Introduction

Throughout history, there has been a never-ending effort to develop materials with higher yield strengths. However, a higher yield strengths 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, thus 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

With elastic deformation, the strains are proportional to the stress, so every level of stress causes some elastic deformation. However, 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 e 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 predicting energy absorption in automobile crashes, designing of dies for stamping parts, and analyzing 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

This book is concerned with the mechanical behavior of materials. The term mechanical behavior refers to the response of materials to forces. Under load, a material may either deform or break. The factors that govern a material's resistance to deforming are quite different than those governing its resistance to fracture. The word strength may refer either to the stress required to deform a material or to the stress required to cause fracture; therefore, care must be used with the term strength.

When a material deforms under small stresses, the deformation may be elastic. In this case, when the stress is removed, the material will revert to its original shape. Most of the elastic deformation will recover immediately. There may be, however, some time-dependent shape recovery. This time-dependent elastic behavior is called anelasticity or viscoelasticity.

Larger stresses may cause plastic deformation. After a material undergoes plastic deformation, it will not revert to its original shape when the stress is removed. Usually, high resistance to deformation is desirable so that a part will maintain its shape in service when stressed. However, it is desirable to have materials deform easily when forming them by rolling, extrusion, and so on. Plastic deformation usually occurs as soon as the stress is applied. At high temperatures, however, time-dependent plastic deformation called creep may occur.

Fracture is the breaking of a material into two or more pieces. If fracture occurs before much plastic deformation occurs, we say the material is brittle. In contrast, if there has been extensive plastic deformation preceding fracture, the material is considered ductile.

Introduction

Tensile properties are used in selecting materials for different applications. Material specifications often include minimum tensile properties to ensure quality so tests must be made to guarantee that materials meet these specifications. Tensile properties are also used in research and development to compare new materials or processes. With plasticity theory (Chapter 5), tensile data 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 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 measures are used, with appropriate caution, in engineering design. A material's ductility may also be of interest. Ductility describes 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.

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. The gauge section should be long compared to the diameter (typically, four times).

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. […]