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Reversible work hardening in alloys of cubic metals
REVERSIBLE
WORK
HARDENING
IN ALLOYS
OF CUBIC
METALS*
D. V. WILSON? During plastic deformation of a polye~stall~e metal, if the direction of straining is reversed, part of the work hardening developed by the first mode of straining is lost in reversed straining. The difference in the flow stresses for reversed deformation and continued forward deformation, (measured at the same absolute strain and beyond the first few percent of reversed deformation), was termed “permanent softening” by Orowan. In this research, the possibility of using the magnitude of “permanent softening”, r8, as a measure of the contribution of long range internal stresses to work hardening is explored in the case of alloys of cubic metals. Dispersion hardened alloys give relatively large values of r8. The internal stress system developed by unidirectional straining in such alloys involves back-stresses in the matrix which are balanced by opposing stresses in the second-phase particles. Using X-ray diffractjon me&surementsof the directional lattice strains, the reverse strain required to reduce the average value of back-stresses in the matrix to zero, E,, was determined in a number of alloys. So,, the value of the “permanent softening” observed at E,, was compared with r2, the average value of back-stresses in the matrix estimated directly from the directional residual lattice strains present after unidirectional plastic deformation. The results support the idea that the removal of back-stresses is the dominant factor contributing to r.,, in two-phase alloys. It is suggested that r6. and r2 provide upper and lower limit estimates of back-stress hardening. On this basis a comparison is made of the contribution of back-stresses to work hardening in a representative group of alloys of cubic metals. CONSOLIDATION
REVERSIBLE
DANS
LES ALLIAGES
DE METAUX
CUBIQUES
Dans la deformation plastique d’un polycristal, si l’on inverse le sens de la deformation, une partie de la consolidation produite par le premier mode de deformation est perdue lors de la di?formation inverse. La difference des contraintrs d’ecoulement entre la deformation inverse et la poursuite de la dbformation initiale (mesuree pour la m&me dbformation absolue et au-de18 des quelques premiers pour cent de deformation inverse) a 6th appel&epar Orowan “adoucissement permanent”. Dans cette etude, l’auteur examine la possibiliti d’utiliser la grandeur de ~~l’adoucissementperm~ent” r*, comme une mesure de la contribution des contraintes internes a longue distance a la consolidation, dans le cas des alliages de metaux cubiquas. Les alliages a durcissement de dispersion donnent des valeurs de r8 relativement &levees. Le systeme de contraintes internes developpe dans ces alliages par une deformation unidirectionnelle met en jue des contraintes d’opposition dans la matrice, lesquelles sont contrebalancees par des contraintes en sens oppose dans les particles de la seconde phase. Utilisant des mesures par diffraction des rayons X pour determiner les d&formations du rbeau, l’auteur a d&ermino la dbformation inverse E++ qu’il est nbcessaire d’appliquer pour r&luire a z&o la valuer moyenne des contraintes d’opposition dans la matrice; dans le cas dun certain nombre d’alliages. La valeur de 1“adoucissement permanent” rsA observ& pour la deformation E, a 4% comparCe aver rl, valeur moyenne des contraintes d’opposition dens la matrice estimee direetement a partir des d8formations du r6seau presentes apr&s une deformation plastique unidirectionnelle. Les r&sultatsobtenus appuyent l’hypothese que la disparition des contraintes d’opposition est le faoteur dominant contribuant a rsn dans les alliages a deux phases. 11 est suggere que rp, et 7, donnent des estimations sup&ieure et infirieure dudurcissement produitpar lescontraintesd’opposition. Sur cette base, l’auteur compare les contributions de ces contraintes a la consolidation dans un groupe repmsentif d’alliages de m&aux cubiques. REVERSIBLE
VERFESTIGUNG
IN LECIERUNGEN
KUBISCHER
METALLE
Bei der plastischen Verformung eines polykristallinen Metalles geht bei Richtengsumkehr der Dehnung ein Teil der bei der ersten Dehnungsart erzeugten Verfestigumg verloren. Orowan bezeichnet den Unterschied der FlieBspannungenfur umgekehrte Verformung und fortgesetzte Vorwiirtsverformung (gemessen bei gleicher absoluter Dehnung und oberhalb der ersten paar Prozent mngekehrter Verformung) als “pcrmanente Entfestigung”. In der vorliegenden Arbeit wird die Miiglichkeit des Gebrauchs der GrijDe der “permanenten Entfestigung”, T*, als em Ma6 fur den Beitrag weitreiehender innerer Spannungen zur Verfestigung im Falle von Legierungen kubischer Metalle untersucht. Legierungen mit Dispersionsh~rt~g zeigen relativ hohe Werte fur rg. Das bei Dehnung in einerRichtung in solchen Legierungen erzeugte System innerer Spannungen schlierjt Riickspannungen in der Matrix ein, die ausgeglichen werden durch entgegengesetzte Spannungen in den Teilchen der zweiten Phase. In einor Reihe von Legierungen wurde mit Hilfe von Rentgenbeugungsmessungen der gerichteten Gittervcrzerrungen die umgekehrte Dehnung E, bestimmt, die notwendig ist, urn den Durchschnittswert der Ruckspannungen in der Matrix auf Null zu reduzieren. Der zu E, gehijrige Wert r,, fur die “permanente Entfestigung” wurde vergliehen mit rZ, dem Durohschnittswert der Riickspannungen in der Matrix, der direkt aus den nach gerichteter plastischer Verformung verbleibenden gerichteten Git~rverzerrun~n abgeschatzt wurde. Die Ergebnisse unterstiitzen die Vorstellung, da6 in zweiphasigen Legierungen die und r2 Entfernung der Riiekspannungen den Hauptbeitrag zu -rsl liefert. Man kann annehmen, da9 z511 Abschatzungen fur die obere und untere Grenze der Rtiokspannungsverfestigung ermiigliohen. Auf diescr Grundlage wird fur eine reprlisentative Gruppe von Legienmgen kubischer Metalle ein Vergleich angestellt iiber den Beitrag von Riickspannungen zur Vorfestigung. * Received July 28, 1964; revised November 30, 1964. t Department of Industrial Metallurgy, University of Birmingham. ACTA M~TALLURGICA,
VOL. 13, JULY
1965
807
805
ACTA
METALLURGICA,
1. INTRODUCTION
Although knowledge of the behaviour of metals in reversed defor~tion has played an essential part in the development of dislocation theories of work hardening, its main contribution has been in showing that the dislocation arrangements must possess a certain stability in unloading and stress reversal, rather than in providing a direct insight into the role of internal stresses. Thus the evidence of back-stress hardening given by the Bausehinger effect is illdefined, and in many cases there are other possible sources of directionality in flow strength. Orowancl) has discussed this problem in terms of the effects of two generalised types of obstacle to slip, namely, strong barriers and permeable obstacles. The former, but not the latter, were expected to promote back-stress hardening due to some form of dislocation pile-up at the barriers. Orowan emphasised that backstress hardening must be wiped out to a large extent by reversed plastic deformation, and that it would therefore give rise to a prominent “permanent” softening effect.* On the other hand, an array of permeable obstacles could give rise to low initial flow strength in reversed deformation without causing appreciable permanent softening. In this Case it was envisaged that mobile dislocations would move forward through the array at the flow stress, breaking through individual obstacles under the pressure of very few piled-up dislocations. At the end of the prestraining most dislocations would have reached a point where they were held up against a particularly effective, or closely spaced row of obstacles. On stress reversal movement of the dislocations back from the obstacles would start at a relatively low applied stress, but in the absence of extensive dislocation pile-ups the flow stress would increase rapidly towards the value obtained in prestraining, as effective groups of obstacles were again encountered. Since most single phase oubic metals show a rather small permanent softening effect, Orowan concluded that work strengthening in such metals is mainly due to obstacle hardening, rather than to back-stresses associated with strong barriers. There is one group of alloys of cubic metals which certainly contain strong dislocation barriers. These are the alloys strengthened by a dispersion of strong
VOL.
13,
1985
non-coherent particles of a second phase. This investigation was made, firstly, to test the prediction implied in Orowan’s argument, that the pe~anent softening effects given by dispersion hardened alloys will be much greater than those in single phase materials and, secondly, to explore how far measurements of permanent softening are useful in making quantitative estimates of the contribution of back-stresses to work hardening. 2.
THE ON
EFFECTS OF MICROSTRUCTURE PERMANENT SOFTENING IN REVERSED STRAINING
~~easurements of the Bauschinger effect were made on each of the alloys described in Table 1, using tubular specimens tested in torsion at a strain rate of 6 x 10M5set-l. Strains were measured optically using mirrors attached at each end of a 2 in. gauge length and torques were measured with a strain-gauge load cell. The tubes were of 0.062 in. wall thickness and either 0.875 or 0.565 in. external dia., depending on the 00.0~strength. Stresses were calculated using the relationship due to Woolley.‘2) Two specimens were tested in each structural condition. Both received a prestrain of 9 T$ shear strain: this was followed by a reverse strain of 4 to 5% in the first specimen and a continued forward strain of the same amount in the second. Fig. 1. gives examples of the measurements made in completed tests. For convenience they are plotted in terms of absolute stress and absolute strain. Thus rj and rr represent the ilow stress in continued forward straining and reverse straining respectively. Orowan estimated that a reverse strain of between 1 and 3% would be sufficient to remove transient softening associated with obstacle hardening. Thus, to allow a comparison of the relative permanent sof~ning effects in all the alloys tested, in Fig. 2 the values of 7, for reverse strains up to 4% are plotted as a fraction of TV,measured at the same total strain. The results summarised in Fig. 2 fall into two groups: the first, concerned with alloys which are effectively single phase, showed small permanent softening effects, the second, concerned with the two phase alloys, showed relatively large effects. The first group contains representative alloys which, in addition to the hardening mechanisms present in pure metals, * Here permanent implies that part of the initia2 work would be strengthened by one or more of the following: hardening is permanently removed by reverse straining. The solution hardening, Cottrell locking, order hardening evidence considered is, essentially, the difference in the and (in alloy 6a), solute clusters formed in a super(absolute) values of flow strength, for continued forward and for reverse straining, measured at the same total strain and saturated solution. In addition, the low carbon steel, beyond the point at which the two flow stress curves become and to a lesser extent the Duralumin Ba, contained a approximately parallel: e.g. in Fig. l(a). the permanent softening 7f is given by the vertical separation of the two small volume fraction of widely dispersed second phase flow stress curves, measured beyond the Grst fow percent of particles (not predominantly at grain boundaries). reversed strain.
WILSON:
REVERSIBLE
WORK
HARDENING
The behaviour of this first group was not essentially different from that of pure polycrystdtlline cubic metals. The largest permanent softener effect in the group was shown by the alloy of low stacking fault energy, u-brass. In contrast, all the dispersion hardened materials showed relatively large permanent softening effects. With comparable particle sizes, permanent softening increased with the volume fraction of precipitate (alloy 3 compared with 2a, 2b snd la). With similar volume fractions it increased with the fineness of dispersion (2a compared with 2b and 6b with 60). However, platelets of 8 precipitate in A.-& (alloy fib) and cementite lamellae in pearlite were possibly more effective than dispersions of relatively equi-d~ensioned particles. In terms of the distinction made by Orowan, the dislocation barriers in the first group are expected to be relatively permeable and those in the second group relatively strong. However, we expect that the extent of piled-up groups of like dislocations at strong barriers will generally be limited by slip processes within the matrix. In most cases of practical interest the effects of differences in barrier strength and barrier spacing will be differences in degree rather than in kind. In this connection results with the age hardened slloy 6 are of p~rtieul~r interest since they can occur in either group, depending on the stage of precipitation. A more detailed study of the effects of pro-
FIG. 1. The effect of precipittltion on the Bauschinger Effect in alloy 7 (Al-4 % Cu): quenched from 525°C and aged at 190°C, for (a) 5 hr and (b) 340 hr, befare straining. (Fig. lb is also used to define T,,.)
IN
ALLOYS
OF
CUBIC
METALS
80%
2. The ratio of flow strength in reversed deformstion to that in continued forwefd deform&ion, merasured at the same total strain. The curves ttre identified with the alloy designations used in Table 1.
FIG.
gressive changes in barrier strength was made in such a system using the coarse grained, high purity Al4% Cu alloy 7. This was quenched from 525% and aged at 190°C for the series of times given in Fig. 3. Complete stress-strain relationships representing extremes of behaviour are given in Fig. 1. From such relationships the initial yield (O.l”/O proof) stress, rW, and the work hardening rate at 3% strain, (dr/&)s, were measured in prestr~~ing. After 9% prestrain and reverse straining the permanent softening, T$ = rt - T?, was measured at a reverse strain of 3% (total strain 12%). Results for the complete series are summarised in Fig. 3. The precipitation sequence in Al-t% Cu is well known and its effects on mechanical properties have recently been reviewed by Kelly and Nicholson.(3) Up to ageing times of about 10 hr the precipitate would be mainly closely spaced 0” particles which are coherent with the matrix. The increase in yield and flow strengths would be determined mainly by the increased stress necessary to force dislocations through the growing 8” precipitates. Despite the large rise in yield strength in this first stage of ageing, neither the initial work hardening rate nor the evidence of back-stress hardening provided by rQwas significantly greater than that obtained with the solid solution. In contrast, within the range of ageing times from about 10 to 100 hr, when the partially coherent 8’ precipitate would progressively replace 8”, the initial work hardening rate and r, increased continuously and in step. Ma,ximum values of (d~lds), and r5.were reached after
ACTA
810
I
,
METALLURGICA,
I
I
I
I
,
I
I
/ 1
j 1
I
I 10 hours
at
lo* 190'C.
I lo3
FIG. 3. Al-4% Cu (dloy 7) quenched from 525°C and aged at 190°C before testing in the manner illustrated in Fig. 1: showing the effect of ageing time on the initial yield strength, on work hardening at 3% strain and on permanent softening measured at 3% reverse strain efter 9 o? prestrain.
about 300 hrs. at 19O”C., a stage ofageing well beyond that giving peak yield strength. The microstructure after 300 hrs. ageing consisted mainly of 8’ platelets, distributed rather inhomogeneously but with an average diameter and spacing of about 500-600 A. The curves for alloys 6b and 6c in Fig. 2 show that 7, for more widely dispersed 0 particles, obtained in more severely overaged conditions, would be appreciably smaller than the values obtained with 8’. The results summarised in Figs. 2 and 3 are promising insofar as the relative magnitudes of the permanent softening effects are qualitatively consistent with expectations of the relative electiveness of dislocation barriers in the different alloys. However, a direct relationship between -r, and the magnitude of back stress hardening remains to be demonstrated and, if quantitative comparisons are to be made, equivalent strains with respect to reversal of the directional component of the internal stress system must be identified. 3.
EVIDENCE OF THE REVERSAL BACK-STRESSES
OF
If back-stresses are developed in prestraining they will at first assist dislocation movement in reverse straining. However, continued reverse straining will progressively annul back stresses inherited from the earlier deformation and eventually a new pattern of such stresses will be developed, of opposite sign to the original ones. The effects of such a re-organisation of the internal stress system on flow strength is, perhaps, better described by the term “reversible work hardening” than by permanent softening. In a real material reversal of the internal stress system will develop inhomogeneously,Y nevertheless there will be a narticuA.
VOL.
13,
1965
lar value of the reverse strain, E,, which reduces the average value of the long range baok-stresses in the advanced up matrix to zero. In terms of the ardent the reversible work hardening to this stage Tag, measured at strains corresponding to E, (Fig. l(b)), should provide a suitable basis for comparison of the contributions of back-stress hardening to total flow strength. However, we should note that a reduction in other sources of work hardening might contribute to TV,,,particularly if some of the obstacles generated by dislocation interactions in prestraining are at first removed by reversed deformation. It has been shown(4) that the dominant long range stress system in a deformed high oarbon steel involves back-stresses in the ferrite matrix which are balanced, for the most part, by opposing stresses in the cementite particles. Such an inter-phase stress system can be investigated using X-ray diffraction measurements of lattice strains. A selection of the main types of alloy described in Table 1 were investigated in this way. Several torsion specimens were prepared from a given material: each was prestrained 976 and then given one of a series of reverse strains in the range O-lOo/o. For examination of the residual back-stresses in the matrix phase, the strained testpieces were sectioned at 45’ to the axis in two mutually perpendicular directions, corresponding to the directions of principal stress. The sections were polished carefully to remove the disturbed layers at the surfaces to be used in the X-ray examinations. A28, the angular difference in the positions of the diffraction line peaks, obtained from the two mutually pe~endicular sections of each testpiece, was measured using a Geiger counter diffractometer. (The method has been described previously(4).) Deformation by simple shear proved to be advantageous because equivalent reflections from the two sections of a testpiece were of the same line profile, allowing small differences in A26 to be measured by simple superpositioning despite severe line broadening. Measurements were made on the steels using the 310 and, in some cases, the 211 and 220 ferrite reflections with Co and Fe radiations. For the ~l~inium alloys the 420 and 511,333 reflections were used with Co and Cu radiations. Fig. 4 shows a complete series of measurements made on the high carbon steel. In this case E, was close to 3% reverse strain. Similar measurements on the low carbon steel la are also given in Fig. 4 although, in the ease of single phase materials, inte~retation of the apparent X-ray evidence of back-stresses is speculative.t4) Table 2 summarises the results of stress measurements made on all the alloys in which E,,~ was estimated in this manner. Apart from the reversible
WILSON:
REVERSIBLE
WORK
HARDENING
IN
ALLOYS
OF
CUBIC
811
METALS
TABLE 1
= Approximate microstructural parameters
i Chemical composition (wt. %)
Alloy
._ la (low carbon steel)
0.11 C; 0.002 N; 0.47 Mn; 0.08 Si; 0.03 P; 0.03 S
Volume fraction of precipitate (f)
Pre-treatments
Recrystallised at 650°C
Meen planar diameter of Imatrix grains (microns)
Mean planar Mean planar diameter of spacing of precipitate particles (microns) (microns) _-
0.016
10
19
0.4 .-
lb (decarburised 0.006 C; 0.001 N; steel other elements as in la ~.
Recrystallised at 630°C
2a
Quenched 82O”C, spheroidised 700°C for 10 hr
2b
0.74 C; 0.29 Mn; 0.19 Si; 0.015 P; 0.01 S
3
(1.14% c&rbon steel).
1.14 C; 0.32 Mn; 0.01 Si; 0.03 P; 0.01 s
Quenched from 9OO”C,tempered 650°C for 2 hr
4
(70/30 Brass)
69.5 Cu; 0.02 Fe; Pb, Sn and Ni < 0.01; Zn balance
Recrystallised at 55OOC
5
(Al/3 Mg)
3.2 Mg in high purity Aluminium
Recrystallised at 350°C
4.05 Cu; 0.72 Mn; 0.69 Mg 0.74 Si; 0.31 Fe
Durelumin
0.5
N2?
0.11
0.7
0.25
Nl?
Lamellar spacing 0.17
0.11
0.17
0.3
0.5
colony size 10 -4
to l?
-. 29
_. 93
_,
Quenched 52O”C, aged 190°C for 150 hr Quenched 52O”C, aged 350°C for 3 hr
6c 7
1.4
Quenched 52O”C, aged 20°C for 90 min
6a
6b
0.11
_
Quenched 82O”C, tempered 650°C for 2 hr Rapidly air cooled from 820°C
2c
33 _
_ (0.74 % carbon steel.)
_
(Al/4 Cu)
4.1 Cu in high purity Aluminium
5
NO.05
~0.06
NO.055
1.0
NO.05
I
0.25
-
Quenched 525°C and aged for times given in Fig.3
(Al/3 Mg)
1.6
2.0
1.2
6.1
-
13.2
0.15
0.09
la (low carbon steel)
1.65
3.6
2.4
15.6
6.0
25.2
0.14
0.09
6b (Duralumin)
3.3
12.4
6.3
20.5
-
28.5
0.44
0.22
2a (0.74 C steel, annealed 700°C)
2.9
13.4
7.5
30.5
-
46.0
0.29
0.16
2b (0.74 C steel, tempered 650°C)
3.1
19.6
12.7
42.5
31.5
56.0
0.35
0.23
3
3.1
26.7
13.4
43.5
33.6
64.1
0.42
0.21
4.7
31.2
16.1
36.8
32.4
67.2
0.46
0.24
(1.1 C steel, tempered 650°C)
2c (0.74 C steel, pearlite) Stresses are in 1000 p.s.i.
80
ACTA
812
METALLURGICA,
0
0 .4
0 0)
[
on
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-0 .a --
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REVERSE
1 lo-
Em. 4. Effect of reverse straining on directional residual lattice strains in the matrix of steels previously strained 9%. A20 is the angular difference in the positions of 310 reflections (using Co radiation) from two mutually perpendicular sections cut in directions of principal stress. Curve 1 refers to the high carbon steel 3 and curve 2 to low carbon steel la.
work hardening rSa, the initial macroscopic yield (or 0.1% proof) stress ry, and the flow strength at 9% prestrain 7-f(9),are given. Estimates of work hardening in unidirectional straining from 75 - 7y are misleading because 7y will generally be affected by transient strengthening effects associated, for example, with initial dislocation locking. For this reason in some cases an estimate was made of T,,, the “friction” stress at zero plastic strain. This was obtained in the usual way, by extrapolation of the later part of the stressstrain curve back to zero plastic strain using a log-log plot. 4.
THE RELATION OF REVERSIBLE HARDENING TO BACK-STRESSES
VOL.
13,
1965
where E, is half the observed total difference in mean strain, calculated from the relative positions of the diffraction line peaks given by the two sections in the directions of principal stress: E is Young’s modulus for the particular crystallographic direction used in the measurements and Y is an average value of Poisson’s ratio. Although the results in Fig. 5 show that T, was close to being proportional to T,,, the ratio of their magnitudes was only about 0.53. 7a must underestimate the back stresses present during plastic deformation, partly because the stresses will relax to some extent on unloading and a further relaxation is expected in the thin layer close to a free surface which must be used in diffraction measurements. To minimise plastic relaxation effects, the specimens used for measurements of 7, were strain aged for at least 24 hr with the flow stress still applied, before unloading and sectioning. Ageing of the aluminium alloys was at room temperature and of the steels at about 60°C. (In the latter case ageing under stress gave an increase of about 15% in the observed residual strains.) Apart from stress relaxation effects, the observed movement of the diffraction line peaks may be reduced, in relation r
!8-
!4-
!O-
6-
WORK
In two phase alloys a direct estimate of the average long range back-stresses developed in plastic straining can be made from X-ray measurements of the directional residual lattice strains in the matrix. In Fig. 5 the results of such estimates, based on X-ray measurements on specimens which had been strained unidirectionally to 9% shear strain, are compared with the values of r,, obtained with similar specimens forward strained 9% and then reversed strained to the appropriate values of E, (Table 2). TV,the estimate of the average back stresses based on the residual lattice strains, was calculated assuming
2-
8-
4-
L
Fro. 5. Relationship between the magnitude of reversible work hardening (T,,) and of back-stresses, estimated from residual lattice strains in the matrix. v alloy 5; 0 alloy la; A alloy 6b using 420 reflection: A alloy 6b using 611, 333 reflection; x alloy 2a; + alloy 2b; p alloy 3 using 220 reflection; n alloy 3 using 310 reflection; 0 alloy 2c using 220 reflection; l alloy 20 using 310 reflection; 0 alloy 2c using 211 reflection.
WILSON:
REVERSIBLE
to the magnitude
of the effective
stresses are partly balanced No allowance lating
7,.
WORK
HARDENING
back-stresses,
was made for this possibility
The consequent
error may
in calcu-
not
of the results with single phase materials
quite uncertain.
support
back-stress
AND
the results
On the question quantitative
in Fig.
5 clearly
given
between
of how far the measurements it has been
allow
noted
previous section that T~, derived from X-ray
be represented
in 7,, and, in this
case, there may be a contribution
from other sources
formation.
For the two-phase
reasonable
to conclude
0.15
fraction
volume
of carbide
FIG. 6. Relationship between reversible work hardening and the volume fraction of spheroidal cementite particles for steels lb, la, 2 and 3. Results for steels 2 and 3 apply to a mean planar particle diameter of about 0.3 p (the value for steel 2 was interpolated). The corresponding diameter for steel la was -0.4 ,u.
in reverse de-
alloys,
it is therefore
that 7,, and 7, provide
upper
proportional
to the volume
fraction
strain of E, was similar to or slightly
measurements
macroscopic
on single phase alloys is much
more
In the stronger alloys the flow strength at a reverse
uncertain
but, in any case, results for reversible work
material.
hardening
in these materials imply that back stresses
estimate
small contribution
to the flow
strength.
yield
tially consistent
alloys follow an essen-
pattern in which back-stress
ing increases with volume fraction with the fineness of its dispersion modulus
of the matrix
shape varied examined
of the precipitate, and with the elastic
(Table 2).
clearly of first importance, sons are restricted
Particle
rather
spacing is
but quantitative
in this respect widely.
the most
harden-
Among
effective
compari-
because
particle
the structures
precipitates
were the
strength
However,
of the fraction rapidly
deformation most
of the alloys
sources of strengthening location
locking).
pearlitic
steel 2c (f=
cementite
0.11).
lamellae
In both these alloys TV
and 7s12indicated that back stresses probably ted between
a quarter
precipitate
the low carbon steel, the most ineffective was
the
coarse
dispersion
with
f = 0.11,
back
stresses
of
rounded
steel 2a. In this accounted
between 0.16 and 0.29 of the total flow strength. far as they go, results for the comparable steels are reasonably
for As
spheroidised
consistent with T,, being directly
initial
be affected
yielding
in
by transient
(e.g. the effects of initial dis-
An approximate
allowance
can be
made for the effects of the second factor by extrapolating the stress-strain
relationship
observed at large Some values
plastic strains back to zero plastic strain. of T,, estimated
in this way are given in Table 2. For
the stronger steels, the mean value of the two estimates stresses represents
about
~~(s) -
3 of the apparent
TV.
An interesting feature of reverse straining in the two phase alloys is that, apart from the special case of the pearlitic steel, the strain E, required to annul the back stresses was rather insensitive
strain.
cementite particles in the spheroidised case,
contribu-
and a half of the total flow
strength after 9% unidirectional Excluding
in the
because
would
of back
the
due
at the outset of large scale plastic
total work hardening,
and
undeformed
of total work hardening
and second,
closely spaced 8’ platelets in A1-Cu (e.g. alloy 6b with 0.05)
less than the
the
it is difficult to make a realistic
f
about
of
to back stresses, first because back stresses build up extremely
Results with the two-phase
of precipitate
(Fig. 6).
and lower limit estimates of the contribution of backInterpretation of the X-ray stress hardening.
must make a relatively
I
I
I
0.10
0.05
measure-
ments of residual strains, must tend to underestimate
of work hardening which are diminished
I
in the
the back-stresses present at the end of the initial straining. On the other hand, the whole of back-stress should
15-
and reversible work hardening.
conclusions,
hardening
20-z2
CONCLUSIONS
view of the relationship
hardening
813
METALS
25-
0 0 c
Orowan’s
CUBIC
be very
pretation
DISCUSSION
OF
if the
in the 2-phase alleys(4) but it makes inter-
Qualitatively
ALLOYS
within the matrix phase.
important
5.
IN
precipitates. tributing
to the spacing
The total number
of dislocations
to the reverse strain is expected
as the barrier spacing is decreased. fore suggests the possibility macroscopic
con-
to increase
This result there-
that, so far as resultant
strain is concerned,
the effects of precipi-
tate spacing on the density of moving approximately
of the
counterbalanced
dislocations
is
by its effects on their
814
ACTA
METALLURCICA,
mean free path. Insofar as this is true E, may provide a measure of the effectiveness of individual barriers. Prom a more practical viewpoint, the results ilh~ trate the directional character of work hardening in two phase alloys, which must be particularly important in applications involving stress reversal. In general, directionality becomes more extreme as dispersion hardening becomes more effective. The results with the pea&tic structure point to the hketihood of even stronger effects of the same kind in some fibre-strengthened materials.
VOL.
13,
1965 ACKNOWLEDGMENTS
Thanks are due particularly to Professor %. C. Rollason for the provision of research facifities and to Dr. G;. R. Ogram, who designed and constructed the torsion apparatus. REFERENCES 1. E. ORow_w, Internal Strassw and Pa&we in Met&, p. 59. EIs&er ff959). 2. R. L. WOOLEY, Phil. &fag. 4.4, 597 (1953). 3, A. KELLY kind R. R. NICHQL.sON, I%?p%% ‘im X&&ls &Sence, Vol. 10,“Precipitatian Hardening”, Pergamorr Press, Oxford (1963). 4. D.V. WILSON andY,A. KoNNAN,A&~C~~~. 12,617(1964).
WORK
HARDENING
IN ALLOYS
OF CUBIC
METALS*
D. V. WILSON? During plastic deformation of a polye~stall~e metal, if the direction of straining is reversed, part of the work hardening developed by the first mode of straining is lost in reversed straining. The difference in the flow stresses for reversed deformation and continued forward deformation, (measured at the same absolute strain and beyond the first few percent of reversed deformation), was termed “permanent softening” by Orowan. In this research, the possibility of using the magnitude of “permanent softening”, r8, as a measure of the contribution of long range internal stresses to work hardening is explored in the case of alloys of cubic metals. Dispersion hardened alloys give relatively large values of r8. The internal stress system developed by unidirectional straining in such alloys involves back-stresses in the matrix which are balanced by opposing stresses in the second-phase particles. Using X-ray diffractjon me&surementsof the directional lattice strains, the reverse strain required to reduce the average value of back-stresses in the matrix to zero, E,, was determined in a number of alloys. So,, the value of the “permanent softening” observed at E,, was compared with r2, the average value of back-stresses in the matrix estimated directly from the directional residual lattice strains present after unidirectional plastic deformation. The results support the idea that the removal of back-stresses is the dominant factor contributing to r.,, in two-phase alloys. It is suggested that r6. and r2 provide upper and lower limit estimates of back-stress hardening. On this basis a comparison is made of the contribution of back-stresses to work hardening in a representative group of alloys of cubic metals. CONSOLIDATION
REVERSIBLE
DANS
LES ALLIAGES
DE METAUX
CUBIQUES
Dans la deformation plastique d’un polycristal, si l’on inverse le sens de la deformation, une partie de la consolidation produite par le premier mode de deformation est perdue lors de la di?formation inverse. La difference des contraintrs d’ecoulement entre la deformation inverse et la poursuite de la dbformation initiale (mesuree pour la m&me dbformation absolue et au-de18 des quelques premiers pour cent de deformation inverse) a 6th appel&epar Orowan “adoucissement permanent”. Dans cette etude, l’auteur examine la possibiliti d’utiliser la grandeur de ~~l’adoucissementperm~ent” r*, comme une mesure de la contribution des contraintes internes a longue distance a la consolidation, dans le cas des alliages de metaux cubiquas. Les alliages a durcissement de dispersion donnent des valeurs de r8 relativement &levees. Le systeme de contraintes internes developpe dans ces alliages par une deformation unidirectionnelle met en jue des contraintes d’opposition dans la matrice, lesquelles sont contrebalancees par des contraintes en sens oppose dans les particles de la seconde phase. Utilisant des mesures par diffraction des rayons X pour determiner les d&formations du rbeau, l’auteur a d&ermino la dbformation inverse E++ qu’il est nbcessaire d’appliquer pour r&luire a z&o la valuer moyenne des contraintes d’opposition dans la matrice; dans le cas dun certain nombre d’alliages. La valeur de 1“adoucissement permanent” rsA observ& pour la deformation E, a 4% comparCe aver rl, valeur moyenne des contraintes d’opposition dens la matrice estimee direetement a partir des d8formations du r6seau presentes apr&s une deformation plastique unidirectionnelle. Les r&sultatsobtenus appuyent l’hypothese que la disparition des contraintes d’opposition est le faoteur dominant contribuant a rsn dans les alliages a deux phases. 11 est suggere que rp, et 7, donnent des estimations sup&ieure et infirieure dudurcissement produitpar lescontraintesd’opposition. Sur cette base, l’auteur compare les contributions de ces contraintes a la consolidation dans un groupe repmsentif d’alliages de m&aux cubiques. REVERSIBLE
VERFESTIGUNG
IN LECIERUNGEN
KUBISCHER
METALLE
Bei der plastischen Verformung eines polykristallinen Metalles geht bei Richtengsumkehr der Dehnung ein Teil der bei der ersten Dehnungsart erzeugten Verfestigumg verloren. Orowan bezeichnet den Unterschied der FlieBspannungenfur umgekehrte Verformung und fortgesetzte Vorwiirtsverformung (gemessen bei gleicher absoluter Dehnung und oberhalb der ersten paar Prozent mngekehrter Verformung) als “pcrmanente Entfestigung”. In der vorliegenden Arbeit wird die Miiglichkeit des Gebrauchs der GrijDe der “permanenten Entfestigung”, T*, als em Ma6 fur den Beitrag weitreiehender innerer Spannungen zur Verfestigung im Falle von Legierungen kubischer Metalle untersucht. Legierungen mit Dispersionsh~rt~g zeigen relativ hohe Werte fur rg. Das bei Dehnung in einerRichtung in solchen Legierungen erzeugte System innerer Spannungen schlierjt Riickspannungen in der Matrix ein, die ausgeglichen werden durch entgegengesetzte Spannungen in den Teilchen der zweiten Phase. In einor Reihe von Legierungen wurde mit Hilfe von Rentgenbeugungsmessungen der gerichteten Gittervcrzerrungen die umgekehrte Dehnung E, bestimmt, die notwendig ist, urn den Durchschnittswert der Ruckspannungen in der Matrix auf Null zu reduzieren. Der zu E, gehijrige Wert r,, fur die “permanente Entfestigung” wurde vergliehen mit rZ, dem Durohschnittswert der Riickspannungen in der Matrix, der direkt aus den nach gerichteter plastischer Verformung verbleibenden gerichteten Git~rverzerrun~n abgeschatzt wurde. Die Ergebnisse unterstiitzen die Vorstellung, da6 in zweiphasigen Legierungen die und r2 Entfernung der Riiekspannungen den Hauptbeitrag zu -rsl liefert. Man kann annehmen, da9 z511 Abschatzungen fur die obere und untere Grenze der Rtiokspannungsverfestigung ermiigliohen. Auf diescr Grundlage wird fur eine reprlisentative Gruppe von Legienmgen kubischer Metalle ein Vergleich angestellt iiber den Beitrag von Riickspannungen zur Vorfestigung. * Received July 28, 1964; revised November 30, 1964. t Department of Industrial Metallurgy, University of Birmingham. ACTA M~TALLURGICA,
VOL. 13, JULY
1965
807
805
ACTA
METALLURGICA,
1. INTRODUCTION
Although knowledge of the behaviour of metals in reversed defor~tion has played an essential part in the development of dislocation theories of work hardening, its main contribution has been in showing that the dislocation arrangements must possess a certain stability in unloading and stress reversal, rather than in providing a direct insight into the role of internal stresses. Thus the evidence of back-stress hardening given by the Bausehinger effect is illdefined, and in many cases there are other possible sources of directionality in flow strength. Orowancl) has discussed this problem in terms of the effects of two generalised types of obstacle to slip, namely, strong barriers and permeable obstacles. The former, but not the latter, were expected to promote back-stress hardening due to some form of dislocation pile-up at the barriers. Orowan emphasised that backstress hardening must be wiped out to a large extent by reversed plastic deformation, and that it would therefore give rise to a prominent “permanent” softening effect.* On the other hand, an array of permeable obstacles could give rise to low initial flow strength in reversed deformation without causing appreciable permanent softening. In this Case it was envisaged that mobile dislocations would move forward through the array at the flow stress, breaking through individual obstacles under the pressure of very few piled-up dislocations. At the end of the prestraining most dislocations would have reached a point where they were held up against a particularly effective, or closely spaced row of obstacles. On stress reversal movement of the dislocations back from the obstacles would start at a relatively low applied stress, but in the absence of extensive dislocation pile-ups the flow stress would increase rapidly towards the value obtained in prestraining, as effective groups of obstacles were again encountered. Since most single phase oubic metals show a rather small permanent softening effect, Orowan concluded that work strengthening in such metals is mainly due to obstacle hardening, rather than to back-stresses associated with strong barriers. There is one group of alloys of cubic metals which certainly contain strong dislocation barriers. These are the alloys strengthened by a dispersion of strong
VOL.
13,
1985
non-coherent particles of a second phase. This investigation was made, firstly, to test the prediction implied in Orowan’s argument, that the pe~anent softening effects given by dispersion hardened alloys will be much greater than those in single phase materials and, secondly, to explore how far measurements of permanent softening are useful in making quantitative estimates of the contribution of back-stresses to work hardening. 2.
THE ON
EFFECTS OF MICROSTRUCTURE PERMANENT SOFTENING IN REVERSED STRAINING
~~easurements of the Bauschinger effect were made on each of the alloys described in Table 1, using tubular specimens tested in torsion at a strain rate of 6 x 10M5set-l. Strains were measured optically using mirrors attached at each end of a 2 in. gauge length and torques were measured with a strain-gauge load cell. The tubes were of 0.062 in. wall thickness and either 0.875 or 0.565 in. external dia., depending on the 00.0~strength. Stresses were calculated using the relationship due to Woolley.‘2) Two specimens were tested in each structural condition. Both received a prestrain of 9 T$ shear strain: this was followed by a reverse strain of 4 to 5% in the first specimen and a continued forward strain of the same amount in the second. Fig. 1. gives examples of the measurements made in completed tests. For convenience they are plotted in terms of absolute stress and absolute strain. Thus rj and rr represent the ilow stress in continued forward straining and reverse straining respectively. Orowan estimated that a reverse strain of between 1 and 3% would be sufficient to remove transient softening associated with obstacle hardening. Thus, to allow a comparison of the relative permanent sof~ning effects in all the alloys tested, in Fig. 2 the values of 7, for reverse strains up to 4% are plotted as a fraction of TV,measured at the same total strain. The results summarised in Fig. 2 fall into two groups: the first, concerned with alloys which are effectively single phase, showed small permanent softening effects, the second, concerned with the two phase alloys, showed relatively large effects. The first group contains representative alloys which, in addition to the hardening mechanisms present in pure metals, * Here permanent implies that part of the initia2 work would be strengthened by one or more of the following: hardening is permanently removed by reverse straining. The solution hardening, Cottrell locking, order hardening evidence considered is, essentially, the difference in the and (in alloy 6a), solute clusters formed in a super(absolute) values of flow strength, for continued forward and for reverse straining, measured at the same total strain and saturated solution. In addition, the low carbon steel, beyond the point at which the two flow stress curves become and to a lesser extent the Duralumin Ba, contained a approximately parallel: e.g. in Fig. l(a). the permanent softening 7f is given by the vertical separation of the two small volume fraction of widely dispersed second phase flow stress curves, measured beyond the Grst fow percent of particles (not predominantly at grain boundaries). reversed strain.
WILSON:
REVERSIBLE
WORK
HARDENING
The behaviour of this first group was not essentially different from that of pure polycrystdtlline cubic metals. The largest permanent softener effect in the group was shown by the alloy of low stacking fault energy, u-brass. In contrast, all the dispersion hardened materials showed relatively large permanent softening effects. With comparable particle sizes, permanent softening increased with the volume fraction of precipitate (alloy 3 compared with 2a, 2b snd la). With similar volume fractions it increased with the fineness of dispersion (2a compared with 2b and 6b with 60). However, platelets of 8 precipitate in A.-& (alloy fib) and cementite lamellae in pearlite were possibly more effective than dispersions of relatively equi-d~ensioned particles. In terms of the distinction made by Orowan, the dislocation barriers in the first group are expected to be relatively permeable and those in the second group relatively strong. However, we expect that the extent of piled-up groups of like dislocations at strong barriers will generally be limited by slip processes within the matrix. In most cases of practical interest the effects of differences in barrier strength and barrier spacing will be differences in degree rather than in kind. In this connection results with the age hardened slloy 6 are of p~rtieul~r interest since they can occur in either group, depending on the stage of precipitation. A more detailed study of the effects of pro-
FIG. 1. The effect of precipittltion on the Bauschinger Effect in alloy 7 (Al-4 % Cu): quenched from 525°C and aged at 190°C, for (a) 5 hr and (b) 340 hr, befare straining. (Fig. lb is also used to define T,,.)
IN
ALLOYS
OF
CUBIC
METALS
80%
2. The ratio of flow strength in reversed deformstion to that in continued forwefd deform&ion, merasured at the same total strain. The curves ttre identified with the alloy designations used in Table 1.
FIG.
gressive changes in barrier strength was made in such a system using the coarse grained, high purity Al4% Cu alloy 7. This was quenched from 525% and aged at 190°C for the series of times given in Fig. 3. Complete stress-strain relationships representing extremes of behaviour are given in Fig. 1. From such relationships the initial yield (O.l”/O proof) stress, rW, and the work hardening rate at 3% strain, (dr/&)s, were measured in prestr~~ing. After 9% prestrain and reverse straining the permanent softening, T$ = rt - T?, was measured at a reverse strain of 3% (total strain 12%). Results for the complete series are summarised in Fig. 3. The precipitation sequence in Al-t% Cu is well known and its effects on mechanical properties have recently been reviewed by Kelly and Nicholson.(3) Up to ageing times of about 10 hr the precipitate would be mainly closely spaced 0” particles which are coherent with the matrix. The increase in yield and flow strengths would be determined mainly by the increased stress necessary to force dislocations through the growing 8” precipitates. Despite the large rise in yield strength in this first stage of ageing, neither the initial work hardening rate nor the evidence of back-stress hardening provided by rQwas significantly greater than that obtained with the solid solution. In contrast, within the range of ageing times from about 10 to 100 hr, when the partially coherent 8’ precipitate would progressively replace 8”, the initial work hardening rate and r, increased continuously and in step. Ma,ximum values of (d~lds), and r5.were reached after
ACTA
810
I
,
METALLURGICA,
I
I
I
I
,
I
I
/ 1
j 1
I
I 10 hours
at
lo* 190'C.
I lo3
FIG. 3. Al-4% Cu (dloy 7) quenched from 525°C and aged at 190°C before testing in the manner illustrated in Fig. 1: showing the effect of ageing time on the initial yield strength, on work hardening at 3% strain and on permanent softening measured at 3% reverse strain efter 9 o? prestrain.
about 300 hrs. at 19O”C., a stage ofageing well beyond that giving peak yield strength. The microstructure after 300 hrs. ageing consisted mainly of 8’ platelets, distributed rather inhomogeneously but with an average diameter and spacing of about 500-600 A. The curves for alloys 6b and 6c in Fig. 2 show that 7, for more widely dispersed 0 particles, obtained in more severely overaged conditions, would be appreciably smaller than the values obtained with 8’. The results summarised in Figs. 2 and 3 are promising insofar as the relative magnitudes of the permanent softening effects are qualitatively consistent with expectations of the relative electiveness of dislocation barriers in the different alloys. However, a direct relationship between -r, and the magnitude of back stress hardening remains to be demonstrated and, if quantitative comparisons are to be made, equivalent strains with respect to reversal of the directional component of the internal stress system must be identified. 3.
EVIDENCE OF THE REVERSAL BACK-STRESSES
OF
If back-stresses are developed in prestraining they will at first assist dislocation movement in reverse straining. However, continued reverse straining will progressively annul back stresses inherited from the earlier deformation and eventually a new pattern of such stresses will be developed, of opposite sign to the original ones. The effects of such a re-organisation of the internal stress system on flow strength is, perhaps, better described by the term “reversible work hardening” than by permanent softening. In a real material reversal of the internal stress system will develop inhomogeneously,Y nevertheless there will be a narticuA.
VOL.
13,
1965
lar value of the reverse strain, E,, which reduces the average value of the long range baok-stresses in the advanced up matrix to zero. In terms of the ardent the reversible work hardening to this stage Tag, measured at strains corresponding to E, (Fig. l(b)), should provide a suitable basis for comparison of the contributions of back-stress hardening to total flow strength. However, we should note that a reduction in other sources of work hardening might contribute to TV,,,particularly if some of the obstacles generated by dislocation interactions in prestraining are at first removed by reversed deformation. It has been shown(4) that the dominant long range stress system in a deformed high oarbon steel involves back-stresses in the ferrite matrix which are balanced, for the most part, by opposing stresses in the cementite particles. Such an inter-phase stress system can be investigated using X-ray diffraction measurements of lattice strains. A selection of the main types of alloy described in Table 1 were investigated in this way. Several torsion specimens were prepared from a given material: each was prestrained 976 and then given one of a series of reverse strains in the range O-lOo/o. For examination of the residual back-stresses in the matrix phase, the strained testpieces were sectioned at 45’ to the axis in two mutually perpendicular directions, corresponding to the directions of principal stress. The sections were polished carefully to remove the disturbed layers at the surfaces to be used in the X-ray examinations. A28, the angular difference in the positions of the diffraction line peaks, obtained from the two mutually pe~endicular sections of each testpiece, was measured using a Geiger counter diffractometer. (The method has been described previously(4).) Deformation by simple shear proved to be advantageous because equivalent reflections from the two sections of a testpiece were of the same line profile, allowing small differences in A26 to be measured by simple superpositioning despite severe line broadening. Measurements were made on the steels using the 310 and, in some cases, the 211 and 220 ferrite reflections with Co and Fe radiations. For the ~l~inium alloys the 420 and 511,333 reflections were used with Co and Cu radiations. Fig. 4 shows a complete series of measurements made on the high carbon steel. In this case E, was close to 3% reverse strain. Similar measurements on the low carbon steel la are also given in Fig. 4 although, in the ease of single phase materials, inte~retation of the apparent X-ray evidence of back-stresses is speculative.t4) Table 2 summarises the results of stress measurements made on all the alloys in which E,,~ was estimated in this manner. Apart from the reversible
WILSON:
REVERSIBLE
WORK
HARDENING
IN
ALLOYS
OF
CUBIC
811
METALS
TABLE 1
= Approximate microstructural parameters
i Chemical composition (wt. %)
Alloy
._ la (low carbon steel)
0.11 C; 0.002 N; 0.47 Mn; 0.08 Si; 0.03 P; 0.03 S
Volume fraction of precipitate (f)
Pre-treatments
Recrystallised at 650°C
Meen planar diameter of Imatrix grains (microns)
Mean planar Mean planar diameter of spacing of precipitate particles (microns) (microns) _-
0.016
10
19
0.4 .-
lb (decarburised 0.006 C; 0.001 N; steel other elements as in la ~.
Recrystallised at 630°C
2a
Quenched 82O”C, spheroidised 700°C for 10 hr
2b
0.74 C; 0.29 Mn; 0.19 Si; 0.015 P; 0.01 S
3
(1.14% c&rbon steel).
1.14 C; 0.32 Mn; 0.01 Si; 0.03 P; 0.01 s
Quenched from 9OO”C,tempered 650°C for 2 hr
4
(70/30 Brass)
69.5 Cu; 0.02 Fe; Pb, Sn and Ni < 0.01; Zn balance
Recrystallised at 55OOC
5
(Al/3 Mg)
3.2 Mg in high purity Aluminium
Recrystallised at 350°C
4.05 Cu; 0.72 Mn; 0.69 Mg 0.74 Si; 0.31 Fe
Durelumin
0.5
N2?
0.11
0.7
0.25
Nl?
Lamellar spacing 0.17
0.11
0.17
0.3
0.5
colony size 10 -4
to l?
-. 29
_. 93
_,
Quenched 52O”C, aged 190°C for 150 hr Quenched 52O”C, aged 350°C for 3 hr
6c 7
1.4
Quenched 52O”C, aged 20°C for 90 min
6a
6b
0.11
_
Quenched 82O”C, tempered 650°C for 2 hr Rapidly air cooled from 820°C
2c
33 _
_ (0.74 % carbon steel.)
_
(Al/4 Cu)
4.1 Cu in high purity Aluminium
5
NO.05
~0.06
NO.055
1.0
NO.05
I
0.25
-
Quenched 525°C and aged for times given in Fig.3
(Al/3 Mg)
1.6
2.0
1.2
6.1
-
13.2
0.15
0.09
la (low carbon steel)
1.65
3.6
2.4
15.6
6.0
25.2
0.14
0.09
6b (Duralumin)
3.3
12.4
6.3
20.5
-
28.5
0.44
0.22
2a (0.74 C steel, annealed 700°C)
2.9
13.4
7.5
30.5
-
46.0
0.29
0.16
2b (0.74 C steel, tempered 650°C)
3.1
19.6
12.7
42.5
31.5
56.0
0.35
0.23
3
3.1
26.7
13.4
43.5
33.6
64.1
0.42
0.21
4.7
31.2
16.1
36.8
32.4
67.2
0.46
0.24
(1.1 C steel, tempered 650°C)
2c (0.74 C steel, pearlite) Stresses are in 1000 p.s.i.
80
ACTA
812
METALLURGICA,
0
0 .4
0 0)
[
on
i
-0 .4.-
-0 .a --
_I -1 .2=
I
I
I
2
4
I
,
REVERSE
1 lo-
Em. 4. Effect of reverse straining on directional residual lattice strains in the matrix of steels previously strained 9%. A20 is the angular difference in the positions of 310 reflections (using Co radiation) from two mutually perpendicular sections cut in directions of principal stress. Curve 1 refers to the high carbon steel 3 and curve 2 to low carbon steel la.
work hardening rSa, the initial macroscopic yield (or 0.1% proof) stress ry, and the flow strength at 9% prestrain 7-f(9),are given. Estimates of work hardening in unidirectional straining from 75 - 7y are misleading because 7y will generally be affected by transient strengthening effects associated, for example, with initial dislocation locking. For this reason in some cases an estimate was made of T,,, the “friction” stress at zero plastic strain. This was obtained in the usual way, by extrapolation of the later part of the stressstrain curve back to zero plastic strain using a log-log plot. 4.
THE RELATION OF REVERSIBLE HARDENING TO BACK-STRESSES
VOL.
13,
1965
where E, is half the observed total difference in mean strain, calculated from the relative positions of the diffraction line peaks given by the two sections in the directions of principal stress: E is Young’s modulus for the particular crystallographic direction used in the measurements and Y is an average value of Poisson’s ratio. Although the results in Fig. 5 show that T, was close to being proportional to T,,, the ratio of their magnitudes was only about 0.53. 7a must underestimate the back stresses present during plastic deformation, partly because the stresses will relax to some extent on unloading and a further relaxation is expected in the thin layer close to a free surface which must be used in diffraction measurements. To minimise plastic relaxation effects, the specimens used for measurements of 7, were strain aged for at least 24 hr with the flow stress still applied, before unloading and sectioning. Ageing of the aluminium alloys was at room temperature and of the steels at about 60°C. (In the latter case ageing under stress gave an increase of about 15% in the observed residual strains.) Apart from stress relaxation effects, the observed movement of the diffraction line peaks may be reduced, in relation r
!8-
!4-
!O-
6-
WORK
In two phase alloys a direct estimate of the average long range back-stresses developed in plastic straining can be made from X-ray measurements of the directional residual lattice strains in the matrix. In Fig. 5 the results of such estimates, based on X-ray measurements on specimens which had been strained unidirectionally to 9% shear strain, are compared with the values of r,, obtained with similar specimens forward strained 9% and then reversed strained to the appropriate values of E, (Table 2). TV,the estimate of the average back stresses based on the residual lattice strains, was calculated assuming
2-
8-
4-
L
Fro. 5. Relationship between the magnitude of reversible work hardening (T,,) and of back-stresses, estimated from residual lattice strains in the matrix. v alloy 5; 0 alloy la; A alloy 6b using 420 reflection: A alloy 6b using 611, 333 reflection; x alloy 2a; + alloy 2b; p alloy 3 using 220 reflection; n alloy 3 using 310 reflection; 0 alloy 2c using 220 reflection; l alloy 20 using 310 reflection; 0 alloy 2c using 211 reflection.
WILSON:
REVERSIBLE
to the magnitude
of the effective
stresses are partly balanced No allowance lating
7,.
WORK
HARDENING
back-stresses,
was made for this possibility
The consequent
error may
in calcu-
not
of the results with single phase materials
quite uncertain.
support
back-stress
AND
the results
On the question quantitative
in Fig.
5 clearly
given
between
of how far the measurements it has been
allow
noted
previous section that T~, derived from X-ray
be represented
in 7,, and, in this
case, there may be a contribution
from other sources
formation.
For the two-phase
reasonable
to conclude
0.15
fraction
volume
of carbide
FIG. 6. Relationship between reversible work hardening and the volume fraction of spheroidal cementite particles for steels lb, la, 2 and 3. Results for steels 2 and 3 apply to a mean planar particle diameter of about 0.3 p (the value for steel 2 was interpolated). The corresponding diameter for steel la was -0.4 ,u.
in reverse de-
alloys,
it is therefore
that 7,, and 7, provide
upper
proportional
to the volume
fraction
strain of E, was similar to or slightly
measurements
macroscopic
on single phase alloys is much
more
In the stronger alloys the flow strength at a reverse
uncertain
but, in any case, results for reversible work
material.
hardening
in these materials imply that back stresses
estimate
small contribution
to the flow
strength.
yield
tially consistent
alloys follow an essen-
pattern in which back-stress
ing increases with volume fraction with the fineness of its dispersion modulus
of the matrix
shape varied examined
of the precipitate, and with the elastic
(Table 2).
clearly of first importance, sons are restricted
Particle
rather
spacing is
but quantitative
in this respect widely.
the most
harden-
Among
effective
compari-
because
particle
the structures
precipitates
were the
strength
However,
of the fraction rapidly
deformation most
of the alloys
sources of strengthening location
locking).
pearlitic
steel 2c (f=
cementite
0.11).
lamellae
In both these alloys TV
and 7s12indicated that back stresses probably ted between
a quarter
precipitate
the low carbon steel, the most ineffective was
the
coarse
dispersion
with
f = 0.11,
back
stresses
of
rounded
steel 2a. In this accounted
between 0.16 and 0.29 of the total flow strength. far as they go, results for the comparable steels are reasonably
for As
spheroidised
consistent with T,, being directly
initial
be affected
yielding
in
by transient
(e.g. the effects of initial dis-
An approximate
allowance
can be
made for the effects of the second factor by extrapolating the stress-strain
relationship
observed at large Some values
plastic strains back to zero plastic strain. of T,, estimated
in this way are given in Table 2. For
the stronger steels, the mean value of the two estimates stresses represents
about
~~(s) -
3 of the apparent
TV.
An interesting feature of reverse straining in the two phase alloys is that, apart from the special case of the pearlitic steel, the strain E, required to annul the back stresses was rather insensitive
strain.
cementite particles in the spheroidised case,
contribu-
and a half of the total flow
strength after 9% unidirectional Excluding
in the
because
would
of back
the
due
at the outset of large scale plastic
total work hardening,
and
undeformed
of total work hardening
and second,
closely spaced 8’ platelets in A1-Cu (e.g. alloy 6b with 0.05)
less than the
the
it is difficult to make a realistic
f
about
of
to back stresses, first because back stresses build up extremely
Results with the two-phase
of precipitate
(Fig. 6).
and lower limit estimates of the contribution of backInterpretation of the X-ray stress hardening.
must make a relatively
I
I
I
0.10
0.05
measure-
ments of residual strains, must tend to underestimate
of work hardening which are diminished
I
in the
the back-stresses present at the end of the initial straining. On the other hand, the whole of back-stress should
15-
and reversible work hardening.
conclusions,
hardening
20-z2
CONCLUSIONS
view of the relationship
hardening
813
METALS
25-
0 0 c
Orowan’s
CUBIC
be very
pretation
DISCUSSION
OF
if the
in the 2-phase alleys(4) but it makes inter-
Qualitatively
ALLOYS
within the matrix phase.
important
5.
IN
precipitates. tributing
to the spacing
The total number
of dislocations
to the reverse strain is expected
as the barrier spacing is decreased. fore suggests the possibility macroscopic
con-
to increase
This result there-
that, so far as resultant
strain is concerned,
the effects of precipi-
tate spacing on the density of moving approximately
of the
counterbalanced
dislocations
is
by its effects on their
814
ACTA
METALLURCICA,
mean free path. Insofar as this is true E, may provide a measure of the effectiveness of individual barriers. Prom a more practical viewpoint, the results ilh~ trate the directional character of work hardening in two phase alloys, which must be particularly important in applications involving stress reversal. In general, directionality becomes more extreme as dispersion hardening becomes more effective. The results with the pea&tic structure point to the hketihood of even stronger effects of the same kind in some fibre-strengthened materials.
VOL.
13,
1965 ACKNOWLEDGMENTS
Thanks are due particularly to Professor %. C. Rollason for the provision of research facifities and to Dr. G;. R. Ogram, who designed and constructed the torsion apparatus. REFERENCES 1. E. ORow_w, Internal Strassw and Pa&we in Met&, p. 59. EIs&er ff959). 2. R. L. WOOLEY, Phil. &fag. 4.4, 597 (1953). 3, A. KELLY kind R. R. NICHQL.sON, I%?p%% ‘im X&&ls &Sence, Vol. 10,“Precipitatian Hardening”, Pergamorr Press, Oxford (1963). 4. D.V. WILSON andY,A. KoNNAN,A&~C~~~. 12,617(1964).