981; Mughrabi1978; Sedlacek 1995; Hecker & Burmeister 1996). As the number of fatigue cyclesincreases, the PSBs are somewhat hardened by the generation of secondary disloca-tions. For a sufficiently large plastic strain accumulation, the PSB structure changesfrom a ladder structure into a more cellular structure that continues to deform plas-tically but with a reduced local strain amplitude (Wang & Mughrabi 1984; Wanget al . 1984). From the point of view of continuum mechanics, the formation andevolution of veins and PSBs can be understood as a consequence of instability tran-sitions accompanied by processes of self-organization (Gregor & Kratochvil 1998;Kratochvil 2000, 2001). In this view the plastically deformed solid is modelled asa highly nonlinear system driven far from thermodynamic equilibrium and the dis-location substructures formed during fatigue are characteristic dislocation patternsproduced to engender system restabilization.

Vacancies and vacancy clusters accumulate during fatigue. The formation of vacan-cies is believed to be due either to jog dragging of dislocations (Kennedy 1961) or


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760 J. H. Cantrell

to the annihilation of edge dislocation dipoles (Antonopoulos & Winter 1976; Ess-mann et al . 1981). The accumulation of vacancies and vacancy clusters in the PSBsis associated with the emergence of extrusions at the material surface (Lukas etal . 1968). Some models for crack nucleation and growth suggest that vacancies andvacancy clusters link up as a precursor to microcrack formation (Wood et al . 1963;Woods 1973). Experimentally, it is found that the nucleation of microcracks occursmost frequently at the intersections of PSBs with a material surface (Brown 1981;Kim & Laird 1978; Mughrabi et al . 1983). For high cycle fatigue a large number of‘detectable’ microcracks (0.1–1.0 µm in length) usually form during the first 20–40%of the fatigue life (Lukas 1996).

Studies of single crystals show that PSBs occur in approximately the same volumefraction in the interior of the material as near the surface (Winter 1978). The forma-tion of PSBs occurs first in the bulk of the material before spreading to the surfaceof the crystal. In polycrystalline solids PSB formation is still a bulk phenomenon.However, it is generally confined to individual grains, although spreading can occuracross low-angle grain boundaries. The random orientation of the grains and theconstraints imposed by neighbouring grains produce variations in the substructuralmakeup of an individual grain during fatigue. In studies of polycrystalline copperWinter et al . (1981) found PSB structures in the interior grains with volume frac-tions comparable with that of surface grains. Other studies suggest that the volumefraction of interior grain PSBs is much smaller than that of the surface grains (Christ1996). Recent acoustic harmonic generation data in fatigued aluminium alloy 2024-T4 suggest that PSBs in the bulk of the material are localized but occur in thoseregions with volume fractions expected of surface grains (Cantrell & Yost 2001).Such localization may be explained as occurring when a PSB is first formed. Thesoftening of the material at the PSB site reduces the cross-section of hard materialto which a given load is applied. This reduction produces a larger stress in the hardmaterial and the generation of even more local PSB structure. The localization ofthe PSBs may explain in part the difference in volume fractions of bulk PSBs foundin different studies (Christ 1996; Winter et al . 1981).

The localization of PSBs suggest that the characterization of metal fatigue usingbulk acoustic waves is reflective of the results that would be expected from the basicdislocation substructures (veins and PSBs) formed at the material surface duringfatigue, providing that the waves propagate through bulk material containing thelocalized substructures. This is not true, however, for the bulk acoustic assessmentof microcrack nucleation and growth, since crack nucleation occurs more easily atthe intersection of a PSB and a free surface. In the interior of the material cracknucleation would be limited to that occurring at the boundaries of appropriately‘PSB-conditioned’ grains and at inclusions or matrix-separated precipitates in alloymaterials. Crack nucleation at such appropriately PSB-conditioned grains and inclu-sions has been observed in material near the surface (Kim & Laird 1978; Mughrabiet al . 1983). However, it is shown in § 6 that crack nucleation and growth do not con-tribute significantly to the microelastic-plastic nonlinearity until the crack length orcrack density reaches a critical value. This critical value is not likely to occur for highcycle fatigue of most metals until late in the fatigue life (ca. 80–90% of fatigue life). Itthus seems more effective to concentrate on the effects of PSBs than of microcracksin order to find a useful diagnostic technique for fatigue damage.


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3. Microelastic-plastic nonlinearity in stressed solids

The above considerations suggest that the material nonlinearity associated withdislocation motion in fatigued metals arises primarily from two basic dislocationarrangements that serve as building blocks for all dislocation substructures formedduring fatigue of wavy slip metals. The first is associated with the displacement(bowing) of a dislocation between pinning points under an impressed shear stressas described by the Koehler string model (Koehler 1952). The second arrangementis that of a pair of dislocations forming a dipole that glides along their slip planesin opposite directions as independent monopoles acting under the influence of theirmutual trapping force and an external shear stress (Hull & Bacon 1984).

It is assumed (1) that fatigued metals possess internal (initial) stresses result-ing from cyclic loading and (2) that a longitudinal stress perturbation applied to theinitially stressed material gives rise to a longitudinal microstrain composed of an elas-tic strain contribution and two independent plastic strain contributions. The plasticstrain contributions result from the independent motions of dislocation monopolesand dislocation dipoles. It is appropriate then to consider the strains referred to someinitially stressed configuration of the material and to denote the positions of the par-ticles of the solid in the initially stressed configuration by the set of particle vectorsX. The total particle displacement u with respect to the initial configuration isassumed to result from the sum of the component displacements as

u = ue + ump + udp, (3.1)

where ue is the elastic contribution, ump is the dislocation monopole componentand udp is the dislocation dipole contribution. Hence, the total longitudinal strainε = ∂u/∂X in a given region of the solid is related to the elastic and two plasticstrain components as

ε = εe + εmp + εdp, (3.2)

where εe = ∂ue/∂X is the longitudinal elastic strain, εmp = ∂ump/∂X is the resolvedlongitudinal plastic strain due to dislocation monopoles, and εdp = ∂udp/∂X is theresolved longitudinal plastic strain due to dislocation dipoles.

It is important to note that the strains defined in equation (3.2) are referred to theinitially stressed configuration of the material. In order to quantify properly the con-tributions of the various fatigue-induced microstructures by a material nonlinearityparameter, it is necessary to refer the parameters and the strains in equation (3.2) tothe zero stress state of the material (i.e. the configuration of the material in the virginstate). It is also necessary to consider the specific contributions from lattice elastic-ity, dislocation monopole plasticity, and dislocation dipole plasticity from which thefatigue-related microstructural contributions to the nonlinearity are derived. Theseconsiderations are addressed below in terms of the transformation of the stress–strain relationships from the initial configuration of the material to a zero stressconfiguration.

(a) Contribution from lattice elasticity

In order to obtain properly the transformation relationships between the initialconfiguration and the zero stress configuration it is appropriate to consider first some


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762 J. H. Cantrell

basic elements of three-dimensional anisotropic elasticity. The anisotropic transfor-mation relationships are then specialized to isotropic materials and the elastic com-ponent of the material nonlinearity parameter referred to the zero stress configurationis derived.

For present purposes an elastic solid is considered to be a material in which thethermodynamic state of equilibrium depends on the fixed mean configuration of theparticles comprising the material and the entropy S of the system. The relevantstate function describing the system of particles is the internal energy per unit massU(x, S). During a deformation of the solid, the present configuration of the materialparticles is denoted by the set of particle vectors x. It is assumed that the presentconfiguration of material particles is obtained by means of a small (infinitesimal)elastic deformation from some arbitrary initial configuration of particles (with ini-tial stress) denoted by the set of particle vectors X. The elastic deformation isdescribed by the set of transformation coefficients αe

ij = ∂xi/∂Xj . The indices i and jtake the values 1, 2, 3, representing three mutually orthogonal spatial reference axes.The resulting particle displacements ue

i are defined by uei = xi − Xi and the transfor-

mation coefficients αeij = δij + ue

ij , where the displacement gradients ueij = ∂ue

i/∂Xj

and δij is the Kronecker delta. The deformation results in a change in the massdensity of the solid described by the relation (Wallace 1970)

ρ1

ρ= J = det[αe

ij ] = 1 + ueii + · · · , (3.3)

where ρ1 = ρ1(X) is the mass density in the initial state, ρ = ρ(x) is the massdensity in the final state, and J is the Jacobian. We assume in all equations theEinstein summation convention over repeated indices, unless indicated otherwise.

The elastic stresses considered here are components of the first Piola–Kirchhofftensor σij defined by (Thurston 1964)

σij = ρ1αik∂U

∂ηkj= ρ1

∂ηkl

∂uij

∂U

∂ηkl= ρ1

∂U

∂ueij

, (3.4)

where ηkj = (αelkαe

lj − δkj)/2 = (uekj + ue

jk + uelkue

lj)/2 are the Lagrangian strains. Itis seen from equation (3.4) that σij can be expressed in terms of derivatives ofthe internal energy U with respect to the Lagrangian strains ηkj or with respectto the displacement gradients ue

ij . Since the internal energy per unit mass dependsonly on the relative positions of the particles comprising the solid, equation (3.4)suggests that U(x, S) may be regarded to be a function of the Lagrangian strainsas U(x, S) = U(X, ηij , S) or to be a function of the displacement gradients asU(x, S) = U(X, ue

ij , S).It will prove expedient to consider that the internal energy per unit mass is

a function of the displacement gradients. A power series expansion of U(x, S) =U(X, ue

ij , S) about X (i.e. about ueij = 0) to third order in ue

ij yields (Huang 1950)

ρ1U(x, S) = ρ1U(X, ueij , S)

= ρ1U(X, 0, S) + Aeiju

eij + 1

2Aeijklu

eiju

ekl + 1

6Aeijklmnue

ijueklu

emn + · · · ,

(3.5)


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where

Aeij = ρ1

(∂U

∂ueij

)X,S,u=0

, (3.6)

Aeijkl = ρ1

(∂2U

∂ueij∂ue

kl

)X,S,u=0

, (3.7)

Aeijklmn = ρ1

(∂3U

∂ueij∂ue

kl∂uemn

)X,S,u=0

. (3.8)

The Aeij , Ae

ijkl and Aeijklmn are the isentropic first-, second- and third-order Huang

coefficients.Substitution of equation (3.5) into equation (3.4) yields the relation between the

first Piola–Kirchhoff stresses σij in the present configuration x and the displacementgradients referred to the initial configuration X as

σij = ρ1∂U

∂ueij

= (σij)X + Aeijklu

ekl + 1

2Aeijklmnue

kluemn + · · · . (3.9)

It is important to note in equation (3.9) that the (σij)X are the initial stresses inthe material at X (i.e. at ue

ij = 0) and that the Huang coefficients also are referred tothe initial stress configuration X. Although the first Piola–Kirchhoff stresses and theHuang coefficients are not symmetrical, they will prove convenient in deriving waveequations referred to material (Lagrangian) coordinates and in providing consistencyin definition via the displacement gradients when comparing microelastic strains withmicroplastic ones.

Equation (3.9) may be simplified by introducing an orthogonal transformation Rdefined by

ai = RijXj (3.10)

that rotates the X1-axis into the transformed axis direction a1 (the direction chosenin § 4 to be the wave propagation direction). The components of the column vectorR1i (i = 1, 2, 3) of the transformation matrix are the direction cosines of the directiona1 with respect to the set of axes Xi (i = 1, 2, 3). Thus R1i = Ni, where Ni are theCartesian components with respect to the Xi-axes of the unit vector N along a1.Application of this transformation to equation (3.9) yields

σ′pq = RpiRqjσij = (σ′

pq)X + A′epqr1u

′er1 + 1

2A′epqr1t1u

′er1u

′et1 + · · · , (3.11)

where

A′epqr1 = RpiRqjRrkR1lA

eijkl, (3.12)

A′epqr1t1 = RpiRqjRrkR1lRtmR1nAe

ijklmn. (3.13)

To simplify equation (3.11) further, it is convenient to choose q = 1 such that thesecond-order transformed Huang coefficient A′e

p1r1 is then symmetrical. This allowsthe introduction of a second orthogonal transformation S defined by

u′ei = SikPk, (3.14)

which diagonalizes A′ep1r1. The row vector Spα (p = 1, 2, 3 for a given α) represents the

direction of particle displacement with respect to the transformed reference frame ai.


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764 J. H. Cantrell

Hence, RpiSpα = Ui are the Cartesian coordinates of the unit particle displacementvector U with respect to the initial reference frame Xi for a given α. Application ofthe transformation S to equation (3.11) yields (no sum on α, β)

σRγ1 = (σR

γ1)X + µαγ,NPα1 + 12ναβγ,NPα1Pβ1 + · · · , (3.15)

where Pα1 = ∂Pα/∂a1, σRγ1 = Spγσ′

p1 and

µαγ,N = R1lR1j(RpiSpγ)(RrkSrα)Aeijkl, (3.16)

ναβγ,N = R1jR1lR1n(RpiSpγ)(RrkSrα)(RtmStβ)Aeijklmn. (3.17)

The terms corresponding to α = β in equation (3.17) are orders of magnitudelarger than those for which α = β. It is thus convenient to keep only the termscorresponding to α = β in equation (3.15). The resulting equation can be writtenin a form that is expedient to the derivation of a wave equation by choosing γ = α.The constitutive stress–strain relation then becomes

σRα1 = (σR

α1)X + µε(Pα1 − 12βe

ψP 2α1 + · · ·), (3.18)

where the material elastic nonlinearity parameter βεψ is defined as

βeψ = −νααα,N

µαα,N= −

AeijklmnNjNlNnUiUkUn

AeijklNjNlUiUk

. (3.19)

The last equality in equation (3.19) results from substituting into equations (3.12)and (3.13) the relations R1i = Ni for the Cartesian components of the unit vec-tor N along a1 and RpiSpα = Ui for the Cartesian coordinates of the unit particledisplacement vector U , all measured with respect to the initial reference frame Xi.The subscripted symbol ψ = (α, N) is an index designating a selected particle dis-placement U (corresponding to the choice α = 1, 2 or 3) for a chosen directionN . Thus, the choice α = 1 corresponds to a quasi-longitudinal strain (stress) andα = 2, 3 correspond to each of two independent quasi-shear strains (stresses) for achosen direction N in a crystal. The material elastic nonlinearity parameter definesthe first-order deviation from the Hookean stress–strain relationship.

In order to assess the effect of an initial stress (or strain) on the magnitude ofthe material elastic nonlinearity parameter defined by equation (3.19), we recastthe Huang coefficients, equations (3.6)–(3.8), used to define the elastic nonlinearityparameter of equation (3.19) in terms of Huang coefficients referred to the zero stressconfiguration. The position of a material particle, which is denoted above by X in theinitial stress configuration and x in the final deformed configuration, is denoted byX in the zero stress configuration. An overbar is used to denote quantities referred tothe zero stress configuration. We thus write Ae

ijkl for the second-order and Aeijklmn for

the third-order Huang coefficients referred to the zero stress state. It is convenient todefine the Huang coefficients of any order n referred to the zero stress configurationby

Aeijkl··· = ρ0

(∂nU

∂ueij∂ue

kl · · ·

)X,S,u=0

, (3.20)

where the mass density ρ0 = ρ(X).


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The relationships between the Huang coefficients referred to the initially stressedstate and those referred to the zero stress state may be obtained by writing

Aeijkl··· = ρ1

(∂(n)U

∂ueij∂ue

kl . . .

)X,s,u=0

=ρ1

ρ0

[ρ0

(∂(n)U

∂uepq∂ue

rs · · ·

)(∂ue

pq

∂ueij

∂uers

∂uekl

· · ·)]

X,s,u=0. (3.21)

For small variations in ueij the derivative of ue

pq with respect to ueij in equation (3.21)

may be approximated as (∂ue

pq

∂ueij

)X,S,u=0

≈ aeipa

ejq, (3.22)

where

aeij = (αe

ij)X =(

∂xi

∂Xj

)X

= δij + (ueij)X =

∂Xi

∂Xj. (3.23)

With respect to the zero stress configuration, the first Piola–Kirchhoff stress tensoris written as

σij = ρ0∂U

∂ueij

= Aeijklu

ekl + 1

2 Aeijklmnue

kluemn + · · · . (3.24)

From equations (3.21)–(3.23) we obtain

Aeijkl··· =

ρ1

ρ0

[ρ0

(∂(n)U

∂uepq∂ue

rs · · ·

)(ae

ipaejqa

ekra

els · · · )

]X

=ρ1

ρ0

[(ae

ipaejqa

ekra

els · · · )

(∂(n−1)σpq

∂uers · · ·

)]X

. (3.25)

Thus, the first three Huang coefficients referred to the initial configuration X maybe expressed in terms of Huang coefficients referred to the zero stress configurationX as

Aeij = (σij)X =

ρ1

ρ0(ae

ipaejq)[A

epqrs(u

ers)X + · · · ], (3.26)

Aeijkl =

ρ1

ρ0(ae

ipaejqa

ekra

els)[A

epqrs + Ae

pqrstu(uetu)X + · · · ], (3.27)

Aeijklmn =

ρ1

ρ0(ae

ipaejqa

ekra

elsa

emta

enu)[Ae

pqrstu + Aepqrstuvw(ue

vw)X + · · · ]. (3.28)

For polycrystalline solids in the zero stress configuration, the random orientationof the grains (and precipitates in alloys) results in a quasi-isotropic elastic symmetry.For simplicity we assume that if a symmetry change occurs at all during uniaxialfatigue of polycrystalline solids the resulting symmetry is cylindrical. Specifically,it is assumed that such cylindrical symmetry results from initial configurationalstresses generated during fatigue that produce an average uniaxial stress havinga value different from the average radial stress. For either isotropic or cylindricalsymmetries, the ψ = (1, N = [100]) mode material elastic nonlinearity parameter


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766 J. H. Cantrell

along the uniaxial direction referred to the initial stress configuration is obtainedfrom equation (3.19) to be

βe = βe1,[100] = −Ae

111

Ae11

, (3.29)

where the Huang coefficients are written using the Voigt contraction of indices.Since experimental data are more commonly reported in terms of zero stress state

elastic constants, it is convenient to express the Huang coefficients Ae11 and Ae

111in equation (3.29) referred to the initial stress configuration in terms of the initialstresses (σi)X (Voigt notation) and the zero stress state Huang coefficients. Thisis accomplished by substituting equation (3.23) into equations (3.26)–(3.28). Therelation ue

ij ≈ Seijkl(σkl)X (full notation), where Se

ijkl are the compliance coefficientsreferred to the zero stress configuration, is then used to obtain

Ae11 =

ρ1

ρ0Ae

11 + [4Ae11S

e11 + Ae

111Se11 + 2Ae

112Se12](σ1)X

+ [4Ae11S

e12 + Ae

111Se12 + Ae

112(Se11 + Se

12)](σ2 + σ3)X (3.30)

and

Ae111 =

ρ1

ρ0Ae

111 + [6Ae111S

e11 + Ae

1111Se11 + 2Ae

1112Se12](σ1)X

+ [6Ae111S

e12 + Ae

1111Se12 + Ae

1112(Se11 + Se

12)](σ2 + σ3)X. (3.31)

Again, the Voigt contraction of indices is used for the elastic constants and thestress in equations (3.30) and (3.31). We also have assumed isotropic symmetry.Substituting equations (3.30) and (3.31) into equation (3.29), expanding the resultingexpression, and keeping only terms to first order in the stress, we obtain that theacoustic nonlinearity parameter referred to the initial stress configuration may beexpressed in terms of the initial stresses and the zero stress state Huang coefficientsas

βe = βe1 + M(σ1)X + N [(σ2)X + (σ3)X ], (3.32)

where

M =(

Ae1111

Ae111

− Ae111

Ae11

+ 2)

Se11 + 2

(Ae

1112

Ae111

− Ae112

Ae11

)Se

12, (3.33)

N =(

Ae1111

Ae111

− Ae111

Ae11

+ 2)

Se12 +

(Ae

1112

Ae111

− Ae112

Ae11

)(Se

11 + Se12), (3.34)

βe = − Ae111

Ae11

. (3.35)

The parameter βe in equations (3.32) and (3.35) is the material elastic nonlinearityparameter referred to the zero stress configuration.

(b) Contribution from dislocation monopoles

We consider some density Λmp of isolated single dislocations (dislocation mono-poles) in the material. In particular, we assume edge dislocations lying in arbitrary


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slip planes in grains of arbitrary orientation in a polycrystalline solid. For a givendislocation a longitudinal stress σ = σ1 (Voigt notation) applied to the solid andreferred to the zero stress configuration will give rise a shear stress τ at the site ofthe dislocation of magnitude τ = Rσ along its slip direction, where R is the Schmidor resolving factor. If the dislocation is pinned at points a distance 2L apart, thelength of dislocation between the pinning points will bow out under the action ofthe resolved shear stress like an arched string. The movement (bowing) of the dis-location produces a plastic shear strain γmp in the material. The superscript ‘mp’denotes monopole here. With respect to the zero stress configuration X, Hikata etal . (1965) have shown that the relationship between the plastic shear strain and theshear stress for the dislocation motion is

γmp =23

(ΛmpL2

G

)τ +

45

(ΛmpL4

G3b2

)τ3 + · · · , (3.36)

where b is the amplitude of the Burgers vector. If we assume that the relationshipbetween the plastic strain γmp and the longitudinal strain εmp is εmp = Ωγmp, whereΩ is the conversion factor from shear strain to longitudinal strain, then we obtainfrom equation (3.36) and the relation τ = Rσ that

εmp =(

∂εmp

∂σ

)X

σ +16

(∂3εmp

∂σ3

)X

σ3 + · · · , (3.37)

where (∂εmp

∂σ

)X

=23

(ΩΛmpL2R

G

), (3.38)

(∂3εmp

∂σ3

)X

=245

(ΩΛmpL4R3

G3b2

). (3.39)

With respect to the initial stress configuration X, we write

εmp =(

∂εmp

∂σ

)X

(σ − σX) +12

(∂2εmp

∂σ2

)X

(σ − σX)2 + · · · , (3.40)

where (∂εmp

∂σ

)X

=(

∂σ

∂εmp

)−1

X

, (3.41)

(∂2εmp

∂σ2

)X

= −[(

∂2σ

∂(εmp)2

)(∂σ

∂εmp

)−3]X

. (3.42)

The coefficients defined in equations (3.41) and (3.42), referred to the initial stressconfiguration, may be expressed in terms of the coefficients defined in equations (3.38)and (3.39), referred to the zero stress configuration, by using a transformation equa-tion analogous to equation (3.21) for the Huang coefficients. The internal energy perunit mass in equation (3.21) is replaced here by the energy per unit mass associatedwith dislocation bowing against the dislocation line tension. We assume that thenetwork of dislocations behaves as a material having isotropic symmetry. We con-sider here only uniaxial longitudinal stresses (σ1)X = σX . Finally, we assume that


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the mass density of material associated with the dislocations does not change signif-icantly as a consequence of dislocation motion. Hence, in very good approximationρ1/ρ0 = 1 for a dislocation network and we obtain from equations (3.41) and (3.42)and the dislocation monopole analogue to equation (3.21) that(

∂εmp

∂σ

)X

= (amp)−4(

∂εmp

∂σ

)X

≈ 23

(ΩΛmpL2R

G

)− 16

9

(ΩΛmpL2R

G

)2

σX + · · · , (3.43)(

∂2εmp

∂σ2

)X

= (amp)−6(

∂2εmp

∂σ2

)X

≈ 245

(ΩΛmpL4R3

G3b2

)σX + · · · , (3.44)

where amp = 1 + εmp is the transformation coefficient for dislocation monopolescorresponding to the elastic transformation coefficients ae

ij in equations (3.25)–(3.28).

(c) Contribution from dislocation dipoles

For edge dislocation pairs of opposite polarity (vacancy or interstitial dipoles) theforce per unit length Fx along the glide plane (shear force per unit length) on a givendislocation referred to the zero stress state due to the other dislocation in the pairis given as (Hull & Bacon 1984)

Fx = − Gb2

2π(1 − ν)x(x2 − y2)(x2 + y2)2

, (3.45)

where G is the shear modulus, b is the Burgers vector, ν is Poisson’s ratio and(x, y) are the Cartesian coordinates of one dislocation in the pair relative to thecoordinates (0, 0) of the second. It is assumed that motion of the dipole pairs occursonly along parallel slip planes (i.e. along the x-direction) separated by a distancey = h, which we shall call, following Antonopoulos et al . (1976), the equilibriumdipole height. At equilibrium, with no residual or applied stresses, equation (3.45)asserts that x = ±y = ±h. The resolution of a longitudinal stress perturbation σalong the slip planes produces a shear force per unit length bRσ on the dipole pair,where again R is the longitudinal-to-shear conversion factor. It is assumed that inequilibrium the total shear force per unit length on the dipole Fx + bRσ = 0. Therelationship between the longitudinal plastic strain ε dp and the relative dislocationdisplacement ζ is given by ε dp = ΩΛdpbζ, where Ω is the conversion factor fromthe dislocation displacement in the slip plane to longitudinal displacement alongan arbitrary direction and Λdp is the dipole dislocation density. Let ζ = (x − h) bethe relative displacement of the dislocations in the dipole pair with respect to theequilibrium position h. Expanding equation (3.45) in a power series in x with respectto h and using the above relationships between Fx and σ, ε dp and ζ, and ζ and x,we obtain to second order in ε dp

σ = Adp2 ε dp + 1

2 Adp3 (ε dp)2 + · · · , (3.46)

where

Adp2 = −

(G

4πΩRΛdph2(1 − ν)

)(3.47)


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and

Adp3 =

(G

4πΩ2R(Λdp)2h3(1 − ν)b

). (3.48)

With respect to the initial stress configuration X, we write in analogy to equa-tion (3.15)

σ = σX + Adp2 εdp + 1

2Adp3 (εdp)2 + · · · . (3.49)

In a manner similar to that for dislocation monopoles, the coefficients in equa-tions (3.49), referred to the initial stress configuration, may be expressed in terms ofthe coefficients defined in equations (3.47) and (3.48), referred to the zero stress con-figuration, by using a transformation equation analogous to equation (3.25) for theHuang coefficients. The internal energy per unit mass in equation (3.25) is replacedhere by the energy per unit mass associated with the relative motion of dislocations inthe dipole pair against their mutual trapping force. Again, we consider only uniaxiallongitudinal stresses and assume that the network of dipole dislocations behaves asa material having isotropic symmetry. We assume that the mass density of materialassociated with the dislocation dipoles does not change significantly as a consequenceof dislocation motion. Hence, ρ1/ρ0 ≈ 1 and we obtain in analogy to equation (3.25)

Adp1 = σX = (adp)2(Adp

2 ε dp + · · · ) ≈ σX = Adp2 ε dp + · · · , (3.50)

Adp2 = (adp)4[Adp

2 + Adp3 (ε dp)X + · · · ], (3.51)

Adp3 = (adp)6[Adp

3 + Adp4 (ε dp)X + · · · ]. (3.52)

With respect to the initial stress configuration X, we obtain from equation (3.49)

εdp =1

Adp2

(σ − σX) − 12

Adp3

(Adp2 )3

(σ − σX)2 + · · · . (3.53)

(d) Material nonlinearity in an initially stressed solid with dislocations

As indicated in equation (3.2), we assume that the total strain in fatigued materialis composed of an elastic strain component and two plastic strain components. Theelastic strain component εe is obtained from equation (3.9) or equation (3.28) byspecializing the equation to longitudinal stresses and strains in a material of isotropicsymmetry, keeping terms to second order in the strain, and solving the resultingquadratic equation for the strain in terms of the stress. Expanding the resultingequation to second order in the stress and writing σ = σ11 = σR

11, εe = ue11 =

∂ue1/∂X1 = ∂ue/∂X = ∂P1/∂a1, Ae

2 = Ae11 and Ae

3 = Ae111, we obtain

εe =1

Ae2(σ − σX) − 1

2Ae

3

(Ae2)3

(σ − σX)2 + · · · . (3.54)

From equations (3.2), (3.40), (3.53) and (3.54) we obtain the total strain as

ε = εe + εmp + εdp = Q(σ − σX) + R′(σ − σX)2 + · · · , (3.55)

where

Q =1

Ae2

+1

Adp2

+(

∂εmp

∂σ

)X

(3.56)


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and

R′ = −12

[Ae

3

(Ae2)3

+Adp

3

(Adp2 )3

−(

∂2εmp

∂σ2

)X

]. (3.57)

The total material nonlinearity parameter is defined from the coefficients of thestress expanded in terms of the total strain in analogy to equations (3.18) and (3.19)for the elastic nonlinearity parameter. We thus keep terms to second order in thestress in equation (3.55) and solve the quadratic expression for the stress in termsof the total strain. Expansion of the resulting equation in terms of the total strainyields

σ = σX +1Q

ε − R′

Q3 ε2 + · · · . (3.58)

From equations (3.56)–(3.58) and a comparison of equations (3.58) and (3.18), thetotal material nonlinearity parameter β is defined as

β =2R′

Q2 = − βe + βmp + βdp

[1 + Γmp + Γ dp]2, (3.59)

where the elastic nonlinearity parameter βe is defined as

βe = −Ae3

Ae2, (3.60)

the dislocation monopole nonlinearity parameter βmp is defined as

βmp = (Ae2)

2(

∂2εmp

∂σ2

)X

, (3.61)

and the dislocation dipole nonlinearity parameter βdp is defined as

βdp = −(Ae2)

2 Adp3

(Adp2 )3

. (3.62)

The monopole and dipole gamma factors in the denominator of equation (3.59) aregiven, respectively, by

Γmp = Ae2

(∂εmp

∂σ

)X

(3.63)

and

Γ dp =Ae

2

Adp2

. (3.64)

It is assumed in the derivation of equation (3.59) that the dislocation monopolesand dipoles are distributed uniformly throughout the material. Generally, neitherthe monopoles nor the dipoles are distributed uniformly but occur within the dis-crete fatigue-generated substructures in the material. We denote the total volumefraction of substructure containing monopoles by fmp and a total volume fraction ofsubstructure containing dipoles by fdp. If it is assumed that the values of the mono-pole and dipole nonlinearity parameters, βmp and βdp, and gamma factors, Γmp and


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Γ dp, are constant within the substructures, then the total nonlinearity parameter βis more appropriately written

β =βe + fmpβmp + fdpβdp

(1 + fmpΓmp + fdpΓ dp)2. (3.65)

Finally, it is noted that the nonlinearity parameters and gamma factors defined inequations (3.60)–(3.64) are those referred to the initial stress configuration. In thefollowing sections it will be convenient to have expressions for βe, βmp, βdp, Γmp andΓ dp that involve parameters referred to the zero stress configuration. The expressionfor βe in terms of parameters referred to the zero stress configuration is given byequations (3.32)–(3.35). The corresponding expression for βmp can be obtained fromequations (3.30), (3.42) and (3.52) to first order in the initial stress σX referred tothe zero stress configuration as

βmp =245

ΩΛmpL4R3(Ae11)

2

G3b2 |σX |, (3.66)

where from equation (3.30) we have written Ae2 = Ae

11 ≈ A11 = Ae2. The absolute

value of σX in equation (3.66) results from the odd symmetry of equation (3.36).The expression for βdp can be obtained to first order in the initial stress σX fromequations (3.30), (3.50)–(3.52) and (3.62) as

βdp = βdp[1 + NσX ], (3.67)

where

βdp =16π2ΩR2Λdph3(1 − ν)2(Ae

11)2

G2b(3.68)

and

N =Adp

4

Adp2 Adp

3

− 3Adp

3

(Adp2 )2

− 6

Adp2

+2

Ae11

(3Ae11S

e11 + Ae

111Se11 + 2Ae

112Se12). (3.69)

For typical values of the fatigue parameters, NσX is much less than unity. Hence, toan excellent approximation we may write from equation (3.67) that βdp = βdp. Thegamma factors are given as

Γmp = Ae2

(∂εmp

∂σ

)X

=23

(ΩΛmpL2R

G

)Ae

11 (3.70)

and

Γ dp =Ae

2

Adp2

= −4πAe11ΩRΛdph2(1 − ν)

G. (3.71)

4. Material nonlinearity and acoustic harmonic generation

The results of § 3 yield results appropriate to the application of a stress perturba-tion to a fatigued wavy slip metal and the resulting microelastic-plastic deformationrepresented by the total strain ε. Measurements of such small deformations requirevery sensitive measurement techniques. One such technique is that of acoustic har-monic generation. A mathematical description of acoustic harmonic generation may


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be obtained by writing the equations of motion, neglecting body forces and refer-ring to Lagrangian coordinates X for a solid of arbitrary crystalline symmetry, as(Wallace 1970)

ρ1∂2ui

∂t2=

∂σij

∂Xj, (4.1)

where t is time. For simplicity we consider only longitudinal wave propagation inan isotropic medium (strictly, quasi-isotropic polycrystalline solids) and drop thesubscripts in equation (4.1). Equation (3.61) gives the relation between the stress andthe total strain in the material. Substitution of equation (3.61) into equation (4.1)yields the nonlinear wave equation

∂2u

∂t2= c2 ∂2u

∂X2 − βc2 ∂u

∂X

∂2u

∂X2 , (4.2)

where c = (ρ1Q)−1/2 is the sound velocity.A solution to equation (4.2), assuming an input wave of the form u0 cos(ka − ωt)

at X = 0, where u0 is the wave displacement amplitude, k is the wavenumber, andω is the angular frequency, is given to second order in the displacement as (Wallace1970; Cantrell 1984)

u = 18βk2u2

0X + u0 cos(kX − ωt) − 18βk2u2

0X cos[2(kX − ωt)] + · · · . (4.3)

It is clear from equation (4.3) that the material nonlinearity parameters providea quantitative measure of acoustic harmonic generation (Wallace 1970) as well asthat of a generated static displacement (Cantrell 1984). The nonlinearity parameterscan be obtained from absolute amplitude measurements of the fundamental andeither the generated second harmonic or static displacement signals. Details of theexperimental arrangement, sample preparation and measurement methods are givenelsewhere (Cantrell & Salama 1991; Yost & Cantrell 1992).

5. Dependence of the nonlinearity parameter of nickelon the state of fatigue

Grobstein et al . (1991) published an extensive study of fatigue damage accumulationin pure polycrystalline nickel as a function of the per cent full life from the virgin stateto fracture. They performed transmission electron microscopy, positron annihilationlifetime measurement spectroscopy and electrical resistivity measurements of thedislocation substructures, vacancies and vacancy clusters formed at various timesduring the fatigue process for a wide range of loading conditions. We consider here theapplication of their findings to a calculation of the material nonlinearity parameterfor specimens fatigued using two different, fully reversed, four-point-bending, loadingconditions: (1) stress-controlled loading at a stress of 241 MPa and (2) total strain-controlled loading at a total strain of 1.75 × 10−3.

(a) Substructural contributions to the nonlinearity parameter

For stress-controlled loading conditions, Grobstein et al . (1991) report that the dis-location vein structure attains a volume fraction of 0.35 at 10% full life and remainsat that volume fraction to fracture. PSBs begin to form at 0.1% full life and mono-tonically grow with decreasing slope to a volume fraction of 0.158 at full life. The


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dislocation density in the PSB walls is measured to be 1.7 × 1015 m−2. According toWinter (1974) and Mughrabi (1978) the dislocation density in the vein structure isapproximately one-half that in the PSB walls or ca. 1×1015 m−2 in the present case.The vacancy concentration is found to grow approximately logarithmically from zeroat the virgin state to a value of 0.55 × 10−4 at 0.1% full life. Thereafter, it growslogarithmically with a reduced slope to a value of 1.19 × 10−4 at full life.

For total strain-controlled loading we estimate from the data of Grobstein et al .(1991) that the dislocation vein structure attains a volume fraction of ca. 0.4 atca. 10% full life. It is assumed that the vein structure remains at that volume frac-tion to fracture. PSBs begin to form at 0.7% full life and monotonically grow withincreasing slope to a volume fraction of 0.217 at full life. The dislocation density in thePSB walls is measured to be 2.1× 1015 m−2. Following Winter (1974) and Mughrabi(1978), we estimate the density in the vein structure to be ca. 1.1 × 1015 m−2. Thevacancy concentration is found to grow approximately logarithmically from zero atthe virgin state to a value of 1.1 × 10−4 at 0.7% full life. Thereafter, it grows loga-rithmically with a reduced slope to a value of 1.28 × 10−4 at full life.

The vein structure is characterized by short-range stresses resulting from the relax-ation of the dislocation substructure into its equilibrium arrangement. The dislo-cation monopole contribution to the nonlinearity parameter from the vein struc-ture βmp

vein is obtained from equation (3.66). For nickel we assume (Brown 1981)G = 73.2 GPa, A11 = 278 GPa, ν = 0.39 and b = 0.249 nm. For polycrystallinesolids it is reasonable to assume Ω = R = 0.33. The loop length L that appearsin equation (3.66) is raised to the fourth power. Antonopolous et al . (1976) reportmeasurements of the loop lengths of dislocations in fatigued copper single crystals.Assuming that their measurements are also valid for polycrystalline nickel, we cal-culate an appropriate value of L by taking the fourth root of the averaged fourthpower of the loop lengths. We obtain L = 8.2 × 10−8 m. The internal (initial) shearstress τ experienced by a given dislocation in the vein structure is estimated tobe in the range 5–20% of the saturation stress (Mughrabi 1981; Brown 2000). Weassume an internal shear stress of 4 MPa for nickel. The effective longitudinal stress|σX | that appears in equation (3.66) is obtained as |σX | = τ/R = 12 MPa. Fromequation (3.66), the dislocation monopole contribution from the vein structure iscalculated to be βmp

vein = 102 for stress-controlled loading and βmpvein = 112 for strain-

controlled loading. The gamma factor for dislocation monopoles in the vein structureis calculated from equation (3.70) to be Γmp

vein = 1.89 for stress-controlled loading andΓmp

vein = 2.08 for strain-controlled loading.As indicated in equation (3.68) the contribution to the nonlinearity parameter

from dislocation dipoles in the vein structure is not dependent on the initial stresses.The dislocation dipole height h that appears in equation (3.68) is raised to the thirdpower. Antonopolous et al . (1976) report measurements of the dipole heights infatigued copper single crystals. Assuming that their measurements are also valid forpolycrystalline nickel, we calculate an appropriate value of h for equation (3.68) bytaking the cube root of the averaged third power of the dipole heights. We obtainhvein = 7.6 nm for vein structure. It is assumed that the dislocation dipole den-sity in mature veins and PSBs is roughly one-half the dislocation monopole densitymeasured in these structures. Hence, Λdp

vein = 5 × 1014 m−2 for the dipole density instress-controlled loading and Λdp

vein = 5.5 × 1014 m−2 for total strain-controlled load-ing. Substitution of the above values into equation (3.68) yields βdp

vein = 27.7 for the


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value of the dipole contribution to the nonlinearity parameter associated with thevein structure in stress-controlled loading and βdp

vein = 30.4 for total strain-controlledloading. The gamma factor for dislocation dipoles in the vein structure is calculatedfrom equation (3.71) to be Γ dp

vein = 0.28 for stress-controlled loading and Γ dpvein = 0.31

for strain-controlled loading.The dipole contribution to the nonlinearity parameter associated with the PSB

wall (ladder) structure is also obtained from equation (3.68) using the same valuesof the nickel material parameters as that for the vein structure except that for PSBwalls the dislocation density Λdp

PSBw = 8.5 × 1014 m−2 for stress-controlled loadingand Λdp

PSBw = 1.1 × 1015 m−2 for total strain-controlled loading. The dipole heighthPSB = 5.4 nm is calculated from data of Antonopoulos et al . (1976), Winter (1974)and Mughrabi (1978). Using these values, we obtain the PSB wall dipole contri-bution to be βdp

PSBw = 16.9 for stress-controlled loading and βdpPSBw = 21.8 for total

strain-controlled loading. The gamma factor for dislocation dipoles in the PSB wallstructure is calculated from equation (3.71) to be Γ dp

PSBw = 0.08 for stress-controlledloading and Γ dp

PSBw = 0.10 for strain-controlled loading.A substantial contribution to the nonlinearity parameter is also obtained from the

action of initial (internal) stresses on secondary dislocations in PSBs (i.e. disloca-tions generated on secondary slip systems as the PSBs mature). According to Brown(1981), the internal stresses in PSBs that are sufficiently large to promote secondaryhardening in the material correspond to an elastic tensile strain of ca. 5×10−4 alongthe primary slip direction in the PSBs. For nickel this strain corresponds to a tensilestress σPSB of ca. 102 MPa. The absolute value of the longitudinal component σ ofthis tensile stress along the wave propagation direction is given by |σ| = |σPSB|| cos φ|,where φ is the angle between the PSB primary slip direction and the direction ofultrasonic wave propagation. For a polycrystalline solid, the slip direction is randomlyoriented and the average value of | cos φ| is 0.5. Thus, the absolute value of the lon-gitudinal component of the internal PSB tensile stress along the wave propagationdirection (i.e. |σX | in equation (3.66))] is ca. 51 MPa.

Finally, according to Brown (1981) the tensile stresses that develop along theslip direction in the PSBs do not interact with the dislocations in the primaryslip plane. Rather, they interact with secondary dislocations. Transmission electronmicrographs obtained by Wang & Mughrabi (1984) and Wang et al . (1984) sug-gest that the density of secondary dislocations approaches that of the PSB wallstructure (ca. 2 × 1015 m−2) as mature wall structures transform into misorientedcells during secondary hardening. We assume here a secondary dislocation density of1.0 × 1015 m−2. Using these values together with the values of the elastic constants,loop length, Burgers vector, R and Ω used in the preceding calculations, we calcu-late from equation (3.66) the contribution to the nonlinearity parameter from PSBstresses to be βmp

PSBσ = 652. The gamma factor for dislocation monopoles associatedwith secondary dislocations is calculated from equation (3.70) to be Γmp

PSBσ = 0.95.

(b) Vacancy contribution to nonlinearity parameter

The contribution to the nonlinearity parameter from vacancies and vacancy clus-ters may be estimated from an expression obtained by Cantrell & Yost (2000) toassess the effects of precipitate-matrix coherency strains on acoustic harmonic gen-eration. For vacancies and vacancy clusters their expression may be approximated


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as

βvac ≈ 248ΛmpΩR3r4

crit(Ae11)

2(1 + ν)|δ|G2b2 f−1/3, (5.1)

where δ is the misfit parameter between the vacancy and the solid material, rcrit isthe radius of a single vacancy (ca. b/2) and f is related to the volume fraction orconcentration of vacancies in the material. At any given time f = f0 + f1, where f1is the actual concentration at that time and f0 is a constant determined from thecritical radius and loop length L as f0 = (4π/3)(rcrit/L)3. Typically, the value of thevacancy misfit parameter lies in the range −0.1 to 0. According to equation (5.1)the maximum value of βvac occurs in the PSBs when f = f0, yielding the valueβvac

PSB = 0.1. Thus, the vacancy contributions to the nonlinearity parameter are con-siderably smaller than those from the dislocation substructures and are neglected inthe following calculations.

(c) Material nonlinearity versus per cent fatigue life

As indicated in equation (3.65), the change in the material nonlinearity param-eter due to fatigue-generated substructures is obtained by summing the individualcontributions from dislocation monopole and dislocation dipole sources weightedby the volume fractions of material making such contributions. The dislocationdipole contribution has a possible vein structure source and a possible PSB sourcedepending on the state of fatigue (i.e. value of per cent full life). We thus writefdpβdp = fveinβdp

vein + fPSBwβdpPSBw and fdpΓ dp = fveinΓ dp

vein + fPSBwΓ dpPSBw, where

fvein and fPSBw, respectively, are the volume fractions of material at a givenper cent full life containing vein structure and PSB wall structure. We assumefPSBw = 0.1fPSB, where fPSB is the volume fraction of PSBs in the material and thefactor 0.1 is an estimate of the fraction of material in the PSB containing wall struc-ture (Antonopoulos et al . 1976; Winter 1974). The dislocation monopole contributionhas a vein structure source and a PSB structure source resulting from the genera-tion of secondary dislocations. Hence, we write fmpβmp = fveinβmp

vein + fPSBσβmpPSBσ

and fmpΓmp = fveinΓmpvein + fPSBσΓmp

PSBσ, where fPSBσ is the volume fraction of mate-rial within the PSBs that contains sufficiently large secondary dislocation densitiesto significantly influence β via the PSB internal stress field. A crude estimate offPSBσ = 0.1fPSB may be obtained from transmission electron micrographs of sec-ondary dislocation structures in PSBs (Wang & Mughrabi 1984; Wang et al . 1984).

We obtain from the above considerations that the total material nonlinearityparameter β for polycrystalline nickel resulting from lattice elasticity and from thegeneration of dislocation substructures associated with the fatigue process at a givenvalue of per cent full life is assessed as

β =βe + fvein(βmp

vein + βdpvein) + 0.1fPSB(βmp

PSBσ + βdpPSBw)

[1 + fvein(Γmpvein + Γ dp

vein) + 0.1fPSB(ΓmpPSBσ + Γ dp

PSBw)]2. (5.2)

A graph of the calculated material nonlinearity parameter β of pure polycrystallinenickel is plotted in figure 1 (upper curve) as a function of per cent full fatigue lifefor specimens subjected to cyclic loading in stress control at 241 MPa. The plot isobtained using equation (5.2), the above-calculated values of βmp

PSBσ, βdpvein, βdp

PSBw,Γmp

PSBσ, Γ dpvein, Γ dp

PSBw, the estimated value βe ≈ 5 from single-crystal data (Cantrell


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776 J. H. Cantrell

5

10

15

20

25

0.001 0.01 0.1 1 10 100

vein structure (strain control)PSB + vein structure (strain control)vein structure (stress control)PSB + vein structure (stress control)

nonl

inea

rity

par

amet

er

per cent total life

Figure 1. Graph of calculated material (acoustic) nonlinearity parameter plotted as a function ofper cent total fatigue life for polycrystalline nickel cyclically loaded in stress control at 241 MPa(upper curve) and in total strain control at a strain of 1.75 × 10−3 (lower curve).

1994) and the data of Grobstein et al . (1991) for the volume fractions of the disloca-tion substructures formed in stress control as a function of per cent life. The modelcalculations show that the nonlinearity parameter increases monotonically with adecreasing slope as a function of the per cent fatigue life, reaching a maximumvalue of approximately 23 at fracture. Hence, β is predicted to increase by ca. 360%at fracture due to the dislocation substructures generated during stress-controlledcyclic loading at 241 MPa.

Figure 1 (lower curve) also shows a graph of the calculated material nonlinearityparameter of pure polycrystalline nickel plotted as a function of per cent full fatiguelife for specimens subjected to cyclic loading in total strain control at a total strainof 1.75×10−3. Again, the plot is obtained using equation (5.2), the above-calculatedparameters and the data of Grobstein et al . (1991) for the volume fractions of thedislocation substructures formed in total strain control as a function of per cent life.As before, the model calculations show that the nonlinearity parameter increasesmonotonically, reaching a maximum value of approximately 24 at fracture. Thus, βis predicted to increase by ca. 375% at fracture due to the dislocation substructuresgenerated during total strain-controlled cyclic loading at a strain of 1.75 × 10−3.

6. Effects of cracks on nonlinearity parameter

Crack nucleation in the interior of pure polycrystalline metals is generally limitedto the boundaries of appropriately ‘PSB-conditioned’ grains (Brown 1981; Kim &Laird 1978; Mughrabi et al . 1983). Nazarov & Sutin (1997) derived an expression forthe nonlinearity parameter associated with non-interacting cracks in bulk material.Using their suggested values of crack-related constants in their model, we write theirresults in terms of the present notation as

βcrk ≈ 5 × 106N0R4crk, (6.1)

where N0 is the concentration of cracks (number of cracks per unit volume) inthe interior of the material and Rcrk is the radius of the crack. A reasonable esti-


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mate of the maximum microcrack concentration in the interior of polycrystallinenickel is 107 m−3 (Nazarov & Sutin 1997). Electron microscopical studies of nickelalloys (Anton & Fine 1983) indicate that the linear dimensions of a microcrackupon nucleation are as small as 45 nm. Equation (6.1) yields an insignificantly smallvalue βcrk = 1.3 × 10−17 for such microcracks, a result attributable the fourth-powerdependence of the crack radius.

However, the R4crk dependence generates the quite measurable value βcrk = 3.1

when the cracks have grown to a radius of Rcrk = 500 µm. This crack size is wellinto the macrocrack regime. The appearance of macrocracks of this size occurs latein the fatigue life of a material, commonly after more than 80–90% total fatigue life.The fourth-power dependence on crack radius produces a dramatic increase in βcrk

as the macrocracks rapidly grow to several millimetres in radius before fracture.The βcrk calculations are based on the assumption that the crack concentration

remains at the estimated value 107 m−3. In general, the concentration of macrocracksof sufficient size to contribute significantly to βcrk is smaller than the maximummicrocrack concentration. The smaller macrocrack concentration results either fromthe coalescence of some microcracks to form macrocracks as in α–Fe (Vasek & Polak1991) or from the growth of relative few isolated microcracks into macrocracks asin Al–Cu–Mg alloys (Sedlacek et al . 1988). In either case the lower macrocrackconcentration is often more than compensated by the R4

crk growth dependence. Thedetection of macrocracks from ultrasonic harmonic generation has been reported inmany materials (Nazarov 1991; Sutin et al . 1995; Yost et al . 1987). However, it isimportant to note that macrocrack-induced changes in β generally arise during thelast 10–20% of full fatigue life and occur in addition to that of the microstructuralcontributions.

7. Conclusion

The above-derived dependence of the material (acoustic) nonlinearity parameter ontotal fatigue life is based on well-documented microstructural features formed inwavy slip, face-centred-cubic, pure metals subjected to cyclic loading. The modelinteraction of a stress perturbation with vein structures, persistent slip band struc-tures, and cracks is generic in character and the results, though applied above topolycrystalline nickel, are expected to apply quite well to other wavy slip metalssubjected to similar loading conditions. Differences in microstructural detail such asvolume fractions of veins and PSBs, the presence of shearable or non-shearable pre-cipitates, amount of cold-working, etc., may alter specific features of the nonlinearityparameter versus per cent fatigue life curve.

Monotonic increases in β of several hundred per cent have been found experi-mentally in aluminium alloy 2024-T4 (Cantrell & Yost 2001), Ti–6Al–4V duplexphase alloy (Frouin et al . 1999), and in 410 Cb stainless steel (Na et al . 1996) as afunction of per cent total fatigue life of specimens subjected to stress-controlled load-ing. The curve for aluminium alloy 2024-T4 subjected to a stress-controlled loadingof 276 MPa is very similar in shape to that calculated for polycrystalline nickel instress-controlled loading at 241 MPa (figure 1), but the value of β is ca. 20% smallerin the aluminium curve. The experimental Al 2024 curve was taken from the virginstate to ca. 30% total fatigue life. In this range the effects of cracks are too small


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778 J. H. Cantrell

to be measured and the curve is dominated by the dislocation substructural organi-zation. The similar shape of the two curves in their range of overlap suggests thatthe microstructural evolution of wavy slip Al 2024-T4 is similar to that of polycrys-talline nickel, although the details of the substructural organization (e.g. dimensions,volume fractions, etc.) are somewhat different from that of pure nickel due to theeffects of precipitation and alloying. The data for 410 Cb stainless steel show a mono-tonic increase in β of approximately 1000% for fatigue in a very high cycle regime(ca. 108–109 cycles).

The analytical model presented here establishes a necessary underpinning to under-standing the experimentally determined increase in the material (acoustic) nonlin-earity parameter with increasing per cent total fatigue life. Although the presentmodel is developed for wavy slip pure FCC metals, it must be emphasized thatthe model displays some degree of flexibility and adaptability stemming from theuse of dislocation monopoles and dipoles as the calculational building blocks of allfatigue-induced dislocation substructures. Although the details of the substructuralorganization may change with increasing material complexity, the basic calculationalbuilding blocks remain dislocation monopoles and dipoles—even in those materialsdominated by a planar slip character. With the high signal to background, the obser-vation of second harmonic generation may well provide an effective way to monitorfatigue damage that develops very early in life, well before less than 1% of total lifehas elapsed.

I thank Professor L. M. Brown of the Cavendish Laboratory, University of Cambridge, for hishelpful comments.

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