- Email: [email protected]
A viscoelastic representation of the mechanical behavior of RVD graphite at elevated temperatures
Carbon
1971. Vol. 9. pp. 327-340.
Pergamon
Press.
Printed
in Great
Britain
A VISCOELASTIC REPRESENTATION OF THE MECHANICAL BEHAVIOR OF RVD GRAPHITE AT ELEVATED TEMPERATURES* C. E. PUGH
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830 (Received 21 October 1970)
Abstract-A set of constitutive equations is developed to describe the time-dependent mechanical behavior of RVD graphite at temperatures from 3500 to 5000°F. The relations are based upon a transversely isotropic linear viscoelastic model with temperaturedependent parameters. This model is shown to be applicable for stress levels up to onehalf the nominal ultimate stress values for the material. Predictions of these equations are compared with experimental measurements from short-time stress-strain tests, creep tests, and relaxation tests. 1. INTRODUCTION Studies various
of
graphites
temperatures ous
time-dependent
authors.
over
behavior
a range
of elevated
have been reported Many
of these
temperature dependent. A transversely isotropic linear viscoelastic model is shown to characterize this mechanical behavior over a limited stress range. A set of temperaturedependent constitutive equations applicable over this temperature range is developed and shown to describe the mechanical response of the material to a variety of loadings. Experimental results are used both to numerically evaluate parameters and to verify the resulting mathematical model. First, the material is characterized as being mechanically transversely isotropic. The time-dependent behavior is next shown to have a linear dependence on stress over a limited stress range. The forms of the constitutive equations for a transversely isotropic linear viscoelastic material are then formulated in terms of hereditary integrals which contain five independent material properties -three strain ratios (Poisson’s ratios) and two creep functions. The three strain ratios are assumed independent of time, stress, and strain, but are temperature dependent. Thus, the time-dependent character of the mechanical behavior is accounted for by two creep functions. The experimental data used in this study
of
by numer-
investigations
are surveyed and discussed by Reynolds [l] and by Greenstreet [2]. The reported studies, generally, have addressed themselves to the mathematical representation of the response of a given material to a single type of mechanical loading, such as uniaxial creep, relaxation, or recovery. The literature contains very little information about efforts to verify these representations through comparisons of analytical predictions with experimental results for tests other than those used in the development of the models. This paper presents a unified mathematical characterization of the mechanical behavior of RVD graphite for temperatures ranging from 3500 to 5000°F. Within this temperature range, creep and relaxation mechanisms for graphite are operative and give rise to a mechanical behavior which is both time and *Research sponsored by the U.S. Atomic Energy Commission under contract with the Union Carbide Corporation. 327
C. E. PUGH
328
include the results of a larger number of tests performed at 4500°F than at any other temperature. Therefore, the parameters appearing in the constitutive equations are first evaluated at that temperature. Only data from with-grain and across-grain tensile tests and from with-grain creep tests are required for this evaluation. A temperature dependence is then introduced into the model by considering data from with-grain tensile and creep tests conducted at 3500, 4000, and 5000°F. The constitutive equations developed in this manner are then used to predict the results of tests other than those used for evaluation of parameters. This includes comparisons of predictions with data for across-grain creep, with-grain and acrossand short-time stressgrain relaxation, strain behavior for various strain rates. 2. CHARACTERIZATION
OF MATERIAL
BEHAVIOR RVD is a fine-grain graphite with a maximum particle size of O-015 in. It is also a molded graphite and, consequently, has
material properties which are orientation dependent. However, material properties are essentially independent of orientation within planes perpendicular to the molding direction. These planes are considered to be planes of mechanical isotropy and directions parallel to these planes are referred to as with-grain directions. Consequently, the direction normal to these planes, the molding direction, is referred to as the across-grain direction. This type of material symmetry defines a transversely isotropic material and places restrictions on the forms of the constitutive equations to be subsequently presented. The experimental program utilized in this study has been described in detail by Woodard [3] and included various uniaxial tests conducted as constant stress creep tests, stress relaxation tests, and short-time constant strain-rate tests. Tests were performed with both with-grain and across-grain specimens at four specific temperatures: 3500, 4000, 4500, and 5000°F. In this temperature range, deformation mechanisms are operative which make mechanical processes time dependent. This is illustrated by Fig. 1, which shows creep
0.0150
1
I----...
0.0125
0
10
20
30
40 PO TIME (mid
60
Fig. 1. With-grain tensile creep at 4500°F.
70
80
90
MECHANICAL
BEHAVIOR
OF RVD GRAPHITE
strains as functions of time for several individual with-grain specimens which were subjected to constant uniaxial stress values at 4500°F. The stress level for each test is indicated in psi by the number adjacent to the corresponding creep curve. The indicated stress values range up to approximately 75 per cent of the nominal with-grain ultimate strength for this temperature. In order to investigate the dependence of these creep curves upon the stress level, the ratio of creep strain per unit stress, which corresponds to a given creep time, is plotted versus stress. For example, Fig. 2 illustrates this type of plot when the creep period is t = 20 min for the tests shown in Fig. 1. This plot shows the ratio of with-grain creep strain to stress to be, within reasonable experimental scatter, constant with stress up to 3700 psi at 4500°F. This stress level is approximately
AT ELEVATED
TEMPERATURES
329
50 per cent of the nominal ultimate stress value for the with-grain direction of RVD graphite at 4500°F. A similar dependence of the creep strain upon the stress level was observed at the other test temperatures. Therefore, for stress levels up to 50 per cent of the nominal ultimate stress value, the creep strains have an approximately linear dependence upon the stress level. Beyond this stress level, the dependence becomes nonlinear. These observations suggest the mechanical behavior of this material can be characterized by a transversely isotropic linear viscoelastic model for this limited stress range. The model is considered to be temperature dependent, and the limiting stress value, below which the model is considered to be applicable, is taken to be one-half the nominal ultimate stress value corresponding to the
.
-I--
-c. -
30 TENSILE
Fig.
2. With-grain
35 STRESS
creep
40 C,,
x IO-*
strain/stress
4500°F.
45
50
(PSI)
vs. stress
at
55
u,,
330
C. E. PUGH
given temperature. Therefore, for both parameter evaluation and model verification, the only test results considered are those from tests conducted at stress levels below this limiting stress. 3. CONSTITUTIVE
EQUATIONS
The time-dependent strains l ij(t), (i, j = 1, which result from successive stress 2,3), increments Alou, A2crij,. . ., Ancrij applied at times tl, tz, . . ., tn, respectively, to a linear viscoelastic material, are equal to the linear superposition of the strains which would correspond to each stress increment if applied separately to the material. Therefore, the components of stress and strain are related in the following manner:*
x3 axis defines the direction normal to the plane of isotropy and is referred to as the across-grain direction. As shown by Rogers and Pipkin[5], there are only five components ofJ&t) which are independent of each other in the description of a transversely isotropic linear viscoelastic material. If the ratios of induced lateral strains to longitudinal strains for uniaxial states of stress are assumed to be independent of time, stress, and strain, then the constitutive equations for such a material at a constant temperature become
-cc13
where Jijrl (t) are creep compliance functions which characterize the time-dependent response of the material and t is the time at which the strains are evaluated. Therefore, as fully described by Fhigge [4], the strains which result from an arbitrary time-varying stress ukl (t) are represented by
lfj(t)
=
'J&t---r) I0
%dr,
of
a7
= ~JII(++$-PI.~ -pIa%]
e&t)
dr,
I’hi(t-d[
=E
(2)
where the integration variable T ranges over the history of the loading program. The initial values of the creep functions Jtirl(o) represent the elastic compliances. Since RVD is characterized to be a transversely isotropic material, the constitutive equations must be expressed in a form which exhibits an axis of rotational symmetry. The plane of isotropy is defined, here, by the rectangular Cartesian coordinates (x1, x2) and the directions parallel to this plane are referred to as with-grain directions. The *A repeated subscript implies summation terms over the range one to three.
Ed
1dr,
aa,,
3
(3)
cz3(t) =;
I0
’Jdd(t-T)
?dr,
where Jll (t) and JM(t) are independent creep functions and ecu are the ratios of the lateral strains induced in the q directions to the longitudinal strains for a uniaxial stress applied in the xi directions.
MECHANICAL
BEHAVIOR
OF RVD GRAPHITE
The inverse constitutive relations which give stresses in terms of strains have the form
AT ELEVATED TEMPERATURES
The relaxation functions creep functions by
are related
331
to the
t all(t) =
P
I
Gl,(FT)
?+a,?
and
[
0
(6) I
(4)
0’.14Wr)G&)
dr=
N(t),
where H(t) is the Heaviside unit step function. Five parameters, 511(t). JM(~). PID ~1~13, and ,u~~,must, therefore, be evaluated for the complete determination of these constitutive equations at a given temperature. In order to establish the constitutive equations over a range of temperatures, each of these five parameters is considered to be temperature dependent. The terms (Y~, cyz, cu,, and 8 are then determined directly from three of the five parameters as shown above. 4. EVALUATION
OF MATERIAL PARAMETERS
Jo where G,,(t) and Gd4(t) are relaxation functions and
*
independent
Jhz+cL13P31 1 I--P13P31
(y2-_
'
P31(1 +hJ l--1113P31
'
(5) P310
-&)
cY3= 11130 -Pl3cL31)'
The experimental data on which this study is based are sufficient for the complete determination of this model except for the creep function Je4(t) which describes the shear behavior in planes normal to the plane of isotropy. This description is sufficient, however, to allow predictions for a wide variety of problems which do not involve this type of shear behavior. Among these are plane strain, plane stress, and axisymmetric problems, when in each case the defining plane is the plane of isotropy. The strain ratios CL_~~ and pS1 were determined as functions of temperature from transverse and axial strain measurements for short-time constant strain-rate tests conducted at 3500, 4000, 45000 and 5000°F. The average values for each of these two strain ratios were fitted, in the sense of least squares, by linear functions of the absolute temperature (“R) and were expressed by /_b,.J = 0*31TX lo++
1.22 x lo-’
(7)
332
C. E. PUGH
and
to prescribed
stress levels which were then held constant for up to 50 minutes. The p31= 0.2663 x 10-4+ 2.25 x 1O-2. (6) loads were applied to these specimens by extending them at a constant strain rate Figure 3 shows a comparison of the average (O-02 min-‘). Therefore, the values for the measured values for pr2 and p31 and the creep functionJll (t) during this brief loading values given by Equations (7) and (8). time interval must be determined from The strain ratio pr3 is determined from constant strain-rate test data. p3i by using Equations (4) and (5) to show By specializing Equations (4) and (5) for a that the ratio of the stresses oll and cr33, constant strain-rate uniaxial loading prowhich correspond to the same strain level gram, the relaxation function G,,(t) can be for identical uniaxial loading programs in the evaluated for the loading time interval from x1 and x3 directions, respectively, takes on a either with-grain or across-grain tests by the constant value: relations
cll(t>
Figure 3 shows a plot of this ratio vs. from the temperature as determined average measured with-grain and acrossgrain stress-strain curves corresponding to constant strain-rate tests (0.02 min-‘) . The values of this ratio for the various temperatures were found to be essentially independent of the strain level. A parabolic fit was obtained to these data and is also shown in Fig. 3. The resulting expression for the strain ratio pi3 is p13 = 17.7T3 X 10-‘2x 10-4+
0.366,
15.2T2 X lo-‘+2.92T
du,,
K=
~13 du33 -cL31 dE33’
(11)
The average stress-strain curves for constant strain-rate (O-02 min-‘) tests for across-grain specimens at the temperatures of interest were described by an equation due to Woolley [7] which has the form u
33
=A(1
-e-kQ$
(12)
where A and k are material constants. For the specific constant strain rate of &33= 0.02 min-‘, Gus = 0*02t, and the relaxation function is given by Equations (11) and (12) to be
(10)
T is the absolute temperature (OR). * The uniaxial creep tests to be used in evaluating the creep function Jn (t) were conducted by loading with-grain specimens
=
G,,(t) = ~lfwy
(13)
where
*Due to the preferred orientation, the layer planes of the coke particles are parallel to the withgrain plane and possess a different chemical bonding mechanism from the interlayer or across-grain direction [6]. These different bonding mechanisms may explain why p13 exhibits a distinctly different temperature dependence from either p12 or pS1, as only p13 involves an induced strain in the across-grain direction.
where
t is measured
Equations
in minutes.
(6) and (13) then give the creep
function
for
this same limited time range as
Jll(t)
= f-J1
1
+0*02kt].
(14)
If t* is the time required to reach the desired stress level for a uniaxial creep test in the with-grain direction, then Jil (t) is given by Equation (14) for t s t*, and as can be shown from Equations (3) fort > TF by
MECHANICAL
BEHAVIOR
OF RVD GRAPHITE
0.2 r 5 r 90 ‘;$
333
TEMPERATURES
&.-
4,/-
N
AT EL.EVATED
-I,---0.i
-LINEAR l
(3500)
4460
4960
(4000)
(4500)
-
LINEAR l
0’
I 3960 (3500)
FIT
MEASURED 5460 (5000)
‘)A (OF)
FIT
MEASURED I I
I
1
l
(2
I
4460
4960
5460
“R
3960
(4000)
(4500)
(5000)
(‘F)
(3500)
TEMPERATURE
MEASURED I 1
i
4460
4960
5460
oR
(4000)
(4500)
(50001
(OF)
TEMPERATURE
Fig. 3. Strain ratios as functions of’temperature.
In Equation (15), lI1 ( t)cfeei, is the creep strain which has occurred from the time t* to the time t. The measured values used in Equation (15) for ~~,(t)~~~~,,/u~,(t*) correspond to the average creep per unit stress for all tests conducted at stress levels below one-half the nominal ultimate stress value. The value of t* is taken as the average time required to load these specimens to the desired stress levels by extending them at the prescribed constant strain rate (0.02 min-I). The entire creep function JII(Q is interpreted in terms of an elastic response, a viscous component, and a retarded elastic portion. At any given temperature, this description takes the mathematical form:
where E, v, Ai, and Ti are temperaturedependent material parameters. Since the data used in this study include the results of more tests conducted at 4500°F than at any other temperature, the creep function is first determined at this temperature. The first parameter E is taken as the initial slope of the with-grain stress-strain
curve corresponding to constant strain-rate tests (0.02 min-‘) and the coefficient of the second term as the slope of the creep strain per unit stress curve at the end of the data collection period (1= 50 min). The remaining three terms were fitted in the sense of least squares to the difference between J,l (t) data and the two contributions mentioned above. The resulting expression for the creep function at 4500°F is .I,, (t) = [0*467 + 0*0055t+
0.202 ( 1 - e-1s#.491)
$- 0~100 ( 1 - e-n’377f) +0=357(1
-e-“‘06(12f)] lO+(psi)-‘. (17)
Figure 4 illustrates the accuracy of this representation of JI1 (t) data by plotting the measured with-grain creep strain per unit stress values along with the results of Equations (15) and (17). The creep function developed for T = 4500°F is taken as a reference function and a temperature dependence introduced such that at the other temperatures (3500, 4000, and 5~OF) the values given by the modei for the initial modulus, for the duration of primary creep, and for the creep strain and creep strain rate at the end of the data collection periods for uith-grain tests all
C. E. PUGH
334 (x 10-9
-i
-
0.7
: ;
0.6
t: R
Ii
0.5
1
0.4
H g
0.3
9 /
rn
0.2
k 5
0.f
l MEASURED _-CURVE DESCRIBED CREEP FUNCTION
(9
BY
0 0
5
IO
(5
20
25 TIME
30
35
40
45
50
(min)
Fig. 4. RVD with-grain tensile creep at 4500°F. agree with the respective experimental observations. The temperature-dependent model developed in this manner subsequently was used to predict the across-grain creep behavior by the method outlined in the next section. At 3500°F considerable discrepancy was found betwen the model predictions and across-grain creep data. Therefore, in order to better represent the overall material behavior the model presented here requires the discrepancies between the model predictions and the average measured creep curves to be equal for both the with-grain and across-grain directions at 3500°F. With this modification, the resulting creep function is found to be J11(t, T) = $+A,&(1
-e-m31’71)
+ & [7)&t + A, ( 1 - e43t’T* ) +A3( 1 -e4at’ra)],
(18)
where the 4’s are functions of temperature and where E, r), Ai, and ri have the values given in Equation (17). Specifically, the 4 functions were determined to be
4,(T)
= 164TX
lo-4+0*186,
,&1(T) = 37.67e-19.6X10”lRT, 4,(T)
= 2.32e-r~58~103/Rr + 3.16 x lOlOe-135.1x103/RT
&(T)
= 63.47e-1s.7?X103’RL0.631;
44(T)
= 12.96 x 106 e-88+5x103/Rr,
(19)
where T is the absolute temperature in degrees Rankine and R is the universal gas constant. Since the reference creep function corresponds to 4500”F, all of the 4 functions have the value of unity at that temperature. Figures 5 and 6 show the comparison of the average measured with-grain creep per unit stress curves for T = 4000 and 5000”F, respectively, and the predictions of the model described above. Figure 7 illustrates the comparisons of the model predictions with both the measured with-grain and acrossgrain creep per unit stress curves at 3500°F. As discussed earlier, it is seen that the discrepancies between the two predicted curves and the two measured curves are equal in magnitude. It should be pointed out that the creep rates at 3500°F are small in comparison with those at the highest temperatures
MECHANICAL
335
BEHAVIOR OF RVD GRAPHITE AT ELEVATED TEMPERATURES
-PREDICTED
0
5
10
15 TIME (mm)
20
/
30
25
Fig. 5. RVD with-grain tensile creep at 4000°F.
studied. The actual discrepancy between the measured and predicted with-grain creep strain at t = 30 min is approximately O*OOS% per 1000 psi. The experimental data shown in Figs. 5 through 7 correspond to the average creep strain per unit stress curves for tests conducted at stress levels below one-half the nominal ultimate stress values for the respective temperatures.
5.COMPARISON
OF THEORY EXPERIMENTS
AND
Using the viscoelastic model developed in the previous section, predictions are made of the mechanical response of RVD graphite to tests other than those used in the model development. First, predictions are made of the uniaxial creep response of across-grain specimens. The constitutive equations predict the across-grain creep strain per unit stress to be E33 tt,
ncm, =
fl33(t*)
/43(T) a .
ull(t?
T)
--J11(t*,
T)
I> (20)
0
5
15
10 TlME
20
25
30
im,nl
Fig. 6. RVD with-grain tensile creep at 5000°F.
where t* is the time required to load the specimen to the prescribed stress level c:~:~(t*) . Figures 8 through 10 show the comparisons of these predictions with average experimental results from tests conducted at 4000, 4500, and 5000°F and at stress levels below one-half the respective nominal ultimate stress values. It should be noted that one of the ten across-grain specimens tested at 4500°F exhibited anomalous behavior. This specimen experienced no detectable creep deformation after about 8 min. The
336
C. E. PUGH
4,.
MEASURED
-
PREDICTED
i 0
5
10
(5 TIME
20
25
30
(min)
Fig. 7. RVD tensile creep at 3500°F. inclusion of the results of this test in the averaging of measured creep behavior account for some of the difference between the results shown in Fig. 9. The constitutive equations are next used to predict the stress decay for both with-grain and across-grain relaxation tests performed at 4500°F. Experimental tests were performed by first extending uniaxial specimens at a constant strain rate (0.02 min-‘) to prescribed strain levels which corresponded to
.-..
0
5
.-_
10
stress levels below one-half the nominal ultimate stress values. The strains were then held constant while the decreasing stresses were recorded. The constitutive equations predict the stress per unit strain for withgrain and across-grain specimens loaded in this manner to be
__
15 l-IME
20
25
iminI
Fig. 8. RVD across-grain tensiie creep at 4000°F.
MECHANICAL
BEHAVIOR
OF RVD GRAPHITE
AT ELEVATED
337
TEMPERATURES
. MEASURED -PREDICTED
_
O
4--
A
0
MEASURED
-
5
10
I5 TIME
--T-f
0
5
PREDICTED
20
25
30
(m(n)
Fig. 10. RVD across-grain tensile creep at 5000°F.
20
I5 TIME
25
30
(mln)
Fig. 9. RVD across-grain tensile creep at 4500°F. and
(21)
Figure 11 shows the comparisons of the predictions of Equations (21) and (22) with the average experimental relaxation curves for both with-grain and across-grain speciments tested at 4500°F. Stress-strain curves for both the withand across-grain directions were grain determined from uniaxial constant strainrate (0.02 min-‘) tests at 3500, 4000, 4500, (x106) 16
1.4
respectively, where tf and tz are the times required to load with-grain and across-grain specimens, respectively, to the prescribed strain levels. The relaxation function G,,(t) for ?’ = 4500°F is determined from Equations (6) and (17) to be
k
z =
0.8
k*
06
K
0.4
2
0.2
C;,,(t) = (0.653e-27’gR’+ 0.254e-0’446t +
0.3~gr-O~101t+
X
IO’psi.
J
0 0
@838e--o.004731)
(22)
2
4
6
8
10
TIME
(mln)
12
14
Fig. 11. RVD stress relaxation at 4500°F.
16
(8
C. E. PUGH
338
and 5QOO°F. The uniaxial stress-strain relations for with-gram and across-grain specimens extended at constant strain rates G.z and Zzr, respectively, are given by the constitutive equations as
and
(23)
ft 0
dr,
Gtl(tv)
respectively. The relaxation functions corresponding to the temperatures of interest are determined from Equations (6) and (18). Time is replaced in the integrated results of Equations (23) by t = e,,/i,, or t = E,,/&~~to yield the actual stress-strain relationships for these uniaxial tests. Figure 12 shows comparisons of the stress-strain curves given by using this temperature-dependent model with the average measured stress-strain curves. At all temperatures, the correlation between these two sets of curves for both the with-grain and
6000
0.2
0
0.4 STRAIN
Fig.
0.6
0.8
0
(%I
12. Tensile
0.2
0.4
0.6
STRAIN
stress-strain graphite.
curves
i%)
for RVD
0.6
MECHANICAL
BEHAVIOR
OF RVD GRAPHITE
across-grain directions is quite good for stress levels up to at least one-half the nominal ultimate stress. The final experimental points shown in Fig. 12 are in the vicinity of the average measured ultimate stress values. In order to investigate the effects of strain rates on the stress-strain behavior of RVD uniaxial graphite at 45OO”F, with-grain specimens were extended at three distinct strain rates (0.001, 0.02, and O*lOmin-I). The strain-rate value has a pronounced effect on the stress-strain curve, as can be seen from Fig. 13. The resistance to deformation increases with increasing strain rate. The stress-strain curves predicted by the analytical modei for these conditions are also shown in Fig. 13 and are obtained from Equations (23) by using the proper strain-rate values. It is seen from Fig. 13 that the strain-rate effect is predicted remarkably well by the model at this temperature viscoelastic and for the stress range to which the model is limited.
AT ELEVATED
TEMPERATURES
339
7000
6000
5000
i/l I
A,.,0 -
MEASURED WEDICTE3
iooo
6. CONCLUSIONS This study has shown that the timedependent mechanical behavior of RVD graphite at temperatures ranging from 3500 to 5000°F can be represented over a limited stress range by a transversely isotropic linear viscoelastic model. For stress levels up to one-half of the nominal ultimate stress values, this model is shown to adequately predict uniaxial creep, relaxation, and stressstrain behaviors of both with-grain and across-grain specimens. The experimental data used in this study allowed the complete evaluation of the viscoelastic model, except for the shear creep function ,jd4(t) which describes the shear behavior in planes normal to the plane of isotropy. The model as developed, however, is sufficient to allow predictions which do not involve this type of shear behavior. A temperature dependence was included in the constitutive equations by considering test results from four temperatures. This
I
0 0
0.2
0.4 STRAIN
06
08
(901
Fig. 13. Effect of strain rate on RVD with-grain tensile stress-strain curves at 4500°F. dependence was found to be too complex to permit interpretation in terms of models, such as a thermorheolo~cally simple model [8]. The complexity of the temperature dependence of the steady-state creep rate would suggest that probably more than one creep mechanism is operative over this temperature range. The types of tests included in this study were diversified enough that some of the tests discussed were independent of the tests used to evaluate the parameters appearing in the constitutive equations. This allowed a completely independent comparison of model predictions with experimental measurements .
340
C. E. PUGH
for these tests. Specifically, model predictions were compared with experimental results for across-grain creep tests, for with-grain and across-grain relaxation tests, and for shorttime stress-strain tests conducted at various strain rates. The generally good correlation of analytical model predictions with experimental results indicates the capability of this model to predict the response of the material to more general loading conditions. Acknowledgements-
The author expresses his appreciation to B. L. Greenstreet and G. T. Yahr, Reactor Division of the Oak Ridge National Laboratory, for many helpful discussions during the course of this investigation. The research reported in this paper was sponsored by the U.S. Atomic Energy Commission under contract with the Union Carbide Corporation. NOMENCLATURE Ai E %cl (t> H(t) Jim (t> R t T
Transient creep coefficients Elastic modulus Relaxation functions Heaviside’s step function Creep compliance functions Real gas constant Time Absolute temperature
U xi efj oij /Ljj q ri
Activation energy Rectangular Cartesian coordinates Strain components Stress components Strain ratios Steady-state creep coefficient Retardation times
REFERENCES 1. Reynolds W. N., Physical Properties of Graphite, Elsevier, New York (1968). W. L., Mechanical Properties of 2. Greenstreet Artificial Graphites-A Survey Report, USAEC Report ORNL-4327, Oak Ridge National Laboratory (1968). 3. Woodard J. D., An Investigation of the Mechanical Behavior of Grades RVD and AGOT Graphites, Final Report, USAEC Report TJD23288, Southern Research Institute (April 20, 1966). Wilhelm, Viscoelasticity, pp. 24-30. 4. Fhigge Blaisdell, Waltham, Massachusetts (1967). 5. Rogers T. G. and Pipkin A. C., Zeitschriifiir angewandte Mathematik und Physik, 14, 334-343 (1963). 6. Nightingale R. E., Nuclear Graphite, pp. 90-91. Academic Press, New York (1962). 7. Woolley R. L., Phil. Mag. 11(114), 799-807 (1965). 8. Moreland L. W. and Lee E. H., Transactions of the Society of Rheology, IV, 233-263 (1960).
1971. Vol. 9. pp. 327-340.
Pergamon
Press.
Printed
in Great
Britain
A VISCOELASTIC REPRESENTATION OF THE MECHANICAL BEHAVIOR OF RVD GRAPHITE AT ELEVATED TEMPERATURES* C. E. PUGH
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830 (Received 21 October 1970)
Abstract-A set of constitutive equations is developed to describe the time-dependent mechanical behavior of RVD graphite at temperatures from 3500 to 5000°F. The relations are based upon a transversely isotropic linear viscoelastic model with temperaturedependent parameters. This model is shown to be applicable for stress levels up to onehalf the nominal ultimate stress values for the material. Predictions of these equations are compared with experimental measurements from short-time stress-strain tests, creep tests, and relaxation tests. 1. INTRODUCTION Studies various
of
graphites
temperatures ous
time-dependent
authors.
over
behavior
a range
of elevated
have been reported Many
of these
temperature dependent. A transversely isotropic linear viscoelastic model is shown to characterize this mechanical behavior over a limited stress range. A set of temperaturedependent constitutive equations applicable over this temperature range is developed and shown to describe the mechanical response of the material to a variety of loadings. Experimental results are used both to numerically evaluate parameters and to verify the resulting mathematical model. First, the material is characterized as being mechanically transversely isotropic. The time-dependent behavior is next shown to have a linear dependence on stress over a limited stress range. The forms of the constitutive equations for a transversely isotropic linear viscoelastic material are then formulated in terms of hereditary integrals which contain five independent material properties -three strain ratios (Poisson’s ratios) and two creep functions. The three strain ratios are assumed independent of time, stress, and strain, but are temperature dependent. Thus, the time-dependent character of the mechanical behavior is accounted for by two creep functions. The experimental data used in this study
of
by numer-
investigations
are surveyed and discussed by Reynolds [l] and by Greenstreet [2]. The reported studies, generally, have addressed themselves to the mathematical representation of the response of a given material to a single type of mechanical loading, such as uniaxial creep, relaxation, or recovery. The literature contains very little information about efforts to verify these representations through comparisons of analytical predictions with experimental results for tests other than those used in the development of the models. This paper presents a unified mathematical characterization of the mechanical behavior of RVD graphite for temperatures ranging from 3500 to 5000°F. Within this temperature range, creep and relaxation mechanisms for graphite are operative and give rise to a mechanical behavior which is both time and *Research sponsored by the U.S. Atomic Energy Commission under contract with the Union Carbide Corporation. 327
C. E. PUGH
328
include the results of a larger number of tests performed at 4500°F than at any other temperature. Therefore, the parameters appearing in the constitutive equations are first evaluated at that temperature. Only data from with-grain and across-grain tensile tests and from with-grain creep tests are required for this evaluation. A temperature dependence is then introduced into the model by considering data from with-grain tensile and creep tests conducted at 3500, 4000, and 5000°F. The constitutive equations developed in this manner are then used to predict the results of tests other than those used for evaluation of parameters. This includes comparisons of predictions with data for across-grain creep, with-grain and acrossand short-time stressgrain relaxation, strain behavior for various strain rates. 2. CHARACTERIZATION
OF MATERIAL
BEHAVIOR RVD is a fine-grain graphite with a maximum particle size of O-015 in. It is also a molded graphite and, consequently, has
material properties which are orientation dependent. However, material properties are essentially independent of orientation within planes perpendicular to the molding direction. These planes are considered to be planes of mechanical isotropy and directions parallel to these planes are referred to as with-grain directions. Consequently, the direction normal to these planes, the molding direction, is referred to as the across-grain direction. This type of material symmetry defines a transversely isotropic material and places restrictions on the forms of the constitutive equations to be subsequently presented. The experimental program utilized in this study has been described in detail by Woodard [3] and included various uniaxial tests conducted as constant stress creep tests, stress relaxation tests, and short-time constant strain-rate tests. Tests were performed with both with-grain and across-grain specimens at four specific temperatures: 3500, 4000, 4500, and 5000°F. In this temperature range, deformation mechanisms are operative which make mechanical processes time dependent. This is illustrated by Fig. 1, which shows creep
0.0150
1
I----...
0.0125
0
10
20
30
40 PO TIME (mid
60
Fig. 1. With-grain tensile creep at 4500°F.
70
80
90
MECHANICAL
BEHAVIOR
OF RVD GRAPHITE
strains as functions of time for several individual with-grain specimens which were subjected to constant uniaxial stress values at 4500°F. The stress level for each test is indicated in psi by the number adjacent to the corresponding creep curve. The indicated stress values range up to approximately 75 per cent of the nominal with-grain ultimate strength for this temperature. In order to investigate the dependence of these creep curves upon the stress level, the ratio of creep strain per unit stress, which corresponds to a given creep time, is plotted versus stress. For example, Fig. 2 illustrates this type of plot when the creep period is t = 20 min for the tests shown in Fig. 1. This plot shows the ratio of with-grain creep strain to stress to be, within reasonable experimental scatter, constant with stress up to 3700 psi at 4500°F. This stress level is approximately
AT ELEVATED
TEMPERATURES
329
50 per cent of the nominal ultimate stress value for the with-grain direction of RVD graphite at 4500°F. A similar dependence of the creep strain upon the stress level was observed at the other test temperatures. Therefore, for stress levels up to 50 per cent of the nominal ultimate stress value, the creep strains have an approximately linear dependence upon the stress level. Beyond this stress level, the dependence becomes nonlinear. These observations suggest the mechanical behavior of this material can be characterized by a transversely isotropic linear viscoelastic model for this limited stress range. The model is considered to be temperature dependent, and the limiting stress value, below which the model is considered to be applicable, is taken to be one-half the nominal ultimate stress value corresponding to the
.
-I--
-c. -
30 TENSILE
Fig.
2. With-grain
35 STRESS
creep
40 C,,
x IO-*
strain/stress
4500°F.
45
50
(PSI)
vs. stress
at
55
u,,
330
C. E. PUGH
given temperature. Therefore, for both parameter evaluation and model verification, the only test results considered are those from tests conducted at stress levels below this limiting stress. 3. CONSTITUTIVE
EQUATIONS
The time-dependent strains l ij(t), (i, j = 1, which result from successive stress 2,3), increments Alou, A2crij,. . ., Ancrij applied at times tl, tz, . . ., tn, respectively, to a linear viscoelastic material, are equal to the linear superposition of the strains which would correspond to each stress increment if applied separately to the material. Therefore, the components of stress and strain are related in the following manner:*
x3 axis defines the direction normal to the plane of isotropy and is referred to as the across-grain direction. As shown by Rogers and Pipkin[5], there are only five components ofJ&t) which are independent of each other in the description of a transversely isotropic linear viscoelastic material. If the ratios of induced lateral strains to longitudinal strains for uniaxial states of stress are assumed to be independent of time, stress, and strain, then the constitutive equations for such a material at a constant temperature become
-cc13
where Jijrl (t) are creep compliance functions which characterize the time-dependent response of the material and t is the time at which the strains are evaluated. Therefore, as fully described by Fhigge [4], the strains which result from an arbitrary time-varying stress ukl (t) are represented by
lfj(t)
=
'J&t---r) I0
%dr,
of
a7
= ~JII(++$-PI.~ -pIa%]
e&t)
dr,
I’hi(t-d[
=E
(2)
where the integration variable T ranges over the history of the loading program. The initial values of the creep functions Jtirl(o) represent the elastic compliances. Since RVD is characterized to be a transversely isotropic material, the constitutive equations must be expressed in a form which exhibits an axis of rotational symmetry. The plane of isotropy is defined, here, by the rectangular Cartesian coordinates (x1, x2) and the directions parallel to this plane are referred to as with-grain directions. The *A repeated subscript implies summation terms over the range one to three.
Ed
1dr,
aa,,
3
(3)
cz3(t) =;
I0
’Jdd(t-T)
?dr,
where Jll (t) and JM(t) are independent creep functions and ecu are the ratios of the lateral strains induced in the q directions to the longitudinal strains for a uniaxial stress applied in the xi directions.
MECHANICAL
BEHAVIOR
OF RVD GRAPHITE
The inverse constitutive relations which give stresses in terms of strains have the form
AT ELEVATED TEMPERATURES
The relaxation functions creep functions by
are related
331
to the
t all(t) =
P
I
Gl,(FT)
?+a,?
and
[
0
(6) I
(4)
0’.14Wr)G&)
dr=
N(t),
where H(t) is the Heaviside unit step function. Five parameters, 511(t). JM(~). PID ~1~13, and ,u~~,must, therefore, be evaluated for the complete determination of these constitutive equations at a given temperature. In order to establish the constitutive equations over a range of temperatures, each of these five parameters is considered to be temperature dependent. The terms (Y~, cyz, cu,, and 8 are then determined directly from three of the five parameters as shown above. 4. EVALUATION
OF MATERIAL PARAMETERS
Jo where G,,(t) and Gd4(t) are relaxation functions and
*
independent
Jhz+cL13P31 1 I--P13P31
(y2-_
'
P31(1 +hJ l--1113P31
'
(5) P310
-&)
cY3= 11130 -Pl3cL31)'
The experimental data on which this study is based are sufficient for the complete determination of this model except for the creep function Je4(t) which describes the shear behavior in planes normal to the plane of isotropy. This description is sufficient, however, to allow predictions for a wide variety of problems which do not involve this type of shear behavior. Among these are plane strain, plane stress, and axisymmetric problems, when in each case the defining plane is the plane of isotropy. The strain ratios CL_~~ and pS1 were determined as functions of temperature from transverse and axial strain measurements for short-time constant strain-rate tests conducted at 3500, 4000, 45000 and 5000°F. The average values for each of these two strain ratios were fitted, in the sense of least squares, by linear functions of the absolute temperature (“R) and were expressed by /_b,.J = 0*31TX lo++
1.22 x lo-’
(7)
332
C. E. PUGH
and
to prescribed
stress levels which were then held constant for up to 50 minutes. The p31= 0.2663 x 10-4+ 2.25 x 1O-2. (6) loads were applied to these specimens by extending them at a constant strain rate Figure 3 shows a comparison of the average (O-02 min-‘). Therefore, the values for the measured values for pr2 and p31 and the creep functionJll (t) during this brief loading values given by Equations (7) and (8). time interval must be determined from The strain ratio pr3 is determined from constant strain-rate test data. p3i by using Equations (4) and (5) to show By specializing Equations (4) and (5) for a that the ratio of the stresses oll and cr33, constant strain-rate uniaxial loading prowhich correspond to the same strain level gram, the relaxation function G,,(t) can be for identical uniaxial loading programs in the evaluated for the loading time interval from x1 and x3 directions, respectively, takes on a either with-grain or across-grain tests by the constant value: relations
cll(t>
Figure 3 shows a plot of this ratio vs. from the temperature as determined average measured with-grain and acrossgrain stress-strain curves corresponding to constant strain-rate tests (0.02 min-‘) . The values of this ratio for the various temperatures were found to be essentially independent of the strain level. A parabolic fit was obtained to these data and is also shown in Fig. 3. The resulting expression for the strain ratio pi3 is p13 = 17.7T3 X 10-‘2x 10-4+
0.366,
15.2T2 X lo-‘+2.92T
du,,
K=
~13 du33 -cL31 dE33’
(11)
The average stress-strain curves for constant strain-rate (O-02 min-‘) tests for across-grain specimens at the temperatures of interest were described by an equation due to Woolley [7] which has the form u
33
=A(1
-e-kQ$
(12)
where A and k are material constants. For the specific constant strain rate of &33= 0.02 min-‘, Gus = 0*02t, and the relaxation function is given by Equations (11) and (12) to be
(10)
T is the absolute temperature (OR). * The uniaxial creep tests to be used in evaluating the creep function Jn (t) were conducted by loading with-grain specimens
=
G,,(t) = ~lfwy
(13)
where
*Due to the preferred orientation, the layer planes of the coke particles are parallel to the withgrain plane and possess a different chemical bonding mechanism from the interlayer or across-grain direction [6]. These different bonding mechanisms may explain why p13 exhibits a distinctly different temperature dependence from either p12 or pS1, as only p13 involves an induced strain in the across-grain direction.
where
t is measured
Equations
in minutes.
(6) and (13) then give the creep
function
for
this same limited time range as
Jll(t)
= f-J1
1
+0*02kt].
(14)
If t* is the time required to reach the desired stress level for a uniaxial creep test in the with-grain direction, then Jil (t) is given by Equation (14) for t s t*, and as can be shown from Equations (3) fort > TF by
MECHANICAL
BEHAVIOR
OF RVD GRAPHITE
0.2 r 5 r 90 ‘;$
333
TEMPERATURES
&.-
4,/-
N
AT EL.EVATED
-I,---0.i
-LINEAR l
(3500)
4460
4960
(4000)
(4500)
-
LINEAR l
0’
I 3960 (3500)
FIT
MEASURED 5460 (5000)
‘)A (OF)
FIT
MEASURED I I
I
1
l
(2
I
4460
4960
5460
“R
3960
(4000)
(4500)
(5000)
(‘F)
(3500)
TEMPERATURE
MEASURED I 1
i
4460
4960
5460
oR
(4000)
(4500)
(50001
(OF)
TEMPERATURE
Fig. 3. Strain ratios as functions of’temperature.
In Equation (15), lI1 ( t)cfeei, is the creep strain which has occurred from the time t* to the time t. The measured values used in Equation (15) for ~~,(t)~~~~,,/u~,(t*) correspond to the average creep per unit stress for all tests conducted at stress levels below one-half the nominal ultimate stress value. The value of t* is taken as the average time required to load these specimens to the desired stress levels by extending them at the prescribed constant strain rate (0.02 min-I). The entire creep function JII(Q is interpreted in terms of an elastic response, a viscous component, and a retarded elastic portion. At any given temperature, this description takes the mathematical form:
where E, v, Ai, and Ti are temperaturedependent material parameters. Since the data used in this study include the results of more tests conducted at 4500°F than at any other temperature, the creep function is first determined at this temperature. The first parameter E is taken as the initial slope of the with-grain stress-strain
curve corresponding to constant strain-rate tests (0.02 min-‘) and the coefficient of the second term as the slope of the creep strain per unit stress curve at the end of the data collection period (1= 50 min). The remaining three terms were fitted in the sense of least squares to the difference between J,l (t) data and the two contributions mentioned above. The resulting expression for the creep function at 4500°F is .I,, (t) = [0*467 + 0*0055t+
0.202 ( 1 - e-1s#.491)
$- 0~100 ( 1 - e-n’377f) +0=357(1
-e-“‘06(12f)] lO+(psi)-‘. (17)
Figure 4 illustrates the accuracy of this representation of JI1 (t) data by plotting the measured with-grain creep strain per unit stress values along with the results of Equations (15) and (17). The creep function developed for T = 4500°F is taken as a reference function and a temperature dependence introduced such that at the other temperatures (3500, 4000, and 5~OF) the values given by the modei for the initial modulus, for the duration of primary creep, and for the creep strain and creep strain rate at the end of the data collection periods for uith-grain tests all
C. E. PUGH
334 (x 10-9
-i
-
0.7
: ;
0.6
t: R
Ii
0.5
1
0.4
H g
0.3
9 /
rn
0.2
k 5
0.f
l MEASURED _-CURVE DESCRIBED CREEP FUNCTION
(9
BY
0 0
5
IO
(5
20
25 TIME
30
35
40
45
50
(min)
Fig. 4. RVD with-grain tensile creep at 4500°F. agree with the respective experimental observations. The temperature-dependent model developed in this manner subsequently was used to predict the across-grain creep behavior by the method outlined in the next section. At 3500°F considerable discrepancy was found betwen the model predictions and across-grain creep data. Therefore, in order to better represent the overall material behavior the model presented here requires the discrepancies between the model predictions and the average measured creep curves to be equal for both the with-grain and across-grain directions at 3500°F. With this modification, the resulting creep function is found to be J11(t, T) = $+A,&(1
-e-m31’71)
+ & [7)&t + A, ( 1 - e43t’T* ) +A3( 1 -e4at’ra)],
(18)
where the 4’s are functions of temperature and where E, r), Ai, and ri have the values given in Equation (17). Specifically, the 4 functions were determined to be
4,(T)
= 164TX
lo-4+0*186,
,&1(T) = 37.67e-19.6X10”lRT, 4,(T)
= 2.32e-r~58~103/Rr + 3.16 x lOlOe-135.1x103/RT
&(T)
= 63.47e-1s.7?X103’RL0.631;
44(T)
= 12.96 x 106 e-88+5x103/Rr,
(19)
where T is the absolute temperature in degrees Rankine and R is the universal gas constant. Since the reference creep function corresponds to 4500”F, all of the 4 functions have the value of unity at that temperature. Figures 5 and 6 show the comparison of the average measured with-grain creep per unit stress curves for T = 4000 and 5000”F, respectively, and the predictions of the model described above. Figure 7 illustrates the comparisons of the model predictions with both the measured with-grain and acrossgrain creep per unit stress curves at 3500°F. As discussed earlier, it is seen that the discrepancies between the two predicted curves and the two measured curves are equal in magnitude. It should be pointed out that the creep rates at 3500°F are small in comparison with those at the highest temperatures
MECHANICAL
335
BEHAVIOR OF RVD GRAPHITE AT ELEVATED TEMPERATURES
-PREDICTED
0
5
10
15 TIME (mm)
20
/
30
25
Fig. 5. RVD with-grain tensile creep at 4000°F.
studied. The actual discrepancy between the measured and predicted with-grain creep strain at t = 30 min is approximately O*OOS% per 1000 psi. The experimental data shown in Figs. 5 through 7 correspond to the average creep strain per unit stress curves for tests conducted at stress levels below one-half the nominal ultimate stress values for the respective temperatures.
5.COMPARISON
OF THEORY EXPERIMENTS
AND
Using the viscoelastic model developed in the previous section, predictions are made of the mechanical response of RVD graphite to tests other than those used in the model development. First, predictions are made of the uniaxial creep response of across-grain specimens. The constitutive equations predict the across-grain creep strain per unit stress to be E33 tt,
ncm, =
fl33(t*)
/43(T) a .
ull(t?
T)
--J11(t*,
T)
I> (20)
0
5
15
10 TlME
20
25
30
im,nl
Fig. 6. RVD with-grain tensile creep at 5000°F.
where t* is the time required to load the specimen to the prescribed stress level c:~:~(t*) . Figures 8 through 10 show the comparisons of these predictions with average experimental results from tests conducted at 4000, 4500, and 5000°F and at stress levels below one-half the respective nominal ultimate stress values. It should be noted that one of the ten across-grain specimens tested at 4500°F exhibited anomalous behavior. This specimen experienced no detectable creep deformation after about 8 min. The
336
C. E. PUGH
4,.
MEASURED
-
PREDICTED
i 0
5
10
(5 TIME
20
25
30
(min)
Fig. 7. RVD tensile creep at 3500°F. inclusion of the results of this test in the averaging of measured creep behavior account for some of the difference between the results shown in Fig. 9. The constitutive equations are next used to predict the stress decay for both with-grain and across-grain relaxation tests performed at 4500°F. Experimental tests were performed by first extending uniaxial specimens at a constant strain rate (0.02 min-‘) to prescribed strain levels which corresponded to
.-..
0
5
.-_
10
stress levels below one-half the nominal ultimate stress values. The strains were then held constant while the decreasing stresses were recorded. The constitutive equations predict the stress per unit strain for withgrain and across-grain specimens loaded in this manner to be
__
15 l-IME
20
25
iminI
Fig. 8. RVD across-grain tensiie creep at 4000°F.
MECHANICAL
BEHAVIOR
OF RVD GRAPHITE
AT ELEVATED
337
TEMPERATURES
. MEASURED -PREDICTED
_
O
4--
A
0
MEASURED
-
5
10
I5 TIME
--T-f
0
5
PREDICTED
20
25
30
(m(n)
Fig. 10. RVD across-grain tensile creep at 5000°F.
20
I5 TIME
25
30
(mln)
Fig. 9. RVD across-grain tensile creep at 4500°F. and
(21)
Figure 11 shows the comparisons of the predictions of Equations (21) and (22) with the average experimental relaxation curves for both with-grain and across-grain speciments tested at 4500°F. Stress-strain curves for both the withand across-grain directions were grain determined from uniaxial constant strainrate (0.02 min-‘) tests at 3500, 4000, 4500, (x106) 16
1.4
respectively, where tf and tz are the times required to load with-grain and across-grain specimens, respectively, to the prescribed strain levels. The relaxation function G,,(t) for ?’ = 4500°F is determined from Equations (6) and (17) to be
k
z =
0.8
k*
06
K
0.4
2
0.2
C;,,(t) = (0.653e-27’gR’+ 0.254e-0’446t +
0.3~gr-O~101t+
X
IO’psi.
J
0 0
@838e--o.004731)
(22)
2
4
6
8
10
TIME
(mln)
12
14
Fig. 11. RVD stress relaxation at 4500°F.
16
(8
C. E. PUGH
338
and 5QOO°F. The uniaxial stress-strain relations for with-gram and across-grain specimens extended at constant strain rates G.z and Zzr, respectively, are given by the constitutive equations as
and
(23)
ft 0
dr,
Gtl(tv)
respectively. The relaxation functions corresponding to the temperatures of interest are determined from Equations (6) and (18). Time is replaced in the integrated results of Equations (23) by t = e,,/i,, or t = E,,/&~~to yield the actual stress-strain relationships for these uniaxial tests. Figure 12 shows comparisons of the stress-strain curves given by using this temperature-dependent model with the average measured stress-strain curves. At all temperatures, the correlation between these two sets of curves for both the with-grain and
6000
0.2
0
0.4 STRAIN
Fig.
0.6
0.8
0
(%I
12. Tensile
0.2
0.4
0.6
STRAIN
stress-strain graphite.
curves
i%)
for RVD
0.6
MECHANICAL
BEHAVIOR
OF RVD GRAPHITE
across-grain directions is quite good for stress levels up to at least one-half the nominal ultimate stress. The final experimental points shown in Fig. 12 are in the vicinity of the average measured ultimate stress values. In order to investigate the effects of strain rates on the stress-strain behavior of RVD uniaxial graphite at 45OO”F, with-grain specimens were extended at three distinct strain rates (0.001, 0.02, and O*lOmin-I). The strain-rate value has a pronounced effect on the stress-strain curve, as can be seen from Fig. 13. The resistance to deformation increases with increasing strain rate. The stress-strain curves predicted by the analytical modei for these conditions are also shown in Fig. 13 and are obtained from Equations (23) by using the proper strain-rate values. It is seen from Fig. 13 that the strain-rate effect is predicted remarkably well by the model at this temperature viscoelastic and for the stress range to which the model is limited.
AT ELEVATED
TEMPERATURES
339
7000
6000
5000
i/l I
A,.,0 -
MEASURED WEDICTE3
iooo
6. CONCLUSIONS This study has shown that the timedependent mechanical behavior of RVD graphite at temperatures ranging from 3500 to 5000°F can be represented over a limited stress range by a transversely isotropic linear viscoelastic model. For stress levels up to one-half of the nominal ultimate stress values, this model is shown to adequately predict uniaxial creep, relaxation, and stressstrain behaviors of both with-grain and across-grain specimens. The experimental data used in this study allowed the complete evaluation of the viscoelastic model, except for the shear creep function ,jd4(t) which describes the shear behavior in planes normal to the plane of isotropy. The model as developed, however, is sufficient to allow predictions which do not involve this type of shear behavior. A temperature dependence was included in the constitutive equations by considering test results from four temperatures. This
I
0 0
0.2
0.4 STRAIN
06
08
(901
Fig. 13. Effect of strain rate on RVD with-grain tensile stress-strain curves at 4500°F. dependence was found to be too complex to permit interpretation in terms of models, such as a thermorheolo~cally simple model [8]. The complexity of the temperature dependence of the steady-state creep rate would suggest that probably more than one creep mechanism is operative over this temperature range. The types of tests included in this study were diversified enough that some of the tests discussed were independent of the tests used to evaluate the parameters appearing in the constitutive equations. This allowed a completely independent comparison of model predictions with experimental measurements .
340
C. E. PUGH
for these tests. Specifically, model predictions were compared with experimental results for across-grain creep tests, for with-grain and across-grain relaxation tests, and for shorttime stress-strain tests conducted at various strain rates. The generally good correlation of analytical model predictions with experimental results indicates the capability of this model to predict the response of the material to more general loading conditions. Acknowledgements-
The author expresses his appreciation to B. L. Greenstreet and G. T. Yahr, Reactor Division of the Oak Ridge National Laboratory, for many helpful discussions during the course of this investigation. The research reported in this paper was sponsored by the U.S. Atomic Energy Commission under contract with the Union Carbide Corporation. NOMENCLATURE Ai E %cl (t> H(t) Jim (t> R t T
Transient creep coefficients Elastic modulus Relaxation functions Heaviside’s step function Creep compliance functions Real gas constant Time Absolute temperature
U xi efj oij /Ljj q ri
Activation energy Rectangular Cartesian coordinates Strain components Stress components Strain ratios Steady-state creep coefficient Retardation times
REFERENCES 1. Reynolds W. N., Physical Properties of Graphite, Elsevier, New York (1968). W. L., Mechanical Properties of 2. Greenstreet Artificial Graphites-A Survey Report, USAEC Report ORNL-4327, Oak Ridge National Laboratory (1968). 3. Woodard J. D., An Investigation of the Mechanical Behavior of Grades RVD and AGOT Graphites, Final Report, USAEC Report TJD23288, Southern Research Institute (April 20, 1966). Wilhelm, Viscoelasticity, pp. 24-30. 4. Fhigge Blaisdell, Waltham, Massachusetts (1967). 5. Rogers T. G. and Pipkin A. C., Zeitschriifiir angewandte Mathematik und Physik, 14, 334-343 (1963). 6. Nightingale R. E., Nuclear Graphite, pp. 90-91. Academic Press, New York (1962). 7. Woolley R. L., Phil. Mag. 11(114), 799-807 (1965). 8. Moreland L. W. and Lee E. H., Transactions of the Society of Rheology, IV, 233-263 (1960).