The theory originally developed by Simmons in the 1960s reported in detail by Hall et al61 relating the polycrystalline dimensional change rate and CTE with crystallite dimensional change rate, and CTE has been further developed95 in an attempt to explain the shape of the graphite irradiation creep behavior at high dose. The proposed theory argues that if the dimensional change rate in polycrystalline graphite can be related to the CTE, and because irradiation creep has been observed to modify CTE of the loaded specimen differently to that seen in unloaded specimens, changing the CTE by creep would be expected to change the dimensional change rate and hence, the dimensional change in the loaded specimen. This leads to the introduction of the so-called ‘interaction strain.’ The theory behind this methodology is described below.
Considering two specimens (a crept specimen and an unloaded control) being irradiated under identical conditions; in the unloaded control specimen, by applying the Simmons equations, the bulk dimensional change rate gx and bulk CTE ax can be defined by
g (1 AX)g! ^
ax (1 Ax)aa ^ Axac [58]
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Therefore, the difference between the dimensional change rates ofthe unloaded and loaded specimen is
gx=gT(y—a: 1621
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Figure 65 High-fluence German and US data.
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where ga and gc are crystal dimensional change rate in the a and c directions, respectively, and aa and ac are the crystal CTE in the a and c directions, respectively. Ax is referred to as the structure factor and by rearrangement
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• Apparent creep = dimensional change in loaded
specimen —dimensional change in control
Thus, the interaction term gr^—г) is included in the finite element analysis of graphite components.
The limited data that exists on irradiated HOPG indicates that the dimensional change rate of graphite crystallites increases with increasing fluence in the ‘c’ direction and decreases in the ‘a’ direction for all measured irradiation temperatures and dose range. For irradiation temperatures of 450 and 600 °C, the data indicates that ac and aa remain invariant to fluence. However, below 300 °C the crystal CTE appears to change. There are no crystal CTE data for higher temperatures.
It should also be noted that Simmons equations imply that
^ ax aa gx ga
Ax
ac —aa gc —ga
Close examination of typical graphite irradiation data, say for Gilsocarbon irradiated in the temperature range where crystal data are available (450 and 600 °C), shows that the relationship given above does not hold. In fact, the Simmons relationship and measured data diverge at low dose. This is attributed to Simmons assuming that polycrystalline graphite can be considered as a loose collection of crystallites with no mechanical interactions. Others95 have added an extra ‘pore generation’ term to the Simmons dimensional change relationship to try and reconcile these issues, but again there is no real validation of these models.
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gx = ga + gT 0х—г [60]
ac aa
where gT is the crystal shape rate factor and is equal to gc —ga. Similarly for the loaded specimen,
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or
gx = gx+gT(y—a;) 1631
where Aa is the change in CTE under load (aX — ax). This leads to the following definitions:
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• The true dimensional change in the loaded specimen = the dimensional change in the control + the interaction term
• True creep = dimensional change in loaded specimen — true dimensional change in loaded specimen
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The use of this interaction term did not gain wide (international) acceptance as it appeared to be using the Simmons relationship beyond its applicability and did not explain the difference between compressive and tensile loading at high fluence.
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of stresses caused by dimensional change. However, it is difficult to envisage such a yield and shear mechanism in crystalline graphite. The second model98 suggests that under load, the crystallite basal planes will slide because of a pinning and unpinning mechanism during irradiation. Such a mechanism is described in detail by Was99 with relation to metals and could explain primary creep and secondary linear creep. However, if irradiation creep in graphite is associated with basal plane slip due to pinning-unpinning, it is surprising that in PGA, irradiation creep is less in the WG or parallel to the basal plane direction than it is in the AG or perpendicular to the basal plane direction (Figure 57).
Another possibility is that stress modifies the crystal dimensional change rate itself. In support of this are X-ray diffraction measurements1O° that showed that the lattice spacing in compressive crept specimens is less than that in the unstressed control specimens (Figure 66). Such a mechanism would explain the PGA data and could be related to the change in CTE and the observed annealing behavior. However, the data and experimental fluence and creep range given are very limited. It is clear that changes to the lattice spacing in crept graphite would be an area worthy of further investigation in future irradiation creep programs.
Irradiation creep in the graphite crystallite will be reflected in the bulk deformations observed in creep specimens and in reactor components. Changes to the bulk microstructure due to radiolytic oxidation would be expected to influence this bulk behavior, as would large crystal dimensional changes at very high fluence (past dimensional change turnaround). It would be expected that at very high fluence the behavior of graphites with differing microstructures would diverge; this appears to be the case from the limited high fluence data available.
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where a, 1 esu (where this is defined by s/EO); ki, 0.0857e(163O.4/T); b, 0.15 esu per 102Oncm—2 EDN; k2, 0.0128e(127O.8/T); o, 5 esu; k-, 0.4066e(—13359/T); s, stress (P); EO, unirradiated SYM (Pa) appropriate to the stress applied (O.84 x DYM); S(y, T), structure term representing structural induced changes to creep modulus (a function offluence and temperature); W(x), oxidation term representing oxidation-induced changes to creep modulus (a function of weight loss, x, which is a function of fluence).
The lateral strain ratio for the primary and recoverable terms is assumed to be equal to the elastic Poisson’s ratio. The lateral strain ratio for secondary creep, nsc, is assumed to follow the relationship
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Sc is a structural connectivity term that the authors have used in model fits for other graphite property changes.57 This model certainly fits the available inert data better than the previous models, although it cannot be tested against radiolytically oxidized — graphite data as there is none.
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Nuclear grade graphite has been used, and is still used, in many reactor systems. Furthermore, it provides an essential moderator and reflector material for the next-generation high-temperature gas-cooled nuclear reactors that will be capable of supplying high-temperature process heat for the hydrogen economy. Hence, nuclear graphite technology remains an important topic. Although there is a wealth of data, knowledge, and experience on the design and operation of graphite-moderated reactors,
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