Polycrystalline uranium undergoes a kind of dimensional instability when sub­jected to repeated heating and cooling (i. e., thermal cycling) in the alpha-phase regime. This phenomenon is known as thermal cycling growth and results in (i) growth (change in length, that is, increase or decrease) and (ii) surface roughen­ing arising out of wrinkling effect. The growth effect comes from a thermal ratchet­ing mechanism involving (i) relative movement between two neighboring grains with different thermal expansion coefficient due to the basic anisotropy of alpha — uranium, and (ii) stress relaxation in one of the grains by plastic deformation or creep. An interesting example of thermal cycling growth is shown in Figure 7.6, where an alpha-uranium rod has elongated over several times due to thermal cycling between 50 and 500 °C. In superplasticity literature, it is known as thermal cycling superplasticity! Thermal cycling growth coefficient (Gt) is expressed by

As thermal cycling is an inherent feature of nuclear reactor kinetics, it can thus have important influence on the thermal stability of the fuel. The gamma phase of

uranium does not show thermal cycling growth phenomenon and may thus be desirable. Thus, suitable amounts of gamma stabilizing alloying additions (Al, Mo, and Mg) help in avoiding thermal cycling growth effect as illustrated with an exam­ple from U-Mo system (Figure 7.7). Note that U-Mo alloys generally contain at least 6 wt% Mo in order to avoid thermal cycling growth.

Helium gas is produced through transmutation of the component elements in aus­tenitic (FCC) stainless steels and other materials. This can lead to embrittlement behavior that cannot be eliminated by high-temperature annealing. Helium is practi­cally insoluble in metals and hence after generation, it tends to precipitate into bub­bles particularly when the temperature is high enough (>0.5Tm) for helium atoms to migrate. Helium may produce severe embrittlement (intergranular cracking) to such an extent that at elevated temperatures even if the yield strength recovers in the irradiated alloy, the ductility is never regained. The extent of helium embrittlement depends on fast neutron fluence, alloy composition, and temperature.

The source of helium in steels is the component elements present in them. Helium is generated in threshold reactions due to the interaction of neutron with the specific isotopes comprising (n, a) reaction. Boron (B10) and Ni58 are such ele­ments important in generating alpha particle (i. e., helium nucleus or He4) through the following reactions:

B10 + n1 ! Li7 + He4. (6.10)

Ni58 + n1 ! Ni59; Ni59 + n1 ! Fe56 + He4. (6.11)

These reactions occur in both thermal and fast neutron spectra. Similarly, helium (n, a) reactions between fast neutrons and Ni, Fe, Cr, and N atoms occur, although with different reaction cross sections. In fast reactors, helium embrittlement is more pronounced simply because the fast neutron flux in the fast reactor is about three to four orders of magnitude higher than the thermal flux, whereas the ratio is 1:1 in the thermal reactors. Helium embrittlement remains a widely studied topic. Olander [11] summarized various theories of helium embrittlement:

i) Woodford, Smith, and Moteff postulated that the increased strength of the matrix material due to the presence of helium bubbles in the grain interiors would lead to stress concentration at the grain boundary triple points during deformation. But as the stress concentration at the grain boundaries would not be able to relax itself, this would induce grain boundary triple point cracks and eventual propagation of cracks along the grain boundaries. So this theory espouses an indirect origin on helium embrittlement.

ii) Kramer and coworkers showed that helium bubbles can form on the grain boundary carbide particles (M23C6), thereby allowing cracks to form. Now

the question is why helium bubbles tend to nucleate on the carbide parti­cles. One aspect of it could be that these carbide particle surfaces reduce the critical nucleation barriers of the helium bubbles. But it is not clear why it happens to be on these M23C6 particles. Boron was found to be asso­ciated with these structures as [M23(CB)6]. So, when boron transmutes through (n, a)-type reaction, helium is produced close to the carbide parti­cles and forms bubbles on the particle itself. Thus, bubbles are formed on or very near to the grain boundary, thus promoting the possibility of helium embrittlement.

iii) Reiff showed that the presence of helium in triple-point cracks permits unstable growth of these cracks at stresses much lower than that required for a gas-free crack to propagate. The presence of helium deteriorates the grain boundary cohesion, thereby leading to weaker grain boundaries that cannot sustain larger loads.

iv) A majority of researchers believe that this phenomenon occurs due to the stress-induced growth of helium bubbles on the grain boundaries that eventu­ally link up and cause intergranular failure. However, as you can see from above, all these theories are related and would explain the behavior depending on the situation. Perhaps all the above factors play a role in helium embrittle­ment. Figure 6.36 shows the various schematic locations of helium bubble for­mation. On the other hand, Figure 6.37 shows the formation of helium bubbles in different 82 series alloys with a base composition of Fe-25Ni-15Cr.

It demonstrates the formation of helium bubbles at the secondary MX-type precipitates, primary MX precipitates, on the grain boundary M23C6 precipi­tates, and on the grain boundaries themselves.

temperature of 650°C): (a) on secondary MX particles, (b) on primary MX particles, (c) on M23C6 particles, and (d) on the grain boundaries [20].

The effect of irradiation temperature on the ductility (percentage elongation to fracture) of the irradiated 304-type stainless steel is shown in Figure 6.38. The fast neutron fluence was kept at >1022 ncm~2 s-1 during irradiation. Tensile tests were conducted at 50 °C. Note the dip in ductility due to helium embrittlement near ~580 °C.

Interestingly, BCC metals/alloys are less vulnerable to helium embrittlement (i. e., no drastic loss in ductility). It is believed that the large diffusion coefficients in BCC materials due to their more open structure help in relaxing stress concentra­tions at the grain boundaries effectively and thus minimize the stress-enhanced helium bubble growth.

6.2.4

As mentioned before, freshly cut thorium is bright with silvery luster, but darkens quickly on exposure to air. The oxidation creates protective oxide film of thorium oxide (ThO2) up to a temperature of ~350 °C. At further higher temperatures, the breakaway transition occurs when the oxide film cracks and the oxidation proceeds almost linearly. At about 1100 °C, the oxidation rate becomes parabolic again. Tho­rium corrodes at a slow rate at around 100 °C in high-purity water with the forma­tion of an adherent oxide film. In the temperature range of 178-200°C in water, the oxide growth rate becomes rapid and eventually starts to spall (i. e., break up). The reaction becomes very rapid at 315 ° C. Thorium has good resistance against most metals barring aluminum below 900 °C.

Although the mechanism by which irradiation affects stress corrosion cracking is not precisely known, existing theories fall in the following categories: (a) radiation-induced

grain boundary chromium depletion (radiation-induced segregation), (b) radiation hardening, (c) localized deformation, (d) selective internal oxidation, and (e) irradiation creep.

Corrosion by liquid metal and molten salts (especially fluorides) is seldom affected by radiation. Liquid metals appear to be unaffected by radiation (except to develop radioactivity). Reader is referred to various other references for further details.

6.5

Principal planes are the planes on which maximum normal stresses act with no shear stresses and these stresses are known as “principal stresses.” These are desig­nated as CTj, o2, and o3 implying no shear stresses or

Sij — Sij dj with dij — 0 for i — j. (A.6)

Such principal stresses can be found for the 2D case using Mohr’s circle; however, for the general 3D case, one can determine them as the three solutions to the deter­minant of the stress tensor:

where the indices 1, 2, and 3 are used in place of x, y, and z. The three solutions for the determinant are the principal stresses and it is common practice to designate them to be s1 > o2 > o3 taking into account the sign as well.

Expansion of the determinant gives

0 — 03 — (on + 022 + a33)o2 + (О11О22 + S22S33 + О33О11 — o22 — o23 — o2Jo

— (o11022033 + 2o12023031 — s11 o23 — s22o21 — s33o22)

(A.7a)

or

o3 — I1o2 — I2o — I3 — 0, (A.7b)

where I’s are invariants of the stress tensor:

11 — (011 + 022 + 033),

12 — -(011022 + 022033 + 033011 — of2 — o:;3 — o|1),

13 — o11o22o33 + 2o12o23o31 — o11o23 — o22°21 — o33o22.

Note that I1 is the trace of the determinant (sum of diagonal terms).

A.3

In certain materials under neutron irradiation, there is no way of turning off the helium production as a means ofcontrolling void formation. So, in materials where helium is produced, helium bubbles are formed and contribute to the swelling behavior. Some details of helium production have been discussed in Section 6.2.3. In reality, void formation takes place under conditions of damage rate, temperature, and sink density at which the vacancy supersaturation is not enough to nucleate voids. This observation led Cawthorne and Fulton [18] to suggest that helium atoms in the metal stabilize the small void nuclei and prevent the collapse of embryo voids into vacancy loops. Helium remains intimately involved in the formation of voids as nucleation process turns heterogeneous. So, voids are essentially partially or

fully vacuum. As the radiation dose increases, more helium is produced to fill up the voids fully and then the features are rather called “bubbles.” However, one important distinction should be remembered — voids do not need helium gas to grow, but bubbles need gas to grow. The morphology of the bubbles tends to be more spherical compared to voids that assume faceted shape.

Radiation growth of uranium is another form of dimensional instability that occurs under irradiation without the need of any stress in a lower temperature (i. e.,

around 300 °C) regime. Since it does not require stress to occur, it is not considered radiation-induced creep. Also, since the volume of the material remains constant during radiation growth, it is not considered as irradiation swelling. Under radia­tion exposure, a single crystal of alpha-uranium expands in the [010] direction, con­tracts along the [100] direction, and remains more or less unaltered in the [001] direction. The result of this characteristic expansion/contraction is that the volume remains essentially constant. However, in order for radiation growth to take place, single crystal of uranium is not essential but polycrystalline uranium strong crystal­lographic texture can also exhibit the effect. Deformation processing and heat treat­ment is important to minimizing or eliminating the radiation growth effect by suitable texture engineering. Minimization of radiation growth can be achieved by processing the material to produce a fine-grained microstructure with randomly ori­ented grains. Also, suitable alloying additions to stabilize isotropic phases can help.

The radiation growth coefficient (Gt) is given by Percentage length increase

Percentage atom burnup

There have been a number of studies to understand radiation growth, but it remains elusive. A leading hypothesis of radiation growth by Buckley [ ] is based on differential directional rates of interstitial atoms and vacancies. The interstitials have a tendency to migrate along the [010] direction and to the vacancies in the [100] or [001] directions, leading to basically removal of mass from one side and plating them on the other side. Figure 7.8 shows the irradiation growth effect in a uranium fuel against the fuel burnup.

Creep is time-dependent, thermally activated plastic deformation process, as described in Chapter 5 where we entirely dealt with the conventional thermal creep

that generally occurs at elevated temperatures (>0.4Tm) under stress. Materials under stress could undergo creep effects (contributes to the dimensional instability of an irradiated material) under energetic particle flux (such as fast neutron expo­sure) even at much lower temperatures where thermal creep is essentially negligi­ble in the absence of neutron irradiation. The generation of point defects is at the heart of the irradiation creep process. One way to understand the process is from the point of vacancy production in materials from two sources — thermal vacancies (C*) and neutron-induced (C*). That is, the total vacancy concentration Cv = C* + C*. Thus, the total irradiation creep rate (eirr) can be expressed in two components:

eirr = e *+ eth (6.12)

where e* is affected by the radiation component and eth is affected by the thermal creep contribution. However, irradiation creep effect could be quite complex and research is ongoing to fully understand the effect in different material systems. Here, we lay out a general discussion on irradiation creep by categorizing it into two types: radiation-induced creep and radiation-enhanced creep. This is a rather simplistic way of describing irradiation creep, although it has substantial pedagogic advantages.

Radiation-induced creep occurs at lower homologous temperatures at which ther­mal creep is negligible (eth), that is, not thermally activated. At these lower tempera­ture regions, the vacancy concentration produced by atomic displacements due to irradiation (that are not in thermal equilibrium, but produced as a function of flu — ence) could be large enough to induce creep deformation under the application of

stress. A simple relation used for describing radiation-induced creep rate (eirr) is given by

eirr и e* = Bop, (6.13)

where B is a constant relatively insensitive to the test temperature, o is the applied stress, and p is the neutron flux. From the above relation, it is clear that the radia­tion-induced creep rate is directly proportional to the stress and the neutron flux. In essence, it means that with increasing stress and neutron flux, the creep effect would accelerate. If one integrates the above equation over time, it can be seen that the creep strain would vary with the neutron fluence (the product of flux and time). In 1967, Lewthwaite and coworkers in Scotland [30] demonstrated irradiation creep in several metals and alloys at 100 ° C and published their findings in Nature (a well — known journal). Later, the radiation-induced creep has been observed in a number of alloys, including ferritic-martensitic steels (HT9 and T91) and austenitic steels as well as zirconium-based alloys. One way to study the irradiation creep behavior has been to irradiate stressed specimens under neutron exposure and study the stress relaxation behavior. There have been several studies using this mode of testing [31]. Irradiated annealed 304-type austenitic stainless steel specimens at 30°C in a reactor under a neutron flux of 1013 ncm-2 s-1 for ~127 days to a total fluence of ~1.1 x 1019ncm-2. The specimens were subjected to torsional strain due to the application of ~30 MPa. It was found that the level of relaxation in the irradiated material was about a factor of 40 more compared to that in the unirradi­ated (control) material under comparable temperature and stress.

Radiation-enhanced creep, as the term suggests, is the creep process enhanced by irradiation. This occurs at homologous temperatures at which thermal creep can also operate. As we know, generation of defects like vacancies at higher tem­peratures increases the thermal vacancy concentration in the material. This translates into the increase of diffusivity. Thermal creep rate can be shown to be proportional to diffusivity (Section 5.1). Now the addition of more vacancies pro­duced through fast neutron irradiation can augment the vacancy concentration further, enhancing the overall creep rate. In this case, total radiation enhanced creep rate is given by

eirr = e * + eth = Bop + ADon, (6.14)

where eth = ADon. Here A is a constant, D is the diffusivity, o is the applied stress, and n is the stress exponent (n could be 5 or some other number depend­ing on the conditions) (see Eq. (5.40)). In Eq. (6.14), D is proportional to the vacancy concentration that comprises contributions from thermal vacancies (e-sf=lT) and atomic displacements due to radiation (/ dpa).

One example of radiation-enhanced creep is shown in Figure 6.39 from the work of J. R. Weir [32]. Weir determined in-reactor stress-rupture properties of hot — pressed beryllium. Figure 6.39 shows the stress versus rupture time for three beryl­lium materials under three conditions. The neutron flux was 9 x 1013 ncm-2 s-1. The results of the unirradiated material are compared with those of two types of irradiated materials. Some specimens were loaded after placing it in-reactor;

Rupture Time (h)

Figure 6.39 In-reactor stress-rupture properties of hot pressed beryllium at 600 °C.

however, a few specimens were loaded later after the specimens were irradiated for 800 h. The temperature was kept at ~600 °C (i. e., a homologous temperature of ~0.56). The stress-rupture life of the specimens got reduced when the specimen was loaded at once. But the specimen that was irradiated for 800 h accumulated more radiation damage leading to less rupture life. Similar reductions in stress — rupture life were found when stress-rupture tests were conducted on an irradiated 316-type austenitic stainless steel in the temperature range of 540, 600, 650, and 760 °C. Prior to stress-rupture tests, the steel specimens were irradiated up to a total neutron fluence of 1.2 x 1022 ncm~2 at an irradiation temperature of 440 °C.

Irradiation creep sometimes operates at the same time as swelling and radiation growth (if applicable). In such situations, it becomes important to distinguish the contribution of irradiation creep to the total strain. Toloczko and Garner [33] have analyzed irradiation creep data from HT-9 and used the concept of creep compliance (B0) to estimate the contribution of irradiation creep independent of swelling. Figure 6.40 shows the inclusion of an irradiation creep regime along with other thermal creep mechanisms on a deformation mechanism map of a 316-type stainless steel.

6.2.5

Many alloying additions have been attempted for improving mechanical properties and corrosion resistance of thorium. Only few elements (like zirconium and

a) Note that the materials were in variously processed initial conditions before tensile testing at room temperature.

hafnium) allow extensive solid solubility. However, many reactive elements form intermetallic compounds instead of forming solid solutions. Addition of two known alloying elements, namely, uranium and indium, to thorium improves the mechanical strength of thorium, whereas three elements, namely, zirconium, tita­nium, and niobium, improve corrosion resistance. Thorium-uranium and tho­rium-plutonium alloys provide opportunities for combining fertile and fissile materials to develop potential thorium-based fuel cycles. Table 7.4 summarizes ten­sile properties of Th-U alloys as a function of uranium content. It clearly shows that uranium addition to thorium increases the yield strength and tensile strength; however, it decreases ductility. Thorium-uranium alloys in excess of 50 wt% U can be readily melted and cast. Powder metallurgy techniques can also be used.

The creep data of Th-9wt% U alloy are shown in Figure 7.17. The plot shows retention of strengthening due to the mechanical effect at 600 °C.

Here, some general aspects of radiation effects are summarized.

• Mechanical Properties

a) Hardness and strength increase (radiation hardening) due to increased defects mainly dislocations, precipitates, and so on.

b) Ductility decreases (radiation embrittlement).

c) Strain hardening exponent decreases (i. e., uniform ductility decreases).

d) Ductile-brittle transition temperature increases (radiation embrittlement).

e) Fracture toughness decreases (i. e., upper shelf energy decreases).

f) Creep enhancement occurs (radiation-induced and radiation — enhanced) because of increased defect concentration and diffusivity.

g) Low cycle fatigue life decreases and high cycle fatigue life increases due to embrittlement and hardening, respectively.

• Physical Properties

a) Density decreases, that is, volume increases (radiation swelling) due to the formation of voids, bubbles, and depleted zones.

b) Electrical resistivity increases (or conductivity decreases).

c) Magnetic susceptibility decreases.

d) Thermal conductivity decreases due to the increased defect concentration.

• Corrosion Properties

a) Corrosion is enhanced by radiolytic dissociation of the environment.

Detailed discussion on radiation effects on materials is impossible in a single chapter. To have more detailed information on this vast topic, readers can refer to the excellent texts mentioned in the reference list [11, 24] or numerous papers pub­lished in nuclear materials specific journals.

Problems

6.1 A pressure vessel steel specimens (A533B, low-alloy ferritic steel) of diame­ter 1 mm and grain size 0.2 mm exhibited a yield point with the lower yield stress equal to 520 MPa. The source hardening term was found to be

150 MPa. For this steel, ky decreases with temperature by 2MNm—3/2 for 50 °C increment. Assume that this linear dependence is valid and also the yield stress is temperature insensitive. The surveillance capsule tests revealed that the DBTT increased from —20 to +90 ° C following radiation exposure in an LWR environment.

a) Estimate the increase in friction hardening (s) required for this change.

b) Assuming that all this increase in friction hardening is due to dislocations, determine the dislocation density of the irradiated steel.

c) During radiation exposure in the reactor, the steel specimens were at around 425 °C that resulted in an increase in grain size from 0.2 to 0.4 mm. Determine the yield stress ofthe irradiated steel.

6.2 What are the effects of radiation on the following: (a) dislocation density,

(b) vacancies, (c) diffusion, (d) corrosion, (e) strength (hardness), (f) ductility,

(g) toughness, (h) DBTT, (i) upper shelf energy, (j) strain hardening, (k) creep (low temperature versus high temperature), (l) low and high cycle fatigue,

(m) burnup, (n) density, and (o) thermal conductivity.

6.3 Describe the source and friction hardening in BCC and FCC alloys and the effects of neutron radiation exposure on them.

6.4 Describe the pellet-cladding interaction in LWR fuels and possible solutions to mitigate PCI.

6.5 In a fusion reactor blanket coolant channel, a copper (fcc) coolant tube (exposed to 14 MeV neutron radiation to a fluence of 5 x 1022 n cm—2) exhibited radiation hardening and embrittlement with twice the tensile strength accom­panied by reduced elongation to fracture (by 1/2 of that before irradiation) with no necking. The following properties were reported on the unirradiated copper: Young’s modulus = 30 x 106psi, nominal tensile strength = 50ksi, fracture strain = 35%, and uniform strain = 25%.

a) Estimate the change in the toughness of the material following radiation exposure.

This material is known to follow the universal slopes equation relating the fatigue life to the applied strain: De = (Sf/E)(Nf)—0,12 + e—0 6, where Sf is the fracture strength, ef is the fracture strain, and Nf is the number of cycles to fatigue failure.

b) What are the effects of radiation exposure on fatigue life in LCF (Nf <

50 000) and HCF (Nf> 106)? Does neutron irradiation always decrease fatigue life (explain your answers)?

c) Calculate the endurance limits before and after radiation exposure?

d) The radiation exposure resulted in 0.15 dpa. If one vacancy survived per million atomic displacements due to recombination and so on, calculate the density of Frenkel defects following radiation exposure? What is the probability (%) that a unit cell contains a vacant lattice site?

e) How does radiation exposure influence self-diffusion?

f) Show the effect on an Arrhenius plot indicating the relevant parameters.

6.6 Both zircaloys and stainless steels exhibit dimensional changes in-service (in-reactor) known as radiation growth and radiation swelling, respectively.

Describe these two phenomena with emphasis on distinctions between them.

6.7 A copper specimen is exposed to 2 MeV monoenergetic neutrons of flux, ф of 2 x 1016 n cm-2 s-1 for 10 h.

a) Determine the number of atomic displacements per atom (dpa).

b) If PKAs were produced with 80keV, calculate the number of atoms dis­placed by a PKA (assume Kinchin-Pease model)

c) The steady-state creep rate (e) of Cu follows a power law: e — ADlS3, with A — 5 x 10-7 (e in h-1, D in cm2 s-1, and s in MPa). If the creep rupture life of the unirradiated Cu at 400 °C at 20MPa is 100 days, estimate the rupture life of the irradiated material at 300 °C under the same stress (assume that the same equation is valid for the irradiated material also).

1 Nuclear Energy, 48, 325-359.