Introduction to laminar boundary layers

Introduction

Chapters 1 through 3 consider conduction heat transfer in a stationary medium. Energy transport within the material of interest occurs entirely by conduction and is governed by Fourier's law. Convection is considered only as a boundary condition for the relatively simple ordinary or partial differential equations that govern conduction problems. Convection is the transfer of energy in a moving medium, most often a liquid or gas flowing through a duct or over an object. The transfer of energy in a flowing fluid is not only due to conduction (i.e., the interactions between micro-scale energy carriers) but also due to the enthalpy carried by the macro-scale flow. Enthalpy is the sum of the internal energy of the fluid and the product of its pressure and volume. The pressure-volume product is related to the work required to move the fluid across a boundary. You were likely introduced to this term in a thermodynamics course in the context of an energy balance on a system that includes flow across its boundary. The additional terms in the energy balance related to the fluid flow complicate convection problems substantially and link the heat transfer problem with an underlying fluid dynamics problem. The complete solution to many convection problems therefore requires sophisticated computational fluid dynamic (CFD) tools that are beyond the scope of this book.

Introduction to radiation

Radiation

From a thermodynamic perspective, thermal energy can be transferred across a boundary (i.e., heat transfer can occur) by only two mechanisms: conduction and radiation. Conduction is the process in which energy exchange occurs due to the interactions of molecular (or smaller) scale energy carriers within a material. The conduction process is intuitive; it is easy to imagine energy carriers having a higher level of energy (represented by their temperature) colliding with neighboring particles and thereby transferring some of their energy to them. Convection is the process in which the surface of a solid material exchanges thermal energy with a fluid. Although convection is commonly treated as a separate heat transfer mechanism, it is more properly viewed as conduction in a substance that is also undergoing motion. The energy transfer by conduction and fluid motion are coupled, making convection problems more difficult to solve than conduction problems. However, convection is still an intuitive process since it can be explained by interactions between neighboring molecules with different energy levels. Radiation is a very different heat transfer process because energy is transferred without the benefit of any molecular interactions. Indeed, radiation energy exchange can occur over long distances through a complete vacuum. For example, the energy that our planet receives from the sun is a result of radiation exchange. The process of radiation heat transfer is not intuitive to most engineers.

Introduction

In Chapter 4, we dealt with point and line defects. There is another class of defects called interfacial, or planar, defects. These imperfections, as the name signifies, occupy an area or surface and so are two-dimensional, as well as being of great importance. Examples of such defects are free surfaces of a material, grain boundaries, twin boundaries, domain boundaries, and antiphase boundaries. Of all these, grain boundaries are the most important from the point of view of the mechanical properties of the material. In what follows, we consider in detail the structure of grain and twin boundaries and their importance in various deformation processes, and, very briefly, the structure of other interfacial defects. Details regarding the strengthening of a material by grain boundaries are given in Section 5.3. Volumetric defects, such as voids, also play a major role in the mechanical properties of materials, affecting the strength and elastic properties of the material significantly. Volumetric defects are briefly described in Section 5.7. In Section 5.8, we present the defects occurring in polymers.

Grain Boundaries

Crystalline solids generally consist of a large number of grains separated by boundaries. Most industrial metals and ceramics are polycrystalline aggregates, and the mechanical properties of these polycrystals can be radically different from those of the monocrystals that form the individual grains. Figure 5.1 illustrates a polycrystalline aggregate, in which each grain has a distinct crystallographic orientation.

Introduction

We can define a composite material as a material consisting of two or more physically and/or chemically distinct phases, suitably arranged or distributed. A composite material usually has characteristics that are not depicted by any of its components in isolation. Generally, the continuous phase is referred to as the matrix, while the distributed phase is called the reinforcement. Three items determine the characteristics of a composite: the reinforcement, the matrix, and the interface between them. In this chapter, we provide a brief survey of different types of composite materials, highlight some of their important features, and indicate their various applications.

Types of Composites

We may classify composites on the basis of the type of matrix employed in them – for example, polymer matrix composites (PMCs), metal matrix composites (MMCs), and ceramic matrix composites (CMCs). We may also classify composites on the basis of the type of reinforcement they employ (see Figure 15.1):

1. Particle reinforced composites.

2. Short fiber, or whisker reinforced, composites.

3. Continuous fiber, or sheet reinforced, MMCs.

4. Laminate composite.

Figure 15.2 shows typical microstructures of some composites: boron fiber/Al (Figure 15.2(a)), short alumina fiber/Al (Figure 15.2(b)), and NbC/Ni–Cr, an in situ (eutectic) composite (Figure 15.2(c)). Examples of microstructure of a silicon carbide particle (three different volume fractions) reinforced aluminium matrix are given in Figure 15.3. These were made by hot pressing of powders followed by hot extrusion. Note the preferential alignment of SiC particles in the extrusion direction.

Introduction

An intermetallic is a compound phase of two or more normal metals (ordered or disordered). Interest in intermetallics waned in the 1960s and 1970s. However, the demand for materials that are strong, stiff, and ductile at high temperatures has led to a resurgence of interest in intermetallics, especially silicides and ordered intermetallics such as aluminides. A testimony to this resurgence was the appearance in 1994 on the subject of a two-volume set by J. H. Westbrook and R. L. Fleischer, Intermetallic Compounds: Principles and Practice (New York: John Wiley). Intermetallic aluminides and silicides can be very oxidation and corrosion resistant, because they form strongly adherent surface oxide films. Also, intermetallics span a wide range of unusual properties. An important example outside the field of high-temperature materials involves the exploitation of martensitic transformations, exotic colors, and the phenomenon of shape memory in gold-based intermetallics in jewelry making. In what follows, we first describe the silicides, then the ordered intermetallics, and finally the basic structure and properties of foams.

Silicides

About 300 intermetallic compounds melt at temperatures above 1,500 °C. A survey of some silicide intermetallics for high-temperature applications showed that, based on criteria such as availability, phase changes in the temperature range of interest, and oxidation resistance, Ti5S3 and MoSi2 seem to be the most promising materials: Ti5Si3 has the lowest density of all intermetallics, and MoSi2 has a superior oxidation resistance.

Introduction

A solution can be defined as a homogeneous mixture of two or more substances. Generally, one thinks of a solution as liquid, but gaseous or solid forms are possible as well. Indeed, we can have solutions of gases in a gas, gases in a liquid, liquids in a liquid, solids in a liquid, and solids in a solid. A solution can have one or more solutes dissolved in a solvent. The solute is the substance that is dissolved; the solvent is the substance in which the solute is dissolved. In a solution, there is always less solute than solvent. There are two kinds of solid solutions: substitutional and interstitial. Figure 10.1 shows examples of each in a schematic manner. Figure 10.1(a) is of brass, which is a substitutional solid solution of zinc (the solute) in copper (the solvent). We call such an alloy substitutional because the solute atoms merely substitute for the solvent atoms in their normal positions. In a substitutional solution, the atomic sizes of the solute and solvent atoms are fairly close. The maximum size difference is approximately 15%. When the atomic sizes of the solute and solvent are very different, as in the case of carbon or nitrogen in iron, we get an interstitial solid solution. Figure 10.1(b) shows such a solid solution of carbon in iron. We call these solutions interstitial solid solutions because the solute atoms occupy interstitial positions in the solvent lattice.

Introduction

Fracture of any material (be it a recently acquired child's toy or a nuclear pressure vessel) is generally an undesirable happening, resulting in economic loss, an interruption in the availability of a desired service, and, possibly, damage to human beings. Besides, one has good, technical reasons to do fracture testing: to compare and select the toughest (and most economical material) for given service conditions; to compare a particular material's fracture characteristics against a specified standard; to predict the effects of service conditions (e.g., corrosion, fatigue, stress corrosion) on the material toughness; and to study the effects of microstructural changes on material toughness. One or more of these reasons for fracture testing may apply during the design, selection, construction, and/or operation of material structures. There are two broad categories of fracture tests; qualitative and quantitative. The Charpy impact test exemplifies the former, and the plane-strain fracture toughness (KIc) test illustrates the latter. We describe briefly important tests in both of these categories.

Impact Testing

We saw in Chapter 7 that stress concentrations, like cracks and notches, are sites where failure of a material starts. It has been long appreciated that the failure of a given material in the presence of a notch is controlled by the material's fracture toughness. Many tests have been developed and standardized to measure this “notch toughness” of a material. Almost all are qualitative and comparative in nature.

Introduction

Everything that surrounds us is matter. The origin of the word matter is mater (Latin) or matri (Sanskrit), for mother. In this sense, human beings anthropomorphized that which made them possible – that which gave them nourishment. Every scientific discipline concerns itself with matter. Of all matter surrounding us, a portion comprises materials. What are materials? They have been variously defined. One acceptable definition is “matter that human beings use and/or process.” Another definition is “all matter used to produce manufactured or consumer goods.” In this sense, a rock is not a material, intrinsically; however, if it is used in aggregate (concrete) by humans, it becomes a material. The same applies to all matter found on earth: a tree becomes a material when it is processed and used by people, and a skin becomes a material once it is removed from its host and shaped into an artifact.

The successful utilization of materials requires that they satisfy a set of properties. These properties can be classified into thermal, optical, mechanical, physical, chemical, and nuclear, and they are intimately connected to the structure of materials. The structure, in its turn, is the result of synthesis and processing. A schematic framework that explains the complex relationships in the field of the mechanical behavior of materials, shown in Figure 1.1, is Thomas's iterative tetrahedron, which contains four principal elements: mechanical properties, characterization, theory, and processing. These elements are related, and changes in one are inseparably linked to changes in the others.

Introduction

Elasticity deals with elastic stresses and strains, their relationship, and the external forces that cause them. An elastic strain is defined as a strain that disappears instantaneously once the forces that cause it are removed. The theory of elasticity for Hookean solids – in which stress is proportional to strain – is rather complex in its more rigorous treatment. However, it is essential to the understanding of micro- and macromechanical problems. Examples of the former are stress fields around dislocations, incompatibilities of stresses at the interface between grains, and dislocation interactions in work hardening; examples of the latter are the stresses developed in drawing, and rolling wire, and the analysis of specimen–machine interactions in testing for tensile strength. This chapter is structured in such a way as to satisfy the needs of both the undergraduate and the graduate student. A simplified treatment of elasticity is presented, in a manner so as to treat problems in an undergraduate course. Stresses and strains are calculated for a few simplified cases; the tridimensional treatment is kept at a minimum. A graphical method for the solution of two-dimensional stress problems (the Mohr circle) is described. On the other hand, the graduate student needs more powerful tools to handle problems that are somewhat more involved. In most cases, the stress and strain systems in tridimensional bodies can be better treated as tensors, with the indicial notation.

Introduction

The relaxation times for the molecular processes in gases and in a majority of liquids are so short, that molecules/atoms are almost always in a well-defined state of complete equilibrium. Consequently, the structure of a gas or liquid does not depend on its past history. In contrast, the relaxation times for some of the significant atomic processes in crystals are so long, that a state of equilibrium is rarely, if ever, achieved. It is for this reason that metals in general (and ceramics and polymers, under special conditions) show the usually desirable characteristic of work-hardening with straining, or strain-hardening. In other words, plastic deformation distorts the atoms from their equilibrium positions, and this manifests itself subsequently in hardening.

In fact, hardening by plastic deformation (rolling, drawing, etc.) is one of the most important methods of strengthening metals, in general. Figure 6.1 shows a few deformation-processing techniques in which metals are work-hardened. These industrial processes are used in the fabrication of parts and enable the shape of metals to be changed. The figure is self-explanatory. Rolling is used to produce flat products such as plates, sheets, and also more complicated shapes (with special rolling cylinders). In forging, the top hammer comes down, and the part is pushed into a die (closed-die forging) or is simply compressed. Extrusion uses a principle similar to that in the use of a tube of toothpaste. The material is squeezed through a die, and its diameter is reduced.

Introduction

The technological developments wrought since the early twentieth century have required materials that resist higher and higher temperatures. Applications of these developments lie mainly in the following areas:

1. Gas turbines (stationary and on aircraft), whose blades operate at temperatures of 800–950 K. The burner and afterburner sections operate at even higher temperatures, viz. 1,300–1,400 K.

2. Nuclear reactors, where pressure vessels and piping operate at 650–750 K. Reactor skirts operate at 850–950 K.

3. Chemical and petrochemical industries.

All of these temperatures are in the range (0.4–0.65) Tm, where Tm is the melting point of the material in kelvin.

The degradation undergone by materials in these extreme conditions can be classified into two groups:

1. Mechanical degradation. In spite of initially resisting the applied loads, the material undergoes anelastic deformation; its dimensions change with time.

2. Chemical degradation. This is due to the reaction of the material with the chemical environment and to the diffusion of external elements into the materials. Chlorination (which affects the properties of superalloys used in jet turbines) and internal oxidation are examples of chemical degradation.

This chapter deals exclusively with mechanical degradation. The time-dependent deformation of a material is known as creep. A great number of high-temperature failures can be attributed either to creep or to a combination of creep and fatigue. Creep is characterized by a slow flow of the material, which behaves as if it were viscous.

Introduction

Upon being mechanically stressed, a material will, in general, exhibit the following sequence of responses: elastic deformation, plastic deformation, and fracture. This chapter addresses the second response: plastic deformation. A sound knowledge of plasticity is of great importance for the following reasons.

• Many projects are executed in which small plastic deformations of the structure are accepted. The “theory of limit design” is used in applications where the weight factor is critical, such as space vehicles and rockets. The rationale for accepting a limited plastic deformation is that the material will work-harden at that region, and plastic deformation will cease once the flow stress (due to work-hardening) reaches the applied stress.

• It is very important to know the stresses and strains involved in deformation processing, such as rolling, forging, extrusion, drawing, and so on. All these processes involve substantial plastic deformation, and the response of the material will depend on its plastic behavior during the processes. The application of plasticity theory to such processes is presented later in this chapter.

• The mechanism of fracture can involve plastic deformation at the tip of a crack. The way in which the high stresses that develop at the crack can be accommodated by the surrounding material is of utmost importance in the propagation of the crack. A material in which plastic deformation can take place at the crack is “tough,” while one in which there is no such deformation is “brittle.”

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Introduction

The separation or fragmentation of a solid body into two or more parts, under the action of stresses, is called fracture. The subject of fracture is vast and involves disciplines as diverse as solid-state physics, materials science, and continuum mechanics. Fracture of a material by cracking can occur in many ways, principally the following:

1. Slow application of external loads.

2. Rapid application of external loads (impact).

3. Cyclic or repeated loading (fatigue).

4. Time-dependent deformation (creep).

5. Internal stresses, such as thermal stresses caused by anistropy of the thermal expansion coefficient or temperature differences in a body.

6. Environmental effects (stress corrosion cracking, hydrogen embrittlement, liquid metal embrittlement, etc.)

The process of fracture can, in most cases, be subdivided into the following categories:

1. Damage accumulation.

2. Nucleation of one or more cracks or voids.

3. Growth of cracks or voids. (This may involve a coalescence of the cracks or voids.)

Damage accumulation is associated with the properties of a material, such as its atomic structure, crystal lattice, grain boundaries, and prior loading history. When the local strength or ductility is exceeded, a crack (two free surfaces) is formed. On continued loading, the crack propagates through the section until complete rupture occurs. Linear elastic fracture mechanics (LEFM) applies the theory of linear elasticity to the phenomenon of fracture – mainly, the propagation of cracks. If we define the fracture toughness of a material as its resistance to crack propagation, then we can use LEFM to provide us with a quantitative measure of fracture toughness.

Introduction

Environment by its omnipresence, except perhaps in space, affects the behavior of all materials. Such effects can range from swelling in polymers to surface oxidation of metals and nonoxide ceramics to catastrophic failure of some materials under a combined action of stress and environment. Environmental degradation of materials is often referred to as corrosion. Such damage is generally time-dependent, i.e., one is able to predict it. Over time, however, envir-onmental damage can become critical. There is, however, a more insidious corrosion problem which is time-independent. Examples of time-independent corrosion include stress corrosion cracking (SCC), environment induced embrittlement, etc. Such damage can occur at anytime, without much warning. There are many examples of such failures resulting in human and economic loss. Corrosion of structural components in aging aircraft is a serious problem. Just to cite one such example, a Boeing 737 belonging to Aloha Airlines, which flew inter island in Hawaii, lost a large portion of its upper fuselage at 7,500 m (24,000 feet) in the air. It turned out that the fuselage panels joined by rivets had corroded, which resulted in the mid-flight failure due to corrosion fatigue.

All materials (metals, ceramics, and polymers) show phenomena of premature failure or mechanical property degradation under certain combinations of stress and environment. We describe below the salient points in regard to environmental effects in different materials. We emphasize the role that the microstructure of a given mater-ial plays in this phenomenon, especially in environmentally assisted fracture.

Courses in the mechanical behavior of materials are standard in both mechanical engineering and materials science/engineering curricula. These courses are taught, usually, at the junior or senior level. This book provides an introductory treatment of the mechanical behavior of materials with a balanced mechanics–materials approach, which makes it suitable for both mechanical and materials engineering students. The book covers metals, polymers, ceramics, and composites and contains more than sufficient information for a one-semester course. It therefore enables the instructor to choose the path most appropriate to the class level (junior- or senior-level undergraduate) and background (mechanical or materials engineering). The book is organized into 15 chapters, each corresponding, approximately, to one week of lectures. It is often the case that several theories have been developed to explain specific effects; this book presents only the principal ideas. At the undergraduate level the simple aspects should be emphasized, whereas graduate courses should introduce the different viewpoints to the students. Thus, we have often ignored active and important areas of research. Chapter 1 contains introductory information on materials that students with a previous course in the properties of materials should be familiar with. In addition, it enables those students unfamiliar with materials to “get up to speed.” The section on the theoretical strength of a crystal should be covered by all students. Chapter 2, on elasticity and viscoelasticity, contains an elementary treatment, tailored to the needs of undergraduate students.

Introduction

In Chapter 7, we described the macroscopic aspects of the fracture behavior of materials. As with other characteristics, the microstructure of a material has a great influence on its fracture behavior. In what follows, we present a brief description of the microstructural aspects of crack nucleation and propagation, as well as the effect of the environment on the fracture behavior of different materials. Figure 8.1 shows, schematically, some important fracture modes in a variety of materials. These different modes will be analyzed in some detail in this chapter. Metals fail by two broad classes of mechanisms: ductile and brittle failure.

Ductile failure occurs by (a) the nucleation, growth, and coalescence of voids, (b) continuous reduction in the metal's cross-sectional area until it is equal to zero, or (c) shearing along a plane of maximum shear. Ductile failure by void nucleation and growth usually starts at second-phase particles. If these particles are spread throughout the interiors of the grains, the fracture will be transgranular (or transcrystalline). If these voids are located preferentially at grain boundaries, fracture will occur in an intergranular (or intercrystalline) mode. The appearance of a ductile fracture, at high magnification (500× or higher) is of a surface with indentations, as if marked by an ice-cream scooper. This surface morphology is appropriately called dimpled. Rupture by total necking is very rare, because most metals contain second-phase particles that act as initiation sites for voids.