During the design and operation of chemical processes, we routinely propose a state for a system by specifying a temperature, pressure, composition, and phase. Then the question is, Can the system be brought to that state? This is a question of observability. In many situations, particularly those involving multicomponent mixtures, the answer is not at all obvious. For example, at certain values for T and P, mixtures of phenol and water can undergo drastic phase changes in response to slight changes in composition: a mixture of phenol in water might be a one-phase vapor, or a one-phase water-rich liquid, or a phenol-rich liquid in equilibrium with a water-rich liquid, or it might be in three-phase vapor-liquid-liquid equilibrium.

In the previous chapter we derived criteria for identifying equilibrium states; for example, in a closed system at fixed T and P, the equilibrium state is the one that minimizes the Gibbs energy. That minimization is equivalent to satisfying the equality of component fugacities. More generally, we derived criteria for thermal, mechanical, and diffusional equilibrium in open systems. But although those criteria can be used to identify equilibrium states, they are not always sufficient to answer the question of observability. Observability requires stability. Thermodynamic states can be stable, metastable, or unstable; a stable equilibrium state is always observable, a metastable state may sometimes be observed, and an unstable state is never observed.

Introduction

Throughout history there has been a never-ending effort to develop materials with higher yield strength. However, a higher yield strength is generally accompanied by a lower ductility and a lower toughness. Toughness is the energy absorbed in fracturing. A high-strength material has low toughness because it can be subjected to higher stresses. The stress necessary to cause fracture may be reached before there has been much plastic deformation to absorb energy. Ductility and toughness are lowered by factors that inhibit plastic flow. As schematically indicated in Figure 13.1, these factors include decreased temperatures, increased strain rates, and the presence of notches. Developments that increase yield strength usually result in lower toughness.

In many ways the fracture behavior of steel is like that of taffy candy. It is difficult to break a warm bar of taffy candy to share with a friend. Even children know that warm taffy tends to bend rather than break. However, there are three ways to promote its fracture. A knife may be used to notch the candy bar, producing a stress concentration. The candy may be refrigerated to raise its resistance to deformation. Finally, rapping it against a hard surface raises the loading rate, increasing the likelihood of fracture. Notches, low temperatures, and high rates of loading also embrittle steel.

There are two important reasons for engineers to be interested in ductility and fracture. The first is that a reasonable amount of ductility is required to form metals into useful parts by forging, rolling, extrusion, or other plastic working processes.

Introduction

It has been estimated that 90% of all service failures of metal parts are caused by fatigue. A fatigue failure is one that occurs under cyclic or alternating stress of an amplitude that would not cause failure if applied only once. Aircraft are particularly sensitive to fatigue. Automobile parts such as axles, transmission parts, and suspension systems may fail by fatigue. Turbine blades, bridges, and ships are other examples. Fatigue requires cyclic loading, tensile stresses, and plastic strain on each cycle. If any of these are missing, there will be no failure. The fact that a material fails after a number of cycles indicates that some permanent change must occur on every cycle. Each cycle must produce some plastic deformation, even though it may be very small. Metals and polymers fail by fatigue. Fatigue failures of ceramics are rare because there seldom is plastic deformation.

There are three stages of fatigue. The first is nucleation of a crack by small amounts of inhomogeneous plastic deformation at a microscopic level. The second is the slow growth of these cracks by cyclic stressing. Finally sudden fracture occurs when the cracks reach a critical size.

Surface observations

Often visual examination of a fatigue fracture surface will reveal clamshell or beach markings as shown in Figure 17.1. These marks indicate the position of the crack front at some stage during the fatigue life. The initiation site of the crack can easily be located by examining these marks.

Introduction

Tensile test data have many uses. Tensile properties are used in selecting materials for various applications. Material specifications often include minimum tensile properties to ensure quality. Tests must be made to ensure that materials meet these specifications. Tensile properties are also used in research and development to compare new materials or processes. With plasticity theory, tensile stress–strain curves can be used to predict a material's behavior under forms of loading other than uniaxial tension.

Often the primary concern is strength. The level of stress that causes appreciable plastic deformation of a material is called its yield stress. The maximum tensile stress that a material carries is called its tensile strength (or ultimate strength or ultimate tensile strength). Both of these measures are used, with appropriate caution, in engineering design. A material's ductility is also of interest. Ductility is a measure of how much the material can deform before it fractures. Rarely, if ever, is the ductility incorporated directly into design. Rather it is included in specifications to ensure quality and toughness. Elastic properties may be of interest, but these are measured ultrasonically much more accurately that by tension testing.

Tensile specimens

Figure 3.1 shows a typical tensile specimen. It has enlarged ends or shoulders for gripping. The important part of the specimen is the gauge section. The cross-sectional area of the gauge section is less than that of the shoulders and grip region, so the deformation will occur here.

Introduction

Necking limits uniform elongation in tension, making it difficult to study plastic stress–strain relationships at high strains. Much higher strains can be reached in compression, torsion, and bulge tests. The results from these tests can be used, together with the theory of plasticity (Chapter 6), to predict stress–strain behavior under other forms of loading. Bend tests are used to avoid the problem of gripping brittle material without breaking it. Hardness tests eliminate the considerable time and effort required to machine tensile specimens. Also, hardness tests are simple to perform and are not destructive.

Compression test

Much higher strains are achievable in compression tests than in tensile tests. However, two problems limit the usefulness of compression tests: friction and buckling. Friction on the ends of the specimen tends to suppress the lateral spreading of material near the ends (Figure 4.1). A cone-shaped region of dead metal (nondeforming material) can form at each end, with the result that the specimen becomes barrel shaped. Friction can be reduced by lubrication and the effect of friction can be lessened by increasing the height-to-diameter ratio, h/d, of the specimen.

Introduction

Plastic deformation of crystalline materials usually occurs by slip, which is the sliding of planes of atoms over one another (Figure 8.1). The planes on which slip occurs are called slip planes and the directions of the shear are the slip directions. These are crystallographic planes and directions that are characteristic of the crystal structure. The magnitude of the shear displacement is an integral number of interatomic distances, so that the lattice is left unaltered. If slip occurs on only part of a plane, there remains a boundary between the slipped and unslipped portions of the plane, which is called a dislocation. Slip occurs by movement of dislocations through the lattice. It is the accumulation of the dislocations left by slip that is responsible for work hardening. Dislocations and their movement are treated in Chapters 9 and 10. This chapter is concerned only with the geometry of slip.

Visual examination of the surface of a deformed crystal will reveal slip lines. The fact that we can see these indicates that slip is inhomogeneous on an atomic scale. Displacements of thousands of atomic diameters must occur on discrete or closely spaced planes to create steps on the surface that are large enough to be visible. Furthermore, the planes of active slip are widely separated on an atomic scale. Yet the scale of the slip displacements and distances between slip lines is small compared to most grain sizes, so slip usually can be considered as homogeneous on a macroscopic scale.

Introduction

With elastic deformation, the strains are proportional to the stress, so every level of stress causes some elastic deformation. On the other hand, a definite level of stress must be applied before any plastic deformation occurs. As the stress is further increased, the amount of deformation increases, but not linearly. After plastic deformation starts, the total strain is the sum of the elastic strain (which still obeys Hooke's law) and the plastic strain. Because the elastic part of the strain is usually much less than the plastic part, it will be neglected in this chapter and the symbol ε will signify the true plastic strain.

The terms strain-hardening and work-hardening are used interchangeably to describe the increase of the stress level necessary to continue plastic deformation. The term flow stress is used to describe the stress necessary to continue deformation at any stage of plastic strain. Mathematical descriptions of true stress–strain curves are needed in engineering analyses that involve plastic deformation, such as prediction of energy absorption in automobile crashes, design of dies for consist stamping parts, and analysis of the stresses around cracks. Various approximations are possible. Which approximation is best depends on the material, the nature of the problem, and the need for accuracy. This chapter will consider several approximations and their applications.

Introduction

The strengths of metals are sensitive to microstructure. Most hardening mechanisms involve making dislocation motion more difficult. These mechanisms include decreased grain size, strain-hardening, solid-solution hardening, and dispersion of fine particles. With finer grain sizes there are more grain boundaries to impede dislocation motion. Metals strain-harden because deformation increases the number of dislocations and each interferes with the movement of others. In solid solutions, solute atoms disrupt the periodicity of the lattice. Fine dispersions of hard particles create obstacles to dislocation motion. Martensite formation and strain aging in steels are sometimes considered separate mechanisms, but both are related to the effects of interstitial solutes on dislocations.

Other factors affecting strength are constraints from neighboring grains, preferred orientations, and crystal structure. This chapter will review all of these mechanisms.

Deformation of polycrystals

In polycrystalline materials, each grain is surrounded by others and must deform in such a way that its change of shape is compatible with its neighbors. Slip on a single system within a grain will not satisfy the need for compatibility. Early attempts to calculate the stress–strain behavior of polycrystals by averaging the stresses to cause single slip in each grain did not meet with success. G. I. Taylor achieved much better agreement by assuming that each grain of a polycrystal undergoes the same shape change (set of strains) as the whole polycrystal.

Introduction

Throughout history, humanity has used composite materials to achieve combinations of properties that could not be achieved with individual materials. The Bible describes mixing of straw with clay to make tougher bricks. Concrete is a composite of cement paste, sand, and gravel. Today poured concrete is almost always reinforced with steel rods. Other examples of composites include steel-belted tires, asphalt blended with gravel for roads, plywood with alternating directions of fibers, and fiberglass-reinforced polyester used for furniture, boats and sporting goods. Composite materials offer combinations of properties otherwise unavailable.

The reinforcing material may be in the form of particles, fibers, or sheet laminates.

Fiber-reinforced composites

Fiber composites may be classified according to the nature of the matrix and the fiber. Examples of a number of possibilities are listed in Table 21.1.

Various geometric arrangements of the fibers are possible. In two-dimensional products, the fibers may be unidirectionally aligned, at 90° to one another in a woven fabric or cross-ply, or randomly oriented (Figure 21.1). The fibers may be very long or chopped into short segments. In thick objects short fibers may be random in three dimensions. The most common use of fiber reinforcement is to impart stiffness (increased modulus) or strength to a matrix. Toughness may also be of concern.

Elastic properties of fiber-reinforced composites. The simplest arrangement is long parallel fibers. The strain parallel to fibers must be the same in both the matrix and the fiber, εf = εm = ε.

Introduction

It was well known in the late 19th century that crystals deformed by slip. In the early 20th century the stresses required to cause slip were measured by tension tests of single crystals. Dislocations were not considered until after it was realized that the measured stresses were far lower than those calculated from a simple model of slip. In the mid-1930s G. I. Taylor, M. Polanyi, and E. Orowan independently postulated that preexisting crystal defects (dislocations) were responsible for the discrepancy between measured and calculated strengths. It took another two decades and the development of the electron microscope for dislocations to be observed directly.

Slip occurs by the motion of dislocations. Many aspects of the plastic behavior of crystalline materials can be explained by dislocations. Among these are how crystals can undergo slip, why visible slip lines appear on the surfaces of deformed crystals, why crystalline materials become harder after deformation, and how solute elements affect slip.

Theoretical strength of crystals

Once it was established that crystals deformed by slip on specific crystallographic systems, physicists tried to calculate the strength of crystals. However, the agreement between their calculated strengths and experimental measurements was very poor. The predicted strengths were orders of magnitude too high, as indicated in Table 9.1.

The basis for the theoretical calculations is illustrated in Figure 9.1. Each plane of atoms nestles in pockets formed by the plane below (Figure 9.1a).

Introduction

Elastic deformation is reversible. When a body deforms elastically under a load, it will revert to its original shape as soon as the load is removed. A rubber band is a familiar example. Most materials, however, can undergo very much less elastic deformation than rubber. In crystalline materials elastic strains are small, usually less than 1/2%. It is safe for most materials, other than rubber to assume that the amount of deformation is proportional to the stress. This assumption is the basis of the following treatment. Because elastic strains are small, it doesn't matter whether the relations are expressed in terms of engineering strains, e, or true strains, ε.

The treatment in this chapter will start with the elastic behavior of isotropic materials, the temperature dependence of elasticity, and thermal expansion. Then anisotropic elastic behavior and thermal expansion will be covered.

Isotropic elasticity

An isotropic material is one that has the same properties in all directions. If uniaxial tension is applied in the x-direction, the tensile strain is ex = σx/E, where E is Young's modulus. Uniaxial tension also causes lateral strains, ey = ez = −vex, where v is Poisson's ratio.