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The dynamic mechanical behavior of titanium in shear
J. Mech. Phys. Solids, 1972, Vol. 20, pp. 65 to 76.
THE DYNAMIC
Pergamon Press.
Printed in Great Britain.
MECHANICAL BEHAVIOR OF TITANIUM IN SHEAR By J. E. LAWSONand T. NICHOLAS
Air Force Materials Laboratory,
Wright-Patterson
Air Force Base, Ohio
(Received 31sr August 1971)
CONSTANT strain-rate
stress-strain data in shear for commer&aIly pure titanium are obtained through the use of a torsional split Hopkinson pressure-bar and a pneumatic-mechanical torsion testing machine. These data, which cover a range in shear strain rates from lo-* to 5 x 103 s-r, are used to obtain the constants in a constitutive equation which expresses the dynamic over-stress as a function of the logarithm of the plastic strain rate. The equation is used to predict the results of torsional plastic wave propagation experiments using the rate-dependent theory of L. E. Malvern. The sensitivity of the results to the constants in the constitutive equation is studied along with predictions using the mt~~de~ndent theory and an elevated static stress+&ain curve.
1.
I~~DLJCTION
STRAIN-RATE effects in metals have been studied extensively by numerous investigators over the past two decades. Both direct and indirect methods have been employed by ex~rnen~~s~ to determine the response of materials to dynamic loading. In the first, constant strain-rate tests on short specimens provide stress-strain curves at high strain-rates which can be compared to the quasi-static behavior of the material. In the second, the propagation of plastic stress waves down long bars or rods is used to deduce a constitutive equation for the material. In either technique there is usually insurgent info~ation genemt~ to characterize uniquely the behavior of the material as a function of strain rate. Constant strain-rate tests in uniaxial stress at high rates-of-strain are performed using the Kolsky technique (KOLSKY,1949), more commonly referred to as the split Hopkinson pressure-bar apparatus. The range in strain rates which is encompassed by the technique is roughly 102-IO4 s--i. Relatively little information has been generated at inte~ediate strain rates below lo2 and above 10 -’ s -’ because of the scarcity of experimental apparatus which operates in this range. Conventional testing machines are used to obtain data at strain rates below 10-l s-l. In order to characterize fully the behavior of a material as a function of strain rate it is necessary to have constant strain-rate data over a wide range of strain rates including intermediate regions. Plastic wave pro~ga~ion studies suffer from the necessity of assuming, a priori, a form for the constitutive equation describing the behavior of the material. Theoretical predictions based on various forms of plastic wave propagation theories have been shown to be quite similar in several respects. RIPPBBGBB and WATSON (1968) demon65
66
J.
E. LAWSONand T. NICHOLAS
strated by computer analysis that the velocity of propagation of plastic waves is not very sensitive to the form of the constitutive equation proposed for the material. WOOD and PHILXZS (1967) have shown analyti~lly that the constant strain plateau at the impact end of a bar as predicted by the rate-independent theory of plastic wave propagation is likewise predicted by the rate-dependent theory of MALVERN (1951). It is apparent, then, that a plastic wave propagation experiment can, at most, verify a proposed constitutive equation; it cannot determine it uniquely. If the rate-independent theory of plastic wave propagation is used, a single dynamic stress-strain curve may be deduced which may, in general, differ from the quasistatic curve. In the rate-dependent theory, as proposed by MALVERN(195t), any one of a number of forms of the constitutive equation may be assumed expressing the plastic strain rate as a function of stress and strain. The linear form, most commonly used, is not particularly realistic in describing the results of constant strain-rate tests although it lends itself to simpler computations. Another point which has raised questions in the interpre~tion of data obtained in dynamic testing is the use of the compression (or tension) mode of deformation where problems such as those of barreling (or necking) and radial inertia and radial shear effects are known to exist. The use of the torsional mode of deformation applied to thin-walled tubular specimens has been introduced 1to eliminate these potential problems (BAKER and YEW, 1966, DUFFY, CAMPBELLand HAWLEY,1971, and NICHOLASand LAWSON,1972). The final point which may be the cause of much of the controversy in dynamic plasticity regarding the existence of strain-rate effects in metals is the widespread use of relatively rate-insentitive materials in experimental investigations. Materials such as pure aluminum have been used extensively to study strain-rate effects or to validate the rate-de~ndent (or rate-i~de~ndent) theory of plastic wave propagation (BELL, 1968). The fact that the rate-dependent and rate-independent theories predict virtually the same wave profiles, especially when the degree of rate-dependence is small, leads to some uncertainty in relating transient wave profiles to a specific constitutive equation. The present investigation describes the application of existing experimental techniques to characterize the behavior of a highly rate-sensitive material under constant strain-rate loading conditions and to study subsequently the propagation of plastic waves in that material using a constitutive equation derived directly from the results of the constant strain-rate tests. The behavior of pure titanium is studied in pure shear using a torsional split Hopkinson pressure-bar apparatus and a mechanicalpneumatic machine which, together, allow stress-strain data to be obtained over A ~onstitutive equation which fits the eight orders of ma~tude in strain-rate. experimental data is used to predict the propagation of plastic waves in long tubes. Finally, the sensitivity of the results of the wave propagation experiments to the constants, and to the form of the constitutive equation, is investigated. 2.
E~~I~~~AL
APPARATUS
The torsional split Hopkinson pressure-bar apparatus used in this investigation to obtain stress-strain dam at high rates of strain has been described in detail by NICHOLASand LAWSON(1972). The device operates by releasing a torque stored
The dynamicmecha.nicalbehaviorof titaniumin shear
67
in a length of bar through the rapid fracture of a bolt which holds a clamp in place.
Specimens of thin-walled tubular cross-section are attached to the input and output bars by means of an epoxy adhesive. Through the use of specimens having gage lengths as short as 0.050 in., reliable stress-strain data can be obtained at shear strain rates up to nearly lo4 s -‘. Quasi-static and intermediate strain-rate tests were performed on a combined mechanical-pneumatic machine described by NICHOLAS(1971). Strain rates from lo4 to 10e2 s-l can be achieved by driving a screw-jack attached to a lever arm with a motor. Strain-rates up to IO2 s -’ are obtained by driving a piston attached to another lever arm through the release of compressed gas by a quick-operating valve. The torque is measured with a conventional strain gage load cell while angle of rotation is recorded through the use of a differential capacitor activated by a 10 kH carrier signal which produces an analog signal proportional to the rotation of the capacitor. Specimens of commercially pure 50A titanium were machined from a single bar to minimize the possible variation in material properties. All specimens were thinwalled hollow tubes having a O-5in. inside diameter and a wall thickness of nominally O-025 in. All wall thicknesses were accurately measured and specimens having variations of more than O*OOlin. were rejected. Gage lengths for specimens varied from O-5 to 0.05 in. Both the torsional split Hopkinson pressure-bar and pneumatic-mechanical machine were calibrated using dead weight loading. The split Hopkinson pressurebar was also calibrated dynamically by propagating pulses of known magnitude down the input and output bars attached to one another using an epoxy adhesive. 3.
CONSTANTSTRAIN-RATEBEHAVIOR
3.1 Experimental results Torsional split Hopkinson bar tests were performed on twelve specimens of 50A titanium covering a range of strain rates from 191 to 5000 s-l. The results of those tests along with the quasi-static and dynamic test results obtained using the mechanical-pneumatic machine are presented in Fig. 1. The amount of scatter in data obtained at any given strain rate was generally less than 5 per cent and was within the experimental accuracy. The results show that this material displays a de&rite strain-rate sensitivity over the entire range of strain rates encountered in the tests. The data are re-plotted as stress vs. log (strain rate) for 10 per cent strain and presented in Fig. 2. The results show that the flow stress at a given strain depends linearly upon the log (strain rate) over the entire range of strain rates. It can also be seen from this plot that very little deviation from this linear law was found in the data. The consistency of the data is clearly shown in not only the small amount of scatter but also from the nearly identical results obtained from both the lowest strain-rate Hopkinson bar tests and the highest strain-rate tests on the pneumatic machine. It is significant to note also that these data obtained at overlapping strain rates were obtained from long gage length specimens on the Hopkinson bar and very short gage length specimens on the pneumatic machine. This is additional evidence of the lack of any apparent gage length effect in the test results (NICHOLASand LAWSON,1972).
J. E.
LAWSON and
T.
NICHOLAS
IO I
0'
2
0
1
I
4
I
6
FIG. 1. Shear stress-strain
8
lo
SILUI
STRAIN.%
12
I4
m
curve-s for commercially
IO
pure titanium
20
50-A.
80
70 ‘; .s ui
60
ii K z5
50
TEST
5
METHOD
HOPKINSON
BAR
PNEUMATIC 40
30 10-4
IO“
10-l
10-2
SHEAR
I STRAIN
IO’
IO RATE,
IO’
IO’
sac-’
FIG.2. Stress vs. log (strain rate) at 10 per cent strain. 3.2 Constitutive
equation
Many authors have proposed various constitutive equations to describe the strain-rate dependent behavior of materials. LUDWIK(1909) postulated a logarithmic equation of the form r = tl + 7. In ($&J, (3.1) where 71, 7,, and qO are material constants and p, is the plastic strain rate. This relation, which has been verified experimentally, has been the basis of numerous proposed constitutive equations to describe rate effects (C~~~TJBCU,1967). MALVERN (1951) proposed the use of an expression 7 =f(y)+
a In (1 + bj,),
(3.2)
The dynamic mechanical behavior of titanium in shear
6P
where a and b are material constants and f(r) is the quasi-static torsional stressstrain relation. Equation (3.2) may be rewritten in the form Gj,=K
[expcq)-l]
=g(r,y)
which implies that the plastic strain rate is a function of the over-stress. (1951) used a simpler form of (3.2) for calculation purposes, viz. g(r, 7) =
a~-.ml
(3.2) MALVERN
(3.3)
which is the widely-used linear form of the over-stress dependence. This equation, although simple in form, is not a realistic representation of experimental data obtained over a wide range of strain rates (LINDHOLMand BESSEY,1969). Equation (3.2) is a better representation of the data, giving a linear relation between stress and log (strain rate) for large strain-rates. This form, equation (3.2), was thus chosen to represent the experimental data obtained in this investigation because of the linear dependence on log (strain rate) as shown in Fig. 2. The values of K and a chosen to fit the data were a = 1860 psi and K = 560 psi s-l.
4.
PLASTICWAVE PROPAGATIONIN TITANIUM
4.1 Experimental results
The objective of this phase of the investigation was to perform a series of experiments on pure titanium in which plastic wave propagation effects would have to be considered and to compare the experimental results with theoretical predictions based on the constitutive equation derived from constant strain-rate data. To accomplish this, long thin-walled tubular specimens were subjected to large incidentstrain pulses in the torsional split Hopkinson bar apparatus. In essence, these amounted to Hopkinson bar tests of very long specimens where the assumptions of stress uniformity and neglect of wave propagation effects were invalidated. Using the largest strain-pulse possible in the input bars, torsional waves were propagated into long tubes of pure titanium machined from the same bar as the specimens used in the constant strain-rate tests. The tubes were attached in the same manner as the standard torsional Hopkinson bar specimens and were either 3.5 or 6.0 in. long and had wall thicknesses of 0.030 and 0.035 in. As with the Hopkinson bar specimens, accurate geometry was determined before each tube was used. The first two tubes were subjected to a torsional pulse in an unstressed initial condition. A static twisting device consisting of a lever arm and pully arrangement to apply dead weight torque was used to pre-stress the remaining four specimens. Two were prestressed to 2.5 x IO4 psi and the last two to 3.2 x IO4 psi before being subjected to a torsional pulse. These values of pre-stress were chosen as points just below and just above the yield point of the niaterial. The static curve was taken to be that obtained at a strain rate of 10m4 s -I as shown in Fig. 1. The stress-time history at the strain gage station on the output bar was displayed on an oscilloscope along with the incident strain pulse and the results recorded with the aid of a Polaroid camera. The experimental data were plotted as stress against time for each of the six tests and are shown as the broken lines in Figs. 3-8.
70
J. E. LAWSONand T. NICHOLAS ~~-r-----, ""I--
1
20
--O-O-
TIME,
Fro. 3. Thor&al
THEORY EXPERIMENT
MlC~O6ECO~DS
and experimentat stress-time profile for transmitted pulse; f = 3.5 in., pR4tM!4.
60
20
-e-e-
I
40
Fto. 4.
I
60
THEORY EXFERIYENT
I
t
I
IM
I60
2op
-
I
240
I
,260
Theoretical and experimental stress-time profile for transmitted pulse; i = 3-5 in., 2.5 x 10’ psi prc-3trcss.
4.2 Theomtical predictions The equations of motion and ~ntin~ty and the ~o~stitutive equation for the propagation of plane torsional waves in a uniform, homogeneous rod in L,angrangian form are dr au z=przP (4.1) (4.2) (4.3)
71
The dynamic mechanical behavior of titanium in shear
--O-O-
01 0
4e
TIME. FIG.
5.
Theoretical
OY
0
40
00
FIG. 6.
where strain, using which
and experimental
200
240
200
YICROSECONOS
and experimental stress-time profile 3.2 x 10’ psi prc-stress.
I20 TIME,
Theoretical
160
120
80
THEORY EXPERIYENT
160
for transmitted
pulse;
I
I
I
200
240
ZOO
I = 3.5 in.,
WCROSECONDS
stress-time profik zero pn3stress.
for transmitted
pulse;
I = 6a
in.,
p is the density, o the angular velocity, r the radius of the tube, y the shear and 9 is given by (3.2). The solution to these equations is obtained numerically the well-known method of characteristics (HOPKINS, 1968). The relations apply along the characteristic directions are dr-pcrdw+gdt=O dt+pcrdw+gdt=O
G dy - dr - g dt = 0
alongdz=+cdr, alongdz=-cdt, along dz = 0.
I
The solution is carried out in a step-by-step numerical integration procedure in the characteristic z-f plane in the usual manner. The mesh size was taken small enough in each case to assure convergence and stability of the numerical scheme.
72
J. E. LAWSON and T. NICHOLAS
60 -
0 FIG.
7.
Theoretical
40
and
60
experimental
I20 160 200 TIME, MICROSECONDS stress-time
profile
for
240
transmitted
260 pulse;
1 = 6.0 in.,
2.5 X 10’ psi pre-stress.
0 8.
Theoretical
and
THEORY EXPERIMENT
I
I
I
I
40
60
I20
160
TIME. FIG.
--e---c--
I
200
I
240
I 260
MICROSECONDS
experimental stress-time profile 3.2 x lo4 psi pre-stress.
for
transmitted
pulse;
1 = 6.0
in.
This was assured by obtaining solutions with one-half the previous mesh-size until the results at the most distant computation point from the origin agreed to within five significant figures. At each computation point the solution had to be obtained by an iterative procedure. Convergence of the iterative scheme was assured when two successive computations agreed to one part in 106. The computations were carried out on a CDC 6600 digital computer. The solution consisted of the values of stress, strain, angular velocity and the function g(z, JJ) at the two ends of the tube. The solution was carried out to 280 us starting from the first deformation of the incident end of the tube. The incident strain pulse was taken to be as close to the actual pulse used in the experiments as possible. A ramp rise of 40 us and an incident strain level of 2.625 x 10m3 (lo4 psi stress in the
The dynamic mechanicalbehavior of titanium in shear
73
incident bar) was used. The specimen tube was subjected to neither a prescribed force nor a prescribed displacement boundary condition but rather the actual material/ geometrical mismatch of elastic bars and specimen was considered in the solution. Thus, the reflection in the incident bar was taken into consideration and the theoretical and experimental results could be compared directly. As in the experiments, tube lengths of 3.50 and 6.00 in. were used. Again, two specimens were assumed to be un-prestressed, two at 2.5 x IO4 psi pre-stress and two at 3.2 x lo4 psi pre-stress. The static stress-strain curvef(y) for the unstressed case was taken to be the lower curve in Fig. 1. For this a linear relation up to 2.5 x lo4 psi was used followed by a power-law function which closely approximated the static curve from this point out to 3.2 x lo4 psi. Beyond this point another straight line was used with the appropriate shallow slope. For the cases where the specimen was pre-stressed, the static stress-strain curve was taken as the portion of the actual curve beyond the prestress, i.e. the point of pre-stress was taken as the origin. A comparison of theoretical and experimental results is presented in Figs. 3-8 for the six cases considered. The stress values plotted are those recorded by the strain gages on the output bar. It can be seen that very good agreement was obtained in all cases. The theoretical solutions show well deGned steps as the stress builds up while the experimental results show rounded-off steps. This is due to the fact that a strict ramp-type pulse was considered in the theoretical study while the actual pulse produced on the Hopkinson bar had a rounded-off rise. The predicted values of the flow stresses agreed extremely well with the experimental results in all cases. 4.3 Sensitivity to variations in constitutive equation From the computer results for the 35 in. specimens the plastic strain rates for the two specimen ends were plotted against time in Fig. 9 for the cases of zero prestress and 3.2 x IO4 psi pre-stress. It can be seen that initially all the deformations are elastic and 9, is zero. After this the plastic strain rate increases rapidly and after several small oscillations remains approximately constant. Thus most of the plastic deformation takes place within a narrow band of strain rates. For the specimen with zero pre-stress, the equilibrium plastic strain rate was approximately 9, = 50 s-l. Referring to Fig. 2 the flow stress corresponding to 9, = 50 s -’ is approximately 6.2 x IO4 psi. It might be expected that changing the slope of the line through this point would not have a very large effect on the plastic wave prediction since most of the plastic deformation takes place in a narrow band of strain rates around this point. In the expression for &, equation (3.2), it can be seen that the parameter a represents the slope of the line on the stress vs. In f, plot. The value of a obtained in Section 4.2 was 1860 psi. To investigate the effect of changing the constitutive equation, this value was doubled and then halved in order to obtain equations for fictitious materials having different degrees of strain rate sensitivity. The equations are shown graphically in Fig. 10 as stress vs. log (strain rate) for a constant strain. The value of K was adjusted in each case so that the new equation would still go through the original point (rs = 6.25 x lo4 psi and f, = 50 s -I). Thus, the equations were matched at one particular strain rate only. In addition, one computation was carried out which approximated the strain-rate-independent theory, but again for a material having the same stress (6.25 x IO4 psi) at 10 per cent strain. In this case, 6
J. E. LAWWIN and T. NICHOLAS
Id-
OI,
2 I
i 1
IO -
a-
I
I
C-
t
4-
1
I
2-
I 0
FIG. 9. Plastic strain rate history for 3.5 in. specimen;
FIG. 10. Stress vs. log (strain rate) at umtant
zero and 3.2 x l(r psi pre4ms.s.
strain for constitutivc equations considered.
The dynamic mechanical behavior of titanium in shear ‘C ?O-
60-
0
FIG. 11. Theoretical
40
stress-time
60
-___-
m.3720
---_
a.930
-----
‘RblE-MDEPENMNT’
I20
160
TIME.
MlCROSiCONDS
K = 4.13 X Id
200
240
260
profiles for transmitted pulse for various constitutive I = 3-S in., zero pre-stress.
equations;
the stress would be the same for all strain rates; thus, the strain-rate-independent theory is represented by a horizontal line in Fig. 10. This would be a limiting case where a + 0 and K -+ co in the rate-dependent theory; however, taking a to be zero would lead to computational difficulties. To avoid such difficulties and to avoid additional rate-independent theory computations, the value of a was taken to be small, K was taken to be large, and the computations were carried out with the ratedependent theory. In essence, this computation was for a material having an extremely small degree of rate dependence with a “static” curve, f(r), corresponding to the dynamic stress-strain curve for titanium at a constant strain rate of 50 s -‘. The results of these computations are presented in Fig. 11 for the 3.5 in. tube in the initially unstressed condition. The solid line represents the stress-time profile of the specimen end adjacent to the output bar using the fitted constitutive equation. The dashed lines which are asymptotic to the solid curve represent results obtained by doubling and by halving respectively the constant a in the equation. It can be seen that the solutions are changed very little by varying the constitutive equation as depicted in Fig. 10. The solution which approximates the strain-rate-independent theory is shown as a dashed line in Fig. 11. Again, it can be seen that there is not a large difference between this and the original solution. This solution is for a rateindependent material having a “static” stress-strain curve nearly twice the amplitude of the actual static curve for the same strain. However, all the constitutive equations predict the same stress at a strain rate of 50 s -’ which is the value near which most of the plastic deformation occurs. Thus, the results appear to be sensitive to the value of stress at only one particular strain-rate or range of strain-rates and not necessarily to the form of the constitutive equation. 5.
DISCUSSION AND CONCLUSIONS
The propagation of torsional plastic waves in pure titanium can be predicted from the rate-dependent theory of plastic wave propagation utilizing a constitutive
J. E. LAWSON and T. NICHOLAS
76
equation which fits experimental data from constant strain-rate tests. The excellent agreement between theoretical and experimental results for stresses at the end of a long tube subjected to a torsional pulse demonstrates the consistency of the entire testing procedure. However, it cannot be concluded that the constitutive equation derived is unique in predicting the results. The investigation has shown that theoretical predictions of experiments involving plastic wave propagation and interaction are not very sensitive to changes in the constitutive equation governing rate-dependent plastic flow. The fact that most of the plastic deformation in these experiments took place within a narrow band of strain rates readily explains the insensitivity of the results to the form of the constitutive equation as long as the stresses at this narrow band of strain rates are the same. Thus, even the rate-independent theory gave good agreement with experiments by using an apparent static stress-strain curve considerably higher than the actual one. One should thus be careful about drawing conclusions concerning the form of the relation describing the material from the results of plastic wave propagation experiments in which the deformations occur largely in a narrow range of strain rates. It can be concluded, therefore, that the only type of experiment involving plastic wave propagation from which one could expect to derive a unique constitutive equation would be one involving a wide range of strain rates during the plastic deformation. It would appear to be more reliable and easier to deduce constitutive equations from the results of constant strain-rate tests obtained over a broad range of strain rates. The torsional split Hopkinson bar has been shown to be a most useful tool for obtaining such data at high strain-rates. REFERENCES BAKER, W. E. BELL, J. F. C~tsrrscu,
and YEW, C. H.
N.
1966 1968 1967
J. appl. Mech. 33, 917. The Physics of Large Deformation of Crystalline Solids. Springer, New York. Dynamic Plasticity, p. 112. Wiley, New York.
DUFFY, J., CAMPBELL,J. D. and HAWLEY, R. H. HOPKINS,H. G.
1971 1968
KOLSKY, H. LINDHOLM,U. S. and BE~~EY,R. L.
1949 1969
LUDWIG, P. MALVERN,L. E. NICHOLAS,T. NICHOLAS,T. and LAWSON,J. E. RIPPERGER,E. A. and WA’ISON,H.
1909 1951 1971 1972 1968
WOOD, E. R. and PHILLIPS,A.
1967
J. appl. Mech. 38, 83. Engineering Plasticity, (Edited by HEXMAN, J. and LECKIE. F. A.), p. 277. bridge University Press.
Cam-
Proc. Phys. Sot. B62, 676. Air Force Materials Laboratory Report No. AFML-TR-69-119, Wright-Patterson AFB, Ohio. Z. Phys. lo,41 1.
J. appl. Mech. 18, 203. Exp. Mech. 11, 370. J. Mech. Phys. Solids Zo, Mechanical Behavior of Materials under Dynamic Loaa!~, (Edited by LMDHOLM, U. S.). Springer, New York.
J. Mech. Phys. Soliris 15, 241.
THE DYNAMIC
Pergamon Press.
Printed in Great Britain.
MECHANICAL BEHAVIOR OF TITANIUM IN SHEAR By J. E. LAWSONand T. NICHOLAS
Air Force Materials Laboratory,
Wright-Patterson
Air Force Base, Ohio
(Received 31sr August 1971)
CONSTANT strain-rate
stress-strain data in shear for commer&aIly pure titanium are obtained through the use of a torsional split Hopkinson pressure-bar and a pneumatic-mechanical torsion testing machine. These data, which cover a range in shear strain rates from lo-* to 5 x 103 s-r, are used to obtain the constants in a constitutive equation which expresses the dynamic over-stress as a function of the logarithm of the plastic strain rate. The equation is used to predict the results of torsional plastic wave propagation experiments using the rate-dependent theory of L. E. Malvern. The sensitivity of the results to the constants in the constitutive equation is studied along with predictions using the mt~~de~ndent theory and an elevated static stress+&ain curve.
1.
I~~DLJCTION
STRAIN-RATE effects in metals have been studied extensively by numerous investigators over the past two decades. Both direct and indirect methods have been employed by ex~rnen~~s~ to determine the response of materials to dynamic loading. In the first, constant strain-rate tests on short specimens provide stress-strain curves at high strain-rates which can be compared to the quasi-static behavior of the material. In the second, the propagation of plastic stress waves down long bars or rods is used to deduce a constitutive equation for the material. In either technique there is usually insurgent info~ation genemt~ to characterize uniquely the behavior of the material as a function of strain rate. Constant strain-rate tests in uniaxial stress at high rates-of-strain are performed using the Kolsky technique (KOLSKY,1949), more commonly referred to as the split Hopkinson pressure-bar apparatus. The range in strain rates which is encompassed by the technique is roughly 102-IO4 s--i. Relatively little information has been generated at inte~ediate strain rates below lo2 and above 10 -’ s -’ because of the scarcity of experimental apparatus which operates in this range. Conventional testing machines are used to obtain data at strain rates below 10-l s-l. In order to characterize fully the behavior of a material as a function of strain rate it is necessary to have constant strain-rate data over a wide range of strain rates including intermediate regions. Plastic wave pro~ga~ion studies suffer from the necessity of assuming, a priori, a form for the constitutive equation describing the behavior of the material. Theoretical predictions based on various forms of plastic wave propagation theories have been shown to be quite similar in several respects. RIPPBBGBB and WATSON (1968) demon65
66
J.
E. LAWSONand T. NICHOLAS
strated by computer analysis that the velocity of propagation of plastic waves is not very sensitive to the form of the constitutive equation proposed for the material. WOOD and PHILXZS (1967) have shown analyti~lly that the constant strain plateau at the impact end of a bar as predicted by the rate-independent theory of plastic wave propagation is likewise predicted by the rate-dependent theory of MALVERN (1951). It is apparent, then, that a plastic wave propagation experiment can, at most, verify a proposed constitutive equation; it cannot determine it uniquely. If the rate-independent theory of plastic wave propagation is used, a single dynamic stress-strain curve may be deduced which may, in general, differ from the quasistatic curve. In the rate-dependent theory, as proposed by MALVERN(195t), any one of a number of forms of the constitutive equation may be assumed expressing the plastic strain rate as a function of stress and strain. The linear form, most commonly used, is not particularly realistic in describing the results of constant strain-rate tests although it lends itself to simpler computations. Another point which has raised questions in the interpre~tion of data obtained in dynamic testing is the use of the compression (or tension) mode of deformation where problems such as those of barreling (or necking) and radial inertia and radial shear effects are known to exist. The use of the torsional mode of deformation applied to thin-walled tubular specimens has been introduced 1to eliminate these potential problems (BAKER and YEW, 1966, DUFFY, CAMPBELLand HAWLEY,1971, and NICHOLASand LAWSON,1972). The final point which may be the cause of much of the controversy in dynamic plasticity regarding the existence of strain-rate effects in metals is the widespread use of relatively rate-insentitive materials in experimental investigations. Materials such as pure aluminum have been used extensively to study strain-rate effects or to validate the rate-de~ndent (or rate-i~de~ndent) theory of plastic wave propagation (BELL, 1968). The fact that the rate-dependent and rate-independent theories predict virtually the same wave profiles, especially when the degree of rate-dependence is small, leads to some uncertainty in relating transient wave profiles to a specific constitutive equation. The present investigation describes the application of existing experimental techniques to characterize the behavior of a highly rate-sensitive material under constant strain-rate loading conditions and to study subsequently the propagation of plastic waves in that material using a constitutive equation derived directly from the results of the constant strain-rate tests. The behavior of pure titanium is studied in pure shear using a torsional split Hopkinson pressure-bar apparatus and a mechanicalpneumatic machine which, together, allow stress-strain data to be obtained over A ~onstitutive equation which fits the eight orders of ma~tude in strain-rate. experimental data is used to predict the propagation of plastic waves in long tubes. Finally, the sensitivity of the results of the wave propagation experiments to the constants, and to the form of the constitutive equation, is investigated. 2.
E~~I~~~AL
APPARATUS
The torsional split Hopkinson pressure-bar apparatus used in this investigation to obtain stress-strain dam at high rates of strain has been described in detail by NICHOLASand LAWSON(1972). The device operates by releasing a torque stored
The dynamicmecha.nicalbehaviorof titaniumin shear
67
in a length of bar through the rapid fracture of a bolt which holds a clamp in place.
Specimens of thin-walled tubular cross-section are attached to the input and output bars by means of an epoxy adhesive. Through the use of specimens having gage lengths as short as 0.050 in., reliable stress-strain data can be obtained at shear strain rates up to nearly lo4 s -‘. Quasi-static and intermediate strain-rate tests were performed on a combined mechanical-pneumatic machine described by NICHOLAS(1971). Strain rates from lo4 to 10e2 s-l can be achieved by driving a screw-jack attached to a lever arm with a motor. Strain-rates up to IO2 s -’ are obtained by driving a piston attached to another lever arm through the release of compressed gas by a quick-operating valve. The torque is measured with a conventional strain gage load cell while angle of rotation is recorded through the use of a differential capacitor activated by a 10 kH carrier signal which produces an analog signal proportional to the rotation of the capacitor. Specimens of commercially pure 50A titanium were machined from a single bar to minimize the possible variation in material properties. All specimens were thinwalled hollow tubes having a O-5in. inside diameter and a wall thickness of nominally O-025 in. All wall thicknesses were accurately measured and specimens having variations of more than O*OOlin. were rejected. Gage lengths for specimens varied from O-5 to 0.05 in. Both the torsional split Hopkinson pressure-bar and pneumatic-mechanical machine were calibrated using dead weight loading. The split Hopkinson pressurebar was also calibrated dynamically by propagating pulses of known magnitude down the input and output bars attached to one another using an epoxy adhesive. 3.
CONSTANTSTRAIN-RATEBEHAVIOR
3.1 Experimental results Torsional split Hopkinson bar tests were performed on twelve specimens of 50A titanium covering a range of strain rates from 191 to 5000 s-l. The results of those tests along with the quasi-static and dynamic test results obtained using the mechanical-pneumatic machine are presented in Fig. 1. The amount of scatter in data obtained at any given strain rate was generally less than 5 per cent and was within the experimental accuracy. The results show that this material displays a de&rite strain-rate sensitivity over the entire range of strain rates encountered in the tests. The data are re-plotted as stress vs. log (strain rate) for 10 per cent strain and presented in Fig. 2. The results show that the flow stress at a given strain depends linearly upon the log (strain rate) over the entire range of strain rates. It can also be seen from this plot that very little deviation from this linear law was found in the data. The consistency of the data is clearly shown in not only the small amount of scatter but also from the nearly identical results obtained from both the lowest strain-rate Hopkinson bar tests and the highest strain-rate tests on the pneumatic machine. It is significant to note also that these data obtained at overlapping strain rates were obtained from long gage length specimens on the Hopkinson bar and very short gage length specimens on the pneumatic machine. This is additional evidence of the lack of any apparent gage length effect in the test results (NICHOLASand LAWSON,1972).
J. E.
LAWSON and
T.
NICHOLAS
IO I
0'
2
0
1
I
4
I
6
FIG. 1. Shear stress-strain
8
lo
SILUI
STRAIN.%
12
I4
m
curve-s for commercially
IO
pure titanium
20
50-A.
80
70 ‘; .s ui
60
ii K z5
50
TEST
5
METHOD
HOPKINSON
BAR
PNEUMATIC 40
30 10-4
IO“
10-l
10-2
SHEAR
I STRAIN
IO’
IO RATE,
IO’
IO’
sac-’
FIG.2. Stress vs. log (strain rate) at 10 per cent strain. 3.2 Constitutive
equation
Many authors have proposed various constitutive equations to describe the strain-rate dependent behavior of materials. LUDWIK(1909) postulated a logarithmic equation of the form r = tl + 7. In ($&J, (3.1) where 71, 7,, and qO are material constants and p, is the plastic strain rate. This relation, which has been verified experimentally, has been the basis of numerous proposed constitutive equations to describe rate effects (C~~~TJBCU,1967). MALVERN (1951) proposed the use of an expression 7 =f(y)+
a In (1 + bj,),
(3.2)
The dynamic mechanical behavior of titanium in shear
6P
where a and b are material constants and f(r) is the quasi-static torsional stressstrain relation. Equation (3.2) may be rewritten in the form Gj,=K
[expcq)-l]
=g(r,y)
which implies that the plastic strain rate is a function of the over-stress. (1951) used a simpler form of (3.2) for calculation purposes, viz. g(r, 7) =
a~-.ml
(3.2) MALVERN
(3.3)
which is the widely-used linear form of the over-stress dependence. This equation, although simple in form, is not a realistic representation of experimental data obtained over a wide range of strain rates (LINDHOLMand BESSEY,1969). Equation (3.2) is a better representation of the data, giving a linear relation between stress and log (strain rate) for large strain-rates. This form, equation (3.2), was thus chosen to represent the experimental data obtained in this investigation because of the linear dependence on log (strain rate) as shown in Fig. 2. The values of K and a chosen to fit the data were a = 1860 psi and K = 560 psi s-l.
4.
PLASTICWAVE PROPAGATIONIN TITANIUM
4.1 Experimental results
The objective of this phase of the investigation was to perform a series of experiments on pure titanium in which plastic wave propagation effects would have to be considered and to compare the experimental results with theoretical predictions based on the constitutive equation derived from constant strain-rate data. To accomplish this, long thin-walled tubular specimens were subjected to large incidentstrain pulses in the torsional split Hopkinson bar apparatus. In essence, these amounted to Hopkinson bar tests of very long specimens where the assumptions of stress uniformity and neglect of wave propagation effects were invalidated. Using the largest strain-pulse possible in the input bars, torsional waves were propagated into long tubes of pure titanium machined from the same bar as the specimens used in the constant strain-rate tests. The tubes were attached in the same manner as the standard torsional Hopkinson bar specimens and were either 3.5 or 6.0 in. long and had wall thicknesses of 0.030 and 0.035 in. As with the Hopkinson bar specimens, accurate geometry was determined before each tube was used. The first two tubes were subjected to a torsional pulse in an unstressed initial condition. A static twisting device consisting of a lever arm and pully arrangement to apply dead weight torque was used to pre-stress the remaining four specimens. Two were prestressed to 2.5 x IO4 psi and the last two to 3.2 x IO4 psi before being subjected to a torsional pulse. These values of pre-stress were chosen as points just below and just above the yield point of the niaterial. The static curve was taken to be that obtained at a strain rate of 10m4 s -I as shown in Fig. 1. The stress-time history at the strain gage station on the output bar was displayed on an oscilloscope along with the incident strain pulse and the results recorded with the aid of a Polaroid camera. The experimental data were plotted as stress against time for each of the six tests and are shown as the broken lines in Figs. 3-8.
70
J. E. LAWSONand T. NICHOLAS ~~-r-----, ""I--
1
20
--O-O-
TIME,
Fro. 3. Thor&al
THEORY EXPERIMENT
MlC~O6ECO~DS
and experimentat stress-time profile for transmitted pulse; f = 3.5 in., pR4tM!4.
60
20
-e-e-
I
40
Fto. 4.
I
60
THEORY EXFERIYENT
I
t
I
IM
I60
2op
-
I
240
I
,260
Theoretical and experimental stress-time profile for transmitted pulse; i = 3-5 in., 2.5 x 10’ psi prc-3trcss.
4.2 Theomtical predictions The equations of motion and ~ntin~ty and the ~o~stitutive equation for the propagation of plane torsional waves in a uniform, homogeneous rod in L,angrangian form are dr au z=przP (4.1) (4.2) (4.3)
71
The dynamic mechanical behavior of titanium in shear
--O-O-
01 0
4e
TIME. FIG.
5.
Theoretical
OY
0
40
00
FIG. 6.
where strain, using which
and experimental
200
240
200
YICROSECONOS
and experimental stress-time profile 3.2 x 10’ psi prc-stress.
I20 TIME,
Theoretical
160
120
80
THEORY EXPERIYENT
160
for transmitted
pulse;
I
I
I
200
240
ZOO
I = 3.5 in.,
WCROSECONDS
stress-time profik zero pn3stress.
for transmitted
pulse;
I = 6a
in.,
p is the density, o the angular velocity, r the radius of the tube, y the shear and 9 is given by (3.2). The solution to these equations is obtained numerically the well-known method of characteristics (HOPKINS, 1968). The relations apply along the characteristic directions are dr-pcrdw+gdt=O dt+pcrdw+gdt=O
G dy - dr - g dt = 0
alongdz=+cdr, alongdz=-cdt, along dz = 0.
I
The solution is carried out in a step-by-step numerical integration procedure in the characteristic z-f plane in the usual manner. The mesh size was taken small enough in each case to assure convergence and stability of the numerical scheme.
72
J. E. LAWSON and T. NICHOLAS
60 -
0 FIG.
7.
Theoretical
40
and
60
experimental
I20 160 200 TIME, MICROSECONDS stress-time
profile
for
240
transmitted
260 pulse;
1 = 6.0 in.,
2.5 X 10’ psi pre-stress.
0 8.
Theoretical
and
THEORY EXPERIMENT
I
I
I
I
40
60
I20
160
TIME. FIG.
--e---c--
I
200
I
240
I 260
MICROSECONDS
experimental stress-time profile 3.2 x lo4 psi pre-stress.
for
transmitted
pulse;
1 = 6.0
in.
This was assured by obtaining solutions with one-half the previous mesh-size until the results at the most distant computation point from the origin agreed to within five significant figures. At each computation point the solution had to be obtained by an iterative procedure. Convergence of the iterative scheme was assured when two successive computations agreed to one part in 106. The computations were carried out on a CDC 6600 digital computer. The solution consisted of the values of stress, strain, angular velocity and the function g(z, JJ) at the two ends of the tube. The solution was carried out to 280 us starting from the first deformation of the incident end of the tube. The incident strain pulse was taken to be as close to the actual pulse used in the experiments as possible. A ramp rise of 40 us and an incident strain level of 2.625 x 10m3 (lo4 psi stress in the
The dynamic mechanicalbehavior of titanium in shear
73
incident bar) was used. The specimen tube was subjected to neither a prescribed force nor a prescribed displacement boundary condition but rather the actual material/ geometrical mismatch of elastic bars and specimen was considered in the solution. Thus, the reflection in the incident bar was taken into consideration and the theoretical and experimental results could be compared directly. As in the experiments, tube lengths of 3.50 and 6.00 in. were used. Again, two specimens were assumed to be un-prestressed, two at 2.5 x IO4 psi pre-stress and two at 3.2 x lo4 psi pre-stress. The static stress-strain curvef(y) for the unstressed case was taken to be the lower curve in Fig. 1. For this a linear relation up to 2.5 x lo4 psi was used followed by a power-law function which closely approximated the static curve from this point out to 3.2 x lo4 psi. Beyond this point another straight line was used with the appropriate shallow slope. For the cases where the specimen was pre-stressed, the static stress-strain curve was taken as the portion of the actual curve beyond the prestress, i.e. the point of pre-stress was taken as the origin. A comparison of theoretical and experimental results is presented in Figs. 3-8 for the six cases considered. The stress values plotted are those recorded by the strain gages on the output bar. It can be seen that very good agreement was obtained in all cases. The theoretical solutions show well deGned steps as the stress builds up while the experimental results show rounded-off steps. This is due to the fact that a strict ramp-type pulse was considered in the theoretical study while the actual pulse produced on the Hopkinson bar had a rounded-off rise. The predicted values of the flow stresses agreed extremely well with the experimental results in all cases. 4.3 Sensitivity to variations in constitutive equation From the computer results for the 35 in. specimens the plastic strain rates for the two specimen ends were plotted against time in Fig. 9 for the cases of zero prestress and 3.2 x IO4 psi pre-stress. It can be seen that initially all the deformations are elastic and 9, is zero. After this the plastic strain rate increases rapidly and after several small oscillations remains approximately constant. Thus most of the plastic deformation takes place within a narrow band of strain rates. For the specimen with zero pre-stress, the equilibrium plastic strain rate was approximately 9, = 50 s-l. Referring to Fig. 2 the flow stress corresponding to 9, = 50 s -’ is approximately 6.2 x IO4 psi. It might be expected that changing the slope of the line through this point would not have a very large effect on the plastic wave prediction since most of the plastic deformation takes place in a narrow band of strain rates around this point. In the expression for &, equation (3.2), it can be seen that the parameter a represents the slope of the line on the stress vs. In f, plot. The value of a obtained in Section 4.2 was 1860 psi. To investigate the effect of changing the constitutive equation, this value was doubled and then halved in order to obtain equations for fictitious materials having different degrees of strain rate sensitivity. The equations are shown graphically in Fig. 10 as stress vs. log (strain rate) for a constant strain. The value of K was adjusted in each case so that the new equation would still go through the original point (rs = 6.25 x lo4 psi and f, = 50 s -I). Thus, the equations were matched at one particular strain rate only. In addition, one computation was carried out which approximated the strain-rate-independent theory, but again for a material having the same stress (6.25 x IO4 psi) at 10 per cent strain. In this case, 6
J. E. LAWWIN and T. NICHOLAS
Id-
OI,
2 I
i 1
IO -
a-
I
I
C-
t
4-
1
I
2-
I 0
FIG. 9. Plastic strain rate history for 3.5 in. specimen;
FIG. 10. Stress vs. log (strain rate) at umtant
zero and 3.2 x l(r psi pre4ms.s.
strain for constitutivc equations considered.
The dynamic mechanical behavior of titanium in shear ‘C ?O-
60-
0
FIG. 11. Theoretical
40
stress-time
60
-___-
m.3720
---_
a.930
-----
‘RblE-MDEPENMNT’
I20
160
TIME.
MlCROSiCONDS
K = 4.13 X Id
200
240
260
profiles for transmitted pulse for various constitutive I = 3-S in., zero pre-stress.
equations;
the stress would be the same for all strain rates; thus, the strain-rate-independent theory is represented by a horizontal line in Fig. 10. This would be a limiting case where a + 0 and K -+ co in the rate-dependent theory; however, taking a to be zero would lead to computational difficulties. To avoid such difficulties and to avoid additional rate-independent theory computations, the value of a was taken to be small, K was taken to be large, and the computations were carried out with the ratedependent theory. In essence, this computation was for a material having an extremely small degree of rate dependence with a “static” curve, f(r), corresponding to the dynamic stress-strain curve for titanium at a constant strain rate of 50 s -‘. The results of these computations are presented in Fig. 11 for the 3.5 in. tube in the initially unstressed condition. The solid line represents the stress-time profile of the specimen end adjacent to the output bar using the fitted constitutive equation. The dashed lines which are asymptotic to the solid curve represent results obtained by doubling and by halving respectively the constant a in the equation. It can be seen that the solutions are changed very little by varying the constitutive equation as depicted in Fig. 10. The solution which approximates the strain-rate-independent theory is shown as a dashed line in Fig. 11. Again, it can be seen that there is not a large difference between this and the original solution. This solution is for a rateindependent material having a “static” stress-strain curve nearly twice the amplitude of the actual static curve for the same strain. However, all the constitutive equations predict the same stress at a strain rate of 50 s -’ which is the value near which most of the plastic deformation occurs. Thus, the results appear to be sensitive to the value of stress at only one particular strain-rate or range of strain-rates and not necessarily to the form of the constitutive equation. 5.
DISCUSSION AND CONCLUSIONS
The propagation of torsional plastic waves in pure titanium can be predicted from the rate-dependent theory of plastic wave propagation utilizing a constitutive
J. E. LAWSON and T. NICHOLAS
76
equation which fits experimental data from constant strain-rate tests. The excellent agreement between theoretical and experimental results for stresses at the end of a long tube subjected to a torsional pulse demonstrates the consistency of the entire testing procedure. However, it cannot be concluded that the constitutive equation derived is unique in predicting the results. The investigation has shown that theoretical predictions of experiments involving plastic wave propagation and interaction are not very sensitive to changes in the constitutive equation governing rate-dependent plastic flow. The fact that most of the plastic deformation in these experiments took place within a narrow band of strain rates readily explains the insensitivity of the results to the form of the constitutive equation as long as the stresses at this narrow band of strain rates are the same. Thus, even the rate-independent theory gave good agreement with experiments by using an apparent static stress-strain curve considerably higher than the actual one. One should thus be careful about drawing conclusions concerning the form of the relation describing the material from the results of plastic wave propagation experiments in which the deformations occur largely in a narrow range of strain rates. It can be concluded, therefore, that the only type of experiment involving plastic wave propagation from which one could expect to derive a unique constitutive equation would be one involving a wide range of strain rates during the plastic deformation. It would appear to be more reliable and easier to deduce constitutive equations from the results of constant strain-rate tests obtained over a broad range of strain rates. The torsional split Hopkinson bar has been shown to be a most useful tool for obtaining such data at high strain-rates. REFERENCES BAKER, W. E. BELL, J. F. C~tsrrscu,
and YEW, C. H.
N.
1966 1968 1967
J. appl. Mech. 33, 917. The Physics of Large Deformation of Crystalline Solids. Springer, New York. Dynamic Plasticity, p. 112. Wiley, New York.
DUFFY, J., CAMPBELL,J. D. and HAWLEY, R. H. HOPKINS,H. G.
1971 1968
KOLSKY, H. LINDHOLM,U. S. and BE~~EY,R. L.
1949 1969
LUDWIG, P. MALVERN,L. E. NICHOLAS,T. NICHOLAS,T. and LAWSON,J. E. RIPPERGER,E. A. and WA’ISON,H.
1909 1951 1971 1972 1968
WOOD, E. R. and PHILLIPS,A.
1967
J. appl. Mech. 38, 83. Engineering Plasticity, (Edited by HEXMAN, J. and LECKIE. F. A.), p. 277. bridge University Press.
Cam-
Proc. Phys. Sot. B62, 676. Air Force Materials Laboratory Report No. AFML-TR-69-119, Wright-Patterson AFB, Ohio. Z. Phys. lo,41 1.
J. appl. Mech. 18, 203. Exp. Mech. 11, 370. J. Mech. Phys. Solids Zo, Mechanical Behavior of Materials under Dynamic Loaa!~, (Edited by LMDHOLM, U. S.). Springer, New York.
J. Mech. Phys. Soliris 15, 241.