梦见猫是什么预兆
International Journal of Fatigue23(2001)有出息的男人
S487–S490
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Nonlinear ultrasonic characterization of fatigue microstructures
John H.Cantrell*,William T.Yost
National Aeronautics and Space Administration,Langley Rearch Center,Mail Stop231,Hampton,VA23681-2199,USA
Abstract
Dislocation dipole substructures formed during metal fatigue are shown to produce a substantial distortion of ultrasonic waves propagating through the fatigued material.A model of ultrasonic wave–dislocation dipole interactions is developed that quantifies the wave distortion by means of a material nonlinearity parameter b.Application of the model to AA2024-T4predicts a value of b approximately300%larger in material cyclically loaded for100kcycles in stress-control at276MPa and R=0than that measured for virgin material.Experimental measurements show a monotonic increa in b as a function of the number of fatigue cycles that cloly approaches the predicted increa.The experiments also suggest that the relevant dislocation substructures are localized in the material.Published by Elvier Science Ltd.
Keywords:Dislocation dipoles;Stress–strain nonlinearity;Ultrasound;Plastic strain;Aluminum alloy2024
1.Introduction
Elastic and plastic nonlinearities in a material lead to ultrasonic wave distortion along the wave propagation path and the generation of harmonics of the initial wave-form.A quantitative measure of the wave distortion is the‘acoustic’nonlinearity parameter.The magnitude of the nonlinearity parameter is highly dependent on the crystalline structure of the solid[1]and on the prence of defect structures[2–4].Such dependence has made measurements of the acoustic nonlinearity parameter a uful materials characterization tool[5].Cyclic loading in metal fatigue promotes the formation of dislocation dipoles as the result of the mutual trapping of dislo-cations moving to-and-fro in respon to the cyclic stress.To asss the potential of acoustic nonlinearity measurements as a characterization tool for metal fatigue,a generic model is developed of the interaction of ultrasonic waves with arrays of dislocation dipoles. The model predictions are compared to experimental measurements of polycrystalline aluminum alloy2024-T4.
奥丁狗粮*Corresponding author.
0142-1123/01/$-e front matter Published by Elvier Science Ltd. PII:S0142-1123(01)00162-12.Interaction of ultrasonic waves with dislocation dipoles
A longitudinal stress perturbation s associated with a propagating ultrasonic wave produces a longitudinal strain in the material.It is assumed that the total longi-tudinal strain e is the sum of an elastic component e e and a plastic component e pl associated with the motion of dislocations in the dipole configuration.Thus,
eϭe eϩe pl(1) The relation between the stress perturbation and elastic strain may be written in the nonlinear Hooke’s law form
sϭA e2e eϩ
1
2
A e3e2eϩ%(2)
where A e2and A e3are the Huang coefficients[6].It is uful to write the inver relation
e eϭͩ∂e e∂sͪsϩ12ͩ∂2e e∂s2ͪs2ϩ%ϭͩ∂s∂e eͪ−1s(3)
Ϫ1
2ͫͩ∂2s∂e2e
ͪͩ∂s∂e eͪ−3ͬs2ϩ%ϭ1A e2sϪ12A e3(A e2)3s2ϩ%
The relation between the stress perturbation and the plastic strain e pl may be obtained from a consideration of dipolar forces.For edge dislocation pairs of opposite
S488J.H.Cantrell,W.T.Yost /International Journal of Fatigue 23(2001)S487–S490
polarity (vacancy or interstitial dipoles)the force per unit length F x along the glide path (shear force per unit length)on a given dislocation due to the other dislo-cation in the pair is given as [7]F x ϭϪ
Gb 22p (1−v )x (x 2−y 2)
(x 2+y 2)2
(4)
where G is the shear modulus,b is Burgers vector,n is Poisson ’s ratio,and (x ,y )are the Cartesian coordinates of one dislocation in the pair relative to the coordinates (0,0)of the cond.It is assumed that the motion of the dipole pairs occurs only along parallel slip planes (i.e.,along the x -direction)parated by the dipole height y =h .At equilibrium,with no residual or applied stress,Eq.(4)asrts that x =±y =±h .
The resolution of a longitudinal stress perturbation s along the slip planes produces a shear force per unit length bR s on the dipole pair,where R is the Schmid factor.It is assumed that in equilibrium the total force per unit length on the dipole (F x +bR s )=0.The relation between the longitudinal plastic strain e pl and the relative dislocation displacement z is given by e pl =⍀⌳dp b z where ⍀is the conversion factor from the dislocation displace-ment in the slip plane to longitudinal displacement along an arbitrary direction and ⌳dp is the dislocation dipole density.Let z =(x Ϫh )be the relative displacement of the dislocations in the dipole pair with respect to the equilib-rium position h .Using the above relationships among F x ,s ,e pl z and an expansion of Eq.(4)in a power ries in x with respect to h ,we obtain the following relation between the stress perturbation and e pl s ϭA dp 2e pl ϩ12A dp 3e 2
pl ϩ%(5)
where A dp 2ϭϪ
ͩ
G
4p ⍀R ⌳dp h 2
(1−v )
ͪ,A dp 3(6)
ϭͩ
G
南瓜尖
4p ⍀2R ⌳2dp h 3
(1−v )b ͪ
.The inver relation to Eq.(5)is e pl ϭ1A dp 2s Ϫ12A dp 3
(A dp 2)
3
s 2ϩ%.(7)
Substitution of Eqs.(3)and (7)into Eq.(1)yields e ϭͩ
1A e 2ϩ1A dp 2ͪs Ϫ12ͩ
A e 3(A e 2)3ϩA dp 3(A dp 2)
3ͪ
s 2
ϩ%.(8)
The inver relation to Eq.(8)may be approximated for typical material parameters as
s ϷA e 2
ͭe Ϫ12ͩA e 3A e 2ϩA dp 3(A e 2)
2
(A dp 2)
3
ͪe 2ϩ%ͮ
.(9)
The derivative of Newton ’s Law with respect to the Lagrangian (material)coordinate ‘a ’is given as [3](r 0=mass density)r 0∂2e ∂t 2ϭ∂2s ∂a
2.(10)
Substitution of Eq.(9)into Eq.(10)yields the nonlinear
acoustic wave equation ∂2e ∂t 2Ϫc 2∂2e ∂a 2ϭϪc 2
b ͫe ∂2e ∂a 2ϩͩ∂e ∂a
ͪ2
ͬ
(11)
where the wave velocity c =(A e 2/r 0)1/2.The parameter b is the total acoustic nonlinearity parameter of the material de fined by the expression
b ϭb e ϩb dp (12)
请珍惜对你好的人
where b e ϭϪA e 3
A e
2
(13)
is the elastic or lattice contribution to b and
b dp ϭϪA dp 3(A e 2)2(A dp 2)3ϭ
16p 2⍀R 2⌳dp h 3(1−v )2(A e 2)2
G 2b
(14)
is the plastic contribution from the dislocation dipoles.It is instructive to note that the ‘acoustic ’nonlinearity parameter also occurs as the t of coef ficients of e 2in Eq.(9),the nonlinear stress –strain relation,quite inde-pendently of its role as an acoustic parameter.Thus,the b parameter is more generally a ‘material ’nonlinearity parameter.
A solution to Eq.(11),assuming a purely sinusoidal input wave of the form e 0sin w t at a =0where e 0is the wave amplitude,k the wavenumber,and w the angular frequency,is given as [8]
e ϭe 0sin(w t Ϫka )Ϫ1
4b k e 2
a sin[2(w t Ϫka )].(15)
感恩的近义词
Eq.(15)shows that,in addition to the fundamental sinusoidal signal of frequency w ,a harmonic signal of frequency 2w is generated with an amplitude that is directly dependent on the magnitude of the total nonlin-earity parameter.
It is instructive to estimate the relative contributions to the total nonlinearity parameter of the elastic and plastic components for aluminum alloy 2024-T4.Prent measurements (e below)and related calculations in virgin material yield the values G Ϸ28.6GPa,A e 2Ϸ109
GPa,A e
3ϷϪ510GPa,n =0.33,and ⍀=R =0.33.The meas-ured elastic component of the nonlinearity parameter b e =4.7.For material fatigued in the high cycle regime reasonable values of the alloy dislocation parameters may be estimated as ⌳dp Ϸ2.5×1015m Ϫ2and h Ϸ5nm
S489 J.H.Cantrell,W.T.Yost/International Journal of Fatigue23(2001)S487–S490
from data on copper[9–11]and bϷ0.4nm from data on
五一休假安排Al–Cu–Mg alloys[12].
It is assumed in the derivation of Eq.(14)that the
dislocation dipoles are distributed uniformly throughout
the material.If the dipole distribution is not uniform, then the dipole contribution b dp to the total nonlinearity
parameter b in Eq.(12)must be replaced by f dp b dp, where f dp is the volume fraction of material containing dipoles.The volume fraction of dipoles in a material is
very nearly equal to the combined volume fractions of vein structure(or pre-persistent slip band structure in
some materials)and persistent slip band wall structure,
since the substructures are comprid predominately of dipoles[9–11].A reasonable estimate of the volume
fraction of dislocation substructures formed for material
fatigued in the high cycle regime is f dp=0.5[13,14]. Using this value for f dp and the material parameters given
above in Eq.(14),we calculate the value f dp b dpϷ14.5 for the plastic dipole component to the total nonlinearity parameter.The model thus predicts that the contribution
to b from dislocation dipoles for material fatigued in the
high cycle regime is roughly3.1times that of the elastic f dp b dp/b eϷ3.1).
3.Experiments
Three ASTM standard‘dogbone’specimens of AA2024-T4were fatigued at a rate of10Hz under uni-axial,stress-controlled load at276MPa and R=0.Each specimen was fatigued for a different number of cycles: 3cycles,10kcycles,and100kcycles.A fourth specimen was unfatigued.Three cylindrical samples each of length 1.9cm were cut from the2.53cm diameter gauge c-tion of each of the four specimens.Optical microscopical examination of the end surfaces of the samples revealed no cracks longer than35–40µm.Such crack dimensions are well below that needed to contribute measurably to the nonlinearity parameter[4].Purely sinusoidal ultra-sonic bulk waves of frequency5MHz were launched into the samples using a1.27cm diameter lithium niob-ate piezoelectric transducer bonded to aflat end of the sample.After propagating through the solid both the fun-damental and harmonic signals were detected by an air gap capacitance transducer at the opposite end as indi-cated in Fig.
1.Measurements of b were made from absolute amplitude measurements of the fundamental and cond harmonic ,from the Fourier spec-trum of the received distorted signal)in a manner pre-viously described[5,15]and illustrated in Fig.1.
The maximum measured values of b are plotted as a function of the number of fatigue cycles in Fig.2.The b parameter increas monotonically with increasing fatigue cycles,although the increa in the range from 10to100kcycles,presumably the range dominated by the growth of persistent slip bands,is relatively
smaller.Fig.1.Schematic showing distortion of initially sinusoidal ultrasonic wave propagating in ctioned sample of specimen(top),the resulting Fourier spectrum of received signal(bottom),and relation to measure-ment of nonlinearity parameter
(bottom).
Fig.2.Graph of maximum measured value of nonlinearity parameter as function of number of fatigue cycles for aluminum alloy2024-T4. This may indicate,as found with single crystal and polycrystalline copper[16],that stress-controlled load-ing of AA2024-T4produces a slow but monotonic increa in the volume fraction of persistent slip bands throughout the fatigue life.The value of b measured at 100kcycles of fatigue,b100kc,is roughly3.9times larger than that obtained for the virgin material,b virgin=b e.
Assuming that f dp b dp=b100kcϪb e,we obtain f dp b dp/b eϷ3.0.This is in good agreement with the model prediction of3.1.
The values shown in Fig.2are the maximum values
S490J.H.Cantrell,W.T.Yost/International Journal of Fatigue23(2001)S487–S490
Table1
Value of material nonlinearity parameter at different locations in AA2024-T4specimens fatigued for10and100kcycles
Number of fatigue cycles10kcycles100kcycles
Location S1S2S3S4S5S6S7S8 Nonlinearity parameter7.117.312.212.513.815.614.918.4
of b obtained in the gauge length of a given specimen. The measured b values are found to be strongly depen-dent on the region(location)of the specimen through which the ultrasonic wave propagates.Table1shows the results taken of waves propagating in specimens fatigued at10and100kcycles.The values shown for locations S1–S3are obtained for waves propagating centrally along the cylindrical axis of each of the three samples cut from the specimen fatigued for10kcycles.The larg-est value of b in this t is that obtained on the sample cut from the center of the gauge ction of the specimen. The values shown for locations S4–S8are obtained only on the sample cut from the center of the gauge ction of the specimen fatigued for100kcycles.In addition to a measurement made centrally along the cyl-inder axis,measurements were also obtained at four locations clor to the edge of theflat cylindrical surface but spaced at different azimuthal angles spanning a range of360°.The largest value of b obtained from the measurements is from an off-axis location.
4.Conclusion
The ultrasonic measurements taken of AA2024-T4 show a substantial monotonic increa of the material nonlinearity parameter with an increasing number of fatigue cycles.Similarly dramatic increas in b have been obrved in fatigued Ti–6Al–4V[17]and in410Cb stainless steel specimens[18].Good agreement is obtained between the value of the nonlinearity parameter measured at100kcycles of fatigue in AA2024-T4and that predicted from the generic ultrasonic wave–dislo-cation dipole interaction model.No cracks of sufficient size to affect the nonlinearity parameter were obrved in the samples.This indicates that the origin of the increa in b is probably associated with the growth and transformation of dislocation dipole substructures for-med in the material during fatigue.The large variation of b with measurement location in the material suggests that the relevant substructural changes are localized.The development of a more detailed model of the interaction of ultrasonic waves with specific,microscopically characterized substructures formed during the fatigue of wavy slip materials is in progress.References
[1]Cantrell JH.Crystalline structure and symmetry dependence of
acoustic nonlinearity parameters.J Appl Phys 1994;76(6):3372–80.
[2]Hikata A,Chick B,Elbaum C.Dislocation contribution to the
cond harmonic generation of ultrasonic waves.J Appl Phys 1965;36(1):229–37.
[3]Cantrell JH,Yost WT.Acoustic harmonic generation from
fatigue-induced dislocation dipoles.Phil Mag A 1994;69(2):315–26.
[4]Nazarov VE,Sutin AM.Nonlinear elastic constants of solids and
cracks.J Acoust Soc Am1997;102(6):3349–54.
[5]Cantrell JH,Salama K.Acoustoelastic characterization of
materials.Int Mater Rev1991;36(4):125–46.
[6]Born M,Huang K.Dynamical theory of crystal lattices.Oxford:
Clarendon,1988.
高压锅[7]Hull D,Bacon DJ.Introduction to dislocations.Oxford:Perga-
mon,1984.
[8]Wallace DC.Thermoelastic theory of stresd crystals and
higher-order elastic constants.In:Ehrenreich H,Seitz F,Turnbull D,editors.Solid state physics,vol.25.New York:Academic Press,1970:301–404.
[9]Antonopoulos JG,Winter AT.Weak-beam study of dislocation
structures in fatigued copper.Phil Mag1976;33(1):87–95. [10]Antonopoulos JG,Brown LM,Winter AT.Vacancy dipoles in
fatigued copper.Phil Mag1976;34(4):549–63.
[11]Brown LM.Dislocations and the fatigue strength of metals.In:
Ashby MF,Bullough R,Hartley CS,Hirth JP,editors.Dislo-cation modelling of physical systems.Oxford:Pergamon, 1981:51–67.
[12]Loffler H,Siebert P,Kroggel R,Heyroth W.On thefit between
the precipitates occurring in Al–Cu and Al–Cu–Mg alloys and the Al-rich matrix.Cryst Res Technol1991;26(7):901–9. [13]Winter AT.A model for the fatigue of copper at low plastic strain
amplitudes.Phil Mag1974;30(4):719–38.
[14]Mughrabi H.The cyclic hardening and saturation behavior of
copper single crystals.Mater Sci Engng1978;33:207–23. [15]Yost WT,Cantrell JH.Absolute ultrasonic displacement ampli-
tude measurements with a submersible electrostatic acoustic transducer.Rev Sci Instrum1992;63(9):4182–8.
[16]Winter AT,Pedern OB,Rasmusn KV.Dislocation micro-
structures in fatigued copper polycrystals.Acta Metall 1981;29:735–48.
[17]Frouin J,Sathish S,Matikas TE,Na JK.Ultrasonic linear and
nonlinear behavior of fatigued Ti–6Al–4V.J Mater Res 1999;14(4):1295–8.
[18]Na JK,Cantrell JH,Yost WT.Linear and nonlinear ultrasonic
properties of fatigued410Cb stainless steel.In:Thompson DO, Chimenti DE,editors.Review of progre
ss in quantitative nonde-structive evaluation,vol.14.New York:Plenum,1996:1347–53.
