Mechanical engineering | Materials expertise » J. W. Morris - A Survey of Materials Science IV., Mechanical Properties

Source: http://www.doksinet A Survey of Materials Science IV. Mechanical Properties J. W Morris, Jr Department of Materials Science and Engineering University of California, Berkeley Fall, 2008 Source: http://www.doksinet Materials Science Fall, 2008 PART IV: MECHANICAL PROPERTIES 655 CHAPTER 19: MECHANICAL BEHAVIOR 658 19.1 Introduction 658 19.2 Uniaxial Tension 19.21 The load-displacement curve 19.22 The engineering stress-strain curve 19.23 True stress and true strain 19.24 Varieties of tensile behavior 659 659 660 665 668 19.3 Creep 19.31 Creep deformation 19.32 The steady-state creep rate 19.33 Superplasticity 19.34 Stress rupture 19.35 Stress relaxation 669 669 670 671 671 671 19.4 Fracture 19.41 Fracture toughness 19.42 Plane strain fracture toughness 19.43 Sources of critical cracks 672 672 673 673 19.5 Fatigue 19.51 The s-n curve 19.52 The fatigue crack growth rate 674 674 675 19.5 Stress Corrosion Cracking 676 19.6 Hardness 678 CHAPTER 20: ELASTIC

PROPERTIES 681 20.1 Introduction 681 20.2 Elastic Properties of Isotropic Materials 20.21 The basic load geometries 20.22 Uniaxial tension; Youngs modulus and Poissons ratio 20.23 Hydrostatic compression; the bulk modulus 20.24 Shear stress and the shear modulus 682 682 683 685 686 20.3 Physical source of the elastic moduli 20.31 The bulk modulus 20.32 The shear modulus 689 689 691 Page 653 Source: http://www.doksinet Materials Science Fall, 2008 20.4 Specific Modulus; Stiffness 692 20.5 Engineering the Elastic Modulus 20.51 Composition 20.52 Microstructure 20.53 Composite materials 694 694 694 694 CHAPTER 21: PLASTIC DEFORMATION 698 21.1 Introduction 21.11 Engineering significance of plastic deformation 21.12 Mechanisms of plastic deformation 698 698 699 21.2 The Yield Strength 21.21 Deformation by dislocation motion 21.22 The critical resolved shear stress 21.23 The tensile yield strength of a single crystal 21.24 The yield strength of a polycrystal 700 700 702

704 705 21.3 Microstructural Control of the Yield Strength 21.31 Structural hardening; the Peierls-Nabarro stress 21.32 Grain refinement 21.33 Obstacle hardening 21.34 Solution hardening 21.35 Dislocation hardening 21.36 Precipitation hardening 707 707 709 710 712 714 717 21.4 The Influence of Temperature and Strain Rate 21.41 The variation of yield strength with temperature 21.42 The influence of strain rate on strength 719 719 721 21.5 Work Hardening 721 21.6 Plastic Instability, Necking and Failure 21.61 The Considere criterion 21.62 Tensile elongation 724 724 726 CHAPTER 22: FRACTURE 729 22.1 Introduction 22.11 Engineering importance 22.12 Why things fall apart 729 729 730 22.2 Fracture Mechanics 22.21 Crack propagation from a pre-existing flaw 22.22 The fracture toughness 22.23 Fracture-sensitive design 731 731 733 736 Page 654 Source: http://www.doksinet Materials Science Fall, 2008 22.3 Microstructural Control of Fracture Toughness 22.31 The fracture mode

22.32 Choice of fracture mode; the ductile-brittle transition 22.33 Suppressing brittle fracture 22.34 Raising the toughness in the ductile mode 22.35 Raising the fracture stress in the brittle mode 738 739 741 743 745 746 22.4 Fatigue 22.41 Fatigue crack growth 22.42 Nucleation-limited fatigue 22.43 Fatigue-safe design 747 748 752 753 Part IV: Mechanical Properties It was Eudoxas and Archytas who were the originators of the now celebrated and highly prized art of mechanics. They used it with great ingenuity to illustrate geometrical theorems, and to support by means of mechanical demonstrations easily grasped by the senses propositions which are too intricate for proof by word or diagram . Plato was indignant at these developments, and attacked both men for having corrupted and destroyed the ideal purity of geometry. He complained that they had caused her to forsake the realm of disembodied and abstract thought for that of material objects, and to employ instruments which required

much base and manual labor . - Plutarch, The Life of Marcellus The final class of material properties we shall consider are the mechanical properties that govern the behavior of materials that are subject to mechanical loads. Almost all engineering systems experience mechanical loads. The materials used in the individual parts of the system must support those loads without deforming so that they lose their fit or breaking so that they cease to function at all. Moreover, the manufacturing operations that make engineering parts are most frequently mechanical processes such as machining, forging, or mechanical forming. The mechanical properties of the materials that make these parts must be compatible with the manufacturing procedures that are to be used. Several qualitatively different kinds of mechanical behavior are pertinent to engineering design, including the following types. (1) Elasticity. A material that experiences a relatively small load deforms elastically, that is, it

undergoes a shape change that is, ordinarily, proportional to the load, and disappears when the load is removed. Elastic deformation must be taken into account in engineering design since it changes the shapes of parts and affects their fit to one another. Elastic deformation may even cause a mechanical instability in which the part buckles or Page 655 Source: http://www.doksinet Materials Science Fall, 2008 breaks. Examples of elastic instability include buckling of the columns that support a building or denting the fender of an automobile. (2) Yielding and plasticity. Under higher loads the material deforms plastically, that is, it undergoes a shape change that is retained when the load is removed. At normal temperatures, plastic deformation begins when the applied load exceeds the yield strength of the material. If the load on a material rises beyond its yield strength, its plastic deformation increases monotonically with the load until it finally breaks at the ultimate

tensile strength. Plastic deformation cannot usually be tolerated in an operating device, since it alters the geometry of the parts. Hence most devices must be designed so that the mechanical loads they support do not reach the yield strength. On the other hand, plastic deformation does provide a margin of safety that acts against catastrophic failure at loads near the yield strength. Consequently, the materials used in most structures are required to have a certain degree of plastic deformation before failure. On the other hand, plastic deformation is the usual method by which parts are formed into shape. The materials that are used in these parts must be capable of significant plastic deformation without failure. (3) Creep. A material that is subjected to a static load deforms at a finite rate even when the magnitude of the load is below the yield strength. This phenomenon is known as creep. The creep rate is sensitive to the temperature as well as the load It is very small when the

temperature is below about half the melting point. Creep deformation is a potential problem when a material is used at high temperature, for example, in the hot section of a gas turbine. Creep can alter the geometry of the part, and will eventually lead to failure by a fracture mechanism that is known as creep rupture. (4) Premature fracture. When a part fractures prematurely, at loads well below its ultimate tensile strength, this is usually because it contains a flaw that concentrates the load so that it exceeds the local strength of the material. The flaw generates a crack that propagates to failure. The fracture toughness of a material is a measure of its ability to tolerate internal flaws. A high toughness provides a margin of safety against premature failure. (5) Fatigue. A well-made part is ordinarily safe from premature fracture because it contains no significant flaws. However, if the part is subject to a cyclic load, it may spontaneously develop a crack through a mechanism

that is known as fatigue, even if the maximum load in the cycle is below the yield strength Cyclic loads cause cracks to grow at an accelerating rate until they reach critical size and propagate to failure. Fatigue failures are a disturbingly common source of the premature failure of engineering devices. Fatigue must be taken into account in the design of any structurally critical part that is subject to a cyclic load. (6) Stress corrosion. A second mechanism that can introduce flaws that lead to fracture is stress corrosion cracking. When a part is loaded in a corrosive environment, it may corrode in a pattern that forms sharp-tipped cracks that propagate into the material. When these grow to critical size, the part breaks. Susceptibility to stress corrosion cracking Page 656 Source: http://www.doksinet Materials Science Fall, 2008 is an important consideration in the design of any part that is to be used in a corrosive environment. (7) Friction and wear. When materials that are

in contact move with respect to one another they generate frictional forces that oppose their relative motion. The magnitude of the friction depends on the normal load that forces the parts together, and also depends on the nature of the materials, including whatever lubricant may be used at the interface. Where there is friction, there is inevitable wear; material is mechanically removed from one or both surfaces. Friction is a problem because it consumes power, and also because it generates heat that may soften or weaken the parts. Wear distorts the geometry of the parts, and eventually leads to failure. Friction and wear are particular problems in the operation of gears and bearings, and in the forming of parts with mechanical dies. We shall discuss each of these mechanical phenomena below. However, to put them into context, it is useful to consider a simple mechanical test, the uniaxial tensile test, in which most of these phenomena are revealed. Page 657 Source:

http://www.doksinet Materials Science Fall, 2008 Chapter 19: Mechanical Behavior The Burgundian-French rivalry promptly precipitated an arms race of the sort so familiar to us today. Metalworkers on both sides aimed at a single goal: to make guns mobile without sacrificing their battering power. Between 1465 and 1477, designers solved their problem brilliantly by resorting to smaller, denser projectiles. They discovered that a comparatively small iron cannonball could strike a more damaging blow than a stone projectile (which fractured on impact) ever could, no matter how large. This meant that guns could be made smaller, but had to be stronger, too . As it happened the Burgundians, though they had taken the lead in mobile siege artillery, were not destined to profit from it. Instead, Charles the Bold, Duke of Burgundy, being too impatient to await the arrival of his guns on the battlefield at Nancy, led a cavalry charge against a massed formation of Swiss pikemen . and met his

death on the point of a pike Burgundian lands were then swiftly partitioned. . If a single ruler had continued to control that region and been able to monopolize the new guns for as along as a generation, he might have been able to use his mobile wall destroyers to achieve hegemony wherever the new siege guns could reach. Instead, the kings of France and the Hapsburgs of Germany divided up the Burgundian guns and gunmakers . assuring a standoff, which became permanent, thanks to the swift improvement in fortifications after 1500. . For Japan as well as the other empires of Asia, the effect of gunpowder weapons was to consolidate central power over comparatively vast territories. Once a ruler had monopolized big guns or, in Japan, disarmed the whole nation, there was no incentive to continue experiments with improvements [while] among the rival states of Europe such experimentation continued, and at a very rapid pace . The amazing imperial success of European nations in the eighteenth

and nineteenth centuries resulted - and all because in 1477 the Burgundian lands were divided between French and German rulers . Truly the course of history, even on a world scale sometimes turns on single events, and on a skilled (or lucky) response in a moment of crisis. - William McNeill, The Age of Gunpowder Empires, 1450-1800 19.1 INTRODUCTION Let a material be made into a round bar and put in a device that grips its two ends and pulls on them, as illustrated in Fig. 191 This is the normal geometry of a tensile test , which is the test used to generate much of the mechanical property data that is used in engineering design. The tensile testing machine can control either the load on the specimen, in which case the test is said to be stress-controlled, or the specimen length, in which case the test is Page 658 Source: http://www.doksinet Materials Science Fall, 2008 strain-controlled. Most commonly, the tensile test is done in strain control; the specimen is gradually

elongated and the load recorded until the specimen fails. Other mechanical properties are measured by modifications of the tensile test For example, creep properties are tested by fixing the load and measuring the elongation as a function of time, or by fixing the length and recording the load decay with time (a creep phenomenon known as stress relaxation). The fracture toughness is measured by replacing the round bar with a plate that is cracked on one side, and measuring the load necessary to propagate the crack to failure. Fatigue resistance is measured by setting the machine to cycle the load in a programmed pattern, and measuring the number of cycles required to fail the specimen, or, in a more detailed test, measuring the rate of growth of a pre-existing crack. Stress corrosion resistance is studied by immersing the specimen in a corrosive medium, usually under fixed load, and measuring the rate of growth of a pre-existing crack. P L A P . Fig. 191: A cylindrical sample of

length, L, and cross-sectional area, A, in uniaxial tension. A final variation measures the hardness of a material. In a hardness test one of the specimen grips is replaced by a flat support, and the other by a small indenter that is impressed into the material by loading it in compression. The size or depth of the indentation made by a standard indenter under a standard load measures the hardness, which is the property that controls the wear resistance of the material in most situations. 19.2 UNIAXIAL TENSION 19.21 The load-displacement curve Let a round bar specimen be placed in a tensile machine and pulled under strain control, so that the elongation is gradually increased and the load recorded. The relation between the applied tensile load (P) and the change in length of the bar (ÎL) defines the load-displacement curve for the sample. (In an actual test the specimen must be gripped at its ends. To avoid end effects the specimen is shaped at its ends and the length is measured

Page 659 Source: http://www.doksinet Materials Science Fall, 2008 between reference marks on the specimen well away from the grips, as shown in Fig. 19.2) P P L ÎL . Fig. 192: A typical form for the load-deflection curve of a ductile metal tested in tension. While the form of the load-displacement curve is characteristic of the material, the quantitative relation between the load and the displacement depends on the geometry of the specimen as well. As you can easily verify by stretching rubber bands or bending paper clips, the load required to achieve a given displacement increases with the cross-sectional area of the bar (a thin rubber band is more easily stretched than a thick one) while the displacement under a given load increases with the length of the specimen (a long rubber band is more easily stretched by a given length than a short one). These phenomena reflect the fact that the total elongation of the specimen is the sum of the elongations of each crosssectional

element along its length, while the total load is the sum of the loads carried by each element across a section through it. 19.22 The engineering stress-strain curve Engineering stress and strain To account for the specimen geometry in a simple way, we measure the load by the engineering stress, s, P s=A 19.1 0 where P is the load, A0 is the cross-sectional area of the specimen, and the stress is force per unit area. We measure the displacement by the engineering strain, e, ÎL e= L 0 19.2 Page 660 Source: http://www.doksinet Materials Science Fall, 2008 where ÎL is the elongation, L0 is the specimen length, and the strain is dimensionless. The variation of s with e is called the engineering stress-strain curve. It is simply the load-displacement curve corrected for the dimensions of the specimen The engineering stress and strain are only approximate measures of the local loads and displacements within the material. As the specimen elongates, its cross-sectional area

decreases, so the average stress across its section is always higher than the engineering stress suggests. Moreover, the local stress (load per unit area) varies through the crosssection; the material in the center of the specimen supports a higher load per unit area than that near the periphery. Nonetheless, the engineering stress-strain curve is a sufficiently good representation of the mechanical properties of the material that it is the one tabulated in most compilations of mechanical properties, and provides the mechanical property data that is used in all but the most sophisticated engineering designs. su sy s I II III e . Fig. 193: The engineering stress-strain curve of a typical ductile metal. The engineering stress-strain curve of a typical polygranular metal is illustrated in Fig. 193 The curve can be divided into three regions that represent qualitatively different kinds of mechanical behavior. The three regions are denoted by roman numerals in the figure Elastic

deformation Region I defines the range of elastic deformation. Elastic deformation ordinarily has two characteristics. (1) Elastic deformation is recoverable. If the stress is removed from the specimen, the strain vanishes; the specimen returns to its original length. (2) In most materials, the elastic stress is proportional to the strain, s = Ee 19.3 Page 661 Source: http://www.doksinet Materials Science Fall, 2008 where the constant of proportionality, E, is called Youngs modulus. Equation 193 is known as Hookes Law, after its discoverer. Materials that obey Hookes Law are called linear elastic materials. Most structural materials are members of this class, including almost all metals and ceramics Stable plastic deformation Region II defines the range of stable plastic deformation. Stable plastic deformation ordinarily has three characteristics. s ep . Fig. 194: ee e Relaxation of a specimen strained slightly into the plastic region. ep is the plastic strain, ee is the

elastic strain (1) A portion of the strain is plastic. If the stress is decreased after the specimen is loaded into the plastic range, the specimen relaxes along a line that is nearly parallel to the linear elastic curve, as illustrated in Fig. 194 The specimen does not return to its original length after the stress is removed. The remnant strain is called the plastic strain, and measures the permanent deformation of the specimen The total strain of a specimen that is loaded into region II is the sum of its elastic strain (ee) and its plastic strain (ep). (2) The stress-strain curve is non-linear. The slope of the stress-strain curve, ds/de, drops below the modulus, E, as the sample begins to deform plastically, and ordinarily decreases with the plastic strain as shown in Fig. 193 However, the slope remains positive in region II. The engineering stress must be raised to increase plastic deformation This behavior is known as work hardening. (3) The plastic deformation is uniformly

distributed along the length of the specimen. This behavior is known as uniform elongation The specimen increases in length, but retains its shape, to within a uniform contraction. Page 662 Source: http://www.doksinet Materials Science Fall, 2008 The yield strength The stress at which plastic deformation begins is called the yield strength of the material, and denoted sy. The yield strength is the boundary between the elastic (I) and plastic (II) ranges of the stress-strain curve. In some materials, the yield strength is welldefined by a yield point at which there is an obvious discontinuity in the stress-strain curve An example is illustrated in Fig. 195a However, most materials do not show a well-defined yield point and there is some uncertainty in the location of the yield strength To remove this uncertainty, it is conventional to define the yield strength as the stress that corresponds to a plastic strain of 02% To measure the yield strength according to this criterion, a

line is drawn parallel to the elastic portion of the stress-strain curve so that it intersects the strain axis at 0.2%, as illustrated in Fig 195b The stress at which this line crosses the stress-strain curve defines the 0.2% offset yield, which is the value of the yield strength that is tabulated in most compilations of mechanical properties. sy s s 0.2% e (a) . Fig. 195: e (b) (a) The stress-strain curve of a material with a well-defined yield point. (b) Illustration of the 02% offset criterion for the yield strength (the elastic region has been exaggerated for clarity). Note that the conventional definition of the yield strength is such that there is some measurable plastic deformation at stresses below yield. Engineers who design parts with very tight tolerances must, hence, make sure that operating stresses are kept well below the tabulated yield strength. If the part is a critical one, they should re-measure the yield strength against the criterion that is pertinent to

the particular application. (On a number of occasions I have been contacted by a puzzled engineer who designed some metal part to operate at stresses below the tabulated yield strength, and was shocked to discover that it nonetheless experienced a measurable plastic deformation. He usually wants to know why the published data is wrong, not why he is.) Page 663 Source: http://www.doksinet Materials Science Fall, 2008 Unstable plastic deformation The engineering stress-strain curve of a ductile material usually passes through a maximum, and enters the region marked III in Fig. 193 In region III the stress decreases monotonically with the strain until the specimen finally fractures. The behavior in region III is due to plastic instability. Rather than being uniformly distributed along the specimen, the plastic deformation concentrates at a particular point, which thins into a neck of the type illustrated in Fig. 196 As the sample necks down, its cross-sectional area decreases. The

actual stress on the material within the neck is P/A, where A is its true cross-sectional area. As A decreases, the stress in the necked region increases, even though the engineering stress, P/A0, decreases The increase in stress causes the sample to neck further, which causes the local stress to rise until the sample finally breaks. P P L P . Fig. 196: L P Necking of a tensile specimen due to unstable plastic deformation. The ultimate tensile strength The maximum value of the engineering stress is called the ultimate tensile strength, and denoted su. When the test is done in strain control, su defines the boundary between stable (region II) and unstable (region III) plastic deformation. It is the value of the engineering stress at which plastic deformation first becomes unstable and forms a neck If the test is done in stress control the ultimate tensile strength defines the highest load a sample with a given initial cross-section can support without breaking. The sample

fractures when the ultimate tensile strength is reached. Hence su is a critical parameter in engineering design The uniform and total elongation In designing an engineering part it is often important to know how much the part can be elongated before it fractures in tension. If the material is a ductile one with a stressstrain curve like that shown in Fig 193, the answer depends on how the test is controlled If the load is controlled, the sample fractures when the ultimate tensile strength is reached. The plastic strain that corresponds to the ultimate tensile strength is called the uniform elon- Page 664 Source: http://www.doksinet Materials Science Fall, 2008 gation, eu, since the plastic strain prior to the ultimate tensile strength is uniformly distributed along the specimen. If the strain is controlled, the sample necks before fracture, and has an additional post-necking elongation. The total strain up to fracture at the neck is called the total elongation, eT Since the

post-necking elongation is non-uniform and concentrated in the region of the neck, the total elongation depends on the length of the specimen (Fig. 196) The value of the total elongation that is usually tabulated is obtained by measuring the engineering strain in a 2 in. section of the specimen (2 in gage length) that includes the neck If some other gage length is used, it must be specified if the value of total elongation is to be meaningful. Reduction in area The final tensile property that is often measured and tabulated is the reduction in area prior to fracture at the thinnest part of the neck: RA = A0 - A A0 19.4 As we shall see below, the reduction in area is a measure of the true strain at fracture, and is, hence, a measure of the useful ductility of the material. It is the tensile property that correlates most closely to the fracture toughness of the material s eT eu e . Fig. 197: The uniform and total elongation in a tensile test. 19.23 True stress and true strain The

engineering stress and strain generate useful numbers for mechanical design. However, except in the limit of small strain, they are not very accurate measures of the stress and strain actually borne by the material. The reason is that the sample thins as it Page 665 Source: http://www.doksinet Materials Science Fall, 2008 elongates and, being an inanimate object, has no reliable memory of what its original crosssectional area was. Definition of the true stress and true strain The actual value of the average stress on a cross-section of area, A, is called the true stress, ß, and is P ß=A 19.5 A 0  ß = s  A 19.6 Since the true stress exceeds the engineering stress by a factor that increases as the sample become longer and thinner. To find a measure of the true strain we use the fact that an infinitesimal increase in the stress on a uniform sample of length, L, produces an infinitesimal strain that is given by dL d‰ = L 19.7 If the stress is increased

so that the length increases from L0 to Lf, the true strain is Lf L L0 L0 f  Lf  dL ⌠ ⌠   ⌡ ⌡ ‰= d‰ = = ln L0 L 19.8 When the strain is uniformly distributed along the length of the sample, as it is for the elastic and stable plastic regions of the stress-strain curve, eq. 198 gives the true strain that is associated with an increase in the overall length from L0 to Lf. However, when the strain is non-uniform, as it is when the sample necks, eq. 198 can only be used by dividing the samples into length increments that are so small that the strain can be assumed uniform within each of them, and applying eq. 198 separately to each One can do this, in practice, by inscribing a fine grid on the surface of the specimen and separately calculating the strain for each element of the grid by measuring the displacement of the grid lines. However, there is an easier way to determine the local value of the true strain in a tensile specimen. It is based on the

fact that plastic strain conserves volume As we shall see later, the elastic strain does not ordinarily conserve volume. But if the total strain is primarily plastic, as it ordinarily is by the time the sample begins to neck, then an element of Page 666 Source: http://www.doksinet Materials Science Fall, 2008 the sample that has length, L, and cross-sectional area, A, has a constant value of the product, V = AL. Then dL dA d‰ = L = A 19.9 and A 0  ‰ = ln A  19.10 f where Af is the final value of the cross-sectional area of a specimen whose original crosssection was A0. Eq 1910 can be used to find the non-uniform strain in a round bar tensile specimen by simply measuring the cross-sectional area along its length. Relation to the engineering stress and strain From the definition of the engineering strain (eq. 192), the relation between the engineering strain and the true strain is e= Lf - L0 L0 = A0 - Af = e‰ - 1 A0 ‰ = ln(1+e) 19.11 19.12 Using

eq. 196, the relation between the engineering stress and the true stress is  Af  s = ßA  = ße-‰ 19.20 ß = s(1+e) 19.22 0 In the limit of small strain (‰ << 1), ‰ « e and ß « s. Most structural materials yield at small values of the elastic strain. Hence we can usually ignore the difference between the engineering strain and the true strain when the plastic strain is small. The true stress-strain curve The relation between the true stress and true strain in tension is called the true stress-strain curve. The true stress-strain curve of a typical polygranular metal has a form like that illustrated in Fig. 198 It is identical to the engineering stress-strain curve for small strains, but differs qualitatively when the strain is large. In particular, there is no obvious feature of the true stress-strain curve that reveals the value of the ultimate tensile strength or the onset of plastic instability. This is because there is no change in the

mechanical behavior of the material at plastic instability Page 667 Source: http://www.doksinet Materials Science Fall, 2008 As we shall discuss in more detail later, the plastic instability that determines the ultimate tensile strength happens because the rate of increase of stress with strain (the work hardening rate, dß/d‰) becomes insufficient to preserve the geometry of the specimen. For a cylindrical tensile specimen, this happens when dß d‰ = ß 19.23 As the specimen deforms, the stress, ß, rises while the work hardening rate, dß/d‰, drops. The slope of the engineering stress-strain curve is ds d(ße-‰ ) = de d(e‰ -1) = dß  e-‰ dß - ße-‰ d‰  -2‰  = e ß  d‰  e‰ d‰ 19.24 which vanishes when eq. 1923 is satisfied Eq 1923 is called the Considere criterion ßf ßy ß ‰ . Fig. 198: ‰f The true stress-strain curve of a typical ductile metal. The true stress and strain at fracture are indicated. While the true

stress-strain curve conceals the ultimate tensile strength, it accurately records the work hardening behavior of the material, correctly shows that hardening continues after the deformation has become unstable (in contrast to the engineering stress-strain curve, which implies that the material softens after instability) and gives the values of the stress and strain across the section at fracture. The design engineer is usually interested in the engineering stress-strain curve, since the ultimate tensile strength is an important consideration in design. The materials scientist is usually more interested in the true stressstrain curve, since it is a more valid representation of the mechanical behavior of the material 19.24 Varieties of tensile behavior Page 668 Source: http://www.doksinet Materials Science Fall, 2008 The engineering stress-strain curve of a structural material usually has one of four characteristic shapes. The stress-strain curves of polygranular metals and

structural plastics usually resemble that shown in Fig. 192, which we have already discussed in some detail Several important metals and alloys, including carbon steel, yield at a well-defined yield point, and, hence, have stress-strain curves that resemble the one shown in Fig. 195a Brittle materials, including most structural ceramics, have stress-strain curves like that shown in Fig. 199a; they do not yield (at least not to noticeable degree) but simply extend elastically until they fracture. Elastomeric materials are capable of large elastic strains, and have non-linear elastic regions like that shown in Fig. 199b A pure elastomer, such a vulcanized rubber, does not yield unless it is strained very slowly; it deforms elastically until fracture. Note that the "ultimate tensile strength" of a brittle or elastomeric material is not due to plastic instability, but to simple fracture. As we shall see, the stress at which an elastic material fractures is determined by the

nature of the flaws the material contains rather than by its inherent mechanical properties. For this reason, the ultimate tensile strength of a brittle or elastomeric material is not a well-defined material property. It varies significantly from sample to sample. While the "ultimate tensile strengths" of such materials are often measured and tabulated, they do not constitute reliable design values. They give only a rough indication of the level of tensile strength one might expect. su su s s e e (a) . Fig. 199: (b) (a) Stress-strain curve of a brittle solid. (b) Stress-strain curve of an elastomer. In both cases the strain is elastic to fracture at s u. 19.3 CREEP 19.31 Creep deformation Consider a tensile specimen that is tested under stress control, and suppose that, instead of elongating it to plasticity and failure, it is loaded to a stress below the yield strength, and held at that stress for a long time. The specimen gradually increases in length This

phenomenon is called creep. The creep rate (the strain rate at constant load) is, ordinarily, almost vanishingly small unless the temperature of the specimen is greater than Page 669 Source: http://www.doksinet Materials Science Fall, 2008 about half of its melting point. However, when the temperature is a significant fraction of the melting point the creep rate becomes appreciable, and leads to significant plastic deformation in a relatively short time. The normal behavior of a specimen that is tested in creep is illustrated in Fig. 1910, where we have plotted the strain as a function of time at constant stress. Creep occurs in three stages. Stage I is primary creep During primary creep the strain rate decreases monotonically with time. Stage II is secondary or steady-state creep This stage, which may last for some time, is characterized by a constant creep rate. Eventually, the sample enters tertiary creep. The creep increases monotonically, often because of progressive void

formation within the specimen, until the sample finally ruptures. ‰ I II III t Fig. 1910: The variation of strain with time in normal creep, showing I: primary creep, II: steady-state creep, III: tertiary creep to rupture. 19.32 The steady-state creep rate The property that is most commonly used to characterize creep for design purposes is the steady-state strain rate, the strain rate in the steady-state creep. The steady-state strain rate governs the creep rate at the moderate strains that usually have the greatest engineering significance. The steady state strain rate, •‰, increases with the applied stress and the temperature Most high-temperature creep data can be fit to relations of the form - Q  •‰ = Aßnexp kT  19.25 The stress exponent, n, has a value that indicates the dominant mechanism of creep. For metals at moderate to high stress, the dominant mechanism is the climb-assisted motion of dislocations through the bulk, and n falls in the

range 4-7, depending on the material. At very low stresses both metals and ceramics creep by diffusional processes with n = 1. The activation energy, Q, is ordinarily the activation energy for the dominant diffusion step in creep. Page 670 Source: http://www.doksinet Materials Science Fall, 2008 19.33 Superplasticity Some materials adopt a creep mechanism at intermediate stresses that produces very high tensile elongation prior to failure. Such materials are said to be superplastic The total strain of a superplastic material may exceed 1000% before failure, in contrast to the total elongations of 30-40% that apply to exceptionally ductile materials in a normal tensile test. Superplastic creep is characterized by a steady state strain rate that obeys eq. 1925 with a stress exponent, n « 2, and an activation energy, Q, near that for grain boundary diffusion. Superplastic creep occurs in metals that are fine-grained, and maintain small grain size during deformation. Their

deformation is dominated by grain boundary sliding Individual grains are displaced with respect to one another by sliding along the boundary between them. Superplastic creep is only observed at intermediate stresses; at both high and low stresses it is superseded by bulk creep mechanisms with higher stress exponents. 19.34 Stress rupture High temperature creep eventually terminates with the breakage of the specimen, usually through the coalescence of voids that nucleate and grow in the material as it deforms. The time to creep failure is an important design parameter for devices that are intended to operate at high temperature under load, such as turbine engines The failure time in creep depends on the stress and the temperature as well as on the material. The property that is most often tabulated as a measure of creep failure is called the rupture strength. The rupture strength is the stress that causes failure in some pre-selected time (often 1000 hr. under load). The rupture

strength is a sharply decreasing function of the temperature 19.35 Stress relaxation ß decreasing temperature t . Fig. 1911: Stress relaxation of a sample loaded and held at fixed strain Finally, suppose that a sample is given a fixed displacement that is maintained for a long time. A common example is a bolt, which is tightened (strained elastically) and then left alone. Creep occurs under the residual load in the sample Since the strain is fixed, the stress decreases with time, as shown in Fig. 1911 This phenomenon is known as stress Page 671 Source: http://www.doksinet Materials Science Fall, 2008 relaxation. The stress relaxation rate decreases as the temperature decreases, as shown in the figure. Stress relaxation is important in engineering since it cause the disappearance of the stresses that are used to bind a system together. Exposure to a sufficiently high temperature for a sufficiently long time causes tightened bolts to loosen and shrunk-fit parts to come free.

19.4 FRACTURE 19.41 Fracture toughness If a specimen that contains a part-through crack is tested in tension, it usually breaks at a load that is significantly below its yield strength. The crack becomes unstable at a critical value of the stress and spontaneously propagates across the sample. The value of the stress at which the crack propagates to failure depends on the size of the crack and on a material property, called the fracture toughness, that measures its resistance to fracture. The fracture toughness of a material is most conveniently studied by using a specimen that is a thick plate with a sharp-tipped crack of known length cut into one side, as illustrated in Fig. 1912 A tensile load is applied perpendicular to the crack, and increased until the crack propagates. The critical stress, ßc, at which the sample breaks is inversely proportional the square root of the crack length, a . The fracture toughness of the plate, Kc, is defined as the product, Kc = Qßc a 19.26 where

Q is a geometric factor that depends on the ratio, a/W, where W is the width of the specimen in the direction of crack propagation (Fig. 1912) When W >> a, Q = 112 π = 1.99 ß a W ß . Fig. 1912: A cracked plate loaded in tension used to measure fracture toughness. Page 672 Source: http://www.doksinet Materials Science Fall, 2008 19.42 Plane strain fracture toughness The fracture toughness, Kc, decreases as the plate thickness (T) increases. This is because the specimen relaxes at its free surfaces, which changes the local distribution of the stress at the tip of the crack. However, when the sample is very thick, the relaxation at its lateral surfaces can be ignored. In this limit the sample is said to be in plane strain When the sample thickness is sufficient to achieve plane strain, the fracture toughness asymptotes at the value, KIc, the plane strain fracture toughness. and becomes independent of thickness Kc KIc plane strain T . Fig. 1913: The normal variation

of fracture toughness (Kc) with the thickness (T) of the test specimen. Kc asymptotes to the plane strain fracture toughness, KIc, when the specimen is thick enough to establish plane strain at the crack tip. The plane strain fracture toughness, KIc, is the property that is ordinarily used to characterize resistance to fracture. It is the preferred measure both because it is a material property and because it provides a conservative design criterion. The critical fracture stress, ßc, of a part that contains a crack of length, a, can be estimated from eq. 1926 by setting Kc = KIc. Since Kc ≥ KIc, the actual value of the critical stress is at least as high, and is ordinarily higher, than the estimated value. 19.43 Sources of critical cracks The critical crack length, ac, at which a crack propagates to fracture under a given value of the applied stress, ß, is, from equation 19.26, Kc2 ac = Q-2 ß  19.27 To ensure the reliability of an engineering structure one must

guarantee that no critical crack is present, and none will form during service. The cracks that cause the failure of engineer- Page 673 Source: http://www.doksinet Materials Science Fall, 2008 ing systems have three principal sources: manufacturing defects, fatigue crack growth and stress corrosion cracking. One guards against manufacturing defects by controlling the manufacturing process and inspecting the finished parts. Fatigue and stress corrosion cracking require independent analysis, since they provide mechanisms that allow innocuous flaws to grow to critical size. 19.5 FATIGUE Suppose that a nominally flaw-free sample of a ductile material is placed in a tensile machine and subjected to a load that oscillates with the time, as illustrated in Fig. 1914 Let the maximum value of the stress, ßmax, be below the yield strength of the material, but above about 1/3 of the ultimate tensile strength. For many cycles of the load, nothing obvious happens The sample stretches

slightly as the stress is increased to ßmax, relaxes to its original length as the stress returns to zero, and compresses slightly as the load decreases to - ßmax. Then, often with no obvious warning, the sample breaks This phenomenon is called fatigue, an anthropomorphic image that suggests that the material grows tired and weak. A closer analysis reveals that cyclic stress provides a mechanism to nucleate cracks and grow them until they reach critical size. Crack nucleation and growth occurs because plastic deformation is never quite reversible. It eventually concentrates at local sites within the specimen, and forms cracks. The irreversibility of plastic deformation at the tip of a crack causes it to grow at an accelerating rate until it finally reaches critical size and propagates to failure. 19.51 The s-n curve The fatigue properties of a material are usually represented in one of two ways. First, nominally flaw-free specimens are tested under reversed, uniaxial loading to

produce the s-n curve of the material. The s-n curve is a plot of the number of cycles to failure (N) as a function of the maximum engineering stress, s, in a cycle between + s and - s. The number of cycles increases dramatically as the stress cycle is decreased. Some materials, including many common structural steels, have a fatigue limit, a value of the stress, sl, below which fatigue cracks do not form. The fatigue limit, sl, is often about one-third of the ultimate tensile strength, su. Most structural materials have no fatigue limit However, the stress that is associated with a life of n = 108 cycles is often used as a practical fatigue limit since 108 cycles corresponds to a long operating time for most systems and since n increases rapidly as s is lowered below this value. When the material is likely to be stressed or strained into the plastic region, it is common to represent its fatigue behavior in terms of a Coffin-Manson plot of n against the cyclic plastic strain, Ήp.

This plot can be easily generated by testing under strain control Page 674 Source: http://www.doksinet Materials Science Fall, 2008 sm su s -sm s t sl log(n) . Fig. 1914: The s-n curve of a typical metal The stress oscillates with time as shown at right. The fatigue limit, sl, is either the asymptote of the curve or the stress that causes failure in 108 cycles. 19.52 The fatigue crack growth rate Most of the life of a flaw-free sample that is fatigued under cyclic load is spent in the nucleation phase of fatigue crack growth. As a consequence, the s-n curve significantly overestimates the life of a specimen that contains pre-existing cracks. This is a serious shortcoming, since it is difficult to prevent small cracks or flaws in real parts that are cast, formed, or machined into shape. For this reason modern engineering design often assumes the presence of a small pre-existing crack, and attempts to ensure that it will not grow to critical size during the life of the part.

The data that is needed to support this design methodology concerns the fatigue crack growth rate. ln(da/dn) ÎK th ln(ÎK) . Fig. 1915: The dependence of the fatigue crack growth rate (da/dn) on the cyclic stress intensity (ÎK). Page 675 Source: http://www.doksinet Materials Science Fall, 2008 The stress cycle that drives the growth of a fatigue crack is the stress cycle at the crack tip. As we shall discuss further below, this stress cycle is proportional to the cyclic stress intensity, ÎK, where ÎK = QÎß a 19.28 where Q is the geometric factor introduced in eq. 1926, Îß is the nominal stress cycle in the material well away from the crack, and a is the crack length. The crack growth per cycle, da/dn, varies with the cyclic stress intensity, ÎK, as illustrated in Fig 1915 The behavior can be conveniently divided into three regions The first is the threshold region. There is a reasonably well-define threshold stress intensity, ÎKth, below which a normal fatigue crack

does not grow. One objective of fatigue-safe design is to ensure that the largest crack is so small, or the cyclic stress is so low, that ÎKth is never exceeded. When the cyclic stress intensity is near ÎKth, da/dn increases rapidly with ÎK. The second region is the power-law region, or Paris region (after the person who first discovered it), in which da m dn = A(ÎK) 19.29 where A and m are constants that are characteristic of the material. Power-law behavior pertains for most of the period of fatigue crack growth to failure, and, hence, this growth law is often employed to estimate the service lives of parts that must be assumed to contain pre-existing flaws. The third region pertains when the crack length is very close to the critical crack length, ac, for unstable fracture. The rate of crack growth increases rapidly as a approaches ac. We shall discuss later how this sort of information is used to estimate the life of an engineering part or to set inspection intervals that

guard against failure. 19.5 STRESS CORROSION CRACKING Let a sample be loaded in tension while immersed in a electrolyte under conditions where general corrosion is relatively slow. The sample may sit quite happily in this environment for some time, and then suddenly fail by catastrophic fracture This phenomenon is known as stress corrosion cracking, and is due to the corrosion-assisted growth of cracks to critical size. Stress corrosion cracking is a troublesome engineering problem since it often leads to failure without warning under conditions in which general corrosion is mild, and perhaps imperceptible. Page 676 Source: http://www.doksinet Materials Science Fall, 2008 Stress corrosion cracking is a crack growth mechanism that causes small flaws to propagate until they reach critical size. The flaws that lead to failure may be either pre-existing manufacturing flaws or corrosion-induced flaws at the base of corrosion pits or crevices. The enhanced stress at the tip of the

flaw leads to the growth of a sharp-tipped crack that propagates into the material. Depending on the material and the environmental conditions, the crack may grow by anodic dissolution at a local anode, or by a hydrogenassisted crack growth process at a local cathode (in this case the phenomenon is often called hydrogen embrittlement). In either case, stress corrosion cracking usually occurs under conditions where general corrosion is mild; otherwise general corrosion would blunt or remove cracks that were growing in from the surface. As in the case of fatigue, two types of test data are used to characterize stress corrosion cracking. The simplest representation is a plot of the time-to-failure as a function of applied stress for a nominally flaw-free specimen in a given corrosive environment. If the environment is such that stress corrosion cracking occurs, these test yield curves of the form shown in Fig. 1916 The failure time increases dramatically as the load is decreased Failure

is not observed at stresses below a threshold stress, which is almost always greater than 0.5 sy, and is usually closer to 08 sy su s sc t . Fig. 1916: Typical variation of time-to-failure in stress corrosion cracking with the applied stress. The threshold stress, sc, depends on the precise environment. Time-to-failure plots are not applicable to samples that contain pre-existing flaws. Since these may be present, and, therefore, must be assumed to be present in most manufactured parts, the more sophisticated analyses that are now used in engineering design depend on direct measurements of the crack growth rate. The crack growth rate, da/dt, in a given material in a given environment is determined by the stress intensity, K, at the crack tip. By analogy to equation 1926, K = Qß a 19.30 Page 677 Source: http://www.doksinet Materials Science Fall, 2008 The relation between da/dt and K usually has a form like that shown qualitatively in Fig. 19.17 As before, it is conveniently

divided into three regions K Iscc ln(da/dt) KIc ln(K) . Fig. 1917: The stress corrosion crack growth rate as a function of the crack tip stress intensity (K). KIscc is a strong function of the environment. Stress-corrosion cracks do not grow at stress intensities less than KIscc, the critical stress intensity for crack growth. KIscc is not a material property; it depends on both the material and the environment. Nonetheless, it is a useful design value if it is known, since stress corrosion will not cause failure if K is held below KIscc, but will inevitably lead to eventual failure if K > KIscc. When K is raised above KIscc, the crack growth rate increases rapidly to a steady state velocity (stage II). The steady-state velocity is maintained until the crack grows to the extent that K approaches Kc, the fracture toughness of the material. When this happens (stage III) the crack growth rate increases monotonically until the sample fails at K = Kc. 19.6 HARDNESS The final simple

mechanical property we shall mention here is the hardness. The hardness of a material measures its resistance to indentation. It is tested with a device like that shown in Fig. 1920 An indenter is pressed into the surface of the material under a pre-selected load. The hardness is associated with the degree of penetration The hardness is a qualitative measure of resistance to indentation that is a particularly good measure of resistance to abrasive wear. It also provides a simple, qualitative measure of resistance to plastic deformation that is widely used as a quality control tool to verify the strengths of heat-treated metal parts. While many different hardness tests have been used over the years, two are commonly used today. The first is the Rockwell hardness test In the Rockwell test the indenter is impressed into the surface to a pre-set load The load is then relaxed, and the depth of penetration is measured. The major advantage of this test is that the impression is made and Page

678 Source: http://www.doksinet Materials Science Fall, 2008 the depth of penetration measured in sequential operations on the same machine. The Rockwell hardness test is quick and easily automated. The result of the Rockwell hardness test is a unitless hardness number that is read from a standard scale. The hardness number decreases with the depth of penetration, that is, materials that are more resistant to penetration have higher hardness. Otherwise, the Rockwell hardness number has no fundamental significance. There are, in fact, several different variations of the Rockwell hardness test that vary in the load used and the shape of the indenter. These are used for materials that have different hardness ranges and thicknesses The Rockwell-C scale is the one that is commonly used for steel parts This test uses a conical, diamond indenter with a 120º included angle and a spherical cap with an 0.2 mm radius, and a 150 kg. major load Rockwell A or B tests are used for softer metals

Rockwell A uses a conical indenter with a smaller load, Rockwell B uses a spherical indenter that makes a more shallow impression. Empirical relations have been proposed to relate the various Rockwell scales to one another and to the yield or tensile strength of the material. These are useful, but are only approximate, particularly when they are applied to materials that differ significantly from the materials for which they were developed P . Fig. 1920: The hardness test The second common hardness test is the Knoop test, which is used to measure the microhardness of small regions of the material. Microhardness readings are used, for example, to determine the case depth of a part whose surface has been hardened by induction heating or carburizing. The Knoop test uses an indenter that has the shape of a distorted pyramid and makes an impression that has the elongated diamond geometry shown in Fig. 19.21 The indenter shape is such that the depth of the indentation is about 1/4 of the

short dimension of the diamond, and only about 1/30 of the long dimension. The Knoop hardness is defined as P K=A P = 14.2 2 l 19.31 where P is the load, A is the projected area of the impression, l is the length of the impression, and the factor, 14.32, applies to the standard geometry of the Knoop indenter In a normal microhardness test the loading device and specimen are mounted in an optical microscope that is used to measure the length, l, of the impression. The impression need be no more than about 10 µm in size, so very local values of the hardness can be measured. Page 679 Source: http://www.doksinet Materials Science Fall, 2008 Empirical relations have been developed that provide approximate conversions between the Knoop and Rockwell scales. The Knoop and Rockwell-C tests are often used together in the quality control of metal parts. For example, the material specifications for a hardened gear, shaft or bearing race often include a specification for the superficial

hardness, given as minimum and maximum acceptable values of RC , and a specification of the minimum case depth, the distance below the surface for which a specified value of RC must be maintained. The superficial hardness specification can be verified non-destructively by performing hardness tests on the surface of the part. To verify the case depth, however, it is necessary to section selected parts and survey the hardness profile below the surface This is ordinarily done with a microhardness tester that uses the Knoop method. The Knoop values are subsequently converted to RC to verify the specification that governs case depth. l . Fig. 1921: The shape of the impression made by a Knoop indenter Page 680 Source: http://www.doksinet Materials Science Fall, 2008 Chapter 20: Elastic Properties Robert Hooke . gave the first rough law of proportionality between the forces and displacements. Hooke published his law first in the form of an anagram "ceiiinosssttuu" in 1676,

and two years later gave the solution of the anagram: "ut tensio sic vis", which can be translated freely as "the extension is proportional to the force." - I.S Sokolnikoff, The Mathematical Theory of Elasticity 20.1 INTRODUCTION Small mechanical loads cause elastic deformation. Elastic deformation is traceable to displacements at the atomic level. The bonds between aotms are stretched by the applied load until the restoring force of the stretched bonds is just sufficient to balance the stretching force of the applied load. Since there is no long-range atom movement, elastic deformation is recoverable; when the applied load is removed the atoms return to their original, equilibrium positions. The elastic properties of a material govern its response to small loads. The elastic moduli relate small stresses and strains. The elastic moduli are the properties that are normally used in the design of engineering structures and devices. Even large structures must be

constrained to very small strains. A 1% strain of a 50 story building changes its height by almost 10 ft. If the building responded to wind gusts with even a fraction of this strain, the occupants of its upper floors would be permanently seasick. The dimensional constraints on mechanical devices are equally stringent. The allowable clearances between moving parts in engines are measured in thousandths or ten-thousandths of an inch. Plastic strains are intolerable, and uncompensated elastic strains of a small fraction of a percent would cause catastrophic interference in many devices. Materials that are subjected to very small strains are linear in their elastic response; the strain is simply proportional to the applied stress, in accordance with Hookes Law. Linear strains are additive; the response of a material to a complex state of stress can be found by expressing the total stress as the sum of individual, simple stresses, and summing the strains associated with each of these. Since

the configuration of atoms changes with direction in a crystal, the elastic response of a crystal depends on the direction of the load; crystals are elastically anisotropic. However, the crystalline materials that are ordinarily used in engineering structures are polycrystalline metals or ceramics whose elastic properties average those of the individual grains and are, hence, nearly isotropic. Glasses and amorphous polymers are automatically isotropic. Isotropic materials require only two elastic moduli to specify their behavior As Page 681 Source: http://www.doksinet Materials Science Fall, 2008 we shall see, these can be chosen to be Youngs modulus, E, and Poissons ratio, ˆ, though other choices are also possible, and often convenient. Since the most important structural materials are nearly isotropic, and since this case is the simplest, we shall confine the following discussion to isotropic materials. Note, however, that there are at least three common situations in which

elastic anisotropy must be taken into account. (1) Single crystals and strongly textured (oriented) crystalline films are widely used in microelectronic devices. They are often subjected to thermal stresses, and their elastic anisotropy must be considered in predicting their response. (2) Fiber composites are widely used in structures that support primarily uniaxial loads, since, as we shall see, they can be engineered to have exceptional stiffness in the fiber direction. Fiber composites have very different elastic properties in the directions parallel and perpendicular to the fiber direction. (3) Even a fine-grained polycrystalline metal is often measurably anisotropic. When the metal is rolled into a sheet or extruded into a bar or wire it develops a crystallographic texture in which particular crystallographic directions align parallel and perpendicular to the direction of the primary load. This texture produces an elastic anisotropy that is sometimes large enough that it cannot be

neglected. This is the reason that tabulations of the elastic properties of metal sometimes list separate moduli for deformation parallel and perpendicular to the rolling direction. 20.2 ELASTIC PROPERTIES OF ISOTROPIC MATERIALS The elastic properties of an isotropic material are best understood by considering its response to four basic stress states: tensile load, hydrostatic pressure, simple shear and balanced shear. The basic stress states are illustrated in Fig 201 20.21 The basic load geometries (a) Fig. 201: (b) (c) (d) The four basic stress states: (a) tension, (b) hydrostatic compression, (c) simple shear, (d) shear through balanced tensile and compressive loads. The simplest loading geometry is uniaxial tension or compression, as illustrated in Fig. 201(a) Uniaxial tension is the load geometry that is ordinarily used to measure elastic Page 682 Source: http://www.doksinet Materials Science Fall, 2008 properties, not only because it is the most common loading

geometry, but also because complex stress states can be regarded as the superposition of uniaxial loads. Uniaxial loading is complicated, however, by the fact that it produces both a shape change (the sample gets longer) and a volume change (the sample expands slightly). At the atomic level, shape and volume changes are qualitatively different. Shape changes alter the atom configuration at constant atomic volume, while volume changes alter the volume at constant configuration. It is, therefore, useful to consider elastic deformations that isolate these two effects. The volume changes at constant shape when the material is subjected to equal loads along each of three perpendicular axes. This stress state is called hydrostatic compression, and is illustrated in Fig. 201(b) A change in shape at constant volume is accomplished by shear loads. There are two different ways in which a cube of isotropic material can be deformed in pure shear The first is illustrated in Fig. 201(c); the cube is

deformed by equal and opposite shear stresses applied to opposite faces, and balanced so that there is no net moment on the cube. The second, balance shear, is illustrated in Fig 201(d); equal and opposite uniaxial loads are applied in two perpendicular directions. Since the displacements of the two perpendicular surfaces are equal and opposite, there is no volume change. As illustrated in Fig. 202, a balanced shear is a simple shear of a square rotated 45º with respect to the square cross-section of the parent cube. Fig. 202: Illustration showing how equal and opposite displacements on the surface of a square cause a simple shear of an inscribed square that is rotated by 45º. 20.22 Uniaxial tension; Youngs modulus and Poissons ratio When an isotropic material is loaded in uniaxial tension it elongates in the direction of the load and contracts in the perpendicular directions, as illustrated in Fig. 203 For small strains from an unstressed initial state, the tensile stress and

tensile strain are related by Hookes Law: ßx = E‰x 20.1 Page 683 Source: http://www.doksinet Materials Science Fall, 2008 where ßx = P/A, the applied load divided by the cross-sectional area, ‰x = ÎLx/Lx is the strain along the x-axis in the figure, and the constant of proportionality, E, is Youngs modulus. y x Fig. 203: Uniaxial tension. The lateral strain that is induced by a stretch in the x-direction is the same in the two perpendicular directions, y and z, and is given by ‰y = ‰z = - ˆ‰x 20.2 where ˆ is Poissons ratio. The fractional change in the volume of the body is, to within terms of order ‰x2, ÎV V = (Lx + ÎLx)(Ly + ÎLy)(Lz + ÎLz) - LxLyLz LxLyLz ~ ‰x + ‰y + ‰z = (1 - 2ˆ)‰x 20.3 When ˆ « 0.5, as it does, for example, for natural rubber and some other elastomers, ÎV « 0, and volume is conserved during tensile deformation. When ˆ < 05, as it is for structural metals and ceramics, the volume increases during tensile

strain. For many structural metals, ˆ « 0.3 Now suppose that stresses are applied in the y- and z-directions as well. So long as the strains are small enough that terms of order ‰2 can be ignored, they can be found by simply summing the strain increments due to each of the stresses. It follows that the total strain in the x-direction is: 1 ˆ ˆ ‰x = E ßx - E ßy - E ßz 1 = E [ßx - ˆ(ßy + ßz)] 20.4 The strains in the y- and z-directions, ‰y and ‰z, are given by similar expressions. The three equations for the strains, ‰x, ‰y and ‰z can be solved for the stresses. The result is Page 684 Source: http://www.doksinet Materials Science Fall, 2008 E ßi = (1-2ˆ)(1+ˆ) [(1-ˆ)‰i + ˆ(‰j + ‰k)] 20.5 where the indices (i,j,k) take the values (x,y,z), (y,z,x) or (z,x,y). It can be shown that any state of stress that can possibly be applied to a material can be created by applying normal stresses, ßx, ßy and ßz, along three perpendicular axes, x, y and z,

which are the principal directions of stress for the particular stress state. Since equations 20.4 and 205 hold for all possible choices of orthogonal x- y- and z-axes, it follows that the two elastic constants, E and ˆ, are sufficient to characterize the elastic behavior of an isotropic material under any state of stress. Youngs modulus and Poissons ratio are not only a sufficient set of properties to determine the elastic behavior of isotropic materials, they are a convenient choice for engineering design. For this reason, E and ˆ are the elastic properties that are tabulated in most compilations of material properties (often only the modulus is given since, as we shall see, the values of Poissons ratio are confined to a narrow range). On the other hand, E and ˆ are not the most convenient choice of elastic properties for the purpose of understanding elastic behavior, since the uniaxial tensile test that defines them mixes volume and shape changes. 20.23 Hydrostatic compression;

the bulk modulus When a material sustains a hydrostatic pressure, P, its volume decreases by an amount that is governed by its bulk modulus, ∫: ÎV P=-∫ V 20.6 The compressibility, ˚, is the inverse of the bulk modulus: 1 ∆V ˚ = - V  ∆P  T = ∫ -1 20.7 To relate the compressibility to Youngs modulus and Poissons ratio, recognize that a material under hydrostatic pressure, P, is subject to the triaxial stress ßx = ßy = ßz = -P 20.8 Then, using equations 20.3 and 204 ÎV V = ‰x + ‰y + ‰z = (1-2ˆ) 3(1-2ˆ) [ß + ß + ß ] = P x y z E E Page 685 20.9 Source: http://www.doksinet Materials Science Fall, 2008 From which ∫ = ˚-1 = E 3(1-2ˆ) 20.10 20.24 Shear stress and the shear modulus Let a cubic element of a solid be subject to a shear stress, †, on opposite faces, as illustrated in Fig. 202(c) The effect of this stress is to distort the square cross-section of the cube into a parallelogram, as illustrated in Fig. 203 The

shear strain is ÎLx = L y 20.11 When the shear strain is small, as we assume it is, it is also equal to the change in the angle at the edge of the cube. The edge, which was originally a right angle, is distorted to the angle ÎLx π π œ = 2 - tan-1 L  ~ 2 - y 20.12 ÎL x † † † Ly π/2 - † Fig. 204: The shear strain, , caused by the balanced shear stress, † The strain, which is assumed small, has been exaggerated for clarity. The shear stress and shear strain are related by the shear modulus, G: † = G 20.13 The shear modulus can be expressed in terms of E and ˆ by the following geometric argument which is, unfortunately, a bit tortuous. Page 686 Source: http://www.doksinet Materials Science Fall, 2008 Let a cube be sheared by the balanced stress illustrated in Fig. 201(d), and measure the shear by the distortion of a square inscribed on the cross-section of the cube, as shown in Fig. 202 As diagrammed in Fig 205, the balanced stress

(ß in the x-direction, -ß in the y-direction) stretches the cube in the x-direction and compresses it in the y-direction, so that ‰x = - ‰y = ‰ 20.14 Before straining the cube, imagine that a square is inscribed on its cross-section. Let the edges of the square lie at 45º angles to the x- and y-axes of the cube, as shown in the figure. After straining, the angle of the inscribed square at the left-hand side of the figure has changed from π/2 to π/2 - The shear strain, , is related to the balanced tensile strain, ‰, by the geometric relation π  tan 4 - 2 1+‰ =  1-‰  20.15 Since both and ‰ are small, this equation simplifies to ~ 2‰ = 2ß[1 + ˆ] E 20.16 where the final form of the equation follows from eq. 204 L x(1+‰) -ß L y(1-‰) ß ß π/2 - -ß Fig. 205: Cross-section of a cube that is subject to the balanced stress, ±ß, which produces the balanced tensile strain ±‰. A square inscribed on the

cross-section of the cube experiences the shear strain, . (Lx = Ly = L) To find the shear stress that accomplishes the shear strain, , recognize that a tensile stress on the surface of a body creates a shear stress on a plane that lies at an angle, œ, to the normal. To find this stress, consider the situation shown in Fig 206 The axial force is Page 687 Source: http://www.doksinet Materials Science Fall, 2008 F = ßA 20.17 where A is the cross-sectional area on which ß acts. The component of this force, Ft, that is tangential to a plane whose normal makes an angle, œ, with the normal is Ft = F sin(œ) 20.18 This force acts on an area A(œ) = A/cos(œ) 20.19 Ft † = A(œ) = ßsin(œ)cos(œ) 20.20 so the shear stress is For œ = 45º, as in Fig. 205, † = ß/2 20.21 Fig. 206: Illustration showing the shear stress, †, induced in a plane whose normal makes an angle, œ, with the plane on which the normal force, ß, acts. In the configuration shown in Fig. 205, the

surface of the inscribed square is at 45º to both the x- and y-faces of the cube. Both normal stresses contribute ß/2 to the shear stress on that plane, so the balance shear stress, †, is † = 2(ß/2) = ß 20.22 Substituting the expression for ß obtained from eq. 2016 then gives E † = 2(1+ˆ) 20.23 Page 688 Source: http://www.doksinet Materials Science Fall, 2008 So the shear modulus is E G = 2(1+ˆ) 20.24 20.3 PHYSICAL SOURCE OF THE ELASTIC MODULI As we discussed in Chapter 3, it is often convenient to divide the energy of a solid into two parts: E = E0(v) + E1({R}) 20.25 where E0(v), the dominant term, is determined by the atomic volume only, and E1({R}) is determined by the configuration of the atoms for given atomic volume. This decomposition of the cohesive energy provides a natural framework for understanding the elastic moduli. The bulk modulus, ∫, governs the response to changes in volume at constant atomic configuration, while (to the extent that we can

ignore anisotropy) the shear modulus, G, governs the response to configurational distortions at constant volume. 20.31 The bulk modulus For a given atom configuration, the energy varies with volume as illustrated in Fig. 20.7 When the volume, and, hence, the interatomic separation is large, the atoms attract and the energy and volume decrease together. When the interatomic separation is small, the atoms repel and the energy decreases as the volume expands. Assuming low temperature, the minimum of the energy determines the equilibrium volume. The pressure required to maintain a given value of the volume is given by the differential ∆E P = - ∆v 20.26 and is also plotted in the figure. The bulk modulus is ∆P ∆2E ∫ = - v ∆v = v 2 dv 20.27 that is, ∫ is given by the slope of the function P(V), or the curvature of the function E(V). For small volume changes about the equilibrium volume, v0, the function P(v) is almost linear, so the bulk modulus is nearly constant. Note

that the bulk modulus is determined by the curvature of the energy function at its minimum value, not the minimum value itself. While there is a general tendency for the bulk modulus to increase with increases in properties that are closely related to the cohesive Page 689 Source: http://www.doksinet Materials Science Fall, 2008 energy, such as the melting point and the sublimation energy, the correspondence is not universal. For example, the diamond modification of carbon, which has the highest known value of the bulk modulus, is not even the preferred phase of carbon at atmospheric pressure. P(V) E V0 V E(V) Fig. 207: The variation of energy and pressure with volume The bulk moduli of most non-transition metals can be understood on the basis of a simple model. In these valence metals the radius of the ion core of filled electron shells is very small compared to the effective radius of the atom in the solid. Most of the space within the solid is filled by a "gas"

of valence electrons, and its bulk modulus is, essentially, the modulus of the electron gas. The isothermal modulus of an ideal gas is ∆P ∫ = - v ∆v  kT = v = nkT T 20.28 where v is the volume per particle, n = v-1 is the number of particles per unit volume, and k is Boltzmans constant. In a metal the valence electron energy is essentially constant with T If we replace kT by the Fermi energy, which has a median value of about 4 eV for the valence metals, then the compressibility is a linear function of the valence electron density (number of valence electrons per unit volume): ∫ ~ 0.04n 20.29 where ∫ is the bulk modulus in units of 1012 dyne/cm2, and n is the density of valence electrons in units of 1022 per cm3. Eq 2029 is a reasonable fit to the data for the valence metals, which is plotted in Fig. 208 Equation 20.29 does not work well for transition metals or covalent solids, both of which tend to have much higher bulk moduli than predicted. The

reason is relatively straightforward. In transition metals, the partly filled d-shells in the inner core extend outward to a significant fraction of the atomic radius It follows that the free volume available for the compression of the valence electron cloud is severely restricted, and its modulus is Page 690 Source: http://www.doksinet Materials Science Fall, 2008 higher. In covalent solids, the valence electron shells are essentially filled by shared electrons The Pauli exclusion principle restricts rearrangements of the distribution of valence electrons, raising the modulus. 1.800 valence metals 1.600 transition metals 1.400 covalent solids bulk modulus 1.200 1.000 0.800 0.600 0.400 0.200 0.000 0 5 10 15 20 25 electron density Fig. 208: The bulk modulus plotted as a function of the valence electron density for a number of valence metals, transition metals and covalent solids. 20.32 The shear modulus The shear modulus, G, measures the resistance of the material to

small changes in atomic configuration at given volume. Those solids whose energies are relatively insensitive to the precise atomic configuration, such as molecular solids like rubber, have relative low shear moduli. Those solids whose configurations are relatively rigid, including particularly covalent solids with fixed bond angles, have relatively high values of G The elastic behavior of the solid is particularly sensitive to the relative values of ∫ and G. To study this, we use eqs 2010 and 2021 to express the tensile elastic constants, Youngs modulus and Poissons ratio, in terms of ∫ and G. the result is 9∫G E = (G+3∫) = 2G(1+ˆ) = 3∫(1-2ˆ) 20.30 3∫ - 2G ˆ = 2(G + 3∫) 20.31 Page 691 Source: http://www.doksinet Materials Science Fall, 2008 Consider four cases: (1) Let ∫ >> G, as it is for molecular solids, like rubber, whose energies are relatively insensitive to the precise molecular configuration. In this case: ˆ « 0.5, E « 3G 20.32 Since ˆ

» 0.5, these materials are almost incompressible in tension, even if their bulk moduli are relatively small. (2) Let ∫ « 3G, as it does for many metals. Then ˆ « 0.35, E « 27G 20.33 The value ˆ « 0.33 is a common and useful choice for structural metals (3) Let ∫ « G, as is the case when the atomic configuration is rigid, as in strongly covalent solids like diamond. In that case ˆ « 0.125, E « 225G 20.34 Such a material prefers a relatively large volume change when it is deformed in tension so that its bond angles are not too severely distorted. (4) Let G >> ∫, as would be the case in a material whose bond angles were much more important than its bond lengths. In that case, ˆ « -1, E « 9∫ 20.35 Since ˆ < 0, such a material would actually expand when strained in tension, since volumetric expansion is necessary to preserve bond angles. There is no natural material with ˆ < 0; the lowest known value is about 0.1 However, researchers have recently

made artificial, foam-like network solids that do have negative ˆ These are interesting curiosities, but have no current engineering applications. 20.4 SPECIFIC MODULUS; STIFFNESS In the design of most engineering structures it is important to choose materials that maximize stiffness, or resistance to elastic displacement. Put in its simplest terms, the stiffness of an engineering structure governs the load, P, required to achieve a certain elongation, ÎL. The smaller ÎL for given P, the higher the stiffness Given Hookes Law, ÎL P ß = A = E‰ = E L  Page 692 20.36 Source: http://www.doksinet Materials Science Fall, 2008 or P EA ÎL = L 20.37 The right-hand side of this equation is a measure of the stiffness, where A is the cross-sectional area of the structural member and L is its length. Two situations are common in the engineering design of structural members of given length, L. In the first, space-limited design, one would like the cross-sectional

area, A, to be as small as possible so the structure will not be excessively bulky. Buildings, bridges and vehicles such as heavy trucks are designed under constraints like this. In this case, the modulus, E, should be as large as possible so that the area, A, can be as small as possible for given stiffness. Buildings, bridges, and transport vehicles are normally made of steel or reinforced concrete, both high-modulus materials, rather than aluminum or plastic, which have relative low elastic moduli. In the second situation one would like the weight of the structural member to be as small as possible for given stiffness. This is the case, for example, for aerospace vehicles, camping equipment, and sporting equipment such as golf clubs, tennis racquets and fishing poles, along with among many other weight-limited structures. The weight of a member of length L is W = ®AL 20.38 where ® is the density. Hence the stiffness is measured by the grouping E W P = ®  2 ÎL

L 20.39 To minimize weight (W) for given length and stiffness the quantity that should be maximized is the specific modulus, E E = ® 20.40 The materials of choice for weight-limited structures are low-density metals such as aluminum and titanium, and reinforced plastics such as fiberglass and graphite fiber composites. While the moduli of these materials are typically well below those of structural steels, their low densities have the consequence that their specific moduli are considerably higher. Page 693 Source: http://www.doksinet Materials Science Fall, 2008 20.5 ENGINEERING THE ELASTIC MODULUS One of the great limitations of materials engineering is its relative inability to control the elastic properties of materials. Since these reflect behavior at the atomic level, they are relatively insensitive to changes in the microstructure. However, there are instances in which beneficial changes can be made, by adjusting the molecular structure, or by altering the

microstructure to introduce fibers or inclusions with exceptionally high moduli. 20.51 Composition Small changes in the chemical composition of a solid do not ordinarily cause significant changes in its modulus. To a first approximation, the modulus of a solid solution is the average modulus of its components, and is not significantly changed until the solute content becomes so large that the nature of the material itself is changed. For example, the moduli of the alloy steels that are used in engineering structures are almost the same, independent of composition. An important exception occurs in the alloying of aluminum with lithium. For reasons that are still not well understood, the addition of about 2 weight percent Li to Al raises its modulus by almost 10%. Since Li is a light element, the Li addition also lowers the density of the alloy, so the improvement in the specific modulus is even greater. Because of this behavior, Al-Li alloys have been under intensive development for a

number of years as candidate alloys for aerospace vehicles. The problem is to achieve improved elastic properties without sacrificing other needed properties, such as strength, fracture toughness and fabricability. 20.52 Microstructure The elastic modulus is significantly affected by crystal structure, and can be significantly altered by choosing the crystal structure when more than one structure is available. For example, high alloy steels can be stabilized in the high-temperature, FCC structure, and have lower moduli than similar steels with the normal BCC structure. While crystal structure modification is not ordinarily a useful option for structural metals, the structures of molecular solids often can be modified to improve elastic properties. The classic examples are the elastomeric solids, whose molecules are kinked and coupled so that they behave like microscopic springs and permit large elastic deflections. As we discussed in Chapter 6, both the internal structure of the

elastomer molecule and the cross-linking between adjacent molecules are controlled to adjust the elastic modulus. Many engineering elastomers are complex copolymers that also contain inorganic fillers to control the complex of mechanical properties. 20.53 Composite materials The materials that offer the most attractive combinations of high modulus and low density are ceramic materials, glasses, or fibers that are inherently brittle and, hence, unPage 694 Source: http://www.doksinet Materials Science Fall, 2008 suitable for many structural applications. To achieve some of the benefit of these materials, they are used as isolated fibers or particles that are embedded in a matrix of more ductile material to inhibit brittle failure. The elastic modulus of a composite material is maximized if the high-modulus component is used as a continuous fiber in the direction of primary load. However, these fiber composites are highly anisotropic, and are best used in situations where the applied

load is uniaxial For structures that are subject to more complex load geometries, it is still possible to obtain some of the benefit of high-modulus fillers by cross-plying fibers in two or three dimensions, or using randomly-oriented, discrete reinforcement particles. Fig. 209: Loading in an idealized fiber composite (load-bearing elements in parallel) and particulate composite (load-bearing elements in series). The difference in behavior between a fiber composite and a particulate composite is due to the manner in which the load is shared between the reinforcing particle and the matrix. The difference is illustrated schematically in Fig 209 To a first approximation, the components of a fiber composite act in parallel, while those of a particulate composite act in series. In a fiber composite, the elements have equal strain, while in a particulate composite they carry equal stress. Fiber composites A fiber composite consists of continuous fibers of one material, embedded in a matrix

of a second material. In the usual case, for example, a graphite-epoxy composite, the fiber is a high-modulus material that has the advantage of being very stiff and strong, but the disadvantage of being brittle and, hence, liable to fracture. The matrix is a relatively soft, ductile material that holds the fiber and shields it from fracture under normal loads. The elastic modulus of a fiber composite is maximized when the fibers are aligned parallel to one another, and the load is applied along the direction of the fibers. To estimate the modulus of in this case, recognize that the fiber and matrix are stretched by almost identical amounts by the applied loads (Fig. 209) Hence, ‰1 = ‰2 = ‰ 20.41 Page 695 Source: http://www.doksinet Materials Science Fall, 2008 where ‰1 is the strain in the fibers and ‰2 is the strain in the matrix. The total load, P, required to achieve the strain, ‰, is P = ß1A1 + ß2A2 = A(ß1f1 + ß2f2) 20.42 where ß1 and ß2 are the tensile

stresses in the fiber and matrix, respectively, and f1 and f2 are their areal fractions in the cross-section. If E1 and E2 are the moduli of the fiber and matrix, and ß is the overall average tensile stress, P ß = A = ‰(f1E1 + f2E2) = E‰ 20.43 where E, the effective modulus of the composite, is the average of the moduli of the elements: E = E1f1 + E2f2 20.44 It follows that the effective modulus of the composite in the fiber direction is most strongly influenced by the high-modulus component, and increases with the areal fraction of that component in the cross-section perpendicular to the applied load. Equation 20.43 shows how the modulus of a material can be dramatically increased, at least along the direction of the load, by introducing fibers of high-modulus material. In particular, the specific modulus, E = E/®, can be engineered to exceptionally high values by inserting fibers of a low-density, high modulus material, such as the graphite fibers described in Chapter 6,

in a matrix of low-density material, such as epoxy. This is the essential feature of the graphite composite materials that are used in many weight-critical structures, such as aircraft parts and golf club shafts. These materials have the disadvantage, however, that their properties are anisotropic; the exceptionally high modulus is only achieved for loads that are applied parallel to the fiber axis. Hence they are most useful in structures whose dominant load is in a single, predictable direction, as it is in the shaft of a golf club or the blades of a rotating fan, among many other applications. Particulate composites To a first approximation, the elements of a particulate composite act in series (Fig. 20.9) so that they support the same stress (to be honest, this is a pretty crude approximation; computing accurate moduli for a particulate composite is a formidable problem in mathematics). Assuming load sharing in series, the total strain of the composite is the sum of the strains in

the two elements: Page 696 Source: http://www.doksinet Materials Science Fall, 2008 ‰ = ‰1f1 + ‰2f2  f1 f2  = ßE + E  1 2 20.45 so the modulus of the composite is obtained by averaging the reciprocals of the moduli of the components f1 f2 1 = + E E1 E2 20.46 It follows that the modulus of a particulate composite depends much less strongly on the modulus of the more rigid component. Nonetheless, one can raise the modulus of a material by filling it with particulates of a second, more rigid phase, and the product has the advantage that it is more nearly isotropic in its elastic properties than a fiber composite of the same materials would be. A familiar example of a particulate composite is common fiberglass, which has chopped fibers of glass distributed through an epoxy matrix. The modulus is less than it would be if continuous glass fibers were used, but the material can be easily molded into sheets with complex shapes, and can be made to be nearly

isotropic in the plane of the sheet, which makes it useful for structures, such as the hulls of ships and the underbodies of aircraft, that sustain two-dimensional loads. A second important class of a particulate composites includes the metal matrix composites that are currently under development for many applications in aircraft. The most widely used are aluminum alloys that contain dispersions of chopped fibers or crystalline whiskers of silicon carbide. The high modulus of the SiC increases the modulus of the Al alloy, while the lower density of SiC lowers the average density, adding a further increment to the specific modulus. Page 697 Source: http://www.doksinet Materials Science Fall, 2008 Chapter 21: Plastic Deformation Steel . is capable of being made the hardest of all metals The way of making it so is this: only to heat it red-hot in the fire, and then throw it all at once into cold water; and this matter of hardening is what is called tempering it, and this makes it

capable of cutting or at least breaking all sorts of bodies without exception, even diamonds themselves: For it is certain they will break in pieces with a small stroke with a hammer if it hits right. - Jacques Rohault, Traite de Physique (1671) [trans. John Clarke (1723)] 21.1 INTRODUCTION An arbitrary deformation of a material can always be described as the sum of a change in volume and a change in shape at constant volume (shear). Assuming constant structure, the change in volume is recovered when the load is removed, since the atoms can simply relax back to their equilibrium sizes. The change in shape, on the other hand, may or may not be recovered, since the atoms can relax into new positions that are configurationally identical to the original ones, but displaced from them. The part of the shear that is recovered is elastic, the part that remains is plastic. Plastic deformation is a permanent change in shape through shear. In the tensile test, a material deforms plastically when

it is loaded to a stress beyond its yield strength. The extent of plastic deformation then increases with the applied stress until plastic instability sets in at the ultimate tensile strength, the largest value of the engineering stress that the material can withstand without failure. If the ultimate tensile strength is exceeded, plastic instability leads to catastrophic necking and fracture. A material also deforms plastically when it is subjected to a constant load. Deformation under constant load is called creep. Creep deformation inevitably requires some mass diffusion and, hence, occurs at a negligible rate unless the temperature is high (T > 0.5 Tm is a rough criterion) When a material creeps under constant load, then, after an initial transient, it deforms at a constant rate (steady-state strain rate) until accumulating microstructural damage creates an instability that is characterized by a rapidly increasing strain rate and eventual failure by creep rupture. 21.11

Engineering significance of plastic deformation Engineers care about plastic deformation for at least four reasons. The first, and most common concern, is to avoid it. Very few engineering structures are designed to tolerate more than infinitesimal plastic distortions. They must, therefore, be designed to Page 698 Source: http://www.doksinet Materials Science Fall, 2008 operate at stresses well below the yield strengths of the structural materials that are used in them. This requires that yield behavior be understood and controlled Second, plastic deformation is used to shape materials. The vast majority of manufactured objects contain parts that are formed by plastic deformation. The successful design and efficient manufacture of these products requires an understanding of the nature and limitations of plastic deformation. Third, plastic instability is a major concern in both manufacturing and product safety. It restricts elongation and, hence, limits the possibility of

manufacturing complex shapes with simple forming operations. It also sets the ultimate tensile strength and, hence, fixes the maximum load that can possibly be supported in a particular engineering structure. A structure that is flawed or brittle may, of course, fail by fracture at loads that are well below the ultimate tensile strength but, however well it is made, a structure can never support a load that exceeds the ultimate tensile strength. Fourth, plastic deformation is used as a diagnostic device to understand the performance and interpret the failures of engineering structures. The pattern of deformation in a structure often provides useful information on the magnitude and geometry of the dominant loads it has experienced. This information may indicate service problems and identify the cause of service failures. 21.12 Mechanisms of plastic deformation There are four generic mechanisms of plastic deformation. While all of them are important, we shall concentrate on the first,

dislocation plasticity, since this is the dominant mechanism of deformation of structural materials. 1. Dislocation plasticity First, and most commonly, planes of atoms can slip over one another like cards in a deck, leading to an overall shear that is localized within specific atom planes. It is always energetically favorable to accomplish this slip a little at a time, as one would move a large rug across a floor. And it is usually favorable to slip in increments that correspond to a lattice displacement, so that the area of the plane that has slipped maintains a perfect crystallographic match with the plane beneath it. In this case the boundary of the slipped area is a linear defect, called a dislocation, that we defined and discussed in Chapter 4. The motion, multiplication and interactions of dislocations are responsible for the yield, work hardening and plastic instability that characterize the stressstrain behavior of structural alloys. 2. Diffusion Second, individual atoms can

move so that the crystal becomes longer in one or more of its dimensions and correspondingly shorter in the others. In a crystal individual atoms move by diffusion, and this process is known as diffusional creep. Diffusional creep is dominant at high temperature, where the rate of diffusion is appreciable, and at low stresses, which are insufficient to produce a significant rate of dislocation plasticity. Page 699 Source: http://www.doksinet Materials Science Fall, 2008 3. Structural transformations Third, all of the atoms in the crystal, or some subvolume of it, can move simultaneously to accomplish the shear. We have already discussed spontaneous structural phase transformations, such as the martensitic transformation in steel, that change the size and shape of the parent crystal. Transformation-induced plasticity can be an important deformation mechanism in materials that undergo martensitic transformations, particularly when the material is used at temperatures where it is

metastable with respect to a martensitic transformations that can be triggered by the applied stress (as is the case with many austenitic (FCC) stainless steels), or when it is subject to thermal cycling through the martensitic transformation (as is done to achieve the shape memory effect in a number of modern alloys). In addition to those transformations that change the basic crystal structure, it is also possible for the stress to drive martensitic-like structural changes that reorient the crystal without changing the basic crystal structure. The most familiar example is mechanical twinning, in which a region of the crystal spontaneously shears into a configuration that has the same crystal structure as its parent, but a different orientation in space. Mechanical twinning is an important lowtemperature deformation mechanism in metals that have non-cubic crystal structures, such as the HCP metals, and is also foiund in cubic metals such as Cu and many steels, under the appropriate

combination of load and temperature. 4. Tumbling Fourth, a material can be deformed by the cooperative, liquid-like reconfiguration of groups of atoms, molecules or small grains This is an important mechanism of deformation in amorphous metals, glasses or polymers, and is also important in the high-temperature creep of fine-grained materials (where it is called grain-boundary sliding). The scientific description of these deformation mechanisms is in its very early stages, and I am coining a term by calling it tumbling, which is the best descriptive term I can think of for what is going on. The deformation of very fine-grained or amorphous materials is an active area of current research. The practical motivation for this research is the recent development of bulk glassy metals and new, nanostructured alloys that have unusual mechanical properties A description of these deformation mechanisms is beyond the scope of this course. In this course we concentrate almost exclusively on

dislocation plasticity. 21.2 THE YIELD STRENGTH The yield strength of a typical crystalline material is related to the stress that is needed to move dislocations to cause significant plastic deformation. To understand the yield strength we must first understand the nature of the deformation caused by a dislocation and the force that impels it to move. 21.21 Deformation by dislocation motion As discussed in Chapter 4, the planar motion of a dislocation causes plastic deformation by slip. The part of the crystal that is above the plane and behind the moving dislocation is slipped by the vector, b, the Burgers vector of the dislocation, with respect to the Page 700 Source: http://www.doksinet Materials Science Fall, 2008 part below the plane (Fig. 211) When the dislocation passes through the crystal it displaces the material above the plane by the Burgers vector, b, with respect to that below it † “ b “ b † Fig. 211: Slip caused by the motion of an edge dislocation As

shown in Fig. 211, the deformation that is accomplished by the dislocation is a shear in the direction of b. It follows that the stress that causes dislocation motion is a shear stress, †, resolved in the direction of b. It can be shown that the net force that acts on the dislocation is F = †b 21.1 where b is the magnitude of the Burgers vector and † is the shear stress that acts in the direction of b. For simplicity, Fig. 211 illustrates the shear caused by an edge dislocation But our conclusion is true whatever the type of the dislocation. For example, consider the dislocation loop shown in Fig 212 The crystallographic character of the dislocation changes from edge (b perpendicular to the dislocation line) to mixed (b angled to the dislocation line) to screw (b parallel to the dislocation line) around the circle of the loop. However, the area enclosed by the loop has been slipped by the constant vector, b. If the loop is expanded to sweep out the whole plane, the final

configuration is identical to that shown in Fig. 211; the material above the plane of the loop is slipped with respect to that below the plane of the loop by the vector, b. † b † edge screw Fig. 212: A planar dislocation loop The material above the area enclosed by the loop is slipped with respect to that below the loop by the Burgers vector, b. Page 701 Source: http://www.doksinet Materials Science Fall, 2008 Since an expansion of the dislocation loop causes a shear displacement in the direction of b, the stress that drives the expansion is the shear stress that acts in the direction of b, as shown in the figure. Since an expansion of the loop is accomplished by displacing the dislocation line normal to itself along the whole periphery of the loop, the force that acts on the dislocation line, F = †b, is everywhere perpendicular to the dislocation line, and oriented so that it tends to expand the loop (Fig. 213) †b b †b †b Fig. 213: The force that acts to

expand a dislocation loop has magnitude, †b, and is perpendicular to the dislocation at every point. It follows that, in general, the force on a dislocation line has the magnitude given by eq. 211, and is directed perpendicular to the dislocation line, whatever the crystallographic character of the dislocation. 21.22 The critical resolved shear stress Plastic deformation is driven by shear stress, though we ordinarily think of it in terms of the tensile stress, ß, and quantify the yield behavior of a material by tabulating its tensile yield strength, ßy. To see the connection between the tensile stresses and yielding behavior, it is necessary to understand how a tensile stress can produce a resolved shear stress that acts on a dislocation. We shall do this for the simple case of uniaxial tension Let a tensile stress, ß, be applied to a cylindrical specimen, as illustrated in Fig. 21.4 A dislocation with Burgers vector, b, lies in a plane whose normal makes the angle, œ, with the

tensile axis. The direction of b makes the angle, ƒ, with the tensile axis The force on the vertical face of the bar is F = ßA 21.2 where A is the area of the face. The component of this force in the direction of b is Ft = Fcos(ƒ) = ßAcos(ƒ) Page 702 21.3 Source: http://www.doksinet Materials Science Fall, 2008 The area, A, of the plane on which Ft acts is A = A/cos(œ) 21.4 Hence the resolved shear stress in the direction of b is † = Ft/A = ßcos(œ)cos(ƒ) 21.5 The resolved shear stress has its maximum value when both œ and ƒ lie at 45º to the tensile axis, in which case † = †max = ß/2 œ 21.6 ƒ ß † b Fig. 214: A uniaxial tension, ß, produces a resolved shear stress, †, along the Burgers vector, b, of a dislocation that lies in a plane whose normal is tilted by œ from the tensile axis. There is always a frictional resistance to the motion of a dislocation through a crystal. In the simplest case, this is due to the need to break and restore

bonds as the dislocation is displaced through an elementary step. As we shall discuss below, several other microstructural mechanisms contribute to the frictional resistance, and these can be controlled to adjust the yield strength of the material. The frictional resistance has the consequence that, at least at low to moderate temperature, a dislocation is stationary until the resolved shear stress on it exceeds a critical value, call the critical resolved shear stress, †c. The critical resolved shear stress depends on the nature of the dislocation, which is characterized by the Burgers vector, b, and the plane in which the dislocation lies. In typical structural materials, †c has its least value when b lies in a close-packed direction (the direction which provides the minimum value of |b| for a lattice dislocation), and when the dislocation line lies in a close-packed plane. Since such dislocations are most easily moved, they tend to dominate plastic deformation; the microscopic

slip that happens during plastic deformation tends to be in close-packed directions on close-packed planes. The combination of the slip plane and slip direction is called the slip system Page 703 Source: http://www.doksinet Materials Science Fall, 2008 21.23 The tensile yield strength of a single crystal To initiate plastic deformation in a crystal that is stressed in uniaxial tension, the tensile stress, ß, must be large enough to produce a resolved shear stress, †, that exceeds the critical resolved shear stress for a least one set of dislocations within the material. From equation 21.5,   †c ßy = mincos(œ)cos(ƒ) 21.6 signifying that the yield strength is the minimum value of the factor in brackets, where †c is the critical resolved shear stress for a dislocation whose slip plane is tilted by the angle, œ, from the tensile axis, and whose slip direction (the direction of b) lies at the angle, ƒ. Two important results follow immediately from eq.

216 First, while the critical resolved shear stress is a material property, the tensile yield strength is not. It depends on the orientation of the crystal. Only its minimum value is a material property, and is realized when the most favorable slip system (minimum †c) has œ = ƒ = 45º. In general, ßy ≥ 2†c 21.7 Second, the preferred slip system may change with the orientation of the crystal. This does not happen in FCC metals. In FCC the preferred slip system is {111}<110>, the close-packed direction in the close-packed plane, and the angle between {111} planes is sufficiently small that there is always a {111}<110> set available for glide at stresses not too far above the minimum yield stress. In BCC metals, on the other hand, several slip systems are used, including {110}<111>, {112}<111>, and {123}<111>. Each of these systems has a <111> glide direction, which is the close-packed direction in BCC. They differ in the choice of glide

plane. The reason is that BCC is not a close-packed structure The geometry of the {110}, {112} and {123} planes is not all that different, and they have comparable values of †c. Glide occurs on the plane that is most favorably oriented Fig. 215: The octahedron of {111} planes in the FCC structure Page 704 Source: http://www.doksinet Materials Science Fall, 2008 Multiple slip systems are also used in HCP metals, but in this case the reason is the low symmetry of the HCP crystal structure. The only close-packed plane in HCP is the basal plane of the HCP cell. While dislocation glide is relatively difficult on the prismatic planes of HCP (those angled to the basal plane), these become favored when the tensile axis is either near (œ « π/2) or perpendicular (ƒ « π/2) to the basal plane. Similar considerations apply to other crystals with non-cubic symmetry Finally, note that while dislocation slip produces a shear strain, shear at an angle to the tensile axis causes a net

elongation along that axis, as illustrated in Fig. 216 The tensile stress that triggers slip is the yield strength that is measured in the tensile test. elongation slip ß ß Fig. 216: Slip on a plane that is angled to the tensile axis causes elongation in the direction of the tensile axis. 21.24 The yield strength of a polycrystal When the material is a polycrystal, the definition of the yield strength is somewhat ambiguous. Most polygranular materials exhibit two distinct kinds of yielding behavior when they are stressed in uniaxial tension: local yielding, and general yielding. ß ß Fig. 217: Local yielding by slip on the most favorable slip system in a polygranular body. Local yielding occurs when the applied stress is sufficient to trigger dislocation motion in the weakest element of the microstructure. In the ideal case, this happens when the applied stress is sufficient to move the most favorably oriented dislocations in the polygranular body, that is, when ß = 2†c

21.8 Page 705 Source: http://www.doksinet Materials Science Fall, 2008 Most real polygranular materials experience local yielding at even smaller stresses, since they contain internal stresses left over from processing or prior service that add to the applied stress, and heterogeneities that cause stress concentrations that magnify the applied stress. Local yielding leads to a situation like that shown in Fig 217; dislocation motion, and, hence, plastic deformation, is confined to a few grains in the interior of the polygranular body. The earliest incidents of local yielding ordinarily do not propagate, since the grains around the yielded grain are unlikely to have equally favorable slip systems. Nonetheless, local yielding produces a net plastic elongation of the overall sample. If a volume, V, in a sample of size, V, undergoes the strain, ‰, then it can be shown that the sample undergoes the overall strain, ‰, where V ‰ =  V  ‰ 21.9 It follows that

early yielding produces overall plastic strains that can be measured (at least with instruments that are sensitive enough), and causes the stress-strain curve to deviate from linearity. As the stress is increased beyond that required to cause local yielding, an increasing volume of the specimen is plastically deformed, and the stress-strain curve deviates more noticeably from its initial, linear slope. Eventually, the stress becomes sufficient to cause general yielding, in which the whole specimen behaves as an essentially plastic body. The general yielding of a polygranular specimen requires that its grains be able to deform simultaneously. This requires that the typical grain have enough active slip systems to accomplish an arbitrary change of shape. It can be shown that at least five independent slip systems are needed to accomplish an arbitrary change of shape. To activate five independent slip systems, the yield stress must exceed the value given in eq. 218 by a factor known as

the Taylor factor. For a cubic crystal, the Taylor factor is approximately 1.5, so the stress required for general yielding, which is often used as the theoretical definition of the yield strength, is ßy ~ 3†c 21.10 Note two features of yielding in polycrystals. First, whether local or general yielding is used as the criterion for the onset of plastic deformation, the material property that is most important is the critical resolved shear stress, †c. The microstructural control of yield strength is accomplished by manipulating the microstructure to adjust †c. Second, the yield of a typical polycrystal is gradual rather than abrupt. The yield strength is, therefore, largely a matter of definition. The usual practice is to define the yield strength as the "0.2% offset load", that is, the stress required to accomplish a plastic strain of 0.2% The method of taking the measurement is illustrated in Fig 218 A line is drawn parallel to the linear, elastic portion of the

stress-strain curve that intersects the strain axis at Page 706 Source: http://www.doksinet Materials Science Fall, 2008 a strain of 0.2%, and the yield stress is defined as the stress at which this line intersects the stress-strain curve. ßy ß } 0.2% ‰ Fig. 218: The method of measuring the 02% offset yield strength The 0.2% offset yield has the dual advantages that it is relatively easy to measure in practice, and, for most materials, corresponds fairly well to the stress required for general yielding. 21.3 MICROSTRUCTURAL CONTROL OF THE YIELD STRENGTH The yield strength is modified by controlling the critical resolved shear stress, †c. The value of †c is controlled by placing obstacles in the plane of the dislocation that make it difficult to move. The inherent value of †c is the Peierls-Nabarro stress that is due to the crystal lattice itself; the atoms near the dislocation core must be moved and reconfigured for the dislocation to move through an elementary

step. Additional increments to †c come from crystal defects that interact with dislocations. A dislocation is both an elastic and a crystallographic defect. Its elastic field interacts with the fields of any other defects that distort the crystal lattice, such as point defects and other dislocations. Its crystallographic defect interacts with any other defects that disturb the crystallography of the lattice, such as dislocations, grain boundaries and precipitate particles. The strength increment produced by a particular kind of hardening defect depends on the intensity of its interaction with dislocations and on its density within the material. If several distinct, independent hardening mechanisms, such as solute atoms, dislocations and grain boundaries operate in the same material, their strength increments are approximately additive. For this reason it is possible and useful to employ several different mechanisms simultaneously to control the yield strength. 21.31 Structural

hardening; the Peierls-Nabarro stress The inherent resistance to dislocation motion is due to the fact that the atoms near the core of the dislocation are displaced and reconfigured as the dislocation moves from one stable position to another. Put simply, atomic bonds must be broken and re-established, as Page 707 Source: http://www.doksinet Materials Science Fall, 2008 illustrated in Fig. 219 for the case of an edge dislocation in a simple cubic solid In order for the dislocation to move, the force that acts on it (†b) must be sufficient to accomplish this. The necessary value of the resolved shear stress, the Peierls-Nabarro stress, †p, has been estimated for a simple dislocation, and is given by the expression  2πd  2G †p = 1-ˆ exp- b(1-ˆ) 21.11 where G is the shear modulus, ˆ is Poissons ratio, b is the Burgers vector, and d is the spacing between adjacent slip planes. While eq. 2111 is based on a simple model that does not account for many of

the important details that influence dislocation behavior in particular crystals, such as the possibility of splitting into partial dislocations that was discussed in Chapter 4, it is qualitatively accurate, and provides a good basis for discussing the relative inherent strengths of various materials. Three of its features are particularly worth noting First, if eq. 2111 is solved for values of the variables that are typical for simple metals, it predicts a critical resolved shear stress of the order 10-4 to 10-3G, which is about the right order of magnitude for typical metals in their purest forms. (a) (b) Fig. 219: The reconfiguration of atoms necessary for an edge dislocation in a simple cubic crystal to glide by an elementary step. Second, eq. 2111 predicts that †p has its minimum value for slip on the atomic planes that are most widely separated. These are the close-packed, {111} planes in FCC metals, and the predicted value of †p for these planes is much less than for any

of the nonclose-packed planes. This explains why slip on close-packed planes dominates in the deformation of FCC metals On the other hand, BCC metals have several separate sets of planes whose spacings are not all that different, which helps to explain why several different slip systems are observed in BCC. Third, eq. 2111 predicts that the inherent strength of a material increases with its shear modulus. In particular, materials that have very high shear moduli should be exceptionally strong This is true High-modulus oxides such as Al2O3 and SiO2, high-modulus nitrides and carbides such as WC, TiC, TiN, SiC and Si3N4 are among the strongest Page 708 Source: http://www.doksinet Materials Science Fall, 2008 materials known. The ultimate case is diamond, which combines the highest known value of G with the lowest value of ˆ, and has, as predicted, the highest inherent strength of any known material. However, despite their exceptional yield strengths, these super-strong materials

are rarely used as structural materials. For reasons we shall discuss in the following chapter, their high yield strength has the consequence that they are exceptionally brittle, and fracture easily under tensile loads. They can support compressive loads, and, for this reason, have exceptionally high hardness. They are used as abrasives, in tool bits and as wear-resistant coatings where their high hardnesses provide exceptional properties. The inherent strengths of the common structural metals also follow eq. 2111, and scale with G. However, their inherent strengths (yield strength of a well-annealed, chemically pure material) are low Microstructural hardening mechanisms are used to make them more useful as structural materials. 21.32 Grain refinement One of the simplest and most useful ways to strengthen a structural metal is by refining its grain size. Grain boundaries are discontinuities in the crystal structure that act as barriers to dislocations. Since the orientations of the

active slip planes change abruptly at grain boundaries, slip must be transmitted indirectly from grain to grain. When a dislocation impinges on a grain boundary its stress field produces shear stresses on the potentially active slip planes of the adjacent grain These add to the applied load and help to propagate plastic deformation by the motion of independent dislocations in the adjacent grain. d Fig. 2110: A dislocation pile-up in a grain that has yielded Large grains are particularly efficient at transmitting strain to their neighbors. When a large grain slips, a number of dislocations glide along the preferred plane (or along closely spaced, parallel planes) and pile up against the grain boundary, as illustrated in Fig. 21.20 The stress at the head of such a dislocation pile-up is magnified; if the resolved shear stress is †, the stress at the head of a pile-up of n dislocations is n†. The larger the grain the more easily extensive pile-ups develop, and the more easily strain

is transmitted Page 709 Source: http://www.doksinet Materials Science Fall, 2008 across the boundary. The consequence is that the yield strength of a material decreases with its grain size. The yield strength of a typical metal varies with its grain size according to the HallPetch relation: ßy = ß0 + K d 21.12 where d is the mean grain size and K is a constant whose value depends on the material and the characteristics of the microstructure. Refining the grain size can lead to a substantial increase in strength It is relatively easy to control the grain size during the manufacture of structural metals and alloys. These are normally processed by hot deformation (rolling or drawing) into plates, sheets or bars. The hot deformation is done at temperatures above the recrystallization temperature of the metal. Recrystallization produces fine grains that grow on further exposure to elevated temperature, so the mean grain size is a function of the temperature of hot deformation and

the holding time after deformation. The grain size is controlled by cooling the material at a pre-selected time after its final hot deformation. 21.33 Obstacle hardening The other common hardening schemes employ microstructural obstacles that inhibit dislocation slip through the grain interiors. These barriers may be solute atoms, forest dislocations that thread through the slip plane, or small second-phase precipitates. In each of these cases, the obstacle is a localized barrier that can be idealized as a point in the slip plane. †b Ls R †b Fig. 2111: A dislocation, modeled as a flexible string, pressing against obstacles that are separated by the distance, Ls. Let a dislocation move over its slip plane under the action of a stress, †, that is significantly larger than the Peierls-Nabarro stress, †p. In this case, the atomic structure of the slip plane is relatively unimportant, and the dislocation behaves roughly like a flexible, extensible string. The dislocation has

an energy per unit length, T, which is an effective line tension, and has the approximate magnitude Page 710 Source: http://www.doksinet Materials Science Fall, 2008 1 T « 2 Gb2 21.13 The line tension opposes any increase in dislocation length. Let the dislocation encounter a linear array of barriers that oppose its motion. The dislocation presses against these barriers to create local configurations like that shown in Fig. 2111 To a reasonable approximation, the dislocation bows out between adjacent obstacles to an equilibrium radius T R = †b 21.14 where †b is the force on the dislocation. Assuming that R > Ls/2, where Ls is the spacing between adjacent obstacles, then the dislocation can only advance by penetrating through the obstacles themselves. The force that the dislocation exerts on the obstacles is due to its line tension, and equal to F = 2Tcos(/2) 21.15 where is the angle between the arms of the dislocation at the obstacle. If the obstacles are equally

spaced along the dislocation line, Ls cos(/2) = 2R 21.16 Let Fc be the force required for the dislocation to cut through the obstacle. As the stress, †, is increased, the angle, , decreases. The obstacle is passed when falls to c, where Fc cos(c/2) = ∫ c = 2T 21.17 In practice, the force, Fc, tends to scale with the factor Gb2 just as T does, and the resistance of the obstacle is characterized by the dimensionless strength, ∫c. Using equations 2113, 21.14, 2116 and 2117, the critical resolved shear stress for dislocation motion through a regular array of obstacles with spacing, Ls, is Gb †c « L ∫ c s 21.18 In the typical real case, the obstacles are distributed over the plane in an approximately random pattern with mean spacing Page 711 Source: http://www.doksinet Materials Science Fall, 2008 Ls = 1 n 21.19 where n is the average number of obstacles per unit area. Research has shown that when the obstacles are randomly distributed, equation 21.18 is changed

to †c « Gb∫ c3/2 n 21.20 Eq. 2120 predicts that the critical resolved shear stress increases with the shear modulus, with the 3/2 power of the obstacle strength, and with the square root of the obstacle concentration. The equation holds reasonably well for hardening by solute atoms (in the limit of small concentration), in which case ∫c is in the range 0.01-005, for hardening by "forest" dislocations that thread through the glide plane, with ∫c in the range 0.1 to 03, and for hardening by small precipitate particles, with ∫c in the range 05-08 Since the different types of microstructural obstacle have significantly different strengths, they make essentially independent contributions to the critical resolved shear stress. 21.34 Solution hardening Solute atoms never "fit" quite properly in the parent lattice, so there is always some local distortion of the lattice in the vicinity of the solute (the misfit defect). Moreover, the bonding around the solute

is never quite the same as that in the parent lattice, so there is also some difference in the local value of the elastic constants near the solute (this is often called the modulus defect). The net result is that a solute atom acts as a barrier to dislocation motion; a dilute distribution of solute atoms act as a distribution of obstacles of the type considered in the previous section. While the obstacle strength of solute atoms (∫c) is relatively small, the areal density in the slip plane is relatively large, even when the solute concentration is much less than 1%. Solution hardening is an effective hardening mechanism that is widely used When the solution is dilute, the yield strength is given by an equation of the form ß = ß0 + SGb c 21.21 where ß0 is the yield strength of a solute-free material with the same microstructure and S is a constant. (When the solute concentration becomes appreciable, the strain fields of the individual solute atoms overlap so they no longer

behave like discrete obstacles. In this regime the strength increases roughly as c2/3.) The strength (∫c) of the solute defect is primarily due to its misfit in the parent lattice. It follows that interstitial solutes strengthen an alloy much more effectively than substitutional ones, as illustrated in Fig 2112 Page 712 Source: http://www.doksinet Materials Science Fall, 2008 interstitial ßy substitutional Ôc Fig. 2112: Solute hardening in the dilute solution limit Interstitial solution hardening is more effective in BCC crystals, where the interstitial sites are small and asymmetric, than in FCC crystals, where they are larger and equiaxed. The reason, as discussed in Chap 4 for the particular case of the BCC structure, is the relatively large lattice strain associated with an interstitial defect in the asymmetric site. Carbon and nitrogen are the common interstitial solutes in structural steels, and they harden conventional, BCC steels much more effectively than stainless

steels that are stabilized in the FCC structure. On the other hand, while interstitials are less effective hardening species in FCC crystals, their solubility is relatively high. High-strength, nitrided stainless steels are high strength steels with many applications. Many commercial Al alloys are strengthened by substitutional solutes. These are even less effective hardening species on an atom-by-atom basis, but have high solubility in the FCC Al lattice. The yield point The diffusional mobility of solute atoms may also affect the strength, particularly when the solute is a mobile interstitial or when the test temperature is relatively high. The reason is that solute atoms are attracted to dislocations, and diffuse so that they accumulate there, forming solute atmospheres. compression tension Fig. 2113: The formation of an impurity atmosphere by migration of oversized solutes to the region of tensile strain near an edge dislocation. This phenomenon is illustrated in Fig. 2113 for

the case of a solute that is oversized for its position in the lattice, such as carbon in iron The lattice just beneath an edge Page 713 Source: http://www.doksinet Materials Science Fall, 2008 dislocation is strained in tension, and an oversized atom has lower energy if it is sited there. By the same argument, an undersized atom is attracted to the compressive region above the dislocation line. If the solute is mobile, it will diffuse to the dislocation line to form an impurity atmosphere. The most important engineering consequence of impurity atmospheres is the yield point observed in the room-temperature stress-strain curves of high-carbon steels. Interstitial carbon atoms have moderate mobility even at temperatures near room temperature, and migrate to form atmospheres around dislocations. These are, essentially, rows of carbon atoms just beneath the dislocation lines. Since there is a significant binding energy between the carbon atoms and the dislocation, the dislocation

cannot move until the resolved shear stress is sufficient to literally rip it away from its atmosphere. As a consequence, carbon steels that have formed dislocation atmospheres have high yield strengths that are very well defined. Plastic strain initiates at a sharp yield point in the stress-strain curve, as illustrated in Fig. 2114 Immediately following yielding a material that exhibits a yield point experiences yield point elongation, a plastic elongation at a lower value of the stress. The yield point elongation is due the fact that, once the dislocations are ripped free of their pinning atmospheres, they are free to move at significantly lower stress until work hardening mechanisms restore the strength. If the stress is controlled, and increased until the sample yields, the yield point elongation may occur rapidly and dramatically, and appear in the form of discrete bands of deformation across the sample. ßy yield point } ß yield point elongation ‰ Fig. 2114: The stress

strain curve of a material that exhibits a yield point If a material is stressed beyond its yield point, unloaded, and re-loaded immediately afterwards, the yield point disappears, since it takes some time for the carbon atmospheres to reestablish themselves by diffusion. If the sample is heated slightly, or simply left at room temperature for a time long enough for diffusion to occur, the yield point reappears. 21.35 Dislocation hardening Dislocations interact with one another. Since they are crystallographic as well as elastic defects, these interactions can be rather complicated, and a detailed discussion of them will not be given here. The two simplest examples are illustrated in Fig 2115 If Page 714 Source: http://www.doksinet Materials Science Fall, 2008 two edge dislocations that have the same sign (same orientation of the extra half-plane) approach one another in the same slip plane, their elastic strain fields repel and drive them apart. If two dislocations of opposite

sign approach one another in the same slip plane, their elastic fields attract, and they annihilate by joining their extra half-planes. compression tension compression tension Fig. 2115: Interaction of edge dislocations in the same plane Like dislocations repel, unlike dislocations attract and annihilate From the perspective of strength, however, the most important dislocation interactions are the interactions between a gliding dislocation and the other dislocations that cut through its glide plane (Fig. 2116) The dislocations that intersect the plane are called forest dislocations, and they provide obstacles to the motion of the gliding dislocation that, to a reasonable approximation, can be treated as point obstacles in the glide plane. Fig. 2116: A mobile dislocation that is resisted by forest dislocations in its glide plane. The dislocation-dislocation interaction is much stronger than the dislocation-solute interaction; the forest dislocations act as point barriers that have

strengths (∫c) that typically lie in the range 0.1-03 If the dislocations are randomly oriented, their density, n, the number of dislocation that intersect a unit area of the glide plane, is one-half of the volumetric density of dislocations, ®, which is called the dislocation density, and is defined as the total length of dislocation line per unit volume. The yield strength of a material increases with its dislocation density according to the relation ß = ß0 + åGb ® 21.22 Page 715 Source: http://www.doksinet Materials Science Fall, 2008 Eq. 2122 is in qualitative agreement with the obstacle model, eq 2120, and is reasonably well obeyed by structural metals and alloys. There are three common methods for controlling the dislocation density in structural materials: heat treatment, mechanical deformation and martensitic phase transformations. 1. Heat treatment A material is annealed at elevated temperature to remove dislocations and lower its strength Dislocations are

non-equilibrium defects, and heat treatment decreases their density by either of two mechanisms. The first is recovery If a material containing a high density of dislocations is annealed at a temperature high enough to permit dislocation climb, dislocations migrate and interact, both with one another and with free surfaces. Some of the dislocations are annihilated, others are gathered into stable, planar configurations, such as low-angle grain boundaries (called subgrain boundaries). The net effect is to leave the bulk of the volume relatively free of dislocations. The second mechanism is recrystallization If the dislocation density is high enough, and the material is heated to a temperature above its recrystallization temperature, then, as discussed in Chap. 11, new, defect-free grains nucleate and grow at the expense of the old, producing a microstructure that is relatively free of dislocations. While heat treatments can decrease the dislocation density, they do not eliminate it

entirely. A typical structural alloy that has been recrystallized and annealed has a dislocation density of 107 - 108/cm2 2. Mechanical deformation When a material is plastically deformed, dislocations slip and interact strongly with one another. Their interactions produce a significant and monotonic increase in the total dislocation line length. Hence the dislocation density, ®, increases with strain, and the material work hardens according to eq. 2122 When the mechanical deformation is done at high temperature, as it is during the hot deformation that is used to roll metal ingots into plates or sheets, work hardening is counterbalanced by recrystallization and recovery. To achieve a high residual dislocation density it is necessary to deform at relatively low temperature. Metal products that are strengthened in this way are said to be cold-worked or cold-rolled. A severely cold-worked metal has a dislocation density of 1010-1011, producing a dislocation hardening that can be two

orders of magnitude greater than that in the annealed condition. 3. Transformation strengthening Transformation strengthening is possible in materials that undergo martensitic transformations on cooling As discussed in Chap 11, the martensitic transformation changes the structure by shearing the parent lattice (FCC in the case of structural steel) into the product (BCC in steel). This mechanical shear produces a highly defective microstructure, and martensitic steels have very high strength in the asquenched condition. However, freshly created martensite is often very brittle The reason is that the martensitic shear is often accommodated crystallographically, by stacking faults and lattice twins, producing high internal stresses that promote fracture. This is a particular problem in high-carbon steels, since the trapped carbon produces a slightly tetragonal structure that makes it difficult for adjacent martensitic grains to fit together without strain. Page 716 Source:

http://www.doksinet Materials Science Fall, 2008 If the steel is properly alloyed, for example, by increasing the Ni content and decreasing the carbon, the mechanism of the martensitic transformation can be changed slightly so that the transformation strains are accommodated by dislocations (the product is called dislocated martensite). Martensitic steels of this kind combine very high strength with reasonably good toughness is the as-quenched condition. (In ancient times the best source of good, high-Ni iron was meteoritic material, which is often rich in Ni. The superior properties of steel that incorporated meteorite iron lends some credence to the legends that say that the storied blades of antiquity were handed down from heaven. If the movie is to be believed (fat chance) there was meteorite iron in the original Bowie knife.) 21.36 Precipitation hardening The final type of hardening obstacle is a small precipitate in the interior of the grain. As described in Chap. 11, such

precipitates are normally introduced by aging a slightly supersaturated material at relatively low temperature, so the precipitates nucleate primarily in the grain interiors. The volume fraction of the precipitates is determined by the phase diagram, and is, hence, fixed by the composition and temperature The size of the precipitates then depends on the aging time. The precipitates form as very small particles, and coarsen with time as the larger particles consume the smaller ones to decrease the total interface area. The yield strength of a precipitation-hardened material varies with the aging time and temperature as illustrated in Fig. 2117 The strength increases to a maximal value, the peak hardness, then decreases on further aging, or overaging. For a given composition, the strength increases more slowly, but rises to a higher value as the aging temperature is lowered. Lowering the aging temperature increases the peak hardness because it increases the volume fraction of precipitate

phase (that is, it increases the obstacle density, n, in eq. 20.20) The time required to reach peak hardness increases because diffusion is slower, which inhibits the particle coarsening that leads to peak strength. lower temperature ßy time Fig. 2117: The variation of yield strength with aging time for a precipitation-hardened material The reason that there is a peak in the hardness is a bit subtle. In the most common case, it is due to the fact that the hardening precipitates increase in strength with their size. When the precipitates are very small, they are generally coherent, and dislocations can cut Page 717 Source: http://www.doksinet Materials Science Fall, 2008 through them. They behave as cuttable obstacles with strengths (∫c) of the order of 05 As they grow, their strength increases, and, since their volume fraction remains constant, their mean separation increases as well. However, in the early stages of coarsening the increase in strength (∫c) generally

outweighs the increase in separation (Ls = n-1/2), so the material hardens. The peak in the yield strength comes from the fact that there is an upper limit to the strength of the obstacles, but no upper limit to their spacing. The maximum force that the arms of a dislocation can exert on an obstacle (Fig. 2111) is 2T, the force when the included angle, , is zero When the force required to cut through the obstacle approaches this value, the dislocation does not cut through the obstacle, but wraps around it, as illustrated in Fig. 2118 If the obstacle cannot be cut by the dislocation (an impenetrable obstacle) the dislocation line is held back until the stress becomes so high that the diameter (2R) of the dislocation loop is comparable to the obstacle spacing. When this is true the arms of the dislocation extend well beyond the obstacles and approach one another (Fig 2118(a)) The two arms of the bowed-out dislocation have opposite sign (note that the Burgers vector, b, points out of the

loops to the immediate left of the obstacles shown in Fig, 21.18(a), but into the loops to their immediate right). Hence the arms attract one another When they meet on the far side of the obstacle, they annihilate to create a continuous dislocation that lies beyond the obstacle row and leave behind loops of dislocation that surround each of the obstacles that was by-passed, as illustrated in Fig. 2118(b) A precipitate particle of maximum possible strength has a breaking angle, c = 0, or a strength, ∫ c = 1 In fact, because the arms of the dislocation attract one another, they help to pull the dislocation past, and an impenetrable obstacle has an effective strength nearer ∫c « 0.8 †b †b †b †b b b (a) (b) Fig. 2118: The interaction of a dislocation with impenetrable obstacles (a) The arms of the dislocation wrap around the obstacle and attract one another. (b) The arms intersect and annihilate, producing a propagating dislocation, and leaving dislocation loops

around the obstacles. Once a precipitate has grown so large that it is impenetrable, its strength is fixed, and is not changed by further growth. But the precipitates that have reached impenetrable size continue to grow; expressed in the terms used in eq. 2118, Ls increases while ∫c remains the same. The result is that †c, and, hence, the yield strength, ßy, decreases with Page 718 Source: http://www.doksinet Materials Science Fall, 2008 further aging. The peak strength is reached when the largest obstacles just become impenetrable The material overages if any further coarsening is allowed Some materials form very hard precipitates, which are uncuttable even when their size is very small. However, even in this case the hardness tends to increase to a maximum as the alloy is aged. The reason is that such precipitates ordinarily have such high interfacial tension with the alloy that nucleation is difficult and their volume fraction is sharply restricted when their size is

small. The increase in hardening with aging time is due to an increase in the volume fraction of the precipitate phase. 21.4 THE INFLUENCE OF TEMPERATURE AND STRAIN RATE 21.41 The variation of yield strength with temperature thermal degradation ßy athermal barriers thermal activation If the yield strength of a typical material is plotted as a function of its homologous temperature (T/Tm, where Tm is the melting temperature), the result is a curve that resembles that in Fig. 2119 The yield strength is relatively insensitive to the temperature over a range of intermediate values of the homologous temperature, but increases dramatically as the temperature is lowered toward zero, and decreases dramatically as it is raised to near the melting point. Ambient temperature is in the intermediate temperature regime for Al and its alloys, is slightly into the low-temperature regime for typical structural steels, and is in the high-temperature regime for low-melting metals like Pb. T/T m

Fig. 2119: Typical variation of yield strength with homologous temperature Low-temperature strength The rapid strength increase at low temperature is due to the strong temperature dependence of hardening by obstacles that can be cut or passed by thermal activation. These particularly include the weak, short-range obstacles created by isolated solute atoms. As the temperature increases, the increased amplitude of atomic thermal vibrations produces an effective vibration of the dislocation line, which permits it to cut through obstacles that could not be bypassed by the stress alone. Page 719 Source: http://www.doksinet Materials Science Fall, 2008 As a rough rule of thumb, the low-temperature strengthening mechanisms are ineffective at temperatures above 1/4 to 1/3 of the melting point. The increase in strength at low temperature is much more pronounced in the typical BCC metal than in the typical FCC metal. A principal reason is the greater strength of interstitial solutes in

BCC, which become increasingly effective as thermal activation becomes more difficult and the full potency of the interstitial solutes is revealed Typical carbon steels have very high strengths at cryogenic temperatures. Austenitic (FCC) stainless steels that are intended for service at very low temperature are also strengthened by adding interstitials, with nitrogen preferred to carbon because of its higher solubility in FCC alloys. While the low-temperature strength increment per unit solute content is lower for FCC, the high nitrogen solubility has the consequence that nitrided austenitic steels can achieve very high strengths at low temperature. Intermediate temperature strength At intermediate temperature the yield strength is a relatively weak function of temperature. The strength in this region tends to be controlled by dislocation-dislocation and dislocation-precipitate interactions. These present obstacles that have relatively large effective sizes; they spread over at least

several atom spacings in the slip plane, and are not easily passed by thermal vibrations of the dislocation Room temperature is well within the intermediate temperature regime for Al alloys, but is slightly below for typical structural steels. High temperature strength At temperature above 0.5Tm the yield strength begins to drop dramatically The principal reason is the increasing rate of solid state diffusion, which affects both the dislocations and the microstructural barriers. The microstructural effect is most important Dislocation configurations recover by climb and recombination, precipitates coarsen and overage, and grains grow; virtually all of the available microstructural barriers become ineffective. At the same time, the high diffusivity makes it possible for dislocations to climb at an appreciable rate, so the obstacles that remain are more easily passed. The only practical mechanism for hardening metals that are to be used at a significant fraction of their melting points

is precipitation hardening by precipitates that are thermodynamically very stable, so that they do not coarsen at a rapid rate, and relatively large, so that they are not easily passed by dislocation climb. The best high-temperature alloys are members of a class of materials known as superalloys. These are based on Fe, or, even better, on Ni, and are strengthened by large, blocky precipitates of Ni3X ordered intermetallics, where X may be Ti, Al, Nb, Ta or Mo. Such alloys are expensive, and they are difficult to process, but they are the workhorse alloys of the high-temperature turbine industry. Even better high-temperature strengths are exhibited by certain oxides (MgO and Al2O3), carbides (SiC), and nitrides (Si3N4), which have high melting points and high Peierls-Nabarro stresses to inhibit plastic flow. However, even at elevated temperature Page 720 Source: http://www.doksinet Materials Science Fall, 2008 these materials are relatively brittle, and are only rarely used in

safety-critical structures such as turbine engines. In addition to the degradation of the tensile yield strength, high temperature also makes the material liable to deform by creep if it is held under load. Creep deformation was discussed briefly in Chap. 19 In assessing the strengths of the common metals and alloys, it is useful to keep in mind the principle that the homologous temperature (T/Tm), rather than the actual temperature, governs the strength. Materials with low melting points, like Pb and eutectic Pb-Sn alloys, are very soft at room temperature. The reason is not so much their low inherent strength (though they do have relatively low shear moduli) but their high homologous temperature, which invalidates conventional deformation processes. Similarly, materials like pure Mo have relatively high room temperature strength, largely because of their high melting points. 21.42 The influence of strain rate on strength The yield strength of a ductile metal or alloy always increases

with the rate at which the sample is strained. However, the magnitude of the rate effect varies dramatically with the material and with the temperature at which the test is done. The reason is that the common rate effects have the same source as the temperature effects If deformation is thermally activated it becomes easier to accomplish as the rate of deformation is lower; more time is provided for thermal activation. To a good first approximation, increasing the strain rate is equivalent to decreasing the temperature. It follows that materials exhibit pronounced strain rate effects when they are tested in either the high-temperature or the low-temperature regimes, and are relatively insensitive to strain rate when they are tested at intermediate temperature. As might be expected, the strengths of structural steels are very sensitive to strain rate, while those of Al and its alloys are almost constant, even when the strain rate is changed by several orders of magnitude. 21.5 WORK

HARDENING When a material deforms plastically it also hardens. The yield strength increases with the strain, a phenomenon known as work hardening. The basic mechanism of work hardening was described above, when we considered how dislocations harden materials. As strain builds up inside the material, dislocations slip, intersect and interact with one another, as illustrated schematically in Fig. 2120 These interactions cause an increase in the dislocation density (total dislocation length per unit volume) that is monotonic in the strain. Page 721 Source: http://www.doksinet Materials Science Fall, 2008 Fig. 2120: Dense dislocation network in a severely strained material The yield strength increases with the dislocation density according to eq. 2122: ß = ß0 + åGb ® 21.22 The associated work hardening rate, œ, is, then dß åGb d® œ = d‰ = 2 ®  d‰  21.23 and depends on the rate at which dislocations multiply with the strain. A major part of the

fundamental research that has been done on work hardening concerns the behavior that is peculiar to single crystals and is, therefore, of limited engineering interest. There are two common representations of the work-hardening behavior of the polycrystalline metals and alloys that are of interest in engineering. The first comes from the parabolic stress-strain relation that is often assumed in continuum mechanics: ß = k‰n 21.24 where k is a constant, giving œ = nk‰n-1 21.25 Equation 21.25 is analytically simple, but does not have any simple association with the mechanisms of dislocation plasticity, and is not, in fact, very accurate. A more useful relation comes from the Kocks-Mecking model of work hardening. Assume that the material has undergone general yielding so that its dislocation distribution is well developed and reasonably homogeneous on the macro scale. Then the instantaneous value of the yield strength is given by eq. 2122, and the work hardening rate is given by

eq. 2123 To complete the model we require an expression for d®/d‰, the increase in dislocation density with strain. A detailed analysis suggests that d®/d‰ is the sum of two terms, one reflecting the rate of dislocation multiplication (hardening), and the second the Page 722 Source: http://www.doksinet Materials Science Fall, 2008 rate of dislocation annihilation (recovery). An analysis of the relation between ® and ‰ suggests that the multiplication rate is related to the frequency of intersection with forest dislocations, and, hence, proportional to the spacing between forest dislocations in the glide plane. It follows that d®   = C1 ®  d‰ + 21.26 where C1 is a constant. The rate of recovery is the rate at which dislocations annihilate, which happens predominantly through the intersection of dislocations that have a common plane. An analysis of this process gives d®   = - C2 ®  d‰ - 21.27 Summing 21.26 and 2127, and

substituting the result into 2123 gives the Kocks-Mecking relation for the work hardening rate œ = œ0 - Cwß 21.28 where œ0 and Cw are constants. œ ß Fig. 2121: The variation of the work hardening rate with the stress for aluminum alloys. The linear Kocks-Mecking relation is reasonably well obeyed by many engineering materials, and is particularly applicable to FCC metals and alloys. However, it does not apply to local yielding, nor to the transient behavior that may appear as local yielding develops into general plasticity. The typical variation of the work hardening rate with the stress for polygranular aluminum alloys is shown in Fig. 2121 It contains three distinguishable regimes The first, to the left-hand side of the figure, is associated with local yielding. The work hardening rate decreases rapidly as more and more local regions within the crystal yield and contribute to plastic deformation. The second regime is a transient stage as re- Page 723 Source:

http://www.doksinet Materials Science Fall, 2008 gions that have yielded locally link together and develop the microstructure an deformation pattern that produces general yielding. The third regime is the linear work hardening behavior that is observed after general yielding 21.6 PLASTIC INSTABILITY, NECKING AND FAILURE The final subject we shall discuss here is plastic instability, which defines the ultimate tensile strength of a tensile specimen, and triggers the necking deformation and failure if the specimen is stressed or strained beyond its ultimate tensile strength. 21.61 The Considere criterion To understand plastic instability, we must first understand how deformation actually happens in a specimen, such as a tensile bar, that is stretched in tension. Plastic deformation is never strictly homogeneous Just as a chain breaks at its weakest link, each increment of strain happens at that volume within the specimen that is instantaneously weakest. The macroscopic deformation

appears uniform because each small element that is strained is work-hardened. Because it is strengthened by work-hardening, the strained element ceases to be the weakest in the body, and the next increment of deformation happens at some other place. In this way local deformation works its way back and forth through the body. Necking happens when this process breaks down, and the strain remains concentrated at a particular site ß ß Fig. 2122: Incipient neck in a tensile bar To find the condition that produces plastic instability in a round tensile bar, consider the tensile specimen shown in Fig. 2122, and let the instantaneous strain thin a narrow region, as illustrated, with considerable exaggeration, in the figure. The cross-sectional area of the strained region of the specimen has decreased by dA, so the stress it must bear under the applied load, P, is increased by the amount dß = d(P/A) = - -dA  P dA = ß A  = ßd‰ 2 A 21.29 But the yield strength of

the material has also increased, and is dß dßy =  d‰  d‰ 21.30 Page 724 Source: http://www.doksinet Materials Science Fall, 2008 If dßy > dß, the material has hardened more than is necessary to support the increased load, and the deformation is stable. The neck will not grow, and the next increment of deformation happens at some other point along the bar If dßy < dß, on the other hand, the incipient neck cannot support its increased load and the subsequent strain is localized there, causing necking and failure. Since dß/d‰ is a decreasing function of strain (at least for the common structural materials), the cross-over from stable to unstable deformation happens when dß d‰ = ß 21.31 Equation 21.31 is known as the Considere criterion It states that plastic instability occurs when the work hardening rate falls below the true stress. œ = dß/d‰ plastic instability : ds/de = 0 su ß plastic instability œ=ß s ‰ e (a) (b)

Fig. 2123: True and engineering stress-strain curves for the same material, showing the points at which plastic instability occurs The slope of the engineering stress-strain curve at plastic instability is, from eq. 19.24 dß  d(ße-‰ ) e-‰ dß - ße-‰ d‰ ds  =0 -2‰  = = = e ß   de d‰ d(e‰ -1) e‰ d‰ 21.32 Hence plastic instability occurs at the maximum of the engineering stress-strain curve and defines the ultimate tensile stress, su. These relations are illustrated in Fig. 2123 Fig 2123(a) shows an example true stress-strain curve with the work hardening rate also plotted. Plastic instability intrudes when these two curves intersect. Note that there is nothing in the behavior of the true stress-strain curve itself to indicate that this has happened. Fig 2123(b) shows the engineering stress-strain curve for the same material In this case, the instability point is obvious Plastic instability intrudes at the peak of the stress-strain curve, which

defines su Page 725 Source: http://www.doksinet Materials Science Fall, 2008 If the stress-strain curve is parabolic, as in eq. 2124, then the uniform elongation is easily found from equation 21.25, and is ‰u = n 21.33 More generally, the stress at which instability occurs can be found graphically from a plot of the work hardening rate, œ, as a function of ß (Fig. 2121) It is the intersection of the work hardening curve with the line œ = ß, as illustrated in Fig. 2124 Finally, note that the Considere criterion matches a material property, the work hardening coefficient, œ, with a geometric criterion, the increase in applied stress with local strain. The latter is sensitive to the geometry of the specimen, and the simple Considere criterion, eq. 2131 only holds when the specimen is a cylindrical bar in uniaxial tension If the geometry of the specimen changes, the locus of plastic instability changes as well. For example, a thin sheet specimen in uniaxial tension

experiences local necking when dß ß = d‰ 2 21.34 Other criteria apply to other geometries. plastic instability œ œ=ß ß Fig. 2124: Location of the instability stress on a plot of œ vs ß 21.62 Tensile elongation In addition to determining the ultimate tensile strength, plastic instability determines the limit of the uniform elongation of a specimen. This is an important parameter in the forming of materials, since a part that is formed to strains beyond the instability limit develops non-uniformities in its cross-section, and is liable to fracture. The instabilities that are of interest in forming are, of course, those that pertain to the material geometry (usually a sheet) and the load configuration (usually biaxial) that is used. Specific forming tests have been designed to define forming limit diagrams that govern particular forming operations. Nonetheless, we can gain significant insight into the behavior of a material in forming by understanding the factors that

limit tensile elongation. Page 726 Source: http://www.doksinet Materials Science Fall, 2008 Fig. 2123(a) illustrates how the work hardening behavior limits uniform elongation The ultimate criterion is the intersection of the work hardening and stress-strain curves, which suggests that the work hardening rate should be made as high as possible to forestall this intersection. But this is deceptive In fact, the tensile elongation is affected by the yield strength, and by the work hardening behavior over the whole range of plastic strain. ß ‰ Fig. 2126: Illustration of the deleterious influence of a high early work hardening rate on the tensile elongation. The simplest way to increase the elongation of a typical material is to decrease its yield strength, particularly that part of the yield strength that is due to dislocations left over from prior deformation. If a material can be softened without significantly changing its work hardening behavior at higher strains, then there

is an added, "free" elongation that occurs before the stress reaches values that approach the work hardening rate. For this reason, alloys that are intended for forming are usually supplied in a well-annealed, soft condition ß ‰ Fig. 2127: An increase in the work hardening rate at high strains increases the total elongation. Second, the elongation can be increased by decreasing the extent of early yielding, where the work hardening rate is relatively high. As illustrated in Fig 2126, a high rate of Page 727 Source: http://www.doksinet Materials Science Fall, 2008 work hardening early in the deformation process raises the stress, and decreases the strain at which the stress becomes equal to the work hardening rate. Since early yielding is associated with heterogeneities in the microstructure, materials that are microstructurally homogeneous tend to have better tensile elongations Finally, one can increase the tensile elongation by increasing the work hardening rate

in the later stages of deformation, as illustrated in Fig. 2127 Several metallurgical tricks have been developed over the years to accomplish this. One of the more clever is incorporated in the so-called dual phase steels These are alloy steels that are processed so that their microstructures contain a fine mixture of relatively soft BCC ferrite, produced by a nucleation-and-growth transformation from the FCC austenite, and relatively hard martensite, produced by quenching a structure that is partly ferrite and partly residual austenite. The ferrite yields at low stress, but its deformation does not significantly affect the much harder martensite until work hardening has raised its yield strength to a value comparable to that of the martensite. The martensite then yields fairly late in the deformation process Its subsequent work hardening adds a significant increment to the large-strain work hardening rate of the alloy so that plastic stability is maintained to high strains. Page 728

Source: http://www.doksinet Materials Science Fall, 2008 Chapter 22: Fracture Stone-cutters fighting time with marble, you foredefeated Challengers of oblivion Eat cynical earnings, knowing rock splits, records fall down, The square-limbed Roman letters Scale in the thaws, wear in the rain. The poet as well Builds his monument mockingly; For man will be blotted out, the blithe earth die, the brave sun Die blind and blacken to the heart: Yet stones have stood for a thousand years, and pained thoughts found The honey of peace in old poems. - Robinson Jeffers, To the Stone-Cutters 22.1 INTRODUCTION 22.11 Engineering importance The most threatening form of mechanical behavior is fracture, in which an intact structure separates into two pieces. Sudden, unexpected fracture is the culprit in many of the more famous engineering disasters, including such tragedies as the Liberty ship crackups, the Ohio River and Kansas City Hyatt bridge collapses, the Tokyo and Sioux City commercial

airplane crashes, the in-flight destruction of the space shuttle Challenger, the Great Boston Molasses Flood of nineteen-ought-whatever-it-was, the crack in the Liberty Bell, and many others. In the background of these widely publicized disasters is a steady stream of crashed vehicles, split pipes, exploded tanks, collapsed structures, broken toys and malfunctioning electronic devices that failed because the materials inside them broke. While unexpected, catastrophic structural failures have plagued engineers for just about as long as structural materials have been used, an accurate understanding of crack propagation to failure has only come about in the last few decades. Fracture mechanics, the now large subfield of engineering that specializes in the study of fracture, was not even taught when I was a graduate student in Materials Science in the 1960s. The discipline was born in the late 1950s, largely as a product of research initiated and funded by the United States Navy. The U.S

Navy had a strong, practical interest in developing a theory of fracture During the Second World War, the standard method of construction of sea-going hulls had changed from riveted steel plates to welded ones. Welding was faster, cheaper and, in theory, produced a more reliable product. But sailors do not float on theory, and the Page 729 Source: http://www.doksinet Materials Science Fall, 2008 welded tankers and transports that were turned out by the hundreds in yards like Henry J. Kaisers shipyard in Richmond, California, had a most annoying habit of splitting in two, right through the center of the hull, with no prior warning. Since a vessel with half a hull is not seaworthy, the Navy was anxious to know what had gone wrong with these "liberty ships" and how the problem might be prevented in the future. The result was the field of Fracture Mechanics. The introduction to the subject that is given below here will necessarily be brief and superficial. It is,

nonetheless, sufficient to understand a good part of what happens in common material failures. 22.12 Why things fall apart Before we turn to the subject of fracture mechanics it is important to note that structural materials fracture for three fundamentally different reasons: buckling, necking, and cracking. The first two of these have virtually nothing to do with the materials resistance to fracture itself The phenomenon of necking to failure was discussed in the previous chapter. If a piece of material is loaded to a stress beyond its ultimate tensile strength, its plastic deformation becomes unstable and concentrates, thinning the cross-section of the piece and raising the true stress on that cross-section until it finally breaks. So long as a material has sufficient toughness (resistance to fracture) that it does not break before plastic instability sets in, its toughness does not matter very much. Once it begins to neck, it will inevitably fracture. Buckling is a similar

phenomenon that was mentioned briefly in Chapter 20. Buckling is an elastic instability. It is a common failure mode in structural columns that are loaded in compression, and in sheets, such as automobile fenders, that are crushed or dented. The simplest case is the column If a column is loaded in compression it bows out into an arc, whose deflection increases with the load. Eventually, the load reaches a value where the bowing angle is so great that increasing the load actually decreases the elastic resistance from the column, and it collapses. When a column buckles it forms a hinge at some point along its length where deformation concentrates. Unless the material is very ductile, it fractures at this hinge at some point during the buckling process. The toughness of the material is almost irrelevant. The elastic instability has made fracture inevitable, whatever the toughness may be. The third mechanism of fracture, unstable crack propagation, is the subject of this chapter. When a

material that contains a crack or geometric flaw is loaded, the applied stress concentrates at the tip of the flaw. When the local value of this stress reaches a critical level the material separates, and the flaw propagates as an unstable crack that splits the material apart. The material property that governs the stress at which the flaw propagates is its fracture toughness. But a knowledge of the fracture toughness is not sufficient to determine the load at which the material fractures. It is also necessary to know Page 730 Source: http://www.doksinet Materials Science Fall, 2008 the size and geometry of the flaws it contains, since these determine the stress concentration factors that magnify the applied stresses and lead to fracture. While buckling and necking are threatening failure modes, they are driven by the macroscopic stress and are, hence, relatively easy to incorporate into engineering design. Cracking is a much messier problem. Unless the material has very high

fracture toughness, the fracture stress is sensitive to the presence of flaws that are small, and difficult to detect, and these may propagate to catastrophic failure at nominal stresses that are well below the yield strength of the material. Compounding the problem, a material that is initially sound may develop critical flaws during service, though fatigue or stress corrosion cracking, or may lose its toughness during service via any one of several embrittling mechanisms. Catastrophic crack propagation can strike without warning in structures that meet all normal macroscopic design criteria, and has on a great many occasions. 22.2 FRACTURE MECHANICS 22.21 Crack propagation from a pre-existing flaw The stress concentration at a flaw Let a material specimen contain an internal crack of length, a, and let it support a load that would produce a nominal tensile stress, ßa, as illustrated in Fig. 221 If the tip of the crack is viewed under a sufficient magnification, it will appear

rounded, as shown in Fig. 222 Let its effective radius be ® ßa ßT ß ßa a r ßa Fig. 221: A section from a material that contains a crack of length, a, and is subjected to an applied stress, ßa. The stress concentration at the crack tip results in the local stress, ßT. Page 731 Source: http://www.doksinet Materials Science Fall, 2008 The stress at the tip of the crack is magnified by the stress concentration factor of the crack. If we assume (or approximate) elastic behavior up to the crack tip, the stress at the crack tip, ßT, is given by the relation ßT = Qßa a ® 22.1 where Q is a geometric factor that depends on the shape of the crack and the geometry of the specimen. A considerable part of the work of analytic fracture mechanics is concerned with the evaluation of the geometric factor, Q, and the specific consequences of local plastic deformation near the crack tip, but we do not need this sophistication to understand the fundamentals of fracture. Moreover,

we need not require that the flaw be a narrow crack, as shown in the figure. Blunt cracks, voids and inclusions in the material also produce stress concentrations that can be represented by equivalent cracks. ® Fig. 222: Magnified view of the tip of the crack in Fig. 221, showing the radius, ®. Equation 22.1 has three important qualitative features First, the crack tip stress, ßT , increases linearly with the applied stress, ßa. Second, it increases with the root of the crack length; the bigger the flaw, the higher the crack tip stress, and the greater the threat to the integrity of the sample. Third, it increases with the inverse root of the crack tip radius, ®. This is a critical factor in determining the relative crack susceptibility of materials If a material is soft and yields easily, then plastic deformation near the crack tip causes the crack to blunt, increasing the effective radius, ®, and decreasing the crack tip stress. In this sense, flaws in ductile materials are

self-healing. If the material is very strong, on the other hand, the cracks that appear in it remain sharp, and produce intense magnifications of the applied stress. This is the basic reason that strong materials, like silica glass, oxide ceramics, carbides and nitrides, and diamond are very brittle. Incipient cracks in such materials remain almost atomicly sharp. The critical fracture stress The simplest fracture criterion is the stress criterion suggested by Orowan. He proposed that the material has a breaking stress, ßB , and spontaneously fractures if the local value of the tensile stress exceeds this value. In this (somewhat oversimplified) model, which we shall adopt here, the crack in Fig. 221 propagates when the crack tip stress ßT = ßB 22.2 Since ßT increases with the crack length, a crack that has started to propagate will ordinarily continue to do so. Eq 222 is the criterion for unstable crack propagation to failure Page 732 Source: http://www.doksinet Materials

Science Fall, 2008 Eq. 222 refers to the local condition at the crack tip The stress that is meaningful to the engineer is the applied stress, ßa. Using eqs 221 and 222, the sample breaks when the applied stress, ßa, reaches the fracture stress, ßc, where ßc = Q-1 ßB ® 22.3 a If a material contains many flaws, as all real materials do, then the fracture stress is determined by the minimum value of the right-hand side of eq. 223 This defines the "weakest link" in the material. As we anticipated above, the tensile stress at which a material breaks, ßc, is not determined by its inherent breaking stress, ßB , but by a combination of ßB and the size and geometry of the worst flaw it contains. 22.22 The fracture toughness Fracture toughness There are four independent parameters on the right-hand side of eq. 223 Two of these, the geometric factor, Q, and the crack length, a, can be controlled or measured, and, in particular, can be controlled in fracture experiments

by fixing the geometry of the test specimen. The other two parameters, the inherent fracture stress, ßB , and the crack tip radius, ®, are extremely difficult to measure or predict. The reason is that both are influenced by the details of processes that occur at the crack tip as it approaches the critical condition. An inherent fracture stress, ßf, can be defined, for example, as the true stress at the final fracture of a tensile specimen. But this stress is sensitive to the geometry of the specimen, and averages behavior over a volume of material that is very large compared to that in which fracture initiates at the tip of a crack. The effective crack tip radius, ®, is influenced by plastic deformation or other relaxation processes that occur as the crack tip evolves into the unstable configuration, and is virtually impossible to measure. While ßB and ® cannot be measured, we can hope that they will have reproducible values for near-critical sharp-tipped cracks in a given

material. On this assumption, we define their product as the fracture toughness, Kc, where, from equation 223 Kc = Q ßc 22.4 a The fracture toughness, Kc, has the advantage that is readily measured in tests that employ samples with cracks of known length and geometry. It has the second advantage that it is an experimental number that is model-independent. There are theoretical alternatives to the Orowan criterion for crack propagation that we used above. In fact, the energy-based Griffith criterion, or modifications of it, is much more widely used. These alternative ap- Page 733 Source: http://www.doksinet Materials Science Fall, 2008 proaches lead to different expressions in the denominator of eq. 223 However, the different fracture criteria do not change the definition of the fracture toughness as given in eq 22.4 (They are also unimportant in the qualitative discussion of fracture behavior that is given below.) Plane strain fracture toughness There is, however, a problem

with the fracture toughness, Kc, that is defined in eq. 22.4 It is not an unique material property In particular, if Kc is measured for a number of specimens that have different thicknesses, T, the value of Kc varies with T as shown in Fig. 22.3 After an initial transient for very thin samples, the toughness decreases with T to asymptote at a constant value, called the plane strain fracture toughness, in the thick-sample limit. Kc KIc plane strain T Fig. 223: The variation of fracture toughness (Kc) with the thickness (T) of the test specimen. Kc asymptotes to the plane strain fracture toughness, KIc. The reason for this behavior can be explained from the basic relation of the Orowan model: Kc = ßB ® 22.5 While the inherent fracture stress, ßB , does not vary with T, the crack tip radius, ®, does. The reason is that value of ® reflects the extent of crack tip blunting by plastic deformation as the crack tip is loaded up to the critical stress. This deformation occurs in

two perpendicular planes (Fig. 224): deformation in the plane of the specimen tends to open the crack, while deformation in the plane perpendicular to the specimen tends to thin the specimen along the line of the crack tip (Fig. 224(b)) The feature that changes with the thickness of the specimen is the thinning in the thickness direction. The reason is illustrated in Fig 225 Plastic deformation is driven by shear, and concentrates along planes at 45º to the tensile axis, where shear is greatest. The high stress at the crack tip is confined to the immediate vicinity of the crack tip, and extends for only a short vertical distance along the sample, as shown schematically in the figure. When the sample is thin, like the sample on the left in Fig. 225, it is possible to extend Page 734 Source: http://www.doksinet Materials Science Fall, 2008 shear along 45º lines across the whole width of the specimen while remaining within the region of high stress. Hence the specimen thins

significantly in the through-thickness direction. This deformation contributes to crack blunting, increasing ® and, hence, Kc When the specimen is thick, like the right-hand sample, deformation can extend only a short distance in from the lateral surfaces of the specimen before the shear planes outreach the region of high stress. The result is that only the edge of the specimen thins Over most of the crack length through the specimen, only the deformation in the sample plane contributes to crack blunting. The crack tip radius, ®, and, hence, Kc are smaller ß ß Fig. 224: T1 Side and front views of a thin fracture toughness specimen showing plastic deformation in the sample plane and thinning in the thickness direction. When the through-thickness deformation can be ignored, only the in-plane deformation contributes to crack blunting, and the sample is said to be in plane strain. The value of the fracture toughness in the plane strain limit is called the plane strain fracture

toughness, and denoted KIc. The plane strain fracture toughness is the asymptotic value for large sample thicknesses (Fig. 223) It is also the minimum value of the fracture toughness The plane strain fracture toughness, KIc, is the "fracture toughness" that is ordinarily tabulated and used. It is the best measure of toughness for two reasons First, it is a material property that can be used with some confidence for design purposes. Second, since it is the minimum value of the toughness, it provides a conservative estimate of the fracture stress. It follows from eq 224 and Fig 223 that ßc ≥ Q-1 KIc 22.5 a A structure is definitely safe with respect to crack propagation so long as the peak stress remains below the value of the right-hand side of eq. 225 (To ensure safety, we should Page 735 Source: http://www.doksinet Materials Science Fall, 2008 also take the geometric constant, Q, from the worst-case flaw geometry.) In the following we shall use this conservative

estimate for ßc. d T1 Fig. 225: ß T2 Front view of thin (T1 ) and thick (T2) fracture specimens (the side view of both would resemble Fig. 224), showing the 45ºC planes of maximum shear and the regions of significant plastic deformation. The crack tip stress varies with height (d) as shown at left. 22.23 Fracture-sensitive design Eq. 225 shows that the critical fracture stress depends not only on the fracture toughness, KIc, a property of the material, but also on the crack length, a, which is the length of the worst flaw in the material. It follows that fracture-sensitive designs must incorporate some method for determining what the worst flaw in the material is The importance of flaw detection and characterization has given rise to a whole subfield of engineering, which is called NDE, the non-destructive examination of materials. Experts in NDE employ a variety of techniques to detect and measure structural flaws. These range from fairly simple methods, such as visual

inspection, often with the aid of dyes or magnetic particles that makes surface cracks more visible, to complex methods that employ x-rays or ultrasonic waves and elaborate computations to visualize results in three dimensions. A detailed discussion is beyond the scope of this course However, as is the case in every field, the more elaborate the inspection, the longer the time and the higher the cost. Engineers try to design structures so that critical flaws will be large and relatively easy to find. Fail-safe design In fail-safe design one uses a material that is so tough that crack propagation is highly unlikely at stresses below the ultimate tensile strength. If we conservatively estimate the toughness by the plane strain fracture toughness, KIc, then the critical crack length, ac, the length of a crack that would propagate to failure under the stress, ß , is Page 736 Source: http://www.doksinet Materials Science Fall, 2008 KIc2 ac = Q-2 ß  22.6 The

critical flaw size increases with the square of KIc. If the fracture toughness is sufficiently large ac is large and easily detectable (for example, a significant fraction of the crosssection of the device) even if ß were to equal the ultimate tensile stress, ßu A material that satisfies this condition will fail, if at all, at its ultimate tensile strength, and the danger of unstable crack propagation can be ignored. A common approach to fail-safe design is to ensure that the critical crack length is so large that the sample will undergo general yielding before fracture. Since common design rules require that the maximum operating stress be well below yield, this rule ensures a good margin of safety against fracture. Under this criterion, fail-safe design suggests the use of materials whose characteristic length KIc2 a* =  ß  22.7 y is a significant fraction of the section thickness of the specimen. The characteristic length, a*, is a material property that is

often used as a "figure of merit" for the quality of a structural material. Flaw-sensitive design The problem with a fail-safe design is that it virtually rules out the use of highstrength materials. Since, as we shall discuss further below, fracture toughness tends to decrease with yield strength, materials with very high values of KIc tend to have very low strengths. While the use of low-strength materials in engineering structures is not limited by fracture, it is limited by yield, since a structure must ordinarily be designed to operate at no more than a fraction of its yield strength. This is a particular problem in weight-critical structures such as aircraft, where low strength and high safety margins translate into high weight and poor performance. Since aircraft structures are also safety-critical, a major part of the effort to produce viable, flaw-sensitive designs has been in the aerospace industry. Flaw-sensitive design is based on the symbiotic use of engineering

design, materials selection and non-destructive inspection. If ßD is the maximum value of the design stress (with appropriate safety factors) then the critical flaw size for unstable crack propagation is KIc2 ac ≥ aD = Q-2 ß  D Page 737 22.8 Source: http://www.doksinet Materials Science Fall, 2008 where the geometric factor, Q, is chosen to represent a realistic "worst case". The structure is safe if the non-destructive testing procedures that are specified can reliably detect flaws that are significantly smaller than aD. The difficulty with this procedure is that one must not only consider flaws that are present in the structure as-manufactured, but must also consider the possibility of crack growth during service by such mechanisms as fatigue and stress corrosion cracking. If crack growth is a threat, as it invariably is in aircraft, then the structure must be periodically re-inspected. Despite these difficulties, flaw-sensitive design is widely

and successfully used in advanced structures. 22.3 MICROSTRUCTURAL CONTROL OF FRACTURE TOUGHNESS The mechanical consideration that most often governs the selection of a structural material for service in a fracture-sensitive structure is the combination of yield strength, ßy, and plane strain fracture toughness, KIc. Both strength and toughness are critical properties since failure may occur through either plastic deformation or fracture. The combination is important since strength and toughness have an inverse relation to one another; an increase in strength at given temperature almost invariably leads to a decrease in fracture toughness (Fig. 226) In the design or selection of materials for cryogenic service it is desirable to maximize the strength-toughness combination or, at least, to achieve values that lie within a "design box" in a strength-toughness plot that is bounded by the minimum acceptable strength and toughness values. Since both strength and toughness vary

with the temperature the only strictly meaningful design box is one that is defined at the intended service temperature. K IC ßy Fig. 226: Typical relation between yield strength and fracture toughness for a ductile material. There is no reliable quantitative theory of the strength-toughness relation of structural alloys. However, research on the mechanisms of yield and fracture combined with specific studies of the behavior of materials at temperatures has produced a qualitative understanding of the strength-toughness combination that is useful for materials selection, quality control and new alloy design. Page 738 Source: http://www.doksinet Materials Science Fall, 2008 22.31 The fracture mode At the micromechanical level the fracture of a material is either ductile, in which case the material is torn apart after considerable local plastic deformation, or brittle, in which case the crack propagates with very little plastic deformation. In most cases there is a first-order

correspondence between the level of toughness and the fracture mode: a change from a ductile to a brittle fracture mode causes a substantial drop in the fracture toughness. It follows that the fracture mode is the first concern in interpreting the strengthtoughness characteristic of a material ß ß Fig. 227: Brittle crack propagation by intergranular fracture. Since the simplest fracture modes are the brittle modes, we shall describe these first. In a polygranular material, brittle fracture can occur in either of two modes. The first is intergranular fracture, in which the crack propagates by separating grains along the grain boundaries, as illustrated in Fig. 227 ß ß Fig. 228: Brittle crack propagation by transgranular cleavage. The second brittle fracture mode is transgranular cleavage, in which the crack propagates by fracturing the individual grains, as illustrated in Fig. 228 Cleavage describes the mechanism of brittle fracture of a crystal. When a crystal cleaves it

separates along specific atomic planes, the preferred cleavage planes whose separation is easiest. In metals and Page 739 Source: http://www.doksinet Materials Science Fall, 2008 covalent crystals, these tend to be the planes with the most dense atomic packing, since fracture along close-packed planes breaks the fewest bonds. In ionic crystals, the preferred cleavage are planes that are both relatively close-packed and electrically neutral, like the {100} planes in the NaCl structure. Because a propagating cleavage crack seeks out particular crystal planes, it must reorient itself when it passes across a grain boundary. This has the consequence that transgranular cleavage cracks follow a somewhat tortuous path through a polycrystal, and tend to be arrested at grain boundaries. Ductile fracture There are, in fact, several fracture mechanisms that differ in micromechanical detail that are properly called ductile. The mechanism that is most important is microvoid coalescence The

mechanism is illustrated in Fig 229 The material does not so much fracture as it rip itself apart; the process is often referred to as ductile rupture. It occurs in two steps Voids nucleate at inclusions, large precipitates or microstructural heterogeneities, then grow to link up with one another. Inclusions, such as oxides and sulfides, are the dominant nucleation sites for microvoids in most structural alloys. These create voids through inclusion fracture or decohesion at the inclusion-matrix interface in the crack-tip strain field. The voids grow by plastic deformation, and link up to propagate the crack. ß ß Fig. 229: Crack propagation by ductile rupture. The fracture surface is left covered with "ductile rupture dimples", as shown in the scanning electron micrograph on the right. Ductile fracture almost invariably produces a high fracture toughness, both because the crack tip is blunted (high ®) and because ductile rupture requires local tensile stresses of the

order of the tensile strength (high ßB ). The ductility of the material at the crack tip may not be apparent to the naked eye; since deformation is confined to the region of the crack tip the macroscopic fracture surface may appear rather flat and brittle. But a closer examina- Page 740 Source: http://www.doksinet Materials Science Fall, 2008 tion, for example, at high magnification in a scanning electron microscope, reveals that the fracture surface is rough, and densely decorated with small craters, called ductile rupture dimples, that are left over from the voids that grew and joined to form the fracture surface. 22.32 Choice of fracture mode; the ductile-brittle transition The choice of fracture mode depends, basically, on the yield strength of the material in the region near the crack tip. If the material undergoes extensive yielding at crack tip stresses that are below the critical stress for the easiest of the brittle fracture modes, then the material will fracture in the

ductile mode. If the material is so strong that critical stress for brittle fracture is reached before the material has yielded over a sufficient volume to blunt the crack tip, then the fracture mode will be brittle. This is the essential reason that soft materials, like pure Al, Cu and Au, invariably fracture in a ductile mode while very hard materials, such as silica, alumina, silicon carbide and diamond, are invariably brittle. Most of the useful structural materials have intermediate strength, and can be either ductile or brittle, depending on the microstructure and the temperature at which they are used. Since strength is a function of temperature, these materials undergo a ductile-brittle transition as the temperature is lowered (Fig. 2210) At high temperature the material fractures in a ductile manner by a microvoid coalescence mechanism and has a relatively high fracture toughness. When the temperature falls below the ductile-brittle transition temperature, TB , the mode of

crack propagation changes to brittle fracture either by transgranular cleavage or intergranular separation, whichever is easiest. Fig. 2210: The ductile-brittle transition; the fracture toughness decreases dramatically over a narrow range of temperatures near TB . Page 741 Source: http://www.doksinet Materials Science Fall, 2008 The qualitative source of the ductile-brittle transition and its relation to the yield strength can be illustrated by the "Yoffee diagram" shown in Fig. 2211, which represents the relative likelihood of plastic deformation and fracture at the tip of a pre-existing crack. As the applied stress is increased toward failure the stress at the crack tip reaches one of two levels first: the "yield" stress, ßY, at which significant plastic deformation occurs, or the brittle fracture stress, ßB , at which the crack propagates in a brittle mode by the most favorable mechanism. Extensive plastic deformation at the crack tip limits the local

stress and inhibits brittle fracture. Hence the fracture mode is ductile and the toughness high if ßY < ßB . The ductile-brittle transition temperature, TB , is that at which ßY rises above ßB ßII ß ßI ßY TB T Fig. 2211: The Yoffee diagram, illustrating the source of the ductile-brittle transition. ßI and ßII are the critical stresses for the transgranular and intergranular fracture modes The "yield" stress in the Yoffee diagram is a qualitative concept that is not precisely defined by any available theory. It is most closely related to the yield strength under plane strain conditions, whose value is significantly above the tensile yield strength, ßy. However, ßY varies with ßy, so the ductile-brittle transition is most pronounced in alloys whose yield strengths increase rapidly at low temperature. The prominent example is carbon steel, in which carbon solutes in the interstices of the BCC structure cause a dramatic rise in strength as temperature is

lowered. (The liberty ships were made of welded carbon steel, and most of those that failed broke up in cold water. The Titanic was also made of carbon steel, and recent tests on recovered samples show that its hull would be brittle in any ocean that was not almost boiling.) The ductile-brittle transition is less commonly observed in FCC materials, such as austenitic steels, largely because of the lower increment to the low-temperature yield strength by solute impurities. As suggested by the Yoffee diagram, the fracture mode below TB is that which provides the smallest fracture stress, ßB . In BCC material this may be either transgranular cleavage or intergranular separation. In FCC material the brittle mode is ordinarily intergranular While there are isolated observations of transgranular cleavage in FCC alloys, the cleavage stress is usually high enough that no brittle transition is observed unless an intergranular fracture mode intrudes. Page 742 Source: http://www.doksinet

Materials Science Fall, 2008 The ductile-brittle transition is significant to design for two reasons. The first is the toughness itself. The materials that are brittle in service are those that are used at temperatures below the ductile-brittle transition It follows that fail-safe design must be done using material properties at the lowest service temperature that matters. Many structural design codes, which are guided by the knowledge that has been acquired since the liberty ship fiasco, require that structures be designed to material properties measured 15º below the lowest intended service temperature. (This provision can lead to fiascoes of its own I was once peripherally involved in the design of a high-field superconducting magnet. The pertinent authorities reasoned that, since the magnet case was a container for liquid helium, the cryogen for the superconductor, it qualified as a pressure vessel. Hence they required that it be designed under the ASME Boiler and Pressure

Vessel Code, which incorporates the "service temperature minus 15º" rule in some of its pertinent sections. The problem was that the temperature of liquid helium is 4.2K We offered a Nobel Prize nomination to anyone who designed a successful fracture test that met the code, but there were no takers.) It also follows that the designer (or, more appropriately, the person immediately responsible for materials selection and quality control) must pay some attention to the metallurgical factors that influence the ductile-brittle transition. A structural material that tests superbly can be a disastrous choice if metallurgical variations that are permitted in the material specifications cause a significant upward movement in TB . 22.33 Suppressing brittle fracture The understanding of the ductile-brittle transition that is gathered in the Yoffee diagram also suggests useful microstructural mechanisms to lower or eliminate the ductilebrittle transition. One obvious method is to lower

the yield strength (Fig 2212(a)) The low-temperature strength increment can be specifically decreased by removing interstitial solutes or by "gettering" them into relatively innocuous precipitates or second phases. For example, ferritic steels that are intended for low-temperature service are often given "intercritical" heat treatments that gather carbon into isolated pockets of retained austenite phase or are alloyed with Ti to getter carbon into precipitates. ß TB ßB ß ßB ßy ßy TB T (a) T (b) Fig. 2212: Lowering TB by (a) lowering the yield stress; (b) raising the brittle fracture stress. Page 743 Source: http://www.doksinet Materials Science Fall, 2008 The second obvious method is to raise the brittle fracture stress (Fig. 222(b)) The best metallurgical method for doing this depends on the microstructural mechanism of the brittle fracture that defines TB . If the fracture is intergranular its source is either a grain boundary contaminant,

such as the metalloid impurities S and P in steel, or an inherent weakness of the grain boundary, as is apparently found in Fe-Mn alloys and in intermetallic compounds such as Ni3Al. In the case of chemical embrittlement the alloy may be purified of deleterious surfactants, alloyed to getter these into relatively innocuous precipitates, or heat treated to avoid the intermediate temperature regime at which these impurities segregate most strongly to the grain boundaries. When the grain boundaries are inherently weak the metallurgical solution is the addition of beneficial grain boundary surfactants that serve to glue them together. The most prominent of the beneficial surfactants is boron, which is extremely effective in suppressing intergranular fracture in Fe-Mn steels and in Ni3Al intermetallics. Carbon is also an effective surfactant in Fe-Mn steels when it is present in low concentration. When the brittle fracture mode is transgranular, as it is in typical ferritic steels, the most

effective microstructural technique is to decrease the grain size of the alloy. This raises the critical stress for transgranular cleavage by decreasing the mean free path of an element of cleavage fracture. Of course, grain refinement also raises the yield strength, which tends to increase the ductile-brittle transition temperature. But in many alloys, and particularly in BCC structural steels, the influence of grain size on the fracture stress significantly outweighs its influence on strength (Fig. 2213) Grain refinement is a widely used method for controlling TB in ferritic (BCC) steels and weldments. ß ßB TB ßy T Fig. 2213: Lowering TB by grain refinement when brittleness is transgranular. Transgranular cleavage can also be suppressed by adding second-phase particles that interrupt crack propagation and blunt cleavage cracks. Interestingly, relatively "dirty" alloys that contain a high density of inclusions may have lower values of TB that alloys with greater

purity. The inclusions lower the toughness in the ductile mode, above TB , but also lower the transition temperature by interrupting propagating cracks. The result may be a surprising increase in toughness when the service temperature of the alloy is slightly below the transition temperature of the "clean" material. (I once consulted with a manufacturer of heavy vehicles who was shocked to encounter a brittle fracture problem Page 744 Source: http://www.doksinet Materials Science Fall, 2008 when his supplier "improved" the chemical quality of the steel he was using for heavy structural members.) 22.34 Raising the toughness in the ductile mode The fracture mode that is conducive to a favorable combination of strength and toughness is ductile rupture with significant plastic deformation before fracture. The characteristic variation of the fracture toughness of a ductile material with the yield strength at constant temperature is shown in Fig. 226 Over the

intermediate strength range of greatest practical interest the toughness decreases monotonically as the strength is raised. Since ductile fracture ordinarily occurs through the nucleation and growth of voids, the fracture toughness can be improved by addressing either void nucleation or void growth. Void nucleation The most effective way to increase the toughness of a high-strength structural alloy is to eliminate void nucleation sites. Since these are, ordinarily, inclusions that result from impurities in the alloy, alloy purification is an effective means for raising the toughness. A change in the inclusion count at constant microstructure causes an increase in the plane strain fracture toughness roughly according to the relation KIc fi ßy 22.9 Nv where NV is the volume density of active inclusions. Interestingly, eq 229 predicts that the inclusion count has a much stronger influence on fracture toughness as the yield stress rises, which suggests that the effect should be most

apparent at the lowest temperatures and in the highest-strength ductile steels. This prediction is in qualitative agreement with a number of recent observations on the behavior of high-strength steels and weldments. When it is impractical to eliminate inclusions, it is often beneficial to change their shape or chemical composition to improve their resistance to void nucleation. For example, the oxides and sulfides of iron that are the dominant inclusions in ordinary structural steels tend to deform into long "stringers" during processing that are particularly efficient void nucleation sites. A common method for improving toughness is "inclusion shape control", in which minor chemical additions are used to gather O and S into stable precipitates that are spherical in shape and less likely to nucleate voids. The common additives are Ca and rare earth metals such as Ce and La. Void growth For a given inclusion distribution the best available ductile fracture theories

lead to predictions of the general form Page 745 Source: http://www.doksinet Materials Science Fall, 2008 KIc fi ‰f Eßy 22.10 where E is Youngs modulus, ßy is the tensile yield strength, and ‰f is the strain to failure, whose precise definition (and power) varies slightly from one model to another. The explicit dependence of the fracture toughness on the yield strength suggests that the two should vary together, in with the experimental data that invariably shows a decrease in toughness as the strength rises (Fig. 226) The resolution of this discrepancy lies in the dependence of the failure strain on the yield strength; ‰f decreases strongly and monotonically with ßy at constant temperature. Its influences dominates, so KIc decreases with ßy as well. Eq. 2210 suggests that the best way to increase the strength-toughness relation, KIc(ßy), is to improve the work-hardening characteristics of the alloy so that the failure strain increases. This is difficult to do in a

controllable way, particularly since the microstructural theory of work hardening is not very well developed. While there has been progress, the microstructural tricks that are used are beyond the scope of this course. 22.35 Raising the fracture stress in the brittle mode There are many engineering structures that must be made out of high-strength materials that fracture in a brittle mode. Examples include optical devices, like windows and optical fibers, semiconducting devices that employ brittle silicon chips, and very high-temperature devices for which strong ceramics are the only viable option. In all of these cases it is important to use materials that maximize toughness in the brittle mode. There are three generic methods for increasing the toughness of brittle materials: eliminating flaws, introducing compressive stresses and increasing crack growth resistance. Eliminating flaws Since a brittle material fractures at the worst flaw it contains, its fracture stress (which is the

apparent value of the "ultimate tensile strength" in a tensile test) is a very sensitive function of its flaw distribution. Any successful method of eliminating the more severe flaws has a very beneficial effect on the fracture stress. The best method for controlling the maximum flaw size is, of course, care in manufacturing and handling. In addition, at least three methods have been used to eliminate flaws after manufacture. The first is based on the fact that most of the severe flaws originate at the surface If the surface is coated with a ductile material that penetrates flaws, such as a transparent, ductile polymeric coating on glass, then the surface flaws can be sealed and, effectively, eliminated. The second method is based on the same approach: the mitigation of surface flaws. The surface is lightly etched with a chemical that preferentially attacks surface cracks and blunts them by removing material from the crack tips. Since flaws in brittle materials are Page 746

Source: http://www.doksinet Materials Science Fall, 2008 ordinarily very sharp, a chemical blunting of this type can significantly increase the fracture stress. A third method, that is imaginative, if it is not generally applicable, is useful for thermoplastic materials like silica glass that can be rewelded after breaking. When a brittle material is stressed to failure it breaks at its most severe flaw, so the remaining pieces of the material necessarily have a higher fracture strength than the original. If the material can be repaired after breaking, it is stronger. By repeating this process, the severe flaws in the material can be eliminated and the strength (fracture resistance) raised. Compressive stress Cracks propagate in tension. If a compressive stress is added across the plane of the crack, the fracture stress is increased by the amount of the compression, since this must be overcome before the crack-tip stress turns tensile. A compressive stress can be added at the

surface of a brittle material by chemical doping or by heat treatments that cause the material near the surface to expand relative to that in the interior. A familiar example is "tempered glass", which is heat treated to produce a compressive surface layer Crack growth resistance The crack growth resistance of brittle materials can be improved in two ways. The first, and most universal, is the use of internal heterogeneities to inhibit crack growth, as is done to inhibit transgranular cleavage in metals. The most effective microstructural heterogeneities are second-phase particles, particularly second-phase particles that are ductile, or that delaminate at the particle-matrix interface to blunt propagating cracks. A much more effective device is transformation toughening, which uses stress-induced martensitic phase transformations to create plastic deformation and blunt propagating cracks. This method is restricted to a few materials, the most important of which is ZrO2

(zirconia) and its alloys with yttrium. The reason is that the material must contain particles that are metastable with respect to martensitic transformation, but so slightly metastable that they transform under load. Moreover, it must be possible to distribute the transforming particles densely through the microstructure, so that they are available when and where they are needed. Despite these restrictions, a number of transformation-toughened ceramics have been developed, and are used in such commercial products as ceramic nails that will not corrode, and ceramic-headed golf clubs that are exceptionally light. 22.4 FATIGUE Fatigue was reviewed briefly in Chapter 19. It is a mechanism by which materials that are subject to cyclic stresses become liable to catastrophic fracture over time. It is a serious problem in engineering, and a major source of unexpected failure. Page 747 Source: http://www.doksinet Materials Science Fall, 2008 The possibility that a nominally sound

structure will deteriorate through fatigue has been recognized for almost two centuries, or virtually as long as engineering structures have been made of metal alloys that may experience fatigue. The word itself is an old one, and expresses the point of view that the metal has somehow become "exhausted" and unable to maintain its integrity. Sometime in the middle of the nineteenth century the "exhaustion" theory was challenged by the "crystallization" theory Fracture surfaces that are produced by fatigue usually appear bright and shiny. Since it was not yet known that metals are crystalline, and since the materials that were known to be crystalline were brittle compounds with shiny surfaces, the notion was promulgated that fatigue resulted from the crystallization and consequent embrittlement of metals that were amorphous in the normal, ductile state. Unfortunately, both the "exhaustion" and the "recrystallization" models have

considerable resilience, and can still be found in what purports to be reliable literature in the field of mechanics. The "exhaustion" theory is especially robust Its modern form suggests that cyclic deformation deposits energy in the material that accumulates until it becomes so great that atomic bonds are disrupted and the material falls apart. In fact, the fatigue process is not all that mysterious. Fatigue is, basically, a mechanism for growing cracks, which, if their growth is not stopped, will eventually reach critical size and propagate to failure While we speak of "fatigue failures", fatigue is not, strictly speaking, a failure mechanism. The failure mechanism is unstable crack growth of the sort we have been discussing. Fatigue is a mechanism for producing cracks that are large enough to propagate under stresses that would otherwise be safe. Since fatigue crack growth requires a cyclic stress, fatigue only happens in structures that experience

time-dependent loads. Since fatigue crack growth takes time, a structure or device may operate successfully for many years before fatigue becomes a problem. Since ultimate failure is due to the unstable propagation of cracks that were stable until the last increment of growth brought them to critical size, fatigue failures are sudden, usually unexpected and often catastrophic. 22.41 Fatigue crack growth The driving force for crack growth Let a sample that contains a crack of length, a, be subject to a cyclic stress, as shown in Fig. 2214 For a given value of the stress, ßa, the crack tip experiences the stress, ßT, where ßT increases with the square root of the crack length, a (eq. 221)  ßT = Q a  ®  ßa 22.1 The cycle in the applied stress causes a cycle in the crack tip stress, ßT, that has the same period. If the crack tip stress exceeds the yield strength, its cycle causes a local plastic deformation, which tends to open the crack as the load increases

and close it again as the load returns to zero. Page 748 Source: http://www.doksinet Materials Science Fall, 2008 There is still some uncertainty about the exact mechanism of crack growth under cyclic stress. In fact, there are probably several mechanisms that differ in detail and vary from one material to another. But it fairly easy to see that cyclic plastic deformation should cause crack growth. Plastic deformation is not completely reversible, and, as follows ultimately from the Second Law of thermodynamics, the consequence of its irreversibility will be to deform the material in the direction that relaxes the applied load, in this case, the direction that propagates the crack. Hence the crack grows, at a rate that should increase with the cyclic crack tip stress, ÎßT. ßa ßT ß ßa a ßa r t ßa Fig. 2214: A material containing a crack is subject to a stress cycle like that shown at right. If we ignore the variation in the effective crack tip radius, ®, over the

cycle, ÎßT is determined by the applied stress cycle, Îßa,  ÎßT = Q a  ®  Îßa 22.11 with a constant of proportionality that varies with the length and sharpness of the crack. While we cannot measure ÎßT directly, it is straightforward to measure the cyclic stress intensity, ÎK, where ÎK = ® ÎßT = ÎßaQ a 22.12 The crack growth rate From eq. 2212, the driving force for crack growth is proportional to ÎK This suggests that the fatigue crack growth rate, da/dn, the crack growth per cycle of the applied load, should be an increasing function of the cyclic stress intensity. It invariably is The crack growth rate, da/dn, increases with ÎK in a pattern that is nearly the same for all ductile materials, and is illustrated in the log-log plot in Fig. 2215 Page 749 Source: http://www.doksinet Materials Science Fall, 2008 As illustrated in Fig. 2215, fatigue crack growth behavior can be divided into three regimes. The most important is the central,

or power law regime, in which the crack growth rate is given by the so-called Paris Law: da m dn = A(ÎK) 22.13 Kmax = K Ic fracture: da/dn power law threshold where the exponent, m, is a constant whose value is not too far from 2 for many materials. The crack growth behavior deviates from this simple constitutive relation at both high and low values of ÎK. ÎK t ÎK Fig. 2215: The dependence of the crack growth rate on the cyclic stress intensity (log-log plot). To understand behavior outside the power law region, recognize that ÎK = Kmax - Kmin = Q-1 a ßmax - Q-1 a ßmin 22.14 In the example shown in Fig. 2214, which we shall use for simplicity, ßmin = 0, and ÎK = Kmax = Q-1 a ßmax 22.15 On the low-stress side, the crack growth decreases dramatically with decreasing ÎK until it vanishes at the threshold stress intensity, ÎKt. The usual reason for this behavior is the phenomenon of crack closure. When the maximum value of K, Kmax, is small, the magnitude of K is

not sufficient to prevent the rough sides of the crack from contacting one another over most of the range of ÎK. As soon as the crack sides contact, the crack is effectively closed, so the stress cycle is truncated and becomes ineffective A certain minimum value of Kmax is needed to force the crack open over a sufficient portion of its cycle to drive growth. On the high-stress side, Kmax increases until it becomes equal to the fracture toughness, KIc. When Kmax = KIc the fatigue crack has reached critical size and propagates to Page 750 Source: http://www.doksinet Materials Science Fall, 2008 failure. For the last few cycles before failure, Kmax is sufficiently close to KIc that the crack growth per cycle is unusually large. The crack length If we focus attention on crack growth in the power law regime, which ordinarily dominates fatigue crack growth, and substitute eq. 2211 into 2213, the result is da m m m/2 dn = A(ÎK) = A[Îßa] [a] 22.16 When the applied stress cycle,

Îßa, is a constant, eq. 2216 can be integrated to find the crack length, a, as a function of the number of cycles, n. When m = 2, which we take as a representative value, the result is simple: a(n) = a0e¬a 22.17 where the constant, ¬, is ¬ = AQÎßa 22.18 and a increases exponentially with n, as illustrated in Fig. 2216 ac fracture: a = a c a a0 n Fig. 2216: Crack length as a function of cycle number The physical reason for the accelerating crack growth that appears in Fig. 2216 is the dependence of the driving force, ÎK, on the crack length, a. As the crack grows under a constant cyclic stress, the driving force, ÎK, increases according to eq. 2215, and the crack grows faster. Fig. 2216 illustrates an important qualitative feature of fatigue that every engineer (and, in fact, every person who uses equipment) should know. Fatigue crack growth accel- Page 751 Source: http://www.doksinet Materials Science Fall, 2008 erates with cycle number. A fatigue crack may

remain small and, apparently, innocuous for a very long time, before suddenly accelerating its growth and failing in a very short time. The real situation is much worse than is suggested by eq. 2216, which represents fatigue crack growth under a constant cyclic load. Real devices experience a spectrum of load cycles, most of which are relatively small. When the crack is small these minor loads contribute almost nothing to fatigue crack growth. Most of them produce cyclic stress intensities that are much less than the threshold value, ÎKt But as the crack grows the cyclic stress intensity produced by a stress cycle of given amplitude, Îßa, increases with a . Stress cycles that are innocuous when the crack is small become damaging when it is large. Fig. 216 illustrates crack growth as a function of the number of cycles The user of a device is more interested in crack growth per unit time in service The mounting significance of minor cycles can increase the number of cycles per unit

time by orders of magnitude, dramatically affecting fatigue failure. For example, I once did an analysis of a fatigue failure of a structural member in an airplane that caused a fatal crash. Since the fracture surface was clean and very well preserved, it was possible to trace out the propagation of the fatigue crack. The analysis showed that the fatigue crack had grown to barely perceptible size over a very long period of time, perhaps several years. It then grew from barely perceptible size to the several inches that were required for instability in approximately 15 minutes of flight time. The reason was that while the crack was microscopic only the most severe loads contributed to its growth. However, once it reached that barely perceptible size, the normal operating loads of the aircraft contributed to fatigue crack growth, with the consequence that the number of damaging cycles per unit time changed from 1-2 per flight hour to 2-3x103 per minute of flight time. 22.42

Nucleation-limited fatigue The cracks that can grow to failure under fatigue conditions are those that have sufficient length to generate cyclic stress intensities greater than the threshold value, ÎKt, under the cyclic stresses that are experienced in service. If there are no such cracks, the material is proof against fatigue unless such cracks can be nucleated Unfortunately, they often can be. If the maximal value of the cyclic load on a material is sufficient to cause plastic deformation anywhere within it, for example, in favorably oriented grains or in the stress concentration fields of inclusions, then the cyclic stress may set up a cyclic plastic deformation that leads to crack nucleation. The mechanism is the formation of persistent slip bands, which are microscopic bands of within which plastic deformation occurs with each stress cycle. These bands can accumulate dislocations, and, eventually, decohere to nucleate potential fatigue cracks Crack nucleation is responsible for

the fatigue of nominally sound structural members that are subject to relatively low stresses. The typical behavior of such a material under a cyclic load that varies from sm to -sm is shown in Fig 2217 The sample fails after a Page 752 Source: http://www.doksinet Materials Science Fall, 2008 number of cycles that decreases with the stress amplitude, sm. The curve that represents the fatigue life as a function of the log of the number of stress cycles is called an s-n curve. The s-n curve terminates at the fatigue limit, sl. In steels and a number of other materials, the fatigue limit is real; the material does not fatigue if sm < sl. In other materials, such as Al and most of its alloys, there is no strict fatigue limit, but the s-n curve nearly asymptotes by n = 108, and this is a good estimate of the practical fatigue limit. sm su s -sm s t sl log(n) . Fig. 2217: The s-n curve of a typical metal The stress oscillates with time as shown at right. The fatigue limit, sl,

is either the asymptote of the curve or the stress that causes failure in 108 cycles. The fatigue limit of a part in service does change if the part is subject to a mean stress in addition to the cyclic stress, or if it contains notches or geometric irregularities that introduce stress concentrations. The effect of there, and other pertinent variables, has been studied, and design criteria are available that take them into account. Studies of s-n fatigue have shown that the bulk of the fatigue life at a given value of the plastic stress is associated with the nucleation of the fatigue cracks. The fatigue limit is a stress so low that fatigue cracks are very unlikely to nucleate. In many structures the fatigue cycle is a cyclic strain rather than a cyclic stress. A simple example is a joint between dissimilar materials that expand by different amounts in response to an elastic stress or a temperature excursion. In this case the number of cycles to failure decreases with the cyclic

plastic strain. 22.43 Fatigue-safe design Fail-safe design A part is safe against fatigue if the cyclic stress, sm, is below the fatigue limit (as adjusted for mean stress and stress concentration effects) and if the material contains no crack of size sufficient to generate a cyclic stress intensity that exceeds the threshold value. Page 753 Source: http://www.doksinet Materials Science Fall, 2008 When a structure is not weight-limited it is usually possible to design it to operate at low values of the cyclic stress, and choose a material that has reasonable values of sl and ÎKt, and can be fabricated and inspected to ensure a starting condition that is free of significant flaws. This is the approach that is usually taken in design Most of the parts that are currently used in operating engineering systems, from microelectronic devices to automobiles to airplanes, are designed to fail-safe criteria. Flaw-sensitive design Fail-safe criteria do, however, impose a significant weight

and performance penalty since they require very low operating stresses. Where higher performance is essential and cost is not limiting, as in critical parts in aerospace vehicles, it is possible to use an alternative, flaw-sensitive design procedure that is based on the combined use of non-destructive and the fatigue crack growth curve. To use a flaw-sensitive design, the fracture toughness and fatigue crack growth curve of the material must be known. The part must be inspectable during service, and the minimum capabilities of the non-destructive inspection procedure must be well-characterized. Let the non-destructive testing procedure be definitively capable of identifying flaws that are larger than some known minimum size. It follows that there is a "worst case" flaw of effective length a* that cannot be detected. If a part is inspected without flaw indications, it is conservative to assume that the "worst case" non-detectable flaw, a0, is present in it. If this

is the case then the initial rate of flaw growth is determined by the cyclic stress intensity, ÎK = QÎß a0 22.19 where Q is a constant (assuming the worst flaw geometry) and Îß is the cyclic applied stress. This flaw size will grow with time at a rate that is predictable from a knowledge of the cyclic operating stress and the fatigue crack growth curve. It is usually conservative to assume that the flaw grows in the power law regime: da m dn = A(ÎK) 22.20 where da/dn is the crack growth per cycle and A and m are material constants. A conservative fracture criterion gives the crack size at fracture as  KIc ac = Q-1ß 2  max 22.21 where KIc is the plane strain fracture toughness and ßmax is the maximum of the cyclic applied stress. The structure is hence safe if the number of cycles is such that, say, a < 01ac If the material has sufficient fracture toughness for a crack of size 0.1ac to be found easily, Page 754 Source: http://www.doksinet Materials

Science Fall, 2008 then the structure can be inspected after it has sustained enough cycles for the worst flaw, a0, to grow to size 0.1ac This criterion (or some other suitable criterion on the flaw size) can be used to set an inspection interval. If a flaw is found on inspection the part is retired or repaired. If no flaw is found, the largest flaw that can be present at that time is a0, the largest flaw that cannot be reliably found by the inspection procedure. One assumes that this flaw is present, and sets a new safe inspection period on this basis. In this approach, periodic inspection is used to insure that the part will not fail in fatigue. Flaw-sensitive design is now used for many critical parts. It is, of course, a very good idea to inspect periodically for fatigue damage even when fail-safe design has been used, since designs are imperfect, flaws may be introduced during operation, and, for many reasons that are difficult to anticipate, parts may be subject to cyclic

stresses that are beyond those anticipated in the design. This is commonly done But it is also important to take appropriate action when a fatigue crack is actually found. Unfortunately, this is not always done, largely because of ignorance of how fatigue cracks behave. A fatigue crack that exists in a structure has grown to its present size, and will continue to grow at a rapidly accelerating rate unless the part is taken out of service or the part is repaired to remove the crack. It makes no sense to assume the crack is innocuous because its present size is below the critical size for fracture. The crack is growing, and it is its future size that one should worry about. A part that contains a visible fatigue crack must be retired or repaired unless it is known that the crack will not grow to critical size before its next inspection, or unless the failure of the part would not pose a threat to safety. Page 755