We are already familiar with the concept of electrolytic cell (recall electrolysis of water into hydrogen and hydroxyl ions). Corrosion occurs in a reverse way. In an electrolytic cell, a voltage is applied between the anode and the cathode to dissociate the electrolyte. In order for corrosion to occur, an electrochemical cell needs to be set up and corrosion current is produced between the electrodes (anode and cath­ode). A schematic electrochemical cell (also known as galvanic cell) is shown in Figure 5.60, showing various cell components. The components of the cell are anode, cathode, electrolyte, and an external conductive circuit (or path) between the anode and the cathode. At anode, metal M loses electron(s) via the half-cell reaction, M ! M+n + ne~, where n is considered the valence of M. However, for reaction to proceed, both the electrons and the ions need to be removed. The

electrons moving through the conductive path go to the cathode and get accepted by it following the half-cell reaction of N+m + me~ ! N (m is the valence of N) or by other metal ions. The metal ions thus created either remain dissolved in the electrolyte or react together to form insoluble surface deposit.

Basically, an oxidation reaction (i. e., loss of electrons) occurs at the anode and a reduction reaction (gain in electrons) occurs at the cathode with the acceptance of electrons. All metals are from their origin at a higher energy level and they are subject to oxidation reaction. In real world, one does not need to have the exact form of an electrochemical cell, as shown in Figure 5.60, in order for corrosion to occur. Anode and cathode can be set up on the same metal piece depending on different conditions. The metal piece itself can provide the conductive path. The chemical environment in which the metal piece exists will serve as the electrolyte. A schematic example of such a situation is shown in Figure 5.61. Generally,

N

corrosion can be stopped if the electrical connection is interrupted or anode reac­tants are depleted, or cathode products are saturated.

Depending on the relative tendency of metals to lose electrons, one can measure the electrode potential of such a half-cell reaction. The electrode potential of any specific half-cell reaction is generally measured with respect to hydrogen reaction (H2 ! 2H+ + 2e—) that is regarded as a reference and considered as 0 V under stan­dard conditions (1 molar solution and 25 °C temperature). Iron is more active (or anodic) than hydrogen and its electrode potential (for the reaction, Fe! Fe2+ + 2e—) is —0.44 V under standard conditions. Based on the tendency of these reactions, electrochemical series as shown in Table 5.4 is developed. Note that gold is on the top of the chart, that is, at the noble end, and lithium is at the bottom of the chart, that is, at the active end. From the chart, one can predict which metal will corrode (i. e., act as anode) preferentially and which will be protected (act as cath­ode) when they are electrically connected.

As already stated, the electrochemical series given in Table 5.4 has been created based on standard conditions, that is, 1 molar solution at 25 °C (298 K). This chart is used by both electrochemists and corrosion engineers alike. There is another varia­tion of this chart where the sign of electrode potential is reversed due to convention. However, the latter form of chart is mostly used by physical chemists and thermody — namists. In this chapter, we base our discussion on the former form of the electro­chemical series. To calculate the actual electrode potential E (i. e., under nonstandard conditions), Nernst equation is used. The relevant expression is given as

E = E298 + (^)(ln X),

252 I 5 Properties of Materials

Table 5.5 A condensed version of a galvanic series applicable in seawater.

Galvanic series in seawater

Noble end 18-8 Stainless steel (type 316) (passive))

18-8 Stainless steel (type 304) (passive))

Titanium

Nickel (passive)

Copper

Brass

Tin

Lead

18-8 Stainless steel (type 316) (active))

18-8 Stainless steel (type 304) (active))

Pb-Sn solder

Case iron

Mild steel

Alclad

Aluminum

Zinc

Active end Magnesium where E298 is the electrode potential under standard condition, X is the concentration in moles per liter of solution (or molar concentration), k is the Boltzmann’s constant (86.1 x 10-6 eV K-1), and n’ is the valence of the metal or species involved. In practice, the concentration of the electrolyte remains quite dilute (X^ 1). For this reason, the electrode potentials tend to shift more to the anodic end. For example, at room temper­ature (25 °C), if the concentration of Fe2+ in solution is 0.01 M, the electrode potential for the anodic reaction would be -0.47 V compared to -0.44 V under the standard condition (i. e., 1 M).

In practice, we are interested to know the relative corrosion tendency of alloys in different environments. For this, the galvanic series is the more practical chart. One example of a galvanic series in seawater is given in Table 5.5. Magnesium is at the end of the series being the most active and 18/8 stainless steels (in passive condition) is on the top of the chart due to their ability to form chromium oxide — based passive film. Note that titanium and nickel that are more active in the electro­chemical series are near the noble end of the galvanic series, by virtue of their abil­ity to form passive films.

As already noted, the anodic reaction is the main reaction for corrosion, which is of the general form M! M+n + ne-. However, electrons released during anodic reaction need to go to the cathode. So, we need to know some specific cathodic reactions, which are of interest. They are given in Table 5.6. Among them, the reaction for hydroxyl ion formation has some important implications. Hydroxyl ion formed would react with Fe3+ ion and form Fe (OH)3 deposit, which is known as the red rust that forms on iron in moist, oxygen-rich environment.

5.3.2

The new reactors that are being built or will be built within a few years are of Gener — ation-III category. These are mainly advanced LWRs. Examples include advanced boiling water reactor (ABWR) and evolutionary or European power reactor (EPR). In the same line, Generation III+ category aims to provide reactor systems that have much improved designs and safety features, and much greater capacities. Notably, all these reactors are thermal in nature. No fast reactor is in the pipeline in these categories. An improved version of the US EPR® is being designed and developed as a Generation III+ reactor by AREVA NP. It is a four-loop plant with a rated ther­mal power of 4590 MWth. Figure 1.13a and b shows the details of a EPR power plant.

The primary system design and main components are similar to those of PWRs currently operating in the United States. However, the US EPR incorporates several new advancements in materials technology as well as new uses for existing materi­als. Some notable examples include M5® fuel cladding and the stainless steel heavy neutron reflector. Some details of these are discussed below:

a) Zircaloy-4 (Zr-4) has been used extensively for many years in PWR fuel cladding applications. Zr-4 is a cold worked stress-relieved (CWSR) alloy with a zirconium base containing tin, iron, chromium, and oxygen. This offers good corrosion resistance and mechanical properties. AREVA NP has developed an advanced zirconium alloy for PWR fuel rod cladding and fuel assembly structural compo­nents, known as M5® (Zr-0.8-0.12Nb-0.09-0.13O). M5® makes high-burnup fuel cycles possible in the increasingly higher duty operating environments of PWRs. Introduced commercially in the 1990s after a rigorous development pro­gram, M5® was a breakthrough in zirconium alloy development. This fully recrystallized, ternary Zr-Nb-O alloy produces improved in-reactor corrosion,

hydrogen, growth, and creep behaviors. The remarkably stable microstructure responsible for these performance improvements is the result of carefully con­trolled ingot chemistry and product manufacturing parameters. M5® helps util­ities achieve significant fuel cycle cost savings and enhanced operating margins

by allowing higher burnups and higher duty cycles. Excellent corrosion resist­ance and low irradiation growth allow higher burnups, extended fuel cycle opera­tion, and fuel assembly design modifications that enhance operating flexibility. The corrosion resistance of M5® allows operation in high pH environments, eliminating the risk of oxide spalling and helping to minimize dose rates. M51 has been proven to withstand severe operating conditions: high neutron flux, heat flux, and high operating temperatures. As of January 2007, over 1 576000 fuel rods in designs from 14 x 14 (Figure 1.14) to 18 x 18 matrices have been irradiated in commercial PWRs to burnups exceeding 80 000 MWd per mtU. In fuel rod cladding with hydrogen concentrations above the solubility limit, excess hydrogen precipitates as brittle hydrides, reducing the ability of the cladding to cope with pellet-to-clad mechanical interactions during reactivity insertion acci­dents. During a LOCA, ( Loss Of Coolant Accident) high hydrogen levels incre­ase the transport and solubility of oxygen at high temperatures, leading to substantial embrittlement of the cladding as it cools and the oxygen precipitates. M5® cladding absorbs approximately 85% less hydrogen than other Zr-4-based alloys. As a result, M5® will not reach hydrogen levels sufficient enough to pre­cipitate hydrides and will not lead to excess oxygen absorption during a LOCA.

b) The space between the multicornered radial periphery of the reactor core and the cylindrical core barrel is filled with an austenitic stainless steel structure, called the heavy reflector. The primary purpose of the heavy reflector is to reduce fast neutron leakage and flatten the power distribution of the core, thus improv­ing the neutron utilization. It also serves to reduce the reactor vessel fluence (~1 x 1019ncm~2, E > 1 MeV after 60 years). The reflector is inside the core barrel above the lower core support plate. The reflector consists of stacked forged slabs (rings) positioned one above the other with keys, and axially restrained by tie rods bolted to the lower core.

The solute atoms whose sizes are much smaller than the parent atom size can occupy the interstitial spaces easily. These solute atoms could be in the alloy and result in increased strength. They are called interstitial impurity atoms (i. e., IIA). However, the same type of point defect would be created if the interstitial atoms act as alloying constituents. In this case, they are more likely to be called as “interstitial alloying atoms.” The presence of interstitial atoms in the host lattice may lead to the creation of “interstitial solid solutions.” These are the homogeneous mixture of two or more kinds of atoms, of which at least one is dissolved in the host lattice by occupying the interstitial spaces. Elements with relatively small atomic radii (less than 100 pm) like hydrogen, oxygen, nitrogen, boron, and carbon are the most com­mon interstitial atoms. As per Hume-Rothery, extensive interstitial solid solubility results if the apparent solute atom diameter is 0.59 smaller than that of the solvent. It has been noted that atomic size factor is not the only sufficient factor to deter­mine the feasibility of interstitial solid solution formation. Transition elements such as iron, titanium, tungsten, molybdenum, thorium, uranium, and so on have preferred solubility for the interstitial atoms than the nontransition metals. It is postulated that the unusual electronic structure of the transition metals (incomplete d subshell) is the reason for higher interstitial solid solubility, which is however almost always smaller than the typical substitutional solute solubility (see the next section).

In plain carbon steels, different solid solution phases can be formed based on the type of the host iron lattice structure. Here, we give an example of the formation of two different phases, austenite and a-ferrite, in steel. In the FCC unit cells, octahe­dral interstitial sites are located at the midpoints of the cube edges and at the body center. The maximum size of an atom that can be accommodated in the confine of this type of octahedral site without causing distortion is 0.414r, where r is the par­ent atom radius surrounding the interstitial space. On the other hand, smaller tet­rahedral sites are located on the body diagonals (Figure 2.15a). The size of an atom that fits the tetrahedral space is only 0.225r. We know that the iron atom radius is 140 pm. Thus, the maximum interstitial atom size that can be accommodated in the octahedral and tetrahedral sites without distortion are ~58.0 and 31.5 pm, respectively. The carbon atom radius is only 29.0 pm. So, the carbon atoms can easily occupy the octahedral sites as well as tetrahedral sites without causing any lattice distortion, and hence it becomes energetically stable configuration. Thus, the austenite phase can be defined as an interstitial solid solution of carbon atoms located at both octahedral and tetrahedral sites of the FCC iron lattice (i. e., y-Fe). Conversely, in a BCC unit cell, the tetrahedral spaces are slightly larger (0.29r) com­pared to the octahedral spaces (0.15r). The maximum interstitial atom sizes to be fitted inside the octahedral and tetrahedral sites in a BCC iron would be 21.0 and

40.6 pm, respectively. Given the carbon atom radius (29 pm), only the tetrahedral sites could be filled without causing any lattice distortion. Thus, the a-ferrite phase can be defined as the interstitial solid solution of carbon atoms (located at the tetra­hedral sites) in a BCC iron (a-Fe). In general, interstitial solutes cannot occupy all the interstitial spaces available in the host lattice because of the possibility of much greater lattice distortion.

In this type of dislocation loops, the Burgers vector is on the same plane as the dislocation loop. Hence, the loop expands or contracts under an applied stress. The loop here has edge, screw, and mixed orientations. Figure 4.11 shows how the glide loop can produce plastic deformation.

4.2.1.1 Prismatic Loop

In this type of dislocation loops, the Burgers vector lies perpendicular to the plane of the dislocation loop, as shown in Figure 4.12a. As the Burgers vector is perpen­dicular to the dislocation loop, the loop is entirely of pure edge orientation. The glide plane is perpendicular to the loop. Thus, the loop cannot expand or contract on the plane of the loop conservatively unlike the glide loop. If the loop movement

does take place, it would produce crystal slip, as shown in Figure 4.12b. In latter sections, we will again discuss about this type of dislocation loops.

4.2.2

Hardness is defined as resistance to indentation. Hardness testing is simple and is a useful technique to characterize mechanical properties of materials. It provides a rapid and economical way of determining the resistance of materials to deforma­tion. It generally does not involve a total destruction of the sample as needed in tension testing, and sometimes considered as a semi-nondestructive technique, and generally needs small test volume of materials. Thus, the hardness values are generally proportional to the strength values as obtained from conventional tension or compression tests. Materials scientists (metallurgists) are mostly interested in the hardness that is defined by the resistance and of a material against indentation. So, in all hardness tests, we use a hard indenter placed vertically against the sample surface and load the indenter with a specific load for a specified time into the sam­ple, and measure the depth or lateral dimension of the indentation. Thus, relatively larger indentations are noted in softer materials.

There are two major kinds of hardness testing: macrohardness and microhard­ness. In the group of macrohardness test techniques, Brinell hardness test, Rock­well test, and Macro-Vickers test are important. Generally, larger loads are used giving rise to larger size indentation on the samples. In the group of microhard­ness test techniques, tests are generally done in two ways — micro-Vickers and Knoop indentation.

We have, so far, discussed various effects that energetic radiations can produce in reactor materials. Void swelling is one of them. Increase in strength and decrease in ductility are some of the commonly observed effects of void formation, which negatively affect effective life of the reactor components, mostly in-core compo­nents (such as fuel cladding). In 1967, using transmission electron microscopy, Cawthorne and Fulton gave the first experimental evidence of radiation-produced voids in LMFBR stainless steel fuel cladding tubes causing void swelling due to the fast neutron irradiation at reactor ambient temperatures (400-600 °C). Figure 6.14a

depicts the microstructure of voids in stainless steel and Figure 6.14b the conse­quent macroscopic swelling of fuel rods. Voids contain some helium gas (gener­ated due to transmutation reactions), but generally do not contain sufficient amounts to be called “bubbles.” But generally their interrelation is very strong, and will be described in the following sections.

In this introductory chapter, we first introduced nuclear energy and discussed its significance in the modern civilization. We also discussed some nuclear physics fundamentals such as half-value thickness for neutron beam attenuation, nuclear cross sections, neutron flux and fluence, and other concepts. A detailed overview of different reactors is presented. The material selection criteria for nuclear compo­nents are also discussed.

Appendix 1.A

Zirconium-based alloys are commonly used in water reactors for cladding UO2, while Zircaloy-2 and Zircaloy-4 are used in BWRs and PWRs, respectively. The fol­lowing are the reasoning and historical development of these cladding materials: The fuel (UO2) is inserted in canning tubes that separate the radioactive fuel from the coolant water. The requirements for cladding materials thus are as follows:

• Low cross section for absorption of thermal neutrons

• Adequate strength and ductility

• Compatibility with fuel

• Adequate thermal conductivity

• Corrosion resistance to water

In order of increasing cross section for absorption } of thermal neutrons,

the various metals can be classified [normalized to Be] as follows:

Be: scarce, expensive, difficult to fabricate, and toxic

Mg: not strong at high temperatures and poor resistance to hot water corrosion Al: low melting point and poor high temperature strength

Zirconium is relatively abundant, is not prohibitively expensive, has good corro­sion resistance, has reasonable high-temperature strength, good fabricability, and can be further improved by proper alloying. Processing of Zr metal from ore requires removal of hafnium [Hf], which is always associated with Zr. Hf has rela­tively high absorption of thermal neutrons. This Kroll process was relatively more expensive ) Mg treated. The elements used in alloying for increasing strength are O, Sn, Fe, and [Cr, Ni] and for improving corrosion are Cr, Ni, and Fe.

Thus, the Zircaloys were developed (mainly from the US Navy in 1950s). For a nice description of the history of Zry development, refer to the following:

• Krishnan, R. and Asundi, M. K. (1981) Zirconium alloys in nuclear technology, in Alloy Design (eds. S. Ranganathan et al.), Indian Academy of Sciences.

• Rickover, H. G. (1975) History of Development of Zirconium Alloys for Use in Nuclear Power Reactors, US ERDA, NR&D.

Problems

1.1 a) What is the percentage of U235 in naturally occurring uranium and what is the rest made of?

b) A nuclear fission reaction of an U235 atom caused by a neutron produces one barium atom, one Krypton atom, and three more neutrons. Evaluate approximately how much energy is liberated by this reaction.

Approximately how much percentage of energy is carried by the fission fragments (no calculation necessary for the last part of the question)?

1.2 What is the difference between fissile and fertile isotopes? Give two examples of each. What is the role of fertile isotopes in a breeder reactor?

1.3 Define a nuclear reactor? What is the basic difference between an atomic bomb and a power-producing reactor?

1.4 What are the prime differences between LWR and CANDU reactors (com­ment mostly on materials aspects)?

1.5 Describe the importance of control materials with respect to reactor safety and control. What are the primary requirements for a control material? Give at least four examples of control materials.

1.6 Categorize neutrons based on their kinetic energy. What is the major differ­ence between a thermal reactor and a fast reactor?

1.7 a) Zirconium and hafnium both have crystal structures (HCP) in the general

operating regimes of LWRs. Naturally occurring Zr always has some Hf (1-3 wt%) in it. Hf-containing Zr alloys are very common in chemical industries but not in nuclear industries. Why? b) What is the main application of Zr alloys in LWRs? What are the various functions of this reactor component? What are the reasons that make Zr alloys suitable for such use?

1.8 What are the two main zirconium alloys used in light water reactors? Give their compositions. Name two recently developed zirconium alloys with their compositions.

1.9 What is neutron economy? What significance does it have? How much influ­ence does it exert in the selection of materials used in nuclear reactors?

1.10 a) Define neutron flux and neutron fluence. What are their units?

b) Define neutron cross section? Briefly comment on the importance of neu­tron cross section from a reactor perspective.

c) Neutrons of 10keV energy are incident on a light water barrier. The neu­tron cross section for hydrogen (protium) at 10keV is about 20 b and that of oxygen is only 3.7 b. Determine the half-value thickness of neutron attenuation for the water barrier (assume that neutron interaction with oxygen in water molecule is negligible). Find out the half-thickness value for 1 MeV neutrons traveling through the water barrier (neutron cross sec­tion for protium is 4.1 b for 1 MeV neutrons). Comment on the signifi­cance of the results.

Additional Reading Materials

1 L’Annunziata, M. F. (1998) Handbook of Radioactivity Analysis, Academic Press, New York.

2 Murray, R. L. (1955) Introduction to Nuclear Engineering, Prentice Hall, New York.

3 Smith, C. O. (1967) Nuclear Reactor Materials, Addison-Wesley,

Massachusetts.

4 Ma, B. (1983) Nuclear Reactor Materials and Applications, Van Nostrand Reinhold Company, New York.

7 Hinds, D. and Maslak, C. (January 2006)

Next Generation Nuclear Energy: The ESBWR, Nuclear News, pp. 35^0.

8 Charit, I. and Murty, K. L. (2008) Structural materials for Gen-IV nuclear reactors: challenges and opportunities. Journal of Nuclear Materials, 383, 189-195.

A German physiologist (not a materials scientist!), Adolf Fick first proposed the laws of diffusion in 1855. These laws are applicable to all three general states (gas, liquid, and solid) of matter. These “macroscopic” laws are basically continuum dif­fusion in nature, and are very akin to equations of heat conduction (Fourier’s laws) or electrical conduction (Ohm’s law), and do not explicitly take into account any defect-assisted atomistic or microscopic mechanisms involved in diffusion processes.

The first law states that the net flux of solute atoms in a solution will occur from the regions of high solute concentrations to the regions of low concentrations, and the solute flux along a particular coordinate axis is directly proportional to the con­centration gradient. In a unidirectional (say, along x-axis) diffusion event, it is expressed as

where J is the diffusion flux, D is the proportionality factor known as diffusion coefficient (or simply diffusivity), c is the concentration of the diffusing species, and x is the diffusion distance. The negative sign on the right-hand side of Eq. (2.17) implies that the diffusion or the mass flow occurs down the concentra­tion gradient. A schematic representation of Fick’s law is shown in Figure 2.42. One assumption is inbuilt with Fick’s first law. It applies to a steady-state (or nearly steady-state) condition only. It is important to understand what the steady-state flow

means. In such a state, the diffusion flux at any cross-sectional plane across the diffusion distance remains constant (i. e., time-independent).

Flux (J) is the net amount atoms diffusing normal to an imaginary plane per unit area in a unit time (much like the definition of heat flux or neutron flux). That is why the unit of diffusion flux is generally given in atoms per m2 s (or mol m-2 s-1). The concentration c is given in unit of atoms per m3 (or mol m-3). If the dimension of diffusion distance (x) is given in m, the unit of diffusivity (m2 s-1) can then be found through dimensional analysis. Diffusivity (D) is a significant parameter that is directly linked to the atomistic and defect mechanisms. D depends on factors such as temperature and composition. Figure 2.43 shows the variation of diffusion coefficients in three gold alloys as a function of composition. The variation in diffusivity with composition is especially marked in Au-Ni alloy, followed by Au-Pd and Au-Pt. However, the diffusivity of species would be practi­cally independent of concentration if present in a trace quantity. Sometimes, D is considered to be independent of composition to preserve the simplicity of the mathematical solutions.

4.3.1

Dislocation Reactions in FCC Lattices

4.3.1.1 Shockley Partials

Let us now turn our attention to a situation in Figure 4.26a. Atomic arrangement (ABCABC. . . ) on the {111} planes can show that slip via the motion of disloca­tions with Burgers vector (b1 o0/2)(110) dislocation may not occur easily. The vec­tor b1 = (o0/2)(101) defines one of such slip directions. On the other hand, if the same shear displacement can be accomplished by a two-step path following b2 and b3, it will be more energetically favorable. That is why the perfect dislocation with b1

Burgers vector dissociates into two partial dislocations of b2 and b3 according to the following dislocation dissociation reaction:

у [10T] ! I [211 ] + у [112]. (4.24)

We have already seen in Example 4.3 how this reaction is feasible (recall Frank’s rule). There could also be similar other reactions maintaining the reaction feasibil­ity criterion intact. Slip via this two-stage process creates a stacking fault, ABCACjABC. Note the appearance of HCP-like stacking sequence (CACA) in the otherwise FCC stacking sequence. The product dislocations (AD and AC with Bur­gers vectors b2 and b3) of this dissociation reaction are known as Shockley partials

(first described by Heidenreich and Shockley in 1948) as shown in Figure 4.26b and c. These are called partial dislocations as these dislocations are imperfect dis­locations since they do not create complete lattice translations. The combination of the two partial dislocations is called extended dislocation. The region between the two partials is the stacking fault that is regarded as a region in a state intermediate between full slip and no slip (for more information on stacking faults, refer to Sec­tion 2.2). Dislocations such as b1 = (a0/2)(101) are known as total or perfect dis­locations and the stacking difference between the total (or perfect) and partial (or imperfect) dislocations is depicted in Figure 4.26d.

Noting that the Shockley partial dislocations lie on close-packed planes with Bur­gers vectors in the CPPs, these Shockley partials are considered glissile dislocations and they can move in the plane of the fault. So, the problem arises when the extended screw dislocation needs to cross-slip. It cannot glide only in the plane of the fault. The extended dislocation moves keeping an equilibrium separation between the two partials. The equilibrium separation (stacking fault width) is brought about by the balance of the repulsive force between two partials and the surface tension of the stacking fault. So, the partials must recombine into a perfect dissociation before it cross-slips. In the recombination process, the wider the stack­ing fault (or the lower the stacking fault energy), the more difficult the formation of constriction and vice versa. The stacking fault energy (ESF) and width (w) are inversely related given by Eq. (4.25):

Hence, materials (like aluminum) with higher stacking fault energy will have a narrow stacking fault width, and thus partial dislocations and stacking faults do not occur. However, for materials (like brass) with lower stacking fault energy, the recombination becomes difficult due to wider stacking fault width. Stacking fault energy is thus an important factor influencing the plastic deformation in different ways through strain hardening more rapidly, twinning easily on annealing, and showing different temperature dependence of flow stress compared to the metals with higher SF energy. Thus, appropriate alloying resulting in decreased stacking fault energy can result in higher strength as seen, for example, in Cu-Al alloys.

Although it is useful to characterize the creep curve, it is often required to find the rupture time (tr); the higher the rupture time, the better the life of structures in­service. Stress rupture tests are similar to creep tests; however, the creep strains are not monitored during the test and tests are carried out to fracture and times to fracture are noted at varied temperatures and stresses (loads). The stress-rupture data are plotted as stress against rupture time (Figure 5.31). The basic information obtained in the stress-rupture test can be used for the design of high-temperature components such as jet turbine components and steam turbine blades where the structure cannot undergo more than an allowable amount of creep deformation during the whole service life.

There are empirical relations and functions that can describe stress-rupture behavior of a material and at the same time provide useful design data. Larson — Miller parameter (LMP) is one of them:

LMP = Tlog (C + tr), (5.43)

where C is a material constant, which is around 20 when T is in K, and tr is in hours. Figure 5.32 shows a plot depicting variation of stress against LMP for nuclear grade zirconium alloys.

Another important empirical relation is the Monkman-Grant relation that fol­lows from the fact that the time to rupture decreases with stress or minimum creep rate:

es ■ tr = constant, (5-44)

where es is the steady-state creep rate. Figure 5.33 shows the data for a commercial pure titanium tubing clearly depicting the validity of Monkman-Grant relationship. This has a significant role in predicting the rupture times for service stresses where relatively low stresses and temperatures are appropriate from the short-term

laboratory tests performed at high stresses/temperatures. Once the stress and tem­perature variations of steady-state creep rates are characterized for a given material and the various constants in Eqs (5.36) and (5.37) are determined from laboratory tests, one can first predict the secondary creep rate at the service temperature and stress following which the application of Monkman-Grant relationship can be used to predict the rupture life in-service.

Another useful parameter is known as the Zener-Holloman parameter that is applicable for a given applied stress:

Z = e eQ/RT. (5.45)

Figure 5.34 depicts such a correlation for a-iron that shows a single curve describing the primary, secondary (minimum creep rate), and tertiary strains at varied temperatures.

Similarly, Sherby-Dorn parameter involves normalization with temperature — compensated time:

Psd = te-Q=RT. (5.46)

The creep curves at three different temperatures at a constant stress merge into a single curve when plotted strain versus compensated time, as depicted in Figure 5.35 for aluminum at 3ksi or ~20.7 MPa.