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Effect of the addition of Cu on irradiation induced defects and hardening in Zr-Nb alloys
Abstract
The irradiation hardening, growth, and creep of Zr alloys are strongly influenced by the presence of alloying elements. Cu is added to some Zr alloys due to its ability to form second phase particles which can cause strengthening. However, how the addition of Cu affects the microstructure and mechanical properties under irradiation remains unclear. In the current study, two Zr alloys, Zr-2.5Nb and Zr-2.5Nb-0.5Cu were selected and irradiated to 0.5 dpa and 5 dpa at 573 K with 5 MeV self-ions. The evolution of microstructure and hardness were characterized by Transmission Electron Microscopy and nanoindentation. Results show that the existence of Cu in the Zr alloys significantly altered the irradiated microstructure and resultant hardness. The addition of Cu resulted in a slight reduction of the irradiation loop size and density in both α and β phases, and its presence in β-Zr notably delayed the precipitation of ω phase. The redistribution of Fe from β phase to α phase was also slower in Zr-2.5Nb-0.5Cu than in Zr-2.5Nb. Irradiation caused hardening in both alloys, however, the hardening in Zr-2.5Nb-0.5Cu was not as significant as in Zr-2.5Nb due to the delayed ω precipitation.
... Zr is called "the first metal of the atomic age" due to its widespread applications [10]. Therefore, there has been extensive research on the Zr alloys such as Zr-Al [11], Zr-Ti [12], Zr-Nb [13], Zr-Nb-Cu [14,15], and Zr-Nb-Mo [16] that exhibit high strength and elongation at different temperatures. It is also necessary to join the structural components of the Zr alloys to increase their stability and lower the difficulty of manufacturing. ...
( ) Research on the diffusion bonding of ( ) similar joints of zirconium (Zr) alloys is limited as compared to that on the diffusion bonding of ( ) dissimilar joints. The similar Zr alloys are difficult to bond together owing to its high melting point; however, an added interlayer can solve this problem. This study demonstrated the successful vacuum diffusion bonding of the Zr-2.5Nb (Zr705) alloy with a Cu interlayer at 900–960 °C and analyzes the microstructure and mechanical properties of the diffusion-bonded joints. The layered morphology of the joints at 900 °C and 920 °C is attributable to the formation of intermetallic compound layers ( ). In contrast, no such formation was observed at 940 °C and 960 °C owing to the complete diffusion of Cu atoms into the Zr substrate ( ). The base material at the bonding temperatures of 900 °C and 920 °C exhibited two different microstructures i.e., a Widmanstätten microstructure near the bonding interface and a duplex microstructure away from the bonding interface. However, the ( ) temperatures at 940 °C and 960 °C exhibited an entirely Widmanstätten microstructure ( ). The tensile strength of the joints increased with the ( ) bonding temperature, from 78 MPa at 900 °C to a maximum of 603 MPa at 960 °C (joint efficiency = 104.8 %), and the elongation first increased and then decreased with the increasing temperature; at 940 °C, it reached 54% that of the original Zr705 alloy.
... range of characteristics such as the matrix composition; grain sizes, shapes and relative orientations; composition and distribution of the phases (including metastable); dislocation density; and texture. It is important to emphasize that all of these parameters critically affect the functional properties of the zirconium alloys (Zavodchikov, 2008;Song et al., 2009;Markelov, 2010;Isaenkova, 2011;Shoesmith and Zagidulin, 2011;Allen et al., 2012;Poletika, 2012;Shishov, 2012;Nikulin et al., 2012;Kim et al., 2013;Adamson et al., 2013;Lumley et al., 2013;Zavodchikov et al., 2012;Evans, 2014;Sarkar and Murty, 2015;Motta et al., 2015;Corvalan et al., 2015;Tian et al., 2015;Sudhakar Rao et al., 2015;Matsunaga et al., 2015;Kumar et al., 2015;Rajpurohit et al., 2016;Jha et al., 2016;Nguyen et al., 2019;Király et al., 2019;Dong et al., 2019). In this section, a brief overview of the structures, the properties, and the applications of the up-to-date commercial zirconium alloys (Tables 1-3) is given based on the data published in (Zaymovskiy et al., (1994) ;Reshetnikov, 1995b;Properties and applicatio, 2020;Zircadyne® 702 and 705 and h, 2020;Sparkovich, 2005;Sutherlin, 2020;Zirconium in sulfuric aci, 2020a;Zirconium in sulfuric aci, 2020b;Zirconium in hydrochloric, 2020;Rudling et al., 2008;Quality and reliability a, 2015;Nikulin, 2007;Adamov, 2005;Kalin, 2008;Nikulina and Malgin, 2008 Rudling et al., 2007). ...
The article provides an overview of publications on arc welding of zirconium and its alloys for seven decades. Their features and peculiarities of the metallurgical phenomenon in relation to welding conditions are shown. The main attention is paid to the parameters of the gas tungsten arc, plasma arc, and gas metal arc welding methods. Preparation of parts for welding, welding zone shielding with inert gases, energy parameters, as well as their effect on the microstructure and the properties of welded joints, are considered. The review also outlines typical operating conditions of the welded joints in the chemical and nuclear industries, requirements for their quality, methods of control, repairing of discontinuities, as well as post weld heat treatment. Considering the fact that it is recommended (in some standards, reference and handbooks) to weld zirconium and its alloys using modes developed for titanium and its numerous alloys (that much better studied), a comparison has been made of their thermophysical properties and used welding conditions. The presented results can be adopted for development of welding procedures for zirconium and its alloys when there are no published data regarding the part thicknesses and the joint designs. However, additional metallographic, strength and corrosion tests are necessary in any case, since even the slightest changes in the chemical or phase compositions of zirconium alloys, as well as their production technology, have a significant effect on the functional properties of the welded joints.
Self-ion irradiation-induced hardening behavior and microstructural evolution in dilute Zr–Nb alloys after exposure to low-dose irradiation have been investigated using nano-indentation measurement, EBSD analysis, and TEM examination. Two types of Zr–Nb alloys, i.e., solute-type Zr0.2Nb with only solute Nb atoms and precipitate-type Zr2Nb with both solute and excessive Nb atoms, are utilized to elucidate how the addition of Nb influences microstructural and mechanical property changes upon irradiation. Significant irradiation-induced hardening was confirmed for both alloys, and the extent of hardening was slightly greater in Zr2Nb relative to Zr0.2Nb. The irradiation-induced hardening at [0001] orientation exhibited the lowest value, which gradually intensified as the orientation deviated from [0001] axis to [101¯0] and [21¯1¯0]axes, irrespective of the concentration of Nb. This tendency is ascribed to the slip-system-dependent loops-dislocation interaction. The signs that the doping of Nb can suppress irradiation defect formation were evidenced by the strong sinking effects verified at the particle–matrix interface and grain boundary. With the application of the dispersed barrier hardening model and quantitative characterization of microstructure features, it is revealed that the -loops are mostly responsible for the hardening in both alloys and a rational obstacle strength of -loop in α-Zr is derived as ∼0.33–0.36.
The effect of proton irradiation on corrosion rate of α-annealed and β-quenched Zr-0.5Nb alloys is investigated. The major focuses of this study are to understand i) if the nucleation of irradiation-induced platelets (IIPs)/nanoclusters requires dissolution of Nb-rich native precipitates, ii) if the irradiated native precipitates and interlaths are stable in the oxide, and iii) how much Nb content in the solid solution is suitable to lower the corrosion rate for Zr-Nb alloys. To answer these questions, the major characterization techniques used in this study are APT and (S)TEM/EDS to study the microstructure and microchemistry evolution following irradiation and oxidation.
The butt welding of Ta and GH3128 plates of 1.5 mm thick by using electron beam is conducted. The effects of the beam offset on the microstructure, defects, mechanical properties and fracture characteristics of the joint are studied. The results show that during direct welding, the major defects in Ta/GH3128 joint are penetrating cracks and gas pores, and the weld is mainly made up of columnar grains. The microhardness of the weld greatly changes and the maximum appears at the fusion line of Ta side. Tensile strength of the welded joint is only 150 MPa. The fracture location occurs near the Ta side by the fusion line, which shows the characteristics of brittle fracture. However, the defect-free sound joint is obtained under the condition of 0.5 mm beam offset to Ta. The microstructure of the weld is turned from the columnar grains into the equiaxial crystals. Hardness of the joint is obviously higher than that of the directly welded joint. Tensile strength of the joint is improved to 277 MPa via a 0.5 mm beam offset to Ta. The explanation for why the welded joint is strengthened under the condition of 0.5 mm deviated to the Ta side is given.
Zr-ion irradiation was performed on pure Zr, Zr-2.5Nb, and Zr-5Nb to a dose of 0.5 or 5 dpa at 573 K. The irradiation induced defects, alloying element redistribution, precipitation, and hardening were analyzed by Transmission Electron Microscopy and nanoindentation. In all the samples, irradiation induced the formation of ⟨a⟩ and ⟨c⟩ component loops in the α phase, and ½〈111〉 and 〈100〉 loops in the β phase. The presence of Nb in the α phase enhanced the formation of ⟨a⟩ component loops. As irradiation dose increased from 0.5 to 5 dpa, there is an apparent increase of the ⟨c⟩ component loop fraction with a corresponding decrease in ⟨a⟩ loops in the α phase. The Nb and Fe were observed to be redistributed from β to α phase. Irradiation significantly refined the size and increased the density of ω precipitates compared to thermal treatment alone. Irradiation caused noticeable hardening; compared with the defects in the α and β phase, the ω precipitates had a much more significant effect on hardening. This study suggests that decreasing the fraction of β phase in Zr-Nb alloys reduces the irradiation hardening.
Advancements in transmission electron microscopy allow us to draw correlations between evolving matrix chemistry environments and the resulting dislocation structures that form. Such phenomena are essential in predicting the lifetime of neutron reactor components, but are not well understood at the fundamental level. We investigate the effect of nano-scale matrix chemical evolution in Zircaloy-2 on dislocation formation after emulating commercial reactor irradiation conditions on a proton beamline. Similarity in the dislocation type, morphology, density and evolution between the different irradiation types establishes proton irradiation in this regard. For the first time, we observe chemical segregation of Fe, Ni and Cr to a-loop positions in basal traces and the segregation of Sn in alternate rows, anticorrelated to the positions of the light transition elements. The resulting layered structure with a periodicity of ∼50 nm creates an even greater anisotropy than that usually associated with HCP materials. Concurrent analysis of chemical effects and dislocation spatial relationships provides evidence that may explain the delayed onset of c-loop nucleation and accelerated dimensional instability regimes in its dependence on the alignment of a-loops parallel to the trace of the basal plane. This demonstrates the applicability of chemical-structural correlations towards key research questions regarding deformation behaviour.
A grand challenge in material science is to understand the correlation between intrinsic properties and defect dynamics. Radiation tolerant materials are in great demand for safe operation and advancement of nuclear and aerospace systems. Unlike traditional approaches that rely on microstructural and nano scale features to mitigate radiation damage, this study demonstrates enhancement of radiation tolerance with the suppression of void formation by two orders magnitude at elevated temperatures in equiatomic single-phase concentrated solid solution alloys, and more importantly, reveals its controlling mechanism through a detailed analysis of the depth distribution of defect clusters and an atomistic computer simulation. The enhanced swelling resistance is attributed to the tailored interstitial defect cluster motion in the alloys from a long-range one-dimensional mode to a short-range three-dimensional mode, which leads to enhanced point defect recombination. The results suggest design criteria for next generation radiation tolerant structural alloys.
Fe and Fe-Cr (Cr = 10-16 at.%) specimens were neutron-irradiated at 300°C to 0.01, 0.1 and 1 dpa. The TEM observations indicated that the Cr significantly reduced the mobility of dislocation loops and suppressed vacancy clustering, leading to distinct damage microstructures between Fe and Fe-Cr. Irradiation-induced dislocation loops in Fe were heterogeneously observed in the vicinity of grown-in dislocations, whereas the loop distribution observed in Fe-Cr is much more uniform. Voids were observed in the irradiated Fe samples, but not in irradiated Fe-Cr samples. Increasing Cr content in Fe-Cr results in a higher density, and a smaller size of irradiation-induced dislocation loops. Orowan mechanism was used to correlate the observed microstructure and hardening, which showed that the hardening in Fe-Cr can be attributed to the formation of dislocation loops and α’ precipitates.
The effect of heavy ion or neutron irradiation on the plasticity of Zr-2.5Nb pressure tube alloy is investigated by a combination of nanoindentation and numerical modeling, with indentation tests along the tube axial (AD) and transverse directions (TD). The true stress-strain curves are calculated from nanoindentation load-displacement curves based on an analytical model. The work hardening behaviors of samples loaded in different orientations are compared before and after irradiation. Heavy ion irradiation up to 1.2 × 10¹⁵ ion/cm² significantly increases the hardness of both AD and TD samples. The analytical model can precisely capture the plastic properties of both unirradiated and irradiated material. The calculated true stress-strain curves, including elasticity, yield strengths, and work hardening behaviors of unirradiated and irradiated samples show reasonably good agreement with experimental stress-strain curves derived from uniaxial tensile tests. The change of work hardening exponent due to irradiation shows strong orientation dependency; it decreases from 0.22 to 0.11 in AD but stays almost the same at ∼0.09 in TD after ion irradiation. The work hardening exponent is about 0.16 and 0.08 for neutron irradiation AD and TD samples, respectively. These orientational differences are related to the anisotropic interactions between plastic-flow dislocations and irradiation induced prismatic dislocation loops.
Tensile properties and microstructure of a Zr–1.8Nb alloy subjected to 140-MeV C⁴⁺ ion irradiation at 573 K up to 5.3 dpa have been investigated by using 0.18 mm-thick tensile specimens. Irradiation damage was introduced into the tensile specimens homogeneously over entire thickness using a variable energy degrader. Irradiation-induced strengthening and embrittlement (loss of ductility) were successfully evaluated. The yield strength of irradiated specimens (3.1- and 5.3-dpa) was 137% and 145% of unirradiated specimen, whereas the total elongation of those irradiated specimens was less than a half of unirradiated specimen. It appears that the rate of embrittlement is fairly faster than the rate of strengthening. Transmission electron microscopy (TEM) observation on the 3.1-dpa specimen revealed that the bcc-Nb/Zr precipitates originally contained in the alloy were stable in terms of crystal structure and size. The size and number density of dislocation loops were 7.2 nm and 1.2 × 10²¹/m³ for the -loops, 15.4 nm and 6.5 × 10²⁰/m³ for the -type loops, respectively. The increase of yield strength expected from these dislocation loops was ∼64% of the actual irradiation-induced strengthening, indicating that the dislocation loops are the main contributors for the irradiation-induced strengthening.
The stability of precipitates in Zr–2.5Nb–0.5Cu alloy under heavy ion irradiation from 100°C to 500°C was investigated by quantitative Chemi-STEM EDS analysis. Irradiation results in the crystalline to amorphous transformation of Zr2Cu between 200°C and 300°C, but the β–Nb remains crystalline at all temperatures. The precipitates are found to be more stable in starting structures with multiple boundaries than in coarse grain structures. There is an apparent increase of the precipitate size and a redistribution of the alloying element in certain starting microstructures, while a similar size change or alloying element redistribution is not detected or only detected at a much higher temperature in other starting microstructures after irradiation.
The thermodynamically equilibrium state was achieved in a Zr-Sn-Nb-Mo alloy by long-term annealing at an intermediate temperature. The fcc intermetallic Zr(Mo, Nb)2 enriched with Fe was observed at the equilibrium state. In-situ 1 MeV Kr²⁺ heavy ion irradiation was performed in a TEM to study the stability of the intermetallic particles under irradiation and the effects of the intermetallic particle on the evolution of type dislocation loops at different temperatures from 80 to 550 °C. Chemi-STEM elemental maps were made at the same particles before and after irradiation up to 10 dpa. It was found that no elemental redistribution occurs at 200 °C and below. Selective depletion of Fe was observed from some precipitates under irradiation at higher temperatures. No change in the morphology of particles and no evidence showing a crystalline to amorphous transformation were observed at all irradiation temperatures. The formation of type dislocation loops was observed under irradiation at 80 and 200 °C, but not at 450 and 550 °C. The loops were non-uniformly distributed; a localized high density of type dislocation loops were observed near the second phase particles; we suggest that loop nucleation is favored as a result of the stress induced by the particles, rather than by elemental redistribution. The stability of the second phase particles and the formation of the type loops under heavy ion irradiation are discussed.
Synchrotron X-ray diffraction, scanning electron microscopy, and transmission electron microscope bright field and high angle annular dark field were employed to investigate the microstructure and precipitates of a Zr-2.5Nb-0.5Cu alloy wire, with potential application as a spacer in the CANDU® reactor design. Results show that three types of microstructure co-exist in the alloy: Widmanstätten structure, α-Zr grains without precipitates and α-Zr grains with precipitates. In the meantime, three types of second phase particles are detected: β-Nb, Zr2Cu and Zr2Fe, which have different distributions within the microstructure. The β-Nb precipitates are observed to be extensively distributed among the α plate boundaries and twin interfaces in the Widmanstätten structure, whereas the Zr2Cu and Zr2Fe precipitates are only found in the α plate boundaries. The orientation relations of the Zr2Cu precipitates with respect to the α-Zr and β-Nb are found to be , and , respectively.
A potential factor dominating the obstacle strength of second phase precipitate particles in dispersion
strengthening is the crystallographic mismatch between the matrix phase and the second phase;
however, yet this concept has not been fully assessed by experiments and simulations. In the present
study, we experimentally investigated the obstacle strength of body centered cubic (bcc) Nb particles and
nanometric Nb clusters embedded in hexagonal close packed (hcp) Zr matrix. The bcc Nb is softer than
the hcp Zr in terms of shear modulus, whereas from a crystallographic viewpoint, the bcc Nb particles
can be nonshearable, strong obstacles because the slip plane inside the particles is not parallel with that
in the matrix. Although the bcc Nb is thermodynamically the stable configuration for Nb atoms
precipitating from the Zr matrix, in the very early stage of solute agglomeration, the crystal structure of
Nb nanoclusters is possibly hcp rather than bcc. The obstacle strength (a) was no greater than 0.5 for the
Nb nanoclusters, whereas 0.85 � a � 1 for the coarse bcc Nb particles; a ¼ 1 was obtained with the
Taylor factor (M) of 5.5 and a ¼ 0.85 with M ¼ 6.5, respectively. These results indicate that the bcc Nb
particles are strong obstacles, and that the Nb nanoclusters are weak obstacles.
Under irradiation, the acceleration of growth of nuclear reactor components made of Zircaloy is clearly correlated to the presence of 〈c〉 component dislocation loops. In the early stages, these loops appear to be essentially located close to the intermetallic precipitates. At higher doses, when 〈c〉 component dislocation loops are observed all over the microstructure, analysis of the Zr matrix reveals an homogeneous iron content due to the dissolution of the precipitates. Thus, iron may play a significant role in the nucleation of 〈c〉 component dislocation loops. A specific study has been performed on Zircaloy thin foils implanted with increasing amounts of iron. The conditions of nucleation and growth of these 〈c〉 component dislocation loops have been followed in a 1-MeV transmission electron microscope. The dose level for 〈c〉 component dislocation loop nucleation is weakly Fe-content dependent. However, their size and density increase with increasing iron implantation. These results are compared to direct observations made on irradiated reactor components. The role of iron on accelerated growth is then discussed.
The displacement damage induced in W and W-5 wt.% Re and W-5 wt.% Ta alloys by 2 MeV W + irradiation to doses 3.3 Â 10 17 – 2.5 Â 10 19 W + /m 2 at temperatures ranging from 300 to 750 °C has been characterised by transmission electron microscopy. An automated sizing and counting approach based on Image J (a Java-based image processing programme developed at the National Institutes of Health) [1] has been performed for all near-bulk irradiation data. In all cases the damage comprised dislocation loops, mostly of interstitial type, with Burgers vectors b = 1/2h1 1 1i (>60%) and b = h1 0 0i. The diameters of loops did not exceed 20 nm with most being 66 nm diameter. The loop number density varied between 10 22 and 10 23 loops/m 3. With increasing irradiation temperature, the loop size distributions shifted towards larger sizes, and there was a substantial decrease in loop number densities. The damage microstructure was less sensitive to dose than to temperature. Under the same irradiation conditions, loop number densities in the W-Re and W-Ta alloys were higher than in pure W but loops were smaller. In grains with normals close to z = h0 0 1i, loop strings developed in pure W at temperatures P500 °C and doses P1.2 dpa, but such strings were not observed in the W-Re or W-Ta alloys. However, in other grain orientations complex structures appeared in all materials and dense dislocation networks formed at higher doses.
The effect of heavy-ion irradiation on deformation mechanisms of a Zr-2.5Nb alloy was
investigated by using the in situ transmission electron microscopy deformation technique. The
gliding behavior of prismatic hai dislocations has been dynamically observed before and after
irradiation at room temperature and 300 �C. Irradiation induced loops were shown to strongly pin
the gliding dislocations. Unpinning occurred while loops were incorporated into or eliminated by
hai dislocations. In the irradiated sample, loop depleted areas with a boundary parallel to the
basal plane trace were found by post-mortem observation after room temperature deformation,
supporting the possibility of basal channel formation in bulk neutron irradiated samples. Strong
activity of pyramidal slip was also observed at both temperatures, which might be another important
mechanism to induce plastic instability in irradiated zirconium alloys. Finally, f01�11gh0�112i
twinning was identified in the irradiated sample deformed at 300 �C.
An atom probe tomography study of the microstructure of a Zircaloy-2 material subjected to 9 annual cycles of BWR exposure has been conducted. Upon dissolution of secondary phase particles, Fe and Cr are seen to reprecipitate in large numbers of clusters and particles of 1–5 nm sizes throughout the Zr metal matrix. Fe and Sn were observed to segregate to ring-shaped features in the metal that are interpreted to be -component vacancy loops. This implies that these two elements play a major role in the irradiation growth phenomenon in Zr alloys, which is believed to be caused by the formation of -loops. Similarly to autoclave-corroded Zr alloys, the formation of a sub-oxide layer of approximate composition ZrO was observed. On the other hand, no oxygen saturated metal phase was detected underneath the oxide scale.
Zirconium alloys used as cladding materials in nuclear reactors can exhibit accelerated irradiation induced growth, often termed linear growth, after sustained neutron irradiation. This phenomenon has been linked to the formation of -component dislocation loops and to the concentration of interstitial solute atoms. It is well documented for the Zircaloys that Fe dissolves from second phase particles (SPPs) during irradiation thus increasing the interstitial solute concentration in the matrix. However, no progress has yet been made into understanding whether a similar process occurs for the newer ZIRLOTM alloys. We aim to overcome this shortcoming here by studying compositional changes in second phase particles in Low Tin ZIRLOTM after neutron and proton irradiation using energy dispersive X-ray (EDX) spectroscopy. Material irradiated to 18 dpa (displacements per atom) using neutrons and to 2.3 and 7 dpa by protons was investigated. The results show that Fe is lost from Zr-Nb-Fe-SPPs during both neutron and proton irradiation. Prior to irradiation, Fe was detected at the interface of β-Nb-SPPs. This Fe enrichment is also dispersed during irradiation. Qualitatively, excellent agreement was found regarding the elemental redistribution processes observed after proton and neutron irradiation.
The effect of the alloying elements Sn, Fe, Cr, Ni, Nb, and O on hydrogen-containing alpha-zirconium and zirconium hydrides is investigated using ab initio quantum mechanical calculations and classical simulations. Cr, Fe, and Ni atoms attract interstitially dissolved H atoms whereas interstitial oxygen atoms show no pronounced interaction with H atoms. The alloying elements destabilize the hydride phases in the order Sn > Fe > Cr > Ni > Nb. Hence, substitutional Sn (if atomically dispersed), Cr and Fe atoms are likely to delay hydride precipitation, effectively increasing the hydrogen solubility. Nb and Sn influence the mobility of Zr self-interstitial atoms (SIA’s), which diffuse rapidly and preferentially parallel to the basal planes forming interstitial dislocations loops perpendicular to the basal planes (a-loops). Nb suppresses this diffusion of SIA’s, thereby reducing the rate of formation of interstitial a-loops. Sn atoms, if present on substitutional sites, have a similar, but smaller effect. If SIA’s approach substitutional Fe, Cr, and Ni atoms, the simulations indicate a spontaneous swap promoting the smaller transition metal atoms into interstitial atoms, which diffuse very rapidly with a preference in the c-direction, thereby facilitating their segregation to energetically more favorable sites such as vacancies, vacancy c-loops, grain boundaries, surfaces, and intermetallic precipitates.
The microstructures of pure Zr and Zr-base alloy (Zircaloy-2) irradiated with Zr ions (self-ions) were studied by transmission electron microscopy. It was found that the c-component dislocation structures formed in neutron-irradiated Zircaloy-2 can also be obtained by self-ion irradiation. This technique can be used to simulate neutron irradiation and is thus expected to deepen our understanding of irradiation damage in pure Zr and Zr-base alloys.
The radiation damage microstructure in hexagonal-close-packed (hcp) metals is more complex than that of cubic materials, primarily because the crystal structure is different (hexagonal rather than cubic symmetry). For pure hcp metals the principal habit plane for dislocation loop nucleation is generally the most close-packed plane and this varies with c/a ratio; {101̄0} prism planes being the most close-packed in materials with c/a < 1.732 and (0001) basal planes being the most close-packed in materials with c/a > 1.732.Recent electron irradiation results indicate that, for hcp metals such as Zr and Mg, with a c/a < 1.732, there is a higher probability for c-component dislocation loops forming on basal planes (in addition to the prism plane loops) when the materials are impure. Both interstitial and vacancy loops are affected by impurities and, once nucleated, loop growth is often very rapid. One can conclude that the most important stage of c-component loop formation is nucleation, the probability of basal loop nucleation increasing with increasing impurity content.
Intermetallic particles of Zr(Cr, Fe)2 and Zr2(Ni, Fe) in Zircaloy-2 and -4 undergo structural, chemical and morphological changes during neutron irradiation which are dependent on temperature, fluence and flux. At low temperatures (< 350 K), both particle types become amorphous. At intermediate temperatures (520–600 K), the Zr2(Ni, Fe) particles remain crystalline and the Zr(Cr, Fe)2 particles become amorphous. The susceptibility of the Zr(Cr, Fe)2 particles to the amorphous transformation increases with increasing flux and Cr content and is also dependent on the depletion of Fe and Cr below the stoichiometric limit. The volume of material transformed is proportional to the neutron fluence. At high temperatures (640–710 K) both the Zr2(Ni, Fe) and Zr(Cr, Fe)2 particles remain crystalline.For all temperatures there is radiation-induced dissolution of the intermetallic particles either as a result of sputtering or diffusion (possibly by an interstitial mechanism). At high temperatures (640–710 K) there is redistribution of the solute resulting in secondary precipitation in the matrix and at grain boundaries and a change in morphology of the intermetallic particles.Sn-rich precipitates are observed following irradiation to moderate fluences () at about 675 K and appear to be the result of radiation-enhanced diffusion.
The effect of fast-neutron damage on the room-temperature tensile properties of zircaloy-2 and zircaloy-4 irradiated at low temperature (< 100 °C) up to a fluence of 1.43 × 1020 n/cm2 (> 1 MeV) and at elevated temperature (320–360 °C) up to 1.53 × 1021 n/cm2 (> 1 MeV) was investigated. The irradiation conditions at elevated temperature simulated those of fuel elements. The behaviour of irradiation-hardening of zircaloy-2 was similar to that of zircaloy-4. Considerable irradiation-hardening and loss of ductil'ty were observed. The rate of hardening for low-temperature irradiation was higher than that for elevated-temperature irradiation. Irradiation-induced changes in yield stress, , showed saturation at a fluence of 4 × 1019 n/cm2 (> 1 MeV) for zircaloy-2 and 5 × 1019 n/cm2 (> 1 MeV) for zircaloy-4 for low temperature, while no saturation was observed after elevated-temperature irradiation up to a fluence of 1.53 × 1021 n/cm2 (>1 MeV). Low-temperature irradiation showed . below saturation while elevated-temperature irradiation showed . (φt)0.1. The amount of irradiation-hardening decreased in zircaloy-4 as the pre-irradiation yield stress was increased either by increasing the amount of cold work or decreasing the grain size. The results are discussed in terms of recent results in electron microscopy of irradiated materials.
The paper describes a novel transmission electron microscopy (TEM) experiment with in situ ion irradiation designed to improve and validate a computer model. TEM thin foils of molybdenum were irradiated in situ by 1 MeV Kr ions up to ∼0.045 displacements per atom (dpa) at 80°C at three dose rates −5 × 10, 5 × 10, and 5 × 10 dpa/s – at the Argonne IVEM-Tandem Facility. The low-dose experiments produced visible defect structure in dislocation loops, allowing accurate, quantitative measurements of defect number density and size distribution. Weak beam dark-field plane-view images were used to obtain defect density and size distribution as functions of foil thickness, dose, and dose rate. Diffraction contrast electron tomography was performed to image defect clusters through the foil thickness and measure their depth distribution. A spatially dependent cluster dynamic model was developed explicitly to model the damage by 1 MeV Kr ion irradiation in an Mo thin foil with temporal and spatial dependence of defect distribution. The set of quantitative data of visible defects was used to improve and validate the computer model. It was shown that the thin foil thickness is an important variable in determining the defect distribution. This additional spatial dimension allowed direct comparison between the model and experiments of defect structures. The defect loss to the surfaces in an irradiated thin foil was modeled successfully. TEM with in situ ion irradiation of Mo thin foils was also explicitly designed to compare with neutron irradiation data of the identical material that will be used to validate the model developed for thin foils.
A study of the deformation behavior of irradiated highly textured Zr-2.5Nb pressure tube material in the temperature range of 30 °C to 300 °C was undertaken to understand better the mechanism for the deterioration of the fracture toughness with neutron irradiation. Strain localization behavior, believed to be a main contributor to reduced toughness, was observed in irradiated transverse tensile specimens at temperatures greater than 100 °C. The strain localization behavior was found to occur by the cooperative twinning of the highly textured grains of the material, resulting in a local softening of the material, where the flow then localizes. It is believed that the effect of the irradiation is to favor twinning at the expense of slip in the early stages of deformation. This effect becomes more pronounced at higher temperature, thus leading to the high-temperature strain localization behavior of the material. A limited amount of dislocation channeling was also observed; however, it is not considered to have a major role in the strain localization behavior of the material. Contrary to previous reports on irradiated zirconium alloys, static strain aging is observed in the irradiated material in the temperature range of 150 °C to 300 °C.
The effect of dislocation self-interaction on the Orowan stress was determined for collinear, impenetrable circular obstacles. In addition, the configuration of a bowing dislocation at the Orowan stress is strongly influenced by the interactions: as the interactions increase in strength, i. e. as the ratio of obstacle size to spacing decreases, the bow-out and area swept are suppressed. The essential features of the interaction can be incorporated into a line tension framework by treating the impenetrable obstacles as if they were penetrable. Based on this line tension analogue, an approximate form for the flow stress of a random distribution of impenetrable obstacles was developed.
The contribution of the elastic interaction between dislocations moving in slip planes and of randomly distributed prismatic dislocation loops and of cavities to the critical shear stress in f.c.c. metals is estimated. The results are applied to quench hardening in aluminium.
Transmission electron microscopy and diffraction have been used to study the isothermal omega transformation in Zr-Nb alloys. A modulated structure of omega particles in a beta matrix is developed when alloys in the composition range 12 to 25% Nb are aged at appropriate temperatures. The omega particles are cuboidal, with {100}β faces, and are aligned in 〈100〉β directions in a three-dimensional array. The average particle size is 1000 Å, and the interparticle spacing is a fraction of this value. Both the particle shape and the alignment can be interpreted in terms of the degree of precipitate-matrix lattice mismatch and elastic strain considerations. The critical factor is the elastic anisotropy of the beta matrix, having soft 〈100〉 directions. The development of the modulated beta-omega microstructure involves selective nucleation and growth of particles at preferred locations with respect to existing particles. The growth kinetics during this selective nucleation and coarsening process obey a t1 3 law. The effect of quenched omega phase formed prior to aging is important only in lean alloys.
In a CANDU reactor, the fuel bundles and primary coolant are contained within Zr–2.5Nb pressure tubes that are approximately 6.3m in length, have an internal diameter of 104mm and a wall thickness of 4.2mm. The Zr–2.5Nb pressure tubes are nominally extruded at 815oC, cold-worked 27%, and stress relieved at 400oC for 24h, resulting in a structure consisting of elongated grains of hexagonal-close-packed α-Zr, partially surrounded by a thin network of filaments of body-centred-cubic β-Zr. These β-Zr filaments are metastable and contain about 20% Nb. The stress-relief treatment results in partial decomposition of the β-Zr filaments with the formation of hexagonal-close-packed ω-phase particles that are low in Nb, surrounded by a Nb-enriched β-Zr matrix. A temperature–time–transformation (TTT) diagram has been developed for the β-phase in Zr–2.5wt%Nb pressure tubes. The results show that the morphology and/or physical state of the β-phase has a significant effect on the transformation behaviour compared with a bulk Zr–20wt%Nb alloy. This means that a specific TTT-diagram is required to describe the behaviour of these engineering components.
The decomposition of the β-phase in Zr-rich Zr-Nb alloys by three processes viz., ω formation, α-formation and hydride precipitation has been examined. In the Zr-20Nb alloy, ω-formation has been examined after thermal treatment as well as after electron irradiation and a comparison has been made between the kinetics of ω-phase formation under these two conditions. The morphology of the α-precipitates and their internal structures has been found to depend upon the type of thermal treatment with step quenching from the β-phase field leading to an allotriomorphic morphology and quenching and aging leading to internally twinned Widmanstätten α. The different morphologies obtained due to change in thermal treatment and composition of the ZrNb alloys has been rationalized. Hydride formation has been examined in α-Zr, β-Zr and in α + β microstructures. A comparison has been made between the mechanism of formation of hydride phase in these three types of microstructures and their morphology and internal structures have been explained.
The effect of alloying additions and impurities on radiation damage in Zr-alloys has been studied using a high voltage electron microscope (HVEM). The electron damage results have been compared with neutron damage results for the same materials. There is some qualtitative agreement between the two techniques in that the relative susceptibility to c-component loop formation is the same in both the neutron and electron case. Cavity formation, however, occurs more readily during electron irradiation. Because of the relationship between c-component loop formation and accelerated growth, these results show that electron irradiation using a HVEM can provide a preliminary indication of the susceptibility of Zr-alloys to accelerated growth prior to their use in nuclear reactors.
An analytical transmission electron microscopy study of two-phase (α-β) structures in a Zr-2.5 wt% Nb pressure tube alloy was used to follow the distribution of Nb and Fe as a function of alloy heat treatment and tube processing. The presence of Fe (~ 0.1 wt%) modifies the (α-β) phase equilibria as Fe is a β-stabilizing element. Significant segregation of Fe to βZr and βNb structures was measured and shown to be in good agreement with the prediction of a graphical method used to find the phase boundaries in the ternary diagram (at the Zr corner) from the binary data. Segregation of Fe at α-α subboundaries was found in the as-extruded and cold-worked plus stress-relieved treatments. The Fe profiles, measured across subboundaries, indicated a strong segregation of Fe to the core or near-core regions of the dislocations comprising the boundary structure.
Recent and current work has shown that self- and substitutional diffusion in α-Zr is enhanced by the presence of the ultra-fast, interstitially-diffusing solute, Fe. The bulk of the data indicates that the diffusion coefficients scale directly with the Fe concentration. This work considers the nature of the Fe-enhancement mechanism in terms of vacancies strongly bound to either substitutional or interstitial Fe atoms. Experimental evidence from diffusion experiments, as well as from electron microscopy and positron annihilation spectroscopy investigations, is considered. It is concluded that the experimentally observed substitutional diffusion enhancements are most readily interpreted in terms of transport via an interstitial Fe/vacancy pair.
The structure of Zr-2.3%Nb and Zr-5.5%Nb alloy martensites on tempering at different temperatures in the range of 350 to 600°C was studied by optical and transmission electron microscopy. The equilibrium β-niobium phase () was found to be the precipitating phase on tempering the Zr-2.3%Nb martensites at temperatures up to 500°C and the Zr-5.5% Nb martensites at temperatures upto 450°C. Precipitation of the metastable phase of the monotectoid composition (Zr20%Nb) was observed to occur on tempering the Zr-2.3%Nb alloy at 550 and 600°C and the Zr-5.5%Nb alloy at 500 and 550°C. On tempering the latter alloy at 600°C, the martensite was found to revert back to the supersaturated β phase, which subsequently decomposed into a mixture of the α and the phases. These observations have been explained on the basis of hypothetical free energy versus composition diagrams. The_orientation relation of the precipitates with respect to the α phase was found to be as follows: ; . It was also seen that a precipitate forming at a twin boundary maintains equivalent orientation relations with the two adjacent twin related portions.
Le zircaloy, utilisé à la fois comme matériau de structure et comme combustible et élément poison servant de revêtement dans les corps de réacteur, refroidi à l'eau est soumis à une déformation contrôlée par l'effort qui, dans de nombreux cas peut atteindre le domaine plastique. Un appareillage a été réalisé et construit pour réaliser des essais de traction en réacteur dans une loupe à eau à 282 °C (540 °F), à des vitesses de déformation comprises entre 5 × 10−6 et 10−3 par heure. Les essais de traction du matériau dans l'état non irradié et irradié étaient réalisés à 282 °C et 315 °C pour des vitesses de déformation comprises entre 3 × 100 et 5 × 10−6 par heure.
Les échantillons étaient prélevés dans le sens longitudinal et le sens transversal par rapport à la direction de laminage d'une plaque de zircaloy-4. Au total, 7 essais en réacteur ont été conduits à 282 °C: 2 échantillons longitudinaux aux vitesses de déformation de 10−5 et 5 × 10−6 par heure, et 5 échantillons transversaux aux vitesses de déformation de 1 × 10−4à 5 × 10−6 par heure. La plupart des échantillons ont été essayés jusqu'à une déformation totale comprise entre 1 et 3%.
Les résultats actuels des essais en réacteurs indiquent que dans un environment neutronique et dans l'intervalle de vitesses de déformation étudiées, le zircaloy-4 a un indice apparent de sensibilité à la vitesse de déformation m = 0,23, au lieu de 0,03 pour le zircaloy-4 non irradié et après irradiation. Comme avec le zircaloy-4 non irradié, la limite élastique à 0,2% en réacteur et dans la direction transverse était plus élevée que dans la direction longitudinale, la limite élastique à 0,2% en réacteur, à la vitesse de déformation de 5 × 10−6 par heure étaient considérablement plus faibles que pour des essais comparables sur un échantillon après irradiation.
The effect of nuclear radiation on the mechanical properties of copper
was studied. It was found that the yield stress, which is substantially
increased by the radiation, increases as the cube roet of the flux. A strong
temperature dependence of the yield stress of irradiated copper was observed with
the yield stress being given by a function similar to sigma = A - BT1/2 above
40 deg K. A Luders band with slip lines of very large step height was associated
with the enhanced yield stress at small strains. At large strains the phenomenon
of overshoot was observed. The annealing kinetics of the radiation hardness were
also studied in the temperature range from 25 to 700 deg K. Little or no
annealing was observed in the region below 80 deg K. In the region from 80 to 300
deg K, approximately 20% of the yield stress was recovered, with the remainder
annealing in the range from 600 to 700 deg K. These results are discussed in
terms of the possible mechanisms by which the hardening can occur. While by no
means conclusive, these duta support a dislocation locking mechanism. On the
other hand a very close analogy exists between radiation hardening and the
hardening which arises from the addition of impurities, e.g., the hardening in
alpha brass. The correlation between work hardening and radiation hardening
appears to be quite small. (auth)
The effects of some heat treatments on the hardness of Zr--2.5 wt% Nb, ; and the effects of neutron irradiation at 50 deg C and at 250 deg C on its ; tensile and impact properties are described. Post-irradiation damage recovery ; data are also presented. It is concluded that the alloy's mechanical properties ; both before and after irradiation are significantly better than those of the ; Zircaloys. The heat-treated Zr-Nb alloy appears to be metallurgically stable ; under irradiation. (auth);
Over forty years of in-reactor testing and over thirty years of operating experience in power reactors have provided a broad understanding of the in-reactor deformation of cold-worked Zr–2.5Nb pressure tubes, and an extensive data-base upon which to base models for managing the life of existing reactors and for designing new ones. The effects of the major operating variables and many of the metallurgical variables are broadly understood. The deformation is often considered to comprise three components: thermal creep, irradiation growth and irradiation creep. Of the three, irradiation growth is best understood – it is thought to be driven by the diffusional anisotropy difference (DAD). It is still not clear whether the enhancement of creep by irradiation is due to climb-plus-glide (CPG), stress-induced preferred absorption (SIPA) or elasto-diffusion (ED). The least understood area is the transition between thermal creep and irradiation where the fast neutron flux may either suppress or enhance the creep rate. The three components are generally treated as additive in the models, although it is recognized that this is only a crude approximation of reality. There are still significant gaps in our knowledge besides the thermal- to irradiation-creep transition, for example, the effect of Mo which is produced from Nb by transmutation in the thermal neutron flux is not known, and on-going work is required in a number of areas. This paper reviews the current state of knowledge of the in-reactor deformation of cold-worked Zr–2.5Nb pressure tubes, and highlights areas for further research.
Annealed Zircaloy-2 specimens irradiated to at ≈425 K were tensile-tested, together with unirradiated material, between 298 and 673 K in vacuum. The surface and microstructure of deformed specimens were observed using projector, optical and transmission electron microscope. Metallographie examinations or irradiated samples showed that localized deformation bands occur at intervals during deformation to the ultimate tensile stress between 473 and 623 K, while in the room temperature deformation, the inhomogeneity in metallographic features was characterized by microscopic dislocation channeling structure, without showing localized band on the surface of deformed specimen. From the shape of the stress-strain curves and metallographic features, it was concluded that the first localized band occurred after a small amount of strain, resulting in yielding and that the flow stress increased monotonically with further yielding. Evidences from temperature dependent mechanical properties of irradiated and unirradiated samples indicate that the radiation-anneal hardening phenomena are of significance in a range of temperature at which localized bands are formed, particularly pronounced at 553–593 K.
Copper single crystals have been simultaneously strengthened by solved gold-atoms and by incoherent SiO2-particles. The critical resolved shear stress (CRSS) τs of a binary copper–gold solid solution and the CRSS τt of the same solid solution additionally strengthened by SiO2-particles have been measured in the temperature range 90–283 K. From the experimentally established temperature dependences of τs and τt and from the theoretically known one of τp (=CRSS if the SiO2-particles are the only strengtheners present), the function τt(τs,τp) has been derived: τt=(τks+τkp)1/k, with k≈1.8. This result is at variance with a linear relationship suggested earlier by Ebeling and Ashby [Phil. Mag. 13 (1966) 805]. The bearings of the present findings on the evaluation of experimental data on dispersion strengthening are evident.
Tracer diffusion of 64Cu in α-Zr single crystals has been measured in the temperature range (615-860)°C. The temperature dependences of the 64Cu diffusion coefficients in directions parallel to and perpendicular to the c axis are given by D∥=0.40e-1.54eV/kT and D⊥=0.25e-1.60eV/kT cm2/sec, respectively. The results are discussed in terms of an interstitial diffusion mechanism.
The irradiation damage configuration and yield stress response obtained in the c.p.h. α’and α phases in a Zr-2·7% Nb alloy after fast neutron irradiation have been studied as a function of pre-irradiation solute distribution. Largest yield stress increments were obtained for specimens irradiated with the solute initially in metastable solution, pre-irradiation ageing treatments reducing the yield stress response. Electron microscope studies have shown that the irradiation damage configuration was also determined by the solute super-saturation in the alloy phases, maximum strengthening being associated with a population of defect clusters or loops of < 30 Å diameter. A mechanism for the defect stabilization effect is proposed.
Microhardness testing is an efficient means of assessing the mechanical properties of many materials, and is especially convenient for irradiated samples because of the small sampling volume requirement. This paper provides correlations between hardness and yield stress for both irradiated austenitic and ferritic steels by combining existing data in the open literature. For austenitic stainless steels, seven data sets were assembled and the change in yield stress was determined to simply be the change in hardness times a factor of 3.03. For the pressure vessels steels, five studies containing both hardness and yield stress data were combined. In ferritic steels, the correlation factor between change in yield stress and change in hardness was found to be 3.06. The similarity in correlation factors for austenitic and ferritic steels is consistent with previous theoretical and experimental results.
The influence of irradiation (3.58 × 1024n/m2; E > 1 MeV at 583 K for 52 days) has been compared to the effects of thermal exposure alone (52 days at 583 K) on the redistribution of Nb and Fe in an as-extruded Zr-2.5 wt% Nb pressure tube alloy. The major change associated with the thermal treatment is a decomposition of the retained β-Zr phase, to form ω. As the ω-phase grows, it rejects Nb and Fe to the surrounding β-Zr. No significant changes were noted in the Nb and Fe levels segregated at α-α grain boundaries on heat treatment. Similar effects were observed in the irradiated samples, but in addition the retained β-phase is depleted of Fe. At the same time Fe is detected in the α-matrix of the irradiated specimen. Although analytical electron microscopy in a dedicated STEM instrument does not lead to reliable estimates of the absolute concentration of Fe in the irradiated sample, a series of experiments are described which show that Fe is present in the α-matrix of the alloy.
The indentation load-displacement behavior of six materials tested with a Berkovich indenter has been carefully documented to establish an improved method for determining hardness and elastic modulus from indentation load-displacement data. The materials included fused silica, soda–lime glass, and single crystals of aluminum, tungsten, quartz, and sapphire. It is shown that the load–displacement curves during unloading in these materials are not linear, even in the initial stages, thereby suggesting that the flat punch approximation used so often in the analysis of unloading data is not entirely adequate. An analysis technique is presented that accounts for the curvature in the unloading data and provides a physically justifiable procedure for determining the depth which should be used in conjunction with the indenter shape function to establish the contact area at peak load. The hardnesses and elastic moduli of the six materials are computed using the analysis procedure and compared with values determined by independent means to assess the accuracy of the method. The results show that with good technique, moduli can be measured to within 5%.
A study of the plastic deformation of metals by hard indenters shows that the indentation hardness is essentially a measure of the plastic yield stress of the metal. With pyramidal (and conical) indenters the hardness, because of geometric similarity, is independent of the size of the indentation. With spherical indenters this is not so, the hardness increasing with size of indentation. From the increase in hardness with load, a semi-quantitative estimate may be made of the work-hardening characteristics of the metal.
In the scratching and indentation of brittle solids such as minerals it is shown that the high hydrostatic pressures developed around the deformed region are often sufficient to inhibit brittle fracture. Under these conditions the deformation is primarily plastic. For this reason there is fairly good correlation between indentation and scratch hardness since both are essentially a measure of the plastic and not the brittle properties of the solid. From this approach it is possible to provide a physical basis for Mohs' scratch-hardness scale and to show that, excluding diamond, there is a reasonable "equality of intervals" between each number on the scale.
Secondary ion mass spectrometry techniques have been used to determine the terminal solid solubility (TSS) of Fe in α-Zr. Single crystals of nominally pure and Fe-doped α-Zr were annealed in the temperature range 770–1100 K to promote equilibration of Fe between surface Zr3Fe precipitates, or β-Zr(Fe), and α-Zr. The results are fair in overall agreement with a recent investigation, based on thermoelectric power measurements, but they differ in detail. In particular this work indicates two regions of temperature dependence: above 930 K the TSS (ppma) is given by CFe = 1.56 × 1010exp(−1.70 ± 0.05 eV/kT), at lower temperatures a weaker temperature dependence is associated with extrinsic effects. In addition, the eutectoid temperature is shown to lie between 1063 and 1068 K.
The current understanding of defect production fundamentals in neutron-irradiated face centered cubic (FCC) and body centered cubic (BCC) metals is briefly reviewed, based primarily on transmission electron microscope observa-tions. Experimental procedures developed by Michio Kiritani and colleagues have been applied to quantify defect cluster size, density, and nature. Differences in defect accumulation behavior of irradiated BCC and FCC metals are discussed. Depending on the defect cluster obstacle strength, either the dispersed barrier hardening model or the Friedel–Kroupa–Hirsch weak barrier model can be used to describe major aspects of radiation hardening. Irradiation at low temperature can cause a change in deformation mode from dislocation cell formation at low doses to twinning or dislocation channeling at higher doses. The detailed interaction between dislocations and defect clusters helps determine the dominant deformation mode. Recent observations of the microstructure created by plastic deformation of quenched and irradiated metals are summarized, including in situ deformation results. Examples of annihilation of stacking fault tetrahedra by gliding dislocations and subsequent formation of mobile superjogs are shown. Ó 2004 Elsevier B.V. All rights reserved.
Strengthening of dislocations by multiple obstacle types occurs in many engineering alloys. Theories have rationalized two
different scaling laws for the total strength, tta = t1a + t2a , \tau_{t}^{\alpha } = \tau_{1}^{\alpha } + \tau_{2}^{\alpha } , with α=1 or 2, where τ
1 and τ
2 are the strengths of the two individual obstacle types. Simulations have clearly demonstrated α=2, while “friction” strengthening must correspond to α=1. Here, line-tension simulations of dislocation glide through two types of point obstacles are performed to examine the
friction limit. One obstacle type is weak (critical angle approaching 180deg) but with very high density, approximately corresponding
to solute strengthening; and the second obstacle type is stronger (smaller critical angle) but with lower density, approximately
corresponding to forest or precipitate strengthening. Additive strengthening α=1 is obtained when the densities of the two obstacle types differ by more than a factor of ~67, while a transition to α=2 occurs with increasing density of the second obstacle. These simulations confirm the long-held metallurgical wisdom regarding
additivity of solute or friction strengthening with other strengthening mechanisms and also demonstrate that apparent intermediate
scaling laws with 1<α<2 can arise for a range of relative obstacle densities. Investigation of several literature experimental studies shows
some agreement with the model here but quantitative comparisons remain difficult.