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.

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 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), plastic, or creep (time-dependent plastic deformation). Fracture may occur suddenly or after repeated application of loads (fatigue). For some materials, failure is time-dependent. Both deformation and fracture are sensitive to defects, temperature, and rate of loading.

The 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 the first chapter. Chapter 2 covers elasticity, including thermal expansion. Chapters 3 and 4 treat mechanical testing. Chapter 5 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 6. Their interplay is emphasized in problem solving. Chapter 7 treats temperature and strain-rate effects and uses an Arrhenius approach to relate them. Defect analysis is used to understand superplasticity as well as strain distribution.

Chapter 8 is devoted to the role of slip as a deformation mechanism. The tensor nature of stresses and strains is used to generalize Schmid's law.

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 to 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 11.1. In this figure, plane 1 undergoes 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, etc. The net effect of the shear between each successive pair of planes 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.

2. […]

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 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

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 to ensure that the deformation is as uniform as possible, in order 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 sheets 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 bulk-forming 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, and of 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 1000.

Introduction

In classic elasticity there is no time delay between the application of a force and the deformation that it causes. For many materials, however, there is additional time-dependent 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 the 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 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 will be developed in the first part of this chapter. Then several damping mechanisms will be described.