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Self-consistent calculations of elastic constants of polycrystalline graphite
Carbon, 1976, Vol. 14, pp. 185-189.
Pergamon Press.
Printed in Oreal Britain
SELF-CONSISTENT CALCULATIONS OF ELASTIC CONSTANTS OF POLYCRYSTALLINE GRAPHITE R. E. SMTH and G. B. SPENCE Union Carbide Corporation, Carbon Products Division, P.O. Box 6116, Cleveland, OH 44101,U.S.A. and J. E. GUBERNAnst and J. A. KRUMHANSL Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853,U.S.A. (Received 7 April 196) Abstract-This paper reports the application of a self-consistent integral equation technique to the calculation of the elastic constants of polycrystalline graphite. The self-consistent expressions used in this paper are restricted to spherical grains and pores and neglect correlation among grains and pores. A model of graphite based upon the assumption that the Brains are cracked crystals was used with the self-consistent expressions to account for the elastic constants of three isotropic or mildly anisotropic graphites to within approximately 6% and for the elastic constants of a more anisotropic Braphite to within approximately 20%. 1.INTRODUCTION
A central problem in the micromechauics
of graphite is the calculation of the auisotropic elastic properties of polycrystalline graphite from properties of siugIe crystals and structural parameters describing porosity and crystal preferred orientation. Previous analytical treatments have
used simple models of the interactions among grains or have, from first principles, placed bounds on the polycrystalline moduli[ld]. These efforts have not been entirely satisfactory due to the overly simplistic nature of the models or to the fact that the upper and lower bounds for graphite are too far apart to be useful. In the last few years, practical computational schemes have been developed for integral equation formulations and applied to the calculation of many properties of polycrystalline materials. Gubernatis and Krumhansl[7] (also see references cited therein) have summarized integral equation methods for elastic properties and applied a selfconsistent formalism to six widely different polycrystalline materials. The agreement between measured and calculated polycrystalline moduli was ,excellent for materials having single crystals which were only moderately anisotropic. For an epoxy-filled graphite, in which the graphite crystals are extremely anisotropic, the agreement was less good but significantly better than that previously obtained by other methods. The present paper gives additional results for extruded and molded fine-grain graphites obtained with the computational scheme of Gubernatis and Krumhansl[7]. The structural model used in [7] included porosity, but neglected cracks (volume-free porosity). The selfconsistent approximation also neglects correlations between the various structural elements. However, most graphites are thought to contain thin microcracks, the orientation of which is highly correlated with the layer
planes of neighboring crystals. Neglect of these structural details may be responsible for a large part of the difference between the values of experimental moduli and the values calculated for porous aggregates of perfect graphite crystals ([7] and Section 4.1 of this paper). As an dternative approach, the calculations were also performed for a model in which grains with the single crystal properties of graphite are replaced by “cracked-crystal” grains with a c-axis Young’s modulus reduced by cracks to values considerably less than that of a perfect single crystal. The obvious shortcoming of this approach is that the proper value for the reduced c-axis modulus is not known initially and is, in effect, an adjustable parameter. Finally, the cracked-crystal model is shown to account qualitatively for the temperature and neutron irradiation dependence of the elastic moduli of an isotropic polycrystalline graphite. Another limitation of the present numerical calculations is that the grains and pores are assumed to be spherical in shape; it is felt that this is an acceptable approximation for the fine-grain graphites studied here. 2. THEORY Since the theory is fully described in [7], only a general outline of the nature of the calculations is given here. Let M(O,4) be a property of a single grain which depends on direction (6, 4). The orientation average of M, (M),for a large number of grains is defined by
(M)=A-'[I, M(B,~)g(e,~)sineded~, (1) where g(8, 4) is the probability distribution function for grain orientation and A is the normalization constant. If
M is a tensor property of graphite, then (M) is taken to be the orientation average of each element of M. Let Co denote the fourth-rank effective elastic stiffness tPresent address: University of California, Los Alamos Scientific Laboratory, P.O. Box 1663, Los Alamos, NM 87544, tensor of the polycrystalline medium and C’ denote the U.S.A. elastic stiffness tensor of the jth phase (pores constitute a 185
R. E. St.urrret al.
186
phase with vanishing elastic stiffness). In the selfconsistent integral equation formalism, the effective stiffness of the polycrystalline aggregate is that tensor Co which satisfies the set of equations
Legendre polynomials: g(0) =
1t a2Pz(cos 0) t a$,(cos
O),
(5)
where a2and a.,are obtained byfittingg(B) to X-ray data. Two models of graphite grains are considered. In the t&Z' (I - G” . St?)-‘) = 0, (2) crack-free-crystal model, the graphite grains are assumed to have the elastic properties which have been measured where on near-perfect graphite crystals (see Table V of [9]). In SC’ = C’ - CO, the cracked-crystal model, the against-grain Young’s modulus (l/S,,) is treated as an adjustable parameter with N is the number of phases, vi is the volume fraction of the a value less than that of the perfect crystal; the with-grain jth phase, Gfi is the integral over the jth grain volume of Young’s modulus (l/S,,) and Poisson’s ratio (-S,,/S,,) the elastic Green’s function for a medium of stiffness CO, are assumed to have the perfect crystal values. In terms of and I is the identity tensor. The tensor G has components stiffness constants, C,, is variable, Co and C,, have perfect crystal values, and C,, is computed from Co = 2.483(C#. This expression for C,, ensures that (l/S,,) and (- S,,/S,,) retain their single crystal values for where the E’s may be computed by explicit volume all values of C,3. In both models, the shear constant CMis integrals defined by Kneer[l]. Equation (3) is the treated as an adjustable parameter with a value between expanded form of the corresponding relation given in [7] those (0.18-4.5 GPa) for dislocation-mobile and which is needed in order to apply the isomorphism given dislocation-pinned single crystals [lo]. Also, in both in [7] between tensor elements and the matrix elements models, all of the porosity is assumed to be in spherical pores and the microcracks in the cracked crystals are used in numerical calculations. The correct (self-consistent) value of Co is found by an assumed to have negligible volume. iterative procedure. Let CnObe the value of Co in the n th 3.2 Samples iteration; the iterations are continued until Four specimens were chosen so that a range of porosity, manufacturing methods (isotropic, molded and IC”O_C”.+,ls clc”OI, (4) extruded), and degree of anisotropy might be investigated. where l is a small positive number. The results reported Tables 1 and 2 list properties of these specimens. The in this paper were obtained with E = 10m3. A smaller value, elastic constants of the isotropic graphite, described in [ll], were not directly measured but were calculated for E = 10m4,resulted, in one case, in greatly increased computing time and in a change in the calculated elements the listed value of porosity from a regression analysis of of Co of approximately 1% or less from the values for the data given in [ll]. Grade ATJSt is a commercial, mildly anisotropic molded graphite. Samples S199 and l = lo-‘. Although the problem has been reduced to quadrature, many double integrals have to be calculated S189-2 are laboratory extruded graphites of moderate and numerically for each iteration and several (typically, ten) strong anisotropy fabricated from petroleum coke flours. Fine-grain graphites were chosen in order that the iterations are needed for convergence; so the calculation is practical only on a high-speed digital computer. (The physical grains might contain fewer single crystals and, thereby, fit the assumptions of the model more closely calculation for bulk isotropy is considerably simpler [7].) than might be the case for coarse-grain graphites.
$
*
3.APPLICATIONS
3.1 Models of graphite The application of the theory requires the specification of the identity, volume fraction, grain shape and properties of each phase. These specifications constitute a model of the microstructure of the material. Failure of predictions to agree with experiment may be a failure either of approximations made in the theory (e.g. neglect of correlation) or of the structural model representing the real material. In this study, the model of polycrystalline graphite consists of two phases: graphite grains of density 2.25 Mg/m3 and pores. To simplify the computation, the shape of both graphite grains and pores is taken to be spherical. The polycrystalline graphite is assumed to have transverse isotropic symmetry, and the crystallite orientation distribution function is expressed in terms of
tProduct of Union Carbide Corporation.
3.3 Best-fit parameter The use of only two variable parameters (C33,C,) does not allow exact prediction of the five polycrystalline elastic constants of transversely isotropic graphite. A fitting parameter, F, was defined to allow an objective determination of the best (C33,C,) pair. If the two-index notation is compressed by ll+l, 33+2, 44+3, 12+4, and 13-5, F = (104/i
i=,
W)2 (WACi/C:)*,
(6)
where AC = CF- CF is the deviation of the predicted polycrystalline CP from the experimentally measured C; and the W are weighting factors selected according to the relative accuracy to which the C; are known. The normalization is such that F is the average RMS error of the prediction in percent. For the isotropic graphite
Self-consistent
calculations of elastic constants of polycrystalline
187
graphite
Table 1. Specimenproperties Material
Pref.
Porosity 1%)
Tylx
Orientation
=2
17.7
=4
IS0
Isotropic
0.0
0.0
ATJS
Molded
17.7
s199
Extruded
28. 0
-0.430
0. 124
5189-Z
Extruded
30. 2
-0.910
0.315
0. 275
-0.01
Table2. Predictedpolycrystallineelasticconstants(in GPa) Polycrystalline
.%iffnesses
Grain
Stiffnesses+
C33
C44
Cl I*
c33*
c44*
Measured Predicted Predicted
36. 5 36. 5
0. 18 4.5
13.54 28. 63 59.19
13.54 28. 63 59.19
4.93 6.75 la. 98
Predicted
4.43
1.37
13.53
13.53
4.93
36. 5 36. 5
0. la 4.5
13.42 29.47 61.95
9.38 26.74 53.76
4.39 6. 52 18.36
1.76
12.41
9.80
0. la 4.5
6. 35 4.50 26. a9
a. a7 4.91 31.87
6. 68 4.07 2. aa 21.48
Material ISO-Isotropic
ATJS-Molded Measured Predicted Predicted Predicted Sl99-Extruded Measured Predicted Predicted Predicted Sla9-Z-Extruded Measured Predicted PredIcted Predicted ‘In the notation
1. 64
36. 5 36. 5 3.96
36. 5 36. 5 1.75 of the text:
2.00
0. Ia 4.5
4. 61
2.76 Cij3
= C;
3.68 15.12 2L.23
c13* 3. 68 15.12 21.23
3. 67
3.67
2. 24 L5.1, 21.55
1.57 15.08 21.07
4.55
2. 17
1.95
2.95 I.08 9. 34
1.53 2. 52 9.97
2. 10 2.46 10.17
8. 62
2.91
1.52
1. ba
LO. 21 3.45 30.21
2.74 0.74 a.05
0. a7 1. 61 8.14
0.40 1.52 a.47
2. 67
0.49
0.83
8.62
for measured
W, = W3= 1 and W2= W4= W5= 0; for the anisotropic graphites W, = Wz = W, = 5, W.,= 2.5, and W5= 1. A computer code was written to determine the value of (C,,, C,) which produces the best fit (minimum F) of predicted CO, to experimental C$ The code uses a polynomial fitting procedure applied to a rectangular array of points in the (&, C,) plane at which complete self-consistent calculations have been made. The code allows the direct plotting of the loci of (Cp,, C,) values which produce the exact prediction of any particular one of the experimental CG; contours of constant F can be plotted to display regions of acceptable fit in the (&, C,) plane.
CIZL
and Cij*
RMS Error I? (%)
59. 221. 0.04
223. 462. 5.9
49. 302. 4.0
69. 426. 20.4
= Csj for predicted.
4.2 Cracked-crystal model
In the cracked-crystal model, the graphite grain stiffness C,, is allowed to be less than the perfect crystal value in order to represent the effect of interlaminar cracks within the grain, and C, is assumed to be within the range 0.18-4.5 GPa. The point in the (C,,, CM)plane at which the RMS error F is a minimum was found by making the self-consistent calculation at a rectangular array of points in the (&, C,) plane. The results of these calculations were used with the polynomial fitting code to determine the coordinates of the point with minimum F and the value of the minimum F. A self-consistent calculation made at the minimum gave the best predicted polycrystalline elastic constants. These results are given in Table 2 on the last line of data for each specimen, 4. RFNJLTS Figure 1 shows the loci of (C,,, C,) points at which 4.1 Crack-free-crystal model each of the polycrystalline stiffnesses is correctly In the crack-free-crystal model, the graphite grain predicted for the Grade ATJS specimen. If all the loci stiffness Csp is set equal to the perfect crystal value of intersected at one point, a unique pair of (CXs,C,) values 36.5 GPa. In Table 2, the measured polycrystalline elastic would exist for which prediction and measurement agree stiffnesses are compared with the predicted stiffness for exactly for all five elastic constants and, therefore, F limiting values of the grain stiffness CU of 0.18 and would vanish. This situation was found for the IS0 4.5 GPa and for C,, = 36.5 GPa. For the ISO, AJTS, and specimen, for which there are only two independent S189-2 specimens, F increases monotonically with CM; elastic constants; however, no exact solution exists for the for specimen S199, additional calculations indicate a other cases. In order to investigate the sharpness of the minimum value of F of 45 at Cu = 0.3 GPa. Although the F-surface near the minimum and to look for local, relative ability to predict the stitInesses of the extruded specimens minima, contours of constant F-values were plotted, as to within a factor of two is encouraging, the results for the shown in Fig. 2 for the Grade ATJS specimen. In view of isotropic and molded graphites are not satisfactory and experimental errors and analytical approximations, probaindicate that the predicted stitInesses are too high or, bly any point within the 10%contour should be considered alternatively, that the assumed stiffness of the graphite as giving acceptable fit of prediction to experiment. In no grain should be reduced. case were relative minima found.
r
188
R. E. SMITHet al. 5. IMPLICATIONS
The results given here show that the self-consistent integral equation technique applied to the cracked-crystal model is remarkably successful in predicting the elastic constants of polycrystalline graphites which are not too strongly anisotropic, although this success is obtained at the expense of introducing grain stiffnesses C31and C, which are not known from first principles. The utility of this approach has been explored further by estimating the effects on the elastic constants of neutron irradiation and of large temperature increases; the isotropic graphite is used in these examples.
Fig. 1. Loci at whichpredictionand measurementagreefor each elasticconstantof GradeATJSgraphite.
C44tGPa)
Fig. 2. Contours of constant RMS error, F, for Grade ATJS graphite.
The results given in Table 2 for the cracked-crystal model show agreement between prediction and measurement on the average to better than 6% for three of the specimens and to within - 20% for the most anisotropic specimen, Sl89-2. The deduced values of the grain stiffness C3J are surprisingly low, but the plausibility of these values cannot be adequately judged, since we lack sufficient knowledge of the extent and effect of microcracks. The deduced values of the gram stiffness CU are only a factor of 2 or 3 less than the value of 4.5 GPa for dislocation-pinned single crystals but are a factor of 8 to 16 greater than the value of 0.18 for large single crystals with mobile dislocations. Since the dislocation content of commercial graphites is known[l2] to be high, these results suggest that the mobility of dislocations in the small crystallites of commercial graphites is low, possibly due to pinning by crystallite boundaries or by c-axis screw dislocations.
5.1 Neutron irradiation efects Neutron irradiation doses of < lOI nV at 50°C have been found[lO] to increase the CM of compressionannealed pyrolytic graphite to 5 4.5 GPa with little effect on the other C’b Therefore, an estimate of the effect of low-temperature neutron irradiation of the IS0 material may be obtained by setting the graphite grain Cc(= 4.5 GPa and keeping C,, at the value 4.43 GPa reported in Table 2. The predicted increase due to irradiation is given in Table 3 for Young’s modulus, E, and the shear modulus, G. Some variation in the experimentally determined modulus increase is found in reports on similar types of isotropic graphites[3,13,14], but the predicted increase appears to be in the right range. 5.2 Temperature effects An increase in temperature is thought to result in the closing of microcracks (see, e.g. [2]). At temperatures near 18WC, all of the microcracks are presumed to have closed, and the effective grain CXlis that of the perfect lattice. The lattice C3, near 1800°Chas been estimated [ 151 to be C,, = 27.4 GPa. Crystals at temperatures near the onset of plastic flow might be expected to shear more easily and, hence, to have a lower value of CU. Table 3 gives for the IS0 material the predicted ratio of the modulus at 1800°Cto the room-temperature modulus for various values of CU, C& = 27.4 GPa, and the previous room temperature values of the other Ci,.The experimental results are for samples 9Q-2 and 9Q-8 in 1163which had the same graphitixation temperature (2900°C)as the IS0 specimen [ 111.Good agreement with experiment would be obtained for a grain C,., of 0.23 GPa. This six-fold decrease in C, is far greater than would be expected for the intrinsic lattice contribution to the shear modulus and suggests that the dislocation system has become quite mobile at 1800°C. Table 3. Predicted effect of neutron irradiation and high temperature on the moduli of isotropic graphite Grain Stiff. C33
Condition Neutron Irradiation
Measured
C3.13,
Predicted High Temperature E::iz:: [‘6’ Predicted Predicted Predicted
141
-
4.43
(GPa) C44 -
Change in Moduli E El 0 G/Go 2. 2-3.3
-3.4
4.5
2. 15
2. 27
(lSOO’C[ 27.4
0. 1
I. 27 14
I.02 1. 24
27.4 27.4 27.4
0. 2 0.4 1.0
I. 25 1.46 1.93
1. 14 1.33 1.79
Self-consistentcalculationsof elasticconstantsof polycrystallinegraphite 6.CONCLUSIONS
A self-consistent integral equation technique has been used to calculate the elastic properties of polycrystalline fine-grain graphites. The present computational technique is restricted to spherical grains and pores and to uncorrelated structures. When the grain properties were taken to be those of the perfect crystal, the average error of the predicted moduli ranged from 45 to several hundred percent. When the grain properties were taken to be those of a cracked crystal (to compensate empirically for correlated microcracks) the average error ranged from 0 to 20%. The small values inferred for the c-axis modulus of the cracked crystal need to be examined further. This examination might include analytical studies to calculate the moduli of crystals containing correlated microcracks or experimental studies to measure the cracked-crystal moduli. Changes in the elastic properties of an isotropic graphite caused by neutron irradiation or by temperature variations were accounted for by plausible changes in the inferred grain properties. However, the further utilization of the techniques reported here would require additional knowledge of the actual changes in graphite grain properties and microstructure due to the irradiation and thermal environment.
189
REFERENCES
1. R. J. Price, Philos. Msg. 12, 561 (1965). 2. A. M. Barrachin, G. Jouquet, G. Micaud and F. Pattou, J. Nucl. Mater. 20, 294 (1966). 3. P. R. Goggin and W. N. Reynolds, Philos. Msg. 16, 317 (1967). 4. 0. D. Slagle, Carbon 6, 111 (1968). 5. D. P. H. Hasselman, 1. Am. Ceram. Sot. 53, 170 (1970). 6. G. M. Jenkins, In Chemistry and Physics of Carbon (Edited by P. L. Walker, Jr. and P. A. Thrower), Vol. 11,pp. 189-242. Marcel Dekker, New York (1973). 7. J. E. Gubernatis and J. A. Krumhansl, J. Appl. Phys. 461875 (1975). 8. G. Kneer, Phys. Status Solidi 9, 825 (1965). 9. 0. L. Blakslee,D. G. Proctor,E. J. Seldin,G. B. Spence and T. Wena, J. AUDI.Phvs. 41. 3373 (1970). 10. E. J. Sei& and C. W. Nezbeda, .I.&pi. khys. 41,3389 (1970). 11. P. Wagner, J. A. O’Rourke and P. E. Armstrong, J. Am. Ceram. Sot. 55, 214 (1972). 12. G. K. Williamson, In Inst. Met., London, Symp. on Uranium and Graphite, Monogr. No. 27, 87 (1%2). 13. R. Taylor, et al., Carbon 5, 519 (1967). 14. A. L. Pitner, Carbon 9, 637 (1971). 15. B. T. Kelly, Carbon 12, 535 (1974). 16. General Electric Company, Hastic Properties of ThreeGrades of Fine Grained Grauhite to 2000°C. Bv M. 0. Marlowe. NASA Contract NASl-9852,Rpt. No. NASA CR-66933(25 June 1970).
Pergamon Press.
Printed in Oreal Britain
SELF-CONSISTENT CALCULATIONS OF ELASTIC CONSTANTS OF POLYCRYSTALLINE GRAPHITE R. E. SMTH and G. B. SPENCE Union Carbide Corporation, Carbon Products Division, P.O. Box 6116, Cleveland, OH 44101,U.S.A. and J. E. GUBERNAnst and J. A. KRUMHANSL Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853,U.S.A. (Received 7 April 196) Abstract-This paper reports the application of a self-consistent integral equation technique to the calculation of the elastic constants of polycrystalline graphite. The self-consistent expressions used in this paper are restricted to spherical grains and pores and neglect correlation among grains and pores. A model of graphite based upon the assumption that the Brains are cracked crystals was used with the self-consistent expressions to account for the elastic constants of three isotropic or mildly anisotropic graphites to within approximately 6% and for the elastic constants of a more anisotropic Braphite to within approximately 20%. 1.INTRODUCTION
A central problem in the micromechauics
of graphite is the calculation of the auisotropic elastic properties of polycrystalline graphite from properties of siugIe crystals and structural parameters describing porosity and crystal preferred orientation. Previous analytical treatments have
used simple models of the interactions among grains or have, from first principles, placed bounds on the polycrystalline moduli[ld]. These efforts have not been entirely satisfactory due to the overly simplistic nature of the models or to the fact that the upper and lower bounds for graphite are too far apart to be useful. In the last few years, practical computational schemes have been developed for integral equation formulations and applied to the calculation of many properties of polycrystalline materials. Gubernatis and Krumhansl[7] (also see references cited therein) have summarized integral equation methods for elastic properties and applied a selfconsistent formalism to six widely different polycrystalline materials. The agreement between measured and calculated polycrystalline moduli was ,excellent for materials having single crystals which were only moderately anisotropic. For an epoxy-filled graphite, in which the graphite crystals are extremely anisotropic, the agreement was less good but significantly better than that previously obtained by other methods. The present paper gives additional results for extruded and molded fine-grain graphites obtained with the computational scheme of Gubernatis and Krumhansl[7]. The structural model used in [7] included porosity, but neglected cracks (volume-free porosity). The selfconsistent approximation also neglects correlations between the various structural elements. However, most graphites are thought to contain thin microcracks, the orientation of which is highly correlated with the layer
planes of neighboring crystals. Neglect of these structural details may be responsible for a large part of the difference between the values of experimental moduli and the values calculated for porous aggregates of perfect graphite crystals ([7] and Section 4.1 of this paper). As an dternative approach, the calculations were also performed for a model in which grains with the single crystal properties of graphite are replaced by “cracked-crystal” grains with a c-axis Young’s modulus reduced by cracks to values considerably less than that of a perfect single crystal. The obvious shortcoming of this approach is that the proper value for the reduced c-axis modulus is not known initially and is, in effect, an adjustable parameter. Finally, the cracked-crystal model is shown to account qualitatively for the temperature and neutron irradiation dependence of the elastic moduli of an isotropic polycrystalline graphite. Another limitation of the present numerical calculations is that the grains and pores are assumed to be spherical in shape; it is felt that this is an acceptable approximation for the fine-grain graphites studied here. 2. THEORY Since the theory is fully described in [7], only a general outline of the nature of the calculations is given here. Let M(O,4) be a property of a single grain which depends on direction (6, 4). The orientation average of M, (M),for a large number of grains is defined by
(M)=A-'[I, M(B,~)g(e,~)sineded~, (1) where g(8, 4) is the probability distribution function for grain orientation and A is the normalization constant. If
M is a tensor property of graphite, then (M) is taken to be the orientation average of each element of M. Let Co denote the fourth-rank effective elastic stiffness tPresent address: University of California, Los Alamos Scientific Laboratory, P.O. Box 1663, Los Alamos, NM 87544, tensor of the polycrystalline medium and C’ denote the U.S.A. elastic stiffness tensor of the jth phase (pores constitute a 185
R. E. St.urrret al.
186
phase with vanishing elastic stiffness). In the selfconsistent integral equation formalism, the effective stiffness of the polycrystalline aggregate is that tensor Co which satisfies the set of equations
Legendre polynomials: g(0) =
1t a2Pz(cos 0) t a$,(cos
O),
(5)
where a2and a.,are obtained byfittingg(B) to X-ray data. Two models of graphite grains are considered. In the t&Z' (I - G” . St?)-‘) = 0, (2) crack-free-crystal model, the graphite grains are assumed to have the elastic properties which have been measured where on near-perfect graphite crystals (see Table V of [9]). In SC’ = C’ - CO, the cracked-crystal model, the against-grain Young’s modulus (l/S,,) is treated as an adjustable parameter with N is the number of phases, vi is the volume fraction of the a value less than that of the perfect crystal; the with-grain jth phase, Gfi is the integral over the jth grain volume of Young’s modulus (l/S,,) and Poisson’s ratio (-S,,/S,,) the elastic Green’s function for a medium of stiffness CO, are assumed to have the perfect crystal values. In terms of and I is the identity tensor. The tensor G has components stiffness constants, C,, is variable, Co and C,, have perfect crystal values, and C,, is computed from Co = 2.483(C#. This expression for C,, ensures that (l/S,,) and (- S,,/S,,) retain their single crystal values for where the E’s may be computed by explicit volume all values of C,3. In both models, the shear constant CMis integrals defined by Kneer[l]. Equation (3) is the treated as an adjustable parameter with a value between expanded form of the corresponding relation given in [7] those (0.18-4.5 GPa) for dislocation-mobile and which is needed in order to apply the isomorphism given dislocation-pinned single crystals [lo]. Also, in both in [7] between tensor elements and the matrix elements models, all of the porosity is assumed to be in spherical pores and the microcracks in the cracked crystals are used in numerical calculations. The correct (self-consistent) value of Co is found by an assumed to have negligible volume. iterative procedure. Let CnObe the value of Co in the n th 3.2 Samples iteration; the iterations are continued until Four specimens were chosen so that a range of porosity, manufacturing methods (isotropic, molded and IC”O_C”.+,ls clc”OI, (4) extruded), and degree of anisotropy might be investigated. where l is a small positive number. The results reported Tables 1 and 2 list properties of these specimens. The in this paper were obtained with E = 10m3. A smaller value, elastic constants of the isotropic graphite, described in [ll], were not directly measured but were calculated for E = 10m4,resulted, in one case, in greatly increased computing time and in a change in the calculated elements the listed value of porosity from a regression analysis of of Co of approximately 1% or less from the values for the data given in [ll]. Grade ATJSt is a commercial, mildly anisotropic molded graphite. Samples S199 and l = lo-‘. Although the problem has been reduced to quadrature, many double integrals have to be calculated S189-2 are laboratory extruded graphites of moderate and numerically for each iteration and several (typically, ten) strong anisotropy fabricated from petroleum coke flours. Fine-grain graphites were chosen in order that the iterations are needed for convergence; so the calculation is practical only on a high-speed digital computer. (The physical grains might contain fewer single crystals and, thereby, fit the assumptions of the model more closely calculation for bulk isotropy is considerably simpler [7].) than might be the case for coarse-grain graphites.
$
*
3.APPLICATIONS
3.1 Models of graphite The application of the theory requires the specification of the identity, volume fraction, grain shape and properties of each phase. These specifications constitute a model of the microstructure of the material. Failure of predictions to agree with experiment may be a failure either of approximations made in the theory (e.g. neglect of correlation) or of the structural model representing the real material. In this study, the model of polycrystalline graphite consists of two phases: graphite grains of density 2.25 Mg/m3 and pores. To simplify the computation, the shape of both graphite grains and pores is taken to be spherical. The polycrystalline graphite is assumed to have transverse isotropic symmetry, and the crystallite orientation distribution function is expressed in terms of
tProduct of Union Carbide Corporation.
3.3 Best-fit parameter The use of only two variable parameters (C33,C,) does not allow exact prediction of the five polycrystalline elastic constants of transversely isotropic graphite. A fitting parameter, F, was defined to allow an objective determination of the best (C33,C,) pair. If the two-index notation is compressed by ll+l, 33+2, 44+3, 12+4, and 13-5, F = (104/i
i=,
W)2 (WACi/C:)*,
(6)
where AC = CF- CF is the deviation of the predicted polycrystalline CP from the experimentally measured C; and the W are weighting factors selected according to the relative accuracy to which the C; are known. The normalization is such that F is the average RMS error of the prediction in percent. For the isotropic graphite
Self-consistent
calculations of elastic constants of polycrystalline
187
graphite
Table 1. Specimenproperties Material
Pref.
Porosity 1%)
Tylx
Orientation
=2
17.7
=4
IS0
Isotropic
0.0
0.0
ATJS
Molded
17.7
s199
Extruded
28. 0
-0.430
0. 124
5189-Z
Extruded
30. 2
-0.910
0.315
0. 275
-0.01
Table2. Predictedpolycrystallineelasticconstants(in GPa) Polycrystalline
.%iffnesses
Grain
Stiffnesses+
C33
C44
Cl I*
c33*
c44*
Measured Predicted Predicted
36. 5 36. 5
0. 18 4.5
13.54 28. 63 59.19
13.54 28. 63 59.19
4.93 6.75 la. 98
Predicted
4.43
1.37
13.53
13.53
4.93
36. 5 36. 5
0. la 4.5
13.42 29.47 61.95
9.38 26.74 53.76
4.39 6. 52 18.36
1.76
12.41
9.80
0. la 4.5
6. 35 4.50 26. a9
a. a7 4.91 31.87
6. 68 4.07 2. aa 21.48
Material ISO-Isotropic
ATJS-Molded Measured Predicted Predicted Predicted Sl99-Extruded Measured Predicted Predicted Predicted Sla9-Z-Extruded Measured Predicted PredIcted Predicted ‘In the notation
1. 64
36. 5 36. 5 3.96
36. 5 36. 5 1.75 of the text:
2.00
0. Ia 4.5
4. 61
2.76 Cij3
= C;
3.68 15.12 2L.23
c13* 3. 68 15.12 21.23
3. 67
3.67
2. 24 L5.1, 21.55
1.57 15.08 21.07
4.55
2. 17
1.95
2.95 I.08 9. 34
1.53 2. 52 9.97
2. 10 2.46 10.17
8. 62
2.91
1.52
1. ba
LO. 21 3.45 30.21
2.74 0.74 a.05
0. a7 1. 61 8.14
0.40 1.52 a.47
2. 67
0.49
0.83
8.62
for measured
W, = W3= 1 and W2= W4= W5= 0; for the anisotropic graphites W, = Wz = W, = 5, W.,= 2.5, and W5= 1. A computer code was written to determine the value of (C,,, C,) which produces the best fit (minimum F) of predicted CO, to experimental C$ The code uses a polynomial fitting procedure applied to a rectangular array of points in the (&, C,) plane at which complete self-consistent calculations have been made. The code allows the direct plotting of the loci of (Cp,, C,) values which produce the exact prediction of any particular one of the experimental CG; contours of constant F can be plotted to display regions of acceptable fit in the (&, C,) plane.
CIZL
and Cij*
RMS Error I? (%)
59. 221. 0.04
223. 462. 5.9
49. 302. 4.0
69. 426. 20.4
= Csj for predicted.
4.2 Cracked-crystal model
In the cracked-crystal model, the graphite grain stiffness C,, is allowed to be less than the perfect crystal value in order to represent the effect of interlaminar cracks within the grain, and C, is assumed to be within the range 0.18-4.5 GPa. The point in the (C,,, CM)plane at which the RMS error F is a minimum was found by making the self-consistent calculation at a rectangular array of points in the (&, C,) plane. The results of these calculations were used with the polynomial fitting code to determine the coordinates of the point with minimum F and the value of the minimum F. A self-consistent calculation made at the minimum gave the best predicted polycrystalline elastic constants. These results are given in Table 2 on the last line of data for each specimen, 4. RFNJLTS Figure 1 shows the loci of (C,,, C,) points at which 4.1 Crack-free-crystal model each of the polycrystalline stiffnesses is correctly In the crack-free-crystal model, the graphite grain predicted for the Grade ATJS specimen. If all the loci stiffness Csp is set equal to the perfect crystal value of intersected at one point, a unique pair of (CXs,C,) values 36.5 GPa. In Table 2, the measured polycrystalline elastic would exist for which prediction and measurement agree stiffnesses are compared with the predicted stiffness for exactly for all five elastic constants and, therefore, F limiting values of the grain stiffness CU of 0.18 and would vanish. This situation was found for the IS0 4.5 GPa and for C,, = 36.5 GPa. For the ISO, AJTS, and specimen, for which there are only two independent S189-2 specimens, F increases monotonically with CM; elastic constants; however, no exact solution exists for the for specimen S199, additional calculations indicate a other cases. In order to investigate the sharpness of the minimum value of F of 45 at Cu = 0.3 GPa. Although the F-surface near the minimum and to look for local, relative ability to predict the stitInesses of the extruded specimens minima, contours of constant F-values were plotted, as to within a factor of two is encouraging, the results for the shown in Fig. 2 for the Grade ATJS specimen. In view of isotropic and molded graphites are not satisfactory and experimental errors and analytical approximations, probaindicate that the predicted stitInesses are too high or, bly any point within the 10%contour should be considered alternatively, that the assumed stiffness of the graphite as giving acceptable fit of prediction to experiment. In no grain should be reduced. case were relative minima found.
r
188
R. E. SMITHet al. 5. IMPLICATIONS
The results given here show that the self-consistent integral equation technique applied to the cracked-crystal model is remarkably successful in predicting the elastic constants of polycrystalline graphites which are not too strongly anisotropic, although this success is obtained at the expense of introducing grain stiffnesses C31and C, which are not known from first principles. The utility of this approach has been explored further by estimating the effects on the elastic constants of neutron irradiation and of large temperature increases; the isotropic graphite is used in these examples.
Fig. 1. Loci at whichpredictionand measurementagreefor each elasticconstantof GradeATJSgraphite.
C44tGPa)
Fig. 2. Contours of constant RMS error, F, for Grade ATJS graphite.
The results given in Table 2 for the cracked-crystal model show agreement between prediction and measurement on the average to better than 6% for three of the specimens and to within - 20% for the most anisotropic specimen, Sl89-2. The deduced values of the grain stiffness C3J are surprisingly low, but the plausibility of these values cannot be adequately judged, since we lack sufficient knowledge of the extent and effect of microcracks. The deduced values of the gram stiffness CU are only a factor of 2 or 3 less than the value of 4.5 GPa for dislocation-pinned single crystals but are a factor of 8 to 16 greater than the value of 0.18 for large single crystals with mobile dislocations. Since the dislocation content of commercial graphites is known[l2] to be high, these results suggest that the mobility of dislocations in the small crystallites of commercial graphites is low, possibly due to pinning by crystallite boundaries or by c-axis screw dislocations.
5.1 Neutron irradiation efects Neutron irradiation doses of < lOI nV at 50°C have been found[lO] to increase the CM of compressionannealed pyrolytic graphite to 5 4.5 GPa with little effect on the other C’b Therefore, an estimate of the effect of low-temperature neutron irradiation of the IS0 material may be obtained by setting the graphite grain Cc(= 4.5 GPa and keeping C,, at the value 4.43 GPa reported in Table 2. The predicted increase due to irradiation is given in Table 3 for Young’s modulus, E, and the shear modulus, G. Some variation in the experimentally determined modulus increase is found in reports on similar types of isotropic graphites[3,13,14], but the predicted increase appears to be in the right range. 5.2 Temperature effects An increase in temperature is thought to result in the closing of microcracks (see, e.g. [2]). At temperatures near 18WC, all of the microcracks are presumed to have closed, and the effective grain CXlis that of the perfect lattice. The lattice C3, near 1800°Chas been estimated [ 151 to be C,, = 27.4 GPa. Crystals at temperatures near the onset of plastic flow might be expected to shear more easily and, hence, to have a lower value of CU. Table 3 gives for the IS0 material the predicted ratio of the modulus at 1800°Cto the room-temperature modulus for various values of CU, C& = 27.4 GPa, and the previous room temperature values of the other Ci,.The experimental results are for samples 9Q-2 and 9Q-8 in 1163which had the same graphitixation temperature (2900°C)as the IS0 specimen [ 111.Good agreement with experiment would be obtained for a grain C,., of 0.23 GPa. This six-fold decrease in C, is far greater than would be expected for the intrinsic lattice contribution to the shear modulus and suggests that the dislocation system has become quite mobile at 1800°C. Table 3. Predicted effect of neutron irradiation and high temperature on the moduli of isotropic graphite Grain Stiff. C33
Condition Neutron Irradiation
Measured
C3.13,
Predicted High Temperature E::iz:: [‘6’ Predicted Predicted Predicted
141
-
4.43
(GPa) C44 -
Change in Moduli E El 0 G/Go 2. 2-3.3
-3.4
4.5
2. 15
2. 27
(lSOO’C[ 27.4
0. 1
I. 27 14
I.02 1. 24
27.4 27.4 27.4
0. 2 0.4 1.0
I. 25 1.46 1.93
1. 14 1.33 1.79
Self-consistentcalculationsof elasticconstantsof polycrystallinegraphite 6.CONCLUSIONS
A self-consistent integral equation technique has been used to calculate the elastic properties of polycrystalline fine-grain graphites. The present computational technique is restricted to spherical grains and pores and to uncorrelated structures. When the grain properties were taken to be those of the perfect crystal, the average error of the predicted moduli ranged from 45 to several hundred percent. When the grain properties were taken to be those of a cracked crystal (to compensate empirically for correlated microcracks) the average error ranged from 0 to 20%. The small values inferred for the c-axis modulus of the cracked crystal need to be examined further. This examination might include analytical studies to calculate the moduli of crystals containing correlated microcracks or experimental studies to measure the cracked-crystal moduli. Changes in the elastic properties of an isotropic graphite caused by neutron irradiation or by temperature variations were accounted for by plausible changes in the inferred grain properties. However, the further utilization of the techniques reported here would require additional knowledge of the actual changes in graphite grain properties and microstructure due to the irradiation and thermal environment.
189
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