The relation between stress-relaxation and creep for some zirconium alloys during neutron irradiation

In-reactor stress-relaxation tests on beam specimens of several zirconium alloys have been performed at 566 K in a fast neutron flux (E>1 MeV) of 2 × 10¹⁷ n/m² · s. The stress-relaxation behaviour is characterized by an initial rapid decrease in the unrelaxed stress ratio followed by a slower steady-state decrease which can be expressed by the relation,

$\text{ln}\text{(σ}\text{σ}{}_0\text{) = ? (AE)t +}\text{ln}\text{D}$

, where $\text{σ}\text{σ}{}_0$ is the unrelaxed stress at a given time t as a fraction of the initial stress σ₀, A is a flux-, material-,temperature-dependent constant, E is Young's modulus, and D is given by $\text{σ}{}_{\mathrm{p}}\text{σ}{}_0$ where σ_p approximates the initial stress decrease.The linear dependence of In $\text{σ}\text{σ}{}_0$ on time and the independence of the stress-relaxation behaviour on the initial stress implies that the creep rate in the steady-state period can be given by the expression, $\text{ε}\text{?}\text{=}\text{A}\text{σ}$. Good agreement was obtained between creep rates derived from stress-relaxation tests on experimental pressure tube materials and creep rates derived from diameter measurements of pressure tubes for identical temperature, flux and stress conditions.     

Irradiation induced creep of ceramic nuclear fuels

This contribution gives a review of the experimental results and accompanying theoretical considerations. The mechanisms considered for irradiation creep are: relaxation of elastic stresses by fission spikes, promotion of dislocation slide by thermal spikes, preferential, stress-orientated nucleation of dislocation loops and preferential growth of dislocation loops. A survey over the irradiation creep rates attributed to steady-state creep shows $\text{ε}{}_{\mathrm{irr}}\text{~ σ · F}$ for oxide fuel in the stress and fission rate ranges of $\text{σ = 10–50}\text{MN/m}{}^2$ and $\text{F = 3 × 10}{}^{12}\text{–1 × 10}{}^{14}\text{f/cm}{}^3\text{· s}$ at burnups < 3%. There seems to be a continuous increase of the irradiation creep rate of oxide fuels with increasing temperature. However, that increase cannot be directly interpreted through a thermally activated process. It seems that the irradiation creep rate will also depend on fuel porosity, on plutonium distribution in mechanically blended UO₂-PuO₂, but not substantially on the plutonium content per se. Some results were already given for carbide and nitride fuels, which show the irradiation creep rate to be lower by about a factor of 10 than for oxide fuel under comparable conditions. Primary irradiation creep has been observed up to $\text{(3–5) × 10}{}^{19}\text{f/cm}{}^3$ and could prevail up to $\text{1 × 10}{}^{20}\text{f/cm}{}^3$.     

Thermally activated deformation of Zircaloy-4

Tensile tests were carried out on Zircaloy-4 over the temperature range 298–798 K. Yield stress values at the strain rates $\text{1.33 × 10}{}^{\mathrm{?4}}\text{s}{}^{\mathrm{?1}}$ and $\text{6.67 × 10}{}^{\mathrm{?4}}\text{s}{}^{\mathrm{?1}}$ were used to determine the activation parameters. A peak in activation volume ($\text{V}{}_{\mathrm{app}}\text{= 3100 b}{}^3$) was observed at about 690 K; outside this temperature range the activation volumes became almost independent of temperature ($\text{V}{}_{\mathrm{app}}\text{= 200?300 b}{}^3$). The peak in activation volume was explained in terms of a basic rate controlling mechanism and dynamic strain aging. This analysis indicated that the peak could be ascribed to the negative value of the strain rate sensitive solute strengthening term $\text{M}$ and that the mechanism based on the non-conservative motion of jogs appeared to be more favored as the basic rate controlling mechanism of Zircaloy-4 than an impurity mechanism     

The effect of high temperature reactor primary circuit helium on the formation and propagation of surface cracks in alloy 800 H and inconel 617

The depths of surface cracks caused by the combined effects of corrosion and creep strain have been measured in specimens of Inconel 617 (Ni-22Cr-9Mo-12Co-1A1) and Alloy 800 H (Fe-32Ni-20Cr). Tests were carried out at 1073–1223 K in impure helium simulating the primary circuit coolant of a high temperature gas-cooled reactor, and in air. A characteristic crack depth $\text{a}{}_{90}$ was derived to represent the observed crack length distribution. In both test environments, an incubation strain was observed for the formation of surface cracks. Although the presence of surface cracks led to an increase in the creep rate, the magnitude of the change was similar for the two test atmospheres. Finally, a method has been developed to allow the depth of surface cracks, characterized by $\text{a}{}_{90}$, to be plotted with the stress rupture curves in order to indicate whether the corrosion effects need to be considered in the derivation of design stresses.     

Quenching studies of high-purity beta and gamma plutonium

A spray quenching system employing liquid nitrogen and high pressure helium was used to investigate the effect of cooling rate on the temperature of the $\text{β}{}_{\mathrm{\gamma \; \to \; \alpha }}$ and $\text{γ → α}$ transformations in high-purity plutonium ($\text{β}{}_{\mathrm{\gamma }}$, $\text{β}{}_{\mathrm{\alpha }}$ is beta phase formed from γ or α phase respectively). Quenching rates up to 1500° C/sec were obtained at 100° C and below. The start temperature for the $\text{β}{}_{\mathrm{\gamma }}\text{→α}$ transformation decreased rapidly with increasing cooling rate and $\text{β}{}_{\mathrm{\gamma }}$ was retained at rates greater than 600°C/sec. The $\text{β}{}_{\mathrm{\gamma }}\text{→α}$ transformation temperatures were higher and $\text{β}{}_{\mathrm{\gamma }}$ could not be retained in lower-purity plutonium. The start temperature for the γ → α transformation decreased slowly with increasing cooling rate, and γ could not be retained even at the maximum rate of 1500°C/sec. The subzero $\text{β → α}$ transformation temperatures for retained $\text{β}{}_{\mathrm{\gamma }}$ and $\text{β}{}_{\mathrm{\alpha }}$ were determined for plutonium quenched to ?196°C and then warmed to room temperature. The retained $\text{β}{}_{\mathrm{\alpha }}$ transformed to a at higher temperatures than the retained $\text{β}{}_{\mathrm{\gamma }}$, except immediately after a $\text{β}{}_{\mathrm{\gamma }}\text{→ α}$ transformation. Further, the subzero $\text{β}{}_{\mathrm{\alpha }}\text{→ α}$ transformation was progressively retarded by prior α α β transformation cycling. These subzero transformation temperature observations are consistent with the difference in $\text{β}{}_{\mathrm{\gamma }}\text{→ α}$ and $\text{β}{}_{\mathrm{\alpha }}\text{→α}$ reaction behavior at high α-phase transformation temperatures.     

Steady-state creep of zircaloy-4 fuel cladding from 940 to 1873 K

Steady-state creep rates of as-received zircaloy-4 fuel cladding have been determined from 940 to 1073 K in the α-Zr range, from 1140 to 1190 K in the mixed (α + β) phase region and from 1273 to 1873 K in the β-Zr phase region. Strain rates of between 10^?6 and 10^?2/s were determined under constant uniaxial load conditions. Assuming that creep rates can be described by a power law-Arrhenius equation, the creep rate for α-phase zircaloy-4 is given by: $\text{ge}{}_{\mathrm{ss}}\text{?}\text{= 2000 σ}{}^{5.32}\text{exp}\text{(?284 600/kT) s}{}^{\mathrm{?1}}$; for the β-phase zircaloy-4 by: $\text{ge}{}_{\mathrm{ss}}\text{?}\text{= 8.1 σ}{}^{3.79}\text{exp}\text{(?142 300/kT) s}{}^{\mathrm{?1}}$; and for the mixed (α + β) phase of zircaloy-4 (for creep rates ?3 × 10^?3 s^?1) by: $\text{ge}{}_{\mathrm{ss}}\text{?}\text{= 6.8 × 10}{}^{\mathrm{?3}}\text{σ}{}^{1.8}\text{exp}\text{(?56 600/kT) s}{}^{\mathrm{?1}}$. For the both the α and β phases, the activation energies for creep are in agreement with those of self-diffusion. For the mixed (α + β) phase region, the low creep rate range is controlled by grain boundary sliding at the α/(α + β) phase boundary.     

An analysis of iodine stress corrosion cracking of zircaloy-4 tubing

A fracture mechanics approach to interpreting iodine-vapor stress-corrosion cracking in unirradiated Zircaloy-4 tubing is presented in which crack velocities are related to the fourth power on the stress intensity factor, $\text{K}{}_{\mathrm{I}}$. The crack growth power law on $\text{K}{}_{\mathrm{I}}$ is shown to predict well the time-to-failure in internally pressurized Zircaloy-4 tubing at 360 and 400°C reported by Busby, Tucker and McCauley. The temperature dependency on iodine stress corrosion cracking in Zircaloy can be described by an Arrhenius-type equation in which the activation energy $\text{Q}$ for recrystallized and cold-reduced Zircaloy was determined to be 42.9 and 35.9 kcal/mole, respectively. It is concluded that the geometry of the initial surface flaw, through its attendant elastic stress field, is directly responsible in controlling the SCC time-to-failure, cold working having a relatively small effect on increasing the susceptibility to SCC. The effects of neutron flux on iodine stress corrosion cracking of Zircaloy-4 tubing in-reactor are still unknown.     

Compressive deformation of polycrystalline uranium at low temperatures

Polycrystalline α-uranium was tested in compression from 20 to 300 K and, with the exception of the 20 K tests, no cracking was observed during extensive plastic deformation ($\text{? ? ? 3.0}$). Alpha uranium was found to be a strongly work hardening material; the dislocation and twinning structures developed, however, are not stable arid resulted in unexpectedly weak material when samples were tested at temperatures above the initial deformation temperature. On the other hand, the results obtained suggest that large strain deformation at warm temperatures should lead to high yield strength uranium at room temperature. The stress-strain curves for annealed polycrystalline α-U from 78 to 300 K can be predicted accurately from the phenomenological relation: $\text{σ = σ}{}_{\mathrm{s}}\text{? exp[-(N?)}{}^{\mathrm{p}}\text{] · (σ}{}_{\mathrm{s}}\text{? σ}{}_{\mathrm{y}}\text{)}$, where $\text{N}$ and $\text{p}$ are material constants equal to 0.613 and 0.58, respectively, and σ_y is the yield strength. The saturation flow stress, σ_s, is predicted to be 6200 MPa (900 ksi) at 78 K and 2900 MPa (420 ksi) at room temperature.     

In-reactor creep of zircaloy-2, zircaloy-4 and Zr-1.15 wt % Cr-0.1 wt % Fe at 568 K derived from their stress-relaxation behaviour

The in-reactor stress relaxation of zirconium alloys can be represented by the equation $\text{σ/σ}{}_0\text{= D exp (?Kt)}$ where the unrelaxed stress ratio is equal to a factor times the exponential of the relaxation rate. Creep rates derived from the bent-beam stress-relaxation data are compared with uniaxial creep results and a good correlation is shown for Zircaloy-2 through a continuous trend curve from low-stress stress relaxation to the high-stress creep behaviour. Specimens quenched from the β-phase had the lowest creep rates compared with those which were annealed or cold-worked, while increasing cold-work increased the creep rates for most specimens. The effects of cold-work and results from post-irradiation annealing of stress-relaxation test specimens suggest that for the low stress régime ($\text{σ <}\text{1}\text{3}\text{σ}$ yield) the operation of a dislocation climb/glide mechanism in addition to the stress-directed loop formation mechanism is required for in-reactor creep.     

The effect of void density on the dislocation bias

Recent experimental work on the void-swelling characteristics of FV548 steel during irradiation is analysed, and it is shown that the dislocation bias for interstitial condensation $\text{p}$ is a function of the void density $\text{N}{}_{\mathrm{v}}$, decreasing from 4&;#x0303;0% when $\text{N}{}_{\mathrm{v}}$ is 10¹⁹/m³ to 0&;#x0303;.5% when $\text{N}{}_{\mathrm{v}}$ is 10²³/m³. This dependence on $\text{N}{}_{\mathrm{v}}$ is responsible for most of the apparent temperature dependence of the bias. In addition to affecting $\text{N}{}_{\mathrm{v}}$, the temperature imposes a maximum void growth rate, which decreases as the temperature is reduced, and because of this, high swelling rates cannot be obtained at low temperatures even when $\text{N}{}_{\mathrm{v}}$ is low. The full effect of the void-density dependence of $\text{p}$ is therefore only visible at high temperatures.     

Creep properties of oxide and carbide fuels under irradiation

The creep behaviour of uranium dioxide and uranium carbide has been examined in both bend and compression experiments in DIDO Materials Test Reactor. In UO₂ no significant variation in creep rate with dose and temperature occured above ~10²⁵ fissions m^?3 between 450°C and 1230°C, the high strain rates measured in compression at low doses being largely attributable to pore sintering. Both a linear rating and stress dependence were observed up to 40 MNm^?2 and creep rates were found to be independent of grain size. At higher doses ($\text{>6 × 10}{}^{26}\text{fissions m}{}^{\mathrm{?3}}$) transient strains were incurred on varying stress and temperature due to the development of grain boundary gas bubbles. This also resulted in a six fold increase in the radiation creep constant between $\text{6 × 10}{}^{26}$ and $\text{1.2 × 10}{}^{27}$ fissions m^?3. A similar pattern of behaviour with respect to rating and stress was observed in hyperstoichiometric UC between 450 and 800°C up to $\text{1 × 10}{}^{27}$ fissions m^?3. However the nominally steady state creep rate was a factor 8 lower than in UO₂ irradiated under the same conditions. The experimental results also suggest that the primary creep contribution to the initial strain in compression is much higher than in UO₂. There was no evidence of either transient strain on changing stress or of an increasing creep rate at high doses. The experimental observations are reported and discussed in relation to models for irradiation induced low temperature creep in ceramic fuels.     

Irradiation creep in Zr single crystals

A single crystal of crystal bar Zr was irradiated, unstressed, at 570 K in a fast (> 1 MeV) neutron flux of $\text{5.5 × 10}{}^{16}\text{n/m}{}^2\text{-s}$. After a dose of $\text{6 × 10}{}^{23}\text{n/m}{}^2$ a tensile stress of 25 MPa was applied during a period of steady reactor power. The loading strain was an order of magnitude smaller than that observed when an identical, unirradiated, crystal was loaded to the same stress. There followed a period of primary creep during which the creep rate decreased to a value of $\text{5 × 10}{}^{\mathrm{?6}}\text{h}{}^{\mathrm{?1}}$ in the first 24 hours of the test. For the final 2000 hours of the test the specimen was observed to creep at a rate of $\text{1 × 10}{}^{\mathrm{?6}}\text{h}{}^{\mathrm{?1}}$ when the reactor was at full power. During shutdowns, the creep rate decreased with time. The results will be discussed and compared with predictions from current theories for the mechanism of irradiation enhanced creep in light of the micro-structures observed.     

Martensitic zirconium alloys: Influence of chemical composition on creep characteristics

Results of an extensive investigation of creep in martensitic zirconium alloys are summarized with the aim to show the influence of chemical composition on the main creep characteristics — the steady state creep rate and the time and strain to fracture. The activation energy of creep and the parameter of stress sensitivity of steady state creep rate are determined and possible creep mechanisms as well as creep strengthening mechanisms are discussed. The time to fracture $\text{t}{}_{\mathrm{f}}$, is related to the steady state creep rate ge_s through the Monkman-Grant relation as modified by Dobe? and Milic?ka. The creep fracture shows features different from those of “classical” intergranular cavitation creep fracture. Most probably the creep fracture is controlled by the same deformation mechanism as the creep.     

Tensile creep rupture at cyclic load variation

Assuming the validity of the life fraction rule the life time is calculated for creep rupture of material subjected to tensile cyclic stress variation. Two cases are considered: a saw tooth stress-time profile with zero and non-zero holding period. The life times and hence the number of cycles at fracture can be predicted by means of values from the static stress-rupture diagram ($\text{n, τ o, τ}{}_{\max }$) and values characterizing the stress cycle ($\text{a, ν, t}{}_{\mathrm{H}}$). The calculations for zero holding period are compared to experiments performed on the stainless steel type: DIN 1.4970 at 700°C. Very good agreement is obtained for loading conditions leading to shorter rupture times.     

Grain-boundary diffusion in uranium dioxide: The correlation between sintering and creep and a reinterpretation of creep mechanism

Previously published data on the final stage sintering kinetics of stoichiometric uranium dioxide are correlated with a reinterpretation of low-stress creep behaviour of identical material (data on both processes by the present authors). For both processes the rate-controlling diffusional flux is considered to be that of uranium ions along grain boundaries. The effective diffusion coefficient for uranium ion diffusion along grain boundaries, $\text{D}{}_{\mathrm{GB}}$, is estimated to be: $\text{D}{}_{\mathrm{GB}}\text{= 1.38(}{}_{\mathrm{÷}}{}^{\mathrm{x}}\text{5) × 10}{}^{\mathrm{?6}}\text{exp ? [(2.39 ± 0.8) × 10}{}^5\text{/8.31T] m}{}^2\text{/s}$. Comparisons are made between this value and those previously measured by radio-tracer methods.     

Observations on the low-temperature creep properties of uranium carbide under irradiation

The creep behaviour of 97% dense hyperstoichiometric UC has been examined during irradiation in three-point bend tests carried out at 450°C up to a dose of $\text{1.65 × 10}{}^{26}\text{fissions/m}{}^3$. A rapid decrease in measured strain rate with dose was observed at each stress level, nominally steady-state creep being established above $\text{≈ 1 × 10}{}^{26}\text{fissions/m}{}^3$ when the creep rate was a factor of 8 lower than that observed in UO₂ irradiated under identical conditions. Creep rates were found to be directly proportional to stress at high doses. Comparison of results from this test with data from other experiments up to $\text{2 × 10}{}^{25}\text{fissions/m}{}^3$ in compression and tension indicates little variation in the radiation-creep constant between 450°C and 800°C. The creep rate for UC, much lower than that observed in UO₂, is consistent with recently reported determinations of the effective uranium self-diffusion coefficients under irradiation in those materials.     

Plasma cooling by metal snow: I. Origin of the effect and elimination procedures

In many plasma devices, steel or inconel walls are exposed to large flux densities ?₁ of atomic hydrogen particles which, soon after the reduction of the surface oxides and carbides has started, penetrate into the lattice. The stationary hydrogen concentration $\text{c}{}_1$ in the lattice is expressed as function of ?₁, of the wall temperature $\text{T}{}_{\mathrm{<mtext>w</mtext>}}$ and of the surface roughness factor σ. It is found to be much larger than in an H₂ surrounding. Dissolved atoms recombining on internal surfaces (e.g. at grain boundaries) within the solid can then build up a considerable pressure within resulting gas pockets; should depend strongly on $\text{T}{}_{\mathrm{<mtext>w</mtext>}}$. Near room temperature, the computed values are such that surface-near pockets should crack open, releasing locally high pressure H₂ gas and some metal dust (impurity source) and increasing σ. The expected distribution of $\text{c}{}_{\mathrm{<mtext>H</mtext>}}$ within the sponge-like structure which is expected to result from a prolonged exposure to ?₁ at low $\text{T}{}_{\mathrm{<mtext>w</mtext>}}$ is derived. Means of avoiding the plasma contamination by dust release are pointed out.     

Fluage du pyrocarbone sous irradiation

Three massive samples of pyrocarbon were irradiated at 1100°C for a maximum fast-neutron dose of $\text{1.6 × 10}{}^{21}$ DNE. They were subjected to stresses in the range $\text{1.33 × 10}{}^2\text{–2 × 10}{}^2$ Kg/cm². The pyrocarbon was deposited from methane in a rotating furnace. Its density, its isotropy, its structure according to X-rays and TEM relate closely to its homologue deposited from methane in fluidised conditions. A study of creep under irradiation showed that a brief stage of primary creep is followed by a stage which is linear with respect to both stress and fast-neutron dose. Creep is thus well represented by an expression of the form $\text{? = Kσφ}$, where $\text{K}$ is $\text{2 × 10}{}^{\mathrm{?25}}$ (Kg · cm^?2. DNE)^?1, which is a value ten times greater than previously estimated. Irradiation is accompanied by densification, a slight increase in anisotropy and a reduction in $\text{L}{}_{\mathrm{c}}$ (apparent crystallite size measured along the $\text{c}$ axis). The variation of these parameters with dose does not, however, differ appreciably between the three creep samples and the unstressed sample.     

Particle re-emission during irradiation

A model of implanted gas concentration leading to surface blistering and re-emission of particles during irradiation is presented. The model assumes that particles diffuse from a Gaussian source which is independent of time and involves the solution of the time-dependent diffusion equation with a source term. The source is calculated assuming a Gaussian form with range and straggling obtained from Lindhard-Scharff-Schiøtt theory. The solution involves the Green's function for a general source term and finally a numerical integration. The re-emitted flux due to diffusion is also calculated as a function of time. The maximum concentration $\text{C}{}_{\mathrm{<mtext>m</mtext>}}\text{(x,t)}$ of particles in the solid at distance $\text{x}$ from the surface at any time $\text{t}$ is the crucial parameter. When $\text{C}{}_{\mathrm{<mtext>m</mtext>}}\text{? p}{}_{\mathrm{<mtext>c</mtext>}}\text{NA}$ anywhere in the implanted material, an interconnected region is assumed to form. $\text{N}$ is the atomic density of the target material and $\text{A}$ the area of implant. The critical concentration $\text{p}{}_{\mathrm{<mtext>c</mtext>}}$ is taken to be the concentration required for the onset of percolation ($\text{p}{}_{\mathrm{<mtext>c</mtext>}}\text{? 0.24}$ for a b.c.c. lattice). An effective diffusion coefficient for He in Nb at $\text{≈400°}\text{C}$ (and Pd at ?170°C) is extracted from experimental data and used to predict the dependence of the critical dose on the current density. The fractional release and rate of re-emission during irradiation are also calculated as a function of implanted dose.     

The tensile deformation behavior of fine-grained α-plutonium

Tensile deformation of extruded monoclinic α-plutonium with an average grain size of 4 μm was studied at stress from 2 500 to 100 000 psi (17.3 to 689 MN/m²) and temperatures from 22 to 108°C. The strain rate varied from 10^?9 to 7 × 10^?3 sec^?1. The relation, $\text{? = 2.86 × 10}{}^{\mathrm{?7}}\text{σ}{}^{4.2}\text{exp}\text{(?25 600/}\text{RT}\text{)}\text{sec}{}^{\mathrm{?1}}$, was obeyed from 12 000 to 60 000 psi (71.7 to 414 MN/m²) for strain rates greater than about 10^?6 sec^?1. Stress and temperature dependences of creep rate over this stress range were in accord with a dislocation climb controlled creep model, although the power law behavior occurred at stresses higher than theory predicts. The value of 25 600 cal/mole proved a reasonable value for the activation energy for self-diffusion in α-plutonium. At lower stresses the apparent activation energy for creep increased with decreasing stress, and the stress exponent $\text{n}\text{(=}\text{d log}\text{?/}\text{d log}\text{σ)}$ increased from 4.2 to 7.9. The high apparent activation energies for creep and high n values at low stresses were attributed to grain growth during creep. Tensile elongation increased with decreasing strain rate and increasing temperature over the entire stress range. Low elongation at high stresses was attributed to lack of grain boundary sliding. Grain size changed during creep toward a size determined by stress. At the highest test temperatures and lowest stresses grain growth occurred during large strains, while at high stresses the average grain size decreased.