Total stress tensor can be divided into two components: hydrostatic or mean stress tensor (om) involving only pure tension or compression and deviatoric stress tensor

(o’j) representing pure shear with no normal components:

It is to be noted that pure hydrostatic stress does not lead to plastic deformation and finite deviatoric stresses are needed for any plastic deformation to take place.

A.4

Normal and Shear Stresses on a Given Plane (Cut-Surface Method)

Given Oj in reference system 12 3, n is the unit vector normal to the plane = njn2n3, m is the unit vector in the plane = m1m2m3, oN is the normal stress along n, and t is the shear stress along m (see Figure A.2).

Note: П ■ m — 0, n2 + n2 + n| = 1, and m1 + m2 + m2 = 1. If П — 1, 2, 5 ) П — 1/^30, 2Д/30, 5Д/30 , n is a unit vector, where /12 + 22 + 52 = /30 so that n1 + n2 + n2 = 1.

First, we find the stress vector (S) {the stress vector is the vector force per unit area acting on the cut}:

that is, S1 = o11n1 + o12n2 + o13n3, and so on. oN and t are given as follows: oN = S ■ n = S1n1 + S2n2 + S3n3 and

t — S ■ m = S1m1 + S2m2 + S3m3,

Figure A.2 Designations for normal and shear stress calculations.

and tmax occurs when n, S, and m are in the same plane:

Thus, given the stress tensor, we need only two elastic constants E and n, since G is related to E and n:

G = E/2(1 + n). (A. 14)

We now note that the volume change or dilatation is given by

Д =(1- + e1)(1 + e2)(1 + e3) — 1 — e1 + e2 + e3

since e’s << 1. Note that Д is the first invariant of the strain tensor and mean strain em = Д/3.

Thus, bulk modulus

(A.17)

Note that the derivative of U0 with respect to any strain component equals the cor­responding stress component:

— 1Д T 2Gex — Sx

and similarly

A.7

Generalized Hooke’s Law

eij Sijklakl:

Here Sjki is the elastic compliance tensor (fourth rank) and

aij Cijklekh

where Cijkl is elastic stiffness (or elastic constants).

Crystal symmetry reduces the number of independent terms: cubic — 3, hexago­nal — 5, and so on, and these are related to E and G.

Details on these compliance and stiffness coefficients are beyond the scope of this book and may be found elsewhere.

Appendix B

B.1

SI Units

B.2

Some Derived Units

B.4

Some Unit Conversion Factors

B.4.1

Length

1 inch = 25.4 mm 1 nm= 10-9m 1 mm = 10-6 m 1A= 10-10m = 0.1 nm

B.4.2

Temperature

T (K) = T (°C) + 273.15 T (°C) = [T (°F) — 32]/1.8

B.4.3

Mass

1 Mg= 103kg 1kg = 10~3 Mg

1kg = 103g 1kg = 2.205 lbm

1g = 10~3kg

B.4.4

Force

1kgf = 9.81 N 1lb = 4.448 N 1 dyne = 10~5 N

B.4.5

Stress

1 ksi (i. e., 103 psi) = 6.89 MPa 1 MPa= 1N mm-2 = 145 psi 1Pa = 10 dyn cm-2 1 ton in.-2 = 15.46 MPa 1 atm = 0.101325 MPa 1bar = 0.1 MPa 1 Torr (mmHg) = 133.3 MPa

B.4.6

Energy, Work, and Heat

1 eV atom 1 = 96.49 kJ mol 1 1 cal = 4.184 J 1 Btu = 252.0 cal 1 erg = 10-7 J

B.4.7

Miscellaneous

1° = rad

57.3

1 g cm-3 = 1000 kg m~3 1 poise = 0.1 Pas 1 ksi in.1/2 = 1.10 MN m~3/2

B.8

Mechanical Properties of Some Important Ceramics

Note: The above values are indicative only. Adapted from Ref. [2].

References

[1] For example, B10 + n1 ! Li7 + He4.

[2] Using these data, evaluate the parameters n and Q in the power-law creep equation.

ii) Identify the underlying creep mechanism.

iii) If in another example, a thin polycrystalline sample exhibited essentially the same n and Q, what would be the controlling creep mechanism?

[3] Relaxation (or excess) volume is generally asso­ciated with a defect leading to some form of internal stress and is expressed in terms of atomic volume (V). Relaxation volume basi­cally measures the distortion volume induced in the lattice due to the insertion of a vacancy or interstitial. Most calculations of relaxation

We have already observed the effect of fluence in Figure 6.18 on the void swelling behavior of austenitic stainless steels like 316 type. It has been observed that the swelling extent increases with increasing neutron fluence (or dpa) and empirical relations have also been developed. It has been noted that there exists an incubation period that represents the neutron dose needed to produce enough helium and point defect concentration to make void nuclea — tion possible. The incubation period may also be needed to develop enough number of interstitial dislocation loops to allow the preferred absorption of interstitials by dislocations to sufficiently bias the point defect population in the material in favor of vacancies as to permit vacancy agglomeration into voids. However, incubation period does depend on temperature and material/ microstructure. Figure 6.20 shows a generalized form of void swelling behav­ior as a function of dose or fluence. The initial transient period is followed by a steady-state swelling period. This steady-state period for FCC-based materials continues to proceed with no sign of saturation. However, void swelling in BCC-based metals/alloys (like F-M steels) shows saturation effect. In these systems, at higher radiation doses, voids/bubbles self-organize following the crystallography of the metal leading to void/bubble lattices.

6.1.3

Radiation-Induced Segregation

Radiation-induced segregation (RIS) involves segregation of alloying elements under fast particle irradiation to certain microstructural locations leading to a situa­tion where otherwise homogeneous alloys become heterogeneous. During fast par­ticle irradiation, significant diffusion fluxes of point defects (vacancies and interstitials) can be set up in the vicinity of the defect sinks like surfaces or internal grain boundaries. Generally, we have come to know from Section 2.3 that the differ­ent atomic species in an alloy migrate at different rates in response to the already set up point defect fluxes so that some species travel toward the said defect sinks while others move away. Thus, RIS can lead to significant alterations in the local composition near the sinks like grain boundaries, and this can have substantial bearing on the macroscopic properties of the materials. This phenomenon has been studied in some detail, particularly in structural materials used in nuclear reactor components. Understanding RIS is of particular importance in chromium containing austenitic stainless steels or nickel base superalloys because these alloys are used in commercial power reactors and potential candidate materials for advanced reactors. RIS can potentially lead to irradiation-induced stress corrosion cracking as chromium segregates away from the grain boundaries where it is most needed. At low temperatures, defect concentration builds up and rather than going to the sinks, point defects tend to recombine. At higher temperatures, thermal dif­fusion dominates and composition becomes equilibrated or homogeneous. At intermediate temperatures, RIS becomes acute due to the operation of a process known as “inverse Kirkendall” effect. Figure 6.21 shows the composition profile in an irradiated 300 series stainless steel, analyzed by energy dispersive spectroscopic measurement conducted with a JEOL 2010F high-resolution transmission electron microscope, showing depletion of Cr and enrichment of Ni, Si, P at the grain boundary due to RIS.

6.1.4

Irradiation swelling is a type of dimensional instability (in the form of volume expansion) that is encountered through the formation of voids/bubbles and fission

20 —

15 —

4000

2000

5000

FUEL BURNUP (MWD/T)

Figure 7.9 The extent of swelling in various uranium-based materials as a function of fuel burnup. Taken from Ref. [2].

not in gamma-uranium, while both phases can exhibit radiation swelling even though alloying may minimize the radiation swelling effect.

Figure 7.9 shows the volume change as a function of fuel burnup for a number ofuranium-based materials. This does show the effect of alloying in suppressing radi­ation swelling. Note that the adjusted uranium shown in the plot is known as the Brit­ish Standard Fuel Produce (contains 400-1200ppm, 300-600ppmC, and small amounts ofMo, Nb, and Fe) and it shows better radiation swelling performance.

Irradiation Creep

Creep is an important time-dependent mechanical property for high-temperature application. In Chapter 6, we have learned how thermal creep is different from irradiation creep. So, here we are not going to repeat the fundamental principles. Figure 7.10 shows the effect of irradiation on the creep behavior of hot rolled uranium.

7.2.2

Here, we will only briefly discuss the effect of radiation on fatigue properties. Recall the universal slopes method in Section 5.1, given by

Де = ANf 0:6 + BNf 0:12, (6.15)

where the first term represents the low cycle fatigue (LCF) that is controlled by ductility, while the second term represents high cycle fatigue (HCF) controlled

0.2 0.4 0.6 0.8

Normalized Temperature, T/TM

Figure 6.41 Effect of neutron radiation exposure on fatigue of 304-type stainless steel [35].

by strength. Now we know the constant A is directly proportional to ductility (i. e., reduction in area), while the constant B is proportional to strength (such as ulti­mate tensile strength). Since radiation exposure leads to hardening and embrittle­ment, fatigue life decreases in LCF and increases in HCF. Figure 6.41 illustrates the features clearly delineated in irradiated stainless steel tested at 325 °C, where it is noted that radiation exposure results in decreased fatigue life in LCF while improved life in HCF.

6.3

Thorium has cubic crystal structure and is thus isotropic. It does not show radia­tion growth effect and thus has better dimensional stability than a-U under irradiation.

7.2.3.1 Pros and Cons ofThorium-Based Fuel Cycles

Thorium is more abundant in nature compared to uranium and naturally there is interest in having an economic fuel cycle based on thorium. Thorium-based fuel cycles offer attractive features that are low level of waste generation along with a less amount of transuranics in the waste and provide a robust diversification option for nuclear fuel supply. Also, the use of thorium in majority of reactors leads to significant additional safety margins. However, the full commercial exploitation of thorium fuels has some significant obstacles in terms ofbuilding an economic case to undertake the necessary developmental work. A great deal of R&D, testing, and qualification work is required before any thorium fuel can be considered for rou­tine commercial application. Other obstacles to the development of thorium fuel cycle are the greater fuel fabrication and reprocessing costs to include the fissile plutonium as a driver material. The high cost of fuel fabrication is partly because of the high level of radioactivity that is involved in the presence of U-233, chemi­cally separated from the irradiated thorium fuel. But the U-233 gets contaminated with traces of U-232 that decays (69-year half-life) to daughter nuclides such as thal­lium-208 that are high-energy gamma-emitters [14]. Even though this improves the proliferation resistance of the fuel cycle, it also makes U-233 hard to handle and easy to detect. Notwithstanding, thorium fuel cycle provides hope for long-term energy security benefits without the need for fast reactors.

Metallic fuels with their high neutron economy, good thermal conductivity, and thermal shock resistance should be the natural choice for fuels. However, they are not adequate for high-temperature reactors due to low strength at high temperatures, phase transformations, and so on. The other ceramics have superior strength at higher temperatures, low thermal expansion, good corro­sion resistance, and good radiation stability. A wide range of compounds are considered as ceramics, which include oxides, carbides, nitrides, borides, sili — cides, sulfides, selenides, and so forth. But ceramics also suffer from brittle­ness especially at lower temperatures. Here, we will discuss few ceramic nuclear fuels and discuss their salient features.

7.3.1

“We believe the substance we have extracted from pitch-blende contains a metal not yet observed, related to bismuth by its analytical properties. If the existence of this new metal is confirmed we propose to call it polonium, from the name of the original country of one of us.”

—Marie Curie

7.1

Introduction

The heart of a nuclear reactor is the “reactor core” that contains nuclear fuels among other components/materials. Nuclear fuel forms consist of radioactive materials that may create the fission chain reaction under suitable conditions creat­ing a large amount of heat that is then utilized for producing the electrical power. The following are the basic requirements of a nuclear fuel:

a) The capital installation costs for nuclear power plants are substantial. In order to maintain profitability in the power production, the fuel costs must be minimal.

b) Adequate thermal conductivity of nuclear fuels is necessary to ensure that they can withstand the thermal gradients generated between the fuel center and periphery.

c) The fuel should be able to resist repeated thermal cycling due to the reactor shutdowns and start-ups.

d) It should have adequate corrosion resistance against the reactor fluids.

e) It should transmit heat quickly out of the fuel center.

f)The fuel should be relatively free from the constituent elements or impurities with high neutron capture cross section in order to maintain adequate neutron economy.

g) It must be able to sustain mechanical stresses.

h) The fuels should be amenable for reprocessing or disposal.

Nuclear fuel materials developed over decades include metals/alloys (uranium, pluto­nium, and thorium) and ceramics (oxides, carbides, nitrides, and silicide compounds containing the former radioactive elements). Nuclear fuels are fabricated in a wide

An Introduction to Nuclear Materials: Fundamentals and Applications, First Edition.

K. Linga Murty and Indrajit Charit.

© 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.

The history of metallic nuclear fuels dates back to the first developmental stages of nuclear reactors. U — and Pu-based fuels were used in the Experimental Breeder Reactor-1 (i. e., EBR-1) that produced useful electricity for the first time in December 1951. In addition, EBR-2, the first-generation Magnox reactors (such as Calder Hall in the United Kingdom), and many other subsequent fast reactors have used metallic fuels. When water-cooled reactors were being devel­oped, the metallic fuels were not chosen mainly because of the compatibility issues between water and metallic fuel at elevated temperatures arising during the event of a cladding breach resulting in the formation of metal hydrides or oxides. However, in the mid-1960s, when the fast reactor development was gain­ing ground, designers chose oxide fuels in place of metallic fuels as they envi­sioned that the metallic fuels would have only limited burnups because of the presumed swelling problems and anticipated creation of liquid phases in fuels during the higher temperature operations. During that time, oxide fuels were recommended for power reactors even though limited information was available on them. Later simple design changes for the fuel elements, widening the gap between the fuels and cladding materials and providing a plenum volume for accumulating fission gases, have shown marked improvement (1% versus 20% burnup) over the earlier designs where there was no gap or very little gap between the fuel and the cladding materials. However, it is important to note that research and test reactors have traditionally used metallic fuels mainly because of the lower temperature operations involved.

Here we highlight three main metallic nuclear fuel materials. Metallic fuels have a number of advantages as well as disadvantages often specific to the fuel types. However, the metallic fuels generally have higher thermal conductivity, high fissile atom density (improved neutron economy), and fabricability as their prime advan­tages, whereas lower melting points, various irradiation instabilities, poor corro­sion resistance in reactor fluids, and various compatibility issues with the fuel cladding materials are some prominent disadvantages. Metallic fuels can also be used in alloy forms to improve corrosion resistance and irradiation performance among others.

7.2.1

At intermediate temperatures and radiation doses of >10dpa, RIS effect may, in fact, lead to phase instability, involving precipitation of new phases or dissolution of existing phases. In Ni-Si system, Si as an undersize solute (atomic radius: 110 pm) compared to Ni (atomic radius: 135 pm) enriches the sinks, and when conditions are ripe, the solubility of Si in Ni exceeds the level forming new phase Ni3Si (y’-type structure). A more illustrative example is shown in Figure 6.22. Nemoto et al. [20] showed evidence of radiation-induced precipitation in various irradiated Mo-Re alloys. Precipitates formed from Mo-4Re alloys after fast neutron irradiation consist of chi and sigma phases.

6.2

Mechanical Properties

Mechanical properties are important to varying degrees in a majority of nuclear reactor components that experience a variety of loading conditions under various temperature and irradiation regimes.

6.2.1

Plutonium (atomic number 94) and its alloys can be used as nuclear fuels in nuclear reactors and space batteries. Notably Pu239 is the major fissile isotope of plutonium. Among them, plutonium (Pu239) serves as a fissile fuel as its fission cross section is high (742.5 b) with thermal neutrons. Pu241 isotope also has signifi­cant fission cross section in the thermal spectrum (1009 b), whereas Pu240 can act as burnable poison allowing reactor to have constant reactivity throughout reactor lifetime. Plutonium is found in natural uranium only in trace quantity. It is mainly produced artificially by the transmutation reaction of U238 isotope (fertile fuel) with a neutron as described in Eq. (1.2). Radioisotopic thermoelectric generators also use plutonium (Pu238) to power spacecrafts. Plutonium appears originally as a bright silvery-white metal, but soon loses its bright color when oxidized in air. Smaller critical mass of plutonium (almost one-third of that of uranium), its high toxicity, and pyrophoricity warrant safe handling of this metal. Plutonium can be recovered from the spent fuel of a thermal reactor through chemical treatment. However, depleted uranium can be kept together with plutonium for fuels used in fast breeder reactors such as LMFBRs. In other cases, separated plutonium can be used in plutonium-burning reactors.

Hecker [5] noted the following in his paper about plutonium:

“PLUTONIUM is a notoriously unstable metal — with little provocation, it can change its density by as much as 25 pct; it can be as brittle as glass or as malleable as alumi­num; it expands when it solidifies; and its silvery freshly machined surface will tar­nish in minutes, producing nearly every color in the rainbow. In addition, plutonium’s continuous radioactive decay causes self-irradiation damage that can fundamentally change its properties over time.”

As we present in the next several sections various characteristics of plutonium, we must bear in mind this fickle nature of plutonium.

Various physical properties such as thermal conductivity, thermal expansion coefficient, density, elastic constant, and so on are of interest for nuclear

applications. Hence, it is important to understand the irradiation effects on the physical properties. Before we embark on discussing these, we must accept that irradiation can cause various changes in the structure of the materials and thus there may not be a general trend — but the effects will depend on particular situa­tions. So, one needs to be prudent while analyzing these conditions and drawing inferences.

6.3.1 Density

Calculations indicate that vacancy-interstitial pairs should cause substantial changes in the density of the irradiated material as they would increase the volume ~1.5 times theoretically. However, experimental observations show very little or no change in the density (which should decrease due to the generation of Frenkel pairs), except in the radiation swelling regime where volume increase occurs through the creation of voids/bubbles (discussed in Section 6.1.2.2). It is thought that due to the greater mobility of interstitials and their clusters, they would diffuse even at homologous temperatures of 0.15-0.20 and get trapped or annihilated. This implies that we would expect to see little or no change in density of metals/alloys irradiated near or below room temperature. Some exceptions have been seen in very high melting metals such as refractory metals. In such cases, at lower temper­atures, the interstitials are not that mobile, resulting in some significant changes in volume and in turn resulting in decreased density, as observed from lattice parame­ter measurements.

6.3.2

Three main ceramic uranium fuels are UO2, UC, and UN (to a smaller extent U3Si and US), i. e. uranium sulfide. The operating experience with UO2 is the greatest even though UN and UC remain the best potential fuels for higher performance in the long term. Improvement in fuel performance and enhanced thermal efficiency require the fuel element temperature to be as high as possible. However, with metallic fuels, two main problems may occur: (a) central fuel melting and (b) excessive irradiation swelling and creep deformation due to irradiation instability at higher temperatures. In this regard, ceramic fuels have certain advan­tages over metallic uranium fuels: (a) higher permissible fuel and plant operating temperatures due to higher melting point, (b) good irradiation stability due to the absence of polymorphic phase transformation, and (c) high corrosion resistance to the environmental attack as a result of its chemical inertness and compatibility with cladding. The basic nuclear properties of competitive ceramic fuels are as follows:

(a) large number of fissile uranium (U235) atoms per unit volume of the fuel in order to avoid necessity for high enrichment, and (b) small neutron absorption cross section of the nonfissile components of the compound for preserving the neutron economy. The following sections discuss various aspects of uranium diox­ide, which is the mainstay of nuclear fuels used in current generation of power reactors. We will briefly discuss UN and UC.