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Work-hardening in cleavage fracture toughness
Acta merall. Vol. 37, No. 8, pp. 2279-2285, 1989 Printed in Great Britain. All rights reserved
WORK-HARDENING
Copyright cr 1989Maxwell
IN CLEAVAGE TOUGHNESS
N. J. PETCH’
OOOI-6160/89 $3.00 + 0.00 Pergamon Macmillan plc
FRACTURE
and R. W. ARMSTRONG2
‘Division of Metallurgy, University of Strathclyde, Glasgow Cl IXN, Scotland and 2Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, U.S.A. (Received 3 June 1988) Ahtract-The contribution of work-hardening to fracture toughness, when the fracture mechanism is cleavage, is considered. This leads to simple relationships for the effect of the dislocation friction stress and of grain size on toughness. There is agreement with published measurements. R&sum-n considtre la contribution de l’tcrouissage 21la r6sistance B la rupture, lorsque le mkcanisme de la rupture est le clivage. Ceci conduit B des relations simples pour l’effet de la contrainte de frottement des dislocations, et de la taille des grains, sur la rtsistance. Ces relations sont en accord avec des mesures publikes. Zusammenfassung-Der Beitrag der Verfestigung zur BruchzSihigkeit wird fiir den Fall betrachtet, da13der Bruchmechanismus Spaltung ist. Hier ergeben sich einfache Zusammenhlnge zwischen Reibungsspannung der Versetzungen und KomgriiDe einerseits und der ZLhigkeit andererseits. Wir finden iibereinstimmung mit verijffentlichten Ergebnissen.
1. INTRODUCTION
In fracture toughness measurements on steels, the fracture mechanisms is usually cleavage at temperatures below about -40°C. There has been extensive analysis of the toughness in terms of the stress distribution ahead of a sharp crack [l-3]. The cleavage may be initiated in carbide or inclusion particles and the influence of the statistical distribution of these has also been extensively examined [4-71. The present paper concentrates directly on the involvement of work-hardening, and this brings out rather simply some features of metallurgical interest. 2. THE DISLOCATION
particle size and local intensified stress come together and cleavage propagation results. If the temperature is raised, go is lowered and further plastic strain is required in the ferrite at the particle to reach the critical local stress. On this basis, a calculation of the dependence of the critical, externally applied, stress intensity, I&, on u0 can be made. True stress-strain curves for steel follow approximately the relationship or = At” where t is the strain, A equals the flow stress at t = 1 and n is a constant. The work-hardening rate is then
FRICTION STRESS
da 2
The yield stress of a ferritic steel depends on two terms, the friction stress, which is the resistance to dislocation movement across a grain, and the grain boundary resistance to the propagation of slip. After yielding, the flow stress or depends on the same two terms, but their values change. The friction stress co arises from the crystal lattice, from solution, precipitation or irradiation hardening and from interaction with other dislocations. The temperature-dependence of the yield stress is entirely due to crOdown to about - 140°C. In a fracture toughness measurement involving cleavage, this is normally initiated in a hard particle by plastic deformation at the experimental sharp crack, and a critical stress is then required to propagate the cleavage on through the ferritic matrix. Thus, as the applied stress is increased, the plastic zone at the crack extends until a suitable combination of
dt
= nA6”
~- ‘,
or approximately
G.
(1)
The value of n is commonly O.lLO.3, with the lower values for the stronger steels of principal interest in relation to fracture toughness. As n decreases, A increases and nA is about constant. Figure 1 shows true stress-strain curves for a number of steels of different strengths. It will be seen that the workhardening rates are quite similar. The nA values are 100-l 10 N mmm2, with a somwhat higher value of 135 N mm-’ for the spheroidised high carbon steel. If there is no alteration in the stress intensification, the increase in strain at the cleavage particles required to compensate for a decrease in the uniaxial (T, there due to a friction stress change is, from (I), approximately
2279
dt = -;da,.
2280
PETCH and ARMSTRONG:
WORK-HARDENING IN FRACTURE TOUGHNESS
305
1
I
I
,
5
10
lb
STRAIN
I
%
Fig. 1. Room temperature stress-strain curves for some
steels, (a) annealed 0.1 C, 0.5 Mn, (b) normal&d 0.19 C, 1.3 Mn, (c) spheroid&d 1.1C, (d) quenched and tempered 0.2C, 0.35Mn, l.SCr, 2.0Ni (e) ditto 0.2C, 0.35 Mn, 1.5Cr, 2.8 Ni. Data from IS-lo].
The friction stress decrease has a closely similar effect on by and a,, so dt = -3
dirY.
Taking & d@ _N--, E @ where @ is the crack opening displacement
particles will then increase by N 35 N mme2. The rate of increase in the cleavage strength falls as the strain increases in a manner similar to the strain-hardening. Thus, the figure of 35 N mm-2 will not vary greatly with the initial strain. Second, a change in the yield stress produces some change in the stress intensification at the sharp crack [1,12,13]. Lowering aY at constant K allows the plastic zone to extend and Fig. 2 shows this will increase the stress intensification factor at the cleavage particles. For example, suppose the initial oY is 500 N mm-* and the intensification factor at the cleavage particle is 3.5. If ay falls by 20 N mmm2, the drop in stress at the particle is 70 N rnrnw2 for constant intensification. However, the increased intensification will partly restore this stress, From Fig. 2, the estimated restoration is about 40 N mm-*. This is not much affected by the initial values of the yield stress or the intensification factor. The third factor is the possibiiity of a change in the particles used to nucleate cleavage as the temperature changes. As aYis lowered, the plastic zone extends, so larger particles may be encountered. However, unless the particle distribution is such that there is a rapid increase in the size of the largest particle in the sample, the effect of larger particles will be counterbalanced by the fact that the new volume is at a lower stress. Thus, there is probably not much change of Kit in this way. The two complications, namely, the increase in the cleavage strength by plastic strain and the restoration of stress at the cleavage particle by increased stress intensi~cation, act in opposite directions and approximately cancel Thus, equations (3) and (4) for the variation of & with friction stress, which consider only the contribution of work-hardening, may well be reasonably accurate.
With a constant stress at the cleavage particles for fracture, the critical stress intensity & is approximately proportional to @ii*, so dK,=
-&da,.
(3)
Thus
where Ki is a constant. lo& & = fog, lug - UY/2nA.
(4)
Three other factors have, so far, been neglected in this calculation of &. First, the strain raises not only the flow stress, but, also, to some extent, the stress required at the particle for cleavage. A strain-hardening of 20 N mmw2 in the uniaxial B, requires a strain of 0.02 from an initial strain of 0.1. This is from (1) with nA N 100 N mm-‘. The measurements of Groom and Knott [l t] for a mild steel show that the cleavage strength at the
X4K/CTy)2 Fig. 2. The dist~bution of the tensile stress rrrrat a distance x ahead of a sharp crack for two values of the strainhardening coefficient n [13].
PETCH and ARMSTRONG:
WORK-HARDENING
IN FRACTURE
TOUGHNESS
IOCI-
\
.
, >\
\ .
\
.
\ I-
\
‘\
‘\
‘v
.
b STRESS
MN G2
4( )-
t
Fig. 3. Kc vs yield stress for three mild steels. Data from [l, 61.
Comparison with experiment shows quite good agreement. Figure 3 gives Kc vs yield stress for three mild steels between -70°C and just below - 140°C. At still lower temperatures (higher a,), Kc decreases
.
i 250
350 STRESS
-2 MN M
Fig. 4. Lo& Kc vs yield stress from
Fig. 3.
650
I
1
1
I
I
I
750 STRESS
>., 1 650
MN ni*
Fig. 5. K, vs yield stress for (a) a quenched and tempered alloy steel (b) a spheroidised steel. Data from [14, IS].
only slowly. This figure is constructed from measurements of the temperature-dependence of cry and Kc [l, 61. It will be noted that the slope progressively increases as Kc increases, as in equation (3). Figure 4 gives log, Kc vs yield stress for these three steels. Apart from the divergence below - 14O”C, there is a linear relationship giving an observed slope of -l/200 to -l/230 l/N mm-*. This is in agreement with equation (4) and with observed values of nA. Figures 5 and 6 give the corresponding data for a quenched and tempered alloy steel (0.16 C, 0.64 MO, 1.78 Cr, 1.65 Ni) and a spheroidised steel (0.96 C, 1.14 Mn). Again, there is agreement in form with (3) and (4). The slopes in Fig. 6 are - l/200 and - l/230 l/N mm-*. Again Kc changes only slowly below about - 140°C. It would seem that the observed effect of a co change on Kc correlates quite well with the analysis in terms of work-hardening as a major factor. Although the change in cr,,by temperature has been used here, the same treatment should give the effect on Kc of solution hardening, precipitation, irradiation or strain rate once the effect of these on (rO is established. Below - 140°C there is clearly some change. For ferritic steels, one factor may be a change in the detail of the deformation process. With a grain size d, a,=a,+k,d-‘:2, where k,d-“* is the stress required to propagate yielding through the grain boundaries. Only crOchanges on going down to about - 140°C.
2282
PETCH
and ARMSTRONG:
WORK-HARDENING IN FRACTURE TOUGHNESS of the concurrent value of T,. In contrast, a change in a,, produces a change in & that is proportional to the concurrent value of K, [equation (3)]. This is because the strain-hardening rate depends on the concurrent strain. The lesson from this is that the higher the value of KC that has been achieved, the more that is lost by a o0 increase. Thus, the achievement of tensile strength through co is disadvantageous for high cleavage fracture toughness, unless the cleavage strength is also increased. From the present viewpoint, a decrease in the cleavage strength resembles an increase in a,. Thus, the higher the value of KC that has been achieved, the greater is the loss if the cleavage stress is lowered. 3. OTHER MICROSTRUCTURES
650
750 =Y
650 MN
ni2
Fig. 6. Lo& & vs yield stress from Fig. 5. Below that k, begins to increase sharply, possibly because of a change to planar slip. The corresponding term for Q, also increases, but more slowly. A temperature change then alters o,, more than a,. Two effects can then be expected from this. First, in Figs 3 and 4, at this low temperature, the change in cr,,overstates the change in 0, at the cleavage particles and, therefore, overestimates the expected change in KC. Also, during a temperature rise, the larger by fall may give sufficient stressrestoration at the particles by extension of the plastic zone, as already discussed, that little additional strain is required to counter-balance the smaller a, fall. Little K, change would then be needed. Other changes such as twinning may also become involved in the deformation and fracture mechanisms at these low temperatures. It should be noted that the concern in this consideration of work-hardening is with the change in K, produced by an alteration in the friction stress. If it were possible to lower the work-hardening rate, that would be advantageous. The other factors that influence KC are the size and distribution of carbide and inclusion particles, the grain size, subgrain size and other metallurgical features that contribute to the yield, flow and cleavage strengths. These affect Kg in (4). There is an interesting contrast in the effect of a,, on the Charpy transition temperature T, and its effect on &. An increase in a, simply raises T, (measured low in the transition range) by an amount that lowers a,, to its original value. Since the temperaturedependence of a, is fairly linear for limited temperature changes, dT,/da,, is fairly constant, independent
Some extremely interesting cleavage measurements are due to Bowen et al. [7]. These are for A533B plate (0.25C, 1.51 Mn, 0.63Ni, 053Mo) in a variety of structures varying from autotempered martensite through bainite to ferrite + pearlite, giving a room temperature proof stress range of 450-1350 N mm-*. Cleavage strengths were obtained by blunt crack measurements, and it was taken that fracture occurred from the largest carbide particles. It was then argued that these were not normally sampled in the small plastic zone at the sharp crack in fracture toughness measurements, so more average particle sizes are relevant, The corresponding cleavage strengths were calculated from the blunt crack measurements using a simple (carbide diameter)-1’2 relationship. For steels with fine average particles, it was concluded that, even at the bottom of the temperature range, the sharp crack would need to use its maximum stress intensification factor (limited by blunting) to achieve cleavage stress. At higher temperatures, lower yield stresses, the maximum intensification would then be too low for cleavage of an average particle, so the plastic zone would have to extend in search of larger particles. The rise in KC with temperature was consequently attributed to the larger plastic zone involved in this search. There is certainly a possibility of complications from blunting of the sharp crack, but there is some uncertainty about the present proposal. This centres on the use of a (diameter)-“* calculation. For ferrite + grain boundary carbide, it has been argued that the cleavage strength is not determined simply by the carbide size and also that, below a certain size, the cleavage may be initiated in a particle, but the cleavage strength reaches a limit determined by the matrix [16]. With spheroidised carbides, Hodgson and Tetelman [17] found the cleavage strength was matrix-controlled when the carbides were smaller than about 1 pm. For bainites and martensites, there is no general agreement on the factors that determine the cleavage strength [l&22]. Thus, it is uncertain that the crack-blunting situation arises and it may be that the temperature-dependence of KC is associated
PETCH and ARMSTRONG:
WORK-HARDENING
Fig. 7. K, vs yield stress for AS33B steel. Data from 171. (a) Ferrite + divorced pear& 9OO”C, transform 610°C. (b) Ferrite + pearl&e; 90&C, transform 400°C. (c) Upper bainite; 125o”C, transform 400°C.
700
600
800a
IN FRACTURE
TOUGHNESS
2283
Fig. 9. K, vs yield stress for A533B steel. Data from [7]. (a) Tempered martensite; 9OO”C,oil quench, temper 615°C. (b) Autotem~red martensite; 9OO”C,oil quench. principally with work-hardening in the manner discussed in the previous section. The measurements of the temperature-dependence of yield stress and & have been used to construct Figs 7-10, givng lu, and log, k;I against o;i for the various microstructures. It witI be seen that these figures have the same form as the corresponding ones in Section 2. The linear slope of log, K, varies from -l/215 to - l/250 l/N mmm2. All this is consistent with a temperature-dependence of Kc determined p~ncipajly by work-hardening. It is quite surprising to find such a close similarity of the log, K, dope over the wide range of microstructures, carbide sizes and distributions examined in the present and previous sections, involving room temperature oYvalues of 250-l 350 N mmez. A dependence principally on work-hardening makes this more understandable. The work-hardening rate represents the balance between the creation and annihilation of dislocations; if more are created, the chance of annihilalition is greater. Although there can certainly be variations in work-hardening rate between steels, in many cases this is not marked, as is evident in Fig. 1. 4.THE INFLUENCE OF GRAIN SIZE IN PLAIN FERRITIC STEELS
0;
MN
n=i*
Fig. 8. Lug, k, vs yield stress from Fig. 7.
A d-“’ dependence of K, has been proposed [23], and Fig. I I shows experimental support [2]. The present approach can be applied to this topic.
2284
PETCH
Fig. 10. Log,
and ARMSTRONG:
WORK-HARDENING IN FRACTURE TOUGHNESS
Kc vsyield stress from Fig. 9.
Suppose there is a Ad-“2 = 1 mm-i’2 refinement in grain size without alteration in the particles involved in cleavage. Various factors then change. (a) Calculation of the cleavage strength for medium grain size ferrite gives an approximately linear dependence on d- ‘/2. For a carbide thickness of 1 pm, the 1 mm-“’ refinement raises the cleavage strength by -40 N rnrnm2 for the post-yield situation [16]. (b) The uniaxial o, at constant strain goes up by - 10 N mm-* in this refinement, so the corresponding stress at the cleavage particles goes up by -35 Nmmm2. (c) The yield stress rises by - 20 N mrnm2 in the refinement, and this will contract the plastic zone at constant K, so lowering the stress at the particles. An estimate of this lowering from Fig. 2, as in Section 2, indicates -40 N mmm2. Thus, (b) and (c) will tend to cancel, and, for cleavage, this grain refinement will need an increase in the stress at the particles of -40 N mm-2, determined principally by the increase in the cleavage strength. This will involve an increase in Us of -11 NmmA2. In the achievement of this by strain-hardening, allowance has also to be made for the concurrent further increase in the cleavage strength with strain. The rate of this is about half the rate of the corresponding strain-hardening at the particles, so an allowance can be made by using an “effective” strainhardening rate of about half normal. For an initial Kc of 30 MPa mi12, the increase in Kc with the refinement can then be calculated from
(3) using do, - 11 N mm-’ in place of -da, and an effective nA of - 55 N mmm2. This gives an increase in Kc of 3.0 MPa ml’* for the 1 mm-Ii2 grain refinement. This is quite an approximate calculation, but the observed change (Fig. 11, - 120°C) is similar at 3.3 MPa m”‘. For the next 1 mm-“’ refinement, the start is from this higher Kc, and, since dK, is proportional to Kc [equation (3)], this will tend to increase dK, in the new refinement. However, there is some counterbalancing, since the increase in the cleavage strength with d-II2 is rather less than truly linear. Thus, the d -‘I’ dependence of Kc is not precise. With ordinary heattreatments, there is the additional factor that changes in grain size tend to be accompanied by some change in carbide size. However, as Fig. 11 shows, the eventual outcome is quite a reasonable linear dependence over the common grain size range of 410 mm-“2. At very fine grains or coarse carbides, there will be little dependence of the cleavage strength, and consequently of Kc, on grain size [ 161.At coarse grain sizes (e.g. <2 mm- “* for 1 pm carbide), when the cleavage strength becomes independent of carbide size, the cleavage strength and Kc should have a steep dependence on d-“2 [16]. Temperature affects the d-II2 dependence of Kc. At each grain size, Kc increases with temperature in the manner discussed in Section 2. Since the rate of increase is a function of the concurrent Kc, equation (3) the increase is greater for fine grains than for coarse. Thus, both the intercept and the slope increase as the temperature is raised. Experimental data from [6] in Fig. 11, although limited to two grain sizes, illustrate this point. 5. CONCLUSION 1. When fracture is by cleavage, the effect on Kc of temperature, or other factors that influence the
1
-1
t
42 d
-1n
mm
Fig. 11.Kcvsd-"2for two mild steels. (a) 0.1 C, 0.44 Mn at - 120°C. (b) 0.08 C, 0.26 Mn at -80°C and - 120°Cand at two carbide sizes. Data from [2,6].
PETCH
and ARMSTRONG:
WORK-HARDENING IN FRACTURE TOUGHNESS
friction stress, can be largely accounted for by the work-hardening required to maintain a critical stress at the cleavage particles. 2. For a change da, in the yield stress due to a change in the friction stress, to a reasonable approximation dK,= dlog, K,=
-&do, -2.
There is good experimental agreement with these expressions. 3. Over a fairly wide range of common grain sizes in plain ferritic steels, K, is linear with d-I/*. This can be accounted for on the work-hardening model. The slope increases with temperature.
REFERENCES I. R. 0. Ritchie, J. F. Knott and J. R. Rice, .I. Mech. Phys. Solids 21, 395 (1973). 2. D. A. Curry and J. F. Knott, Metal Sci. 10, 1 (1976). 3. D. A. Curry, Mater. Sci. Engng 43, 135 (1980). 4. D. A. Curry and J. F. Knott, Metal Sci 13, 341 (1979).
2285
5. Tsann Lin, A. G. Evans and R. 0. Ritchie, J. Mech. Phys. Solids 34, 417 (1986). 6. Tsann Lin, A. G. Evans and R. 0. Ritchie, Metall. Trans. MA, 641 (1987). 7. P. Bowen, S. G. Druce and J. F. Knott, Acta metall. 35, 1735 (1987). 8. R. W. Armstrong, I. Codd, R. M. Douthwaite and N. J. Petch, Phil. Mag. 7, 45 (1962). 9. C. T. Liu and J. Gurland, Trans. metall. Sot. A.I.M.E. 242, 1535 (1968). 10. J. Harding, J. Iron Steel Inst. 210, 425 (1972). 11. J. D. G. Groom and J. F. Knott, Mefal Sci. 9, 390 (1975). 12. D. M. Tracey, J. Engng Mater. Tech. A.S.M.E. 98, 146 (1976). 13. K. Wallin, T. Saario and K. Tiirriien Metal Sci. 18, 13 (1984). 14. D. A. Curry and J. F. Knott, Metal Sci. 13, 341 (1979). 15. A. J. Krasowsky. Yu. A. Kashtalyan and V. N. Krasiko, Int. J. Fract. 23, 297 (1983). 16. N. J. Petch, Acta metall. 34, 1387 (1986). 17. D. E. Hodgson and A. S. Tetelman, Proc. 2nd. Int. Conf. Fract., p. 266. Chapman & Hall, (1969). 18. P. Brozzo, G. Buzzichalli, A. Mascanzoni and M. Mirabile, Metal Sci. 11, 123 (1977). 19. D. A. Curry, Metal Sci. 16, 435 (1982); ibid 18, 67 (1984). 20. D. A. Curry, Metal Sci. 14, 327 (1982). 21. P. Bowen and J. F. Knott, Metal Sci. 18, 225 (1984). 22. P. Bowen, S. G. Druce and J. F. Knott, Acta metall. 34, 1121 (1986). 23. R. W. Armstrong, Engng Fract. Mech. 28, 529 (1987).
WORK-HARDENING
Copyright cr 1989Maxwell
IN CLEAVAGE TOUGHNESS
N. J. PETCH’
OOOI-6160/89 $3.00 + 0.00 Pergamon Macmillan plc
FRACTURE
and R. W. ARMSTRONG2
‘Division of Metallurgy, University of Strathclyde, Glasgow Cl IXN, Scotland and 2Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, U.S.A. (Received 3 June 1988) Ahtract-The contribution of work-hardening to fracture toughness, when the fracture mechanism is cleavage, is considered. This leads to simple relationships for the effect of the dislocation friction stress and of grain size on toughness. There is agreement with published measurements. R&sum-n considtre la contribution de l’tcrouissage 21la r6sistance B la rupture, lorsque le mkcanisme de la rupture est le clivage. Ceci conduit B des relations simples pour l’effet de la contrainte de frottement des dislocations, et de la taille des grains, sur la rtsistance. Ces relations sont en accord avec des mesures publikes. Zusammenfassung-Der Beitrag der Verfestigung zur BruchzSihigkeit wird fiir den Fall betrachtet, da13der Bruchmechanismus Spaltung ist. Hier ergeben sich einfache Zusammenhlnge zwischen Reibungsspannung der Versetzungen und KomgriiDe einerseits und der ZLhigkeit andererseits. Wir finden iibereinstimmung mit verijffentlichten Ergebnissen.
1. INTRODUCTION
In fracture toughness measurements on steels, the fracture mechanisms is usually cleavage at temperatures below about -40°C. There has been extensive analysis of the toughness in terms of the stress distribution ahead of a sharp crack [l-3]. The cleavage may be initiated in carbide or inclusion particles and the influence of the statistical distribution of these has also been extensively examined [4-71. The present paper concentrates directly on the involvement of work-hardening, and this brings out rather simply some features of metallurgical interest. 2. THE DISLOCATION
particle size and local intensified stress come together and cleavage propagation results. If the temperature is raised, go is lowered and further plastic strain is required in the ferrite at the particle to reach the critical local stress. On this basis, a calculation of the dependence of the critical, externally applied, stress intensity, I&, on u0 can be made. True stress-strain curves for steel follow approximately the relationship or = At” where t is the strain, A equals the flow stress at t = 1 and n is a constant. The work-hardening rate is then
FRICTION STRESS
da 2
The yield stress of a ferritic steel depends on two terms, the friction stress, which is the resistance to dislocation movement across a grain, and the grain boundary resistance to the propagation of slip. After yielding, the flow stress or depends on the same two terms, but their values change. The friction stress co arises from the crystal lattice, from solution, precipitation or irradiation hardening and from interaction with other dislocations. The temperature-dependence of the yield stress is entirely due to crOdown to about - 140°C. In a fracture toughness measurement involving cleavage, this is normally initiated in a hard particle by plastic deformation at the experimental sharp crack, and a critical stress is then required to propagate the cleavage on through the ferritic matrix. Thus, as the applied stress is increased, the plastic zone at the crack extends until a suitable combination of
dt
= nA6”
~- ‘,
or approximately
G.
(1)
The value of n is commonly O.lLO.3, with the lower values for the stronger steels of principal interest in relation to fracture toughness. As n decreases, A increases and nA is about constant. Figure 1 shows true stress-strain curves for a number of steels of different strengths. It will be seen that the workhardening rates are quite similar. The nA values are 100-l 10 N mmm2, with a somwhat higher value of 135 N mm-’ for the spheroidised high carbon steel. If there is no alteration in the stress intensification, the increase in strain at the cleavage particles required to compensate for a decrease in the uniaxial (T, there due to a friction stress change is, from (I), approximately
2279
dt = -;da,.
2280
PETCH and ARMSTRONG:
WORK-HARDENING IN FRACTURE TOUGHNESS
305
1
I
I
,
5
10
lb
STRAIN
I
%
Fig. 1. Room temperature stress-strain curves for some
steels, (a) annealed 0.1 C, 0.5 Mn, (b) normal&d 0.19 C, 1.3 Mn, (c) spheroid&d 1.1C, (d) quenched and tempered 0.2C, 0.35Mn, l.SCr, 2.0Ni (e) ditto 0.2C, 0.35 Mn, 1.5Cr, 2.8 Ni. Data from IS-lo].
The friction stress decrease has a closely similar effect on by and a,, so dt = -3
dirY.
Taking & d@ _N--, E @ where @ is the crack opening displacement
particles will then increase by N 35 N mme2. The rate of increase in the cleavage strength falls as the strain increases in a manner similar to the strain-hardening. Thus, the figure of 35 N mm-2 will not vary greatly with the initial strain. Second, a change in the yield stress produces some change in the stress intensification at the sharp crack [1,12,13]. Lowering aY at constant K allows the plastic zone to extend and Fig. 2 shows this will increase the stress intensification factor at the cleavage particles. For example, suppose the initial oY is 500 N mm-* and the intensification factor at the cleavage particle is 3.5. If ay falls by 20 N mmm2, the drop in stress at the particle is 70 N rnrnw2 for constant intensification. However, the increased intensification will partly restore this stress, From Fig. 2, the estimated restoration is about 40 N mm-*. This is not much affected by the initial values of the yield stress or the intensification factor. The third factor is the possibiiity of a change in the particles used to nucleate cleavage as the temperature changes. As aYis lowered, the plastic zone extends, so larger particles may be encountered. However, unless the particle distribution is such that there is a rapid increase in the size of the largest particle in the sample, the effect of larger particles will be counterbalanced by the fact that the new volume is at a lower stress. Thus, there is probably not much change of Kit in this way. The two complications, namely, the increase in the cleavage strength by plastic strain and the restoration of stress at the cleavage particle by increased stress intensi~cation, act in opposite directions and approximately cancel Thus, equations (3) and (4) for the variation of & with friction stress, which consider only the contribution of work-hardening, may well be reasonably accurate.
With a constant stress at the cleavage particles for fracture, the critical stress intensity & is approximately proportional to @ii*, so dK,=
-&da,.
(3)
Thus
where Ki is a constant. lo& & = fog, lug - UY/2nA.
(4)
Three other factors have, so far, been neglected in this calculation of &. First, the strain raises not only the flow stress, but, also, to some extent, the stress required at the particle for cleavage. A strain-hardening of 20 N mmw2 in the uniaxial B, requires a strain of 0.02 from an initial strain of 0.1. This is from (1) with nA N 100 N mm-‘. The measurements of Groom and Knott [l t] for a mild steel show that the cleavage strength at the
X4K/CTy)2 Fig. 2. The dist~bution of the tensile stress rrrrat a distance x ahead of a sharp crack for two values of the strainhardening coefficient n [13].
PETCH and ARMSTRONG:
WORK-HARDENING
IN FRACTURE
TOUGHNESS
IOCI-
\
.
, >\
\ .
\
.
\ I-
\
‘\
‘\
‘v
.
b STRESS
MN G2
4( )-
t
Fig. 3. Kc vs yield stress for three mild steels. Data from [l, 61.
Comparison with experiment shows quite good agreement. Figure 3 gives Kc vs yield stress for three mild steels between -70°C and just below - 140°C. At still lower temperatures (higher a,), Kc decreases
.
i 250
350 STRESS
-2 MN M
Fig. 4. Lo& Kc vs yield stress from
Fig. 3.
650
I
1
1
I
I
I
750 STRESS
>., 1 650
MN ni*
Fig. 5. K, vs yield stress for (a) a quenched and tempered alloy steel (b) a spheroidised steel. Data from [14, IS].
only slowly. This figure is constructed from measurements of the temperature-dependence of cry and Kc [l, 61. It will be noted that the slope progressively increases as Kc increases, as in equation (3). Figure 4 gives log, Kc vs yield stress for these three steels. Apart from the divergence below - 14O”C, there is a linear relationship giving an observed slope of -l/200 to -l/230 l/N mm-*. This is in agreement with equation (4) and with observed values of nA. Figures 5 and 6 give the corresponding data for a quenched and tempered alloy steel (0.16 C, 0.64 MO, 1.78 Cr, 1.65 Ni) and a spheroidised steel (0.96 C, 1.14 Mn). Again, there is agreement in form with (3) and (4). The slopes in Fig. 6 are - l/200 and - l/230 l/N mm-*. Again Kc changes only slowly below about - 140°C. It would seem that the observed effect of a co change on Kc correlates quite well with the analysis in terms of work-hardening as a major factor. Although the change in cr,,by temperature has been used here, the same treatment should give the effect on Kc of solution hardening, precipitation, irradiation or strain rate once the effect of these on (rO is established. Below - 140°C there is clearly some change. For ferritic steels, one factor may be a change in the detail of the deformation process. With a grain size d, a,=a,+k,d-‘:2, where k,d-“* is the stress required to propagate yielding through the grain boundaries. Only crOchanges on going down to about - 140°C.
2282
PETCH
and ARMSTRONG:
WORK-HARDENING IN FRACTURE TOUGHNESS of the concurrent value of T,. In contrast, a change in a,, produces a change in & that is proportional to the concurrent value of K, [equation (3)]. This is because the strain-hardening rate depends on the concurrent strain. The lesson from this is that the higher the value of KC that has been achieved, the more that is lost by a o0 increase. Thus, the achievement of tensile strength through co is disadvantageous for high cleavage fracture toughness, unless the cleavage strength is also increased. From the present viewpoint, a decrease in the cleavage strength resembles an increase in a,. Thus, the higher the value of KC that has been achieved, the greater is the loss if the cleavage stress is lowered. 3. OTHER MICROSTRUCTURES
650
750 =Y
650 MN
ni2
Fig. 6. Lo& & vs yield stress from Fig. 5. Below that k, begins to increase sharply, possibly because of a change to planar slip. The corresponding term for Q, also increases, but more slowly. A temperature change then alters o,, more than a,. Two effects can then be expected from this. First, in Figs 3 and 4, at this low temperature, the change in cr,,overstates the change in 0, at the cleavage particles and, therefore, overestimates the expected change in KC. Also, during a temperature rise, the larger by fall may give sufficient stressrestoration at the particles by extension of the plastic zone, as already discussed, that little additional strain is required to counter-balance the smaller a, fall. Little K, change would then be needed. Other changes such as twinning may also become involved in the deformation and fracture mechanisms at these low temperatures. It should be noted that the concern in this consideration of work-hardening is with the change in K, produced by an alteration in the friction stress. If it were possible to lower the work-hardening rate, that would be advantageous. The other factors that influence KC are the size and distribution of carbide and inclusion particles, the grain size, subgrain size and other metallurgical features that contribute to the yield, flow and cleavage strengths. These affect Kg in (4). There is an interesting contrast in the effect of a,, on the Charpy transition temperature T, and its effect on &. An increase in a, simply raises T, (measured low in the transition range) by an amount that lowers a,, to its original value. Since the temperaturedependence of a, is fairly linear for limited temperature changes, dT,/da,, is fairly constant, independent
Some extremely interesting cleavage measurements are due to Bowen et al. [7]. These are for A533B plate (0.25C, 1.51 Mn, 0.63Ni, 053Mo) in a variety of structures varying from autotempered martensite through bainite to ferrite + pearlite, giving a room temperature proof stress range of 450-1350 N mm-*. Cleavage strengths were obtained by blunt crack measurements, and it was taken that fracture occurred from the largest carbide particles. It was then argued that these were not normally sampled in the small plastic zone at the sharp crack in fracture toughness measurements, so more average particle sizes are relevant, The corresponding cleavage strengths were calculated from the blunt crack measurements using a simple (carbide diameter)-1’2 relationship. For steels with fine average particles, it was concluded that, even at the bottom of the temperature range, the sharp crack would need to use its maximum stress intensification factor (limited by blunting) to achieve cleavage stress. At higher temperatures, lower yield stresses, the maximum intensification would then be too low for cleavage of an average particle, so the plastic zone would have to extend in search of larger particles. The rise in KC with temperature was consequently attributed to the larger plastic zone involved in this search. There is certainly a possibility of complications from blunting of the sharp crack, but there is some uncertainty about the present proposal. This centres on the use of a (diameter)-“* calculation. For ferrite + grain boundary carbide, it has been argued that the cleavage strength is not determined simply by the carbide size and also that, below a certain size, the cleavage may be initiated in a particle, but the cleavage strength reaches a limit determined by the matrix [16]. With spheroidised carbides, Hodgson and Tetelman [17] found the cleavage strength was matrix-controlled when the carbides were smaller than about 1 pm. For bainites and martensites, there is no general agreement on the factors that determine the cleavage strength [l&22]. Thus, it is uncertain that the crack-blunting situation arises and it may be that the temperature-dependence of KC is associated
PETCH and ARMSTRONG:
WORK-HARDENING
Fig. 7. K, vs yield stress for AS33B steel. Data from 171. (a) Ferrite + divorced pear& 9OO”C, transform 610°C. (b) Ferrite + pearl&e; 90&C, transform 400°C. (c) Upper bainite; 125o”C, transform 400°C.
700
600
800a
IN FRACTURE
TOUGHNESS
2283
Fig. 9. K, vs yield stress for A533B steel. Data from [7]. (a) Tempered martensite; 9OO”C,oil quench, temper 615°C. (b) Autotem~red martensite; 9OO”C,oil quench. principally with work-hardening in the manner discussed in the previous section. The measurements of the temperature-dependence of yield stress and & have been used to construct Figs 7-10, givng lu, and log, k;I against o;i for the various microstructures. It witI be seen that these figures have the same form as the corresponding ones in Section 2. The linear slope of log, K, varies from -l/215 to - l/250 l/N mmm2. All this is consistent with a temperature-dependence of Kc determined p~ncipajly by work-hardening. It is quite surprising to find such a close similarity of the log, K, dope over the wide range of microstructures, carbide sizes and distributions examined in the present and previous sections, involving room temperature oYvalues of 250-l 350 N mmez. A dependence principally on work-hardening makes this more understandable. The work-hardening rate represents the balance between the creation and annihilation of dislocations; if more are created, the chance of annihilalition is greater. Although there can certainly be variations in work-hardening rate between steels, in many cases this is not marked, as is evident in Fig. 1. 4.THE INFLUENCE OF GRAIN SIZE IN PLAIN FERRITIC STEELS
0;
MN
n=i*
Fig. 8. Lug, k, vs yield stress from Fig. 7.
A d-“’ dependence of K, has been proposed [23], and Fig. I I shows experimental support [2]. The present approach can be applied to this topic.
2284
PETCH
Fig. 10. Log,
and ARMSTRONG:
WORK-HARDENING IN FRACTURE TOUGHNESS
Kc vsyield stress from Fig. 9.
Suppose there is a Ad-“2 = 1 mm-i’2 refinement in grain size without alteration in the particles involved in cleavage. Various factors then change. (a) Calculation of the cleavage strength for medium grain size ferrite gives an approximately linear dependence on d- ‘/2. For a carbide thickness of 1 pm, the 1 mm-“’ refinement raises the cleavage strength by -40 N rnrnm2 for the post-yield situation [16]. (b) The uniaxial o, at constant strain goes up by - 10 N mm-* in this refinement, so the corresponding stress at the cleavage particles goes up by -35 Nmmm2. (c) The yield stress rises by - 20 N mrnm2 in the refinement, and this will contract the plastic zone at constant K, so lowering the stress at the particles. An estimate of this lowering from Fig. 2, as in Section 2, indicates -40 N mmm2. Thus, (b) and (c) will tend to cancel, and, for cleavage, this grain refinement will need an increase in the stress at the particles of -40 N mm-2, determined principally by the increase in the cleavage strength. This will involve an increase in Us of -11 NmmA2. In the achievement of this by strain-hardening, allowance has also to be made for the concurrent further increase in the cleavage strength with strain. The rate of this is about half the rate of the corresponding strain-hardening at the particles, so an allowance can be made by using an “effective” strainhardening rate of about half normal. For an initial Kc of 30 MPa mi12, the increase in Kc with the refinement can then be calculated from
(3) using do, - 11 N mm-’ in place of -da, and an effective nA of - 55 N mmm2. This gives an increase in Kc of 3.0 MPa ml’* for the 1 mm-Ii2 grain refinement. This is quite an approximate calculation, but the observed change (Fig. 11, - 120°C) is similar at 3.3 MPa m”‘. For the next 1 mm-“’ refinement, the start is from this higher Kc, and, since dK, is proportional to Kc [equation (3)], this will tend to increase dK, in the new refinement. However, there is some counterbalancing, since the increase in the cleavage strength with d-II2 is rather less than truly linear. Thus, the d -‘I’ dependence of Kc is not precise. With ordinary heattreatments, there is the additional factor that changes in grain size tend to be accompanied by some change in carbide size. However, as Fig. 11 shows, the eventual outcome is quite a reasonable linear dependence over the common grain size range of 410 mm-“2. At very fine grains or coarse carbides, there will be little dependence of the cleavage strength, and consequently of Kc, on grain size [ 161.At coarse grain sizes (e.g. <2 mm- “* for 1 pm carbide), when the cleavage strength becomes independent of carbide size, the cleavage strength and Kc should have a steep dependence on d-“2 [16]. Temperature affects the d-II2 dependence of Kc. At each grain size, Kc increases with temperature in the manner discussed in Section 2. Since the rate of increase is a function of the concurrent Kc, equation (3) the increase is greater for fine grains than for coarse. Thus, both the intercept and the slope increase as the temperature is raised. Experimental data from [6] in Fig. 11, although limited to two grain sizes, illustrate this point. 5. CONCLUSION 1. When fracture is by cleavage, the effect on Kc of temperature, or other factors that influence the
1
-1
t
42 d
-1n
mm
Fig. 11.Kcvsd-"2for two mild steels. (a) 0.1 C, 0.44 Mn at - 120°C. (b) 0.08 C, 0.26 Mn at -80°C and - 120°Cand at two carbide sizes. Data from [2,6].
PETCH
and ARMSTRONG:
WORK-HARDENING IN FRACTURE TOUGHNESS
friction stress, can be largely accounted for by the work-hardening required to maintain a critical stress at the cleavage particles. 2. For a change da, in the yield stress due to a change in the friction stress, to a reasonable approximation dK,= dlog, K,=
-&do, -2.
There is good experimental agreement with these expressions. 3. Over a fairly wide range of common grain sizes in plain ferritic steels, K, is linear with d-I/*. This can be accounted for on the work-hardening model. The slope increases with temperature.
REFERENCES I. R. 0. Ritchie, J. F. Knott and J. R. Rice, .I. Mech. Phys. Solids 21, 395 (1973). 2. D. A. Curry and J. F. Knott, Metal Sci. 10, 1 (1976). 3. D. A. Curry, Mater. Sci. Engng 43, 135 (1980). 4. D. A. Curry and J. F. Knott, Metal Sci 13, 341 (1979).
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5. Tsann Lin, A. G. Evans and R. 0. Ritchie, J. Mech. Phys. Solids 34, 417 (1986). 6. Tsann Lin, A. G. Evans and R. 0. Ritchie, Metall. Trans. MA, 641 (1987). 7. P. Bowen, S. G. Druce and J. F. Knott, Acta metall. 35, 1735 (1987). 8. R. W. Armstrong, I. Codd, R. M. Douthwaite and N. J. Petch, Phil. Mag. 7, 45 (1962). 9. C. T. Liu and J. Gurland, Trans. metall. Sot. A.I.M.E. 242, 1535 (1968). 10. J. Harding, J. Iron Steel Inst. 210, 425 (1972). 11. J. D. G. Groom and J. F. Knott, Mefal Sci. 9, 390 (1975). 12. D. M. Tracey, J. Engng Mater. Tech. A.S.M.E. 98, 146 (1976). 13. K. Wallin, T. Saario and K. Tiirriien Metal Sci. 18, 13 (1984). 14. D. A. Curry and J. F. Knott, Metal Sci. 13, 341 (1979). 15. A. J. Krasowsky. Yu. A. Kashtalyan and V. N. Krasiko, Int. J. Fract. 23, 297 (1983). 16. N. J. Petch, Acta metall. 34, 1387 (1986). 17. D. E. Hodgson and A. S. Tetelman, Proc. 2nd. Int. Conf. Fract., p. 266. Chapman & Hall, (1969). 18. P. Brozzo, G. Buzzichalli, A. Mascanzoni and M. Mirabile, Metal Sci. 11, 123 (1977). 19. D. A. Curry, Metal Sci. 16, 435 (1982); ibid 18, 67 (1984). 20. D. A. Curry, Metal Sci. 14, 327 (1982). 21. P. Bowen and J. F. Knott, Metal Sci. 18, 225 (1984). 22. P. Bowen, S. G. Druce and J. F. Knott, Acta metall. 34, 1121 (1986). 23. R. W. Armstrong, Engng Fract. Mech. 28, 529 (1987).