Category Archives: An Introduction to Nuclear Materials

Corundum Structure

One of the most common oxide crystal structures is of corundum or a-alumina (Al2O3). The stoichiometry of the corundum-type structures is M2X3. These com­pounds are also known as “sesquioxides.” The basic structure contains hexagonal close-packed layers of the anion (i. e., O2~) sublattice. If the cations need to fill in all the octahedral sites, the stoichiometric ratio becomes MX as the number of the octahedral sites is equal to the number of the regular lattice sites in a HCP struc­ture. Because of the particular stoichiometry of Al2O3 (i. e., cation/anion = 2:3), the cations only fill in two-thirds of the available octahedral interstitial sites, however,

image065

Figure 2.23 The crystal structure of zincblende (the gray circles represent S2 and the black circles represent Zn2+).

maintaining maximum cation separation in an orderly manner. A simpler way to understand the corundum structure is to see the ion positions from both the top and a vertical section. A basal plane of a corundum structure is shown in Figure 2.24a. The picture shows a layer of cations (indicated by filled circles) forming a hexagonal network with the same ion spacing between the two anion layers (only one layer is delineated by the larger open circles in this figure). Note that x positions indicate the unfilled octahedral sites. The next cation layer would have the same hexagonal configuration, but shifted to one atom spacing in the direction of “vector 1,” as shown in Figure 2.24a. When an another close- packed oxygen layer is added on top of this cation layer, an another cation layer will sit on it, although shifted by yet another atom spacing in the direction of “vector 2” while maintaining the similar hexagonal configuration. The structure becomes clearer if we take a vertical plane {1010} across the corundum struc­ture, as shown in Figure 2.24b. This shows that each octahedral column of cat­ions contains two cations in a row followed by one missing cation and so on. If one takes into account the periodic spacing of both the cations and anions at the same time, we find that the structure repeats itself after six such layers (shown as ABABAB), thus making the lattice constant in the vertical direction (c) as 1.299 nm. As the octahedral sites in the corundum share faces and two out of every three octahedral sites are occupied, the electrostatic repulsion between cations helps move them slightly into the unfilled octahedral site, whereas oxy­gen ions also shift slightly from their ideal close-packed configuration, thus making the corundum structure bit inherently distorted.

The corundum structure is exhibited by a multiple number of oxides, such as Fe2O3, Cr2O3, Ti2O3, V2O3, Ga2O3, Rh2O3, La2O3, Nd2O3, and so on. There are many ternary compounds such as ilmenite (FeTiO3) and lithium niobate (LiNbO3)

image066

Figure 2.24 (a) Filling oftwo-third octahedral sites in the basal plane of corundum. (b) Two-third occupancy of octahedral site columns (the plane shown is the dashed line in part (a)) (Ref. 10)

that assume “derivative” corundum structures. The Cr-containing steels in contact with water generally have corrosion layers composed of Fe2O3 and Cr2O3, both of which have a corundum-type crystal structure. Furthermore, the rare earth-based fission products, such as Nd2O3 and La2O3, have corundum-type crystal structure.

Table 2.3 A summary of important characteristics in some crystal structures.

Important parameters

SC

BCC

FCC

HCP

Effective number of atoms per unit cell

1

2

4

6

Coordination number

6

8

12

12

Atomic radius (r) versus lattice constant (a)

a = 2r

f3a = 4r

p2a = 4r

a = 2r

Atomic packing fraction

0.52

0.68

0.74

0.74

Closest-packed direction

(100)

(111)

(110)

(1120)

Closest-packed plane

{100}

{110}

{111}

{0002}

Close-packed stacking sequence

ABCABC

ABABAB

2.1.9

Determination of Burgers Vector Magnitude

So far, we have come across a couple of equations that include the magnitude of Burgers vector in this chapter. The concept of Burgers vector was first introduced in Section 2.2. Now we need to understand how the magnitude of the Burgers vec­tors can be determined. Burgers vector is basically the shortest lattice vector joining one lattice point to another. This type of Burgers vector is associated with disloca­tions termed as perfect or unit dislocation. For example, in a BCC crystal, the shortest lattice vector is the distance between a corner atom and the body center. We can resolve the vector by a0/2 length from the origin along X-, Y-, and Z-axes, where a0 is the lattice constant. The standard notation for Burgers vector is then [(a0/2), (a0/2), (a0/2)] or (a0/2)[111]. The magnitude (strength) of the Burgers vector ofa perfect dislocation in a BCC crystal is

b — ^4 + 0° + a0 (i’6’’ half °f the body diagonal).

Similarly, in FCC metals, the Burgers vector is (a0/2) [110], that is, half of the face diagonal. The magnitude of the vector is a0//2. In a cubic crystal, we can follow the following procedure to find out the Burgers vector of a dislocation (does not need to be a perfect dislocation). If a dislocation has Burgers vector of q[uvw] with x being a fraction or a whole number, the magnitude of the Burgers vector is q-(u2 + v2 + w2)1/2. For instance, Burgers vector of a0/3 [112] dislocation has a mag­nitude of (a0/3)(12 + 12 + 22)1/2 — (a0/3)p6.

Determination of the Burgers vector of a perfect dislocation in a HCP crystal is much simpler than the cubic metals. The strength of the Burgers vector is given by the edge of the base, that is, a0.

■ Example 4.3

An FCC crystal (lattice constant a0 — 0.286 nm) contains a total dislocation density of 109 m~2, of which only half are mobile (i. e., glissile). If we assume that the Burgers vector of all the mobile dislocations is a0/2 [110] and the shear strain rate is 10-1 s_1, what is the average dislocation velocity?

Solution

To solve the above problem, we need to use Eq. 4.8.

Given are the shear strain rate (у) = 10-1 s-1, mobile dislocation density (pm) = 0.5 x 109 cm-2, and the magnitude of the Burgers vector = a0//2 = 0.202 nm. Therefore, the average dislocation velocity is given by

■ 1ms 1.

Подпись:10-1 s—1

Vd =

4.1.5

Stress-Strain Curves

Any material would deform under an applied load. If all the deformation may recover upon unloading, this type of deformation is known as elastic deformation

image315

Figure 5.2 (a) A schematic of a tensile tester. Adapted from Ref. [1]. (b) An Instron universal tester that can perform tensile test among many other tests. Courtesy: Instron.

as we defined earlier. However, if the load is large enough, plastic deformation sets in so that the material undergoes a permanent deformation, that is, upon unload­ing only the elastic portion of the deformation recovers but the rest remains.

Even though the raw data obtained from the tension test are load and elongation, they need to be converted into stress and strain, respectively, to have meaningful use in engineering considerations. First, let us talk about the engineering stress — strain curve that is of prime importance in many load-bearing applications. Engi­neering stress and strain are determined based on the original dimensions of the specimens. Engineering stress (ae), also known as conventional or nominal stress,

image316

Figure 5.3 Design of an ASTM standard round tensile specimen.

image317186 I 5 Properties of Materials thus is given by P

Se = —, (5.5)

where P is the instantaneous load and A0 is the initial diameter of the gauge sec­tion. Engineering strain is defined as the ratio of the elongation of the gauge length (d = L — L0) to the original gauge length (L0) as shown in the equation below:

Подпись:_ d _ L — L0 L0 L0

where L is the instantaneous gauge length. Engineering strain is generally expressed in terms of percent elongation. As the engineering stress-strain curves are obtained based on the original dimensions (constant), the engineering stress — strain curve and load-elongation curve have the similar shape. A schematic engi­neering stress-strain curve is shown in Figure 5.4.

Let us talk about the general shape of the engineering stress-strain curve.

a) In the initial linear portion of the curve, stress is proportional to strain. This is the elastic deformation (instantaneously recoverable) regime where Hooke’s law is applicable. The modulus of elasticity or Young’s modulus (Eq. (5.3)) can be determined from the slope of the straight portion on the stress-strain curve. Young’s modulus is determined by the interatomic forces that are difficult to change significantly without changing the basic nature of the materials. Hence, this is the most structure-insensitive mechanical property. However, any signifi­cant change in the crystal structure (such as material undergoing polymorphic transformation) would also change the elastic modulus. It is only affected to a small extent by alloying, cold working, or heat treatment. Most aluminum alloys have Young’s moduli close to 72 GPa. Modulus of elasticity does decrease with increasing temperature as the interatomic forces become weaker.

Strain to fracture

Uniform strain

Offset yield strength

Tensie strength

fracture stress

Elongation (%)

Figure 5.4 A typical engineering stress-strain curve.

b) image319When the deformation proceeds past a point, it becomes nonlinear, and the point at which this linearity ends is known as the proportional limit and the stress at which plastic deformation or yielding starts is known as yield stress or yield strength that depends on the sensitivity of the strain measurement. A majority of materials show a gradual transition from the elastic to the plastic regime and it becomes difficult to determine exactly what the yield stress actu­ally is. That is why yield strength is generally taken at an offset strain of 0.2%, as shown in Figure 5.4. In cases, where there is no straight portion in the stress — strain curve (such as for soft copper and gray cast iron), the yield strength is defined as the stress that generates a total strain of0.5%.

Another shape of stress-strain curve that is commonly observed in some spe­cific materials represents discontinuous yielding. Figure 5.5 shows such an engineering stress-strain curve. Some metals/alloys (in particular, low carbon steels) exhibit a localized, heterogeneous type of transition from elastic to plastic deformation. As shown in the figure, there is a sharp stress drop as the stress decreases almost immediately after the elastic regime. This maximum point on the linear stress-strain curve is known as “upper yield point.” Following this, the stress remains more or less constant, and this is called “lower yield point.”

Yield point phenomenon occurs because the dislocation movement gets impeded by interstitial atoms such as carbon, nitrogen, and so on forming sol­ute atmospheres around the dislocations. However, at higher stress, the disloca­tions break away from the solutes and thus requires less stress to move. That is why a sharp drop in load or stress is observed. The elongation that occurs at constant load or stress at the lower yield point is called “yield point elongation.” During yield point elongation, a type of deformation bands known as Liiders bands (sometimes known as Hartmann lines or stretcher strains) are formed across this regime. This particular phenomenon is called Piobert effect. After Liiders bands cover the whole gauge length of the specimen, the usual strain hardening regime sets in, as shown in Figure 5.5. However, temperature of test­ing could change the behavior drastically.

Upper yield

Yield elongation

Lower yield

Подпись: pointStress

•Liiders

Wr band

Unyielded metal

Elongation

Figure 5.5 Typical discontinuous stress-strain curve with distinct yield point phenomenon. From Ref.[2].

c) After proportional limit or yielding, the material enters a strain hardening regime where stress increases with increase in strain until it reaches a stress where non­uniform plastic deformation (or necking starts). This stress is known as tensile strength or ultimate tensile stress (UTS) corresponding to the maximum load in the load-elongation curve. So, basically UTS is given by the maximum load divided by the original cross-sectional area of the specimen. UTS in itself is not a property of fundamental significance. But this has long been used in the design of materials with a suitable safety factor (~2). Nowadays yield strength rather than UTS is used for the purpose of designing. But still it can serve as a good quality control indicator and in specifications of the products. However, since UTS is easy to determine and is quite a reproducible property, it is still used in practice. For brittle materials, UTS is considered a valid design criterion.

d) Ductility as determined by the tension testing is a subjective property; however, it does have great significance: (i) Metal deformation processing needs a mate­rial to be ductile without fracturing prematurely. (ii) A designer is interested to know whether a material will fail in service in a catastrophic manner or not. That information may come from ductility. (iii) If a material is impure or has undergone faulty processing, ductility can serve as a reliable indicator even though there is no direct relationship between the ductility measurement and service performance.

image321 Подпись: (5.7) (5-8)

Generally, two measures of ductility as obtained from a tension test are used — total fracture strain (ef) and the reduction of area at fracture (q). These properties are obtained after fracture during tension test using the following equations:

where Lf and Af are the final gauge length and cross-sectional area, respectively. Generally total fracture strain is composed of two plastic strain components (prenecking deformation that is uniform in nature, and postnecking deforma­tion that is nonuniform in nature). Both these properties are expressed in per­centage. We note from Eq. (5.7) that the elongation to fracture depends on the gauge length (L0). That is why it is customary to mention the gauge length ofthe tension specimen while reporting the elongation values; 2 in. gauge lengths are generally used. On the other hand, percentage reduction in area is the most structure-sensitive property that can be measured in a tension test even though it is difficult to measure very accurately or in situ during testing.

e) Fracture stress or breaking stress (Eq. (5.9)) are often used to define the engi­neering stress at which the specimen fractures. However, the parameter does not have much significance as necking complicates its real value.

Подпись: (5.9)Pf

A ’

where Pf is the load at fracture.

f)We discuss two more properties that are of importance — resilience and tough­ness. From dictionary meaning, they would be considered synonyms. But in the context of material properties, they are bit different. Resilience is the ability of a material to absorb energy when deformed elastically. Modulus of resilience (U0) is used as its measure and is given by the area under the stress-strain curve up to yielding:

Подпись:U0 = ^. 0 2E

where se0 is the yield strength and E is the modulus ofelasticity.

image325

Toughness is the ability of a material to absorb energy in the plastic range. As we will see later, there are two more types of toughness that are often used: frac­ture toughness and impact toughness. Hence, the toughness one obtains from stress-strain curves is known as tensile toughness. The area under the stress — strain curve generally indicates the amount of work done per unit volume on the material without causing its rupture. Toughness is generally described as a parameter that takes into account both strength and ductility. There are empiri­cal relations that express toughness; however, as they are based on original dimensions, they do not represent the true behavior in the plastic regime.

Even though we use the parameters obtained from engineering stress-strain curves for engineering designs, they do not represent the fundamental deforma­tion behavior of the material as it is entirely based on the original dimensions of the tensile specimen. In the engineering stress-strain curve, the stress falls off after the maximum load due to the gradual load drop and calculation based on the original cross-sectional area of the specimen. But in reality, the stress does not fall off after maximum load. Actually, the strain hardening effect (the stress in fact increases) remains in effect until the fracture, as shown in Figure 5.6, but the cross-sectional area of the specimen decreases more compared to the load drop, thus increasing the stress. This happens because the true stress (st) is based on the

Подпись: Figure 5.6 A comparison between an engineering stress-strain curve and a true stress-true strain curve.

instantaneous cross-sectional area (A) and is expressed as P

St = A = Se(e + 1). (5.12)

Подпись: E = ln Подпись: LL Подпись: ln (e + 1). Подпись: (5.13)

Before necking, it is better to calculate the true stress from the engineering stress and engineering strain based on the constancy of volume and a homogeneous dis­tribution of strain across the gauge length. However, after necking, these assump­tions are hardly valid and the true stress should then be calculated by using the actual load and cross-sectional area in the postnecking regime. True strain, on the other hand, is generally calculated from the expression as given below:

Подпись: E = ln Подпись: A0 Подпись: 2ln Подпись: (5.14)

However, the second equality of the equation does not hold valid after necking. That is why beyond the maximum load, true strain should be calculated based on actual area or diameter of the specimen measurement following the relation given below:

For the true stress-strain curves, some parameters such as true stress at maxi­mum load, true fracture stress, true fracture strain, true uniform strain, and so on can be calculated using appropriate relations.

One important parameter that can be obtained from the true stress-true strain curve is the strain hardening exponent (n). This is generally described by Hollo — mon’s equation:

Подпись: (5.15)S = Ken,

Figure 5.7 Double logarithmic plot illustrating the method of determining strain hardening exponent and strength coefficient.

where K is the strength coefficient and n is known as strain hardening (or work hardening) exponent or parameter or coefficient. As the above equation is in the form of a power law, the true stress and true strain data when plotted on a double logarithmic scale and fitted to a straight line will yield a slope that is equal to strain hardening exponent, and the strength coefficient (i. e., the true stress at e = 1.0) can be calculated from the extrapolated line, as shown in Figure 5.7. Note that the data used for the calculation of n should not be taken beyond the maximum load (i. e., at or after maximum load). Theoretically, n value can range between 0 (elastic solid, follows Hooke’s law) and 1 (perfectly plastic solid). For most metals/alloys, n values are found to be 0.1-0.5. Note that these aspects are valid for stresses and strains beyond the elasticity (i. e., in the plastic deformation regime). Strain hardening is sensitive to the microstructure, which in turn depends on processing. The rate of strain hardening is obtained by differentiating Eq. (5.15) with respect to e and then replacing Ken by st, the following equation is obtained:

Подпись:Подпись:

Подпись: n
image337
Подпись: K
Подпись: 1

t St

n— .

e

Note that often the data at low strains and high strains tend to deviate from Hollomon’s equation. That is why many other techniques, such as Ludwik’s equa­tion, Ludwigson’s equation, Ramberg-Osgood relation, and so on have been pro­posed over the past years to obtain a better estimate of strain hardening exponents.

One important aspect of strain hardening exponent is that it represents the true uniform strain, and is thus related to the ductility of a material. In order to derive the relation, we need to first discuss about the instability effect that occurs in tension. In tension, necking (i. e., localized deformation) occurs at

the maximum load (generally in a ductile material since brittle materials frac­ture well before reaching that point). The load-carrying capacity of the material increases as the strain increases; however, as noted before, the cross-sectional area decreases as an opposing effect. However, at the onset of necking, the increase in stress due to decrease in cross-sectional area becomes higher than the concomitant increase in the load-carrying ability of the material because of strain hardening. The condition of instability can be written in a differential form as dP = 0 where P is a constant (the maximum load). Now replacing P with (st ■ A) in this instability equation, we obtain

dP — d(at ■ A) — A ■ dat + at ■ dA — 0

or

Подпись: (5.17)dA dat

A at

Due to the constancy of volume (V) during plastic deformation, we can also write

dV — 0, V — AL; hence dV — d(AL) —A ■ dL + L ■ dA — 0

Подпись: dL T Подпись: dA — de. A Подпись: (5.18)

or

image343 Подпись: (5.19)

By comparing Eqs (5.17) and (5.18), we can write

which is valid at the condition oftensile instability (or at maximum load). Now comparing Eqs (5.16) and (5.19), we obtain

n — eu, (5.20)

where eu is the true uniform strain, that is, true strain at the maximum load. Hence, it can be noted that the higher the strain hardening exponent, the greater the uniform elongation, which in turn helps in improving the overall ductility. Note that Considere’s criterion (da/de — a or da/de — a/(1 + e)) can be used to estimate the strain hardening exponent from a graphical plot, this is however out­side the scope of this book. For further details on these relations, readers are referred to the suggested texts enlisted in Bibliography.

5.1.1.1 Effect of Strain Rate on Tensile Properties

The rate at which strain is imposed on a tensile specimen is called strain rate (e — de/dt). The unit is generally expressed in s_1. It is instructive to know the ranges of generic strain rates used in different types of mechanical testing and deformation processing. Generally, quasi-static tension testing involves strain rates in the range of 10_5-10_1 s_1. Generally, tension testing is done by placing the tension specimen in the cross-head fixture and running the test at a constant

cross-head speed. However, it is important to know what the strain rate is in the specimen. Following Nadai’s analysis, the nominal strain rate is expressed in terms of cross-head speed (v) and original gauge length (L0):

Подпись: (5.21)v

Lo

image344 Подпись: (5.22)

However, the true strain rate changes as the gauge length changes:

Therefore, it must be noted that most tensile tests are not conducted at constant true strain rates. However, specific electronic feedback (open loop before necking and closed loop after necking) system can be set up with the tension tester where the cross-head speed is continuously increased as the test progresses in order to maintain the same true strain rate throughout the test. But the test becomes more complicated without much benefit.

The flow stress increases with increasing strain rate. The effect of strain rate becomes more important at elevated temperatures. The following equation shows the relation between the flow stress and true strain rate at a constant strain and temperature:

S = Cem (5.23)

where C is a constant and m is the strain rate sensitivity (SRS). The exponent m can be found out from the slope of the double logarithmic plot of true stress versus true strain rate. The value of strain rate sensitivity is quite low (<0.1) at room tempera­ture, but it increases as the temperature becomes higher with a maximum value of 1 when the deformation is known as viscous flow. Figure 5.8 shows flow stress (at 0.2% strain) versus strain rate on a double logarithmic plot for an annealed 6063 Al-Mg-Si alloy. In superplastic materials, the strain rate sensitivity is higher (0.4-0.6). But these materials require finer grain diameter (<10 pm) and tempera­tures at or above half the melting temperature (in K). Superplastic materials exhibit higher than normal ductility (as rule of thumb more than 200%) and utilize strain rate hardening instead of strain hardening.

Cluster Formation

The fraction of defects produced in a cascade is between 20-40% of that that is predicted by the NRT model because of intracascade recombination. If the clusters are stable, they may migrate away from the cascade region and can be absorbed at various sinks such as dislocations and grain boundaries. In general, vacancy clus­ters and interstitial clusters should be treated separately. Interstitial clusters are sta­ble, whereas vacancy clusters are not. Interstitial clusters possess higher mobility than their vacancy counterparts.

Incascade clustering is important as it helps promote nucleation of extended defects. Interstitial cluster occurs either in the transition phase between the colli — sional and thermal spike stages or during the thermal spike stage. The probability of clustering is enhanced with increase in the PKA energy with interstitial cluster­ing being predominant, as shown in Figure 6.6.

image507

Figure 6.6 The fraction of SIAs that survive as clusters containing at least two interstitials in several metals and Ni3Al — MD simulation results Ref. [1].

image508

Figure 6.7 Weak beam dark field TEM images of defect clusters in neutron-irradiated molybdenum at different dose levels: (a) 7.2 ■ 10~5 dpa, (b) 7.2 ■ 10~4 dpa, (c) 7.2 ■ 10~3 dpa, (d) 0.072 dpa, and (e) 0.28 dpa Ref. [2, 3].

The structure of clusters is generally a strong function of the crystal structure. In a-Fe (BCC crystal structure), the most stable configuration is small clusters (<10 SIAs), a set of (111) crowdions. The next in stability is the (110) crowdions. As the cluster size grows, only two configurations become stable: (111) and (110). These crowdions can act as the precursor for the formation of perfect interstitial loops. Figure 6.7 shows a few TEM weak beam dark field images of small interstitial type loops in fast neutron irradiated molybdenum (BCC).

In copper (FCC crystal structure), the (100) dumbbell configuration is the stable configuration of the SIA; the smallest cluster may contain only two such dumbbells (di-interstitials). Larger clusters could be a set of (100) dumbbells or a set of (110) crowdions each with {111} habit plane. During growth, the clusters change to faulted Frank loops with Burgers vector (1/3)(111) and to perfect loops with (1 /2)(110). Figure 6.7 shows a few TEM weak beam dark field images of small interstitial type loops in fast neutron irradiated molybdenum.

Подпись: Note Crowdion Crowdion takes place when an atom is added to a lattice plane, yet it does not stay in an interstitial position. To accommodate the atom, lattice atoms numbering over 10 or more in a particular direction are all shifted with respect to their lattice sites. A crowdion configuration is shown in Figure 6.8. Also, see crowdion configuration in Seeger’s model as illustrated in Figure 3.1(b). The configuration can resemble a dumbbell spread over 10 atoms along a row. This phenomenon is a regular feature in focusons (i.e., focusing collisions). However, these configurations are not stable and attempt to go back to the original configuration as the knock-on atom energy is dissipated.

Clustering of the vacancies occurs within the core of the cascade and the extent of clustering varies with the host lattice. Based on size and density measurements of vacancy clusters, the fraction of vacancies in clusters is estimated to be less than 15%. The stability of vacancy clusters is low relative to the interstitial clusters.

Alpha-Fe (BCC): A set of divacancies on two adjacent {100} planes (that can transform to a dislocation loop of Burgers vector (100)) or a set of first nearest neighbor vacancies on a {110} plane (that can change to a perfect dislocation loop with (1/2)(111) Burgers vector).

image510

Figure 6.8 A schematic configuration of a crowdion.

image511

Figure 6.9 Dark field TEM images showing the defect structures of (a) gold at 3.5 x 1020 n/m2,

(b) silver at 2.1 x 1020 n/m2 and (c) copper at 1.5 x 1021 n/m2, irradiated as thin foils with 14 MeV fusion neutrons at room temperature Ref. [4].

Copper (FCC): The most stable configurations are the stacking fault tetrahedron (SFT) and faulted clusters on {111} planes that form Frank loops with Burgers vector (1/3)(111). The binding energy per defect in vacancy cluster is much less than that for interstitial clusters.

Figure 6.9(a), (b) and (c) show the defect structures (consisting of primarily SFTs) in fusion neutron-irradiated gold, silver and copper (all FCC), respectively. The majority of vacancy clusters in FCC metals and alloys have the shape of stacking fault tetrahedra and they appear as white triangles when viewed along the [110] direction with a proper weak beam dark field imaging condition. SFTs are created from Frank dislocation loops and subsequent dislocation reactions. An SFT consists of four triangular {111} planes as faces and 1/6<110> type stair-rod dislocations as six sides.

6.1.2

Susceptibility to Induced Radioactivity

The materials in the reactor can absorb fast/thermal neutrons and undergo reactions that may lead to the production of different radioactive isotopes of the constituent elements of the materials. These reactions can induce radio­activity as these isotopes would decay by emitting gamma rays, beta rays, and alpha rays of different energy levels. While selecting an alloy, we should be concerned about the following factors: (a) quantity of the impurities/alloying elements, (b) abundance of the isotopes and corresponding cross section, (c) half-life of the product nuclide, and (d) the nature of the radiation produced.

If the produced isotope has a short half-life and emit radiation of low energy, it should not be a cause for great concern. However, if the isotope is long-lived and produces radiation of high energy, all precautions must be taken. For

Подпись: Note The development of reduced activation steels comes from the consideration of the induced radioactivity. In the mid-1980s, the international fusion reactor program initiated the development of these steels first in Europe and Japan and later in the United States. The rationale behind developing these materials stems from the easier hands-on maintenance and improved safety of operation requirements that the materials used to build the fusion reactor would not activate when irradiated by neutrons or even if it gets activated may develop only low level of activation and would decay fast. Thus, the program to produce reduced activation steels that would require only shallow burial as opposed to putting them in deep geologic repository was pursued. Researchers found out that replacing or minimizing the amount of molybdenum, niobium, nickel, copper, and nitrogen in the alloy steels would help in developing reduced activation steels. Tungsten, vanadium, and/or tantalum (low activating) have been added to these steels. Table 1.6 shows the nominal compositions of three reduced activation steels. Although the approach has originated in the fusion reactor program, it can be equally applicable to fission reactor systems.

example, the main isotope of iron (Fe56) that accounts for almost 92% of the natural iron forms a stable isotope (Fe57) upon absorbing neutrons. The absorption of neutrons in Fe54 and Fe58 yielding Fe55 (half-life: 2.9 years) and Fe59 (half-life: 47 days) results in activation. However, the impurities or alloy­ing elements cause more induced radioactivity than iron itself. Generally, the test samples irradiated in a reactor are not examined immediately after taking out from the reactor because they remain literally hot and continue to be hot due to the decay heat produced by various reactions even if the fission chain reaction no longer occurs. The Fukushima Daiichi accident in Japan did show the severity of the heat produced due to these decay reactions even after the emergency shutdown of the reactor, leading to very high temperatures (in the absence of proper coolant) and eventually resulting in the cladding breach and perhaps some form of core melting.

Table 1.6 Nominal compositions of reduced activation steels (in wt%, balance Fe).

Steel

Region

C

Si

Mn

Cr

W

V

Ta

N B

JLF-1

Japan

0.1

0.08

0.45

9.0

2.0

0.2

0.07

0.05 —

Eurofer

Europe

0.11

0.05

0.5

8.5

1.0

0.25

0.08

0.03 0.005

9Cr-2WVTa

USA

0.10

0.30

0.40

9.0

2.0

0.25

0.07

— —

Other Boundaries

When the crystal structures of the two contiguous phases are similar and the lattice parameters are nearly equal, the boundary between the two crystals is called a coher­ent interface with one-to-one correspondence of atoms at the interface, as illustrated in Figure 2.40a. The surface energy of the coherent interface is 0.01-0.05 J m~2. When the phase partially loses the coherence with the matrix, the interface is called a semicoherent interface (Figure 2.40b). When there is no such similarity or no matching exists, the interface is called incoherent, which is quite similar in structure and energy to high-angle grain boundaries. An incoherent interface is shown in Figure 2.40c.

2.2.4

Volume Defects

Volume defects are three-dimensional in nature, and include precipitates, disper — soids, inclusions, voids, bubbles, and pores that can occur in materials under dif­ferent environmental conditions or processing conditions. They do have various important effects on the properties of materials. A TEM picture is shown in Figure 2.41 showing various types of precipitates present in a 2024 Al alloy. We will discuss these vital roles in Chapters 4 and 7 in detail.

2.2.5

Intersection of Dislocations

A direct result of dislocation intersection is a contribution of strain hardening, that is, the increase in flow stress with increasing strain. In reality, dislocations need to intersect forest dislocations (dense existing dislocation networks) in order for the plastic deformation to continue. These dislocation intersection phenomena could

image261

Figure4.19 Intersection oftwo edge dislocations (AB and CD) with Burgers vectors perpendicular to each other. (a) An edge dislocation AB is moving toward an edge dislocation line CD. (b) A jogJJ’ is produced on dislocation line CD.

be quite complex. Here, we discuss only the cases of intersection between straight edge and/or screw dislocations. The intersection of dislocations may create two types of sharp breaks, only a few atoms wide — (a) Jog. A jog is a sharp break on the dislocations moving it out of the slip plane. (b) Kink. It is a sharp break in the dislo­cation line but lies in the same slip plane.

To understand the dislocations intersection event better, let us consider first the case of two edge dislocations with their Burgers vectors perpendicular to each other, as shown in Figure 4.19a. An edge dislocation AB (with Burgers vector b1) is gliding on the slip plane PAB. The edge dislocation CD (Burgers vector b2) lies on its slip plane PCD. The dislocation AB cuts through the dislocation CD and creates a sharp break JJ) on dislocation CD. The sharp break produced on dislocation CD is called a jog, as depicted in Figure 4.19b. Hence, the jog has a Burgers vector of b2, but with a length (or height) of b1. Hence, the strain energy of the jog (Ej) would be aGb^. If b1« b2 = b, we can write Ej = aGb2b1 = aGb3. However, the jog is a very small dislocation seg­ment, no long-range elastic strain energy is possible. The jog energy mostly consists of the core energy. So, instead of a = 1, a with a value of 0.1-0.2 is more appropriate.

Now another example of two orthogonal edge dislocations with parallel Burgers vectors is discussed (Figure 4.20a). Dislocation XY is moving toward dislocation WV

image262

Figure4.20 Intersection oftwo edge dislocations (XYand WV) with parallel Burgers vectors.

(a) Dislocation XY is moving toward dislocation WV. (b) A jog SS’ is produced on dislocation XY and QQ’ on dislocation WV. These jogs are actually kinks.

In this case, jogs SS’ and QQ are created on dislocations XY and WV, respectively, as illustrated in Figure 4.20b. But these are not real jogs as they are on the same slip plane as the dislocation lines. They are called kinks as we have already defined. These kinks are generally not stable and disappear as the dislocations glide.

The intersection of a screw dislocation with an edge dislocation creates a jog with an edge orientation on the edge dislocation and a kink with an edge orientation on the screw dislocation (right-handed). Intersection between two like screw disloca­tions would produce jogs of edge orientation on both the dislocations involved. Here, all jogs have height in the order of atomic spacing. These are called elemen­tary or unit jogs. There are many cases where jogs with height of more than one atom spacing have been found. They are called superjogs. However, the discussion on superjogs is outside the scope of this book.

Подпись: „O Figure 4.21 Movement of a jog on a screw dislocation [1].

Jogs produced by the intersection of edge dislocations are of edge orientations and they lie on the original slip planes of the dislocations. They can glide with the edge dislocations on the stepped surface instead of a single slip plane. Hence, jogs found on such edge dislocations do not hinder their motion. However, jogs pro­duced by the intersection of screw dislocations are of edge orientation. As we know, edge dislocations can only glide in the plane that contains both the disloca­tion line and its Burgers vector. The screw dislocation can slip (i. e., move conserva­tively) with its jog if it glides on the same plane. However, if the screw dislocation tries to move to a different plane (MNN’O) as illustrated in Figure 4.21, it can take its jog with it only via nonconservative motion such as dislocation climb. As the dislocation climb process requires higher temperatures (i. e., thermally activated), the movement of jogged screw dislocations becomes temperature dependent. That is why at lower temperatures, the movement of screw dislocations is sluggish com­pared to edge dislocations as their motion is impeded by the existing jogs. At high stress regimes, the movement of jogs would leave behind a trail of vacancies or interstitials based on the dislocation sign and the direction of its movement. If the jog leaves behind vacancies, it is called vacancy jog, and if the jog goes in the oppo­site direction producing a trail of interstitials, it is called interstitial jog.

4.2.7

Test Procedure

image381

The following discussion on the fracture test procedure is partly adapted from the book by Dieter [2]. Different types of specimens are used for measuring KIc (Figure 5.24). The compact test specimen (CT) is quite popular. After the notch is machined into the specimen, the sharpest possible crack is produced at the notch root by fatiguing specimen in low cycle, high-strain mode (typically 1000 cycles). Plane strain fracture toughness is quite unusual in that there is no advance assur­ance that a valid KIc can be measured in a particular test. Equation (5.33) is used with an estimate of the expected KIc to determine the specimen thickness required for plane strain loading condition.

>r

P

Figure5.24 Three specimen designs (CT, three-bend, and notched round specimen) for Kc measurement. From Ref. [2].

Подпись: A Figure 5.25 Load-displacement curves (type-II and type-III show “pop-in” behavior). From Ref. [2].

The test should be carried out in a testing machine that provides for a con­tinuous autographic record of load (P) and relative displacement across the open end of the notch (proportional to crack displacement). The three types (type-1, type-II, and type-III) of load-crack displacement curves that can be obtained depending on the type of material are shown in Figure 5.25. The ASTM procedure requires to first draw a secant line OPs from the origin with a slope that is 5% less than tangent OA. This determines the Ps value. Now draw a horizontal line at a load equal to 80% of Ps and measure the distance along this line from the tangent OA to the actual curve. If the value of x1 exceeds one-fourth of the corresponding distance xs at Ps, the material is too ductile to obtain a valid KIc. If the material is not too ductile, the load Ps is then designated as Pq and used in the calculation.

image383 Подпись: 29.6 image385 Подпись: 185.5 image387 image388

The value of PQ determined from the load-displacement curve is used to calcu­late a conditional value of fracture toughness denoted by Kq using the following equation (for CTspecimen):

The crack length (a) used in this equation is measured after fracture. Then, calculate the factor 2.5(Kq/ct0)2. If this quantity is less than both the thickness and the crack length of the specimen, then Kq is actually KIC. Otherwise, it is necessary to use a thicker specimen to determine KIc. The measured value of Kq can be used to estimate the new specimen thickness through Eq. (5.33). Table 5.2 shows fracture toughness and tensile strength values of three engineering alloys. It demonstrates that maraging steel that has the highest plain strain fracture toughness would be able to tolerate crack for a given applied stress or with an existing crack of known length and will withstand the highest stress.

Table 5.2 Fracture toughness and tensile strength of two steels and an aluminum alloy.

Property

Ni-Cr-Mo steel

Maraging steel

7075 Al alloy

KIc (MPa m1/2)

46

90

32

Tensile strength (MPa)

1820

1850

560

Example Problem

A tensile stress of 300 MPa acts on a sheet sample of an alloy (FCC: E = 200 GPa, oy = 350 MPa, KIC = 40 MPa m1/2) with a central crack of length 5 mm. Is the sam­ple going to fail in a brittle manner?

Solution

We find KI for the case and compare it with KIC.

KI = oppa = 300/p x 2.5 x 10~3 = 26.59 MPam1/2 taking Y = 1 and this is less than KIC = 40 MPa m1/2, meaning the sample does not fail in a brittle manner.

5.1.6

Types of Nuclear Energy

Nuclear energy can be derived from many forms such as nuclear fission energy, fusion energy, and radioisotopic energy.

1.2.1

Nuclear Fission Energy

The essence of nuclear fission energy is that the heat produced by the splitting of heavy radioactive atoms (nuclear fission) during the chain reaction is used to gener­ate steam (or other process fluid) that helps rotate the steam turbine generator, thus producing electricity. Nuclear fission energy is the most common mode of produc­ing the bulk of the nuclear energy.

1.2.2

Nuclear Fusion Energy

A huge amount of energy (much higher than fission) can be produced using the nuclear fusion reaction (deuterium-tritium reaction). There is currently no com­mercial fusion reactors and is not envisioned to be set up for many years. A proto­type fusion reactor known as ITER (International Thermonuclear Experimental Reactor) is being built in France and scheduled to produce the first plasma by 2018.

1.2.3

Radioisotopic Energy

Either radioactive isotopes (e. g., 238Pu, 210Po) or radioactive fission products (e. g.,

85 Kr, 90 Sr) can produce decay heat that can be utilized to produce electric power. These types of power sources are mainly used in remote space applications.

1.3

Polymorphism

Many metals, nonmetals, and minerals exhibit an interesting feature called poly­morphism (poly means “many” and morphism means “structure”). Polymorphism (sometimes called allotropism) is known as the ability of a material to be present in more than one type of crystal structure as determined by temperature and/or pres­sure. Many phase transformations are based on this unique feature of the materi­als. As a general trend, more close-packed crystal structures are favored at lower temperatures, whereas open structure such a BCC crystal structure is most favored at higher temperatures. So, many metals show polymorphic transformation from the HCP to BCC crystal structure as the temperature increases and these phases are commonly referred to by Greek alphabets (a, b, y, etc.). For example, zirconium (Zr) assumes a HCP structure (a-Zr) at <865 °C, but becomes BCC (b-Zr) above that temperature. Similarly, hafnium (Hf) exhibits HCP structure below 1950 °C, but assumes BCC crystal structure above 1950 °C until its melting point (2233 °C). The example of iron (Fe) is interesting and bit of an exception. a-Fe (BCC) trans­forms to y-Fe (FCC) at about 912 °C, and then back to a BCC allotrope known as 6-Fe (with slightly larger lattice constant than that of a-Fe) above about 1394 °C. Thus, with the example of a-Fe and 6-Fe, polymorphism may not necessarily mean the presence of entirely different crystal structures, but the definition needs to be more related to the difference in crystal parameters such as lattice constants. There is also a HCP crystal structure (є-Fe) that is formed only under very high pressures. Polymorphic transformations of iron as a function of temperature make the heat treatment of steels possible leading to multitude of resulting microstructures and properties. In general, no FCC or BCC metal transforms to a HCP phase without an exception. Nuclear fuels like uranium and plutonium show multiple crystal structures at different temperature regimes as a result of polymorphism, and will be mentioned in more detail in Chapter 7.

54 I 2 Fundamental Nature of Materials

2.1.5