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Dynamic deformation of shock prestrained copper
Materials Science and Engineering, A 138 ( 1991 ) 23-31
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Dynamic deformation of shock prestrained copper P. S. Follansbee and G. T. Gray III Materials Science and Technology Division, Los Alamos National Laboratory, Los A lamos, NM 87545 (U.S.A.) (Received March 22, 1990; in revised form September 21, 1990)
Abstract Measurements of yield stress and strain hardening behavior at strain rates in the range 10-3_ 104 S - 1are reported for copper that has been previously shock deformed at a shock pressure of 10 GPa. Characterization of the mechanical properties and microstructure, using transmission electron microscopy, showed that the hardening induced by the shock wave is equivalent to that achieved when copper is deformed at low strain rates to strains between 20% and 30%. Deformation in the shock-deformed material is analyzed using the internal-state-variable model applied by Follansbee and Kocks. The model adequately describes the stress levels, the strain hardening rate, and the dependence of the flow stress on strain rate measured for high strain rate compression tests on the shock-deformed material.
I. Introduction Measurements reported by several investigators of copper [1], nickel [2], aluminum [3], and other pure f.c.c, metals [4] have shown that the strength of these metals appears to increase dramatically when the strain rate is raised above approximately 103 s - 1 . An example showing the compressive strength of copper determined at a constant strain e of 0.15 as a function of strain rate is given in Fig. 1 [5]. Follansbee showed that the strain-rate sensitivity of strain hardening, rather than the strain-rate sensitivity of the yield stress, on samples in a given microstructural state, is responsible for this behavior [6]. This is consistent with the work of several previous investigators who reported that copper exhibits a large strain-rate history effect [7-9]. Follansbee and Kocks [5] showed that the high strain rate results in copper could be modeled using the deformation theory proposed by Kocks [10] and Mecking and Kocks [11]. One of the predictions of the Follansbee and Kocks analysis of deformation in copper is that the strain-rate sensitivity found at high strain rates in a plot of stress at constant strain vs. strain rate should be lower in heavily deformed material than in annealed material. This, in fact, was shown by Follansbee [12] to be the case in material which was first strained 15% in compres0921-5093/91/$3.50
sion at a low strain rate, then reloaded with a further strain of 10% (for a total strain of 25%) at various (high) strain rates. The purpose of this paper is to investigate the dynamic deformation (i.e. between strain rates of 103 and 104 s -~) behavior of copper which is initially shock deformed. The interest in prior shock deformation on subsequent material response is two-fold. First, the shock wave leads to very high strain harden-
400
I
STRAIN
I
t
- 0.15
I
.
j
300 --
200--
100 10-5
J
10 3
!
I
10-1 10I STRAIN RATE, s-1
I
103
105
Fig. 1. Flow stress measurements (compression) at e = 0.15 in OFE copper as a function of strain rate. © Elsevier Sequoia/Printed in The Netherlands
24 ing in materials [13, 14]. (This is true even though the total strains in a shock are actually quite small.) Thus, the use of shock deformed material for dynamic deformation studies is convenient for testing predictions of the Follansbee and Kocks analysis. Second, dynamic deformations are often produced by the detonation of explosives or by high velocity impact events which generate shock waves. Because shock waves travel at high velocity and can lead to significant hardening which precedes the massive, large strain deformations resulting from the explosive or impact event, the behavior of shock deformed material is of real technological interest. The procedures used to generate the shock deformed copper and stress-strain measurements on samples machined from the shock prestrained material are presented in the following section. Microstructural characterization using transmission electron microscopy (TEM) of the shock prestrained material is reported in Section 3. Of particular interest is the comparison between the dislocation substructures observed in the shock prestrained material and those observed in material quasistatically deformed to similar strain levels. The Follansbee and Kocks analysis is briefly reviewed in Section 4 and predictions of the model are compared with experimental results. Section 5 contains a summary of the experiments and analysis.
2. Experimental details Shock recovery experiments were performed using an 80 mm single-stage gas gun. The specimen shock assembly consisted of a sample 5.08 mm thick and 38 mm in diameter, sandwiched behind a cover plate 38 mm in diameter and 2.54 mm thick, which were both tightly fitted into a similarly sized bored recess in the inner momentum disc 12.7 mm thick and tapered by 7 °. The sample was protected from spallation by backing the central momentum disc with a spall plate 3.8 mm thick. The central disk and spall plate were in turn surrounded by two concentric momentum trapping rings with outside diameters of 69.8 and 82.5 mm. The sample and flyer plate were fabricated from oxygen-free electronic (OFE) copper, annealed in vacuum at 600 °C for 1 h, which yielded an average grain diameter of 40 /~m. The surrounding tings and plates were fabricated from phosphor bronze. While in previous work [15, 16], OFE copper was used for
the surrounding components as well, we have found that the higher strength of the phosphor bronze helped to ensure "soft" recovery (see next paragraph). A schematic diagram of the sample assembly, in cross-section, is shown in Fig. 2. The copper samples were shocked at 10 GPa for a pulse duration of 1/~s by impacting a copper flyer plate 2.36 mm thick, fixed to a low impedance glass microbaUoon-filled projectile, at 518 m swith the specimen assembly in a vacuum (less than 2 Pa). The sample assembly was placed on top of a steel impact cylinder that allowed the passage of the central sample disk through a central hole but stopped the projectile. Samples were "soft" recovered and simultaneously cooled by decelerating the central sample disk in a water catch chamber positioned immediately behind the impact area. Through the use of the described shock assembly and recovery techniques, the residual plastic sample strain (defined here as the change in sample thickness divided by the starting sample thickness) in the copper was 2%. A detailed description of the importance of "soft" shock recovery techniques to post-shock structure-property behavior and the actual techniques utilized have been previously published [15, 16]. Samples for optical metallography and TEM were sectioned, through thickness, from the shock deformed disc. TEM observations concentrated on characterizing the substructure of the bulk portions of the shock recovered samples; TEM samples were purposely not sectioned
LEGEND SAMPLE & FLYER PLATE Zu RINGS & PLATES : PHOSPHORBRONZE [~"--I PLEXIGLASS
~]
GLASS MICROBALLOONS
Fig. 2. Schematic diagram of the "soft" shock recovery assembly showing the copper sample (the copper flyer plate is not shown), the phosphor bronze momentum tings and spall plate, and the Plexiglass and glass microballon supporting structures.
25
parallel to the sample surfaces because of the known influences of contact shear stresses on the dislocation substructure directly adjoining the surface [15]. A qualitative estimate of the dislocation cell size in the shock-loaded substructure was made by measuring the distance between the centers of the cell walls on several micrographs taken in different grains using a [110] zone axis. (A constant zone axis was used to keep the number of visible dislocations contributing to the image fixed.) Discs 3 mm in diameter were punched and electropolished in a solution of 25% H3PO4 and 75% H 2 0 at 0°C, with a current density of 80 mA mm -2. Foils were observed with a J E O L 2000EX at 200 kV. Compression specimens (5 mm long by 5 mm in diameter) were electro-discharge machined (EDM) from the shock loaded and recovered disk. The compression axis was parallel to the shock direction. One of these compression specimens was reloaded at a strain rate of 0.001 s-1 using an electro-mechanical testing machine at room temperature, while another compression specimen was reloaded at the same strain rate and at 77 K (using a compression cage in a liquid nitrogen bath). The two resulting stress-strain curves are shown in Fig. 3. Also included in Fig. 3 is the stress-strain curve measured in the starting condition. All three stress-strain curves exhibit smooth yielding with no evidence of a yield drop. The strengthening achieved through shock deformation is evident from a comparison of the strength levels measured in the quasistatic reloads to those measured in material in the initial
5OO
i
i
annealed condition. Six other specimens machined from the shock loaded disk were reloaded at room temperature and strain rates in the range 2750-9000 s- 1 using a split Hopkinson pressure bar (SHPB). The experimental procedures for these dynamic compression tests have been described previously [17]. Two stress-strain curves from these six measurements are shown in Fig. 4. Data obtained from SHPB compression tests contain large oscillations from elastic wave dispersion which make precise stress measurements difficult [18]. In previous SHPB studies [6], we performed multiple tests to offset the inherent scatter in SHPB measurements. While the limited number of samples available from the shockprestrained disk precluded this possibility, we note that the scatter in these six dynamic measurements is not larger than that typically observed (see Fig. 9). Table 1 summarizes the test results. Included in the table are the yield stresses from the quasistatic reloads and the flow stress at e = 0.10 for all of the reloads. The oscillations in the dynamic stress-strain curves and the lack of stress equilibration in the specimen at low strains make the determination of yield inaccurate at high strain rates. An estimate of the yield stress at the lowest strain rate tested using the SHPB (where the errors are lowest [6]) is given in Table 1. Of particular interest in Table 1 is the yield stress of 232 MPa measured in the shock recovered specimen at room temperature and low strain rate. This corresponds roughly to the stress achieved in an annealed specimen deformed to a strain of 20%
500
i
400
I
I
I
I
I
400
~" 0_300
~
100
~
Shock deformed, I 0 G Pa
|
J
300
09 295 K, Starting condition
200
! 100
0 0.00
I 0.05
= 0.001 s °1 I 0.10 Strain
1 0.15
0.20
Fig. 3. Reload stress-strain curves measured in the starting material and in compression samples electro-discharge machined from the shock deformed (10 GPa, 1 ~s) and recovered disk.
0
0.00
I
0.05
r
0.10
I
0.15
1
0.20
I
0.2s
o.3o
Strain
Fig. 4. Dynamic reload stress-strain curves measured in a split Hopkinson pressure bar. The oscillations in the data arise from elastic wave dispersion in the pressure bars.
26 TABLE 1 Summary of compression test results Sample condition
Reload strain rate
(s-l)
Annealed Annealed Shock recovered Shock recovered Shock recovered Shock recovered Shock recovered Shock recovered Shock recovered Shock recovered
0.001 0.001 0.001 0.001 2750 4000 6000 7000 7000 9000
Reload temperature
Yield stress (Mea)
(K) 295 77 295 77 295 295 295 295 295 295
Flow stress (e=O.lO)
(Mea) 31 232 302 266
171 209 255 387 327 316 328 319 320 333
strain at g = 0.001 s-1. Thus, although the total strain in the shock wave was only 8.25%, the strain hardening during shock deformation is seen to exceed dramatically that during quasistatic deformation. 3. Microstructural characterization
The substructure of the 10 GPa shock loaded copper was observed to consist of equiaxed dislocation cells, approximately 0.4 ~ m in diameter, with numerous dislocation loops interspersed within the cell interiors and cell walls (Fig. 5). The observation of a cellular dislocation substructure in this study is consistent with several previous substructural observations on shock loaded, high stacking-fault-energy (SFE), single crystal and polycrystalline f.c.c, metals (copper and nickel) [13-22]. A previous analysis of the substructural evolution in copper, with an initial grain size of 120 /~m, shock loaded at 10 GPa with a 2 /~s duration also reported an equiaxed cell substructure with a 0.5 /~m cell size [13]. The relatively minor variation in cell size between the current and the previous studies is consistent with the documented influence of grain size on shock loading and the overriding control of peak shock pressure as the major variable controlling shock hardening [ 13]. The substructure of quasi-statically (g = 1 0 - 3 s -1) and dynamically (g ~ 500-5000 s -1) deformed copper exhibits a well defined cellular dislocation substructure following deformations to low strains (approximately 10%)[23-26]. Copper quasi-statically deformed exhibits a cellular dislocation morphology with a 1 # m cell size following 10% strain and a 0.6/~m cell size follow-
Fig. 5. Transmission electron micrograph of the substructure of 10 GPa shock loaded copper showing typical cellular arrangement of dislocations.
ing 25% strain [23]. A previous study of dynamically deformed copper, using material identical with that used in the present study, showed that copper deformed at a strain rate of 5000 s- ~to a strain of 0.16 also exhibited a dislocation cell substructure [24]. The decreasing dislocation cell size in quasi-statically deformed copper with increasing strain has been shown to correlate with the applied shear stress until a minimum cell size is attained; the minimum cell size being observed to occur simultaneously with dislocation saturation [24]. Other studies on polycrystalline copper have shown that with increasing plastic strain (20%-30%) under quasi-static conditions, the substructure of copper, while still consisting of primarily dislocation cells, also begins to exhibit a limited number of microbands [23]. The disparity in the observed cell size
27
between the shock loaded and quasi-statically deformed samples at approximately equivalent total strains is consistent with the roughly ten orders of magnitude difference in strain rate. It is well documented that shock deformation results in the generation and storage of a larger dislocation density than that which occurs during the quasi-static process [13, 16, 22, 27]. A study of the reload yield strengths of cold rolled and shock loaded copper deformed to equivalent strains graphically showed that shock loading leads to considerably more strengthening than does cold rolling [13, 22]. This unique feature of the shock process has repeatedly been recognized since the early shock recovery experiments of Smith [13, 28].
pie, internal state variable, constitutive model to a relatively complex deformation process.
4. Analysis of deformation in shock recovered copper
#
The mechanical property measurements have shown that the rate of strain hardening during shock deformation exceeds that during quasistatic deformation. Similarly, microstructural characterization of shock deformed material has shown that the rate of dislocation accumulation (as evidenced by the cell size) during shock deformation exceeds that during quasi-static deformation. Clearly, rate dependent dislocation accumulation and hardening is fundamental to the deformation process and should be incorporated into any modeling procedure used to describe deformation over a wide range of strain rates. Follansbee and Kocks analyzed deformation in copper, over the strain rate range l 0- 4_ 104 s- 1, using the Kocks-Mecking deformation, or the mechanical threshold stress (MTS) model [5]. As an internal state variable model, the underlying premise of the analysis is that strength (in a pure f.c.c, metal of a single grain size) should first be related to the dislocation density. While additional microstructural features (e.g. detailed dislocation debris arrangement, point defect density etc.) may also influence strength, their effects remain poorly defined and are likely to be of secondary importance. In principle it should be possible to account for these effects through the use of additional state variables, but they are neglected in the analysis described below. In this section we will review the MTS model and test its predictions for deformation in shock prestrained copper. One of the objectives of this study has been to test the applicability of a sim-
4.1. The Kocks-Mecking deformation analysis The Kocks-Mecking model considers the yield stress o to be the sum of an athermal stress o a arising from the interaction of dislocations with long-range obstacles (e.g. grain boundaries) and a thermally activated contribution, s(~, T)6, arising from the interaction of dislocations with short-range obstacles (e.g. other dislocations). In the Kocks-Mecking model the internal state variable is 6, also termed the mechanical threshold stress, which represents the yield stress at 0 K (in the absence of any thermal activation). The governing relation is written as
O--Oad'-S(,~,T) O#
(1)
kt
It should be noted in eqn. (1) that the stresses are normalized by the temperature dependent shear modulus/~. (For copper we use/~(GPa) = 47.095 - (0.1429 + 0.0002763 T2) 1/2 for T> 100 K but/~ = 45.455 GPa at T= 77 K.) At temperatures greater than 0 K the yield stress (minus the athermal stress) is less than 6 by the factor s(g, T ), because thermal activation energy combines with the applied stress to allow the dislocation to overcome the obstacle. One expression for s given by thermal activation theory [29] is s = 1-;~g01Og[~)l
1
(2)
where go is the normalized total activation energy, k is the Boltzmann constant, b is the Burgers vector, and go, P and q are constants. In copper, k/b3=0.823 MPa K -1, p = 2 / 3 , q = l , and /~0 = 10 7 S -1.
The strain rate and temperature dependence defined by eqns. (1) and (2) apply to the yield stress in material whose state or structure is specified by the mechanical threshold stress 6. A complementary set of equations is required to describe the evolution of the state. In the Kocks-Mecking model, the incremental change in the threshold stress with strain is written as a function of the ratio of the current threshold stress to a temperature and strain rate dependent saturation threshold stress 6 s d e6= 00(~) 1 - F
(3)
28 where 00 is the Stage II hardening rate, which in the original copper work showed a linear strain rate dependence [5] that became important at strain rates exceeding approximately 1 0 3 S - 1 , given by 00 (MPa) = 2390 + 12 log g + 0.034g
0.04
I
I
Shock deformed 10 GPa 1 ps
(4)
oo3!
The function F in eqn. (3) was chosen as [5]
I
Finally, the strain rate and temperature dependence of the saturation threshold stress is defined by [5]
( ~ l =lzb3 g log 6-~ kT trso
I 0.05
I 0.010
0.015
(6)
l°g\eso /
Fig. 6. Reload yield stress
where A , eso, and d~o are constants. The separation of the temperature and strain rate dependence of the yield stress (eqns. (1) and (2)) from that of strain hardening (eqns. (3)-(6)) is an essential feature of the Kocks-Mecking analysis. Equations (1)-(6) were shown to provide a good description of the low temperature deformation of copper over a wide strain-rate range to strains as high as 100% [5]. In the next section we will investigate the application of these equations to the deformation of shock deformed copper.
4.2. Measurement of the mechanical threshold stress In order to apply eqns. (1)-(6) to model deformation in the shock deformed copper, it is first necessary to determine the state or mechanical threshold stress introduced by the shock event. This estimate is made by plotting the yield stress measured in samples machined from the shock recovered sample vs. test temperature and strain rate. Equation ( 1 ) with eqn. (2) inserted for s(g, T) gives an equation describing the dependence of yield stress on temperature and strain rate from material in a given state. Rearranging the combined equations yields
The three yield stress Table 1 are plotted in ordinates suggested by through the data in Fig.
0.02
measurements listed in Fig. 6 according to coeqn. (7). The line drawn 6 is a linear least squares
vs.
reload test temperature and
strain rate for the shock loaded copper. The mechanical threshold stress is given by the intercept of the line with the ordinate at T= 0.
fit. The intercept of this line at T = 0 gives an estimate of O a + 6 = 2 9 2 MPa ( d = 2 5 2 MPa), while the slope of this line gives go = 0.8. This estimate of go agrees well with the experimental results reported earlier which showed that, over a strain rate range 1 0 - 4 - 1 0 4 s - 1 and for strains to unity, the mean value of go equaled 1.6 but that go decreased with increasing threshold stress [5]. (The variation of go with dis problematic because it is not supported by theory [5].) The estimated mechanical threshold stress can be compared with previous estimates which we have made in similar material deformed under nearly identical shock conditions. Follansbee and Gray first measured d = 2 6 2 MPa for copper shock deformed at 10 GPa [30]. In subsequent experiments, Gray and Follansbee reported d = 203 MPa for the same shock conditions [31]. These results, including the go values reported in the earlier work, are summarized in Table 2. The variation in the results listed in Table 2 illustrates (i) the difficulty in performing uniaxial strain shock recovery experiments (particularly in soft copper), and (ii) the uncertainty ifi the precise d estimate because of the nature of the data and fit in Fig. 6. The estimates of d in copper deformed at 10 GPa are plotted vs. strain in Fig. 7. Included in Fig. 7 are data previously reported for strain rates of 10 -4 s -1 and 104 S - 1 [5]. The equivalent strain for a 10 GPa shock is computed from
29 400
TABLE 2 Summary of threshold stress measurements in copper shock deformed at 10 GPa
Predicted300
Reference
Residual strain (%)
6 (MPa)
g0
This work 28 29
2% 2% 1.5%
252 262 203
0.8 0.9 1.3
i
i
-
i
-
-
-
~
~
,¢o;o01
200 Q. ~300
The data in ref. 28 were reanalyzed according to p = 2/3 and q=l.
.2750 s-1 200
400
"~ 300 13.
"D -5 ,t-
i
8
i
(
i
100
0
•
0.00
I
O.10 Strain
I
O.15
0.20
Fig. 8. Comparison of predicted and measured dynamic reload stress-strain curves for the shock loaded material.
200
100
•
10GPashock I
•
~ = 10"4s
•
~ = 1 0 4 s -1
-1
I
0 0.00
I
0.05
I 0.10
0.20 0.30 Strain
0.40
0,50
Fig. 7. Mechanical threshold stress vs. strain for the shock loaded material and for samples deformed at lower strain rates [5].
e = ~ l o g ( V J V o ) , which at 10 GPa gives e~ = 0.0825. Even though there is scatter in the data for the shock deformed material, it is clear from the data in Fig. 7 that shock deformation leads to a higher rate of strain hardening than deformation at strain rates of 10- 4 s- 1 or 104 s - 1. This is consistent with the microstructural observations described earlier and with the analysis presented in eqns. (3)-(6). 4.3. Calculating the reload stress-strain curves With the estimate of 6 described above it is straightforward to apply eqns. (1)-(6) to calculate the stress-strain behavior on reloading for any specified strain rate and initial temperature. The initial threshold stress is taken as 6 = 252 MPa and the total normalized activation energy is taken as g0 = 0.8. (The average value of go defined in ref. 5 would lead to slightly (around 5%) higher estimates for the reload stresses in the shock deformed material.) The differential change in 6
is computed from eqns. (3)-(6) whereas the current stress is calculated from eqns. (1)-(2). For tests below a strain rate of roughly 1 s- 1, isothermal conditions apply and the temperature of the specimen as it deforms remains the initial temperature. At high strain rates, however, adiabatic conditions apply and the temperature increases during straining. The temperature increase can be computed from A T = ~p / o ( e ) d e pCp
(8)
where Cp is the heat capacity, p is the density (pcp = 3.43 MPa K-I), and ~p is the fraction of work in deforming the specimen that is transferred to heat. We will assume that ~p= 0.95. Stress-strain curves computed for the tests at strain rates of 2750 s- ] and 7000 s- l are shown in Fig. 8 along with the measured curves. Although the computed curves do not provide perfect fits to the measured curves (particularly at the higher strain rate), the computed curves do reproduce the stress levels as well as the hardening rates found experimentally. The temperature rise due to adiabatic heating at e = 0 . 2 0 for g = 7000 s- l is computed to be only 19 K. Thus, for the data in Fig. 8 (as well as in Fig. 9) the adiabatic temperature rise can essentially be neglected. The data in Table 1 summarizing the measured flow stress at a strain of 10% for various reload
30 350
I
5. Summary and conclusions
I
10 GPa Shock prestrain
325 0 13.
85oo [] [] 275
No prestrain ¢ - 0.15
250
,~.
,
[] []
I
,
4000
,
~]
,
t
aooo
,
12000
Strain rate (s-1) Fig. 9. Comparison of the measured (open symbols) and predicted (smaller, solid symbols) flow stresses for strain rates between 103 s i and 1.2 × 104 s 1. Results are shown for the shock loaded material (flow stress at e = 0.10) and for samples with no previous prestrain history (i.e. flow stress at e = 0.15 for monotonic loading at the strain rate indicated).
strain rates are plotted in Fig. 9 (open circles) along with the data shown in Fig. 1 for the flow stress at a strain of 15% vs. strain rate for the starting, annealing material (open squares). Although the scatter in both sets of measurements is high, the strain rate dependence of the flow stress, defined as m = Oo/Og, is lower for the shock deformed and recovered material than for the initially annealed material. This is opposite to the trend expected at constant structure, where from eqns. (1) and (2) the strain rate sensitivity should increase with increasing strain (or 6). Because the variation in 6, evaluated at a constant strain, with strain rate is small at low strain rates (which follows directly from eqn. (4)), one typically finds that at low strain rates m increases with increasing strain. The behaviour noted in Fig. 9 illustrates that at high strain rates the strain rate dependence of strain hardening begins to dominate the rate dependence, particularly at low strains. The smaller, solid symbols and associated smooth curves in Fig. 9 show the predicted behavior for both sets of measurements. While predictions for the shock deformed and recovered material are slightly (approximately 5 MPa) higher than the measurements, the agreement between the predicted and measured results demonstrates that the model described by eqns. (1)-(6) is able to capture the trends observed experimentally.
The main conclusions from this work are listed below. (1) Post mortem mechanical property measurements and microstructural characterization in copper shock deformed to 10 GPa show that shock deformation produces a dislocation cell substructure with strength level equivalent to that produced through quasi-static deformation to a strain of between 20% and 30% (see Fig. 7). (2) The measurement of a single internal state variable, the mechanical threshold stress 6, in the shock deformed material is sufficient to allow a prediction of the dynamic reload stress-strain behavior. (3) The results of this study offer further support for the description of the rate dependence of the flow stress in copper through a combination of the rate dependence of strain hardening and the rate dependence of the yield stress. The separation of these two contributions is implicit in the model described by eqns. ( 1 )-(6). In this work we have analyzed the deformation induced by the shock wave by performing measurements on samples machined from the shock deformed material. A remaining challenge is to predict the hardening produced by the shock wave using a model which includes the plastic constitutive behavior described by eqns. (1)-(6). Recent work to this end by Tonks and Johnson has met with partial success [32].
Acknowledgment The authors are grateful for the technical assistance of M. F. Lopez, W. J. Wright and C. P. Trujillo in performing the experiments described herein. This work was performed under the auspices of the United States Department of Energy.
References 1 E. A. Ripperger, in N. J. Huffington (ed.), Behavior of Materials Under Dynamic Loading, American Society of Mechanical Engineers, Chicago, 1965, p. 62. 2 T. Muller, J. Mech. Eng. Sci., 14 (1972) 3. 3 E E. Hauser, J. A. Simmons and J. E. Dorn, in P. G. Shewmon and V. E Zackay (eds.), Response of Metals to High Velocity Deformation, Interscience, New York, 1961, p. 93. 4 M. Malatynski and J. Klepaczko, Int. J. Mech. Sci., 22 (1980) 173.
31 5 P. S. FoUansbee and U. E Kocks, Acta Metall., 36 (1988) 81. 6 P. S. Follansbee, in L. E. Murr, K. E Staudhammer and M. A. Meyers (eds.), Metallurgical Applications of ShockWave and High-Strain-Rate Phenomena, Marcel Dekker, New York, 1986, p. 451. 7 T. Glenn and W. Bradley, Metall. Trans., 4 (1973) 2343. 8 J. R. Klepaezko, Mater. Sci. Eng., 18(1975) 121. 9 P. E. Senseny, J. Duffy and H. Hawley, J. Appl. Mech., 45 (1978) 60. 10 U. E Kocks, J. Eng. Mater. Technol., 98(1976) 76. 11 H. Mecking and U. E Kocks, Acta Metall., 29 (1981) 1865. 12 P. S. Follansbee, in J. Harding (ed.), The Mechanical Pro-
perties of Materials at High Rates" of Strain, Inst. Phys. Conj. Ser., 102 (1989) 237. 13 L.E. Murr, in M. A. Meyers and L. E. Murr (eds.), Shock Waves and High-Strain-Rate Phenomena in Metals, Plenum, New York, 1980, p. 607. 14 W. C. Leslie, in R. W. Rohde, B. M. Butcher, J. R. Holland and C. H. Karnes (eds.), Metallurgical Effects at High Strain Rates, Plenum, New York, 1973, p. 571. 15 G. T. Gray Ill, P. S. Follansbee and C. E. Frantz, Mater. Sci. Eng., A III (1989) 9. 16 G. T. Gray llI, in S. C. Schmidt, J. N. Johnson and L. W. Davison (eds.), Shock Compression of Condensed Matter-1989, Elsevier, New York, 1990, p. 407. 17 C. E. Frantz, E S. Follansbee and W. J. Wright, in I. Berman and J. W. Schroeder (eds.), High Energy Rate Fabrication, American Society of Mechanical Engineers, New York, 1984, p. 229. 18 E S. Follansbee and C. E. Frantz, J. Eng. Mater. Teehnol., 105(1983)61. 19 J. George, Philos. Mag., 15 (1967) 497.
20 R. L. Nolder and G. Thomas, Acta Metall., 12 (1964) 227. 21 P. S. Follansbee and G. T. Gray III, J. Plasticity, in the press. 22 A. S. Appleton and J. S. Waddington, Philos. Mag., 12 (1965) 273. 23 E R. Swann, in G. Thomas and J. Washburn (eds.), Electron Microscopy and Strength of Crystals, Wiley-Interscience, New York, 1963, p. 131. 24 M. F. Stevens and P. S. Follansbee, in G. W. Bailey (ed.),
Proc. 44th Annu. Meet. of the Electron Microscopy Society of America, San Francisco Press, San Francisco, CA, 1986, p. 420. 25 Y.J. Chang, A. J. Shume and M. N. Bassim, Mater. Sci. Eng., A 103 ( 1988) L 1 -L4. 26 E Cheval and L. Priester, Scripta Metall., 23 (1989) 1871. 27 E S. Follansbee, in S. C. Schmidt and N. C. Holmes (eds.), Shock Waves in Condensed Matter--1987, Elsevier Science Publishers, New York, 1988, p. 249. 28 C.S. Smith, Trans. TMS-AIME, 212 (1958) 574. 29 U. E Kocks, A. S. Argon and M. E Ashby, Thermo-
dynamics and Kinetics of Slip, Progress in Materials" Science, Vol. 19, Pergamon, New York, 1975. 30 P. S. Follansbee and G. T. Gray, in Y. M. Gupta (ed.), Shock Waves in Condensed Matter. Plenum, New York, 1986, p. 371. 31 G.T. Gray Ill and E S. Follansbee, in C. Y. Chiem, H.-D. Kunze and L. W. Meyer (eds.), Impact Loading and Dynamic Behaviour of Materials, lnformationsgesellschaft Verlag, Oberursel, 1988, p. 541. 32 D. L. Tonks and J. N. Johnson, in S. C. Schmidt, J. N. Johnson and L. W. Davison (eds.), Shock Compression of Condensed Matter--1989, Elsevier Science Publishers, New York, 1990, p. 333.
23
Dynamic deformation of shock prestrained copper P. S. Follansbee and G. T. Gray III Materials Science and Technology Division, Los Alamos National Laboratory, Los A lamos, NM 87545 (U.S.A.) (Received March 22, 1990; in revised form September 21, 1990)
Abstract Measurements of yield stress and strain hardening behavior at strain rates in the range 10-3_ 104 S - 1are reported for copper that has been previously shock deformed at a shock pressure of 10 GPa. Characterization of the mechanical properties and microstructure, using transmission electron microscopy, showed that the hardening induced by the shock wave is equivalent to that achieved when copper is deformed at low strain rates to strains between 20% and 30%. Deformation in the shock-deformed material is analyzed using the internal-state-variable model applied by Follansbee and Kocks. The model adequately describes the stress levels, the strain hardening rate, and the dependence of the flow stress on strain rate measured for high strain rate compression tests on the shock-deformed material.
I. Introduction Measurements reported by several investigators of copper [1], nickel [2], aluminum [3], and other pure f.c.c, metals [4] have shown that the strength of these metals appears to increase dramatically when the strain rate is raised above approximately 103 s - 1 . An example showing the compressive strength of copper determined at a constant strain e of 0.15 as a function of strain rate is given in Fig. 1 [5]. Follansbee showed that the strain-rate sensitivity of strain hardening, rather than the strain-rate sensitivity of the yield stress, on samples in a given microstructural state, is responsible for this behavior [6]. This is consistent with the work of several previous investigators who reported that copper exhibits a large strain-rate history effect [7-9]. Follansbee and Kocks [5] showed that the high strain rate results in copper could be modeled using the deformation theory proposed by Kocks [10] and Mecking and Kocks [11]. One of the predictions of the Follansbee and Kocks analysis of deformation in copper is that the strain-rate sensitivity found at high strain rates in a plot of stress at constant strain vs. strain rate should be lower in heavily deformed material than in annealed material. This, in fact, was shown by Follansbee [12] to be the case in material which was first strained 15% in compres0921-5093/91/$3.50
sion at a low strain rate, then reloaded with a further strain of 10% (for a total strain of 25%) at various (high) strain rates. The purpose of this paper is to investigate the dynamic deformation (i.e. between strain rates of 103 and 104 s -~) behavior of copper which is initially shock deformed. The interest in prior shock deformation on subsequent material response is two-fold. First, the shock wave leads to very high strain harden-
400
I
STRAIN
I
t
- 0.15
I
.
j
300 --
200--
100 10-5
J
10 3
!
I
10-1 10I STRAIN RATE, s-1
I
103
105
Fig. 1. Flow stress measurements (compression) at e = 0.15 in OFE copper as a function of strain rate. © Elsevier Sequoia/Printed in The Netherlands
24 ing in materials [13, 14]. (This is true even though the total strains in a shock are actually quite small.) Thus, the use of shock deformed material for dynamic deformation studies is convenient for testing predictions of the Follansbee and Kocks analysis. Second, dynamic deformations are often produced by the detonation of explosives or by high velocity impact events which generate shock waves. Because shock waves travel at high velocity and can lead to significant hardening which precedes the massive, large strain deformations resulting from the explosive or impact event, the behavior of shock deformed material is of real technological interest. The procedures used to generate the shock deformed copper and stress-strain measurements on samples machined from the shock prestrained material are presented in the following section. Microstructural characterization using transmission electron microscopy (TEM) of the shock prestrained material is reported in Section 3. Of particular interest is the comparison between the dislocation substructures observed in the shock prestrained material and those observed in material quasistatically deformed to similar strain levels. The Follansbee and Kocks analysis is briefly reviewed in Section 4 and predictions of the model are compared with experimental results. Section 5 contains a summary of the experiments and analysis.
2. Experimental details Shock recovery experiments were performed using an 80 mm single-stage gas gun. The specimen shock assembly consisted of a sample 5.08 mm thick and 38 mm in diameter, sandwiched behind a cover plate 38 mm in diameter and 2.54 mm thick, which were both tightly fitted into a similarly sized bored recess in the inner momentum disc 12.7 mm thick and tapered by 7 °. The sample was protected from spallation by backing the central momentum disc with a spall plate 3.8 mm thick. The central disk and spall plate were in turn surrounded by two concentric momentum trapping rings with outside diameters of 69.8 and 82.5 mm. The sample and flyer plate were fabricated from oxygen-free electronic (OFE) copper, annealed in vacuum at 600 °C for 1 h, which yielded an average grain diameter of 40 /~m. The surrounding tings and plates were fabricated from phosphor bronze. While in previous work [15, 16], OFE copper was used for
the surrounding components as well, we have found that the higher strength of the phosphor bronze helped to ensure "soft" recovery (see next paragraph). A schematic diagram of the sample assembly, in cross-section, is shown in Fig. 2. The copper samples were shocked at 10 GPa for a pulse duration of 1/~s by impacting a copper flyer plate 2.36 mm thick, fixed to a low impedance glass microbaUoon-filled projectile, at 518 m swith the specimen assembly in a vacuum (less than 2 Pa). The sample assembly was placed on top of a steel impact cylinder that allowed the passage of the central sample disk through a central hole but stopped the projectile. Samples were "soft" recovered and simultaneously cooled by decelerating the central sample disk in a water catch chamber positioned immediately behind the impact area. Through the use of the described shock assembly and recovery techniques, the residual plastic sample strain (defined here as the change in sample thickness divided by the starting sample thickness) in the copper was 2%. A detailed description of the importance of "soft" shock recovery techniques to post-shock structure-property behavior and the actual techniques utilized have been previously published [15, 16]. Samples for optical metallography and TEM were sectioned, through thickness, from the shock deformed disc. TEM observations concentrated on characterizing the substructure of the bulk portions of the shock recovered samples; TEM samples were purposely not sectioned
LEGEND SAMPLE & FLYER PLATE Zu RINGS & PLATES : PHOSPHORBRONZE [~"--I PLEXIGLASS
~]
GLASS MICROBALLOONS
Fig. 2. Schematic diagram of the "soft" shock recovery assembly showing the copper sample (the copper flyer plate is not shown), the phosphor bronze momentum tings and spall plate, and the Plexiglass and glass microballon supporting structures.
25
parallel to the sample surfaces because of the known influences of contact shear stresses on the dislocation substructure directly adjoining the surface [15]. A qualitative estimate of the dislocation cell size in the shock-loaded substructure was made by measuring the distance between the centers of the cell walls on several micrographs taken in different grains using a [110] zone axis. (A constant zone axis was used to keep the number of visible dislocations contributing to the image fixed.) Discs 3 mm in diameter were punched and electropolished in a solution of 25% H3PO4 and 75% H 2 0 at 0°C, with a current density of 80 mA mm -2. Foils were observed with a J E O L 2000EX at 200 kV. Compression specimens (5 mm long by 5 mm in diameter) were electro-discharge machined (EDM) from the shock loaded and recovered disk. The compression axis was parallel to the shock direction. One of these compression specimens was reloaded at a strain rate of 0.001 s-1 using an electro-mechanical testing machine at room temperature, while another compression specimen was reloaded at the same strain rate and at 77 K (using a compression cage in a liquid nitrogen bath). The two resulting stress-strain curves are shown in Fig. 3. Also included in Fig. 3 is the stress-strain curve measured in the starting condition. All three stress-strain curves exhibit smooth yielding with no evidence of a yield drop. The strengthening achieved through shock deformation is evident from a comparison of the strength levels measured in the quasistatic reloads to those measured in material in the initial
5OO
i
i
annealed condition. Six other specimens machined from the shock loaded disk were reloaded at room temperature and strain rates in the range 2750-9000 s- 1 using a split Hopkinson pressure bar (SHPB). The experimental procedures for these dynamic compression tests have been described previously [17]. Two stress-strain curves from these six measurements are shown in Fig. 4. Data obtained from SHPB compression tests contain large oscillations from elastic wave dispersion which make precise stress measurements difficult [18]. In previous SHPB studies [6], we performed multiple tests to offset the inherent scatter in SHPB measurements. While the limited number of samples available from the shockprestrained disk precluded this possibility, we note that the scatter in these six dynamic measurements is not larger than that typically observed (see Fig. 9). Table 1 summarizes the test results. Included in the table are the yield stresses from the quasistatic reloads and the flow stress at e = 0.10 for all of the reloads. The oscillations in the dynamic stress-strain curves and the lack of stress equilibration in the specimen at low strains make the determination of yield inaccurate at high strain rates. An estimate of the yield stress at the lowest strain rate tested using the SHPB (where the errors are lowest [6]) is given in Table 1. Of particular interest in Table 1 is the yield stress of 232 MPa measured in the shock recovered specimen at room temperature and low strain rate. This corresponds roughly to the stress achieved in an annealed specimen deformed to a strain of 20%
500
i
400
I
I
I
I
I
400
~" 0_300
~
100
~
Shock deformed, I 0 G Pa
|
J
300
09 295 K, Starting condition
200
! 100
0 0.00
I 0.05
= 0.001 s °1 I 0.10 Strain
1 0.15
0.20
Fig. 3. Reload stress-strain curves measured in the starting material and in compression samples electro-discharge machined from the shock deformed (10 GPa, 1 ~s) and recovered disk.
0
0.00
I
0.05
r
0.10
I
0.15
1
0.20
I
0.2s
o.3o
Strain
Fig. 4. Dynamic reload stress-strain curves measured in a split Hopkinson pressure bar. The oscillations in the data arise from elastic wave dispersion in the pressure bars.
26 TABLE 1 Summary of compression test results Sample condition
Reload strain rate
(s-l)
Annealed Annealed Shock recovered Shock recovered Shock recovered Shock recovered Shock recovered Shock recovered Shock recovered Shock recovered
0.001 0.001 0.001 0.001 2750 4000 6000 7000 7000 9000
Reload temperature
Yield stress (Mea)
(K) 295 77 295 77 295 295 295 295 295 295
Flow stress (e=O.lO)
(Mea) 31 232 302 266
171 209 255 387 327 316 328 319 320 333
strain at g = 0.001 s-1. Thus, although the total strain in the shock wave was only 8.25%, the strain hardening during shock deformation is seen to exceed dramatically that during quasistatic deformation. 3. Microstructural characterization
The substructure of the 10 GPa shock loaded copper was observed to consist of equiaxed dislocation cells, approximately 0.4 ~ m in diameter, with numerous dislocation loops interspersed within the cell interiors and cell walls (Fig. 5). The observation of a cellular dislocation substructure in this study is consistent with several previous substructural observations on shock loaded, high stacking-fault-energy (SFE), single crystal and polycrystalline f.c.c, metals (copper and nickel) [13-22]. A previous analysis of the substructural evolution in copper, with an initial grain size of 120 /~m, shock loaded at 10 GPa with a 2 /~s duration also reported an equiaxed cell substructure with a 0.5 /~m cell size [13]. The relatively minor variation in cell size between the current and the previous studies is consistent with the documented influence of grain size on shock loading and the overriding control of peak shock pressure as the major variable controlling shock hardening [ 13]. The substructure of quasi-statically (g = 1 0 - 3 s -1) and dynamically (g ~ 500-5000 s -1) deformed copper exhibits a well defined cellular dislocation substructure following deformations to low strains (approximately 10%)[23-26]. Copper quasi-statically deformed exhibits a cellular dislocation morphology with a 1 # m cell size following 10% strain and a 0.6/~m cell size follow-
Fig. 5. Transmission electron micrograph of the substructure of 10 GPa shock loaded copper showing typical cellular arrangement of dislocations.
ing 25% strain [23]. A previous study of dynamically deformed copper, using material identical with that used in the present study, showed that copper deformed at a strain rate of 5000 s- ~to a strain of 0.16 also exhibited a dislocation cell substructure [24]. The decreasing dislocation cell size in quasi-statically deformed copper with increasing strain has been shown to correlate with the applied shear stress until a minimum cell size is attained; the minimum cell size being observed to occur simultaneously with dislocation saturation [24]. Other studies on polycrystalline copper have shown that with increasing plastic strain (20%-30%) under quasi-static conditions, the substructure of copper, while still consisting of primarily dislocation cells, also begins to exhibit a limited number of microbands [23]. The disparity in the observed cell size
27
between the shock loaded and quasi-statically deformed samples at approximately equivalent total strains is consistent with the roughly ten orders of magnitude difference in strain rate. It is well documented that shock deformation results in the generation and storage of a larger dislocation density than that which occurs during the quasi-static process [13, 16, 22, 27]. A study of the reload yield strengths of cold rolled and shock loaded copper deformed to equivalent strains graphically showed that shock loading leads to considerably more strengthening than does cold rolling [13, 22]. This unique feature of the shock process has repeatedly been recognized since the early shock recovery experiments of Smith [13, 28].
pie, internal state variable, constitutive model to a relatively complex deformation process.
4. Analysis of deformation in shock recovered copper
#
The mechanical property measurements have shown that the rate of strain hardening during shock deformation exceeds that during quasistatic deformation. Similarly, microstructural characterization of shock deformed material has shown that the rate of dislocation accumulation (as evidenced by the cell size) during shock deformation exceeds that during quasi-static deformation. Clearly, rate dependent dislocation accumulation and hardening is fundamental to the deformation process and should be incorporated into any modeling procedure used to describe deformation over a wide range of strain rates. Follansbee and Kocks analyzed deformation in copper, over the strain rate range l 0- 4_ 104 s- 1, using the Kocks-Mecking deformation, or the mechanical threshold stress (MTS) model [5]. As an internal state variable model, the underlying premise of the analysis is that strength (in a pure f.c.c, metal of a single grain size) should first be related to the dislocation density. While additional microstructural features (e.g. detailed dislocation debris arrangement, point defect density etc.) may also influence strength, their effects remain poorly defined and are likely to be of secondary importance. In principle it should be possible to account for these effects through the use of additional state variables, but they are neglected in the analysis described below. In this section we will review the MTS model and test its predictions for deformation in shock prestrained copper. One of the objectives of this study has been to test the applicability of a sim-
4.1. The Kocks-Mecking deformation analysis The Kocks-Mecking model considers the yield stress o to be the sum of an athermal stress o a arising from the interaction of dislocations with long-range obstacles (e.g. grain boundaries) and a thermally activated contribution, s(~, T)6, arising from the interaction of dislocations with short-range obstacles (e.g. other dislocations). In the Kocks-Mecking model the internal state variable is 6, also termed the mechanical threshold stress, which represents the yield stress at 0 K (in the absence of any thermal activation). The governing relation is written as
O--Oad'-S(,~,T) O#
(1)
kt
It should be noted in eqn. (1) that the stresses are normalized by the temperature dependent shear modulus/~. (For copper we use/~(GPa) = 47.095 - (0.1429 + 0.0002763 T2) 1/2 for T> 100 K but/~ = 45.455 GPa at T= 77 K.) At temperatures greater than 0 K the yield stress (minus the athermal stress) is less than 6 by the factor s(g, T ), because thermal activation energy combines with the applied stress to allow the dislocation to overcome the obstacle. One expression for s given by thermal activation theory [29] is s = 1-;~g01Og[~)l
1
(2)
where go is the normalized total activation energy, k is the Boltzmann constant, b is the Burgers vector, and go, P and q are constants. In copper, k/b3=0.823 MPa K -1, p = 2 / 3 , q = l , and /~0 = 10 7 S -1.
The strain rate and temperature dependence defined by eqns. (1) and (2) apply to the yield stress in material whose state or structure is specified by the mechanical threshold stress 6. A complementary set of equations is required to describe the evolution of the state. In the Kocks-Mecking model, the incremental change in the threshold stress with strain is written as a function of the ratio of the current threshold stress to a temperature and strain rate dependent saturation threshold stress 6 s d e6= 00(~) 1 - F
(3)
28 where 00 is the Stage II hardening rate, which in the original copper work showed a linear strain rate dependence [5] that became important at strain rates exceeding approximately 1 0 3 S - 1 , given by 00 (MPa) = 2390 + 12 log g + 0.034g
0.04
I
I
Shock deformed 10 GPa 1 ps
(4)
oo3!
The function F in eqn. (3) was chosen as [5]
I
Finally, the strain rate and temperature dependence of the saturation threshold stress is defined by [5]
( ~ l =lzb3 g log 6-~ kT trso
I 0.05
I 0.010
0.015
(6)
l°g\eso /
Fig. 6. Reload yield stress
where A , eso, and d~o are constants. The separation of the temperature and strain rate dependence of the yield stress (eqns. (1) and (2)) from that of strain hardening (eqns. (3)-(6)) is an essential feature of the Kocks-Mecking analysis. Equations (1)-(6) were shown to provide a good description of the low temperature deformation of copper over a wide strain-rate range to strains as high as 100% [5]. In the next section we will investigate the application of these equations to the deformation of shock deformed copper.
4.2. Measurement of the mechanical threshold stress In order to apply eqns. (1)-(6) to model deformation in the shock deformed copper, it is first necessary to determine the state or mechanical threshold stress introduced by the shock event. This estimate is made by plotting the yield stress measured in samples machined from the shock recovered sample vs. test temperature and strain rate. Equation ( 1 ) with eqn. (2) inserted for s(g, T) gives an equation describing the dependence of yield stress on temperature and strain rate from material in a given state. Rearranging the combined equations yields
The three yield stress Table 1 are plotted in ordinates suggested by through the data in Fig.
0.02
measurements listed in Fig. 6 according to coeqn. (7). The line drawn 6 is a linear least squares
vs.
reload test temperature and
strain rate for the shock loaded copper. The mechanical threshold stress is given by the intercept of the line with the ordinate at T= 0.
fit. The intercept of this line at T = 0 gives an estimate of O a + 6 = 2 9 2 MPa ( d = 2 5 2 MPa), while the slope of this line gives go = 0.8. This estimate of go agrees well with the experimental results reported earlier which showed that, over a strain rate range 1 0 - 4 - 1 0 4 s - 1 and for strains to unity, the mean value of go equaled 1.6 but that go decreased with increasing threshold stress [5]. (The variation of go with dis problematic because it is not supported by theory [5].) The estimated mechanical threshold stress can be compared with previous estimates which we have made in similar material deformed under nearly identical shock conditions. Follansbee and Gray first measured d = 2 6 2 MPa for copper shock deformed at 10 GPa [30]. In subsequent experiments, Gray and Follansbee reported d = 203 MPa for the same shock conditions [31]. These results, including the go values reported in the earlier work, are summarized in Table 2. The variation in the results listed in Table 2 illustrates (i) the difficulty in performing uniaxial strain shock recovery experiments (particularly in soft copper), and (ii) the uncertainty ifi the precise d estimate because of the nature of the data and fit in Fig. 6. The estimates of d in copper deformed at 10 GPa are plotted vs. strain in Fig. 7. Included in Fig. 7 are data previously reported for strain rates of 10 -4 s -1 and 104 S - 1 [5]. The equivalent strain for a 10 GPa shock is computed from
29 400
TABLE 2 Summary of threshold stress measurements in copper shock deformed at 10 GPa
Predicted300
Reference
Residual strain (%)
6 (MPa)
g0
This work 28 29
2% 2% 1.5%
252 262 203
0.8 0.9 1.3
i
i
-
i
-
-
-
~
~
,¢o;o01
200 Q. ~300
The data in ref. 28 were reanalyzed according to p = 2/3 and q=l.
.2750 s-1 200
400
"~ 300 13.
"D -5 ,t-
i
8
i
(
i
100
0
•
0.00
I
O.10 Strain
I
O.15
0.20
Fig. 8. Comparison of predicted and measured dynamic reload stress-strain curves for the shock loaded material.
200
100
•
10GPashock I
•
~ = 10"4s
•
~ = 1 0 4 s -1
-1
I
0 0.00
I
0.05
I 0.10
0.20 0.30 Strain
0.40
0,50
Fig. 7. Mechanical threshold stress vs. strain for the shock loaded material and for samples deformed at lower strain rates [5].
e = ~ l o g ( V J V o ) , which at 10 GPa gives e~ = 0.0825. Even though there is scatter in the data for the shock deformed material, it is clear from the data in Fig. 7 that shock deformation leads to a higher rate of strain hardening than deformation at strain rates of 10- 4 s- 1 or 104 s - 1. This is consistent with the microstructural observations described earlier and with the analysis presented in eqns. (3)-(6). 4.3. Calculating the reload stress-strain curves With the estimate of 6 described above it is straightforward to apply eqns. (1)-(6) to calculate the stress-strain behavior on reloading for any specified strain rate and initial temperature. The initial threshold stress is taken as 6 = 252 MPa and the total normalized activation energy is taken as g0 = 0.8. (The average value of go defined in ref. 5 would lead to slightly (around 5%) higher estimates for the reload stresses in the shock deformed material.) The differential change in 6
is computed from eqns. (3)-(6) whereas the current stress is calculated from eqns. (1)-(2). For tests below a strain rate of roughly 1 s- 1, isothermal conditions apply and the temperature of the specimen as it deforms remains the initial temperature. At high strain rates, however, adiabatic conditions apply and the temperature increases during straining. The temperature increase can be computed from A T = ~p / o ( e ) d e pCp
(8)
where Cp is the heat capacity, p is the density (pcp = 3.43 MPa K-I), and ~p is the fraction of work in deforming the specimen that is transferred to heat. We will assume that ~p= 0.95. Stress-strain curves computed for the tests at strain rates of 2750 s- ] and 7000 s- l are shown in Fig. 8 along with the measured curves. Although the computed curves do not provide perfect fits to the measured curves (particularly at the higher strain rate), the computed curves do reproduce the stress levels as well as the hardening rates found experimentally. The temperature rise due to adiabatic heating at e = 0 . 2 0 for g = 7000 s- l is computed to be only 19 K. Thus, for the data in Fig. 8 (as well as in Fig. 9) the adiabatic temperature rise can essentially be neglected. The data in Table 1 summarizing the measured flow stress at a strain of 10% for various reload
30 350
I
5. Summary and conclusions
I
10 GPa Shock prestrain
325 0 13.
85oo [] [] 275
No prestrain ¢ - 0.15
250
,~.
,
[] []
I
,
4000
,
~]
,
t
aooo
,
12000
Strain rate (s-1) Fig. 9. Comparison of the measured (open symbols) and predicted (smaller, solid symbols) flow stresses for strain rates between 103 s i and 1.2 × 104 s 1. Results are shown for the shock loaded material (flow stress at e = 0.10) and for samples with no previous prestrain history (i.e. flow stress at e = 0.15 for monotonic loading at the strain rate indicated).
strain rates are plotted in Fig. 9 (open circles) along with the data shown in Fig. 1 for the flow stress at a strain of 15% vs. strain rate for the starting, annealing material (open squares). Although the scatter in both sets of measurements is high, the strain rate dependence of the flow stress, defined as m = Oo/Og, is lower for the shock deformed and recovered material than for the initially annealed material. This is opposite to the trend expected at constant structure, where from eqns. (1) and (2) the strain rate sensitivity should increase with increasing strain (or 6). Because the variation in 6, evaluated at a constant strain, with strain rate is small at low strain rates (which follows directly from eqn. (4)), one typically finds that at low strain rates m increases with increasing strain. The behaviour noted in Fig. 9 illustrates that at high strain rates the strain rate dependence of strain hardening begins to dominate the rate dependence, particularly at low strains. The smaller, solid symbols and associated smooth curves in Fig. 9 show the predicted behavior for both sets of measurements. While predictions for the shock deformed and recovered material are slightly (approximately 5 MPa) higher than the measurements, the agreement between the predicted and measured results demonstrates that the model described by eqns. (1)-(6) is able to capture the trends observed experimentally.
The main conclusions from this work are listed below. (1) Post mortem mechanical property measurements and microstructural characterization in copper shock deformed to 10 GPa show that shock deformation produces a dislocation cell substructure with strength level equivalent to that produced through quasi-static deformation to a strain of between 20% and 30% (see Fig. 7). (2) The measurement of a single internal state variable, the mechanical threshold stress 6, in the shock deformed material is sufficient to allow a prediction of the dynamic reload stress-strain behavior. (3) The results of this study offer further support for the description of the rate dependence of the flow stress in copper through a combination of the rate dependence of strain hardening and the rate dependence of the yield stress. The separation of these two contributions is implicit in the model described by eqns. ( 1 )-(6). In this work we have analyzed the deformation induced by the shock wave by performing measurements on samples machined from the shock deformed material. A remaining challenge is to predict the hardening produced by the shock wave using a model which includes the plastic constitutive behavior described by eqns. (1)-(6). Recent work to this end by Tonks and Johnson has met with partial success [32].
Acknowledgment The authors are grateful for the technical assistance of M. F. Lopez, W. J. Wright and C. P. Trujillo in performing the experiments described herein. This work was performed under the auspices of the United States Department of Energy.
References 1 E. A. Ripperger, in N. J. Huffington (ed.), Behavior of Materials Under Dynamic Loading, American Society of Mechanical Engineers, Chicago, 1965, p. 62. 2 T. Muller, J. Mech. Eng. Sci., 14 (1972) 3. 3 E E. Hauser, J. A. Simmons and J. E. Dorn, in P. G. Shewmon and V. E Zackay (eds.), Response of Metals to High Velocity Deformation, Interscience, New York, 1961, p. 93. 4 M. Malatynski and J. Klepaczko, Int. J. Mech. Sci., 22 (1980) 173.
31 5 P. S. FoUansbee and U. E Kocks, Acta Metall., 36 (1988) 81. 6 P. S. Follansbee, in L. E. Murr, K. E Staudhammer and M. A. Meyers (eds.), Metallurgical Applications of ShockWave and High-Strain-Rate Phenomena, Marcel Dekker, New York, 1986, p. 451. 7 T. Glenn and W. Bradley, Metall. Trans., 4 (1973) 2343. 8 J. R. Klepaezko, Mater. Sci. Eng., 18(1975) 121. 9 P. E. Senseny, J. Duffy and H. Hawley, J. Appl. Mech., 45 (1978) 60. 10 U. E Kocks, J. Eng. Mater. Technol., 98(1976) 76. 11 H. Mecking and U. E Kocks, Acta Metall., 29 (1981) 1865. 12 P. S. Follansbee, in J. Harding (ed.), The Mechanical Pro-
perties of Materials at High Rates" of Strain, Inst. Phys. Conj. Ser., 102 (1989) 237. 13 L.E. Murr, in M. A. Meyers and L. E. Murr (eds.), Shock Waves and High-Strain-Rate Phenomena in Metals, Plenum, New York, 1980, p. 607. 14 W. C. Leslie, in R. W. Rohde, B. M. Butcher, J. R. Holland and C. H. Karnes (eds.), Metallurgical Effects at High Strain Rates, Plenum, New York, 1973, p. 571. 15 G. T. Gray Ill, P. S. Follansbee and C. E. Frantz, Mater. Sci. Eng., A III (1989) 9. 16 G. T. Gray llI, in S. C. Schmidt, J. N. Johnson and L. W. Davison (eds.), Shock Compression of Condensed Matter-1989, Elsevier, New York, 1990, p. 407. 17 C. E. Frantz, E S. Follansbee and W. J. Wright, in I. Berman and J. W. Schroeder (eds.), High Energy Rate Fabrication, American Society of Mechanical Engineers, New York, 1984, p. 229. 18 E S. Follansbee and C. E. Frantz, J. Eng. Mater. Teehnol., 105(1983)61. 19 J. George, Philos. Mag., 15 (1967) 497.
20 R. L. Nolder and G. Thomas, Acta Metall., 12 (1964) 227. 21 P. S. Follansbee and G. T. Gray III, J. Plasticity, in the press. 22 A. S. Appleton and J. S. Waddington, Philos. Mag., 12 (1965) 273. 23 E R. Swann, in G. Thomas and J. Washburn (eds.), Electron Microscopy and Strength of Crystals, Wiley-Interscience, New York, 1963, p. 131. 24 M. F. Stevens and P. S. Follansbee, in G. W. Bailey (ed.),
Proc. 44th Annu. Meet. of the Electron Microscopy Society of America, San Francisco Press, San Francisco, CA, 1986, p. 420. 25 Y.J. Chang, A. J. Shume and M. N. Bassim, Mater. Sci. Eng., A 103 ( 1988) L 1 -L4. 26 E Cheval and L. Priester, Scripta Metall., 23 (1989) 1871. 27 E S. Follansbee, in S. C. Schmidt and N. C. Holmes (eds.), Shock Waves in Condensed Matter--1987, Elsevier Science Publishers, New York, 1988, p. 249. 28 C.S. Smith, Trans. TMS-AIME, 212 (1958) 574. 29 U. E Kocks, A. S. Argon and M. E Ashby, Thermo-
dynamics and Kinetics of Slip, Progress in Materials" Science, Vol. 19, Pergamon, New York, 1975. 30 P. S. Follansbee and G. T. Gray, in Y. M. Gupta (ed.), Shock Waves in Condensed Matter. Plenum, New York, 1986, p. 371. 31 G.T. Gray Ill and E S. Follansbee, in C. Y. Chiem, H.-D. Kunze and L. W. Meyer (eds.), Impact Loading and Dynamic Behaviour of Materials, lnformationsgesellschaft Verlag, Oberursel, 1988, p. 541. 32 D. L. Tonks and J. N. Johnson, in S. C. Schmidt, J. N. Johnson and L. W. Davison (eds.), Shock Compression of Condensed Matter--1989, Elsevier Science Publishers, New York, 1990, p. 333.