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Constitutive laws for steady state deformation of metals, a microstructural model
Pergamon 0956-716x(9s)O0103-4
CONSTITUTIVE LAWS FOR STEADY STATE DEFORMATION OF METALS, A MICROSTRUCTURAL MODEL Erik Nes The Norwegian Institute of Technology
Department of Metalhugy, 7034 Trondheim, Norway (Received January 3,1995) (Revised January 20,199s) Introduction
Based on extensive investigations of the steady state conditions for creep and hot working, Sellars and McG. Tegart (1) demonstrated that the flow stress was well represented by the following relation:
(1)
where a’, A’, n are temperature independent constants and Z is the Zener-Hollomon parameter, Z = i exp U/kT, whereC is the applied strain rate and U is an activation energy. This relationship is widely used in the modelling of metal forming precesses such as hot rolling, forging and extrusion. The good fit between experimental observations and theoretical predictions is illustrated in Fig. 1, where the saturation stress for two aluminum alloys, tested in plane strain compression (2,3), are plotted as a fuction of Z. In calculating the Z values both Ref. 2 and 3 have used the commonly selected activation energy U = 156 kJ/mol. It is pointed out, however, that Eq. 1 is a purely empirical relationship from which no basic physical mechanism cau be inferred, this applies also to the selection of the activation energy, as a given such value de&es a set of con&m& a’, A’ and n. To &s&ate this point, consider the plane strain compression results given in Fig. 1: Although the results obtained on both the AlMnlMgl alloy (2) and the AlFeO.3SiO.05 alloy (3) are well represented by Eq. 1, it is not possible, based on the result i?om one of these alloys, to predict the behaviour of the other. Or in other words, Eq. 1 has no predictive power outside the alloy conditions tested, to which data the parameters U, a’, n and A’ have to be fitted for each case examined. The objective of this work is to explore the possibilities for deriving an alternative, physically based, relationship which is capable of correlating the steady state flow stress both to the alloy condition and the microstructural characteristics of steady state deformation. . DefUution
:
In order to derive an expression for the steady state saturation stress equations of the form: z, = f (ti, p,) and p,=f(i!T)~tobe~~andsolvedincombination. Thefirstequationdefinestheflowstressinterms of the stored dislocation density p,and the tictim stress T+ The stored dislocation density is not a single, well 225
Vol. 33, No. 2
STEADYSTATEDEFORMATIONOF METALS
is@-
U =1Sbid/mal
140 ,30 _ o Al0.3FeO.lSSi ,20
_ . AllMnlMp
110 3
ton
L zz v
x0 P
b
-
YO 70 60 so
-
40
-
l/n=fl.22 A’=b.b*lO’O
30 20 10
-
. IO“
Id’
IO”
Figure1. Steadyrateflow stressas a h&ion of theZmer-Hohmon parametex forthe.alloysgiven(2,3).
defined quantity, but a more complex parameter , i.e. p, = f(p,i3,8,...) where p represents an elementary dislocation density term, 6 a cell size, 8 a cell or sub-boundary misorientation etc. The second equation is even less well defmed, except being a solution of a di&rential equation of the form:
(2)
This relation simply reflects that during steady state, the defibrmationinduced dislocations, dplldt, are balanced by the dislocations lost due to dynamic recovery, dp;/dt. The dp :/dt-term is derived Corn the well established differential equation dy = bLdp, where p, refers to the mobile dislocations and L is the average slip distance. This equation is an important element in all theories of work hardening. However, except for the standard Stage II-solution, no solution of this equation has been derived which is applicable in a general case where the slip distance, L, relates to a complex substmctum, i.e. L=f(pJ,B,x). Properly solved, this equation will provide a solution: p,=f;(y ,fX, from which the storage term dprdt is obtained by simple differentiation. This short communication will not attempt such an ambitious approach. However, based on the well established observation that during low temperature de&mation, metals display a linear work hardening at large strains (Stage IV), a general expression for the dpi/dt-term can be derived, as wiIl be demonstrated below. During steady state condition this storage term is balanced by the annihilation effect due to dynamic recovery, Eq. 2. The dynamic recovety treatment will be based on the recent work on static recovery by Nes (4) and Fur-uet al. (5). As will be demonstrated in the following, the result becomes a general solution to the problem of deriving an expression for the steady state flow stress as defined here. Model
The following theory rests on the assumption that during steady state deformation of a pure metal, or stable solid sohnion, the substructure can be adequately described by a few microstructural elements, the two most importaut ones being the steady state cell/subgrain size, 8,, and the dislocation density in the celI interior, pti Based on this microstructural description several possible approaches can be taken in order to calculate the flow stress, as discussed in detail in Ref. 4. Two interesting treatments being the composite theory due to Mughrabi (6,7) and Pedersen et al. (8) and a modification (4) of the original Kuhhnann-Wilsdorf (9,10) link
Vol. 33, No. 2
STEADY STATE DFJOFMATION
OF METALS
227
length model. However, as shown in Ref. 4 both approaches predict a relationship of the form ‘c
= %i +
a1 Gb
pi + d--
a2
Gb
8;’
(3)
where ti is a friction stress, G is the shear modulus, b is the Burgers vector and a i, a, are constants. Based
on an extensive investigation of the substructure evolution during hot deformation Sellars and coworkers (2,ll) have demonstrated that during steady state the principle of similitude applies in the sense that the separation of dislocations within the cells (1WpJ scales with the cell size 8,: Jpi = C,/8,, where C, is a constant of typical value of the order 10. Combining this relationship with Eq. 3, it follows that the steady state flow stress can be expressed in terms of only one microstructural parameter, namely the cell/subgrain size 8,, i.e. 7
= T1 +
a,Gb8i1
(4)
where a3 = (C,a , + CQ),with a, expected to be of the order 3. This relationship is comirmed by CastroFemandez et al. (2), see also the following paper (12). The next step is to solve the diIferential Eq. 2, a solution which will provide the steady state subgrain size as a function of strain rate and temperature: 8, = f(q,T). The Steady State Subnrain Size
On the assumption that during steady state defbmMion the principle of similitude applies (i.e. Jp, = C,/8) Eq. 2 can be rewritten in the form
The lirst term detines the rate at which new sub-boundaries are created while the second term gives the subgrain growth rate due to dynamic recovery, balancing these terms defines steady state. As already mentioned in the introduction above, an empirical approach will be taken in order to define the d8~/dt-term. If metals are deformed at room temperature one finds that the variation in the average subgrain size with strain follows a 8 vs E relationship, which is of a similar type for a range of metals, and independent of the mode of deformation as shown in Fig. 2 (taken from the work of Gil Sevillano et al. (13)). In Fig. 2 the variation in the inverse cell size is also given (broken line), and it can be seen that this quantity varies linearly with strain fall. This is au interesting observation in view of the fact that the flow stress commonly is correlated with the inverse cell size. Accordingly, this result is consistent with a linear Stage IV work hardening behaviour. From Fig. 2 it follows that deformation at room temperature for strains &>y/M= 1 (M is the Taylor factor) gives l/8 = (0.7+0.09y)18,,,,5 which in combination with Eq. 3 gives a Stage IV work hardening rate of 0,=0.09a,Gb/8,,,,-2=10-‘G (for Cu, 8 r_,,5=0.29uwork m (13), a,=2.5 and b=2.56*1O%r) which is a very reasonably RT-value indeed. In general the Stage IV hardening rate scales with the initial Stage IV shear stress (t,J which again scales with the steady state saturation stress (r,), i.e. 8,=0.lt,=O.O7t, (see Gil Sevillano (14)). Combining this expression for BWwith Eq. 3 gives the following approximate solution: t+O.O7a,Gb/8,. The d8-/dt-term then takes the form:
l-1 d8-
dt
I -
P, T
ON+
b*
a2Gb
t 8* = - 0.07 8,
(6)
228
STEADYSTATEDEFORMATIONOF METALS
-
xii
z ox.,’
Vol. 33, No. 2
2
-05
zOZ.’ 2 v) 1
5TdN
r,
-0
5
6
s
Figure2. Subgaiakell size as a fimctionof equivalentstrain The size is normalizedto the value found atT~1.5 (13).
Recovery during static annealing of deformed metals has recently been treated by Nes (4) and Furu et al. (5). These works demonstrate that static recovery can be satisfactorily treated based on an internal state variable approach comprising two variables, i.e. the cell/subgrain size and the dislocation density in the cell interior. A similar approach will be tried here in handling the dynamic case. In terms of the basic physics involved, the two types of recovery reactions (static and dynamic) are expected both to be driven by forces derived f?om the fi-eeenergy changes associatedwith a reduction in the stored energy (4,5). No attempts will be made here in accounting for the growth reactions in mechanistic terms, for such details see Refs. 5 and 15. In the following treatment of the steady state flow stress, dynamic recovety will be incorporated by assuming a situation where the sub-boundary migration is controlled by solute drag (5), in which case the dynamic growth rate becomes: db’ dt
= 2bvDCB exp
(7)
where v, isthe Debye tkzquency and Ui is the interaction energy between the solute atom and the boundary dislocation core. P is the driving pressure for subgrain growth (P=2yda where ysB is the sub-boundary where v is the Poisson’s ratio). V, is the activation energy, i.e. ysB=a4 Gb0ln(eeJB) with a,=lNx(l-v) volume (V, = lsb, where 1”is the separation of solute atoms along the boundary dislocation line and h=b/g is the separation of boundary dislocations where 0 is the average sub-boundary misorientation. C, is a constant which needs to be determined experimentally. A general pure-metal-solution, incorporating both vacancy bulk and core dBitsion, requires a more extensive treatment which will be the topic of a subsequent paper. However, such a treatment gives a subgrain growth rate expression of the same form as that of Eq. 7, but with other activation energies and differently defined activation volumes. A main point here is that the physics of the present approach is adequatly ilhtstrated by using Eq. 7. The steady state subgrain size is now obtained by combining Eqs. 5,6 and 7, i.e.
229
STEADY STATE DEZFORMATIONOF METALS
Vol. 33, No. 2
where C, = 6 a,G b3 and C, = 0.07 M/v,bC,. This equation has no analytieal solution with respect to the subgrain size, but can easily be solved numerically. Depending on the value of the PV&T-ratio, Eq. 8 takes the forms: PV(jb
1 -.-+ 6,
-),l: W
Ufb
W ln (ie b,C,) C,lEb
C,lSb
n
-K Cl
In (Z5&)
(9)
Ip
In aluminium slloys, the variation in subgrain size with T and I: during steady state hot deformation has been
studied by several workers (2,16,17), the result becoming an empirical relationship of the form: 1 --.A’hZ-B’ 6,
(11)
Where A* and B’ are constants. The observations by Castro-Femandez et al. (2) are plotted in Fig. 3 and as canbeseen~thisdi~thedataareequallywellaccountedforbybothEqs.8and 11. Inbothcasesau activation energy of II! = 156 kJ/mol is used. It follows corn this treatment (Eqs. 8- 10) that we have no simple relationship between the steady state substructure and the Zener-Hollomon parameter. This is ill& in Fig. 3 by solving 8 for di&rent strain rates. It follows from Fig. 3 that within the investigated Zrauge the empirical Eq. 11 gives a very good fit. However, outside this range care must be exercised in using Eq. 11. This point is discussed in more detail in Ref. 12.
In solute containing metals the interactions between the mobile dislocations and the solute atoms may result in a friction stress which may contribute signiticantly to the flow stress. For instance, it follows from the obervations by Castro-Femadez et al. (2) that the o,/o,-ratio is about 0.15 at 300°C and 0.5 at 500°C (+
1.6 E j_ 1.4
2 . -
1.2 I 0.8 0.6 0.4 0.2 23
a’-,
25
’
’ 27
’
’ 29
’
’ 31
’
’ 33
’
’ 35
’
’ 37
’
In Z Figure 3. Depedme
of subgraia size on Zmer-Hollom~n parameta. Experihd
data, +5 (2)
230
STEADY STATE DEFORMATION OF METALS
Vol. 33, No. 2
5sec~‘).An expression for the friction stress is obtained as follows: Consider a metal crystal subjected to a shear stress z. If this crystal contait~~ a uniform density of mobile dislocations, pm, then, if we ignore the intluence of long rang elastic stresses, these dislocations will be exposed to the short range interaction with the solute atoms only, and the applied stress will cause the mobile dislocations to move at a rate (15)
(12)
where Q’is the interaction energy between the solute and the mobile dislocations, y is the activation volume W&J’, where 1”is the separation of solute atoms along the mobile dislocations) and C, is a constant which needs to be de&m&d experimentally. In this artificial case (i.e. ignoring the long range stresses) the applied flow stress becomes identical to the lktional stress, z = q. Further, this stress will cause the crystal to deform at a rate: ?j=bp,v, which in combination with Eq. 12 gives:
(13)
In deriving these expressions the density of mobile dislocations is assumed to scale with the total dislocation density, i.e. J’p,,,= C&3, and C’s=M/CiC,. Like Eq. 8, this equation also needs to be solved numerically with respect to tP Strain Rate Sensitivity
A practical approach in order to measure the activation volume (y=l;bz) becomes to perform an instantaueous change in stmin rate f?om b, to d 2’ It follows from Eqs. 13 that such a change in strain rate will result in a corresponding m - eauilibrium change in the flow stress, A ri, given by (for T ti > kT):
(14)
This equation provides an alternative physical interpretation of the mechanical-equation-of-state introduced by Urcola and Sellars (18). Equation 14 states that if we suddenly change strain rate, the available number of mobile disltions, p& will have to accomodate the rate change, i.e. they will have to change speed from p: to v, = yrjbpk. This higher speed requires an extra stress, AT:, a flow stress change which Vl = Ij,/b involves no change in microstmcture. This response, by nature, becomes a quasi elastic reaction. When a new steady state situation has been reached the new steady state frictional stress becomes: t’, = (kT/P’,J In (8, p I,,,Id, p2,,Jwhich may be only insigniticantly ditferent from t t. For applications and further discussion see Ref. 12. The Steadv State Flow Stress is now obtained by combining Eq. 4 with the solution of Eqs. 9 and 13 for l/B, and zi: Since these two equations refer to di&rent mechanisms of thermal activation it follows that no unique activation energy can be assigned to this t, vs d/T relationship. However, over considerable ranges in Caud T, a constant friction stress becomes a reasonable approximation (see following paper (12)), and in this case an expression for the flow stress can be written on the form:
231
STEADY STATE DEFORMATION OF METALS
Vol. 33, No. 2
160 150
I
_
I
I
I
I
1
I
I
1
I
I
I
I
U,=l56Wlmol
,4() _
'Z
tz5
130 -
.----
e=so
I20 110 2
100 -
E
90 -
G
80 TO60 SO 40 30"""""""'_ 22 24
.
26
2X
311
32
34
36
38
In z
Figure 4. Steady state flow stress vs Zaer-Hollomon
0, - 0, =
Ma3Gb Cl
u,
l
kT In
‘e (a,iUG @pjl
b)C,
pammter.
1 w
Ma,GbkT C,
This relationship can be nicely fitted to the observations of Castro-Femandez et al. (2) as illustrated in Fig. 4. For a further analysis of the application of this model to the deformation of aluminium alloys, see the following paper by Nes and FLU-U (12).
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
CM. Sehrs aud W.J. McG. Tept, Met. Rev. n 1(1972) F.R CastnF& CM. Sellars and J.A. Whkmm, Mat. Sci. and Tech., 9,453 (1990) RShahmi.Brit&mmProject,tobeplbliPhed E.Nea,Adah4etetM&r.,inpres. T.Furu,R0mundadE.Nes,ActaMet.etMater.,inpress. H.MughrabiiaStrrngthofMetalsandAUoys(ICSMAS)(eds. P.Haaseaetal.), PeqamonPreqGxfkd, 1615(1988) H. Mu&rabi, Mater. Sci. Eag, 8i, 15 (1987) O.B. Pedemem, LM. Brown awl WM. Stobbs, Acta Met& Mater., & 1843 (1981) D. Kubhaon Wii Met Tram., L 3173 (1970) D. Kublmam Wilsdoti in Work Hardming in Tension and Fatique (ed A.W. Thompson). The Metallurgical Society of AIME, New York (1977) C.M. Sellara,MateriaisScienceForum, m-u 29 (1993) E.Ne~~mdT.Fum,SubmittedtoSaiptaMetto~withthispaper. J. Gil Sevillano, P. Van Houtte ad E. Aenmdt, Prog. Materi& Sci., u 69 (1980) J. Gil Sevillauo, “Mat&al S&ace and Tedmology. A Compdensive Treatment”, vol. 6 (edited by H. Mu&r&i), VCH, weinbeim, 19, (1993). J.P. Hirth ad J. Lothe, ““hay of Dislocati~“, McGraw-Hill, New York (1968) J.J. Joms,C.M. Sellarsmd W.J. McG.TegarfMet Rev.,& 1(1969) M.A Zaidi ad T. Sheppad. Metal Scieplce. fi 229 (1982) J.J. Urcola and C.M. Sellers, Acta Met et Mater., a (1987), 2659.
CONSTITUTIVE LAWS FOR STEADY STATE DEFORMATION OF METALS, A MICROSTRUCTURAL MODEL Erik Nes The Norwegian Institute of Technology
Department of Metalhugy, 7034 Trondheim, Norway (Received January 3,1995) (Revised January 20,199s) Introduction
Based on extensive investigations of the steady state conditions for creep and hot working, Sellars and McG. Tegart (1) demonstrated that the flow stress was well represented by the following relation:
(1)
where a’, A’, n are temperature independent constants and Z is the Zener-Hollomon parameter, Z = i exp U/kT, whereC is the applied strain rate and U is an activation energy. This relationship is widely used in the modelling of metal forming precesses such as hot rolling, forging and extrusion. The good fit between experimental observations and theoretical predictions is illustrated in Fig. 1, where the saturation stress for two aluminum alloys, tested in plane strain compression (2,3), are plotted as a fuction of Z. In calculating the Z values both Ref. 2 and 3 have used the commonly selected activation energy U = 156 kJ/mol. It is pointed out, however, that Eq. 1 is a purely empirical relationship from which no basic physical mechanism cau be inferred, this applies also to the selection of the activation energy, as a given such value de&es a set of con&m& a’, A’ and n. To &s&ate this point, consider the plane strain compression results given in Fig. 1: Although the results obtained on both the AlMnlMgl alloy (2) and the AlFeO.3SiO.05 alloy (3) are well represented by Eq. 1, it is not possible, based on the result i?om one of these alloys, to predict the behaviour of the other. Or in other words, Eq. 1 has no predictive power outside the alloy conditions tested, to which data the parameters U, a’, n and A’ have to be fitted for each case examined. The objective of this work is to explore the possibilities for deriving an alternative, physically based, relationship which is capable of correlating the steady state flow stress both to the alloy condition and the microstructural characteristics of steady state deformation. . DefUution
:
In order to derive an expression for the steady state saturation stress equations of the form: z, = f (ti, p,) and p,=f(i!T)~tobe~~andsolvedincombination. Thefirstequationdefinestheflowstressinterms of the stored dislocation density p,and the tictim stress T+ The stored dislocation density is not a single, well 225
Vol. 33, No. 2
STEADYSTATEDEFORMATIONOF METALS
is@-
U =1Sbid/mal
140 ,30 _ o Al0.3FeO.lSSi ,20
_ . AllMnlMp
110 3
ton
L zz v
x0 P
b
-
YO 70 60 so
-
40
-
l/n=fl.22 A’=b.b*lO’O
30 20 10
-
. IO“
Id’
IO”
Figure1. Steadyrateflow stressas a h&ion of theZmer-Hohmon parametex forthe.alloysgiven(2,3).
defined quantity, but a more complex parameter , i.e. p, = f(p,i3,8,...) where p represents an elementary dislocation density term, 6 a cell size, 8 a cell or sub-boundary misorientation etc. The second equation is even less well defmed, except being a solution of a di&rential equation of the form:
(2)
This relation simply reflects that during steady state, the defibrmationinduced dislocations, dplldt, are balanced by the dislocations lost due to dynamic recovery, dp;/dt. The dp :/dt-term is derived Corn the well established differential equation dy = bLdp, where p, refers to the mobile dislocations and L is the average slip distance. This equation is an important element in all theories of work hardening. However, except for the standard Stage II-solution, no solution of this equation has been derived which is applicable in a general case where the slip distance, L, relates to a complex substmctum, i.e. L=f(pJ,B,x). Properly solved, this equation will provide a solution: p,=f;(y ,fX, from which the storage term dprdt is obtained by simple differentiation. This short communication will not attempt such an ambitious approach. However, based on the well established observation that during low temperature de&mation, metals display a linear work hardening at large strains (Stage IV), a general expression for the dpi/dt-term can be derived, as wiIl be demonstrated below. During steady state condition this storage term is balanced by the annihilation effect due to dynamic recovery, Eq. 2. The dynamic recovety treatment will be based on the recent work on static recovery by Nes (4) and Fur-uet al. (5). As will be demonstrated in the following, the result becomes a general solution to the problem of deriving an expression for the steady state flow stress as defined here. Model
The following theory rests on the assumption that during steady state deformation of a pure metal, or stable solid sohnion, the substructure can be adequately described by a few microstructural elements, the two most importaut ones being the steady state cell/subgrain size, 8,, and the dislocation density in the celI interior, pti Based on this microstructural description several possible approaches can be taken in order to calculate the flow stress, as discussed in detail in Ref. 4. Two interesting treatments being the composite theory due to Mughrabi (6,7) and Pedersen et al. (8) and a modification (4) of the original Kuhhnann-Wilsdorf (9,10) link
Vol. 33, No. 2
STEADY STATE DFJOFMATION
OF METALS
227
length model. However, as shown in Ref. 4 both approaches predict a relationship of the form ‘c
= %i +
a1 Gb
pi + d--
a2
Gb
8;’
(3)
where ti is a friction stress, G is the shear modulus, b is the Burgers vector and a i, a, are constants. Based
on an extensive investigation of the substructure evolution during hot deformation Sellars and coworkers (2,ll) have demonstrated that during steady state the principle of similitude applies in the sense that the separation of dislocations within the cells (1WpJ scales with the cell size 8,: Jpi = C,/8,, where C, is a constant of typical value of the order 10. Combining this relationship with Eq. 3, it follows that the steady state flow stress can be expressed in terms of only one microstructural parameter, namely the cell/subgrain size 8,, i.e. 7
= T1 +
a,Gb8i1
(4)
where a3 = (C,a , + CQ),with a, expected to be of the order 3. This relationship is comirmed by CastroFemandez et al. (2), see also the following paper (12). The next step is to solve the diIferential Eq. 2, a solution which will provide the steady state subgrain size as a function of strain rate and temperature: 8, = f(q,T). The Steady State Subnrain Size
On the assumption that during steady state defbmMion the principle of similitude applies (i.e. Jp, = C,/8) Eq. 2 can be rewritten in the form
The lirst term detines the rate at which new sub-boundaries are created while the second term gives the subgrain growth rate due to dynamic recovery, balancing these terms defines steady state. As already mentioned in the introduction above, an empirical approach will be taken in order to define the d8~/dt-term. If metals are deformed at room temperature one finds that the variation in the average subgrain size with strain follows a 8 vs E relationship, which is of a similar type for a range of metals, and independent of the mode of deformation as shown in Fig. 2 (taken from the work of Gil Sevillano et al. (13)). In Fig. 2 the variation in the inverse cell size is also given (broken line), and it can be seen that this quantity varies linearly with strain fall. This is au interesting observation in view of the fact that the flow stress commonly is correlated with the inverse cell size. Accordingly, this result is consistent with a linear Stage IV work hardening behaviour. From Fig. 2 it follows that deformation at room temperature for strains &>y/M= 1 (M is the Taylor factor) gives l/8 = (0.7+0.09y)18,,,,5 which in combination with Eq. 3 gives a Stage IV work hardening rate of 0,=0.09a,Gb/8,,,,-2=10-‘G (for Cu, 8 r_,,5=0.29uwork m (13), a,=2.5 and b=2.56*1O%r) which is a very reasonably RT-value indeed. In general the Stage IV hardening rate scales with the initial Stage IV shear stress (t,J which again scales with the steady state saturation stress (r,), i.e. 8,=0.lt,=O.O7t, (see Gil Sevillano (14)). Combining this expression for BWwith Eq. 3 gives the following approximate solution: t+O.O7a,Gb/8,. The d8-/dt-term then takes the form:
l-1 d8-
dt
I -
P, T
ON+
b*
a2Gb
t 8* = - 0.07 8,
(6)
228
STEADYSTATEDEFORMATIONOF METALS
-
xii
z ox.,’
Vol. 33, No. 2
2
-05
zOZ.’ 2 v) 1
5TdN
r,
-0
5
6
s
Figure2. Subgaiakell size as a fimctionof equivalentstrain The size is normalizedto the value found atT~1.5 (13).
Recovery during static annealing of deformed metals has recently been treated by Nes (4) and Furu et al. (5). These works demonstrate that static recovery can be satisfactorily treated based on an internal state variable approach comprising two variables, i.e. the cell/subgrain size and the dislocation density in the cell interior. A similar approach will be tried here in handling the dynamic case. In terms of the basic physics involved, the two types of recovery reactions (static and dynamic) are expected both to be driven by forces derived f?om the fi-eeenergy changes associatedwith a reduction in the stored energy (4,5). No attempts will be made here in accounting for the growth reactions in mechanistic terms, for such details see Refs. 5 and 15. In the following treatment of the steady state flow stress, dynamic recovety will be incorporated by assuming a situation where the sub-boundary migration is controlled by solute drag (5), in which case the dynamic growth rate becomes: db’ dt
= 2bvDCB exp
(7)
where v, isthe Debye tkzquency and Ui is the interaction energy between the solute atom and the boundary dislocation core. P is the driving pressure for subgrain growth (P=2yda where ysB is the sub-boundary where v is the Poisson’s ratio). V, is the activation energy, i.e. ysB=a4 Gb0ln(eeJB) with a,=lNx(l-v) volume (V, = lsb, where 1”is the separation of solute atoms along the boundary dislocation line and h=b/g is the separation of boundary dislocations where 0 is the average sub-boundary misorientation. C, is a constant which needs to be determined experimentally. A general pure-metal-solution, incorporating both vacancy bulk and core dBitsion, requires a more extensive treatment which will be the topic of a subsequent paper. However, such a treatment gives a subgrain growth rate expression of the same form as that of Eq. 7, but with other activation energies and differently defined activation volumes. A main point here is that the physics of the present approach is adequatly ilhtstrated by using Eq. 7. The steady state subgrain size is now obtained by combining Eqs. 5,6 and 7, i.e.
229
STEADY STATE DEZFORMATIONOF METALS
Vol. 33, No. 2
where C, = 6 a,G b3 and C, = 0.07 M/v,bC,. This equation has no analytieal solution with respect to the subgrain size, but can easily be solved numerically. Depending on the value of the PV&T-ratio, Eq. 8 takes the forms: PV(jb
1 -.-+ 6,
-),l: W
Ufb
W ln (ie b,C,) C,lEb
C,lSb
n
-K Cl
In (Z5&)
(9)
Ip
In aluminium slloys, the variation in subgrain size with T and I: during steady state hot deformation has been
studied by several workers (2,16,17), the result becoming an empirical relationship of the form: 1 --.A’hZ-B’ 6,
(11)
Where A* and B’ are constants. The observations by Castro-Femandez et al. (2) are plotted in Fig. 3 and as canbeseen~thisdi~thedataareequallywellaccountedforbybothEqs.8and 11. Inbothcasesau activation energy of II! = 156 kJ/mol is used. It follows corn this treatment (Eqs. 8- 10) that we have no simple relationship between the steady state substructure and the Zener-Hollomon parameter. This is ill& in Fig. 3 by solving 8 for di&rent strain rates. It follows from Fig. 3 that within the investigated Zrauge the empirical Eq. 11 gives a very good fit. However, outside this range care must be exercised in using Eq. 11. This point is discussed in more detail in Ref. 12.
In solute containing metals the interactions between the mobile dislocations and the solute atoms may result in a friction stress which may contribute signiticantly to the flow stress. For instance, it follows from the obervations by Castro-Femadez et al. (2) that the o,/o,-ratio is about 0.15 at 300°C and 0.5 at 500°C (+
1.6 E j_ 1.4
2 . -
1.2 I 0.8 0.6 0.4 0.2 23
a’-,
25
’
’ 27
’
’ 29
’
’ 31
’
’ 33
’
’ 35
’
’ 37
’
In Z Figure 3. Depedme
of subgraia size on Zmer-Hollom~n parameta. Experihd
data, +5 (2)
230
STEADY STATE DEFORMATION OF METALS
Vol. 33, No. 2
5sec~‘).An expression for the friction stress is obtained as follows: Consider a metal crystal subjected to a shear stress z. If this crystal contait~~ a uniform density of mobile dislocations, pm, then, if we ignore the intluence of long rang elastic stresses, these dislocations will be exposed to the short range interaction with the solute atoms only, and the applied stress will cause the mobile dislocations to move at a rate (15)
(12)
where Q’is the interaction energy between the solute and the mobile dislocations, y is the activation volume W&J’, where 1”is the separation of solute atoms along the mobile dislocations) and C, is a constant which needs to be de&m&d experimentally. In this artificial case (i.e. ignoring the long range stresses) the applied flow stress becomes identical to the lktional stress, z = q. Further, this stress will cause the crystal to deform at a rate: ?j=bp,v, which in combination with Eq. 12 gives:
(13)
In deriving these expressions the density of mobile dislocations is assumed to scale with the total dislocation density, i.e. J’p,,,= C&3, and C’s=M/CiC,. Like Eq. 8, this equation also needs to be solved numerically with respect to tP Strain Rate Sensitivity
A practical approach in order to measure the activation volume (y=l;bz) becomes to perform an instantaueous change in stmin rate f?om b, to d 2’ It follows from Eqs. 13 that such a change in strain rate will result in a corresponding m - eauilibrium change in the flow stress, A ri, given by (for T ti > kT):
(14)
This equation provides an alternative physical interpretation of the mechanical-equation-of-state introduced by Urcola and Sellars (18). Equation 14 states that if we suddenly change strain rate, the available number of mobile disltions, p& will have to accomodate the rate change, i.e. they will have to change speed from p: to v, = yrjbpk. This higher speed requires an extra stress, AT:, a flow stress change which Vl = Ij,/b involves no change in microstmcture. This response, by nature, becomes a quasi elastic reaction. When a new steady state situation has been reached the new steady state frictional stress becomes: t’, = (kT/P’,J In (8, p I,,,Id, p2,,Jwhich may be only insigniticantly ditferent from t t. For applications and further discussion see Ref. 12. The Steadv State Flow Stress is now obtained by combining Eq. 4 with the solution of Eqs. 9 and 13 for l/B, and zi: Since these two equations refer to di&rent mechanisms of thermal activation it follows that no unique activation energy can be assigned to this t, vs d/T relationship. However, over considerable ranges in Caud T, a constant friction stress becomes a reasonable approximation (see following paper (12)), and in this case an expression for the flow stress can be written on the form:
231
STEADY STATE DEFORMATION OF METALS
Vol. 33, No. 2
160 150
I
_
I
I
I
I
1
I
I
1
I
I
I
I
U,=l56Wlmol
,4() _
'Z
tz5
130 -
.----
e=so
I20 110 2
100 -
E
90 -
G
80 TO60 SO 40 30"""""""'_ 22 24
.
26
2X
311
32
34
36
38
In z
Figure 4. Steady state flow stress vs Zaer-Hollomon
0, - 0, =
Ma3Gb Cl
u,
l
kT In
‘e (a,iUG @pjl
b)C,
pammter.
1 w
Ma,GbkT C,
This relationship can be nicely fitted to the observations of Castro-Femandez et al. (2) as illustrated in Fig. 4. For a further analysis of the application of this model to the deformation of aluminium alloys, see the following paper by Nes and FLU-U (12).
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
CM. Sehrs aud W.J. McG. Tept, Met. Rev. n 1(1972) F.R CastnF& CM. Sellars and J.A. Whkmm, Mat. Sci. and Tech., 9,453 (1990) RShahmi.Brit&mmProject,tobeplbliPhed E.Nea,Adah4etetM&r.,inpres. T.Furu,R0mundadE.Nes,ActaMet.etMater.,inpress. H.MughrabiiaStrrngthofMetalsandAUoys(ICSMAS)(eds. P.Haaseaetal.), PeqamonPreqGxfkd, 1615(1988) H. Mu&rabi, Mater. Sci. Eag, 8i, 15 (1987) O.B. Pedemem, LM. Brown awl WM. Stobbs, Acta Met& Mater., & 1843 (1981) D. Kubhaon Wii Met Tram., L 3173 (1970) D. Kublmam Wilsdoti in Work Hardming in Tension and Fatique (ed A.W. Thompson). The Metallurgical Society of AIME, New York (1977) C.M. Sellara,MateriaisScienceForum, m-u 29 (1993) E.Ne~~mdT.Fum,SubmittedtoSaiptaMetto~withthispaper. J. Gil Sevillano, P. Van Houtte ad E. Aenmdt, Prog. Materi& Sci., u 69 (1980) J. Gil Sevillauo, “Mat&al S&ace and Tedmology. A Compdensive Treatment”, vol. 6 (edited by H. Mu&r&i), VCH, weinbeim, 19, (1993). J.P. Hirth ad J. Lothe, ““hay of Dislocati~“, McGraw-Hill, New York (1968) J.J. Joms,C.M. Sellarsmd W.J. McG.TegarfMet Rev.,& 1(1969) M.A Zaidi ad T. Sheppad. Metal Scieplce. fi 229 (1982) J.J. Urcola and C.M. Sellers, Acta Met et Mater., a (1987), 2659.