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 much less elastic deformation than rubber. In crystalline materials, elastic strain is small, usually less than ½%. 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 does not 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 uni-axial tension is applied in the x-direction, the tensile strain is εx = σx/E, where E is Young's modulus. Uniaxial tension also causes lateral strains, εy = εz = −νεx, where ν is Poisson's ratio.

Introduction

The treatment of fracture in Chapter 13 was descriptive and qualitative. In contrast, fracture mechanics provides a quantitative treatment of fracture. It allows measurements of the toughness of materials and provides a basis for predicting the loads that structures can withstand without failure. Fracture mechanics is useful in evaluating materials, in the design of structures, and in failure analysis.

Early calculations of strength for crystals predicted strengths far in excess of those measured experimentally. The development of modern fracture mechanics started when it was realized that strength calculations based on assuming perfect crystals were far too high because they ignored preexisting flaws. Griffith reasoned that a preexisting crack could propagate under stress only if the release of elastic energy exceeded the work required to form the new fracture surfaces. However, his theory based on energy release predicted fracture strengths that were much lower than those measured experimentally. Orowan realized that plastic work should be included in the term for the energy required to form a new fracture surface. With this correction, experiment and theory were finally brought into agreement. Irwin offered a new and entirely equivalent approach by concentrating on the stress states around the tip of a crack.

Theoretical Fracture Strength

Early estimates of the theoretical fracture strength of a crystal were be made by considering the stress required to separate two planes of atoms. Figure 14.1 shows schematically how the stress might vary with separation.

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 a 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.

Planes

Planes in a crystal are identified by their Miller indices. The system involves;

1. Writing the intercepts on the three axes;

2. Taking the reciprocals;

3. Reducing these to the lowest set of integers in the same ratio;

4. Enclosing in parentheses.

EXAMPLE PROBLEM AI.1: Write the Miller indices of the two planes shown in Figure AI.1.

Solution: For plane a:

1. The intercepts on the three axes are 1, ∞, and 1. Note that the y intercept is taken as ∞ because the plane does not intercept the y axis.

2. Reciprocals are 1, 0, and 1.

3. These are already reduced to lowest set of integers in same ratio.

4. The plane is (101).

For plane b:

1. The intercepts on the three axes are 1, −1, and 1/2. Note that the plane has to be extended out of the cubic element to intercept the y axis at −.

2. Reciprocals are 1, −1, and 2.

3. These are already reduced to lowest set of integers in same ratio.

4. The plane is (112). Note that the minus sign is indicated by an over bar and that no commas are used.

Directions

Directions are indicated by their components parallel to the three axes reduced to the lowest set of integers and enclosed in brackets, [].

Introduction

The strengths of metals are sensitive to microstructure. Most hardening mechanisms involve making dislocation motion more difficult. These 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 these are related to the effects of interstitial solutes on dislocations.

Crystal Structure

Crystal structure strongly affects hardness and yield strength. Fcc metals tend to have higher strengths than bcc metals and lower strengths than hcp metals. The effect of crystal structure is most apparent in metals that transform from one crystal structure to another. Figure 11.1 shows how the hardness of nearly pure iron changes with temperature. The general decrease of hardness with increasing temperature is interrupted by the α → γ transformation at 910°C. At 910°C, the fcc structure is harder than the bcc structure. The reason the bcc structure is softer is at least partially explained by its greater number of slip systems. Similarly, the hardness of titanium would be expected to undergo a sharp drop as it transforms from hcp because there are fewer slip systems in hcp crystals than bcc ones.

Introduction

Throughout history, mankind 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 also 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 degrees 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 the matrix. Toughness may also be of concern.

Introduction

Most crystals can deform by twinning. Twinning is particularly important in hcp metals because hcp metals do not have enough easily activated slip systems to produce an arbitrary shape change.

Mechanical twinning, like slip, occurs by shear. A twin is a region of a crystal in which the orientation of the lattice is a mirror image of that in the rest of the crystal. Normally, the boundary between the twin and the matrix lies in or near the mirror plane. Twins may form during recrystallization (annealing twins), but the concern here is formation of twins by uniform shearing (mechanical twinning), as illustrated in Figure 10.1. In this figure, plane 1 undergoes a shear displacement relative to plane 0 (the mirror plane). Then plane 2 undergoes the same shear relative to plane 1, and plane 3 relative to plane 2, and so on. The net effect of the shear between each successive plane is to reproduce the lattice, but with the new (mirror image) orientation.

Both slip and twinning are deformation mechanisms that involve shear displacements on specific crystallographic planes and in specific crystallographic directions. However, there are important differences.

1. With slip, the magnitude of the shear displacement on a plane is variable, but it is always an integral number of interatomic repeat distances, nb, where b is the Burgers vector. Slip occurs on only a few of the parallel planes separated by relatively large distances. With twinning, on the other hand, the shear displacement is a fraction of an interatomic repeat distance, and every atomic plane shears relative to its neighboring plane.

2. […]

Introduction

It was well know in the late nineteenth century that crystals deformed by slip. In the early twentieth 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 before dislocations were 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 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 poor. The predicted strengths were orders of magnitude too high, as indicated in Table 8.1.

Introduction

Plastic deformation of crystalline materials usually occurs by slip, which is the sliding of planes of atoms over one another (Figure 7.1). 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 8 and 9. 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 are small compared to most grain sizes, so slip usually can be considered as homogeneous on a macroscopic scale.

Introduction

In classic elasticity, there is no time delay between application of a force and the deformation that it causes. For many materials, however, there is additional timedependent deformation that is recoverable. This is called viscoelastic or anelastic deformation. When a load is applied to a material, there is an instantaneous elastic response, but the deformation also increases with time. This viscoelasticity should not be confused with creep (Chapter 16), which is time-dependent plastic deformation. Anelastic strains in metals and ceramics are usually so small that they are ignored. In many polymers, however, viscoelastic strains can be very significant.

Anelasticity is responsible for damping of vibrations. A high damping capacity is desirable where vibrations might interfere with the precision of instruments or machinery and for controlling unwanted noise. A low damping capacity is desirable in materials used for frequency standards, in bells, and in many musical instruments. Viscoelastic strains are often undesirable. They cause the sagging of wooden beams, denting of vinyl flooring by heavy furniture, and loss of dimensional stability in gauging equipment. The energy associated with damping is released as heat, which often causes an unwanted temperature increase. Study of damping peaks and how they are affected by processing has been useful in identifying mechanisms. The mathematical descriptions of viscoelasticity and damping are developed in the first part of this chapter. Then several damping mechanisms are described.

The term mechanical behavior encompasses the response of materials to external forces. This text considers a wide range of topics. These include mechanical testing to determine material properties; plasticity, which is needed for FEM analyses of automobile crashes; means of altering mechanical properties; and treatment of several modes of failure.

The two principal responses of materials to external forces are deformation and fracture. The deformation may be elastic, viscoelastic (time-dependent elastic deformation), or plastic and creep (time-dependent plastic deformation). Fracture may occur suddenly or after repeated applications of loads (fatigue). For some materials, failure is time dependent. Both deformation and fracture are sensitive to defects, temperature, and rate of loading.

Key to understanding these phenomena is a basic knowledge of the three-dimensional nature of stress and strain and common boundary conditions, which are covered in Chapter 1. Chapter 2 covers elasticity, including thermal expansion. Chapter 3 treats mechanical testing. Chapter 4 is focused on mathematical approximations to stress-strain behavior of metals, and how these approximations can be used to understand the effect of defects on strain distribution in the presence of defects. Yield criteria and flow rules are covered in Chapter 5. Their interplay is emphasized in problem solving. Chapter 6 treats temperature and strain rate effects and uses an Arrhenius approach to relate them. Defect analysis is used to understand both superplasticity and strain distribution.

Introduction

The shapes of most metallic products are achieved by mechanical working. The exceptions are those produced by casting and by powder processing. Mechanical shaping processes are conveniently divided into two groups, bulk forming and sheet forming. Bulk-forming processes include rolling, extrusion, rod and wire drawing, and forging. In these processes, the stresses that deform the material are largely compressive. One engineering concern is to ensure that the forming forces are not excessive. Another is ensuring that the deformation is as uniform as possible so as to minimize internal and residual stresses. Forming limits of the material are set by the ductility of the work piece and by the imposed stress state.

Products as diverse as cartridge cases, beverage cans, automobile bodies, and canoe hulls are formed from flat sheet by drawing or stamping. In sheet forming, the stresses are usually tensile, and the forming limits usually correspond to local necking of the material. If the stresses become compressive, buckling or wrinkling will limit the process.

Bulk-Forming Energy Balance

An energy balance is a simple way of estimating the forces required in many bulkforming processes. As a rod or wire is drawn through a die, the total work, Wt, equals the drawing force, Fd, times the length of wire drawn, ΔL, Wt = FdΔL.

Introduction

A separate chapter is devoted to polymers because of their engineering importance and because their mechanical behavior is so different from that of metals and ceramics. The mechanical response of polymers is far more time dependent than that of crystalline materials. Viscoelastic effects (Chapter 15) are much more important in polymers than in metals or ceramics. The properties of polymers are also much more sensitive to temperature than those of other materials. Changes of molecular orientation with deformation cause large changes in properties and a much greater degree of anisotropy than is observed in metals or ceramics. The phenomena of crazing and rubber elasticity have no analogs in crystalline materials. Some polymers exhibit very large tensile elongations. Although a few alloys exhibit shape-memory behavior, the effect is much greater in polymers, more common, and of greater technological importance.

Elastic Behavior

Elastic strains in metals and ceramics occur by stretching of primary metallic, covalent, or ionic bonds. The elastic modulus of most crystals varies with direction by less than a factor of 3. The effects of alloying, thermal, and mechanical treatments on the elastic moduli of crystals are relatively small. As the temperature is increased from absolute zero to the melting point, Young's modulus usually decreases by a factor of no more than 5. For polymers, however, a temperature change of 30°C may change the elastic modulus by a factor of 1, 000. Elastic deformation of polymeric involves stretching of the weak van der Waals bonds between neighboring molecular chains and rotation of covalent bonds.

Stick-Slip Phenomena

Stick-slip phenomena involve intervals of motion separated by periods of rest. The model in Figure 12.1 illustrates the basic elements: a constantly moving driver (A), a spring (B), and a block (C) in frictional contact with a surface (D). If the static coefficient of friction between the block and the surface is greater than the coefficient of friction when the block is sliding, the motion of the block will be sporadic.

As the driver moves at a constant rate, the spring will elongate causing the force to increase. Once the force is great enough to overcome the static friction, the block will slide and will continue sliding until the spring shortens enough so that the force will drop below that needed to overcome sliding friction. At that point, the block will remain motionless until the force rises high again to overcome static friction. This is illustrated in Figure 12.2.

If plastic deformation behaves in this manner, the result is a serrated stress-strain curve. There are several possible causes of this phenomenon. One cause is dynamic strain aging, which results from the attraction of atoms in solid solution to dislocations. Because this attraction lowers the energy of the system, the solute atoms pin the dislocations. A larger force is required to break the dislocations free from the atoms pinning them than is required for them to continue moving. If the temperature is sufficiently high and the rate of deformation sufficiently slow, the solute atoms may diffuse to the new location of the dislocation and pin it again. This will result in a serrated stress–strain curve (Figure 12.3).

Introduction

Once the concept of dislocations was accepted, there were three important questions to be answered. First, when a single crystal is deformed, slip occurs with shear offsets of thousands of atom distances on relatively widely spaced planes, rather than uniformly throughout the crystal. See Figure 9.1. Why does slip not occur uniformly at an atomic scale?

Second, cold working increases the dislocation content of crystals even though dislocations must run out of the crystals. See Figures 9.2 and 9.3. Where do the additional dislocations come from?

Third, the yield stress increases as the number of dislocations increases even though without any dislocations, the strength would be even higher. See Figure 9.4. Why does the yield stress increase with dislocation density?

Frank-Read Sources

The first two questions can be answered in terms of the Frank-Read source, which generates dislocations. Suppose that there is a finite length of a dislocation, AB, in a slip plane (Figure 9.5). The dislocation leaves the plane at A and B, but the end points are pinned at A and B. A shear stress, τ, acting on the plane, will create a force that causes dislocation to bow. This bowing is resisted by the line tension of the dislocation. As the shear stress is increased, the dislocation will bow out until it spirals back on itself. The sections that touch annihilate each other, leaving a dislocation loop that can expand under the stress and a restored dislocation segment between the pinning points. The process can repeat itself, producing many loops.