The response of synthetic graphite to a stress (its elastic behavior) is dominated by the bond anisotropy in the graphite single crystal lattice, the preferred crystal orientation, and the presence of defects (porosity) in the structure. The elastic response of the strong covalent in-plane bonds of the carbon atoms in the graphene sheets will be vastly different from the graphene sheets held in stacks with weak van der Waals forces. Definition of the stress-strain relationship for the hexagonal graphite crystal requires five independent elastic constants.23 These constants are identified as (using a Cartesian coordi­nate system with the z-axis parallel to the hexagonal axis of the crystal or c-axis):

Txx C11exx T C12eyy T C13ezz

Tyy C12exx T C11eyy T C13ezz

Tzz C13 exx T C13 eyy T C33ezz

Tzx C44ezx

Tzy C44ezy

Txy = 1/2(C11 — Cu)Sxy = C66£xy [9]

where the stresses T/m are defined as the force acting on the unit area parallel to the /th direction; the normal to the unit area is the mth direction. The parameters Cjare the elastic moduli and their inverse Sj are the elastic compliances. The various measure­ments of compliances made on single crystals and highly oriented pyrolytic graphites have been reviewed by Kelly12 and are reported to be the best available estimates (Table 2).

Table 2 reports the Young’s modulus parallel to the hexagonal axis of the crystal Ec = S—31 ~ 36.4GPa, the Young’s modulus parallel to the basal planes, Ea = S—11 ~ 1020 GPa, and the shear modulus paral­lel to the basal planes G = S—1 = C44 ~ 4.5 GPa. The very low value of Ec results from the very weak interlayer van der Waals bonding, while the value of Ea reflects the magnitude of the in-plane (sp2) C-C covalent bonds. Kelly12 reviewed the literature data for

Elastic moduli (GPa) Elastic compliances

(10-13 Pa-1)

Table 2 Elastic constants of single crystal graphite On 1060±20 S11 9.8 ± 0.3 C12 180 ± 20 S12 -1.6 ± 0.6 C13 15 ± 5 S13 -3.3 ± 0.8 C33 36.5 ± 1 S33 275 ± 10 C44 4.0-4.5 S44 2222-2500 Data from Kelly, B. T. Physics of Graphite; Applied Science: London, 1981.

experimental values of C44 and noted that while many reported values are lower than those reported in Table 2, the presence of glissile basal plane disloca­tions can reduce C44 by one or two orders of magnitude.

The Young’s modulus, E, of the single crystal will thus depend upon orientation of the crystal to the Measurements (stress) axis. Thus, E-1, may be writ­ten as a function of the angle, ‘, with respect to the crystal hexagonal axis as

E-1 = Sn(1 — g2)2 + S’33g4

+ (2S13 + S44)g2 (1 — g2) [10]

where g = cos’. Taking values for the elastic com­pliances from Table 2, and allowing S44 = 2.4 x 10- Pa-1, the variation of the reciprocal modulus with the angle between the measurement direction and the crystal c-axis, ‘, can be calculated using eqn [10] (Figure 16). Also plotted in Figure 16 is an approxi­mate value of E-1 calculated, allowing all Sj to be zero except S44. The agreement between the two is good over a wide range of values of ‘, clearly demon­strating the dominance of S44 in controlling the elastic modulus and other mechanical properties in polygranular (synthetic) graphites.

The graphite single crystal shear modulus, G (Pa-1), can also be calculated12 from the elastic compliances (Sj) given in Table 2 (allowing S44 = 2.4 x 10-10Pa-1) from eqn [11]:

G-1 = S44 + (s„ — S12 — (1 — g2)

+ 2(S11 + S33 — 2S13 — S44)g2 (1 — g2) [11]

Values of G-1 as a function of ‘ are plotted in Figure 17. The value of the crystal shear modulus at ‘ = 0, that is, parallel to the basal planes, is the smallest shear modulus and corresponds to C44 for crystal basal plane shear (weak van der Waals forces), again demonstrating the dominance of S44 in controlling the elastic moduli of the crystal. It is

Angle between direction of measurement and crystal c-axis, f (rad) Figure 16 Variation of the reciprocal Young’s modulus with angle of miss-orientation between the c-axis and measurement axis.

interesting to note that while G-1 displays a mini­mum (largest G) at ~я/4, the reciprocal Young’s modulus value displays a maximum (smallest E) at ~я/4, reflecting the different modes of bond stretching and the different bonding nature in these cases.

The crystals in synthetic graphite are not as per­fect as discussed above. Moreover, while both the filler-coke and the binder phase exhibit crystallinity, the alignment of the crystallite regions within the filler and binder is not uniform, although it may display preferred orientation because of filler-coke calcination and the formation of the synthetic arti­fact. Texture also arises because of the alignment of the filler particle during formation. Consequently, the single crystal values of moduli are not realized in synthetic (polygranular) graphite. Typical values of Young’s modulus for synthetic graphite are given in Table 1. There is considerable variation ofYoung’s modulus with density. However, the effect of texture is clearly seen in the anisotropic values for grade AGX, 6.9 GPa (WG) and 4.1 GPa (AG), confirming the tendency of the filler grains to align on extrusion such that WG orientation displays more of the c-axis Young’s moduli (0 and p/2 in Figure 16) and is thus greater than the AG value. Grades with a greater density (^1.8gcm-) possess a Young’s modulus value, ~10 GPa, far less than the single crystal values (except C44). Measurements of the shear modulus derived from the velocity of shear (transverse) waves propagating through the graphite give values ranging from 3 to 4 GPa for various grades all with density ^1.8gcm- . The values of shear modulus (WG) were slightly greater than shear modulus (AG)

for extruded and molded grades, as expected from the theory and textural effects. The measured values of shear modulus for synthetic graphites (3-4 GPa) are in reasonable agreement with the single crystal value of C44 (4-4.5 GPa) reported in Table 2. The presence of glissile basal plane dislocations would be expected to reduce the single crystal value of shear modulus substantially, but the defective nature of the crystals in synthetic graphite assures a high density of dislocation pinning sites, which will increase the value of shear modulus. For a wide range of synthetic graphites, the ratio of E/G ~ 3 is in agreement with previous observations.12 In addition to the effects of texture, such as the crystal/filler preferred orientation, on the elastic moduli of synthetic graphite, the existence of porosity with a wide size, shape, and orientation distribution (refer to Section 2.10.2) makes the interpretation of the elastic properties more complicated. Typical Young’s moduli for syn­thetic graphites range from 5 to 15 GPa (Table 1 ), with lower modulus being exhibited by coarser-grain graphite. The combined effect of porosity and texture in synthetic graphite causes anisotropy of the moduli, with the anisotropy ratio for Young’s modulus being as large as 2. Finer-grain graphites are far more isotropic with respect to Young’s modulus.

Various workers have studied the changes in the elastic compliances of single crystal graphite with temperature, as reviewed by Kelly.12 The single crystal elastic constants decrease with increasing temperature.2 The value of C33 decreases linearly
by more than a factor of 2 over the temperature range 0-2000 K.12 Synthetic polygranular graphite, because of the influence of texture and porosity, shows a completely different temperature depen­dence of their Young’s moduli. The thermal closure of cracks/pores aligned along the a-axis (between the basal planes — the ‘Mrozowski’ cracks15) will cause an increase4,21,24 in Young’s modulus of ^50% from room temperature to ^2300 K. Above this tempera­ture up to ^3000 K, Young’s modulus is reported to decrease slightly.21 Typical high-temperature behav­ior of Young’s modulus is shown in Figure 18 for pitch-coke and petroleum-coke graphite.

Poisson’s ratio for fine-grained isotropic graph­ite (ATJ) has been reported to be 0.1-0.16, and for coarser-grained extruded graphite (AGOT) 0.04-0.09, the value being dependent upon the mea­surement direction relative to the forming axis.21 In a large study of the room-temperature elastic proper­ties of 15 graphite grades measured using the velocity of sound waves (both longitudinal and transverse), the Poisson’s ratio value was seen to be between 0.17 and 0.24 for fine-grain, isotropic grades and between 0.28 and 0.32 for medium-grain, molded, or extruded grades.

2.10.4.2 Strength and Fracture

Typical compressive and tensile stress-strain curves for medium-grain, extruded graphite are shown in Figures 19 and 20, respectively. The stress-strain curve is nonlinear and typically shows hysteresis on

Figure 20 Typical tensile stress-strain curves for medium-grain extruded graphite (WG).

reloading after loading below the fracture stress, with a permanent set.12,23,25 The nonlinearity has been widely attributed to pseudo-plastic events such as basal plane shear and subcritical cracking.12,23-27 Table 1 reports the tensile, flexure, and compressive strength of a range of synthetic graphites. Tensile strengths vary with texture from as low as <5 MPa for coarser-grain, extruded grades to >60 MPa for fine-grain, isotropic grades and can be >80 MPa for some isostatically molded, ultra-fine-grain and micro-fine-grain synthetic graphite. Compressive strengths range from <20 MPa to >140 MPa and typically, the ratio of compressive strength to tensile strength is in the range 2-4. Kelly12 reports that there are two major factors that control the stress-strain behavior of synthetic graphite, namely, the mag­nitude of the constant C44, which dictates how the crystals respond to an applied stress, and the defect/ crack morphology and distribution, which controls

the distribution of stresses within the body and thus the stress that each crystallite experiences.

Generally, the strength increases as the modulus increases but is also greatly influenced by factors such as texture and density (total porosity). The strength (or Young’s moduli) is related to the frac­tional porosity through a relationship of the form

s = ff0e~bP [12]

where P is the fractional porosity, b is an empirical constant, and s0 represents the strength at zero porosity. Figure 21 shows the correlation between flexure strength and fractional porosity for a wide range of synthetic graphite27 varying from fine — grain, high-density, isomolded grades to large-grain, low-density, extruded grades. The data are fitted to an equation of the form of eqn [12] with s0 = 179 MPa and b = 9.62. The correlation coefficient, R2 = 0.80. Significantly, the same flexure strength data (Figure 22) is better fitted when plotted against the mean filler-particle size27 (R2 = 0.87). In synthetic graphite, the filler-particle size is indicative of the defect size, that is, larger filler-particle graphite con­tains larger inherent defects. Thus, the correlation in Figure 22 is essentially one between critical defect size and strength. The importance of defects in controlling fracture behavior and strength in syn­thetic polygranular graphite is well understood, and despite the pseudoplasticity displayed by graphite, it is best characterized as a brittle material with its fracture behavior described in terms of linear elastic fracture mechanics.28 Synthetic graphite criti­cal stress-intensity factor, KIc, values are between 0.8 and 1.3 MPam~1/2 dependent upon their texture

Figure 21 The correlation between mean 3-pt flexure strength and fractional porosity for a wide range of synthetic graphite representing the variation of textures. Reproduced from Burchell, T. D. Ph. D. Thesis, University of Bath, 1986.

Figure 22 The correlation between mean 3-pt flexure strength and mean filler-coke particle size for a wide range of synthetic graphite representing the variation of textures. Reproduced from Burchell, T. D. Ph. D. Thesis, University of Bath, 1986.

and the method of determination.27-29 Such is the importance of the fracture behavior of synthetic graphite that there have been many studies of the fracture mechanisms and attempts to develop a pre­dictive failure model.

An early model was developed by Buch30 for fine — grain aerospace graphite. The Buch model was fur­ther developed and applied to nuclear graphite by Rose and Tucker.31 The Rose and Tucker model assumed that graphite consisted of an array of cubic particles representative of the material’s filler — particle size. Within each block or particle, the graphite was assumed to have a randomly oriented crystalline structure, through which basal plane cleavage may occur. When a load was applied, those cleavage planes on which the resolved shear stress exceeded a critical value were assumed to fail.

If adjacent particles cleaved, the intervening bound­ary was regarded as having failed, so that a contigu­ous crack extending across both particles was formed. Pickup et at32 and Rose and Tucker31 equated the cleavage stress with the onset stress for acoustic emission (AE), that is, the stress at which AE was first detected. In applying the model to a stressed component, such as a bar in tension, cracks were assumed to develop on planes normal to the axis of the principal stress. The stressed component would thus be considered to have failed when sufficient particles on a plane have cleaved such that together they formed a defect large enough to cause such a fracture as the brittle Griffith crack. Pores were trea­ted in the Rose and Tucker model as particles with zero cleavage strength. The graphite’s pore volume was used to calculate the correct number of zero cleavage strength particles in the model. Hence, the Rose and Tucker model took into account the mean size of the filler particles, their orientation, and the amount of porosity but was relatively insensitive to the size and shape distributions of both microstruc­tural features.

Rose and Tucker applied their fracture model to Sleeve graphite, an extruded, medium-grain, pitch-coke nuclear graphite used for fuel sleeves in the British AGR. The performance of the model was disappointing; the predicted curve was a poor fit to the experimental failure probability data. In an attempt to improve the performance of the Rose and Tucker model, experimentally determined filler-particle dis­tributions were incorporated.33 The model’s predic­tions were improved as a result of this modification and the higher strength of one pitch-coke graphite compared with that of the other was correctly pre­dicted. Specifically, the predicted failure stress distribution was a better fit to the experimental data than the single grain size prediction, particularly at lower stresses. However, to correctly predict the mean stress (50% failure probability), it was found necessary to increase the value of the model’s stress — intensity factor (KIc) input to 1.4MPam1/2, a value far in excess of the actual measured KIc of this graph­ite (1.0MPam1/2). The inclusion of an artificially high value for KIc completely invalidates one of the Rose and Tucker model’s major attractions, that is, its inputs are all experimentally determined material parameters. A further failing of the Rose and Tucker fracture model is its incorrect prediction of the buildup of AE counts. Although the Rose and Tucker model considered the occurrence of subcritical dam­age when the applied stress lay between the cleavage and failure stresses, the predicted buildup of AE was markedly different from that observed experimen­tally.34 First, the model failed to account for any AE at very low stresses. Second, at loads immediately above the assumed cleavage stress, there was a rapid accumulation of damage (AE) according to the Rose and Tucker model but very little according to the AE data. Moreover, the observation by Burchell et al.34 that AEs occur immediately upon loading graphite completely invalidated a fundamental assumption of the Rose and Tucker model, that is, the AE onset stress could be equated with the cleavage stress of the graphite filler particles.

Recognizing the need for an improved fracture model, Tucker et al?5 investigated the fracture of polygranular graphites and assessed the performance of several failure theories when applied to graphite. These theories included the Weibull theory, the Rose and Tucker model, fracture mechanics, critical strain energy, critical stress, and critical strain theories. While no single criteria could satisfactorily account for all the situations they examined, their review showed that a combination of the fracture mechanics and a microstructurally based fracture criteria might offer the most versatile approach to modeling frac­ture in graphite. Evidently, a necessary precursor to a successful fracture model is a clear understanding of the graphite-fracture phenomena. Several approaches have been applied to examine the mechanism of fracture in graphite, including direct microstructural observations and AE monitoring.34,36-38

When graphite is stressed, micromechanical events such as slip, shear, cleavage, or microcracking may be detected in the form of AE. In early work, Kaiser39 found that graphite emitted AE when stressed, and upon subsequent stressing, AE could only be detected when the previous maximum stress had been exceeded — a phenomenon named the Kaiser effect. Kraus and Semmler40 investigated the AE response of industrial carbon and polygranular graphites subject to thermal and mechanical stresses. They reported significant AE in the range 2000­1500 °C on cooling from graphitization temperatures, the amount of AE increasing with the cooling rate. Although Kraus and Semmler offered no explanation for this, Burchell et al.34 postulated that it was asso­ciated with the formation of Mrozowski cracks.15 In an extensive study, Burchell et al?4 monitored the AE response of several polygranular graphites, ranging from a fine-textured, high-strength aero­space graphite to a coarse-textured, low-strength extruded graphite. They confirmed the previous results of Pickup et al.,32 who had concluded that the pattern of AE was characteristic of the graphite microstructure. Burchell et al?4 showed that the development of AE was clearly associated with the micromechanical events that cause nonlinear stress — strain behavior in graphites and that postfracture AE was indicative of the crack propagation mode at frac­ture. For different graphites, both the total AE at fracture and the proportion ofsmall amplitude events tended to increase with increasing filler-particle size (i. e., coarsening texture). Ioka et al.41 studied the behavior of AE caused by microfracture in polygra­nular graphites. On the basis of their data, they described the fracture mechanism for graphite under tensile loading. Filler particles, whose basal planes were inclined at 45° to the loading axis deformed plastically, even at low stresses. Slip deformation along basal planes was detected by an increased root mean square (RMS) voltage of the AE event amplitude. The number of filler particles that deform plastically increased with increasing applied tensile stress. At higher applied stress, slip within filler particles was accompanied by shearing of the binder region. Filler grains whose basal planes were perpendicular to the applied stress cleaved, and the surrounding binder sheared to accommodate the deformation. At higher stress levels, microcracks propagated into the binder region, where they coa­lesced to form a critical defect leading to the eventual failure of the graphite. The evidence produced through the numerous AE studies reviewed here suggests a fracture mechanism consisting of crack initiation, crack propagation, and subsequent coales­cence to yield a critical defect resulting in fracture.

A microstructural study of fracture in graphite27,42 revealed the manner in which certain microstructural features influenced the process of crack initiation and propagation in nuclear graphites (Figure 23); the principal observations are summarized below.