The stretched surface of a cavity can relieve its resid­ual strain e* to some degree by reducing the surface area at the expense of creating stresses in the sur­rounding material.

In the absence of externally applied stresses, this creates a spherically symmetric deformation field that can be derived from a radial displacement function

u(r) = A/r2 [119]

where r is the distance from the cavity center. The following bulk strains and stresses can then be obtained:

err 2A/ r ; eyy e» A/r

ffrr = -4mMA/r3, See = S» = 2mMA/r3 [120]

The surface stresses, on the other hand, are deter­mined from

g = gee = g" = 2(ms T As)(e;T a/r3) [121]

The constant A is chosen such as to satisfy the bound­ary condition for the radial bulk stress at the cavity surface:

Srr (R)=2g — P [122]

This boundary condition replaces the incorrect one stated in eqn [110]. The cavity must always satisfy this mechanical equilibrium condition expressed by eqn [122], regardless of whether the thermodynamic equilibrium condition is satisfied or not. In other words, thermodynamic equilibrium and mechanical equilib­rium obey two different and separate conditions.

From eqns [121] and [122] it follows that the surface strains eyy and are both equal to

r A = pR — 4(ms + As)e* r3 4mMR + 4(ms + 1s)

4mMR + 4(ms + 1s)

Using this result we can determine from eqn [118] the surface energy of the cavity as a function of its radius R:

g(e(R)) — g0 + 2 (mS + 1S)[2e* + e(R)]e(R) [124]

Associated with the surface strain e(R) is a stress and a strain field in the surrounding material given by eqn [120]. It gives rise to the strain energy

Sj Ejd 3r — 8pR3mMe2(R)

The reference cavity radius R defined by eqn [106] undergoes a small change as the surface strains adjust to their mechanical equilibrium values given by eqn [120]. As a result, the change in cavity volume, its relaxation volume, is

A Vr(R) — 4tcR2«(R) — 4pR3 e(R) [126]

When gas is present in the cavity at a pressure p, it performs the work —pAV~ when the surface relaxes. Therefore, the total free energy associated with the creation of a cavity is

Fc (R) — 4pR2y(e(R)) + 8nR3 mMe2 (R) — 4pR3pe(R) [127]

Fq(R(N)) replaces now the surface free energy Fs(N) used in eqn [105] to arrive at the cavity surface tension 2g0/R. The latter is now given by Fc(N + 1) — Fc(N). It will be evaluated in the next section and compared with 2g0/R.

At temperatures where both SIAs and vacancies are mobile, the defect cluster evolution is complex due to the wide range of defect cluster geometries that can be nucleated.8,47,92,93 The predominant visible fea­tures in this temperature regime are vacancy and interstitial loops and SFTs for irradiated fcc mate­rials and vacancy and interstitial loops and voids for irradiated bcc materials. For medium — to high- atomic number fcc metals exposed to energetic displacement cascades (e. g., fast neutron and heavy ion irradiation), most of the vacancies are tied up in

50 nm

Figure 8 Weak beam microstructure of dislocation loops in AlN after 2 MeV Si ion irradiation to ~5 dpa at 80 K. The TEM figure is based on irradiated specimens described in Zinkle etal.91

sessile vacancy clusters (SFTs, vacancy loops) that are formed directly in the displacement cascades. As a consequence, the majority of observed dislocation loops in fcc metals in this temperature and PKA regime are extrinsic (interstitial type), and void nucle — ation and growth is strongly suppressed. For bcc metals, the amount of in-cascade clustering into ses­sile defect clusters is less pronounced, and therefore, vacancy loop and void swelling are observed in addi­tion to interstitial dislocation loop evolution. Due to the typical high sink strength of interstitial clusters in this temperature regime, the magnitude of void swelling is generally very small (< 1% for doses up to 10 dpa or higher). The loop density and nature in bcc metals is strongly dependent on impurity content in this temperature regime.5,8,55 For example, the loop concentration in molybdenum irradiated with fission neutrons at 200 °C is much higher in low-purity Mo with ^99% of the loops identified as interstitial type, whereas ^90% of the loops were identified to be vacancy type in high-purity Mo irradiated under the same conditions.8

The dose dependence of defect cluster accumula­tion in this temperature regime is dependent on the material and defect cluster type. For dislocation loops and SFTs in fcc metals, the defect accumulation is initially linear and may exhibit an extended inter­mediate regime with square root kinetics before reaching a maximum concentration level. The maxi­mum defect cluster density is largely determined by displacement cascade annihilation of preexisting defect clusters. In fcc metals, the defect cluster density may approach 1024m~3, which corresponds to a defect cluster spacing of less than 10 nm and is approximately equal to the maximum diameter of subcascades during the collisional phase in neutron- irradiated metals. As with irradiation near recovery Stage II, the critical dose for transition in defect cluster accumulation kinetics is dependent on the overall defect sink strength. With continued irradia­tion, the loops may unfault and evolve into network dislocations, particularly if external stress is applied. Figure 9 summarizes the dose-dependent defect cluster densities in neutron-irradiated copper and nickel.94-96 In both of these materials, the predomi­nant visible defect cluster was the SFT over the entire investigated dose and temperature regime. Depending on the purity of the copper investigated, the transition from linear to square root accumula­tion behavior may or may not be evident (cf. the differing behavior for Cu in Figure 9). The visible defect cluster density in irradiated copper reaches a constant saturation value (attributed to displacement cascade overlap with preexisting clusters) for damage levels above ~0.1 dpa. The lower visible defect clus­ter density in Ni compared to Cu at doses up to 1 dpa has been attributed to a longer thermal spike lifetime of the Cu displacement cascades due to inefficient
coupling between electrons and phonons (thereby promoting more complete vacancy and interstitial clustering within the displacement cascade).97,98

Figure 10 compares the defect cluster accumula­tion behavior for two fcc metals (Cu, Ni) and two bcc metals (Fe, Mo) following fission neutron irra-

30,95,96,99-101

diation near room temperature. For all

four materials, the increase in visible defect cluster density is initially proportional to dose. The visible defect cluster density is highest in Cu over the

investigated damage range of 10~4-1dpa. The irra­diated Fe has the lowest visible density at low doses, whereas Ni and Mo have comparable visible cluster densities. At doses above ^0.01 dpa, the visible loop density in Mo decreases due to loop coalescence in connection with the formation of aligned ‘rafts’ of loops. Partial formation of aligned loop rafts has also been observed in neutron-irradiated Fe for doses near 0.8 dpa, as shown in Figure 11.100 The individual loops within the raft aggregations in neutron — irradiated Fe exhibited the same Burgers vector. The maximum visible cluster density in the fcc metals is about one order of magnitude higher than in the bcc metals (due in part to loop coalescence associated with raft formation). Positron annihilation spectros­copy analyses suggest that submicroscopic cavities are present in the two irradiated bcc metals, with cavity densities that are about two orders of magnitude higher than the visible loop densities.99-

Stable metallic glasses may be produced, com­monly in intermetallic compounds. Interest in the

irradiation properties of this class of materials resulted from preliminary tests that showed that these materials actually became more ductile upon irradiation.69 Other intermetallic compounds have been shown to become amorphous upon irradiation. Although semiconductors such as Si and Ge are susceptible to amorphization under irradiation, the phenomenon is almost exclusively restricted to inter­metallic compounds.70 To mention only a few, Zr3Al, Mo3Si, Nb3Ge, and Fe2Mo are compounds that have been studied in the amorphous state. Results and a detailed review of mechanisms and theories of amor — phization have been published by Motta.70 In simple terms, the lattice disruption and defect generation from irradiation disrupts long-range order in the system. Thermal annealing competes with the disor­dering so that there is a critical temperature above which amorphization is not possible. Figure 30 shows a plot of the irradiation exposure necessary for amorphization as a function of temperature for Zr3Fe.71 The critical temperatures and the necessary exposures are both functions ofthe material as well as the impinging particle. Once formed, the amorphous phases are stable under irradiation, but the critical temperatures are typically lower than would be expe­rienced for structural materials in nuclear systems. They are of interest, however, because some inter­metallic phases, such as Fe2Mo and Fe3B found in commercial alloys, become amorphous under irradiation.70,72 In the example of Zr3Fe, the critical temperature under argon ion irradiation is approxi­mately 250 °C, a temperature too low for most, but not all, reactors.

The intermetallic alloys that can be produced in the amorphous state before irradiation are of more interest as potential structural materials, although they remain in the category of research interest at the present time. In addition to the increase in duc­tility upon irradiation, the absence of a crystalline structure with interacting dislocations was further incentive to investigate the irradiation properties of this class of materials. Metallic glasses containing boron, such as Fe40Ni40B20 and (Mo.6Ru.4)82B18 are a few examples, with the former receiving the most attention in terms of mechanical properties.69,73-75 Amorphous alloys are complex systems where changes in free volume and segregation into clusters of differing composition result in changes in behavior as irradiation proceeds. Investigation of the Fe-Ni-B alloy has shown that ductility first decreases and then increases with increasing fluence due to the compet­ing effects of free volume and formation of regions of boron-depleted and boron-rich clusters.73 For suffi­ciently high fluences, the result is severe embrittle­ment. In the case of alloys based on the intermetallic Zr3Al, very severe embrittlement upon irradiation is attributed to the formation of new amorphous

phases.76

Even though a crystal structure is absent, the atoms may be dislodged from their locations, creating additional free volume. Without the bonds present from a crystal lattice, the low binding energy results in high displacement levels for fluence levels that what would be considered low in crystalline alloys. Fluence levels in the range of 1016—1021 ncm~2 have been investigated resulting in displacement levels exceeding 100 dpa. However, simply having similar displacement levels does not permit a true compari­son with crystalline materials. Much research is nec­essary before this class of materials becomes of commercial importance.

1.04.8 Conclusions

The relationship between the irradiation-induced microstructure and tensile properties has been briefly presented using representative classes of alloys. The austenitic stainless steels are an important class of alloys, and they are less complex than the martensitic steels. In the unirradiated condition, the austenitic alloys are primarily hardened by dislocation reactions leading to conventional work hardening, and the martensitic steels are hardened by phase transforma­tions requiring careful heat treatments. The primary irradiation effects are similar, but they influence microstructure and, therefore, behavior in different ways. Both types of alloys have important applica­tions in the nuclear field. Helium embrittlement might be the most important, considering the use of alloys in a neutron environment at high tempera­tures. For the proper conditions, helium can nearly always cause catastrophic failure. Repair welding of alloys with as little as 1-10 appm helium can lead to severe intergranular cracking.

The refractory metals are useful for space reactor application because of their liquid metal compatibil­ity and their high-temperature strength. Space reac­tors can lose heat only by thermal radiation, necessitating high temperatures. However, this class of alloys is most susceptible to embrittlement by interstitial impurities, and synergism of impurities with irradiation-induced defects is an area that must be addressed further.

Amorphous alloys are a research curiosity in that they have interesting properties with respect to irradiation but little application at the present time. Any new class of alloys must be understood before it can be engineered, so

The CBM concept can also be applied to the effects of grain boundary He on creep rupture properties. Stress-induced dislocation climb also results in gen­eration excess vacancies that can accumulate at
growing voids. In particular, tensile stresses normal to GBs (s) generate a flux of vacancies to boundary cavity sinks, if present, and an equal, but opposite, flux of atoms that plate out along the boundary as illustrated in Figure 20(a). The simple capillary con­dition for the growth of empty cavities is the s > 2g/ r In this case of cavities containing He, the growth rate is given by

dr / dt = [(Dgbd)/(4TCr2)]

{1 — exp[(2g/r — P — s)Q/kT ]g [12]

Here Dgb and d are the grain boundary diffu­sion coefficient and thickness, respectively. The corresponding dr/dt = 0 conditions also lead to a stable bubble (rb) and unstably growing creep cavity (r*) roots. As noted previously, a vacancy supersaturation, L, produces a chemical stress that is equivalent to a mechanical stress s = kTln(L)/O. Thus, replacing ln(L) in eqn [8a] and [8b] with sO/kT directly leads to expressions for m* and r* for creep cavities

m* = [32FvTCg3 ]/[27kT s2] [13a]

r* = 0.75g/s [13b]

This simple treatment can also be easily modified to account for a real gas equation of state. Note that it is usually assumed that GBs are perfect sinks for both vacancies and SIA. Thus, it is generally assumed that displacement damage does not contribute to the for­mation of growing creep cavities.

Understanding HTHE requires a corresponding understanding of the basic mechanisms of creep rup­ture in the absence of He. At high stresses and short rupture times, the normal mode of fracture in AuSS is transgranular rupture, generally associated with power law creep growth of matrix cavities.181,182 However, at lower stresses IG rupture occurs in a

wide range of austentic and ferritic alloys. Although space does not permit proper citation and review, it is noted that a large body of literature on IG creep rupture emerged in the late 1970s and early 1980s. Briefly, this research showed that under creep condi­tions a low to moderate density of grain boundary cavities forms (10-1012m- ), usually in associa­tion with second-phase particles and triple-point junctions.183-184a Grain boundary sliding results in transient stress concentrations at these sites, and interface energy effects at precipitates also reduce the critical cavity volume (Fv ^ 4я/3) relative to matrix voids, as illustrated in Figure 20(b).

Once formed, however, creep cavities can rapidly grow and coalesce if unhindered vacancy diffusion and atom plating take place along clean GBs. Such rapid cavity growth rates lead to short rupture times in low creep strength, single-phase alloys. Thus, use­ful high-temperature multiphase structural alloys must be designed to constrain creep cavity nucle — ation and growth rates by a variety of mechanisms. For example, grain boundary phases can inhibit dis­location climb and atom plating.185

As illustrated in Figure 20(a), growth cavities, which are typically not uniformly distributed on all grain boundary facets, can be greatly inhibited by the con­straint imposed by creep in the surrounding cage of grains, which is necessary to accommodate the cavity swelling and grain boundary displacements.186 Creep
stresses in the grains impose back stresses on the GBs that result in compatible deformation rates. Thus, it is the accommodating matrix creep rate that actually controls the rate of cavity growth, rather than grain boundary diffusion itself. Creep-accommodated, constrained cav­ity growth rationalizes the Monkman-Grant relation187 between the creep rate (e0), the creep rupture time (tr), and a creep rupture strain (ductility) parameter (er) as

e0tr = er [14a]

Thus, in high-strength alloys, low dislocation creep rates (e0) lead to long tr. The typical form of e0

e0 = Acrexp(-Qcr/kT) [14b]

The effective stress power r for dislocation creep is typically much greater than 5 for creep-resistant alloys, and the activation energy for matrix creep of Qcr к 250-350 kJ mol-1 is on the order of the bulk self-diffusion energy.1 1 These values are much higher than those for unconstrained grain boundary cavity growth, with r к 1-3 and Qgb к 200 kJ mol — .

A number of creep rupture and grain boundary cavity growth models were proposed based on these concepts.186,188,189 Note that there are also conditions, when grain boundary vacancy diffusion and atom plating are highly restricted and cavities are well separated, where matrix creep enhances, rather than constrains, cavity growth. As noted above, power law creep controls matrix cavity growth at high stress,
leading to transgranular fracture.181’182 Models of the individual, competing, and coupled creep and cavity growth processes have been used to construct creep and creep rupture maps that delineate the boundaries between various dominant mechanism regimes. How­ever, further discussion of this topic is beyond the scope of this chapter.

Accumulation of significant quantities of grain boundary He has a radical effect on creep rupture, at least in extreme cases. First, at high He levels, the number density of grain boundary bubbles (Ngb) and creep cavities (Nc) is usually much larger than the corresponding number of creep cavities in the absence of He; the latter is of the order 1010-1012m~2.181,190 Figure 21 shows the evolution of He bubbles and grain boundary cavities under stress.191 Indeed, Ngb of more than 1015m~2 have been observed in high-dose He implantation studies.10 , Although Ngb is not well known for neutron-irradiated AuSS, it has been esti­mated to be of the order 1013 m~2 or more.193,194

At high He levels, a significant fraction of the grain boundary bubbles convert to growing creep cavities, resulting in high Nc. Of course, both Ngb and Nc depend on stress as well as many material parameters and irradiation variables, especially those that control the amount of He that reaches and clus­ters on GBs. As less growth is required for a higher density of cavities to coalesce, creep rupture strains,

er roughly scale with N~1/2. Bubble-nucleated creep cavities are also generally more uniformly distributed on various grain boundary facets. More uniform distributions and lower er decrease accommodation constraint, thus, further reducing rupture times asso­ciated with cavity growth.

Equation [13a] suggests that m* scales with 1/s2. If the GB bubbles nucleate quickly and once formed the creep cavities rapidly grow and coalesce, then creep rupture is primarily controlled by gas-driven bubble growth to r* and m*.93-95,97 In the simplest case, assuming a fixed number of grain boundary bubbles Ngb and flux of He to the grain boundary, JHe, the creep rupture time, tr is approximately given by

tr = {[Fv32py3]/[27krff2]}[Ngb/jHe]~Ngb/[GHeff2] [15]

Note that this simple model, predicting tr / 1/s2 scal­ing, is a limiting case primarily applicable at (a) low stress; (b) when creep rupture is dominated by He bubble conversion to creep cavities by gas-driven bub­ble growth to r*; and (c) when diffusion (or irradiation) creep-enhanced stress relaxations are sufficient to pro­duce compatible deformations without the need for thermal dislocation creep in the grains. More gener­ally, scaling of tr / 1/sr, r > 2 is expected for bubbles containing a distribution of m He atoms. For example, if Ngb scales as m~q, then Nc would scale as s2q.194,195 Further, at higher s, hence lower tr, there is less time for He to collect on GBs. Thus in this regime, intragranular dislocation creep, with a larger stress power, r, may return as the rate-limiting mechanism controlling the tr — s relations.

Equation [15] also provides important insight into the effect of both the grain boundary and matrix microstructures. Helium reaches the GBs (JHe) only if it is not trapped in the matrix. Matrix bubbles are, by far, the most effective trap for He.95,97 If it is assumed that the number of matrix bubbles, Nb, is proportional to V GHe while the grain boundary bub­ble number density (Ngb) is fixed, a scaling relation for tr can be approximated as

tr / Ngb/[v/GHes2] [16]

if the number of grain boundary bubbles also scales with VjHe, tr scales with G^7"4. At the other extreme, if Nb and Ngb are both independent of the He con­centration, tr scales with Gj^ (eqn [15]). Thus, micro­structures with high Nb (and Zb) that are resistant to void swelling are also likely to be resistant to HTHE.

HTHE models for AuSS were developed based on these concepts and various elabora — tions93-96,182,190,194,196,197 as well as methane
gas-driven constrained growth of grain boundary cavities.198 The HTHE models developed by Trinkaus and coworkers were closely integrated with the extensive He implantation and creep rupture studies discussed further below. It should be empha­sized that the HTHE models cited above are only qualitative and primarily represent simple scaling concepts that must be validated and calibrated using microstructural, creep rate, and creep rupture data. For example, more quantitative models require detailed treatment of He accumulation and redistri­bution at the GBs into a stably growing population of bubbles, with a time-dependent fraction that ultimately converts to growing creep cavities.

‘Nuclear materials’ denotes a field of great breadth and depth, whose topics address applications and facilities that depend upon nuclear reactions. The major topics within the field are devoted to the materials science and engineering surrounding fission and fusion reactions in energy conversion reactors. Most of the rest of the field is formed of the closely related materials science needed for the effects of energetic particles on the targets and other radiation areas of charged particle accelerators and plasma devices. A more complete but also more cumbersome descriptor thus would be ‘the science and engineering of materials for fission reactors, fusion reactors, and closely related topics.’ In these areas, the very existence of such technologies turns upon our capabilities to understand the physical behavior of materials. Performance of facilities and components to the demanding limits required is dictated by the capabilities of materials to withstand unique and aggressive environments. The unifying concept that runs through all aspects is the effect of radiation on materials. In this way, the main feature is somewhat analogous to the unifying concept of elevated temperature in that part of materials science and engineering termed ‘high-temperature materials.’

Nuclear materials came into existence in the 1950s and began to grow as an internationally recognized field of endeavor late in that decade. The beginning in this field has been attributed to presentations and discussions that occurred at the First and Second International Conferences on the Peaceful Uses of Atomic Energy, held in Geneva in 1955 and 1958. Journal of Nuclear Materials, which is the home journal for this area of materials science, was founded in 1959. The development of nuclear materials science and engineering took place in the same rapid growth time period as the parent field of materials science and engineering. And similarly to the parent field, nuclear materials draws together the formerly separate disciplines of metallurgy, solid-state physics, ceramics, and materials chemistry that were early devoted to nuclear applications. The small priest­hood of first researchers in half a dozen countries has now grown to a cohort of thousands, whose home institutions are anchored in more than 40 nations.

The prodigious work, ‘Comprehensive Nuclear Materials, captures the essence and the extensive scope of the field. It provides authoritative chapters that review the full range of endeavor. In the present day of glance and click ‘reading’ of short snippets from the internet, this is an old-fashioned book in the best sense of the word, which will be available in both electronic and printed form. All of the main segments of the field are covered, as well as most of the specialized areas and subtopics. With well over 100 chapters, the reader finds thorough coverage on topics ranging from fundamentals of atom movements after displacement by energetic particles to testing and engineering analysis methods of large components. All the materials classes that have main application in nuclear technologies are visited, and the most important of them are covered in exhaustive fashion. Authors of the chapters are practitioners who are at the highest level of achievement and knowledge in their respective areas. Many of these authors not only have lived through a substantial part of the history sketched above, but they themselves are the architects. Without those represented here in the author list, the field would certainly be a weaker reflection of itself. It is no small feat that so many of my distinguished colleagues could have been persuaded to join this collective endeavor and to make the real sacrifices entailed in such time-consuming work. I congratulate the Editor, Rudy Konings, and the Associate Editors, Roger Stoller, Todd Allen, and Shinsuke Yamanaka. This book will be an important asset to young researchers entering the field as well as a valuable resource to workers engaged in the enterprise at present.

Dr. Louis K. Mansur Oak Ridge, Tennessee, USA

Suppose that in a spherical domain O = 4pa3 the transformation strain is uniform while it vanishes outside this domain. Then, in the above eqns [B20] to [B22], the transformation strain tensor can be taken outside the integral, and it remains to solve integrals of the following type:

d3 RRaRb • • • Rin

O

A A A dfflR Ra Rj • • • R2n

Here, Ra = Ra/|R| is the a-th component of the unit vector of R, and the remaining integral in eqn [B24] is over the surface of the unit sphere. The value for these surface integrals can be found from the general formula

d°RRaRb. •••R2n dab • •• d2n— 1,2n [B25]

4p (2n + 1)!!

The sum extends over all possible combinations of the indices, and hence it contains (2n-1)!! terms. The double factorial is defined as

(2n + 1)!! = 1*3*5*- • •* (2n + 1)

From these relations, one then finds the following mul­tipole tensors for a spherical inclusion:

Pjk O Cjkmn emn

a2„

Pjkpq = ~5djk Ppq

a4

Pjkpqrs 35 (djk + djp dkp + dkp ) Prs [B26]

We see that all multipoles tensors of higher rank than two are given in terms of the dipole tensor, and all tensors with an odd rank are zero.

When the transformation strain is that associated

Inclusion

Self-interstitials may aggregate into planar clusters with their dumbbell axes aligned in parallel. We may

view them as plate-like inclusions of thickness h and with a normal vector n. If we transform this plate by displacing one of its faces by b relative to the other face, then the transformation displacement field throughout the unconstrained plate is

«T = X-h-b, n, [B29]

where x j are the components of the position vector. The transformation strain is obtained by differentia­tion and found to be

etj = 2( + «£■) = + hn<) [B30]

Inserting these transformation strains into eqn [B21], we find for this plate-like inclusion the dipole tensor

Pjk CjkmnbmnnA [B31]

where A is the area of the plate. Note that this result is independent of the thickness h and of the shape of the plate. However, to evaluate the higher order multi­pole tensors, one must specify the shape of the plate.

Let us consider a circular plate of radius c, vanish­ing thickness h!0, and with its normal unit vector n pointing in the X3-direction.

Then, all tensors Pjkpq… vanish that have one or more indices that are equal to 3. For all other cases, the tensor components can be obtained with the formula

f fx2Mx2N, , c2(M+N+1) (2M — 1)!!(2N- 1)!!

JJXl X2 dX1dX2 = 2(M + N + 1) 2(M+N)(m + N) [B32]

All multipole tensors of odd rank vanish, and the remaining can again be expressed in terms of the dipole tensor. For example, the quadrupole tensor can be written as

Pjhpq = Pjk Qpq

where

c2 1 0 0

Qpq = 4 0 1 0

4 0 0 0

For an infinite isotropic material, the far-field dis­placement components of this circular platelet are given by « (r) ffi — G“ (r)Pjk

A1

_________

8%(1 — n) r2

3

+ 3 Xi r3

where the distance vector r is from the center of the platelet. A comparison of this displacement field with
the one for a circular dislocation loop by Kroupa36 reveals that the dipole approximation gives an accu­rate representation for the dislocation loop for dis­tances r 2 c.

Irradiation of metals and alloys at temperatures below recovery Stage V typically produces pronounced radiation hardening, as discussed in Chapter 1.04, Effect of Radiation on Strength and Ductility of Metals and Alloys. The matrix hardening is typically accompanied by reduction in tensile elongation and in many cases lower fracture toughness.21 -222 The uniform elongation measured in tensile tests for

metals and alloys irradiated in this temperature regime usually decreases to <1% for damage levels above 0.1-1 dpa, which may require use of more con­servative engineering design rules for the allowable stress of structural materials.223 The hardening is largely due to the creation of high densities of sessile defect clusters, which act as obstacles to dislocation motion in the matrix. The defect cluster densities decrease rapidly with increasing temperature above recovery Stage V. Figure 25 compares the

temperature-dependent defect cluster densities224,225
observed in neutron-irradiated Cu, austenitic stainless steel, and V-4Cr-4Ti. Stage V annealing of defect clus­ters is evident for temperatures above 150, ^200, and ^275 °C for Cu, stainless steel, and V-4Cr-4Ti, respectively. The mechanical properties in irradiated nonmetals at temperatures below recovery Stage V exhibit variable behavior, with observations of increased hardness,226,227 unchanged strength,228 and decreased hardness or flexural strength.229-232

For this particular dislocation loop, it is thought that rather than unfaulting, 1/6 (111) {111} dislocations simply dissolve back into the lattice, in favor of the more stable 1/4 (110) {110} loops.12 As discussed earlier, the 1/6 (111) {111} dislocation can be pre­sumed to be relatively unstable because it possesses both anion and cation faults, and in addition, it cannot preserve stoichiometry or charge balance in either normal or inverse spinel.12 Counter to this argument is the idea that if a 1/6 (111) {111} dislocation loop incorporates a partial inversion of its cation content, then this loop could be made both stoichiometric and charge neutral. Such a dislocation would argu­ably be more stable. However, {111} loops are never observed to grow very large (<100nm) and are alto­gether absent in spinel samples irradiated at 1100 K.17 Therefore, it is likely that ‘disordered’ {111} intersti­tial loops are not an important aspect of radiation damage evolution in spinel.

The overall approach to modeling He behavior in complex structural materials, such as FMS and NFA, that accounts for real microstructures was illustrated in Figure 3 and is briefly described in Section 1.06.6.2 below. The atomistic simulation results detailed above were used to inform the higher spatial — and temporal-scale microstructure evolution models of iron-based structural alloys.

1.06.5.4 Dislocation-Cavity Interactions

MD techniques have also been used to study dislocation-cavity interactions.238 The results of this work, where the cavities range from voids, to underpressurized, equilibrium, and overpressurized He bubbles, can be described in terms of an obstacle strengthening parameter (a) defined as

a = tc(L — d) / Gb [21]

Here, tc is the MD critical resolved shear stress for cavities with a diameter d that is spaced L apart,

G is the shear modulus, and b is the Burgers vector.

In summary, a

• Depends on the helium to vacancy ratio, m/n, and is highest for m/n for near-equilibrium bubbles and is lower for overpressurized bubbles.

• Decreases with increasing temperature.

• Increases with cavity size and at 300 K the peak a increases from about 0.2 to 0.4 in the diameter range of 1-4 nm.

This interaction arises not from the strain field of other defects or from applied loads but is caused by the changing strain field of the point defect itself as it approaches an interface or a free surface of the finite solid. We have shown in Section 1.01.4 that the strain energy associated with a point defect is given by

r _ 2Kmo /Vrel2 _ 2m(1 + v) (Vrel)2 U0 = 3K + 4m Q = 9(1 — v) Q-

when the defect is in the center of a spherical body with isotropic elastic properties or when the defect is sufficiently far removed from the external surfaces of a finite solid. This strain energy has been obtained by integrating the strain energy density of the defect over the entire volume of the solid, and since this density diminishes as r-6, where r is the distance from the defect center, it is concentrated around the defect. Nevertheless, close to a free surface, the strain field of the defect changes, and with it the strain energy. This change is referred to as the image interaction energy Uim, and the actual strain energy of the defect becomes

U (h) = U0 + Uim(h) [64]

Here, h is the shortest distance to the free surface. The strain energy of the defect, U(h), changes with h for two reasons. First, as the defect approaches the surface, the integration volume over regions of high strain energy density diminishes, and second, the strain field around the defect becomes nonspherical and also smaller. The evaluation of both of these contributions requires advanced techniques for solv­ing elasticity problems.

Eshelby 1 has shown that the strain energy of a defect, modeled as a misfitting inclusion of radius r0, in an elastically isotropic half-space, is given by

(1 + n) £0 4 h3

where h is the distance from the center of the defect to the surface. Equation [65] clearly demonstrates that the strain energy of the defect decreases as it approaches the surface. The minimum distance h0 is obviously that for which U(h) = 0, and it is given by (1 + n)

Another case for the image interaction has been solved by Moon and Pao,32 namely when a point defect approaches either a spherical void of radius R or, when inside a solid sphere of radius R, approaches its outer surface.

For a defect in a sphere, its strain energy changes with its distance r from the center of the sphere according to 1 + n гЛ3

4 RJ (n + 1)(n + 1)(2n + 1)(2n + 3) r

n2 + (1 + 2n)n + 1 + n while the strain energy of a defect at a distance r from the void center is given by 1 + n Г0 3

4 R)

1 n(n — 1)(2n — 1)(2n + 1) ^2n+2 [68]

n 2 n2 + (1 — 2n)n + 1 — n r

Again, at a distance of closest approach to the void, h0(R), the strain energy of the defect vanishes. The numerical solutions of Us(R + h0) = 0 and of UV(R—h0) = 0 gives the results for h0/r0 shown in Figure 17. There is a modest dependence on the radius of curvature of the surface. Approximately, however, the defect strain energy becomes zero about halfway between the top and first subsurface atomic layer, assuming that r0 is equal to the atomic radius.