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. Figure 5.16a shows one example of microcrack formation in iron after tensile deformation at —140 ° C. The appearance of clevage fracture surface in a low carbon steel is shown in Figure 5.16(b). 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) contains cup-like depressions (dimples) 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. The different stages of this process are shown in Figure 5.17(a) and an SEM fractograph with dimples shown in Figure 5.17(b).

Ductility (as expressed in true strain to fracture) of a material may depend on the volume fraction of second-phase particles present, as shown in Figure 5.18 with examples in steel.

5.1.4

A typical plot of void swelling as a function of temperature in an irradiated 304 austenitic stainless steel is shown in Figure 6.15. It is clear from this figure that the peak in swelling happens at the intermediate temperature. Lower defect mobility limits void growth at lower temperatures, whereas at higher tempera­tures the irradiation-produced defect concentration reaches the value close to thermal equilibrium value and loses the level of vacancy supersaturation (i. e., a lack of excess vacancy concentration) that is required to sustain void growth. These two opposing effects give the shape of the swelling-temperature curve as shown in Figure 6.15.

With increasing temperature, the void number density falls and the average size increases, which is a typical behavior for a process that is dominated by nucleation at lower temperatures (i. e., the void growth is slow) and by growth at higher

temperatures where the driving force for void growth is small. Figure 6.16 shows the void distribution function as a function of void diameter at different irradiation temperatures in a 316-type stainless steel irradiated to a fluence of 6 x 1022 ncm~2. The void size distribution function is narrow and average void size small at low temperatures. However, with increasing temperature, the distribution becomes wider and the average void size also increases. This occurs due to the enhanced mobility of point defects at elevated temperatures.

In the BCC unit cell, an atom is located at the body center of a cubic unit cell in addition to the eight atoms on the cube corners. The coordination number of this type of crystal structure is 8, that is, each atom is surrounded by eight equidistant neighboring atoms. Figure 2.5a shows a schematic of a BCC unit cell. The effective number of atoms in such a unit cell is (8 x 1/8) +1 = 2, because each corner atom is shared by eight unit cells and the body center atom is fully inside the unit cell to be counted as one. We note that each atom is surrounded by eight nearest neigh­bors and this factor is known as coordination number defined as the number of equivalent nearest neighbors. The packing efficiency or atomic packing factor is

defined as the volume of the space occupied by atoms in a unit cell. Below we pres­ent a detailed procedure for calculating the atomic packing factor of a BCC unit cell. Note that the same method can be applied to calculate the packing factors of virtu­ally any kind of known unit cells.

The packing efficiency PE of atoms in the BCC unit cell is derived as follows:

PE Volume of atoms in the unit cell 2 x (4/3)pr3 Volume of the unit cell a3

^2 x (4/3)p((/3/4)a)3 ^ 0 68 a3 ‘ ‘

Many metals such as a-Fe, b-Zr, Mo, Ta, Na, K, Cr, and so on have a BCC crystal structure. This structure is relatively loosely packed (only 68% of the crystal volume is occupied by atoms) compared to the best packing that can be achieved using hard incompressible solids spheres. Two corner atoms touch the body center atom and create the body diagonal that is closest-packed direction in the BCC unit cell. This also gives us an opportunity to calculate the relation between the atom radius (r) and the lattice constant (a) using the simple Pythagorean rule. The body diagonal length is given by (r + 2r + r) = 4r. The body diagonal makes the hypotenuse of the triangle that a base of a face diagonal (/2a) and a cube edge (a), as shown in Figure 2.5b.

Hence, (4r)2 = (p2a)2 + (a)2 or (4r)2 = 3a2 = (p3a)2.

Or, 4r = P3a. (2.1)

& Example 2.1

Calculate the theoretical density of a-Fe at room temperature.

Solution

The density of any crystal can be calculated from the first principles.

Mass of atoms in the unit cell Г Volume of the unit cell

First, we need to know the crystal structure (BCC) and lattice parameter (a = 0.287 nm) of a-Fe.

Mass of an Fe atom = 55.85 atomic mass unit (or amu) from the defini­tion of atomic weight.

1 amu = 1.66 x 10-27 kg.

Hence,

It is, in general, intuitive that diffusion will enhance with increasing temperature. Here, we attempt to derive diffusivity as a function of temperature from atomic theories as applicable to the vacancy mechanism of diffusion. For diffusion of atoms to occur, there should be availability of vacant neighboring sites around the atom. If Г (as defined before) is proportional to the number of nearest neighbors (b) and the probability of finding neighboring lattice sites vacant or vacancy concen­tration (Cv), the following mathematical relation can be written:

Г = bC, rn, (2.39)

where v is the atomic jump frequency. Hence, the number of successful jumps by an atom can be given by

v — nD exp (—AGm/RT), (2.40)

where nD is the lattice vibration frequency (same as the Debye frequency, typically -1013 s—1, which is defined as the theoretical maximum frequency of vibration that make up the diffusion medium crystal), DGm is the free energy maximum (in calo­ries per mole) along the diffusion path or activation barrier to vacancy migration (also known as free energy for migration), and T is the temperature (in K).

From Eq. (2.35), we know D — (1/6)12Г. By replacing the Г expression using Eqs. (2.39) and (2.40), we obtain

1 2

D — — l2bCv VDexp(—DGm/RT). (2.41)

6

We know from earlier chapters

Cv = exp(— AGf/RT), where AGf is the free energy of vacancy formation (see Section 2.2).

Using the general relation from the second law of thermodynamics, AG = AH — TAS, we can write

Rearranging Eq. (2.42), we get

The above equation can also be written as

D = Do exp (—Qsd/rt) ;

1 D 1 l2o (ASf + AS,

where Do — 61 bvD ex^l——- r—-

D0 is called the frequency factor and Qsd is called the activation energy for self­diffusion. D0 and QsD can be obtained by measuring D at different temperatures from experiments. A plot of ln (D) versus 1/T yields a straight line and the slope equals —QJR and the intercept on the y-axis is ln (D0). It is interesting to note that the final diffusivity term does not contain the defect concentration term, rather it has the activation enthalpy for vacancy formation. The derivation is equally applica­ble for substitutional vacancy diffusion and interstitial diffusion mechanisms. On assigning approximate values to the terms in Eq. (2.45) and considering a small positive value of the entropy term, D0 is generally found to be between 10-3 and 10 cm2 s-1.

Strain hardening is possible both in single crystals and polycrystalline materials. When metallic materials are cold worked, their strength increases. Generally, an annealed crystal contains a dislocation density of about 108 m~2. However, moder­ately cold worked materials may contain 1010-1012 m~2 and heavily cold worked materials 1014-1016 m~2. Plastic deformation carried out in a temperature regime and over a time interval such that the strain hardening is not relieved is called cold working. At the first sight, one might think that with increase in the dislocation density, the material would be more ductile. But that is not the case. As the disloca­tion density increases, the movement of dislocation becomes increasingly difficult due to the interfering effect of the stress fields of other dislocations. In polycrystalline metals/alloys, multiple slips occur more readily due to the mutual interference of adjacent grains, leading to significant strain hardening. In the early stages of plastic deformation, slip is generally limited on primary glide planes and the dislocations tend to form coplanar arrays. However, as the deformation proceeds, cross-slip takes place and dislocation multiplication processes start to operate. The cold worked structure then forms high dislocation density regions or tangles that soon develop into tangled networks. Thus, the characteristic structure of the cold worked state is a cellular substructure. The cell structure is schematically shown in Figure 4.31.

With increasing strain, the cell size decreases at initial deformation regime. However, the cell size tends to reach a fixed size implying that as the plastic defor­mation proceeds, the dislocations sweep across the cells and join the tangles into the cell walls. The exact nature of the cold worked structure depends on the mate­rial, strain, strain rate, and deformation temperature. The development of cell structure, however, is less pronounced for low temperature and high strain rate deformation and in materials with low stacking fault energy.

Strain hardening is an important process that is used in metals/alloys that do not respond to heat treatment easily. Generally, the rate of strain hardening is lower in HCP metals compared to FCC metals and decreases with increasing temperature.

Furthermore, the final strength of the cold worked solid solution alloy is always greater than that of the pure metal cold worked to the same extent. Cold working may slightly decrease density and electrical conductivity, whereas thermal expan­sion coefficient and chemical reactivity increase (i. e., corrosion resistance decreases). Figure 4.32 shows the variation of tensile parameters (strength parame­ters of tensile strength, yield strength, and ductility parameters of reduction in area and elongation to be discussed in the Chapter 5) as a function of percentage cold

reduction. Torsteels are used as reinforcing bars for concrete is cold twisted to increase the strength. However, it acts upon the effective ductility of material.

A high rate of strain hardening implies mutual obstruction of dislocations glid­ing on intersecting slip systems. This can arise from three sources: (a) through interaction of the stress fields of dislocations, (b) through interactions that produce sessile locks, and (c) through the interpenetration of one slip system by another (like cutting trees in a forest) that results in the formation of jogs. The basic equa­tion relating flow stress to structure is

Gb

t — a —-,

where ‘ is the obstacle-obstacle spacing with a having a value between 0.5 and 1. If dislocations are acting as obstacles like trees in dislocation forest, ‘ = 1Д/р, so strengthening due to dislocations is given by

t — t0 + aGbp1/2, (4.33)

where t is the stress needed to move a dislocation in a matrix of dislocation density q, t0 is the stress to move the dislocation in the same matrix with no dislocation density, G is the shear modulus, and b is the magnitude of Burgers vector. The constant a may change value, but may be considered as 0.5.

4.4.2

1) Shot peening introduces compressive residual stresses near the surface layers. The surface plastically deformed by repeated impingement of hard steel balls can improve fatigue life since the compressive residual stresses first need to be compensated before crack propagation can take place.

2) Carburizing and nitriding introduce compressive residual stresses at the surface while increasing also the strength/hardness. Decarburization lowers the fatigue strength.

3) A fine grain size improves fatigue life.

4) The crack initiation almost always happens from a site of stress concentration. In design, fillets of adequate radius of curvature should be provided at places where a sudden change of cross section exists. Shafts are usually highly polished to reduce the chance of surface irregularities.

5.1.8

Even though a nuclear reactor can be defined in different ways, almost all reactors except fusion reactor (commercially nonexistent) can be defined as follows: “A nuclear reactor is a device, designed to produce and sustain a long term, controlled fission chain reaction, and made with carefully selected and strategically placed collection of various materials.” The classification of reactors vary and are generally based on the following: type of fission reaction (thermal, epithermal, and fast reactors), purpose of the reactor (power reactors, research reactors, and test reactors), type of the coolant present (such as light/heavy water reactors, gas-cooled reactors, and liquid metal-cooled reactors), type of core construction (cubical, cylindrical, octagonal, and spheri­cal reactors), and so forth.

1.8.1

The CsCl structure contains cations and anions with same valencies similar to those of rock salt-type structure. However, the CsCl crystal structure is essentially different from the NaCl structure. The CsCl structure can be viewed as two interlac­ing, equal-sized simple cubic cation and anion sublattices. It can also be shown as a simple cubic unit cell of cations (Cs+) with an anion (CT) in the body center of the cube and vice versa. The lattice constant (a) remains the same regardless of the way to view the structure (Figure 2.21). Each ion in this structure is surrounded by eight ions of the opposite type (i. e., coordination number 8 for both cations and anions similar to that of BCC structure) as the first nearest neighbors. There is only one ion pair per unit cell of CsCl as per simple cubic structure. Whether a solid of MX type would assume a NaCl-type structure or CsCl-type structure depends largely on the cation/anion radii ratio. For CsCl, the radius ratio is about 0.92, whereas for NaCl it is 0.61. Structures with the radius ratio higher than 0.732 tend to have eight coordination and may assume CsCl-type structure. Other examples of this type of structure are CsI and CsBr. Note that CsI may be found in the spent fuels among other compounds as Cs and I are generated as fission products in the reactor fuels. However, during the reactor operation, it stays in gaseous form (not crystalline)

inside the fuel. In accident situations, CsI can escape the cladding containment and form solid phases (crystalline) if it comes into contact with the cooling water.

We already know that shear stress causes slip in crystals. The resolved shear com­ponent based on the magnitude of the external load, geometry of the crystal

structure, and orientation of the active slip systems needs to reach or exceed a criti­cal value in order for the slip to occur. This critical shear stress is called critical resolved shear stress. The CRSS is the single crystal equivalent of yield stress of a polycrystalline material as obtained in a standard stress-strain curve. The CRSS value for a slip system in a crystal depends on purity and temperature. The following derivation of resolved shear stress makes clear how a single crystal can undergo slip.

Erich Schmid [15] was the first to analyze the problem, one of his contributions to understanding crystal plastic deformation. Let us take a cylindrical single crystal under tensile force F with a cross-sectional area A, as illustrated in Figure 4.4. We need to find out the resolved component of stress.

The normal stress (o) is given by F/A. Now we can define the slip system with respect to the tensile axis and normal to the slip plane. Slip plane contains the slip direction along which the resolved shear stress (tR) would act. As shown in Figure 4.4, the included angle between the tensile axis (along F) and the slip plane normal (along N) is 0, and the angle between the tensile axis and the slip direction is ф. The slip plane is inclined at 0 to the tensile axis, and thus its area (A0) is given by A/cos 0. The normal force F can be resolved along the slip direction as F-cos ф. Hence,

Equation (4.1) is also known as Schmid’s law, while the term cos 0 cos ф is called Schmid factor. Schmid factor gives a measure of the slip system orientation. The resolved shear stress is the maximum when 0 = ф = 45° so that tR = o — (1Д/2) — (1Д/2) = (1/2)o. In the extreme cases, where either 0 or ф is 90°, there will be no resolved shear stress in the slip plane, and the material tend to fracture rather than

deform. A numerical example on this topic is worked out in Example 4.2. Note that even though the derivation of resolved shear stress has been made in a uniaxial tensile situation, it is equally applicable to plastic deformation under compression.

■ Example 4.2

Calculate the critical resolved shear stress for a metallic single crystal (BCC) that starts deforming in the slip system (110)[111] when the tensile stress along [010] is 100 MPa.

Solution

To solve this problem, let us first refer to Eq. (4.1).

Given the tensile stress (o) = 100 MPa; we need to calculate cos 0 and cos ф. Refer to Figure 4.4 to locate the slip plane, slip direction, and tensile axis with their Miller indices. Note that the orientation of slip plane and slip direction can be taken as arbitrary with respect to the tensile axis. It is just a representation of the problem statement so that the steps below can be followed easily.

Before we start calculation of the cosines, let us turn our attention to Section 2.1. We know that in a cubic system the normal direction to a crys­tallographic plane bears the same digits of the Miller indices. So, the normal direction to the slip plane, (110), is [110].

Following Eq. (2.4), the cosine of the angle between the tensile axis [010] and slip plane normal [110] is given by the dot product of the unit vectors along these two directions:

u1u2 + v1v2 + w1w2
sj u2 + vj + w2^/ u2 + v2 + w2

(0)(-1) + (1) ( 1 ) + (0) (0)
p02 + 12 + 0^/ (-1)2 + 12 + 02

Furthermore, the cosine of the angle between the tensile axis [010] and the slip direction [111] is

cos ф = 1 .

V02 + 12 + 0V12 + 12 + 12

Hence, using Eq. (4.1), we obtain the resolved shear stress: tR = (100MPa)(1/P2)(1/p3) =~ 40.8MPa.

The problem statement states that the single crystal starts deforming at the tensile stress of 100 MPa on the given slip system. So, the resolved shear stress must reach the corresponding critical value to accomplish that. Hence, it is clear that the resolved shear stress of 40.8 MPa calculated above is indeed the CRSS.

CRSS value in a single crystal is determined by the interactions between various dislocations and their interactions with other types of defects present. However, this value is quite smaller than the theoretical shear strength of the crystal. Table 4.1 lists comparative values of the CRSS for four different metals. The theoretical shear strength values for respective metals are calculated from G/10 (where G is the shear modulus of the metal), and CRSS values are all experimentally determined [1].

4.1.2

“Inventing is a combination of brains and materials. The more brains you use, the less material you need.”

—Charles F. Kettering

In the previous chapters, we did refer to materials properties without discussing them in details. It is difficult to cover all the aspects of materials properties in a single chapter. Hence, the readers are encouraged to refer to texts enlisted in Bibli­ography. Here, three broad categories of properties — mechanical, physical, and chemical are identified. From the viewpoint of nuclear reactor applications, all these properties are important. Hence, in this chapter, we will learn some basics of these properties, the test techniques used for their evaluation, and relevant test data analysis techniques.

5.1