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The effect of pressure on the flow stress of metals
Acre metal/. Vol. 32. No. 3, pp. 457463. 1984 Printed in Great Britain. All rights reserved
Copyright
c
OOOI-6160184 $3.00 +O.OO 1984 Pergamon Press Ltd
THE EFFECT OF PRESSURE ON THE FLOW STRESS OF METALS? W. A. SPITZIG and 0. RICHMOND U.S. Steel Corporation,
Research Laboratory, (Recehed
14 September
Monroeville, PA 15146, U.S.A. 1983)
Abstract-The classical theory of plastic flow of metals assumes that any pressure dependence of yielding is associated with irreversible dilatation through the normality flow rule. Experimental results presented here on iron-based materials and on aluminum show that although the flow stress is a linear function of pressure, the plastic dilatancy is not related by the normality flow rule and is in fact negligible by comparison. The pressure dependence of the flow stresses in these materials results from the effect of pressure on dislocation motion, whereas the observed plastic volume changes are in accord with expected increases due to dislocation generation. Thus. there is no need for the pressure dependency of yielding to be associated with irreversible plastic dilatancy. R&um&La thCorie classique de l’&oulement plastique des mttaux suppose qu’une dilatation irriversible est associ&e P toute variation de la limite elastique en fonction de la pression, par la rigle de l’tcoulement normal. Les r&ultats exptrimentaux que nous prCsentons ici sur des mat&iaux i base de fer et sur de l’aluminium montrent que bien que la contrainte d’Ccoulement varie lintairement en fonction de la pression, la dilatabilitb plastique n’est pas IiCe par la rkgle de I’Ccoulement normal et qu’elle est en fait negligeable. Dans ces materiaux, la variation des contraintes d’&coulement en fonction de la pression provient de l’effet de la pression sur le mouvement des dislocations, alors que les changements du volume plastique que l’on observe sont en accord avec les augmentations que l’on attend par suite de la g&n&ration des dislocations. Ainsi. il n’est done pas nCcessaire que la dilatabiliti plastique irrtversible soit associCe B une variation de la limite elastique en fonction de la pression.
Zusammenfassung-In der klassischen Theorie des plastischen FlieDens von Metallen wird angenommen, daB jede DruckabhIngigkeit des FlieDens mit irreversibler Dilatation iiber die Nottnalitiits-FlieDregel verbunden ist. Die hier fiir Materialien auf Eisenbasis und fiir Aluminium vorgelegten experimentellen Ergebnisse zeigen, daL3 trotz der linearen Abhingigkeit der FlieDspannung vom Druck die plastische Dilatation nicht iiber die Normalitiits-FlieDregel bestimmt ist; sie ist vergleichsweise sogar vernachliissigbar. Die Druckabhgngigkeit der FlieBspannung dieser Materialien riihrt von der Druckeinwirkung auf die Versetmngsbewegung her, wobei die beobachteten plastischen Anderungen im Volumen mit dem erwarteten Anstieg durch die Versetzungserzeugung iibereinstimmen. Folglich besteht keine Ursache, die Druckabhlngigkeit des FlieBens mit irreversibler plastischer Dilatation zu verkoppeln.
1. INTRODUCTION Previous studies on the effect of superimposed static pressure on the deformation behavior based materials show that the flow stress, 6, linearly on hydrostatic pressure, p, in accord relation u = a,(1 + 35rp)
hydroof irondepends with the (1)
where u0 is the value of u at p = 0 (1 atm) and is strain dependent, whereas 2 is a pressure coefficient and is
Whe material in this paper is intended for general infonnation only. Any use of this material in relation to any specific application should be based on independent examination and verification of its unrestricted availability for such use, and a determination of suitability for the application by professionally qualified personnel. No license under any United States Steel Corporation patents or other proprietary interest is implied by the publication of this paper. Those making use of or relying upon the material assume all risks and liability arising from such use or reliance. 457
a constant for all iron-based materials [l-3]. The results of these previous studies showed that although the flow stress was a linear function of hydrostatic pressure, the plastic dilatancy was negligible and, therefore, not related to the normality flow rule [4]. The increase in flow stress with pressure appeared to result from the effect of pressure on the basic Aow events, such as dislocation motion [5], whereas the small plastic volume change was consistent with expected changes in dislocation density [6]. In the present study it is shown that aluminum satisfies equation (1) but with a different pressure coefficient than iron. In addition it is shown that the results both for the iron-based materials and for aluminum are quantitatively consistent with a recent model of the effect of hydrostatic pressure on dislocation motion through the lattice [I. This appears to firmly establish the mechanism of the observed pressure dependence of the flow stress, and is consistent with the experimental result that this pressure dependence does not require an associated plastic volume change.
458
SPITZIG and RICHMOND: 1900
EFFECT OF PRESSURE ON FLOW STRESS
r
-2OCQ-I603 -1200 -600 -400
0
400
600
1200 1600 2000
I, (MPa)
Fig. 1.
2. EXPERIMENTAL The material used in this study was 254cm-thick plate of grade 1100 commercial aluminum. Tension and compression specimens were machined from the plate and then annealed at 425°C for 1 h. The tension specimens had a gage length of 4.45 cm and a diameter of 0.95 cm, whereas the compression specimens were cylinders 1.9 cm high with a diameter of 0.95 cm. These specimens were tested in a Harwood hydrostatic pressure unit at 1 atm (0.1 MPa) and at pressures of 138, 414 and 828MPa at a nominal strain rate of 2.0 x 10-4s-‘. Teflon films were used on the ends of the compression specimens to reduce friction. The compression specimens showed no significant barreling at strains up to about 10% where the tests were terminated. Descriptions of the hydrostatic testing apparatus and of the procedures used for testing under hydrostatic pressure have been outlined previously [ 1,8]. The density changes resulting from plastic deformation were determined by using the apparatus and procedures developed and described previously by Garofalo and Wriedt [9]. These measurements were made both before and after straining specimens up to 10% at 1 atm and under a hydrostatic pressure of 828 MPa. The tensile specimens had their grip ends removed after straining, and the final density determinations were made on the uniform gage sections.
3. REVIEW OF EXPERIMENTAL RESULTS ON IRON-BASED MATERIALS Previous results on high-strength steels showed that the yield and flow stresses were adequately described by the relation Iz = c - al,
(2)
where I, = u - 3p and I2 = f a are stress invariants, Q is the yield or flow stress, p is the hydrostatic pressure, a is a pressure coefficient, and c is a strength coefficient. In equation (2) u is taken as positive in tension, and the sign for I2 is taken so as to make the
resultant value positive, that is, plus for tension, minus for compression. The coefficients a and c in equation (2) are strain dependent [l-3] as shown by the results for 4330 steel in Fig. 1, which are typical of all the steels studied. It was initially thought that only the strength coefficient, c, was strain dependent, but further analysis of the data on the high-strength steels showed that the pressure coefficient, a, also was strain dependent. Substituting for I, and I, in equation (2) results in the relation C
a=lfa
(1+3ap> C
where the plus and minus signs in front of the coefficient a are for tension and compression, respectively. The quantity c/(1 f a) is the value of CJ when p = 0 (1 atm) and is given the designation a, in equation (1). The coefficient ratio a/c is replaced by a in equation (1). The values obtained for the coefficients a and c in equation (3) for high-strength steels [l, 21, spheroidized 1045 steel [lo], and Fe [8] and Fe-Ti-Mn [l l] single crystals are summarized in Table 1. These data for the iron-based materials are plotted in Fig. 2, and it appears that the ratio a/c or a is a constant and therefore, like the elastic constants, a property of the bulk iron lattice. From equation (3) it is possible to calculate the strength differential (S - D) which is defined as the ratio of the difference between the absolute values of the yield stress in tension and in compression to the average value [l]. The magnitude of the strength differential is thus given by S - D = 2a.
(4)
Table 1 shows the good agreement between the S - D calculated from equation (4) and the experimentally observed S - D [l, 21. Equation (4) shows that the S - D is independenr of hydrostatic pressure in accord with the prior experimental observations [ 1,2]. Using equation (2) it is also possible to calculate the permanent volume change predicted by the nor-
SPITZIG
and RICHMOND:
EFFECT
OF PRESSURE
ON
FLOW
459
STRESS
Table I. Summary of experimental results and comparison of the experimental and predicted values for the strength differential (S - D) and volume expansion (y) during yielding S-P
d =0/c Maferial
(TPa-‘)
Fe single crystal Spheroidized 1045 steel Fe-Ti-Mn single crystal HY 80 steel Maraging steel (unaged) 4310 steel 4330 steel Maraging steel (aged) 1100aluminum
OOOO8 0.009 0.013 0.008 0.017 0.025 0.029 0.037 0.0014
35 470 570 606 1005 1066 1480 1833 2s
23 19 23 13 17 23 20 20 56
.,Ic
(“J
Observed
Predicted
1.5 3.5 5.5 6.0 7.0 0
0.2 I .8 2.6 1.6 3.4 5.0 5.8 7.4 0.3
Observed 0.002 0.001 0.004 OOO4 0.007 oOOO5
Predicted 0.002 0.02 0.04 0.02 0.05 0.07 0.07 0.1 I 0.004
‘Coefficients in equation 1: = c -al,. ‘Predicted from equation I: = c - al,. ‘Predicted by normality flow rule.
mality flow rule [4]. The ratio of the permanent volume expansion to the axial strain, 7 is given by [l] g=&*30 -
(forael)
where the plus sign applies to tension and the minus sign to compression. The values of y predicted from equation (5) are compared with the values measured in high-strength steels in Table 1. The previously reported experimental result that the permanent volume expansion is similar in tension and compression is in accord with equation (5) as is the observed approximate linearity between plastic volume expansion and strain [l, 21. However, the observed values of y are at least an order of magnitude less than the predicted values from equation (5) showing that the normality flow rule is not strictly satisfied in highstrength steels. 4. EXPERIMENTAL RESULTS ALUMINUM
ON
Results of the tension and compression tests at 1 atm (0.1 MPa) and at superimposed pressures of 138,414 and 828 MPa are shown in Fig. 3. There was no apparent difference between the absolute values of the tension and compression yield or flow stresses at all the pressures which indicated a zero S - D. The compression data at 138 MPa were not included in Fig. 3 for clarity of presentation. The effect of hydrostatic pressure on the deformation characteristics of aluminum in tension and compression was to increase the flow stress and to slightly increase the initial work-hardening behavior. A tension and a compression specimen were pressurized at 828 MPa for 30 min and then tested at 1 atm. These specimens showed identical behavior to that obtained in specimens that did not receive the pressurization treatment, showing that pressurization itself did not change the specimen structure so that any pressure dependence of the deformation is a real effect. The effect of hydrostatic pressure on the flow stress of aluminum is shown in Fig. 4 in terms of the stress invariants I, and 12. These data show that the yield stress of aluminum (0.2% offset) like that of iron-
based materials obeys equation (2). The pressure coefficient, a, and the strength coefficient, c, for the yield stress of aluminum had values of 0.0014 and 25 MPa, respectively. These values are included in Table 1 and show that a or the ratio a/c for aluminum is about three times that for iron-based materials. Therefore, from equation (l), the effect of hydrostatic pressure on the increase in flow stress is about three times greater for aluminum than for iron-based materials. Using the value obtained for the pressure coefficient, a, for aluminum in equation (4) shows an expected S -D of only about 0.3% (Table 1). An S -D this small could not be detected and is in accord with the observed apparent absence of an S - D in aluminum. In Fig. 4 the curves for the stress invariants I, and I, are also shown at various offset strains. These data show quite clearly that both the pressure coefficient, a, and the strength coefficient, c, were strain dependent, in accord with the more limited data on the high-strength steels (Fig. l), which was a consequence of the occurrence of appreciable barreling in the compression at about 4% strain. The results in Fig. 4 also confirm that equation (2) is valid for describing the flow stress of aluminum.
005
o Fe single crystal 0 SpheroldlZed 1045 l
004
Fe-Ti-Mn
stngle crystal
t 0 HY 80
I 400
600
1200
c ( MPa ) Fig. 2.
1600
I 2000
SPITZIG and RICHMOND:
460
EFFECT
OF PRESSURE
ON FLOW
STRESS
r
120
‘0.l -
MPO
Tension Compression
@ 20 c
i
f
0
0.02
1
I
1
0.04
006
0.06
True
I
I
0.10
0.12
1
I
014
0,16
1
0.16
I 0.20
strain
Fig. 3.
The values for the pressure coefficient, a, and the strength coefficient, e, are shown in Fig. 5 for strains between 0.002 and 0.2. Also shown for comparison are the a and c values for 4330 steel from Fig. 1I The lines through the data points were obtained using a linear regression analysis. The values obtained for the slopes, a/c or Q, of the lines are in good agreement with those listed in Table 1 for the values at the yield strength. The linear fit of the data for both aluminum and 4330 steel shows that the stope, c(,in equation (1) is independent of strain. The permanent volume increase after plastic deformation of aluminum was measured on tension and compression specimens deformed up to about 10% at I atm and at a pressure of 828 MPa. In all cases the permanent volume increase was about O.OOSo/giving a value for the ratio of the permanent volume expansion to the axial strain, y, of about 0.0005. The value
obtained for the permanent volume expansion of aluminum is very small and could actually be less because it represents the rn~~jrnurn vahre capable of being obtained from the density determination due to the accuracy of the scale readings. Table 1 shows that the observed value of y is about an order of magnitude less than the predicted vafue from the normality flow rule [equation (S)] using the pressure coefficient, a, for aluminum. Therefore, the discrepancy between the observed and predicted permanent volume expansion resulting from deformation of aluminum is similar to what is observed for ironbased materials.
For the iron-based observed permanent @astie deformation, sonable estimates of
materials it was shown that the volume increases, resulting from couid be accounted for by reathe increased dislocation density
120 r
Camprassiotl
l
c f MPO 114?t30
I
6”
--..__ b
ltOO aluminum
3
9330
’
1
I
3teei
0.02
/
-2400
I
I
0
-a
201
3teet
1
I
I
-800
0
400
I - 1600 I,
f MPo
Fig.
4.
f
m96.5 TPa*’
006
SPITZIG and RICHMOND:
EFFECT
OF PRESSURE
ON FLOW
STRESS
461
Table 2. Shear modulus, pressure depcndcnc! of shear modulus, and commessibilitv of iron and ~lummum Material
G,, (MPa)
dG/dp
y0 (MPa-‘)
Fe Al
7.70 x 10’ 2.50 x l@
1.9 2.2
%02 x 10-b 1.37 x lo-’
where 7. and Go are the shear stress and shear modulus at 1 atm. Jung’s model (71 is based on thermodynamics and considers the additional work required in the presence of pressure to induce the generation and motion of dislocations because of the associated change in the crystal volume. The equation obtained for the effect of pressure on the shear stress required to move a screw dislocation has the form
resulting from the deformation [l, 2,6]. A similar conclusion can be reached for aluminum because the relative change in density after a strain of 10% is about O.OOS”~and this corresponds to a dislocation density of about 3 x 10’0cm-2 [6]. This value is in good agreement with estimated dislocation densities in annealed and plastically deformed aluminum [ 121. Therefore, the measured values of the plastic volume increases resulting from plastic deformation in both iron-based materials and in aluminum correspond with those expected from increases in dislocation density. The results on the iron-based materials and on aluminum demonstrate that the pressure dependence of the flow stresses in these materials does not arise from an associated plastic dilatancy as required by the normality flow rule [4]. Rather it appears to be primarily an effect of hydrostatic pressure on the basic flow events, for example, dislocation glide [8, 131. Several dislocation models have been presented to account for the effect of hydrostatic pressure on dislocation motion [5,7, 14). The model of Shmatov [5] incorporates the change in the volume of a specimen as a result of an increase in dislocation density to obtain expressions for the effect of pressure on the forces experienced by Peirls-Nabarro dislocations. The resulting equation for the shear stress, r, required to move a screw dislocation is given by
‘5 =ro(l -YOP)
(1+$z)
where y. is the compressibility at 1 atm. Using the known values given in Table 2 for Go, y. [16] and dG/dp [17], ratios of the flow stress at pressure, up, to the flow stress at 1 atm, co, were calculated from equations (6), (7) and (8) at various pressures for iron-based materials and for aluminum. These results are tabulated in Table 3. Also shown in Table 3 are the experimental values which were obtained from equation (1) using the average a value of 19.2 TPaa’ for the iron-based materials (Fig. 2) and the a value of 56.0 TPa-’ obtained for aluminum (Table 1). Comparison of the data in Table 3 shows that both the Shmatov [5] and the Jung [7] models give good agreement with the experimental observations with Jung’s model showing the best agreement, especially at the higher pressures. The Ashby and Verrall model [14] predicts about one-half of the observed values. The good agreement between the Jung model and the experimental observations is shown more clearly in Fig. 6. The differences between the three models [5,7, 141 on the effect of hydrostatic pressure on dislocation motion can be seen more clearly if they are expressed in a similar form. In Shmatov’s model, equation (6), the ratio of the flow stress at pressure, crp, to the flow stress at 1 atm, uo, can be written as
r=2G
l+i exp(-2&t) ( > where G is the shear modulus and p the pressure. A similar expression is obtained for the stress to move an edge dislocation. The model of Ashby and Verrall [14] is based on thermally activated flow [15], and incorporates the effect of pressure on the activation energy for plastic deformation. The expression obtained for the effect of pressure on the shear stress required to move a screw dislocation is given as
“=$t 00
( > I,$_
0
P
where Gp and Go are the shear moduli at pressure p and 1 atm, respectively. Rewriting G,, as
Table 3. Increases in yield stress at pressure (a,,) as compared to atmospheric pressure (CT,,)predicted by different models of the effect of hydrostatic pressure on dislocation motion
Pressure
Shmatov [5]
Jung 171
Ashby and V&l [14]
Experimental
WPa)
Fe
Al
Fe
Al
Fe
Al
Ft
Al
138 276 414 552 828 1104
1.006 1.010
1.018
1.007 1.014
1.024
1.003 1II07
1.012
1.008 1.016
1.023
1.025 1.020 1.032 1.042
1.105
1.073 1.027 1.041 1.054
(8)
1.146
1.036 1.014 1.020 1.027
1.073
1.070 1.032 1.048 1.064
1.139
462
SPITZIG and RICHMOND:
EFFECT OF PRESSURE ON FLOW STRESS
I.16 -
I 14 -
_
0
1100 aluminum
.
Iron-based
0
materials
112-
the hypothesis that the pressure dependence of the flow stress in metals arises primarily from the effect of hydrostatic pressure on dislocation motion. 6. CONCLUDING
DISCUSSION
P t G r 0. \
I IO
-
I.06
-
1.06
-
1.04
-
I02
-
b” b”
. 0
1.02
I 04
op
106
I
I
I
J
I 06
I IO
1.12
I.14
(observed
/u,
1
Fig. 6.
It has been shown in both iron-based materials and in aluminum that the flow stress is linearly dependent on the hydrostatic pressure in accordance with equation (l), where r is a material constant representative of the elastic constants of the lattice structure. Comparison of dislocation models of the effect of hydrostatic pressure on dislocation motion showed that ti was inversely proportional to the shear modulus and directly proportional to the pressure dependence of the shear modulus, equation (12). The linearity of the pressure dependence of the flow stress deserves further mention. If it is assumed to hold even out to very large values of hydrostatic tension [ 181then equation (2) can be used to estimate the ultimate tensile strength. This occurs when I2 is zero so that the ultimate tensile strength, cr,, is given by [31 1
G,= G,,+
(13)
al=%, and taking Ap =p since p,, = 0.1 MPa _ 0 allows equation (9) to be expressed as
;=(l+E.)(l +gf).
(10)
GO
The first term in equation (10) accounts for the greater predicted pressure dependency for Shmatov’s model as compared to that predicted by equation (7) from the model of Ashby and Verrall (Table 3). The quantity p/G, in the first term gives rise to a 50% increase in the predicted pressure dependency of the flow stress as compared to the second term alone, which is the only term in equation (7). In Jung’s model, equation (8), the quantity y,,p in the first term is negligible for the iron-based materials and aluminum, and the larger predicted pressure dependency results from the factor 2 in the second term of equation (8) as compared to equations (7) and (10). Neglecting the yap term in equation (8) results in the relation Q
=4(1
+$O)
Using the result from Jung’s model [7] on the effect of hydrostatic pressure on dislocation motion to express a in terms of the shear modulus results in
(11)
which is equation (1) with
c’= 2(dG/dp)’ For the iron-based materials and aluminum dG/dp w 2 (Table 2) and G 1~E/2.6. Therefore, equation (14) becomes E 0, =
E
which is in good agreement with classical estimates of the ultimate tensile strength of metals [19]. The results on the iron-based materials and on aluminum demonstrate that the pressure dependence of the flow stress in these materials does not arise from an associated plastic dilatancy as required by the normality flow rule [4]. Rather, it is primarily an effect of pressure on dislocation motion. The plastic volume change, which is in accord with expected increases in dislocation density, is negligible by comparison with that predicted by the normality flow rule.
(12) Acknowledgemenrs-The authors gratefully acknowledge the assistance of R. J. Sober in the experimental work. Using the values for Go and dG/dp from Table 2 gives values of OLof 16.5 and 58.7TPa-’ for the iron-based materials and aluminum, respectively. These values are in good agreement with the experimental observations (Table 1 and Figs 2 and 5). This good agreement between Jung’s model [7] and the experimental observations gives strong support to
REFERENCES 1. W. A. Spitzig, R. J. Sober and 0. Richmond, Acfa merall. 23, 895 (1975). 2. W. A. Spitzig, R. J. Sober and 0. Richmond. Melall. Trans. 7A, 1703 (1976).
SPITZIG and RICHMOND:
EFFECT OF PRESSURE
3. 0. Richmond and W. A. Spitzig. Proc. 15th Int. Conf. Theoretical and Applied Mechanics (edited by F. P. J. Rimrott and B. Tabarrok), p. 377. North-Holland, New York (1980). 4. R. von Mises, Z. Anger. Math. Mech. 8, 161 (1928). 5. V. T. Shmatov, Phys. Met. Metall. 35, 47 (1973). 6. H. Stehle and A. Seeger, Z. Phys. 146, 217 (1956). 7. J. Jung, Phil. Mag. 43, 1057 (1981). 8. W. A. Spitzig, Acta metall. 21, 523 (1979). 9. F. A. Garofalo and H. A. Wriedt, Acra metall. 10, 1007 (1962). 10. A. Brownrigg, W. A. Spitzig, 0. Richmond. D. Teirlinck and J. D. Emburv. Acta metall. 31. 1141 (1983). 11. W. A. Spitzig, Acta metall. 29, 1359 (1981).
ON FLOW STRESS
463
12. M. J. Hordon and B. L. Averbach. Acra metall. 9, 237 (1961). 13. S. Yoshida and A. Oguchi, Trans. Japan. Inst. Metals 11, 424 (1970). 14. M. F. Ashby and R. A. Verrall, Phil. Trans. R. Sot. A288, 59 (1971). 15. U. F. Kocks, A. S. Argon and M. F. Ashby. Prog. Mater. Sci. 19, 1 (1976). 16. C. Kittel, Introduction to Solid State Physics, pp. 99 and 538. Wiley, New York (1961). 17. F. Birch, Handbook of Physical Constants (edited by S. P. Clark Jr), p. 97. Geol. Sot. Am. New York (1966). 18. D. C. Drucker. Metall. Trans. 4, 667 (1973). 19. A. H. Cottrell, The Mechanical Properties o/‘Matter, p. 234. Wiley, New York (1964).
Copyright
c
OOOI-6160184 $3.00 +O.OO 1984 Pergamon Press Ltd
THE EFFECT OF PRESSURE ON THE FLOW STRESS OF METALS? W. A. SPITZIG and 0. RICHMOND U.S. Steel Corporation,
Research Laboratory, (Recehed
14 September
Monroeville, PA 15146, U.S.A. 1983)
Abstract-The classical theory of plastic flow of metals assumes that any pressure dependence of yielding is associated with irreversible dilatation through the normality flow rule. Experimental results presented here on iron-based materials and on aluminum show that although the flow stress is a linear function of pressure, the plastic dilatancy is not related by the normality flow rule and is in fact negligible by comparison. The pressure dependence of the flow stresses in these materials results from the effect of pressure on dislocation motion, whereas the observed plastic volume changes are in accord with expected increases due to dislocation generation. Thus. there is no need for the pressure dependency of yielding to be associated with irreversible plastic dilatancy. R&um&La thCorie classique de l’&oulement plastique des mttaux suppose qu’une dilatation irriversible est associ&e P toute variation de la limite elastique en fonction de la pression, par la rigle de l’tcoulement normal. Les r&ultats exptrimentaux que nous prCsentons ici sur des mat&iaux i base de fer et sur de l’aluminium montrent que bien que la contrainte d’Ccoulement varie lintairement en fonction de la pression, la dilatabilitb plastique n’est pas IiCe par la rkgle de I’Ccoulement normal et qu’elle est en fait negligeable. Dans ces materiaux, la variation des contraintes d’&coulement en fonction de la pression provient de l’effet de la pression sur le mouvement des dislocations, alors que les changements du volume plastique que l’on observe sont en accord avec les augmentations que l’on attend par suite de la g&n&ration des dislocations. Ainsi. il n’est done pas nCcessaire que la dilatabiliti plastique irrtversible soit associCe B une variation de la limite elastique en fonction de la pression.
Zusammenfassung-In der klassischen Theorie des plastischen FlieDens von Metallen wird angenommen, daB jede DruckabhIngigkeit des FlieDens mit irreversibler Dilatation iiber die Nottnalitiits-FlieDregel verbunden ist. Die hier fiir Materialien auf Eisenbasis und fiir Aluminium vorgelegten experimentellen Ergebnisse zeigen, daL3 trotz der linearen Abhingigkeit der FlieDspannung vom Druck die plastische Dilatation nicht iiber die Normalitiits-FlieDregel bestimmt ist; sie ist vergleichsweise sogar vernachliissigbar. Die Druckabhgngigkeit der FlieBspannung dieser Materialien riihrt von der Druckeinwirkung auf die Versetmngsbewegung her, wobei die beobachteten plastischen Anderungen im Volumen mit dem erwarteten Anstieg durch die Versetzungserzeugung iibereinstimmen. Folglich besteht keine Ursache, die Druckabhlngigkeit des FlieBens mit irreversibler plastischer Dilatation zu verkoppeln.
1. INTRODUCTION Previous studies on the effect of superimposed static pressure on the deformation behavior based materials show that the flow stress, 6, linearly on hydrostatic pressure, p, in accord relation u = a,(1 + 35rp)
hydroof irondepends with the (1)
where u0 is the value of u at p = 0 (1 atm) and is strain dependent, whereas 2 is a pressure coefficient and is
Whe material in this paper is intended for general infonnation only. Any use of this material in relation to any specific application should be based on independent examination and verification of its unrestricted availability for such use, and a determination of suitability for the application by professionally qualified personnel. No license under any United States Steel Corporation patents or other proprietary interest is implied by the publication of this paper. Those making use of or relying upon the material assume all risks and liability arising from such use or reliance. 457
a constant for all iron-based materials [l-3]. The results of these previous studies showed that although the flow stress was a linear function of hydrostatic pressure, the plastic dilatancy was negligible and, therefore, not related to the normality flow rule [4]. The increase in flow stress with pressure appeared to result from the effect of pressure on the basic Aow events, such as dislocation motion [5], whereas the small plastic volume change was consistent with expected changes in dislocation density [6]. In the present study it is shown that aluminum satisfies equation (1) but with a different pressure coefficient than iron. In addition it is shown that the results both for the iron-based materials and for aluminum are quantitatively consistent with a recent model of the effect of hydrostatic pressure on dislocation motion through the lattice [I. This appears to firmly establish the mechanism of the observed pressure dependence of the flow stress, and is consistent with the experimental result that this pressure dependence does not require an associated plastic volume change.
458
SPITZIG and RICHMOND: 1900
EFFECT OF PRESSURE ON FLOW STRESS
r
-2OCQ-I603 -1200 -600 -400
0
400
600
1200 1600 2000
I, (MPa)
Fig. 1.
2. EXPERIMENTAL The material used in this study was 254cm-thick plate of grade 1100 commercial aluminum. Tension and compression specimens were machined from the plate and then annealed at 425°C for 1 h. The tension specimens had a gage length of 4.45 cm and a diameter of 0.95 cm, whereas the compression specimens were cylinders 1.9 cm high with a diameter of 0.95 cm. These specimens were tested in a Harwood hydrostatic pressure unit at 1 atm (0.1 MPa) and at pressures of 138, 414 and 828MPa at a nominal strain rate of 2.0 x 10-4s-‘. Teflon films were used on the ends of the compression specimens to reduce friction. The compression specimens showed no significant barreling at strains up to about 10% where the tests were terminated. Descriptions of the hydrostatic testing apparatus and of the procedures used for testing under hydrostatic pressure have been outlined previously [ 1,8]. The density changes resulting from plastic deformation were determined by using the apparatus and procedures developed and described previously by Garofalo and Wriedt [9]. These measurements were made both before and after straining specimens up to 10% at 1 atm and under a hydrostatic pressure of 828 MPa. The tensile specimens had their grip ends removed after straining, and the final density determinations were made on the uniform gage sections.
3. REVIEW OF EXPERIMENTAL RESULTS ON IRON-BASED MATERIALS Previous results on high-strength steels showed that the yield and flow stresses were adequately described by the relation Iz = c - al,
(2)
where I, = u - 3p and I2 = f a are stress invariants, Q is the yield or flow stress, p is the hydrostatic pressure, a is a pressure coefficient, and c is a strength coefficient. In equation (2) u is taken as positive in tension, and the sign for I2 is taken so as to make the
resultant value positive, that is, plus for tension, minus for compression. The coefficients a and c in equation (2) are strain dependent [l-3] as shown by the results for 4330 steel in Fig. 1, which are typical of all the steels studied. It was initially thought that only the strength coefficient, c, was strain dependent, but further analysis of the data on the high-strength steels showed that the pressure coefficient, a, also was strain dependent. Substituting for I, and I, in equation (2) results in the relation C
a=lfa
(1+3ap> C
where the plus and minus signs in front of the coefficient a are for tension and compression, respectively. The quantity c/(1 f a) is the value of CJ when p = 0 (1 atm) and is given the designation a, in equation (1). The coefficient ratio a/c is replaced by a in equation (1). The values obtained for the coefficients a and c in equation (3) for high-strength steels [l, 21, spheroidized 1045 steel [lo], and Fe [8] and Fe-Ti-Mn [l l] single crystals are summarized in Table 1. These data for the iron-based materials are plotted in Fig. 2, and it appears that the ratio a/c or a is a constant and therefore, like the elastic constants, a property of the bulk iron lattice. From equation (3) it is possible to calculate the strength differential (S - D) which is defined as the ratio of the difference between the absolute values of the yield stress in tension and in compression to the average value [l]. The magnitude of the strength differential is thus given by S - D = 2a.
(4)
Table 1 shows the good agreement between the S - D calculated from equation (4) and the experimentally observed S - D [l, 21. Equation (4) shows that the S - D is independenr of hydrostatic pressure in accord with the prior experimental observations [ 1,2]. Using equation (2) it is also possible to calculate the permanent volume change predicted by the nor-
SPITZIG
and RICHMOND:
EFFECT
OF PRESSURE
ON
FLOW
459
STRESS
Table I. Summary of experimental results and comparison of the experimental and predicted values for the strength differential (S - D) and volume expansion (y) during yielding S-P
d =0/c Maferial
(TPa-‘)
Fe single crystal Spheroidized 1045 steel Fe-Ti-Mn single crystal HY 80 steel Maraging steel (unaged) 4310 steel 4330 steel Maraging steel (aged) 1100aluminum
OOOO8 0.009 0.013 0.008 0.017 0.025 0.029 0.037 0.0014
35 470 570 606 1005 1066 1480 1833 2s
23 19 23 13 17 23 20 20 56
.,Ic
(“J
Observed
Predicted
1.5 3.5 5.5 6.0 7.0 0
0.2 I .8 2.6 1.6 3.4 5.0 5.8 7.4 0.3
Observed 0.002 0.001 0.004 OOO4 0.007 oOOO5
Predicted 0.002 0.02 0.04 0.02 0.05 0.07 0.07 0.1 I 0.004
‘Coefficients in equation 1: = c -al,. ‘Predicted from equation I: = c - al,. ‘Predicted by normality flow rule.
mality flow rule [4]. The ratio of the permanent volume expansion to the axial strain, 7 is given by [l] g=&*30 -
(forael)
where the plus sign applies to tension and the minus sign to compression. The values of y predicted from equation (5) are compared with the values measured in high-strength steels in Table 1. The previously reported experimental result that the permanent volume expansion is similar in tension and compression is in accord with equation (5) as is the observed approximate linearity between plastic volume expansion and strain [l, 21. However, the observed values of y are at least an order of magnitude less than the predicted values from equation (5) showing that the normality flow rule is not strictly satisfied in highstrength steels. 4. EXPERIMENTAL RESULTS ALUMINUM
ON
Results of the tension and compression tests at 1 atm (0.1 MPa) and at superimposed pressures of 138,414 and 828 MPa are shown in Fig. 3. There was no apparent difference between the absolute values of the tension and compression yield or flow stresses at all the pressures which indicated a zero S - D. The compression data at 138 MPa were not included in Fig. 3 for clarity of presentation. The effect of hydrostatic pressure on the deformation characteristics of aluminum in tension and compression was to increase the flow stress and to slightly increase the initial work-hardening behavior. A tension and a compression specimen were pressurized at 828 MPa for 30 min and then tested at 1 atm. These specimens showed identical behavior to that obtained in specimens that did not receive the pressurization treatment, showing that pressurization itself did not change the specimen structure so that any pressure dependence of the deformation is a real effect. The effect of hydrostatic pressure on the flow stress of aluminum is shown in Fig. 4 in terms of the stress invariants I, and 12. These data show that the yield stress of aluminum (0.2% offset) like that of iron-
based materials obeys equation (2). The pressure coefficient, a, and the strength coefficient, c, for the yield stress of aluminum had values of 0.0014 and 25 MPa, respectively. These values are included in Table 1 and show that a or the ratio a/c for aluminum is about three times that for iron-based materials. Therefore, from equation (l), the effect of hydrostatic pressure on the increase in flow stress is about three times greater for aluminum than for iron-based materials. Using the value obtained for the pressure coefficient, a, for aluminum in equation (4) shows an expected S -D of only about 0.3% (Table 1). An S -D this small could not be detected and is in accord with the observed apparent absence of an S - D in aluminum. In Fig. 4 the curves for the stress invariants I, and I, are also shown at various offset strains. These data show quite clearly that both the pressure coefficient, a, and the strength coefficient, c, were strain dependent, in accord with the more limited data on the high-strength steels (Fig. l), which was a consequence of the occurrence of appreciable barreling in the compression at about 4% strain. The results in Fig. 4 also confirm that equation (2) is valid for describing the flow stress of aluminum.
005
o Fe single crystal 0 SpheroldlZed 1045 l
004
Fe-Ti-Mn
stngle crystal
t 0 HY 80
I 400
600
1200
c ( MPa ) Fig. 2.
1600
I 2000
SPITZIG and RICHMOND:
460
EFFECT
OF PRESSURE
ON FLOW
STRESS
r
120
‘0.l -
MPO
Tension Compression
@ 20 c
i
f
0
0.02
1
I
1
0.04
006
0.06
True
I
I
0.10
0.12
1
I
014
0,16
1
0.16
I 0.20
strain
Fig. 3.
The values for the pressure coefficient, a, and the strength coefficient, e, are shown in Fig. 5 for strains between 0.002 and 0.2. Also shown for comparison are the a and c values for 4330 steel from Fig. 1I The lines through the data points were obtained using a linear regression analysis. The values obtained for the slopes, a/c or Q, of the lines are in good agreement with those listed in Table 1 for the values at the yield strength. The linear fit of the data for both aluminum and 4330 steel shows that the stope, c(,in equation (1) is independent of strain. The permanent volume increase after plastic deformation of aluminum was measured on tension and compression specimens deformed up to about 10% at I atm and at a pressure of 828 MPa. In all cases the permanent volume increase was about O.OOSo/giving a value for the ratio of the permanent volume expansion to the axial strain, y, of about 0.0005. The value
obtained for the permanent volume expansion of aluminum is very small and could actually be less because it represents the rn~~jrnurn vahre capable of being obtained from the density determination due to the accuracy of the scale readings. Table 1 shows that the observed value of y is about an order of magnitude less than the predicted vafue from the normality flow rule [equation (S)] using the pressure coefficient, a, for aluminum. Therefore, the discrepancy between the observed and predicted permanent volume expansion resulting from deformation of aluminum is similar to what is observed for ironbased materials.
For the iron-based observed permanent @astie deformation, sonable estimates of
materials it was shown that the volume increases, resulting from couid be accounted for by reathe increased dislocation density
120 r
Camprassiotl
l
c f MPO 114?t30
I
6”
--..__ b
ltOO aluminum
3
9330
’
1
I
3teei
0.02
/
-2400
I
I
0
-a
201
3teet
1
I
I
-800
0
400
I - 1600 I,
f MPo
Fig.
4.
f
m96.5 TPa*’
006
SPITZIG and RICHMOND:
EFFECT
OF PRESSURE
ON FLOW
STRESS
461
Table 2. Shear modulus, pressure depcndcnc! of shear modulus, and commessibilitv of iron and ~lummum Material
G,, (MPa)
dG/dp
y0 (MPa-‘)
Fe Al
7.70 x 10’ 2.50 x l@
1.9 2.2
%02 x 10-b 1.37 x lo-’
where 7. and Go are the shear stress and shear modulus at 1 atm. Jung’s model (71 is based on thermodynamics and considers the additional work required in the presence of pressure to induce the generation and motion of dislocations because of the associated change in the crystal volume. The equation obtained for the effect of pressure on the shear stress required to move a screw dislocation has the form
resulting from the deformation [l, 2,6]. A similar conclusion can be reached for aluminum because the relative change in density after a strain of 10% is about O.OOS”~and this corresponds to a dislocation density of about 3 x 10’0cm-2 [6]. This value is in good agreement with estimated dislocation densities in annealed and plastically deformed aluminum [ 121. Therefore, the measured values of the plastic volume increases resulting from plastic deformation in both iron-based materials and in aluminum correspond with those expected from increases in dislocation density. The results on the iron-based materials and on aluminum demonstrate that the pressure dependence of the flow stresses in these materials does not arise from an associated plastic dilatancy as required by the normality flow rule [4]. Rather it appears to be primarily an effect of hydrostatic pressure on the basic flow events, for example, dislocation glide [8, 131. Several dislocation models have been presented to account for the effect of hydrostatic pressure on dislocation motion [5,7, 14). The model of Shmatov [5] incorporates the change in the volume of a specimen as a result of an increase in dislocation density to obtain expressions for the effect of pressure on the forces experienced by Peirls-Nabarro dislocations. The resulting equation for the shear stress, r, required to move a screw dislocation is given by
‘5 =ro(l -YOP)
(1+$z)
where y. is the compressibility at 1 atm. Using the known values given in Table 2 for Go, y. [16] and dG/dp [17], ratios of the flow stress at pressure, up, to the flow stress at 1 atm, co, were calculated from equations (6), (7) and (8) at various pressures for iron-based materials and for aluminum. These results are tabulated in Table 3. Also shown in Table 3 are the experimental values which were obtained from equation (1) using the average a value of 19.2 TPaa’ for the iron-based materials (Fig. 2) and the a value of 56.0 TPa-’ obtained for aluminum (Table 1). Comparison of the data in Table 3 shows that both the Shmatov [5] and the Jung [7] models give good agreement with the experimental observations with Jung’s model showing the best agreement, especially at the higher pressures. The Ashby and Verrall model [14] predicts about one-half of the observed values. The good agreement between the Jung model and the experimental observations is shown more clearly in Fig. 6. The differences between the three models [5,7, 141 on the effect of hydrostatic pressure on dislocation motion can be seen more clearly if they are expressed in a similar form. In Shmatov’s model, equation (6), the ratio of the flow stress at pressure, crp, to the flow stress at 1 atm, uo, can be written as
r=2G
l+i exp(-2&t) ( > where G is the shear modulus and p the pressure. A similar expression is obtained for the stress to move an edge dislocation. The model of Ashby and Verrall [14] is based on thermally activated flow [15], and incorporates the effect of pressure on the activation energy for plastic deformation. The expression obtained for the effect of pressure on the shear stress required to move a screw dislocation is given as
“=$t 00
( > I,$_
0
P
where Gp and Go are the shear moduli at pressure p and 1 atm, respectively. Rewriting G,, as
Table 3. Increases in yield stress at pressure (a,,) as compared to atmospheric pressure (CT,,)predicted by different models of the effect of hydrostatic pressure on dislocation motion
Pressure
Shmatov [5]
Jung 171
Ashby and V&l [14]
Experimental
WPa)
Fe
Al
Fe
Al
Fe
Al
Ft
Al
138 276 414 552 828 1104
1.006 1.010
1.018
1.007 1.014
1.024
1.003 1II07
1.012
1.008 1.016
1.023
1.025 1.020 1.032 1.042
1.105
1.073 1.027 1.041 1.054
(8)
1.146
1.036 1.014 1.020 1.027
1.073
1.070 1.032 1.048 1.064
1.139
462
SPITZIG and RICHMOND:
EFFECT OF PRESSURE ON FLOW STRESS
I.16 -
I 14 -
_
0
1100 aluminum
.
Iron-based
0
materials
112-
the hypothesis that the pressure dependence of the flow stress in metals arises primarily from the effect of hydrostatic pressure on dislocation motion. 6. CONCLUDING
DISCUSSION
P t G r 0. \
I IO
-
I.06
-
1.06
-
1.04
-
I02
-
b” b”
. 0
1.02
I 04
op
106
I
I
I
J
I 06
I IO
1.12
I.14
(observed
/u,
1
Fig. 6.
It has been shown in both iron-based materials and in aluminum that the flow stress is linearly dependent on the hydrostatic pressure in accordance with equation (l), where r is a material constant representative of the elastic constants of the lattice structure. Comparison of dislocation models of the effect of hydrostatic pressure on dislocation motion showed that ti was inversely proportional to the shear modulus and directly proportional to the pressure dependence of the shear modulus, equation (12). The linearity of the pressure dependence of the flow stress deserves further mention. If it is assumed to hold even out to very large values of hydrostatic tension [ 181then equation (2) can be used to estimate the ultimate tensile strength. This occurs when I2 is zero so that the ultimate tensile strength, cr,, is given by [31 1
G,= G,,+
(13)
al=%, and taking Ap =p since p,, = 0.1 MPa _ 0 allows equation (9) to be expressed as
;=(l+E.)(l +gf).
(10)
GO
The first term in equation (10) accounts for the greater predicted pressure dependency for Shmatov’s model as compared to that predicted by equation (7) from the model of Ashby and Verrall (Table 3). The quantity p/G, in the first term gives rise to a 50% increase in the predicted pressure dependency of the flow stress as compared to the second term alone, which is the only term in equation (7). In Jung’s model, equation (8), the quantity y,,p in the first term is negligible for the iron-based materials and aluminum, and the larger predicted pressure dependency results from the factor 2 in the second term of equation (8) as compared to equations (7) and (10). Neglecting the yap term in equation (8) results in the relation Q
=4(1
+$O)
Using the result from Jung’s model [7] on the effect of hydrostatic pressure on dislocation motion to express a in terms of the shear modulus results in
(11)
which is equation (1) with
c’= 2(dG/dp)’ For the iron-based materials and aluminum dG/dp w 2 (Table 2) and G 1~E/2.6. Therefore, equation (14) becomes E 0, =
E
which is in good agreement with classical estimates of the ultimate tensile strength of metals [19]. The results on the iron-based materials and on aluminum demonstrate that the pressure dependence of the flow stress in these materials does not arise from an associated plastic dilatancy as required by the normality flow rule [4]. Rather, it is primarily an effect of pressure on dislocation motion. The plastic volume change, which is in accord with expected increases in dislocation density, is negligible by comparison with that predicted by the normality flow rule.
(12) Acknowledgemenrs-The authors gratefully acknowledge the assistance of R. J. Sober in the experimental work. Using the values for Go and dG/dp from Table 2 gives values of OLof 16.5 and 58.7TPa-’ for the iron-based materials and aluminum, respectively. These values are in good agreement with the experimental observations (Table 1 and Figs 2 and 5). This good agreement between Jung’s model [7] and the experimental observations gives strong support to
REFERENCES 1. W. A. Spitzig, R. J. Sober and 0. Richmond, Acfa merall. 23, 895 (1975). 2. W. A. Spitzig, R. J. Sober and 0. Richmond. Melall. Trans. 7A, 1703 (1976).
SPITZIG and RICHMOND:
EFFECT OF PRESSURE
3. 0. Richmond and W. A. Spitzig. Proc. 15th Int. Conf. Theoretical and Applied Mechanics (edited by F. P. J. Rimrott and B. Tabarrok), p. 377. North-Holland, New York (1980). 4. R. von Mises, Z. Anger. Math. Mech. 8, 161 (1928). 5. V. T. Shmatov, Phys. Met. Metall. 35, 47 (1973). 6. H. Stehle and A. Seeger, Z. Phys. 146, 217 (1956). 7. J. Jung, Phil. Mag. 43, 1057 (1981). 8. W. A. Spitzig, Acta metall. 21, 523 (1979). 9. F. A. Garofalo and H. A. Wriedt, Acra metall. 10, 1007 (1962). 10. A. Brownrigg, W. A. Spitzig, 0. Richmond. D. Teirlinck and J. D. Emburv. Acta metall. 31. 1141 (1983). 11. W. A. Spitzig, Acta metall. 29, 1359 (1981).
ON FLOW STRESS
463
12. M. J. Hordon and B. L. Averbach. Acra metall. 9, 237 (1961). 13. S. Yoshida and A. Oguchi, Trans. Japan. Inst. Metals 11, 424 (1970). 14. M. F. Ashby and R. A. Verrall, Phil. Trans. R. Sot. A288, 59 (1971). 15. U. F. Kocks, A. S. Argon and M. F. Ashby. Prog. Mater. Sci. 19, 1 (1976). 16. C. Kittel, Introduction to Solid State Physics, pp. 99 and 538. Wiley, New York (1961). 17. F. Birch, Handbook of Physical Constants (edited by S. P. Clark Jr), p. 97. Geol. Sot. Am. New York (1966). 18. D. C. Drucker. Metall. Trans. 4, 667 (1973). 19. A. H. Cottrell, The Mechanical Properties o/‘Matter, p. 234. Wiley, New York (1964).