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The growth of artificial voids during superplastic deformation of a ZnAlCu alloy
Materials Science and Engineering, A128 (1990) 147-154
147
The Growth of Artificial Voids during Superplastic Deformation of a Zn-AI-Cu Alloy C. H. CACERES* and D. E. LESCANO Facultad de Matematica, Astronomia y Fisica, Universidad Nacional de Cordoba, 5000 Cordoba (Argentina) (Received December 7, 1989; in revised form February 8, 1990)
Abstract
A series of experiments has been performed on samples of Zn-22wt.%Al-O.5wt.%Cu alloy at 523 K, at low and intermediate strain rates, with either one or three drilled holes. Similar tests were run for polystyrene, to compare the alloy with a Newtonian viscous material. A t low strains and the higher strain rate, the alloy was less sensitive to the presence of the holes than the polymer was. The results are analysed following a modified Hart model of tensile stability, including the effect of hardening due to the grain growth. The instability points, i.e. first when the voids start to grow and then when they start to grow more rapidly in the alloy than in the polymer, are predicted accurately. The main effect of grain growth hardening is to delay the onset of accelerated void growth. From the point of view of microcavitation, the latter implies that the transition between diffusion and plasticity-controlled growth may be delayed. 1. Introduction
Superplastic materials are unique in their ability to sustain extremely large volume fractions of cavitation without fracturing. The reluctance of voids to become unstable cracks indicates that the growing cavities are very stable. This behaviour cannot be accounted for by the strain rate sensitivity of the flow stress alone, because very often samples with extremely large volume fractions of cavitation (up to 30%-35%) reach large elongations [1, 2], even when the strain rate sensitivity is very low (m ~ 0.2). The high tensile stability has been rationalized, at least to first order, by including in the models for the plastic hole *Present address: Department of Materials Science and Engineering, McMaster University, Hamilton, Ontario L8S 4L7, Canada. 0921-5093/90/$3.50
growth [3, 4] the effect of strain hardening resulting from strain-induced grain growth which usually accompanies superplastic deformation [5-7]. The effect of grain growth hardening on hole growth can be understood in the following terms: since the flow stress increases with increasing grain size, grain growth results in a sort of strain hardening [3, 4]. Because the strain tends to concentrate on cavitated regions, there the grains grow faster, increasing locally the resistance to plastic flow. Thus the strain rate is decreased and the cavitation does not develop as fast as it would otherwise. The process, however, cannot be easily modelled following, for instance, the method of Cocks and Ashby [8, 9] because, once the void grows, different strains develop in the cavitated and uncavitated regions. Thus the constitutive relationships in the two regions are different, and the strain rate in the absence of the void, needed to compute the rate of hole growth, cannot be determined [4]. A qualitative picture of the effect of grain growth can be obtained as follows [4, 8]. The volumetric growth rate l? of a structure containing a void is proportional to the difference between the strain rate e~s away from the void, and the strain rate eh in the void region:
¢=
(1)
The strain rate far from the void can be expressed as
es~= \Oo1 ~b] where o 0 and e0 are a reference stress and strain rate respectively, b is a normalizing parameter, n is the creep stress exponent, d is the grain size and p defines the grain size dependence of the flow stress. © Elsevier Sequoia/Printed in The Netherlands
148
Similarly, the strain rate in the void region is given by eh = ~o
(1 -f)-"
(3)
where d h is the grain size near the void. The term ( l - f ) -n, where f is the hole volume fraction, accounts for the larger stress in the void region due to the reduced cross-section. In general, the strain is larger in the voidcontaining region, and thus dh is larger than d. Therefore the difference in strain rates is smaller than it would be if no grain growth occurred, decreasing the volumetric growth rate in eqn. (1). Following this approach, an approximate description of the effect of grain growth hardening on void growth has been developed [4] and compared with the evolution of artificial holes in a copper alloy. Artificial holes drilled in sheet samples provide a simple means of studying the plastic growth of voids. The growth stage of the holes can be separated from both the nucleation stage and the coalescence stage. The latter is especially important because coalescence of microcavitation can occur along several hundred per cent in strain, complicating the analysis of void growth at large strains. It may be argued that artificially introduced holes are much larger than those typically encountered in actual materials. However, once the voids become much larger than the grain size, the voids grow in a continuum material and the scale becomes unimportant. In the present work the evolution of artificial holes in a Zn-A1 alloy is studied. This work has many points in common with the previous study on the CDA638 copper alloy [4]. However, the copper alloy cavitates rapidly. Thus, fracture occurs at very low strains, restricting the study to a rather small deformation range [10]. To avoid this, a damage-resistant Z n - A I alloy was chosen, in order to enable a detailed study to be made up to very large strains. The tensile and fracture behaviours of this material are known from earlier work [11-13]. In addition, a different analytical approach was attempted in the present study, in order to enable a more quantitative assessment to be made of the effect of grain growth hardening on void growth. As voids elongate with little widening during superplastic straining and necking is very diffuse, it may be safely assumed that only one significant stress exists in either the uniform section or the
local inhomogeneity. The latter is the basic assumption in Hart's [14] analysis of tensile deformation [15]. On the assumption that voids are a perturbation in the tensile sample, Hart's [14] model may be used to analyse their growth, provided that the effect of grain growth hardening be considered. Since the strain rate sensitivity of the flow stress is known and the strain-hardening coefficient can be computed from the change in grain size with strain [3, 12], the predictions of the model can be readily contrasted with the experimental data.
2. Experimental details The experiments were performed using tensile samples of Zn-22wt.%A1-0.5wt.%Cu, machined out of a rolled sheet. The samples had a gauge region 12 mm long, 4 mm wide and 2.5 mm thick. In one set of samples, three holes, each 0.45 mm in diameter, were drilled as shown in Fig. 1. In a second set of samples, only one hole was drilled in order to force the fracture to occur in a single void region. Once machined, the samples were annealed for i h at 523 K to obtain a (mean linear intercept) phase size d of 1.2/~m. Tests were run with a table model Instron machine, at 473 K, in air, using a three-zone split furnace. The temperature was kept constant to within _+2 K with two thermocouples attached to the sample extremes. Testing was carried out at strain rates of 1.4 x 10 -3 and 1.4 x 10 -5 s -1. The deforming sample was observed during the tests through a small gap left open in the split furnace. By means of a long telephoto lens fitted with a bellows for close focusing (lens-to-sample distance, 35 cm) the holes were photographed at predetermined intervals. Lateral magnification on the film was about I x. The evolution of the voids
-,,t
4
,-
12
Fig. 1. Schematic diagram of tensile specimen showing positioning of holes (dimensions in millimetres).
149
was studied on the negatives under an optical microscope or on enlarged pictures. After testing to failure, the outer profile of each sample was determined by measuring the width and thickness along the gauge length. This gives information on the strain gradient along the samples and in the holes region and enables one to compute the local reduction in cross-sectional areas by multiplying the width read on the film by the corresponding thickness obtained from the profile of the sample. A series of tests on equally drilled samples of polystyrene were also run, at a strain rate of 1.4x10 -3 s -1 and at T = 3 7 3 K , in order to compare the behaviour of the metallic material with that of a truly Newtonian viscous material. Finally, several additional tests were run in smooth samples of the alloy in order to measure the strain rate sensitivity and the strain-induced grain growth at the studied strain rates. In the latter case, once tested, sections of the samples were polished down to fine alumina and etched with Nital to reveal the phase structure. Phase sizes were determined by linear intersection. Quoted values refer to the average mean linear intersect of phase size.
voids, the strain at fracture was about 3.3 at the higher strain rate and about 3.9 at the lower one. The grain size measured along deformed samples was similar to earlier published data [12] shown in Fig. 3. The strain rate sensitivity index rn, determined by strain rate changes in smooth samples at strains of the order of 0.3, was m = 0 . 2 4 at g= 1.4 x 10 -5 s -1 and m = 0 . 5 5 at g = 1.4 x 1 0 - 3 s- 1. For the polymer, m = 1 at g = 1.4 x 10 - 3 s- 1. In Fig. 4 the change in the dimensions of the holes with far-field strain is shown. A lateral contraction of the hole was observed in the metallic samples at low strains followed by a relatively small widening. It should be noted that the voids elongate some ten times more than they widen, supporting the assumption of unidirectional hole growth. For the polymer, not included in Fig. 4, voids elongated monotonically with strain, but their width remained essentially cosntant, for either single or double voids. Figure 5 shows the strain dependence of the logarithm of fractional decrease in sample area f,
3. Results In order to analyse the experimental data, several quantities will be introduced: (i) the farfield strain, defined as em ln(Ao/A), where A 0 is the initial and A the current cross-sectional area far from the voids; (ii) the local strain, defined as el=ln(Ato/Al), where At0 and A l are the initial and cross-sectional areas in the region of the holes, both measured on the plane of maximum hole width (it should be noted that A~ embodies only the load-beating section); (iii) the initial and current hole area, defined as Ah0 = A 0 -A~0 and A h --A - A t respectively (it should now be noted that the "hole area" includes any external necking); (iv) the initial and current fractional decrease in cross-sectional area due to the presence of the voids, defined as f0 = Aho/Ao and f= Ah/A. Figure 2 shows typical sets of pictures for metallic samples tested to fracture, for single and double voids. In the case of double voids, fracture occurred first in the inner ligament, at a local strain of about 2.5 for both strain rates. The outer ligament failed later, at a local strain of about 3 at the higher strain rate and at a strain of about 4.5 for the lower strain rate. In the case of single =
Fig. 2. A series of pictures showing the deformation of samples with one or three holes (tensile axis vertical). 5
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Fig. 3. T h e increase in grain size as a function o f strain, f o r
different strain rates [12].
150
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£m Fig. 5. The relative change in the area fraction of damage as a function of far-field strain, for single voids: measured; - - - , computed from eqn. (7).
.2
;o
normalized to the initial value f0, as a function of the far-field strain for single voids in both metallic and plastic samples. Figure 6 shows the same results for double voids. Prominent features in Figs. 5 and 6 are the steady increase in the fifo ratio for the polymer and the delay in when the voids start to grow at the higher strain rate, for both single and double voids for the metallic samples. Finally, in Figs. 7 and 8 the experimentally determined local strain (measured at the maximum void width) vs. the far-field strain is shown for single and double voids respectively.
;.
.6
.8
1.4 gm Fig. 7. The strain in the voids region (local strain) as a function of far-field strain, for single voids.
1.2
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El
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° e
4. Discussion
4.1. The growth of plastic instabilities in a tensile bar Let us assume a tensile bar deforming in steady state creep, with constant strain rate sensitivity index m and no hardening (the strain-hardening coefficient y---0; see Appendix A). Hart [14]
1.0
.2
'~"
u o Zn-AI- Cu ~ Polystyrene
1.4"10-3 1.4,10-5 1.4,10"3
O
£:m Fig. 8. The strain in the voids region (local strain) as a function of far-field strain, for double voids.
151
showed that a small inhomogeneity 6A 0 in a tensile bar should grow as
+A (A/"" 6Ao
~Ao]
(4)
where 6A 0 represents an initial inhomogeneity in a sample of initial cross-sectional area A0, and 6A and A represent the same parameters after some strain. Equation (4) applies in the present case for polystyrene (~,= 0), but not for the metallic material owing .to the grain-growth-induced hardening. Nichols [15] and Duncombe [16] extended Hart's analysis to hardening materials. Both predict that, once inhomogeneities start to grow (i.e. when ~+ m < 1), their growth can be described by [15]
6A0
(5)
where I = ( 1 - ~ , - m ) / m . Keeping in mind that e = ln(A o/A ), after replacing in (5), one may easily show that 6A = 6A 0 exp(le)
(6)
The value of I controls the rate of instability growth. 4.2. The growth of voids The preceding analysis can be extended to describe the growth of artificial voids equating the voids to the inhomogeneity in the bar, i.e. 6+4o = Ah0 and 6A = A h. Using the definitions of f and f0 given in the preceding section, eqn. (6) becomes f=fo exp(ke)
growth of damage at all. This is analysed in more detail in Appendix A. The parameter I in eqns. (6) and (7) depends on the strain and may be determined from the strain dependence of the grain size [3, 12] (see Appendix A). The values for L determined in previous work [12], are shown in Fig. 9 as a function of the local strain for three strain rates. The strain e in eqn. (7) is the far-field strain, while the parameter I is a function of the strain in the hole region. Therefore, to compute f from eqn. (7) for any given far-field strain £m, the corresponding local strain el was determined from Fig. 7 or Fig. 8 and then the respective value for I was read from Fig. 9. The values for f so obtained were plotted in Figs. 5 and 6 (broken lines) as functions of em. A very good agreement between predicted and measured values exists at low strains. Agreement is best for single voids at low strains. Certainly this is not surprising because the linearized analysis of Hart assumes that 6A is very small and single holes constitute the smallest perturbation in the tensile sample. At the higher strain rate the agreement covers the important region where the metallic material is more stable than the polymer (in Fig. 9, I < 0, k < 1), for both single and double voids. Departure of experimental data from computed values increases steadily with increasing strain. At the lower strain rate the model overestimates the growth rate; the experimental data points lay below the computed curve for both single and double voids. The opposite occurs at the higher strain rate.
(7)
where k = I + 1. It should be noted that I = 0 implies that 6+4 = 6A o = constant in eqn. (6), i.e. the hole area Ah = Aho = constant. However, as the crosssection A~ decreases with increasing strain, the relative hole size Aho/A ~increases steadily. From eqn. (7), setting k -- 1, f=f0 exp e. This is the case of the Newtonian material. Negative values o f / , possible for hardening materials, decrease the rate of growth of damage and, for I~< - 1, k~< 0, which in practical terms means that f=fo--constant. In simple words, for a hardening material, k < 1 implies that the rate of damage growth is lower than for a Newtonian material while, for k > 1, the opposite is true. When k-- 0, there is no
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1~
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-22AI-05Cu = 12~m = 473K : ~ = 141~Ss ~ : t = 141~3$ " : ~ • 5 l r 2s ~
-4
Fig. 9. The instability parameter I, as a function of local strain, for three different strain rates [12].
152
4.3. The onset of unstable growth Equations (A4) and (A6) in Appendix A allow one to compute the strain at which the instability starts growing, according either to Duncombe ( k = 0 ) or to Hart ( k = l ) respectively. These strains, computed for the Zn-A1-Cu alloy, are given in Table 1. The values quoted in Table 1 are local strains. The negative values obtained for e~0 for the two strain rates and for eH at the lower strain rate indicate that the holes start to grow from the beginning of the deformation. This is observed in Figs. 5 and 6. The positive value obtained for en at the higher strain rate indicates that a delay in strain for unstable growth of holes (k > 1 ) should occur. The corresponding far-field strain, read from Fig. 7 for single holes, is emL-- 0.63 and, read from Fig. 9 for double holes, is er,L= 0.38 both for the higher strain rate. From Figs. 5 and 6, on the contrary, it is seen that the slope of the experimental curve becomes greater than unity (i.e. larger than the slope of the line for the polymer) at a strain of about 0.50 for single holes and at about 0.35 for double holes. Thus the experimental values are very close to the computed values. The plastic growth rates are very similar, once the voids start to grow rapidly, for both high and low strain rates (see Figs. 5 and 6). Therefore the main effect of grain growth hardening is simply to shift the onset of damage towards higher strains. The shift is larger for single voids. By way of comparison, the same calculations have been performed for the CDA638 alloy with data from ref. 3. In this case, eqns. (A4) and (A6) yield positive values in all cases (Table 2), owing to the higher strain hardening which results from the faster grain growth shown by this material. The local strain-far-field strain relationship is not known for the copper alloy, so that curves of the type of Figs. 5 and 6 cannot be plotted for this material. However, it is clear that a large delay in strain required for the holes to start to grow in the copper alloy should be observed at both low and high strain rates every time that a finite strain is needed to reach the point where k >0. The latter implies that the hardening in this material is strong enough to prevent the plastic growth of voids completely for a significant strain at all strain rates. 4.4. Instability criteria and microcavitation The present results show that, depending on the degree of strain hardening, for a finite positive
T A B L E 1 Local strains el0 and e n (computed from eqn. (A4) and eqn. (A6) respectively) for unstable growth in Z n - A I - C u alloy
(s-') 1 . 4 x 10 - s 1.4 x 1 0 - s
el0(k =0)
t. (k= 1)
-0.82 - 0.60
+0.81 - 0.47
For g = 1 . 4 x 1 0 -3 s -n, C = 0 . 6 5 , p = 2 . 4 4 a n d m = 0 . 5 5 . For g = 1.4 x 1 0 - 5 s - l , C = 1.25, p = 2.44 a n d m = 0.21. (Material p a r a m e t e r s are taken f r o m ref. 12.) T A B L E 2 Local strains en0 and e n (computed from eqn. (A4) a n d eqn. (A6) respectively) for the onset of instability growth in CDA638 alloy g
% (k = 0)
e,a ( k = 1)
+0.19 +0.25
+0.32 +0.91
(s-') 5.6 × 10 -6 2.8 × 10 -4
For g = 5 . 6 x 1 0 -6 s -Z, C = 4 . 4 , p = 2 . 2 a n d m = 0 . 2 2 . For = 2.8 x 1 0 - 4 s - 1, C = 1.8, p = 2.44 a n d m = 0.40. (Material p a r a m e t e r s are taken f r o m ref. 3.)
strain the plastic growth of holes can be either prevented (k~<0), as occurs in the copper alloy, or at least retarded (k< 1) as occurs in the Z n - A I alloy at the higher strain rate. This has an important meaning regarding microcavitation because, in mechanistic terms, a delay before plastic growth can start implies that the transition between diffusion-controlled and plasticitycontrolled growth may not occur or will occur gradually until the instability points for strain localization (k>0, k > l ) are surpassed. The growth of cavities will thus be dominated by diffusion over longer times or up to higher strains than if there were no grain growth hardening. 5. Conclusions
The effect of strain hardening due to grain growth on the evolution of artificial voids in superplastic materials can be taken into account using a modified version of Hart's tensile stability model due to Nichols and to Duncombe. The predictions of the model are in good agreement with experimental data at moderate strains and for the lower volume fractions of damage (single holes) for the Zn-A1-Cu alloy. Agreement is best for the strains where the metallic material is more stable (k< 1) than the polymer. The onset of unstable flow (k--1) is predicted fairly accurately for the two void configurations and strain rates studied.
153
The main effect of grain growth hardening is to delay the onset of plastic growth of holes or at least to decrease their rate of growth at low strains. The delaying effect is small for the zincbased alloy but may be much more noticeable in materials with faster strain-induced grain growth, as the CDA638 copper alloy, for which a range of strain with k = 0 (no growth of damage) is predicted. From the point of view of microcavitation, the latter implies that the transition between diffusion-controlled and plasticity-controlled growth may be delayed. Acknowledgment
The author is grateful to Dr. D. S. Wilkinson for critically reading the manuscript. References 1 M.J. StoweU, Met. Sei., 17(1983) 1. 2 M. Sueri, in M. Sued and B. Baudelet (eds.), Superplasticity, Editions du CNRS, Paris, 1985, Paper 9. 3 C. H. Caceres and D. S. Wilkinson, Acta Metall., 32 (1984) 415. 4 D. S. Wilkinson and C. H. Caceres, Mater. Sci. Technol., 2 (1986) 1086. 5 D. S. Wilkinson and C. H. Caceres, J. Mater. Sci. Lett., 3 (1984) 395. 6 D. S. Wilkinson and C. H. Caceres, Acta Metall., 32 (1984) 1335. 7 E. Nes, in M. Sued and B. Boudelet (eds.), Superplasticity, Editions du CNRS, Pads, 1985, Paper 7. 8 A.C. E Cocks and M. E Ashby, Met. Sci., 14 (1980) 395. 9 A. C. E Cocks and M. F. Ashby, Met. Sci., 16 (1982) 465. 10 C. H. Cace~;es and D. S. Wilkinson, Acta Metall., 32 (1984) 423. 11 F. A. Mohamed, M. M. I. Ahmed and T. G. Langdon, Metall. Trans. A, 8 (1977) 933. 12 C.H. Caceres and D. S. Wilkinson, Metall. Trans. A, 17 (1986) 1873. 13 C. H. Caceres and S. P. Silvetti, Acta Metall., 35 (1987) 897. 14 E.W. Hart, Acta Metall., 15 (1967) 351. 15 F.A. Nichols, Acta Metall., 28 (1980) 663. 16 E. Duncombe, Int. J. SolidsStruct., 10 (1974) 1445.
Appendix A A. 1. Instability criteria for damage based on microstructural parameters
Superplastic deformation is usually accompanied by grain growth. Grain growth is driven by the deformation and, since the flow stress increases with increasing grain size, the result is a form of strain hardening. The effect can be incorporated into the constitutive equations as follows
[A1, A2]. A simplified constitutive equation for superplastic flow is (A1)
o = Kg mdpm
where K and p are material parameters. The strain dependence of grain size can be expressed, to a first-order approximation, in the linear form (A2)
d = d o +Ce
Here C is a parameter that depends on the material and the strain rate (the slope of the lines in Fig. 3 for each strain rate). Equations (A1) and A2) can be combined to yield the strain dependence of the strain hardening coefficient y: =
1 do ode
Cpm
- - -
do + Ce
(A3)
The instability rate of growth is controlled by the parameter I = ( 1 - y - m ) / m in eqn. (5) or, in terms of eqn. (7) for damage, by the parameter k = 1 + L The larger the values of I or k, the faster the damage grows. Three possible cases are of interest, depending on whether k~<0, k~< 1 or k>l. Since m > 0, the parameter k ~<0 only when y>~l. Imposing y~>l in eqn. (A3) and solving for the strain,
do
(A4)
e) ~ el()= p m - - -
C
Equation (A4) expresses the maximum strain at which k<~ 0 (i.e. no growth of damage) in terms of material parameters. According to Duncombe's [A3] analysis, the onset of unstable growth occurs when Y= 1, i.e. when k = 0. A more interesting case appears when 0 < k ~<1, which occurs when 1 > ~ >11 - m. From (i3), (A5)
1 - m<~ Cpm do+Ce
Solving for the strain gives el ~
pm
do
1-m
C
(A6)
Equation (A6) gives the maximum strain for which the rate of damage growth is lower than for a Newtonian viscous solid. When e)= en, I = 0 and k = 1, which, for a hardening material occurs
154 when y + m = 1. This condition marks the onset of unstable growth in Hart's model. It should be noted that, for a Newtonian viscous material, k = 1 as well, but in this case m = 1 while y = O. Finally, when k > 1 the damage grows exponentially, at a faster rate than for a Newtonian viscous material.
R e f e r e n c e s to A p p e n d i x A
A1 C. H. Caceres and D. S. Wilkinson, Acta Metall., 32 (1984) 415. A2 C. H. Caceres and D. S. Wilkinson, Metall. Trans. A, 17 (1986) 1873. A3 E. Duncombe, Int. J. SolidsStruct., 10 (1974) 1445.
147
The Growth of Artificial Voids during Superplastic Deformation of a Zn-AI-Cu Alloy C. H. CACERES* and D. E. LESCANO Facultad de Matematica, Astronomia y Fisica, Universidad Nacional de Cordoba, 5000 Cordoba (Argentina) (Received December 7, 1989; in revised form February 8, 1990)
Abstract
A series of experiments has been performed on samples of Zn-22wt.%Al-O.5wt.%Cu alloy at 523 K, at low and intermediate strain rates, with either one or three drilled holes. Similar tests were run for polystyrene, to compare the alloy with a Newtonian viscous material. A t low strains and the higher strain rate, the alloy was less sensitive to the presence of the holes than the polymer was. The results are analysed following a modified Hart model of tensile stability, including the effect of hardening due to the grain growth. The instability points, i.e. first when the voids start to grow and then when they start to grow more rapidly in the alloy than in the polymer, are predicted accurately. The main effect of grain growth hardening is to delay the onset of accelerated void growth. From the point of view of microcavitation, the latter implies that the transition between diffusion and plasticity-controlled growth may be delayed. 1. Introduction
Superplastic materials are unique in their ability to sustain extremely large volume fractions of cavitation without fracturing. The reluctance of voids to become unstable cracks indicates that the growing cavities are very stable. This behaviour cannot be accounted for by the strain rate sensitivity of the flow stress alone, because very often samples with extremely large volume fractions of cavitation (up to 30%-35%) reach large elongations [1, 2], even when the strain rate sensitivity is very low (m ~ 0.2). The high tensile stability has been rationalized, at least to first order, by including in the models for the plastic hole *Present address: Department of Materials Science and Engineering, McMaster University, Hamilton, Ontario L8S 4L7, Canada. 0921-5093/90/$3.50
growth [3, 4] the effect of strain hardening resulting from strain-induced grain growth which usually accompanies superplastic deformation [5-7]. The effect of grain growth hardening on hole growth can be understood in the following terms: since the flow stress increases with increasing grain size, grain growth results in a sort of strain hardening [3, 4]. Because the strain tends to concentrate on cavitated regions, there the grains grow faster, increasing locally the resistance to plastic flow. Thus the strain rate is decreased and the cavitation does not develop as fast as it would otherwise. The process, however, cannot be easily modelled following, for instance, the method of Cocks and Ashby [8, 9] because, once the void grows, different strains develop in the cavitated and uncavitated regions. Thus the constitutive relationships in the two regions are different, and the strain rate in the absence of the void, needed to compute the rate of hole growth, cannot be determined [4]. A qualitative picture of the effect of grain growth can be obtained as follows [4, 8]. The volumetric growth rate l? of a structure containing a void is proportional to the difference between the strain rate e~s away from the void, and the strain rate eh in the void region:
¢=
(1)
The strain rate far from the void can be expressed as
es~= \Oo1 ~b] where o 0 and e0 are a reference stress and strain rate respectively, b is a normalizing parameter, n is the creep stress exponent, d is the grain size and p defines the grain size dependence of the flow stress. © Elsevier Sequoia/Printed in The Netherlands
148
Similarly, the strain rate in the void region is given by eh = ~o
(1 -f)-"
(3)
where d h is the grain size near the void. The term ( l - f ) -n, where f is the hole volume fraction, accounts for the larger stress in the void region due to the reduced cross-section. In general, the strain is larger in the voidcontaining region, and thus dh is larger than d. Therefore the difference in strain rates is smaller than it would be if no grain growth occurred, decreasing the volumetric growth rate in eqn. (1). Following this approach, an approximate description of the effect of grain growth hardening on void growth has been developed [4] and compared with the evolution of artificial holes in a copper alloy. Artificial holes drilled in sheet samples provide a simple means of studying the plastic growth of voids. The growth stage of the holes can be separated from both the nucleation stage and the coalescence stage. The latter is especially important because coalescence of microcavitation can occur along several hundred per cent in strain, complicating the analysis of void growth at large strains. It may be argued that artificially introduced holes are much larger than those typically encountered in actual materials. However, once the voids become much larger than the grain size, the voids grow in a continuum material and the scale becomes unimportant. In the present work the evolution of artificial holes in a Zn-A1 alloy is studied. This work has many points in common with the previous study on the CDA638 copper alloy [4]. However, the copper alloy cavitates rapidly. Thus, fracture occurs at very low strains, restricting the study to a rather small deformation range [10]. To avoid this, a damage-resistant Z n - A I alloy was chosen, in order to enable a detailed study to be made up to very large strains. The tensile and fracture behaviours of this material are known from earlier work [11-13]. In addition, a different analytical approach was attempted in the present study, in order to enable a more quantitative assessment to be made of the effect of grain growth hardening on void growth. As voids elongate with little widening during superplastic straining and necking is very diffuse, it may be safely assumed that only one significant stress exists in either the uniform section or the
local inhomogeneity. The latter is the basic assumption in Hart's [14] analysis of tensile deformation [15]. On the assumption that voids are a perturbation in the tensile sample, Hart's [14] model may be used to analyse their growth, provided that the effect of grain growth hardening be considered. Since the strain rate sensitivity of the flow stress is known and the strain-hardening coefficient can be computed from the change in grain size with strain [3, 12], the predictions of the model can be readily contrasted with the experimental data.
2. Experimental details The experiments were performed using tensile samples of Zn-22wt.%A1-0.5wt.%Cu, machined out of a rolled sheet. The samples had a gauge region 12 mm long, 4 mm wide and 2.5 mm thick. In one set of samples, three holes, each 0.45 mm in diameter, were drilled as shown in Fig. 1. In a second set of samples, only one hole was drilled in order to force the fracture to occur in a single void region. Once machined, the samples were annealed for i h at 523 K to obtain a (mean linear intercept) phase size d of 1.2/~m. Tests were run with a table model Instron machine, at 473 K, in air, using a three-zone split furnace. The temperature was kept constant to within _+2 K with two thermocouples attached to the sample extremes. Testing was carried out at strain rates of 1.4 x 10 -3 and 1.4 x 10 -5 s -1. The deforming sample was observed during the tests through a small gap left open in the split furnace. By means of a long telephoto lens fitted with a bellows for close focusing (lens-to-sample distance, 35 cm) the holes were photographed at predetermined intervals. Lateral magnification on the film was about I x. The evolution of the voids
-,,t
4
,-
12
Fig. 1. Schematic diagram of tensile specimen showing positioning of holes (dimensions in millimetres).
149
was studied on the negatives under an optical microscope or on enlarged pictures. After testing to failure, the outer profile of each sample was determined by measuring the width and thickness along the gauge length. This gives information on the strain gradient along the samples and in the holes region and enables one to compute the local reduction in cross-sectional areas by multiplying the width read on the film by the corresponding thickness obtained from the profile of the sample. A series of tests on equally drilled samples of polystyrene were also run, at a strain rate of 1.4x10 -3 s -1 and at T = 3 7 3 K , in order to compare the behaviour of the metallic material with that of a truly Newtonian viscous material. Finally, several additional tests were run in smooth samples of the alloy in order to measure the strain rate sensitivity and the strain-induced grain growth at the studied strain rates. In the latter case, once tested, sections of the samples were polished down to fine alumina and etched with Nital to reveal the phase structure. Phase sizes were determined by linear intersection. Quoted values refer to the average mean linear intersect of phase size.
voids, the strain at fracture was about 3.3 at the higher strain rate and about 3.9 at the lower one. The grain size measured along deformed samples was similar to earlier published data [12] shown in Fig. 3. The strain rate sensitivity index rn, determined by strain rate changes in smooth samples at strains of the order of 0.3, was m = 0 . 2 4 at g= 1.4 x 10 -5 s -1 and m = 0 . 5 5 at g = 1.4 x 1 0 - 3 s- 1. For the polymer, m = 1 at g = 1.4 x 10 - 3 s- 1. In Fig. 4 the change in the dimensions of the holes with far-field strain is shown. A lateral contraction of the hole was observed in the metallic samples at low strains followed by a relatively small widening. It should be noted that the voids elongate some ten times more than they widen, supporting the assumption of unidirectional hole growth. For the polymer, not included in Fig. 4, voids elongated monotonically with strain, but their width remained essentially cosntant, for either single or double voids. Figure 5 shows the strain dependence of the logarithm of fractional decrease in sample area f,
3. Results In order to analyse the experimental data, several quantities will be introduced: (i) the farfield strain, defined as em ln(Ao/A), where A 0 is the initial and A the current cross-sectional area far from the voids; (ii) the local strain, defined as el=ln(Ato/Al), where At0 and A l are the initial and cross-sectional areas in the region of the holes, both measured on the plane of maximum hole width (it should be noted that A~ embodies only the load-beating section); (iii) the initial and current hole area, defined as Ah0 = A 0 -A~0 and A h --A - A t respectively (it should now be noted that the "hole area" includes any external necking); (iv) the initial and current fractional decrease in cross-sectional area due to the presence of the voids, defined as f0 = Aho/Ao and f= Ah/A. Figure 2 shows typical sets of pictures for metallic samples tested to fracture, for single and double voids. In the case of double voids, fracture occurred first in the inner ligament, at a local strain of about 2.5 for both strain rates. The outer ligament failed later, at a local strain of about 3 at the higher strain rate and at a strain of about 4.5 for the lower strain rate. In the case of single =
Fig. 2. A series of pictures showing the deformation of samples with one or three holes (tensile axis vertical). 5
i
i
Zn-22%Al-O.5%Cu (~ (s-~) L~d 4 -_.._., 7 x,O"2 (vm) - - - " 1.4xlO-z 3--._B
1.4xlO-3
---o
1.4~lO - 4
d o- 1.2p.m T = 573K
.,.--5.~:~:T- ~
LO
• -
i
..~.-':..~...':"~
~.~/" S . . ~" ' ~/' .
.%):~
2 - .......• 5.2xlO-SAtI~:: ~ _
i
•
2.0
.
~
.i
~
"I"
3.0
4.0
Fig. 3. T h e increase in grain size as a function o f strain, f o r
different strain rates [12].
150
10
10 Zn-22AI-0.5Cu 473K
o
E(S-1) Hole 8 • 1.4"I0-3 S ,ell'#t • 1.4~10-3 6. o 1.4~10-5
(2)
a/ ~1
/ e
f/fo
a /
d
/
s
/
/
• (11 Zn-Al-Cu 1.4,10-3 o (2) Zn-AI-Cu 1.4,10-5 z~(3) Polystyrene 1.4.10-3
~"
0
.2
.4
1.
.6
£m Fig. 4. Diameter of single (s) and double (d) voids, G 0 initial ~l parallel and O t perpendicular to the tensile axis: points for double voids; ~ = 1.4 x 10 -3 s-i.
20-
12) ]
+
%
: "
lIt
3'
13)
i
2'
5
i(1) I I I
/;;
£s
Crn Fig. 6. The relative change in the area fraction of damage, for double voids: - - , measured; - - - , computed from eqn. (7). 10£
I
,11)
/
/
/
/
;7
~ •
1 ~
o
o
.)zn-A -cu
(3)
£1
2.0 CIs-1) 14,;0-3
1.0
~:Is-;l
• (3) Polystyrene1.4,,10 -3 •
Zn-AI-Cu 1.4"10-3 o Zn-AI-Cu 1.4.10-5 • Polystyrene 1.4.10-3
i
£m Fig. 5. The relative change in the area fraction of damage as a function of far-field strain, for single voids: measured; - - - , computed from eqn. (7).
.2
;o
normalized to the initial value f0, as a function of the far-field strain for single voids in both metallic and plastic samples. Figure 6 shows the same results for double voids. Prominent features in Figs. 5 and 6 are the steady increase in the fifo ratio for the polymer and the delay in when the voids start to grow at the higher strain rate, for both single and double voids for the metallic samples. Finally, in Figs. 7 and 8 the experimentally determined local strain (measured at the maximum void width) vs. the far-field strain is shown for single and double voids respectively.
;.
.6
.8
1.4 gm Fig. 7. The strain in the voids region (local strain) as a function of far-field strain, for single voids.
1.2
10.0
El
1.0
• ~(S-1)
° e
4. Discussion
4.1. The growth of plastic instabilities in a tensile bar Let us assume a tensile bar deforming in steady state creep, with constant strain rate sensitivity index m and no hardening (the strain-hardening coefficient y---0; see Appendix A). Hart [14]
1.0
.2
'~"
u o Zn-AI- Cu ~ Polystyrene
1.4"10-3 1.4,10-5 1.4,10"3
O
£:m Fig. 8. The strain in the voids region (local strain) as a function of far-field strain, for double voids.
151
showed that a small inhomogeneity 6A 0 in a tensile bar should grow as
+A (A/"" 6Ao
~Ao]
(4)
where 6A 0 represents an initial inhomogeneity in a sample of initial cross-sectional area A0, and 6A and A represent the same parameters after some strain. Equation (4) applies in the present case for polystyrene (~,= 0), but not for the metallic material owing .to the grain-growth-induced hardening. Nichols [15] and Duncombe [16] extended Hart's analysis to hardening materials. Both predict that, once inhomogeneities start to grow (i.e. when ~+ m < 1), their growth can be described by [15]
6A0
(5)
where I = ( 1 - ~ , - m ) / m . Keeping in mind that e = ln(A o/A ), after replacing in (5), one may easily show that 6A = 6A 0 exp(le)
(6)
The value of I controls the rate of instability growth. 4.2. The growth of voids The preceding analysis can be extended to describe the growth of artificial voids equating the voids to the inhomogeneity in the bar, i.e. 6+4o = Ah0 and 6A = A h. Using the definitions of f and f0 given in the preceding section, eqn. (6) becomes f=fo exp(ke)
growth of damage at all. This is analysed in more detail in Appendix A. The parameter I in eqns. (6) and (7) depends on the strain and may be determined from the strain dependence of the grain size [3, 12] (see Appendix A). The values for L determined in previous work [12], are shown in Fig. 9 as a function of the local strain for three strain rates. The strain e in eqn. (7) is the far-field strain, while the parameter I is a function of the strain in the hole region. Therefore, to compute f from eqn. (7) for any given far-field strain £m, the corresponding local strain el was determined from Fig. 7 or Fig. 8 and then the respective value for I was read from Fig. 9. The values for f so obtained were plotted in Figs. 5 and 6 (broken lines) as functions of em. A very good agreement between predicted and measured values exists at low strains. Agreement is best for single voids at low strains. Certainly this is not surprising because the linearized analysis of Hart assumes that 6A is very small and single holes constitute the smallest perturbation in the tensile sample. At the higher strain rate the agreement covers the important region where the metallic material is more stable than the polymer (in Fig. 9, I < 0, k < 1), for both single and double voids. Departure of experimental data from computed values increases steadily with increasing strain. At the lower strain rate the model overestimates the growth rate; the experimental data points lay below the computed curve for both single and double voids. The opposite occurs at the higher strain rate.
(7)
where k = I + 1. It should be noted that I = 0 implies that 6+4 = 6A o = constant in eqn. (6), i.e. the hole area Ah = Aho = constant. However, as the crosssection A~ decreases with increasing strain, the relative hole size Aho/A ~increases steadily. From eqn. (7), setting k -- 1, f=f0 exp e. This is the case of the Newtonian material. Negative values o f / , possible for hardening materials, decrease the rate of growth of damage and, for I~< - 1, k~< 0, which in practical terms means that f=fo--constant. In simple words, for a hardening material, k < 1 implies that the rate of damage growth is lower than for a Newtonian material while, for k > 1, the opposite is true. When k-- 0, there is no
I! Ii
IH
1o
1~
21o
2~
31o
31s
-I -2 -3
Zn do T Ix Ii Ill
-22AI-05Cu = 12~m = 473K : ~ = 141~Ss ~ : t = 141~3$ " : ~ • 5 l r 2s ~
-4
Fig. 9. The instability parameter I, as a function of local strain, for three different strain rates [12].
152
4.3. The onset of unstable growth Equations (A4) and (A6) in Appendix A allow one to compute the strain at which the instability starts growing, according either to Duncombe ( k = 0 ) or to Hart ( k = l ) respectively. These strains, computed for the Zn-A1-Cu alloy, are given in Table 1. The values quoted in Table 1 are local strains. The negative values obtained for e~0 for the two strain rates and for eH at the lower strain rate indicate that the holes start to grow from the beginning of the deformation. This is observed in Figs. 5 and 6. The positive value obtained for en at the higher strain rate indicates that a delay in strain for unstable growth of holes (k > 1 ) should occur. The corresponding far-field strain, read from Fig. 7 for single holes, is emL-- 0.63 and, read from Fig. 9 for double holes, is er,L= 0.38 both for the higher strain rate. From Figs. 5 and 6, on the contrary, it is seen that the slope of the experimental curve becomes greater than unity (i.e. larger than the slope of the line for the polymer) at a strain of about 0.50 for single holes and at about 0.35 for double holes. Thus the experimental values are very close to the computed values. The plastic growth rates are very similar, once the voids start to grow rapidly, for both high and low strain rates (see Figs. 5 and 6). Therefore the main effect of grain growth hardening is simply to shift the onset of damage towards higher strains. The shift is larger for single voids. By way of comparison, the same calculations have been performed for the CDA638 alloy with data from ref. 3. In this case, eqns. (A4) and (A6) yield positive values in all cases (Table 2), owing to the higher strain hardening which results from the faster grain growth shown by this material. The local strain-far-field strain relationship is not known for the copper alloy, so that curves of the type of Figs. 5 and 6 cannot be plotted for this material. However, it is clear that a large delay in strain required for the holes to start to grow in the copper alloy should be observed at both low and high strain rates every time that a finite strain is needed to reach the point where k >0. The latter implies that the hardening in this material is strong enough to prevent the plastic growth of voids completely for a significant strain at all strain rates. 4.4. Instability criteria and microcavitation The present results show that, depending on the degree of strain hardening, for a finite positive
T A B L E 1 Local strains el0 and e n (computed from eqn. (A4) and eqn. (A6) respectively) for unstable growth in Z n - A I - C u alloy
(s-') 1 . 4 x 10 - s 1.4 x 1 0 - s
el0(k =0)
t. (k= 1)
-0.82 - 0.60
+0.81 - 0.47
For g = 1 . 4 x 1 0 -3 s -n, C = 0 . 6 5 , p = 2 . 4 4 a n d m = 0 . 5 5 . For g = 1.4 x 1 0 - 5 s - l , C = 1.25, p = 2.44 a n d m = 0.21. (Material p a r a m e t e r s are taken f r o m ref. 12.) T A B L E 2 Local strains en0 and e n (computed from eqn. (A4) a n d eqn. (A6) respectively) for the onset of instability growth in CDA638 alloy g
% (k = 0)
e,a ( k = 1)
+0.19 +0.25
+0.32 +0.91
(s-') 5.6 × 10 -6 2.8 × 10 -4
For g = 5 . 6 x 1 0 -6 s -Z, C = 4 . 4 , p = 2 . 2 a n d m = 0 . 2 2 . For = 2.8 x 1 0 - 4 s - 1, C = 1.8, p = 2.44 a n d m = 0.40. (Material p a r a m e t e r s are taken f r o m ref. 3.)
strain the plastic growth of holes can be either prevented (k~<0), as occurs in the copper alloy, or at least retarded (k< 1) as occurs in the Z n - A I alloy at the higher strain rate. This has an important meaning regarding microcavitation because, in mechanistic terms, a delay before plastic growth can start implies that the transition between diffusion-controlled and plasticitycontrolled growth may not occur or will occur gradually until the instability points for strain localization (k>0, k > l ) are surpassed. The growth of cavities will thus be dominated by diffusion over longer times or up to higher strains than if there were no grain growth hardening. 5. Conclusions
The effect of strain hardening due to grain growth on the evolution of artificial voids in superplastic materials can be taken into account using a modified version of Hart's tensile stability model due to Nichols and to Duncombe. The predictions of the model are in good agreement with experimental data at moderate strains and for the lower volume fractions of damage (single holes) for the Zn-A1-Cu alloy. Agreement is best for the strains where the metallic material is more stable (k< 1) than the polymer. The onset of unstable flow (k--1) is predicted fairly accurately for the two void configurations and strain rates studied.
153
The main effect of grain growth hardening is to delay the onset of plastic growth of holes or at least to decrease their rate of growth at low strains. The delaying effect is small for the zincbased alloy but may be much more noticeable in materials with faster strain-induced grain growth, as the CDA638 copper alloy, for which a range of strain with k = 0 (no growth of damage) is predicted. From the point of view of microcavitation, the latter implies that the transition between diffusion-controlled and plasticity-controlled growth may be delayed. Acknowledgment
The author is grateful to Dr. D. S. Wilkinson for critically reading the manuscript. References 1 M.J. StoweU, Met. Sei., 17(1983) 1. 2 M. Sueri, in M. Sued and B. Baudelet (eds.), Superplasticity, Editions du CNRS, Paris, 1985, Paper 9. 3 C. H. Caceres and D. S. Wilkinson, Acta Metall., 32 (1984) 415. 4 D. S. Wilkinson and C. H. Caceres, Mater. Sci. Technol., 2 (1986) 1086. 5 D. S. Wilkinson and C. H. Caceres, J. Mater. Sci. Lett., 3 (1984) 395. 6 D. S. Wilkinson and C. H. Caceres, Acta Metall., 32 (1984) 1335. 7 E. Nes, in M. Sued and B. Boudelet (eds.), Superplasticity, Editions du CNRS, Pads, 1985, Paper 7. 8 A.C. E Cocks and M. E Ashby, Met. Sci., 14 (1980) 395. 9 A. C. E Cocks and M. F. Ashby, Met. Sci., 16 (1982) 465. 10 C. H. Cace~;es and D. S. Wilkinson, Acta Metall., 32 (1984) 423. 11 F. A. Mohamed, M. M. I. Ahmed and T. G. Langdon, Metall. Trans. A, 8 (1977) 933. 12 C.H. Caceres and D. S. Wilkinson, Metall. Trans. A, 17 (1986) 1873. 13 C. H. Caceres and S. P. Silvetti, Acta Metall., 35 (1987) 897. 14 E.W. Hart, Acta Metall., 15 (1967) 351. 15 F.A. Nichols, Acta Metall., 28 (1980) 663. 16 E. Duncombe, Int. J. SolidsStruct., 10 (1974) 1445.
Appendix A A. 1. Instability criteria for damage based on microstructural parameters
Superplastic deformation is usually accompanied by grain growth. Grain growth is driven by the deformation and, since the flow stress increases with increasing grain size, the result is a form of strain hardening. The effect can be incorporated into the constitutive equations as follows
[A1, A2]. A simplified constitutive equation for superplastic flow is (A1)
o = Kg mdpm
where K and p are material parameters. The strain dependence of grain size can be expressed, to a first-order approximation, in the linear form (A2)
d = d o +Ce
Here C is a parameter that depends on the material and the strain rate (the slope of the lines in Fig. 3 for each strain rate). Equations (A1) and A2) can be combined to yield the strain dependence of the strain hardening coefficient y: =
1 do ode
Cpm
- - -
do + Ce
(A3)
The instability rate of growth is controlled by the parameter I = ( 1 - y - m ) / m in eqn. (5) or, in terms of eqn. (7) for damage, by the parameter k = 1 + L The larger the values of I or k, the faster the damage grows. Three possible cases are of interest, depending on whether k~<0, k~< 1 or k>l. Since m > 0, the parameter k ~<0 only when y>~l. Imposing y~>l in eqn. (A3) and solving for the strain,
do
(A4)
e) ~ el()= p m - - -
C
Equation (A4) expresses the maximum strain at which k<~ 0 (i.e. no growth of damage) in terms of material parameters. According to Duncombe's [A3] analysis, the onset of unstable growth occurs when Y= 1, i.e. when k = 0. A more interesting case appears when 0 < k ~<1, which occurs when 1 > ~ >11 - m. From (i3), (A5)
1 - m<~ Cpm do+Ce
Solving for the strain gives el ~
pm
do
1-m
C
(A6)
Equation (A6) gives the maximum strain for which the rate of damage growth is lower than for a Newtonian viscous solid. When e)= en, I = 0 and k = 1, which, for a hardening material occurs
154 when y + m = 1. This condition marks the onset of unstable growth in Hart's model. It should be noted that, for a Newtonian viscous material, k = 1 as well, but in this case m = 1 while y = O. Finally, when k > 1 the damage grows exponentially, at a faster rate than for a Newtonian viscous material.
R e f e r e n c e s to A p p e n d i x A
A1 C. H. Caceres and D. S. Wilkinson, Acta Metall., 32 (1984) 415. A2 C. H. Caceres and D. S. Wilkinson, Metall. Trans. A, 17 (1986) 1873. A3 E. Duncombe, Int. J. SolidsStruct., 10 (1974) 1445.