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Fatigue-induced dissolution of shearable particles during subcritical crack growth
Engineeringbiamm ~edranic.~Vol. 46. No. 1. pp. 151-156. 1993 Printed in Great Britain.
0013-7944/93 $6.00 + 0.00 ri’ 1993 Pergamon Press Ltd.
FATIGUE-INDUCED DISSOLUTION OF SHEARABLE PARTICLES DURING SUBCRITICAL CRACK GROWTH AR1 VARSCHAVSKY Univarsidad de Chile, Facuitad de Ciencias Fisicas y Matem&icas, lnstituto de lnv~ti~cion~ y Ensayes de Materiales, IDIEM, Casilla 1420, Santiago, Chile Abstract-A model based on thermodynamic and fracture mechanics considerations is proposed to compute the range of the stress intensity factor within which shearable particles dissolve during fatigue crack propagation. The model is applied to a cold rolled Cu-9 wt% AI alloy containing disperse-ordered domains penetrable by dislocations.
INTRODUCTION DURING low cycle fatigue in several precipitation hardened alloys, deformation is localized in a certain type of persistent slip band (PSB) in which ordered particles are dissolved [l]. There is still discussion as to whether the precipitates’ absence arises from repeated scrambling (disorde~ng) or from dissolution thereof. According to several observations, dissolution should occur during fatigue crack propagation (FCP) because increasing deformation is generated while the process zone propagates [2]. This work uses thermodynamic and fracture mechanics considerations to evaluate particular values of the stress intensity factor AK, and to thus establish suitable relationships that predict precipitate dissolution during fatigue crack propagation.
THERMODYNAMIC
ANALYSIS
When a spherical particle of radius t is formed, the energy barrier for nucleation has been surpassed. This barrier is the free energy maximum of the following function: AG = -(4/3)nr3yc + 4+,,
(1)
where yL’is the gain in free energy per unit volume of precipitate and y, is the specific free energy of the particle/matrix interface. This maximum is reached when r,, = ~YJY,..
(2)
If the stability of ordered particles is considered when they are sheared by dislocations, another term must be included in the energetic balance of eq. (1). When a dislocation shears the precipitate, this produces an antiphase boundary [energy y0 on an average surface (2/3)d’j and a step in the remaining interfaces [energy ySon average surface (1f4)arb, where b is Burgers’ vector] as shown in Fig. 1. It should be noted that the precipitate mean radius is r, = (2/3)% in an arbitrary slip plane. Furthermore, following the treatment made by Brechet et al. [3] and considering that average shear strength of the particles is y, the number, n, of dislocations cutting the particle of radius r as shown in Fig. 2 is given by dn = pa dt(2r), where p is the mobile dislocation density and v is the average velocity of these dislocations. Here Orown’s relationship dy = pub dt yields n = (2r/b)y.
(3)
Therefore, considering mobile dislocation effects, AG becomes AG = -(4/3)nr3y,
+ 4zr2yS + 1/2n*r*yy, + (4/3)nr3yy,/b.
(4)
The last two terms must be weighted by the number of shearing events. Since b Q r and ySe y., eq. (4) can be written as follows: AG(r, y) = 4mzY; - (4/3)~r3~~,, 151
(5)
152
A. VARSCHAVSKY
where yf=~,(l
+7~/g)=:y,
and
% = rJ1 - rr,/(&.)l.
In this way, (aG(r, r)/ar) = 0 produces r,.(y) = rro(l + V/W1
- 8/Yc),
(6)
where 7 and y< correspond to the mean values of y and yc, respectively, in a particle of diameter 2r. jj x y as a first approximation provided y is not too large. yc is given by ?<= by,.iv,.
17)
Hence, if 7 < jrc precipitates with radius r > F<(T)will not dissolve when they are sheared, but if y = Tc then rf(Tc) becomes infinite, which implies that AG(r) is an increasing function of r and all the precipitates should dissolve. Now eqs (2) and (7) give X = 2by~i~r~~,~“).
FRACTURE
MECHANICS
(8)
CONSIDERATIONS
If a fatigue crack is propagating in the process zone 1, as shown in Fig. 3, and assuming that the plastic strain range eP follows the Hult-McClinton relationship [4] f&l= @,.{rR;t(x +
c)l -
1)fE
z o~R$/E(x -I- c),
(9)
where o,, is the yield stress, E is Young’s modulus, R.i is the fatigue plastic zone, x is the distance measured from the crack tip, and c is a constant that does not altow cp to reach an infinite value for x = 0. The average value of cP in the process zone is ZP= a,.R{ ln[(l/c) -t- l]/(E/).
(10)
Then, if Lois the plastic fracture strain and since EP= cf if x = 0, it is justified to assume I = c. Thus: ” rp = 0.7&/(Ei).
(11)
On the other hand [5] R.; = ~1~(8~)~(A~/~~)*
(121
Cp= 0.03 AK2J(q,.El).
(13)
and therefore
In order to evaluate a dynamic Young’s modulus E,, a linear cyclic plastic stress-strain curve is assumed as a first approximation and the elastic cyclic strain is neglected here. These features
b
I
0
1, b iIEiYl3 Antiphase borrndary
0
step
Fig. I. Precipitate shearing showing step and antiphase boundary development.
v.dt
-’
Fig. 2. Scheme used to calculate the number of dislocations shearing on a particle during a time dl.
Dissolution
of shearable
particles during subcritical
153
crack growth
Fztlgne plutic zone
Roceu zone
/ Monotonic
piutic zone
Fig. 3. Schematic representation of a propagating crack. The different zones are indicated.
fatigue
Fig. 4. Schematic stabilized hysteresis loops for different values of the imposed plastic strain. The (a, 6,) points define the cyclic plastic stress-strain curve.
are schematically shown in Fig. 4. Hence, E = Ed in the present model, which also implies that the total plastic strain E z Zp. THE FRACTURE MECHANICS-THERMODYNAMIC
LINK
Since the average plastic shear strain is given by f,, = (E,/3G,)E,, where G, is the dynamic shear modulus, Ed= 2GJl + v), where v is the Poisson ratio and eq. (6) becomes r,(C) = r,,
1 + KE&,/(~~G~) 1 -@Jr‘)
1.
Thus eqs (13) and (14) yield r, (AK) = rco In this way the catastrophic unstable is
1 + 2.4 x 10..‘n AKZ/(aJEdl) 1 - 0.03 AK’/(a,. EJ)
(14)
1’
(15)
stress intensity factor AK,, for which all particle radii become AK,.,= 5.8(a,.E,,1S,,)‘i2,
(16)
where
(17)
c,, = (4/3)(1 + v)W(r,.,JJ,). Hence eq. (15) becomes
1’
(18)
-1 + O.O83n(l + v)(AK,./AK,,)~ rc(AK,) = rm 1 - (AK,lAK,.,)’ [ where AK, is the critical stress intensity factor range. That is, all particles with radius smaller than the average particle radiusLwil1 dissolve within fatigue striations in the fatigue plastic zone at a level AK for which r < r,(AK,). It has been shown experimentally [4] and theoretically [6] that if a fatigue crack propagates then the process zone size must be of the order of the interparticle spacing s. Thus, for a cubic lattice with the particles lying in latticecorners, I= s = 2r(61’,/n)-‘J3, where V, is the volume fraction of the precipitate. Making r = r,(AK,) in the above expression for I, eqs (16), (17) and (18) give the following E value: 4.63[a,Ebpjs(l + ~)/(r,r,)]‘!‘(l/V/)‘~~ [1 + O.l7bV,(l + V)l(V,)l
(19)
154
A. VARSCHAVSKY
If, in a simplified scheme, it is considered that the particle is repeatedly sheared, for an eff‘ective radius r,,, a certain AK, can be calcuIated, so that if this radius is reached before the particle leaves the fatigue process zone it becomes critical. In fact, the average surface in an arbitrary slip plane is [7] F = r*{2 arccos ytV- [ 1 - (2$,,.- 1)2]t’2),
120)
where y,, is the average shear strain in this plane. This gives an effective radius r,, = (F/n)’ ‘~ When using eq. (20) an efficiency factor v] = 0.5 seems reasonable for Y,~,since this equation was derived for a unidirectiona1 shearing process. Therefore r,. = r{(2/n)[2
_ _._. arccos[0.82(1 + v)(AK,/AK,.,)~]] - (1 - [0.03(1 + 2v)(c,., AK,/AK,,j’ -- 11’)’‘1’ ‘, (31)
-If r,. = r,(AKd), AKd and r,,, can be calculated using eqs (17j, (18) and (21 j.
APPLICATION
OF THE MODEL TO FCP IN DISPERSE-ORDERED Cu-9 WT% Al ALLOYS
The model developed in this work was applied for the estimation of AK,,, AK, and AK, during fatigue crack growth in a Cu-9 wt% Al alloy. The alloy was prepared in a Balzers VSG 10 vacuum induction furnace from electrolytic copper (99.97 wt%> in a graphite crucible. The ingots were subsequently hot forged at 650°C to a thickness of 10 mm, pickled to remove oxide from the surface and annealed in a vacuum furnace at 850°C for 36 hr to achieve complete homogeneity followed by cooling to room tem~rature in the furnace. The alloy was than cold rolled to 1.5 mm thickness with intermediate annealings at 630°C for 1 hr. After the last anneal the material was water quenched prior to a final cold rolling to 0.75 mm thickness (50% reduction). To promote ordering, part of the alloy was heat treated at 250-C for 30min. Single edge notched specimens (SEN) were cut by a Servomet spark machine using a razor blade. The specimen was then heavily electropolished in a 50% phosphoric acid and 50% ethyl alcohol solution. FCP tests were carried out under stress control in a two ton Amsler Vibrofore. The experiments were performed in duplicate at stress ratios (minimax stress) R = 0. Crack length was monitored as a function of the number of cycles using a traveling microscope. In order to calculate FCP rates from the plot of crack length versus number of cycles. a linear least squares line was fitted through three consecutive data points and its slope supplied the FCP rate of the central point. FCP rates are shown in Fig. 5 as a function of stress intensity factor for the ordered and disordered materials. It can be seen that for low AK fatigue crack propagation rates are lower for the ordered alloy while at higher AK the behavior is the same for both conditions of the material. It is reasonable to infer that this feature can be attributed to the greater resistance to dislocation motion in the ordered alloy, since the domains act as obstacles that must be previously sheared before the formation of fatigue striations takes place. The above effect is more pronounced for lower values of AK, where the slip is more homogeneous. For medium and high ranges of AK, the ordered/disordered FCP rates are similar, since, as can be expected, disordering of the ordered alloy eliminates the difference between the two materials due to the increase in the cyclic plastic strain ahead of the crack. In fact, for higher AK, the crystallographic sliding within the slip steps should be greater in order to accommodate the larger amount of plastic defo~ation [S]. Once formed, these paths are preferred for further dislocation motion and the sheared domains may revert into solid solution. In order to estimate the value of AK ( = AK,) at which this fatigue-induced dissolution occurs, characteristic parameters of the alloy are required. For the alloy under study, (TV= 410 MPa was measured after the heat treatment. Besides vr= 0.44 and r = 12 nm f9], yS= 20 mJm_’ (corrected) [IO], = 70mJm-* [l l] and Ed = 97 GPa [12]. If rr- 1 nm, b = 0.26 nm and v = 0.33, then Y *Kc, = 38 MNmm3’* and dK,. = 26 MNm- 3/2. This AK, value is a good prediction since it is located in the FCP curves region where the rates become similar for both materials. This in turn strongly
Dissolution
of shearable particles during subcritical crack growth
6810
20
155
40
Fig. 5. FCP rates for ordered and disordered alloy. Values for AK,, AK,, AK,and also an estimated value for AK,* are indicated.
--
suggests that ordered domains do dissolve for AK > AK,. The value AKd = 18.5 MNm-3’2 was also calculated. It can be observed that AK, is somewhat larger than AKd, as expected, which means that the shearing process is a controlling factor of crack growth rate until the critical AK range is reached. For larger AK the model predicts that the microstructure is not a controlling factor of crack growth rate. Such behavior suggests that for AK > AK, the particles will dissolve when they meet early the fatigue plastic zone within fatigue bands. Between AK, and AK,, the region where the curves become closer to each other, the particles dissolve inside the fatigue plastic-- zone. If AK,* is an estimated threshold value for AK in the ordered alloy, when A& < AK < AKd, the precipitate will not dissolve. In conclusion, the model can explain the fatigue propagation curves in terms of a precipitate dissolution process confirmed from -the sequence and position of the different AKs, namely AK,h < rKd < AK, < AK,.,, as expected. Acknowledgements-The author wishes to thank the Fondo National de Desarrollo Cientifico y Tecnol6gico (FONDECYT). for financial support through Project 1930945. He is also indebted to the Institute de Investigaciones y Ensayes de Materiales. Faculatad de Ciencias Fisicas y Matenraticas, Universidad de Chile, for financial support and facilities provided in connection with this research project.
REFERENCES [I] J. Lendvai, H. J. Gudlandt and V. Gerold. The deformation-induced dissolution of 6’ precipitates in AI-Li alloys. Scripta Metall. 22, 1755-1760 (1988). [2] A. Saxena and S. Antolovich, Low cycle fatigue, fatigue crack propagation and substructures in a series of polycrystalline Cu-AI alloys. Metoll. Trans. 6A, 1809-1828 (1975). [3] Y. Brechet, F. Louchet, C. Marchionni and J. L. Verger-Gaugry. Experimental (TEM and STEM) investigation and theoretical approach to the fatigue-induced dissolution of 6’ precipitates in a 2.5 wt% AI-U alloy. Phi/. Msg. 56A, 353-366 (I 987). [4] S. Antolovich and A. Saxena, A model for fatigue crack propagation. Engng Fracture Mech. 7, 649452 (1987). [5] R. W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, p. 475. John Wiley, New York (1976). [6] J. R. Rice and M. A. Johnson, The role of large crack tip geometry changes in plain strain fracture, in fnehric Behaviorof Solids (Edited by Kanninen),pp. 641-672. McGraw-Hill, New York (1970). [7] H. P. Von Klein, Die Gleitverteilung bei legierungen mit kohiirenten ausschidungen. Z. Metal/k. 61, 564572 (1970). [8] E. Hombogen and K. Zum Garhr, Microstructure and fatigue crack growth in a y-Fe-Ni-Al alloy. Acra Metal/. 24, 581-592 (1975). [9] A. Varschavsky and E. Donoso, Ordered domain characterization in aCu-Al alloys from dissolution kinetics studies. J. Mater. Sci. 21, 3873-3882 (1986).
156
A. VARSCHAVSKY
[IO] A. Varschavsky, Non-isothermal calorimetric determination of precipitate interfacial energtes. J. M~trr. Ser. 20, 3881-3889 (1985). [I I] A. Varschavsky and E. Donoso, Antiphase boundary energies of aCu-Al ordered domains. Muter. Sri. Engn~ A104, 141-147 (1988). [12] C. E. Feltner and C. Laird, Cyclic stress-strain response of fee. metals and alloys--l. Phenomenological experiments. Acta Metal/. 15, 1621-1632 (1967). (Received
6 February
1992)
0013-7944/93 $6.00 + 0.00 ri’ 1993 Pergamon Press Ltd.
FATIGUE-INDUCED DISSOLUTION OF SHEARABLE PARTICLES DURING SUBCRITICAL CRACK GROWTH AR1 VARSCHAVSKY Univarsidad de Chile, Facuitad de Ciencias Fisicas y Matem&icas, lnstituto de lnv~ti~cion~ y Ensayes de Materiales, IDIEM, Casilla 1420, Santiago, Chile Abstract-A model based on thermodynamic and fracture mechanics considerations is proposed to compute the range of the stress intensity factor within which shearable particles dissolve during fatigue crack propagation. The model is applied to a cold rolled Cu-9 wt% AI alloy containing disperse-ordered domains penetrable by dislocations.
INTRODUCTION DURING low cycle fatigue in several precipitation hardened alloys, deformation is localized in a certain type of persistent slip band (PSB) in which ordered particles are dissolved [l]. There is still discussion as to whether the precipitates’ absence arises from repeated scrambling (disorde~ng) or from dissolution thereof. According to several observations, dissolution should occur during fatigue crack propagation (FCP) because increasing deformation is generated while the process zone propagates [2]. This work uses thermodynamic and fracture mechanics considerations to evaluate particular values of the stress intensity factor AK, and to thus establish suitable relationships that predict precipitate dissolution during fatigue crack propagation.
THERMODYNAMIC
ANALYSIS
When a spherical particle of radius t is formed, the energy barrier for nucleation has been surpassed. This barrier is the free energy maximum of the following function: AG = -(4/3)nr3yc + 4+,,
(1)
where yL’is the gain in free energy per unit volume of precipitate and y, is the specific free energy of the particle/matrix interface. This maximum is reached when r,, = ~YJY,..
(2)
If the stability of ordered particles is considered when they are sheared by dislocations, another term must be included in the energetic balance of eq. (1). When a dislocation shears the precipitate, this produces an antiphase boundary [energy y0 on an average surface (2/3)d’j and a step in the remaining interfaces [energy ySon average surface (1f4)arb, where b is Burgers’ vector] as shown in Fig. 1. It should be noted that the precipitate mean radius is r, = (2/3)% in an arbitrary slip plane. Furthermore, following the treatment made by Brechet et al. [3] and considering that average shear strength of the particles is y, the number, n, of dislocations cutting the particle of radius r as shown in Fig. 2 is given by dn = pa dt(2r), where p is the mobile dislocation density and v is the average velocity of these dislocations. Here Orown’s relationship dy = pub dt yields n = (2r/b)y.
(3)
Therefore, considering mobile dislocation effects, AG becomes AG = -(4/3)nr3y,
+ 4zr2yS + 1/2n*r*yy, + (4/3)nr3yy,/b.
(4)
The last two terms must be weighted by the number of shearing events. Since b Q r and ySe y., eq. (4) can be written as follows: AG(r, y) = 4mzY; - (4/3)~r3~~,, 151
(5)
152
A. VARSCHAVSKY
where yf=~,(l
+7~/g)=:y,
and
% = rJ1 - rr,/(&.)l.
In this way, (aG(r, r)/ar) = 0 produces r,.(y) = rro(l + V/W1
- 8/Yc),
(6)
where 7 and y< correspond to the mean values of y and yc, respectively, in a particle of diameter 2r. jj x y as a first approximation provided y is not too large. yc is given by ?<= by,.iv,.
17)
Hence, if 7 < jrc precipitates with radius r > F<(T)will not dissolve when they are sheared, but if y = Tc then rf(Tc) becomes infinite, which implies that AG(r) is an increasing function of r and all the precipitates should dissolve. Now eqs (2) and (7) give X = 2by~i~r~~,~“).
FRACTURE
MECHANICS
(8)
CONSIDERATIONS
If a fatigue crack is propagating in the process zone 1, as shown in Fig. 3, and assuming that the plastic strain range eP follows the Hult-McClinton relationship [4] f&l= @,.{rR;t(x +
c)l -
1)fE
z o~R$/E(x -I- c),
(9)
where o,, is the yield stress, E is Young’s modulus, R.i is the fatigue plastic zone, x is the distance measured from the crack tip, and c is a constant that does not altow cp to reach an infinite value for x = 0. The average value of cP in the process zone is ZP= a,.R{ ln[(l/c) -t- l]/(E/).
(10)
Then, if Lois the plastic fracture strain and since EP= cf if x = 0, it is justified to assume I = c. Thus: ” rp = 0.7&/(Ei).
(11)
On the other hand [5] R.; = ~1~(8~)~(A~/~~)*
(121
Cp= 0.03 AK2J(q,.El).
(13)
and therefore
In order to evaluate a dynamic Young’s modulus E,, a linear cyclic plastic stress-strain curve is assumed as a first approximation and the elastic cyclic strain is neglected here. These features
b
I
0
1, b iIEiYl3 Antiphase borrndary
0
step
Fig. I. Precipitate shearing showing step and antiphase boundary development.
v.dt
-’
Fig. 2. Scheme used to calculate the number of dislocations shearing on a particle during a time dl.
Dissolution
of shearable
particles during subcritical
153
crack growth
Fztlgne plutic zone
Roceu zone
/ Monotonic
piutic zone
Fig. 3. Schematic representation of a propagating crack. The different zones are indicated.
fatigue
Fig. 4. Schematic stabilized hysteresis loops for different values of the imposed plastic strain. The (a, 6,) points define the cyclic plastic stress-strain curve.
are schematically shown in Fig. 4. Hence, E = Ed in the present model, which also implies that the total plastic strain E z Zp. THE FRACTURE MECHANICS-THERMODYNAMIC
LINK
Since the average plastic shear strain is given by f,, = (E,/3G,)E,, where G, is the dynamic shear modulus, Ed= 2GJl + v), where v is the Poisson ratio and eq. (6) becomes r,(C) = r,,
1 + KE&,/(~~G~) 1 -@Jr‘)
1.
Thus eqs (13) and (14) yield r, (AK) = rco In this way the catastrophic unstable is
1 + 2.4 x 10..‘n AKZ/(aJEdl) 1 - 0.03 AK’/(a,. EJ)
(14)
1’
(15)
stress intensity factor AK,, for which all particle radii become AK,.,= 5.8(a,.E,,1S,,)‘i2,
(16)
where
(17)
c,, = (4/3)(1 + v)W(r,.,JJ,). Hence eq. (15) becomes
1’
(18)
-1 + O.O83n(l + v)(AK,./AK,,)~ rc(AK,) = rm 1 - (AK,lAK,.,)’ [ where AK, is the critical stress intensity factor range. That is, all particles with radius smaller than the average particle radiusLwil1 dissolve within fatigue striations in the fatigue plastic zone at a level AK for which r < r,(AK,). It has been shown experimentally [4] and theoretically [6] that if a fatigue crack propagates then the process zone size must be of the order of the interparticle spacing s. Thus, for a cubic lattice with the particles lying in latticecorners, I= s = 2r(61’,/n)-‘J3, where V, is the volume fraction of the precipitate. Making r = r,(AK,) in the above expression for I, eqs (16), (17) and (18) give the following E value: 4.63[a,Ebpjs(l + ~)/(r,r,)]‘!‘(l/V/)‘~~ [1 + O.l7bV,(l + V)l(V,)l
(19)
154
A. VARSCHAVSKY
If, in a simplified scheme, it is considered that the particle is repeatedly sheared, for an eff‘ective radius r,,, a certain AK, can be calcuIated, so that if this radius is reached before the particle leaves the fatigue process zone it becomes critical. In fact, the average surface in an arbitrary slip plane is [7] F = r*{2 arccos ytV- [ 1 - (2$,,.- 1)2]t’2),
120)
where y,, is the average shear strain in this plane. This gives an effective radius r,, = (F/n)’ ‘~ When using eq. (20) an efficiency factor v] = 0.5 seems reasonable for Y,~,since this equation was derived for a unidirectiona1 shearing process. Therefore r,. = r{(2/n)[2
_ _._. arccos[0.82(1 + v)(AK,/AK,.,)~]] - (1 - [0.03(1 + 2v)(c,., AK,/AK,,j’ -- 11’)’‘1’ ‘, (31)
-If r,. = r,(AKd), AKd and r,,, can be calculated using eqs (17j, (18) and (21 j.
APPLICATION
OF THE MODEL TO FCP IN DISPERSE-ORDERED Cu-9 WT% Al ALLOYS
The model developed in this work was applied for the estimation of AK,,, AK, and AK, during fatigue crack growth in a Cu-9 wt% Al alloy. The alloy was prepared in a Balzers VSG 10 vacuum induction furnace from electrolytic copper (99.97 wt%> in a graphite crucible. The ingots were subsequently hot forged at 650°C to a thickness of 10 mm, pickled to remove oxide from the surface and annealed in a vacuum furnace at 850°C for 36 hr to achieve complete homogeneity followed by cooling to room tem~rature in the furnace. The alloy was than cold rolled to 1.5 mm thickness with intermediate annealings at 630°C for 1 hr. After the last anneal the material was water quenched prior to a final cold rolling to 0.75 mm thickness (50% reduction). To promote ordering, part of the alloy was heat treated at 250-C for 30min. Single edge notched specimens (SEN) were cut by a Servomet spark machine using a razor blade. The specimen was then heavily electropolished in a 50% phosphoric acid and 50% ethyl alcohol solution. FCP tests were carried out under stress control in a two ton Amsler Vibrofore. The experiments were performed in duplicate at stress ratios (minimax stress) R = 0. Crack length was monitored as a function of the number of cycles using a traveling microscope. In order to calculate FCP rates from the plot of crack length versus number of cycles. a linear least squares line was fitted through three consecutive data points and its slope supplied the FCP rate of the central point. FCP rates are shown in Fig. 5 as a function of stress intensity factor for the ordered and disordered materials. It can be seen that for low AK fatigue crack propagation rates are lower for the ordered alloy while at higher AK the behavior is the same for both conditions of the material. It is reasonable to infer that this feature can be attributed to the greater resistance to dislocation motion in the ordered alloy, since the domains act as obstacles that must be previously sheared before the formation of fatigue striations takes place. The above effect is more pronounced for lower values of AK, where the slip is more homogeneous. For medium and high ranges of AK, the ordered/disordered FCP rates are similar, since, as can be expected, disordering of the ordered alloy eliminates the difference between the two materials due to the increase in the cyclic plastic strain ahead of the crack. In fact, for higher AK, the crystallographic sliding within the slip steps should be greater in order to accommodate the larger amount of plastic defo~ation [S]. Once formed, these paths are preferred for further dislocation motion and the sheared domains may revert into solid solution. In order to estimate the value of AK ( = AK,) at which this fatigue-induced dissolution occurs, characteristic parameters of the alloy are required. For the alloy under study, (TV= 410 MPa was measured after the heat treatment. Besides vr= 0.44 and r = 12 nm f9], yS= 20 mJm_’ (corrected) [IO], = 70mJm-* [l l] and Ed = 97 GPa [12]. If rr- 1 nm, b = 0.26 nm and v = 0.33, then Y *Kc, = 38 MNmm3’* and dK,. = 26 MNm- 3/2. This AK, value is a good prediction since it is located in the FCP curves region where the rates become similar for both materials. This in turn strongly
Dissolution
of shearable particles during subcritical crack growth
6810
20
155
40
Fig. 5. FCP rates for ordered and disordered alloy. Values for AK,, AK,, AK,and also an estimated value for AK,* are indicated.
--
suggests that ordered domains do dissolve for AK > AK,. The value AKd = 18.5 MNm-3’2 was also calculated. It can be observed that AK, is somewhat larger than AKd, as expected, which means that the shearing process is a controlling factor of crack growth rate until the critical AK range is reached. For larger AK the model predicts that the microstructure is not a controlling factor of crack growth rate. Such behavior suggests that for AK > AK, the particles will dissolve when they meet early the fatigue plastic zone within fatigue bands. Between AK, and AK,, the region where the curves become closer to each other, the particles dissolve inside the fatigue plastic-- zone. If AK,* is an estimated threshold value for AK in the ordered alloy, when A& < AK < AKd, the precipitate will not dissolve. In conclusion, the model can explain the fatigue propagation curves in terms of a precipitate dissolution process confirmed from -the sequence and position of the different AKs, namely AK,h < rKd < AK, < AK,.,, as expected. Acknowledgements-The author wishes to thank the Fondo National de Desarrollo Cientifico y Tecnol6gico (FONDECYT). for financial support through Project 1930945. He is also indebted to the Institute de Investigaciones y Ensayes de Materiales. Faculatad de Ciencias Fisicas y Matenraticas, Universidad de Chile, for financial support and facilities provided in connection with this research project.
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