houses are made of stones, so is science made of is not a house and a collection of facts is not neces-

—Henri Poincare

The dislocation concept has already been introduced in Section 2.2 dealing with crystal defects. Now we need to develop the concept further. The importance of dislocations in plastic deformation (i. e., permanent deformation) is well docu­mented. But the question arises as to why we should be concerned about them in a textbook on nuclear materials. We will see in a later chapter how dislocation loops can form from the primary radiation damage; the dislocation loops can either stay as loops or join the overall dislocation networks in the irradiated materials. Indeed, dislocations are the major microscopic defects that are created during irradiation. Hence, this chapter serves as a prelude to understanding these different aspects of dislocations and their significance.

4.1

With the increase in size (10 nm or so depending on the particular alloy system) in coherent precipitate, the coherency is eventually lost. This happens during

overaging of precipitation hardenable alloys (such as Al-4.5 wt% Cu). The disloca­tion line is repelled due to the incoherency of the particles. Orowan (1947) pro­posed that the stress (At) required to bow a dislocation line between two particles separated by a distance 1 is

where G is the shear modulus of the matrix material and b is the Burgers vector of dislocation.

Every dislocation gliding over the slip plane adds one dislocation loop around the particle (Figure 4.36) and these loops exert back stress on the dislocation sources that must be overcome to cause further slip. A real example of Orowan looping is shown in Figure 4.36b. Generally, Orowan bypassing leads to fine and wavy slip compared to the coarse and planar slip due to the particle cutting mechanism.

Cu alloy (GP zones are the primitive particles formed during aging followed by transitional precipitates в" and в’ and finally equilibrium precipitate в.

If we now take the appropriate models of the particle shearing and bypassing and try to see how the strength increments due to these mechanisms contribute to the overall particle strengthening, an interesting observation can be made. Figure 4.37a shows such a prediction. Particle cutting (underaged material) increases strength­ening as the precipitate size increases, whereas Orowan bypassing increases with decreasing particle size (in the overaging regime). The crossover (the peak aging regime) gives the highest strength. Figure 4.37b shows why the aging experiments on aluminum alloys lead to such yield strength profiles as a function of aging time. With the aging time, the precipitate size increases and it follows first the particle cutting mechanism when the particles remain coherent, but when particles lose their coherency, the Orowan bypassing mechanism becomes important. Thus, the trend can be explained.

Stress cells do not involve compositional differences, but involve regions with stress gradients such as dislocations, grain boundaries, and highly stressed areas that become prone to corrosion. One example is the chemical etching of as — polished surface of a metallographic sample. The grain boundaries being the stressed regions act as anodes and get attacked by the etchant preferentially.

Another example would be a cold bent rod when submerged in water gets attacked more at the bent region, which acts an anode because of the high residual stress present in it. There are corrosion mechanisms (stress corrosion cracking (SCC) and corrosion fatigue) that are aided by the presence of stress.

1.9.1.1 General Mechanical Properties

Important general mechanical properties include tensile strength, ductility, and toughness. The material should be strong enough to bear the loads of the structure and also sustain any internal or external stresses generated during service. Also, the material should have enough ductility (a measure of percent­age elongation or reduction in area in standard tensile specimens) to avoid any catastrophic failure. Usually, as a rule of thumb, a percentage elongation of 5% is considered a minimum requirement for a load-bearing engineering struc­ture. But one must admit that this often changes with the type of application at hand. In some cases, the materials should have sufficient ductility in order to be formed into different components. Toughness is defined as the ability of a material to absorb energy without failure, and that dictates how tough a

material is for use. Generally, tensile strength and ductility combined is referred to as toughness. However, generally impact tests and fracture tough­ness tests are conducted to evaluate toughness properties of materials. All these affect the mechanical integrity of the reactor components.

The example of a line defect is dislocation, and it has a very important role to play in plastic deformation of crystalline materials. In this section, we introduce the con­cept of dislocations; however, the majority of the dislocation theories will be

discussed in detail in Chapter 4. Dislocations are not equilibrium defects like point defects because the associated energy is higher than the increase in the enthalpy. They are generally created during solidification, cooling, and mechanical working, and sometimes just by handling. Hence, they are introduced in the crystal in a non­equilibrium way due to the action of mechanical stresses, thermal stresses, collapse of vacancies, or during the precipitate growth and different other events such as exposure to high-energy radiation.

Orowan, Taylor, and Polanyi first conceptualized crystal dislocations during 1930s without directly observing them. The direct evidence of the presence of dis­locations was obtained later (during 1950s) using X-ray topography and transmis­sion electron microscopy (TEM) techniques. Some other indirect techniques (such as etch pit method and decoration method) were also used, but none was viable in comparison to TEM (Figure 2.30).

Under normal conditions, plastic deformation occurs through the relative shear­ing of two crystal parts on a particular plane, called slip plane, along certain crystal­lographic direction (recall close-packed planes and directions in Section 2.1). It would have taken a lot of energy to create the deformation if the atoms needed to jump all at the same time. It has been estimated from the theories of rigid body shear and some other simple assumptions that the shear stress required to initiate plastic deformation (shear yield stress) in a crystal should be ~G/2p, where G is the shear modulus of the crystal. This leads to a huge yield strength number for real metals, which is physically never observed. That is why it led to the belief that the deformation remains localized in a narrow region and propagates through the

crystal in a wave-like fashion (Figure 2.31a). Thus, there are some defects that aid in the plastic deformation such that the stress required to initiate plastic deforma­tion becomes quite less. Let us try the case of aluminum. Aluminum has a shear modulus of ~27 GPa; thus, a shear yield stress of ~4.3 GPa. But we know that the shear yield stress of pure aluminum is in the range of 3-10 MPa. There is a differ­ence of several orders of magnitude between the predicted and observed values. This dichotomy can be solved by the inclusion of dislocations in the conversation. Dislocations can be defined as the line defect (AB) that demarcates the unslipped (D) and slipped (C) regions of a crystal (Figure 2.31b).

The dislocation line perpendicular to the slip direction is called edge or Orowan- Taylor dislocation (Figure 2.32a), and that parallel to the slip direction is called screw or Burgers dislocation (Figure 2.32b). But most dislocations remain in a mixed con­figuration as the dislocation line is typically curved. An important characteristic of dislocation is Burgers vector (b) that represents the unit slip distance and is always along the slip direction. An edge dislocation is illustrated by inserting an extra half­plane of atoms, thus creating a large disturbance in the atomic configuration in a region just below the extra half-plane. If the half-plane is above the slip plane, the dislocation is called a positive edge dislocation (represented by?), and if it is situated below the slip plane, it is called a negative edge dislocation (represented by T). On the other hand, the situation of screw dislocation is little different. The screw disloca­tion moves in a single surface helicoid, much like a spiral staircase. If we look

down on the dislocation and the helix appears to advance in a clockwise circuit, the dislocation is called a right-handed screw dislocation (or positive screw), and if it is anti­clockwise, it is called a left-handed screw dislocation (or negative screw).

The Burgers vector can be found out by constructing a Burgers circuit around the dislocation. Burgers circuit is any atom-to-atom path taken in a crystal that forms a closed path, while the circuit passes through the good part of the crystal. In the presence of a dislocation, the vector needed to close the circuit is the Burgers vec­tor, as illustrated for a schematic edge dislocation configuration in Figure 2.33a. Construction of Burgers circuit for a screw dislocation is shown in Figure 2.33b. Dislocations can end at the crystal surface, internal interfaces (grain boundaries), and so on, but never within the grain unless it forms a node (the sum of the dislo­cation Burgers vectors is zero at the node, Sb = 0) or a closed loop.

A pure edge dislocation can glide or slip perpendicular to its line vector (t) on the slip plane. As the edge dislocation line is normal to its Burgers vector, it remains confined to a specific plane along with its Burgers vector, and hence its glide is limited to a specific plane that contains both the Burgers and line vectors. But it can move vertically leaving the slip plane via a process known as climb. The climb process requires addition or subtraction of atoms from the end edge

mechanism. Since the climb process requires diffusion, the process is likely only at a higher homologous temperature, and it is generally slower than the glide. Because the climb process requires new atoms coming into or going out of the region, the local mass is not conserved, and that is why it is called nonconservative movement (unlike glide that is called conservative movement). If the dislocation moves vertically upward, the process is called positive climb (Figure 2.34a); if the dislocation moves vertically down, it is called negative climb. On the other hand, if a screw dislocation having the Burgers vector is parallel to its line vector, it does not have a preferred slip plane and thus the glide of screw dislocation is much less restricted. It is worth noting that screw dislocation cannot climb, but it can leave its slip plane by a process known as cross-slip or cross-glide (Figure 2.34b). The seg­ments of z and x are the screw components of the dislocation loop shown in Figure 2.34b, and thus are able to cross-slip. Table 2.4 summarizes some interest­ing features of the edge and screw dislocations in a comparative way. More discus­sion on this topic will be initiated in Chapter 4.

An important aspect of dislocation microstructure is the dislocation density (gd) so that the number of dislocations in a given volume can be quantified to be used in a variety of relations describing mechanical behavior of crystalline materials. Thus,

dislocation density is primarily defined as “the total line length of dislocations per unit volume.” Hence, the unit of dislocation density would be cm cm~3, that is, cm~2. Based on the derived unit, there is another way to define dislocation density: the number of dislocation lines that intersect a unit area. Carefully prepared crystal tends to have a low dislocation density, ~102cm~2. Some single crystal whiskers can be made nearly free of any dislocation. On the other hand, heavily deformed metals (cold worked) may contain dislocation density in the range of 1010­1012 cm~2 or more, whereas an annealed crystal may contain 106-108 cm~2.

2.2.3

The stress field of an edge dislocation is more complex than that of a screw disloca­tion. However, in a similar way, a Volterra-type edge dislocation can be created as done for screw dislocation. In Figure 4.13b, such a case is shown. But in this case, the cut has been made perpendicular to the z-axis and a displacement ofb has been made. Thus, the magnitude of the simulated edge dislocation is b that is perpendic­ular to the dislocation line, and hence an edge dislocation. Here also the solution

breaks down when x and y approach zero. So, even in the edge dislocation, the core region does not follow linear elastic solutions. Without getting into the details of derivation, the stress field of an edge dislocation is shown below. It contains both dilatational (oxx, oyy, and ozz) and shear components (txy and tyx).

where C = Gb/(2p(1 — n)).

Here, n is the Poisson’s ratio. At y = 0 where the slip plane lies, all but the shear stress components given in Eq. (4.11c) are zero and the maximum com­pressive stress acts just above the slip plane with maximum tensile stress acting immediately below the slip plane. The stress field of mixed dislocations could also be found out.

4.2.3

This section gives a general discussion of fracture. Fracture is the separation or fragmentation of a solid body into two or more parts under the action of force/ stress. The process of fracture consists of two components: crack initiation and crack propagation. Fractures can be classified into two broad categories: ductile and brittle. A ductile fracture is characterized by appreciable plastic deformation with stable crack growth. A brittle fracture is characterized by a rapid rate of crack propa­gation (unstable) with no gross deformation. Stable crack growth implies that once the load is taken off, the crack does not propagate further. Figure 5.13 shows various fracture types observed in metals/alloys subjected to uniaxial tension.

Fractures are classified with respect to different factors, such as strain to fracture, crystallographic mode of fracture, and the appearance of fracture. A shear fracture occurs as the result of extensive slip on the active slip plane. This type of fracture is

promoted by shear stresses. The cleavage mode of fracture is promoted by tensile stresses acting normal to a crystallographic cleavage plane. In many cases, the frac­ture surfaces are a mixture of fibrous and granular fracture, and it is customary to report the percentage of the surface area. Fractures in polycrystalline samples are transgranular (the crack propagates through the grains) or intergranular (the crack propagates along the grain boundaries).

5.1.3.1 Theoretical Cohesive Strength

Engineering materials typically exhibit fracture stresses that are 10-100 times lower than the theoretical value. This observation leads to the conclusion that flaws or cracks are responsible for the lower-than-ideal fracture strength. Due to Inglis [4], an approach assumes that the theoretical cohesive stress can be reached locally at the tip of a crack, while the average stress is at much lower value. Then, the nominal fracture stress is given by the expression (for the sharpest possible crack):

. <5-28>

where E is the elastic modulus, cs is the surface energy, and c is the crack length.

Microcracks act as precursors for crack propagation in brittle fracture. The pro­cess of cleavage fracture involves three steps: (a) plastic deformation to create dislo­cation pileups, (b) crack initiation, and (c) crack propagation. The initiation of microcracks can be affected by the presence of second-phase particles. Cleavage cracks can also be initiated at mechanical twins.

The ductile fracture starts with the initiation of voids, most commonly at second — phase particles. The particle geometry, size, and bonding play an important role. Dimpled rupture surface (ductile fracture) consists of cup-like depressions that may be equiaxial, parabolic, or elliptical, depending on the precise stress state. Microvoids are generally nucleated at second-phase particles, and the voids grow and eventually the ligaments between the microvoids fracture.

Griffith [5], established the following criterion for crack propagation: “A crack will propagate when the decrease in elastic strain energy is at least equal to the energy required to create new crack surface.” The stress required to propagate a crack in a brittle material is a function of crack length (as shown in Figure 5.14) and is given by

Figure 5.14 The crack configuration used in Griffith’s equation.

Sohncke suggested that fracture occurs when the resolved normal stress (ac) reaches a critical value (Figure 5.15). The critical normal stress for brittle fracture is

ac — P cos ф/(A/cos ф) — (P/A) cos2 ф. (5.31)

The presence of voids in irradiated materials can be primarily measured from transmission electron microcopy studies. Only this kind of examination can pro­vide information on the void size and density. The extent of swelling is generally measured by Archimedes principle by immersing the irradiated material in a fluid of known physical density and compared with the original volume (V), that is, it is measured in terms of the ratio of swelling-induced volume change (Д V/V) to the

original volume. For narrow void distribution, the term is given by the following relation:

DV 4 _3 , N

— = 3 pR 3Nv, (6.2)

where R is the average void radius and Nv is the number of voids per cm3.

A majority of elements (about three-fourths) in the Periodic Table are metals, and of all the metals, more than two-thirds of them possess relatively simple cubic or hexagonal crystal structures. Metal atoms are bonded by a chemical bond known as the metallic bond. The metallic bonds are nondirectional and there is no constraint regarding bond angles (which is not true for covalently bonded materials). Even though a simple cubic system is relatively unknown in metals (or even among ele­ments), a-polonium exhibits a simple cubic crystal structure. The most common crystal structures in metals are discussed in detail in the following sections.

Here, a number of diffusion mechanisms from an atomistic viewpoint are dis­cussed. The simplest picture of diffusion can be visualized through diffusion of interstitial atoms through a lattice. In this mechanism, interstitial atoms move from one interstitial site to another. In the dilute interstitial solid solutions, the probability of finding an interstitial site is very high and close to unity. Note that Figure 2.45a shows a two-dimensional schematic illustration of a monatomic crys­tal with a very few number of impurity atoms that are of much smaller size than the interstitial sites themselves. So the impurity atom can squeeze past through the host lattice atoms to fall into another interstitial site. While it tries to go past the host atoms, repulsive forces would act on the impurity atom and so energy would be needed to surmount the barrier. That energy is supplied by the thermal energy of the interstitial atom. Figure 2.45b shows such an activated state. Following this state, the interstitial atom falls into a new interstitial position completing one jump (Figure 2.45c). If we calculate the energy of an atom as a function of position, we would see that energy is minimum when the impurity atom is at normal position (as shown in Figure 2.45a and b) and the maximum is at the midway between the two positions (i. e., at the activated state). The situation is shown in Figure 2.45d. The amount of this energy barrier is given by the difference between the energy at the activated state and that at the normal state, and is referred to as the activation energy for interstitial diffusion. The real event may consist of a series of such unit atomic jumps. Examples of such interstitial mechanism may comprise diffusion of C inside any allotropic form of iron (alpha iron, gamma iron, or delta iron) or hydrogen diffusion in zirconium, and so on.

Now, let us consider the atomic mechanism by which self-diffusion may occur. In self-diffusion, like atoms exchange lattice positions leaving the lattice identical before and after diffusion. One of the simplest modes of this is the direct exchange mechanism. In this mode, atom X can move to the site of lattice atom Y and at the same time, atom Y moves to the site of atom X (Figure 2.46). But such a direct exchange of atoms is not at all energetically favorable. This may seem implausible even in a very open structure as there are other mechanisms that can actually achieve the same result without expending that much energy. Ring mechanism is one such example. A four-ring mechanism is also depicted in Figure 2.46 (right). As is evident, a greater coordination between atoms is essential for this mechanism to have any consequence. Some evidences suggest that self-diffusion in chromium and sodium may occur by ring mechanism. However, most metals and other engi­neering materials are, in general, too close-packed for that mechanism to occur. That is to say that self-diffusion activation energy associated with direct exchange

and four-ring mechanisms will be always higher than what has commonly been observed such as through vacancy mechanism.

The predominant way by which the self-diffusion or self-substitutional atoms can diffuse is by changing its position with a neighboring vacant site. Figure 2.47a-c shows the different steps involved in substitutional atom diffusion. The same applies to self-diffusion, but for clarity ofthe process, it is shown in terms of substi­tutional atoms. Repetition of this process can result in the transfer of matter over

past two host lattice atoms to the neighboring interstitial site. (c) Configuration of the interstitial atom after diffusion. (d) Energyofan impurity atom as a function of position (E* is the activation barrier).

large atomic distances. This is known as the “vacancy mechanism of diffusion.” There is always an equilibrium number of vacant lattice sites (thermal vacancies) present at any particular temperature and their concentration increases with increasing temperature, as seen in Section 2.2 (Eq. (2.7)). Likewise, self-diffusion also takes place through this mechanism. It is also instructive to note that atom jump can occur into divacancies. However, larger vacancy agglomerates like triva­cancy and quadrivacancy are relatively immobile, and do not take part in general diffusion.

In the interstitialcy mechanism, an atom from a regular lattice site jumps into a neighboring interstitial site that is too small to accommodate it fully. As a result, it displaces another atom from a regular lattice site. Hence, both the atoms share a common site, although displaced from their original lattice sites.

2.3.4