Chapter 7 Mechanical behaviour of materials 7.1 Mechanical testing procedures The load-elongation curves for both polycrystalline 7.1.1 Introduction mild steel and copper are shown in Figures 7.1a and 7.Ib, The corresponding stress (load per unit area, Real crystals, however carefully prepared, contain P/A)versus strain( change in length per unit length, lattice imperfections which profoundly affect those dl/1) curves may be obtained knowing the dimensions properties sensitive to structure. Careful examination of the test piece. At low stresses the deformation is of the mechanical behaviour of materials can give elastic, reversible and obeys Hooke's law with stress information on the nature of these atomic defects. linearly proportional to strain The proportionality con- In some branches of industry the common mechan stant connecting stress and strain is known as the cal tests, such as tensile, hardness, impact, creep and elastic modulus and may be either (a)the elastic or fatigue tests, may be used, not to study the'defect Young's modulus, E, (b) the rigidity or shear modulus duced against a standard specification. Whatever its the strain is tensile, shear or hydrostatic compressive, purpose, the mechanical test is of importance in the respectively. soissons ratio v, the ratio of lateral con- development of both materials science and engineer- tractions to longitudinal extension in uniaxial tension It is different machines for performing the tests are in gen- are related according to eral use. This is because it is often necessary to know E E the effect of temperature and strain rate at different 2(1-2u) μ=2(1+) levels of stress depending on the material being tested. 3K+ Consequently, no attempt is made here to describe the (71) details of the various testing machines. The elements but for impure iron and low carbon steels the onset of plastic deformation is denoted by a sudden drop in 7.1.2 The tensile test load indicating both an upper and lower yield point. This yielding behaviour is characteristic of many met In a tensile test the ends of a test piece are fixed into als, particularly those with bcc structure containing grips, one of which is attached to the load-measuring small amounts of solute element(see Section 7. 4.6) evice on the tensile machine and the other to the For materials not showing a sharp yield point, a co straining device. The strain is usually applied by means ventional definition of the beginning of plastic filow of a motor-driven crosshead and the elongation of the 0. 1 %o proof stress, in which a line is drawn parallel the specimen is indicated by its relative movement The load necessary to cause this elongation may be Load relaxations are obtained only on'hard'beam obtained from the elastic deflection of either a beam olanyi-type machines where the beam deflection is small or proving ring, which may be measured by usil hydraulic, optical or electromechanical methods. The hich the load-measuring device is a soft spring, rapid load last method (where there is a change in the resistance variations are not recorded because the extension re too large, while in dead-loading machines no lo of strain gauges attached to the beam) is, of course, relaxations are possible. In these latter machines sudden easily adapted into a system for autographically record lding will show as merely an extension under constant ing the load-elongation curve
Chapter 7 Mechanical behaviour of materials 7.1 Mechanical testing procedures 7.1.1 Introduction Real crystals, however carefully prepared, contain lattice imperfections which profoundly affect those properties sensitive to structure. Careful examination of the mechanical behaviour of materials can give information on the nature of these atomic defects. In some branches of industry the common mechanical tests, such as tensile, hardness, impact, creep and fatigue tests, may be used, not to study the 'defect state' but to check the quality of the product produced against a standard specification. Whatever its purpose, the mechanical test is of importance in the development of both materials science and engineering properties. It is inevitable that a large number of different machines for performing the tests are in general use. This is because it is often necessary to know the effect of temperature and strain rate at different levels of stress depending on the material being tested. Consequently, no attempt is made here to describe the details of the various testing machines. The elements of the various tests are outlined below. 7.1.2 The tensile test In a tensile test the ends of a test piece are fixed into grips, one of which is attached to the load-measuring device on the tensile machine and the other to the straining device. The strain is usually applied by means of a motor-driven crosshead and the elongation of the specimen is indicated by its relative movement. The load necessary to cause this elongation may be obtained from the elastic deflection of either a beam or proving ring, which may be measured by using hydraulic, optical or electromechanical methods. The last method (where there is a change in the resistance of strain gauges attached to the beam) is, of course, easily adapted into a system for autographically recording the load-elongation curve. The load-elongation curves for both polycrystalline mild steel and copper are shown in Figures 7.1a and 7.lb. The corresponding stress (load per unit area, P/A) versus strain (change in length per unit length, dl/l) curves may be obtained knowing the dimensions of the test piece. At low stresses the deformation is elastic, reversible and obeys Hooke's law with stress linearly proportional to strain. The proportionality constant connecting stress and strain is known as the elastic modulus and may be either (a)the elastic or Young's modulus, E, (b) the rigidity or shear modulus /z, or (c) the bulk modulus K, depending on whether the strain is tensile, shear or hydrostatic compressive, respectively. Young's modulus, bulk modulus, shear modulus and Poisson's ratio v, the ratio of lateral contractions to longitudinal extension in uniaxial tension, are related according to E E 9K/.t K= 2(1-2v)' /.t= 2(1+v)' E= 3K+/z (7.1) In general, the elastic limit is an ill-defined stress, but for impure iron and low carbon steels the onset of plastic deformation is denoted by a sudden drop in load indicating both an upper and lower yield point. 1 This yielding behaviour is characteristic of many metals, particularly those with bcc structure containing small amounts of solute element (see Section 7.4.6). For materials not showing a sharp yield point, a conventional definition of the beginning of plastic flow is the 0.1% proof stress, in which a line is drawn parallel 1Load relaxations are obtained only on 'hard' beam Polanyi-type machines where the beam deflection is small over the working load range. With 'soft' machines, those in which the load-measuring device is a soft spring, rapid load variations are not recorded because the extensions required are too large, while in dead-loading machines no load relaxations are possible. In these latter machines sudden yielding will show as merely an extension under constant load
198 Modern Physical Metallurgy and Materials Engineering Upper Unirradiated td point re (19x20° neutrons Luders strain per cm2) 100 Uniform elongatIon 60 Figure 7.1 Stress-elongation curves for (a)impure iron, (b)copper, (c) ductile-brittle transition in mild steel(after Churchman, Morford and Cottrell, 1957 to the elastic portion of the stress-strain curve from than the strain to fracture measured along the gauge For control purposes the tensile test gives valuable length the point of 0. 1 strain True stress-true strain curves are often plotted to information on the tensile strength(TS= maximum show the work hardening and strain behaviour at large ad/original area)and ductility(percentage reduction strains. The true stress o is the load P divided by the in area or percentage elongation) of the material. When area A of the specimen at that particular stage of strain it is used as a research technique, however, the exact and the total true strain in deforming from initial length ape and fine details of the curve, in addition to the lo to length Ii is e=Ji(dl/I)=In(11/lo). The true way in which the yield stress and fracture stress vary stress-strain curves often fit the Ludwig relation a with temperature, alloying additions and grain size, are ke" where n is a work-hardening coefficient 0. 1-0 robably of greater significance and k the strength coefficient. Plastic instability, or The increase in stress from the initial yield up to the necking, occurs when an increase in strain produces TS indicates that the specimen hardens during defor- no increase in load supported by the specimen, i.e mation (i.e. work-hardens). On straining beyond the dP=0, and hence since P= aA. then TS the metal still continues to work-harden, bi ll to compensate for the reduction in ross-sectional area of the test piece. The deforma- defines the instability condition. During deformation, tion then becomes unstable, such that as a localized the specimen volume is essentially constant (i.e. dv= region of the gauge length strains more than the rest, 0)and from it cannot harden sufficiently to raise the stress for fur- ther deformation in this region above that to cause dv =d(IA)= AdI+ldA=0 further strain elsewhere. A neck then forms in the we obtain lis region until fracture. Under these conditions. the da d/ eduction in area(Ao- At)/Ao where Ao and A, are the initial and final areas of the neck gives a mea- Thus, necking at a strain at which the slope sure of the localized strain, and is a better indication of the true stress-true strain curve equals the true
198 Modern Physical Metallurgy and Materials Engineering "~ 200 Fracture Z ~- Lower y~etd point i ' --<.. I00- Luders strain C .... 1 t 1 1 1 ..... __ (a) Max tmum stress 200 I00 Totat etonga ion 1 . L- 0 in -1 10 Etongation ~ % (b) 6050 ~'f~ radiated 40 i Irradiated --, S 30 2O ,o ~ ....... '40 0 40 8'o Temperature ~ ~ (c) Figure 7.1 Stress-elongation curves for (a) impure iron, (b) copper, (c) ductile-brittle transition in mild steel (after Churchman, Mogford and Cottrell, 195 7). to the elastic portion of the stress-strain curve from the point of 0.1% strain. For control purposes the tensile test gives valuable information on the tensile strength (TS- maximum load/original area) and ductility (percentage reduction in area or percentage elongation) of the material. When it is used as a research technique, however, the exact shape and fine details of the curve, in addition to the way in which the yield stress and fracture stress vary with temperature, alloying additions and grain size, are probably of greater significance. The increase in stress from the initial yield up to the TS indicates that the specimen hardens during deformation (i.e. work-hardens). On straining beyond the TS the metal still continues to work-harden, but at a rate too small to compensate for the reduction in cross-sectional area of the test piece. The deformation then becomes unstable, such that as a localized region of the gauge length strains more than the rest, it cannot harden sufficiently to raise the stress for further deformation in this region above that to cause further strain elsewhere. A neck then forms in the gauge length, and further deformation is confined to this region until fracture. Under these conditions, the reduction in area (A0- A l)/Ao where A0 and A l are the initial and final areas of the neck gives a measure of the localized strain, and is a better indication than the strain to fracture measured along the gauge length. True stress-true strain curves are often plotted to show the work hardening and strain behaviour at large strains. The true stress o is the load P divided by the area A of the specimen at that particular stage of strain and the total true strain in deforming from initial length Io to length ll is e---- f/o' (dl/l)= ln(l~/lo). The true stress-strain curves often fit the Ludwig relation a = ke" where n is a work-hardening coefficient ~0.1-0.5 and k the strength coefficient. Plastic instability, or necking, occurs when an increase in strain produces no increase m load supported by the specimen, i.e. dP = 0, and hence since P -- oA, then dP = Ado + odA = 0 defines the instability condition. During deformation, the specimen volume is essentially constant (i.e. dV = 0) and from dV = d(/a) = Adl + ldA -- 0 we obtain do dA dl a -- A -- 1 --dE (7.2) Thus, necking occurs at a strain at which the slope of the true stress-true strain curve equals the true
its deformation behaviour. The hardness tester forces small sphere, pyramid or cone into the surface of the metals by means of a known applied load, and the hardness number( Brinell or Vickers diamond pyramid) then obtained from the diameter of the impres Instability The hardness may be related to the yield or tensile trength of the metal, since during the indentation, the Nominal strain En material around the impression is plastically deformed to a certain percentage strain. The vickers hardness Figure 7.2 Considere's construction number(VPN) is defined as the load divided by the pyramidal area of the indentation, in kgf/mm, and is about three times the yield stress for materials which stress at that strain, i.e. do/dE= o. Alternatively, since do not work harden appreciably. The Brinell hardness ke"= a= do/de= nke- then e=n and necking number(BHN) is defined as the stress P/A, in kgf/mm occurs when the true strain equals the strain-hardening where P is the load and A the surface area of the exponent. The instability condition may also be spherical cap forming the indentation. Thus expressed in terms of the conventional(nominal strain) dl/o do l BHN=P/D2)(-(-(d/D)21 /2) dl/I where d and d are the indentation and indentor diam =+(1+En)=a eters respectively. For consistent results the ratio d/D (7.3) should be maintained constant and small. Under these conditions soft materials have similar values of BHN which allows the instability point to be located using and VPN. Hardness testing is of importance in both Considere's construction(see Figure 7. 2), by plotting control work and research, especially where informa the true stress against nominal strain and drawing the tion on brittle materials at elevated temperatures tangent to the curve from En=-I on the strain axis. required The point of contact is the instability stress and the tensile strength is a/(1+En) Tensile specimens can also give information on the 7. 1. 4 Impact testing pe of fracture exhibited. Usually in polycrystalline A material may have a high tensile strength and yet metals transgranular fractures occur (i.e. the fracture be unsuitable for shock loading conditions. To deter urface cuts through the grains )and the cup and cone mine this the impact resistance is usually measured by type of fracture is extremely common in really duc- means of the notched or un-notched Izod or Charpy tile metals such as copper. In this, the fracture starts impact test. In this test a load swings from a given at the centre of the necked portion of the test piece height to strike the specimen, and the energy dissi s measured. The test is parti xis, so forming the cup,, but then, as it nears the ularly useful in showing the decrease in ductility and outer surface, it turns into a ' by fracturing along impact strength of materials of bcc structure at mod surface at about 45 to the tensile axis. In detail erately low temperatures. For example, carbon steels the itself consists of many irregular surfaces at have a relatively high ductile-brittle transition tem- about 45to the tensile axis, which gives the fracture a perature(Figure 7. Ic)and, consequently, they may be fibrous appearance. Cleavage is also a fairly common sed with safety at sub-zero temperatures only if the type of transgranular fracture, particularly in materi- transition temperature is lowered by suitable alloy als of bcc structure when tested at low temperatures ons or by refining the grain size. No The fracture surface follows certain crystal planes (e.g. increasing importance is given to defining a fracture [100) planes), as is shown by the grains revealing toughness parameter K for an alloy, since many alloys large bright facets, but the surface also arpels where critical stress, propagate; K, defines the critical com- contain small cracks which, when subjected to some cleavage planes have been tom apart. Intercrystallir fractures sometimes occur, often without appreciable discussed more fully in Chapter 8 deformation. This type of fracture is usually caused by a brittle second phase precipitating out around the 7.1.5 Creep testing grain boundaries, as shown by copper containing bis muth or antimony Creep is defined as plastic flow under constant stress and although the majority of tests are carried out under 7.1.3 Indentation hardness testing constant load conditions, equipment is available for reducing the loading during the test to compensate The hardness of a metal defined as the resistance to for the small reduction in cross-section of the spec- penetration,gives a conveniently rapid indication of imen. At relatively high temperatures creep appears to
Mechanical behaviour of materials 199 / )l/ : .Instobility / ff'y i/strain -1 0 Nominal strain E n Figure 7.2 Considbre's construction. stress at that strain, i.e. do"/de -- o.. Alternatively, since ke" = o. = dot/de = nke "-I then e = n and necking occurs when the true strain equals the strain-hardening exponent. The instability condition may also be expressed in terms of the conventional (nominal strain) do. do. de,, de de,, de do. (dl/lo) do. 1 de,, dl/)i = de, l0 do -- ~(1 + e,,) : o. (7.3) den which allows the instability point to be located using Consid6re's construction (see Figure 7.2), by plotting the true stress against nominal strain and drawing the tangent to the curve from e,, = -1 on the strain axis. The point of contact is the instability stress and the tensile strength is o./(1 + e,, ). Tensile specimens can also give information on the type of fracture exhibited. Usually in polycrystalline metals transgranular fractures occur (i.e. the fracture surface cuts through the grains) and the 'cup and cone' type of fracture is extremely common in really ductile metals such as copper. In this, the fracture starts at the centre of the necked portion of the test piece and at first grows roughly perpendicular to the tensile axis, so forming the 'cup', but then, as it nears the outer surface, it turns into a 'cone' by fracturing along a surface at about 45 ~ to the tensile axis. In detail the 'cup' itself consists of many irregular surfaces at about 45 ~ to the tensile axis, which gives the fracture a fibrous appearance. Cleavage is also a fairly common type of transgranular fracture, particularly in materials of bcc structure when tested at low temperatures. The fracture surface follows certain crystal planes (e.g. {100} planes), as is shown by the grains revealing large bright facets, but the surface also appears granular with 'river lines' running across the facets where cleavage planes have been torn apart. Intercrystalline fractures sometimes occur, often without appreciable deformation. This type of fracture is usually caused by a brittle second phase precipitating out around the grain boundaries, as shown by copper containing bismuth or antimony. 7.1.3 Indentation hardness testing The hardness of a metal, defined as the resistance to penetration, gives a conveniently rapid indication of its deformation behaviour. The hardness tester forces a small sphere, pyramid or cone into the surface of the metals by means of a known applied load, and the hardness number (Brinell or Vickers diamond pyramid) is then obtained from the diameter of the impression. The hardness may be related to the yield or tensile strength of the metal, since during the indentation, the material around the impression is plastically deformed to a certain percentage strain. The Vickers hardness number (VPN) is defined as the load divided by the pyramidal area of the indentation, in kgf/mm 2, and is about three times the yield stress for materials which do not work harden appreciably. The Brinell hardness number (BHN) is defined as the stress P/A, in kgf/mm 2 where P is the load and A the surface area of the spherical cap forming the indentation. Thus BUN --- P/(---~D2)2 / {1 -[1 -(d/O)2] '/2} where d and D are the indentation and indentor diameters respectively. For consistent results the ratio diD should be maintained constant and small. Under these conditions soft materials have similar values of B HN and VPN. Hardness testing is of importance in both control work and research, especially where information on brittle materials at elevated temperatures is required. 7.1.4 Impact testing A material may have a high tensile strength and yet be unsuitable for shock loading conditions. To determine this the impact resistance is usually measured by means of the notched or un-notched Izod or Charpy impact test. In this test a load swings from a given height to strike the specimen, and the energy dissipated in the fracture is measured. The test is particularly useful in showing the decrease in ductility and impact strength of materials of bcc structure at moderately low temperatures. For example, carbon steels have a relatively high ductile-brittle transition temperature (Figure 7.1c) and, consequently, they may be used with safety at sub-zero temperatures only if the transition temperature is lowered by suitable alloying additions or by refining the grain size. Nowadays, increasing importance is given to defining a fracture toughness parameter Kc for an alloy, since many alloys contain small cracks which, when subjected to some critical stress, propagate; Kc defines the critical combination of stress and crack length. Brittle fracture is discussed more fully in Chapter 8. 7.1.5 Creep testing Creep is defined as plastic flow under constant stress, and although the majority of tests are carded out under constant load conditions, equipment is available for reducing the loading during the test to compensate for the small reduction in cross-section of the specimen. At relatively high temperatures creep appears to
200 Modern Pl Metallurgy and Materials Engineering tension or compression, but all involve the same prin ciple of subjecting the material to constant cycles of stress. To express the characteristics of the stress sys tem, three properties are usually quoted: these include (I)the maximum range of stress, (2)the mean stress, and(3)the time period for the stress cycle. Four dif ferent arrangements of the stress cycle are shown in Figure 7. 4, but the reverse and the repeated cycle tests (e.g. push-pull")are the most common, since they are L。wr,Low the easiest to achieve in the laboratory. and to subject them to tests using a different rang of stress, S, on each group of specimens. The num Figure 7.3 Typical creep curves. ber of stress cycles, N, endured by each specimen at a given stress level is recorded and plotted, as shown occur at all stress levels, but the creep rate increase in Figure 7.5. This S-N diagram indicates that some with increasing stress at a given temperature. For the metals can withstand indefinitely the application of a accurate assessment of creep properties, it is clear that large number of stress reversals, provided the applied special attention must be given to the maintenance of limit. for certain ferrous materials when they are used surement of the small dimensional changes involved. in the absence of corrosive conditions the assumption This latter precaution is ne of a safe working range of stress seems justified, bu rials a rise in temperature by a few tens of degrees for non-ferrous materials and for steels when they are used in corrosive conditions a definite endurance limit shows the characteristics of a typical creep curve and cannot be defined. Fatigue is discussed in more detail following the instantaneous strain caused by the sud divided into three stages, usually termed primary or 7. 1.7 Testing of ceramics transient creep, second or steady-state creep and ter- Direct tensile testing of ceramics is not generally tiary or accelerating creep. The characteristics of the favoured, mainly because of the extreme sensitivity of creep curve often vary, however, and the tertiary stage ceramics to surface flaws. First, it is difficult to apply of creep may be advanced or retarded if the tempera- a truly uniaxial tensile stress: mounting the specimen ture and stress at which the test is carried out is high in the machine grips can seriously damage the surface or low respectively(see Figure 7.3, curves b and c). and any bending of the specimen during the test will Creep is discussed more in Section 7.9 cause premature failure. Second, suitable waisted spec- imens with the necessary fine and flawless finish are 7.1.6 Fatigue testing expensive to produce. It is therefore common practice to use bend tests for engineering ceramics and glasses The fatigue phenomenon is concerned with the prema - (They have long been used for other non-ductile mate- ture fracture of metals under repeatedly applied low rials such as concretes and grey cast iron. In the three- stresses, and is of importance in many branches of and four-point bend methods portrayed in Figure 7.6,a engineering (e.g. aircraft structures ). Several differ- beam specimen is placed between rollers and carefully ent types of testing machines have been constructed loaded at a constant strain rate. The fexural strength in which the stress is applied by bending, torsion, at failure, calculated from the standard formulae, is Figure 7.4 Alternative forms of stress cycling:(a)reversed;(b)alternating(mean stress zero),(c) fluctuating and
200 Modern Physical Metallurgy and Materials Engineering s t c e~ u~ c/N,gh /', / i| Low r, Low d Ttrn~ Figure 7.3 Typical creep curves. occur at all stress levels, but the creep rate increase with increasing stress at a given temperature. For the accurate assessment of creep properties, it is clear that special attention must be given to the maintenance of the specimen at a constant temperature, and to the measurement of the small dimensional changes involved. This latter precaution is necessary, since in many materials a rise in temperature by a few tens of degrees is sufficient to double the creep rate. Figure 7.3, curve a, shows the characteristics of a typical creep curve and following the instantaneous strain caused by the sudden application of the load, the creep process may be divided into three stages, usually termed primary or transient creep, second or steady-state creep and tertiary or accelerating creep. The characteristics of the creep curve often vary, however, and the tertiary stage of creep may be advanced or retarded if the temperature and stress at which the test is carried out is high or low respectively (see Figure 7.3, curves b and c). Creep is discussed more fully in Section 7.9. 7.1.6 Fatigue testing The fatigue phenomenon is concerned with the premature fracture of metals under repeatedly applied low stresses, and is of importance in many branches of engineering (e.g. aircraft structures). Several different types of testing machines have been constructed in which the stress is applied by bending, torsion, tension or compression, but all involve the same principle of subjecting the material to constant cycles of stress. To express the characteristics of the stress system, three properties are usually quoted: these include (1) the maximum range of stress, (2) the mean stress, and (3) the time period for the stress cycle. Four different arrangements of the stress cycle are shown in Figure 7.4, but the reverse and the repeated cycle tests (e.g. 'push-pull') are the most common, since they are the easiest to achieve in the laboratory. The standard method of studying fatigue is to prepare a large number of specimens free from flaws, and to subject them to tests using a different range of stress, S, on each group of specimens. The number of stress cycles, N, endured by each specimen at a given stress level is recorded and plotted, as shown in Figure 7.5. This S-N diagram indicates that some metals can withstand indefinitely the application of a large number of stress reversals, provided the applied stress is below a limiting stress known as the endurance limit. For certain ferrous materials when they are used in the absence of corrosive conditions the assumption of a safe working range of stress seems justified, but for non-ferrous materials and for steels when they are used in corrosive conditions a definite endurance limit cannot be defined. Fatigue is discussed in more detail in Section 7.11. 7.1.7 Testing of ceramics Direct tensile testing of ceramics is not generally favoured, mainly because of the extreme sensitivity of ceramics to surface flaws. First, it is difficult to apply a truly uniaxial tensile stress: mounting the specimen in the machine grips can seriously damage the surface and any bending of the specimen during the test will cause premature failure. Second, suitable waisted specimens with the necessary fine and flawless finish are expensive to produce. It is therefore common practice to use bend tests for engineering ceramics and glasses. (They have long been used for other non-ductile materials such as concretes and grey cast iron.) In the threeand four-point bend methods portrayed in Figure 7.6, a beam specimen is placed between rollers and carefully loaded at a constant strain rate. The flexural strength at failure, calculated from the standard formulae, is (8) (b) (c) (d) Figure 7.4 Alternative forms of stress cycling: (a) reversed; (b) altet~ating (mean stress ~ zero), (c) fluctuating and (d) repeated
centre of an electric furnace heated by Sic elements 435 250 pe of the routine testing of graphite electrode samples and Carburized gives a useful indication of their ability to withstand accidental lateral impact during service in steel melting furnaces e Proof-testing is a long-established method of test g certain engineering components and structures In a typical proof test, each component is held at a Decarburized certain proof stress for a fixed period of time; load iron(0.004%C) ng and unloading conditions are standardized. In the ase of ceramics, it may involve bend-testing, inter- withstand the proof test are, in the simplest analysi udged to be sound and suitable for long-term service at the lower design stress. The underlying philosophy as been often questioned, not least because there is a Figure 7.5 S-N curve for carburized and decarburized iron risk that the proof test itself may cause incipient crack g. Nevertheless, proof-testing now has an important ole in the statistical control of strength in ceramics. 7.2 Elastic deformation 7. 2.1 Elastic deformation of metals 3 F(L-L It is well known that metals deform both elastically MoR= and plastically. Elastic deformation takes place at low stresses and has three main characteristics, namely 3-Point bend 4-Point b (1)it is reversible, (2)stress and strain are linearly proportional to each other according to Hooke's Law ons, MoR= modulus of and (3)it is usually small (i.e. <1%elastic strain) The stress at a point in a body is usually defined b=breadth of specimen, d by considering an infinitesimal cube surrounding that point and the forces applied to the faces of the cube by he surrounding material. These forces may be resolved known as the modulus of rupture(MoR)and expresses into components parallel to the cube edges and when the maximum tensile stress which develops on the con- divided by the area of a face give the nine stress vex face of the loaded beam. Strong ceramics, such as mponents shown in Figure 7.7. a given component silicon carbide and hot-pressed silicon nitride, have i is the force acting in the j-direction per unit ery high MoR values. The four-point loading method area of face normal to the i-direction. Clearly, when is often preferred because it subjects a greater volume i=j we have normal stress components(e.g.oux and area of the beam to stress and is therefore more which may be either tensile(conventionally positive) arching MoR values from four-point tests are often or compressive(negative), and when i+j(e.g oxy) substantially lower than those from three-point tests the stress components are shear. These shear stresses on the same material. Similarly, strength values tend exert couples on the cube and to prevent rotation of the to decrease as the specimen size is increased. To pro- cube the couples on opposite faces must balance and vide worthwhile data for quality control and design hence o =oi. Thus, stress has only six independent ctivities, close attention must be paid to strain rate components nd environment, and to the size, edge finish and sur. When a body is strained, small elements in that ace texture of the specimen. with oxide ceramics and body are displaced. If the initial position of an elem silica glasses, a high strain rate will give an appr iably higher flexural strength value than a low strain I'The nine components of stress ou form a second-rank rate, which leads to slow crack growth and delayed tensor usually written ction The bend test has also been adapted for use at high temperatures. In one industrial procedure, specimens of magnesia (basic) refractory are fed individually from a magazine into a three-point loading zone at the and is known as the stress tensor
Mechanical behaviour of materials 201 35 r- : 30 ,rx 25 ~ 20 15 Carburized i c' - C)~...... ~.004% Decarburized _ _ i _ i i ..... I I 104 I 0 S I 06 107 a 08 Cycles ~ N 250 200 E z if) L 150 m 100 Figure 7.5 S-N curve for carburized and decarburized iron. ~F ,,, [-i,- ....... a r~-/,,a d I , 2 2 3 FL MoR- 2 bd 2 I 32P0~nt bend I F F - | 2 2 3 F (L - L,) MoR= 2 bd 2 I 4iPo'nibend i Figure 7.6 Bend test configurations. MoR = modulus of rupture, F = applied force, L = outer span, Li = inner span, b = breadth of specimen, d = depth of specimen. known as the modulus of rupture (MoR) and expresses the maximum tensile stress which develops on the convex face of the loaded beam. Strong ceramics, such as silicon carbide and hot-pressed silicon nitride, have very high MoR values. The four-point loading method is often preferred because it subjects a greater volume and area of the beam to stress and is therefore more searching. MoR values from four-point tests are often substantially lower than those from three-point tests on the same material. Similarly, strength values tend to decrease as the specimen size is increased. To provide worthwhile data for quality control and design activities, close attention must be paid to strain rate and environment, and to the size, edge finish and surface texture of the specimen. With oxide ceramics and silica glasses, a high strain rate will give an appreciably higher flexural strength value than a low strain rate, which leads to slow crack growth and delayed fracture (Section 10.7). The bend test has also been adapted for use at high temperatures. In one industrial procedure, specimens of magnesia (basic) refractory are fed individually from a magazine into a three-point loading zone at the centre of an electric furnace heated by SiC elements. A similar type of hot-bend test has been used for the routine testing of graphite electrode samples and gives a useful indication of their ability to withstand accidental lateral impact during service in steel melting furnaces. Proof-testing is a long-established method of testing certain engineering components and structures. In a typical proof test, each component is held at a certain proof stress for a fixed period of time; loading and unloading conditions are standardized. In the case of ceramics, it may involve bend-testing, internal pressurization (for tubes) or rotation at high speed ('overspeeding' of grinding wheels). Components that withstand the proof test are, in the simplest analysis, judged to be sound and suitable for long-term service at the lower design stress. The underlying philosophy has been often questioned, not least because there is a risk that the proof test itself may cause incipient cracking. Nevertheless, proof-testing now has an important role in the statistical control of strength in ceramics. 7.2 Elastic deformation 7.2.1 Elastic deformation of metals It is well known that metals deform both elastically and plastically. Elastic deformation takes place at low stresses and has three main characteristics, namely (1) it is reversible, (2)stress and strain are linearly proportional to each other according to Hooke's Law and (3) it is usually small (i.e. <1% elastic strain). The stress at a point in a body is usually defined by considering an infinitesimal cube surrounding that point and the forces applied to the faces of the cube by the surrounding material. These forces may be resolved into components parallel to the cube edges and when divided by the area of a face give the nine stress components shown in Figure 7.7. A given component cr~j is the force acting in the j-direction per unit area of face normal to the /-direction. Clearly, when i = j we have normal stress components (e.g. crx~) which may be either tensile (conventionally positive) or compressive (negative), and when i ~ j (e.g. r the stress components are shear. These shear stresses exert couples on the cube and to prevent rotation of the cube the couples on opposite faces must balance and hence o" U = crji. 1 Thus, stress has only six independent components. When a body is strained, small elements in that body are displaced. If the initial position of an element 1The nine components of stress tTij form a second-rank tensor usually written tTxx O'x y tTx z tY y x tT y y tY y z tYzx tTzy tYZZ and is known as the stress tensor
02 Modern Physical Metallurgy and Materials Engineering Figure 7. 8 De trallelogram PQRS involving(i)a rigid body tro oP allowed fc rotation allowed for by rotating the axes to x" r and(ii) Figure 7. 7 Normal and shear hange of shape involving both tensile and shear strains is defined by its coordinates (x, y, z) and its final is simply e= Exr. However, because of the strains position by (x +u.y+U, 2+ w)then the displacemen introduced by lateral contraction, Ery =-ve and E is (u, v, w). If this displacement is constant for all -ve, where v is Poissons ratio; all other components elements in the body, no strain is involved, only a of the strain tensor are zer rigid translation. For a body to be under a condition At small elastic deformations, the stress is linearly of strain the displacements must vary from element proportional to the strain. This is Hooke's law and in to element. A uniform strain is produced when the its simplest form relates the uniaxial stress to the uni displacements are linearly proportional to distance In axial strain by means of the modulus of elasticity For ne dimension then u=ex where e= du/dx is the a general situation, it is necessary to write Hooke's law coefficient of proportionality or nominal tensile strain. as a linear relationship between six stress components For a three-dimensional uniform strain, each of the and the six strain components, i.e. three components u, v, w is made a linear function in terms of the initial elemental coordinates, i.e Orr=CuErx +CeRv+C13E: +C14y C15yer c16yrv u=exrx+ery y+exz Jv=C21 Err +C2Ery + C33+ C24Y:+ C25yer +C26y eurx Orz=C31 Exr+C32Eyy+C33Ezz+C34yx t C3syer +C36yr The strains exr =du/dx, er du/dy, e=dw/d the tensile strains along the x, y and z axes, respec- trv=c61Ex+C62Ew +C63E2+C64yx+c65Yer+c66yev ely. The strains ery, ew, etc, produce shear strains and in some cases a rigid bcdy rotation. The rotation The constants cIl, C12,,,, Cii are called the elastic produces no strain and can be allowed for by rotat- stiffness constants ing the reference axes (see Figure 7.8). In general Taking account of the symmetry of the crystal, many therefore, e=Ei+Oij with Ei the strain compo- of these elastic constants are equal or become zero nents and ai the rotation components. If, however, Thus in cubic crystals there are only three indepen the shear strain is defined as the angle of shear, this dent elastic constants Cu, CI2 and c4 for the three is twice the cot g shear strain component, 1e. independent modes of deformation. These include the y= 2Et. The ke the stress tensor, has application of (1)a hydrostatic stress p to produce a nine components which are usually written as dilatation e given by Er Ezy Ez where K is the bulk modulus, (2)a shear stress on a face in the direction of the cube axis defining sile strains and yrr, etc the shear modulus u= C4, and (3)a rotation about shear strains, All the simple types of strain can be cubic axis defining a shear modulus u1 =2(C11-C12) roduced from the strain tensor by setting some of The ratio u/uI is the elastic anisotropy factor and e components equal to zero. For example, a pure in elastically isotropic crystals it is unity with 2c44 dilatation (i.e. change of volume without change of shape)is obtained when Ex Eyy= E and all other Alternatively, the strain may be related to the stress, e.g components are zero. Another example is a uniaxial Ex=5110cr +$120xx +5130+..., in which case the tensile test when the tensile strain along the x-axis constants st, 512,.... i are called elastic compliances
202 Modern Physical Metallurgy and Materials Engineering y y,, yr , \ o-z, \ c a\: 0 A X /. Figure 7.7 Normal and shear stress components. X r Figure 7.8 Deformation of a square OABC to a parallelogram PQRS involving (i) a rigid body translation OP allowed for by redefining new axes XtY ', (ii) a rigid body rotation allowed for by rotating the axes to X"Y", and (iii) a change of shape involving both tensile and shear strains. is defined by its coordinates (x, y,z) and its final position by (x + u, y + v, z + w) then the displacement is (u, v, w). If this displacement is constant for all elements in the body, no strain is involved, only a rigid translation. For a body to be under a condition of strain the displacements must vary from element to element. A uniform strain is produced when the displacements are linearly proportional to distance. In one dimension then u = ex where e = du/ck is the coefficient of proportionality or nominal tensile strain. For a three-dimensional uniform strain, each of the three components u, v, w is made a linear function in terms of the initial elemental coordinates, i.e. u = exxX + exy y + exzZ v = eyxX d- eyyy -I- eyzZ w = ezxX + ezyy + ezzZ The strains exx = du/dx, err = dv/dy, e=: = dw/dz are the tensile strains along the x, y and z axes, respectively. The strains exy, e yz, etc., produce shear strains and in some cases a rigid bedy rotation. The rotation produces no strain and can be allowed for by rotating the reference axes (see Figure 7.8). In general, therefore, eij "-Eij ~-o.)ij with Eij the strain components and wij the rotation components. If, however, the shear strain is defined as the angle of shear, this is twice the corresponding shear strain component, i.e. y -- 2eij. The strain tensor, like the stress tensor, has nine components which are usually written as: 1 1 exx 2 yxv 2 Yxz F,.r.x F-, x y E xz I " 1 Eyx F, yy Eyz or 5 Y:x eyy ~ yy: Ez.t EZy EZZ ! I 2 Yzr 2 YZy EZZ where exx etc. are tensile strains and Yxy, etc. are shear strains. All the simple types of strain can be produced from the strain tensor by setting some of the components equal to zero. For example, a pure dilatation (i.e. change of volume without change of shape) is obtained when e~x = eyy--ez: and all other components are zero. Another example is a uniaxial tensile test when the tensile strain along the x-axis is simply e = exx. However, because of the strains introduced by lateral contraction, eyy = -re and e:= = -re, where v is Poisson's ratio; all other components of the strain tensor are zero. At small elastic deformations, the stress is linearly proportional to the strain. This is Hooke's law and in its simplest form relates the uniaxial stress to the uniaxial strain by means of the modulus of elasticity. For a general situation, it is necessary to write Hooke's law as a linear relationship between six stress components and the six strain components, i.e. tYxx=CllExx -I- C126vv + Cl3Ezz "~- CI4Yyz + Cl5)"z.r + C16~xy O'yy =C21Exx W C22Evv -~-" ~-~'~"l'- C24ffvz "~- C25~'zx + C26}/xy tYzz=C31Exx + C326vv -]- C336zz + C34Yyz -~- C35Y=r + C36Yxy "t'yz =C41Exx "~- C42Evv "Jr- C43Ezz "~ C44 )/yz + C45 Yzr + C46)/xy Zzr =CsIexr + C52E'vv + C53Ezz + C54Yyz "-~ C55~/z.x + C56)/xy ~xy=C61Exx + C62Evv -'1- C63Ezz q- c64Yyz + C65 }"zr "3 I- C66~xy The constants c~,c~2 ..... cij are called the elastic stiffness constants. ~ Taking account of the symmetry of the crystal, many of these elastic constants are equal or become zero. Thus in cubic crystals there are only three independent elastic constants cl~, ci2 and c44 for the three independent modes of deformation. These include the application of (1) a hydrostatic stress p to produce a dilatation | given by p = - 3 (cll + 2c12)(9 = -• where x is the bulk modulus, (2) a shear stress on a cube face in the direction of the cube axis defining the shear modulus # = c44, and (3) a rotation about a l cubic axis defining a shear modulus ~l =~ (cll - cl2). The ratio lz/#~ is the elastic anisotropy factor and in elastically isotropic crystals it is unity with 2c44 -- 1Alternatively, the strain may be related to the stress, e.g. 8x -- Sl lOxx --I- Sl2Oyy --I- Sl3Ozz %- .... in which case the constants sll, sl2 ..... sij are called elastic compliances
Mechanical behaviour of materials 203 Table 7.1 Elastic constants of cubic crystals(GN/m2) 7.3 Plastic deformation Metal Cr C4 2cu/len -cn2) 7.3.1 Slip and twinning The limit of the elastic range cannot be defined 006000460059 004.6003.7002.6 exactly but may be considered to be that value of the stress below which the amount of plasticity (irre- 501.019801510 versible deformation) is negligible, and above which 0.77 the amount of plastic deformation is far greater than 620 the elastic deformation If we consider the deforma- C tion of a metal in a tensile test, one or other of two types of curve may be obtained. Figure 7. la shows the 3.9 stress-strain curve characteristic of iron from which it can be seen that plastic deformation begins abruptly B-brass 824 it a and continues initially with no increase in stress. at which it occurs is the yield stress. Figure 7. Ib show Cl-C12: the constants are all interrelated with cil= a stress-strain curve characteristic of copper, from which it will be noted that the transition to the plastic Table 7.1 shows that most metals are far from sotropic and, in fact, only tungsten is isotropic; the this case the stress required to start alkali metals and B-compounds are mostly anisotropic. How is known as the flow stress Generally, 2c44>(C1l -C1)and hence, for most elas Once the yield or fiow stress has been exceeded plastic or permanent deformation occurs, and this is tically anisotropic metals E is maximum in the(11 1) found to take place by one of two simple processes, and minimum in the (100) directions. Molybde num and niobium are unusual in having the reverse slip(or glide) and twinning. During slip, shown in anisotropy when E is greatest along(100)directions. the bottom half along certain crystallographic planes, moves over Most commercial materials are polycrystalline, and known as slip planes, in such a way that the atoms onsequently they have approximately isotropic prop- move forward by a whole number of lattice vectors rties. For such materials the modulus value is usuall. independent of the direction of measurement because as a result the continuity of the lattice is maintained. the value observed is an average for all directions, in are not whole lattice vectors and the lattice generated ugh the same as the ing manufacture a preferred orientation of the grains in lattice, is oriented in a twin relationship to the polycrystalline specimen occurs, the material will also be observed that in contrast to slip, the directionality'will take place region in twinning occurs over many atom planes, the atoms in each plane being moved forward by the same amount relative to those of the plane below them 7. 2.2 Elastic deformation of ceramics At ambient temperatures the profile of the stress ver- 7.3.2 Resolved shear stress sus strain curve for a conventional ceramic is similar All working processes such as rolling, extrusion, forg- to that of a non-ductile metal and can be described ing etc, cause plastic deformation and, consequently, as linear-elastic, remaining straight until the point of these operations will involve the processes of slip or fracture is approached. The strong interatomic bonding twinning outlined above. The stress system applied of engineering ceramics confers mechanical stiffness. during these working operations is often quite com- Moduli of elasticity(elastic, shear)can be much higher plex, but for plastic deformation to occur the presence than those of metallic materials. In the case of sin- of a shear stress is essential. The importance of shear gle ceramic crystals, these moduli are often highl anisotropic (e.g. alumina). However, in their polycrys talline forms, ceramics are often isotropic as a resul of the randomizing effect of processing(e.g. isostat serve anisotropic tendencies(e.g extrusion).(Glasses are isotropic, of course. Moduli are greatly influ- enced by the presence of impurities, second phases poin l ceramic is lowered as porosity is increased. As the temperature of testing is raised, elastic moduli usually show a decrease, but there are exceptions Figure 7.9 Slip and twinning in a crystal
Mechanical behaviour of materials 203 Table 7.1 Elastic constants of cubic crystals (GN/m 2) Metal Cll Ci2 C44 2 C44/(Cll -- C12 ) Na 006.0 004.6 005.9 8.5 K 004.6 003.7 002.6 5.8 Fe 237.0 141.0 116.0 2.4 W 501.0 198.0 151.0 1.0 Mo 460.0 179.0 109.0 0.77 AI 108.0 62.0 28.0 1.2 Cu 170.0 121.0 75.0 3.3 Ag 120.0 90.0 43.0 2.9 Au 186.0 157.0 42.0 3.9 Ni 250.0 160.0 118.0 2.6 fl-brass 129.1 109.7 82.4 8.5 Cll --Cl2" the constants are all interrelated with c~ -- 4 2 tc + 5/z, c12 = tc- 5/z and c44 = #. Table 7.1 shows that most metals are far from isotropic and, in fact, only tungsten is isotropic; the alkali metals and/3-compounds are mostly anisotropic. Generally, 2C44 > (C|l -- Cl2) and hence, for most elastically anisotropic metals E is maximum in the (1 1 l) and minimum in the (1 00) directions. Molybdenum and niobium are unusual in having the reverse anisotropy when E is greatest along (1 0 0) directions. Most commercial materials are polycrystalline, and consequently they have approximately isotropic properties. For such materials the modulus value is usually independent of the direction of measurement because the value observed is an average for all directions, in the various crystals of the specimen. However, if during manufacture a preferred orientation of the grains in the polycrystalline specimen occurs, the material will behave, to some extent, like a single crystal and some 'directionality' will take place. 7.2.2 Elastic deformation of ceramics At ambient temperatures the profile of the stress versus strain curve for a conventional ceramic is similar to that of a non-ductile metal and can be described as linear-elastic, remaining straight until the point of fracture is approached. The strong interatomic bonding of engineering ceramics confers mechanical stiffness. Moduli of elasticity (elastic, shear) can be much higher than those of metallic materials. In the case of single ceramic crystals, these moduli are often highly anisotropic (e.g. alumina). However, in their polycrystalline forms, ceramics are often isotropic as a result of the randomizing effect of processing (e.g. isostatic pressing); nevertheless, some processing routes preserve anisotropic tendencies (e.g. extrusion). (Glasses are isotropic, of course.) Moduli are greatly influenced by the presence of impurities, second phases and porosity; for instance, the elastic modulus of a ceramic is lowered as porosity is increased. As the temperature of testing is raised, elastic moduli usually show a decrease, but there are exceptions. 7.3 Plastic deformation 7.3.1 Slip and twinning The limit of the elastic range cannot be defined exactly but may be considered to be that value of the stress below which the amount of plasticity (irreversible deformation) is negligible, and above which the amount of plastic deformation is far greater than the elastic deformation. If we consider the deformation of a metal in a tensile test, one or other of two types of curve may be obtained. Figure 7.1 a shows the stress-strain curve characteristic of iron, from which it can be seen that plastic deformation begins abruptly at A and continues initially with no increase in stress. The point A is known as the yield point and the stress at which it occurs is the yield stress. Figure 7.1 b shows a stress-strain curve characteristic of copper, from which it will be noted that the transition to the plastic range is gradual. No abrupt yielding takes place and in this case the stress required to start macroscopic plastic flow is known as the flow stress. Once the yield or flow stress has been exceeded plastic or permanent deformation occurs, and this is found to take place by one of two simple processes, slip (or glide) and twinning. During slip, shown in Figure 7.9a, the top half of the crystal moves over the bottom half along certain crystallographic planes, known as slip planes, in such a way that the atoms move forward by a whole number of lattice vectors; as a result the continuity of the lattice is maintained. During twinning (Figure 7.9b) the atomic movements are not whole lattice vectors, and the lattice generated in the deformed region, although the same as the parent lattice, is oriented in a twin relationship to it. It will also be observed that in contrast to slip, the sheared region in twinning occurs over many atom planes, the atoms in each plane being moved forward by the same amount relative to those of the plane below them. 7.3.2 Resolved shear stress All working processes such as rolling, extrusion, forging etc. cause plastic deformation and, consequently, these operations will involve the processes of slip or twinning outlined above. The stress system applied during these working operations is often quite complex, but for plastic deformation to occur the presence of a shear stress is essential. The importance of shear 1" 1" I I I I I I ,' 'i;iiI:I:I; ".~ I I I I I I I I I I I -plane IIIIII IIIIII (a) Z Z Z Cb) z z xzz z Figure 7.9 Slip and twinning in a crystal
204 Modern Physical Metallurgy and Materials Engineering tresses becomes clear when it is realized that these A consideration of the tensile test in this way shows stresses arise in most processes and tests even when that it is shear stresses which lead to plastic defor the applied stress itself is not a pure shear stress. This mation, and for this reason the mechanical behaviour may be illustrated by examining a cylindrical crys- exhibited by a material will depend, to some extent tal of area A in a conventional tensile test under a on the type of test applied. For example, a ductile uniaxial load P. In such a test, slip occurs on the material can be fractured without displaying its plastic slip plane, shown shaded in Figure 7. 10 les if tested in a state of hydrostatic or tria ch is a/cosφ, whereφ is the ang normal to the plane oh and the axis of The shear stress on any plane is zero. Conversely, materials ane y be which normally exhibit a tendency to brittle behaviour resolved into a force normal to the plane along OH, in a tensile test will show ductility if tested under con Pcos and a force along OS, P sin o. Here, OS is the ditions of high shear stresses and low tension stresses line of greatest slope in the slip plane and the force In commercial practice, extrusion approximates closely Psin g is a shear force. It follows that the applied stress to a system of hydrostatic pressure, and it is common (force/area)is made up of two stresses, a normal stress for normally brittle materials to exhibit some ductility (P/A)coso tending to pull the atoms apart, and a when deformed in this way(e.g. when extruded) over each other In general, slip does not take place down the line 7.3. 3 Relation of slip to crystal structure of greatest slope unless this happens to coincide with An understanding of the fundamental nature of plastic the crystallographic slip of direction. It is necessary, deformation processes is provided by experiments on plane and in the slip direction. Now, if OT is taken is used the result obtained is the average behaviour of o represent the slip direction the resolved shear stress all the differently oriented grains in the material. Such ll be given by σ=Pcosφsinφcosx/A the resolved shear stress is a maximum along lines of greatest slope in planes at 45 to the tensile axis, slip where x is the angle between OS and OT. Usually this occurs preferentially along certain crystal planes and formula is written more simply as directions. Three well-established laws governing the slip behaviour exist, namely:(1)the direction of slip d= Pcos cosλ/A (7.4) is almost always that along which the atoms are most a is the angle between the slip direction OT and closely packed, (2)sI sually occurs on the most xis of tension. It can be seen that the resolved closely packed plane, and(3)from a given set of slip stress has a maximum value when the slip plane planes and directions the crystal operates on that sys- is inclined at 45 to the tensile axis, and becomes tem(plane and direction) for which the resolved shear smaller for angles either greater than or less than 4 stress is largest. The slip behaviour observed in fc metals shows the general applicability of these laws ular to the tensile axis (d> 45) it is easy to imagine since slip occurs along (1 I o)directions in (lI l) the applied stress has a greater tendency to pull planes. In cph metals slip occurs along (1 120) direc- atoms apart than to slide them. When the slip tions, since these are invariably the closest packed, plane becomes more nearly parallel to the tensile axis but the active slip plane depends on the value of the ( <45 )the shear stress is again small but in this axial ratio. Thus, for the metals cadmium and zinc, case it is because the area of the slip plane, A/cos c/a la is 1.886 and 1.856, respectively, the planes of greatest atomic density are the 1000 1l basal planes and slip takes place on these planes. When the axial atio is appreciably smaller than the ideal value of c/a=1.633 the basal plane is not so closely packed, nor so widely spaced, as in cadmium and zinc, and other slip planes operate. In P and titanium(c/a= 1.587), for example, slip takes place on the (1010 prism pla and on the (101 1) pyramidal planes at higher tem- peratures. In magnesium the axial ratio(c/a= 1. 624) Slp direction pproximates to the ideal value, and although only occurs at room temperature, at temperatures above 225C slip on the (101 1 planes has also been
204 Modern Physical Metallurgy and Materials Engineering stresses becomes clear when it is realized that these stresses arise in most processes and tests even when the applied stress itself is not a pure shear stress. This may be illustrated by examining a cylindrical crystal of area A in a conventional tensile test under a uniaxial load P. In such a test, slip occurs on the slip plane, shown shaded in Figure 7.10, the area of which is A~ cos 4~, where ~b is the angle between the normal to the plane OH and the axis of tension. The applied force P is spread over this plane and may be resolved into a force normal to the plane along OH, P cos 4~, and a force along OS, P sin 4). Here, OS is the line of greatest slope in the slip plane and the force P sin 4~ is a shear force. It follows that the applied stress (force/area) is made up of two stresses, a normal stress (P/A)cosZep tending to pull the atoms apart, and a shear stress (P/A) cos 4~ sin 4~ trying to slide the atoms over each other. in general, slip does not take place down the line of greatest slope unless this happens to coincide with the crystallographic slip of direction. It is necessary, therefore, to know the resolved shear stress on the slip plane and in the slip direction. Now, if OT is taken to represent the slip direction the resolved shear stress will be given by a = P cos 4~ sin 4~ cos x/A where X is the angle between OS and OT. Usually this formula is written more simply as o = P cos q~ cos Z/A (7.4) where )~ is the angle between the slip direction OT and the axis of tension. It can be seen that the resolved shear stress has a maximum value when the slip plane is inclined at 45 ~ to the tensile axis, and becomes smaller for angles either greater than or less than 45 ~ . When the slip plane becomes more nearly perpendicular to the tensile axis (4~ > 45 ~ it is easy to imagine that the applied stress has a greater tendency to pull the atoms apart than to slide them. When the slip plane becomes more nearly parallel to the tensile axis (~b < 45 ~ the shear stress is again small but in this case it is because the area of the slip plane, A/cos ~b, is correspondingly large. H ~) Sl~p plane P ._._.__.. T Figure 7.10 Relation between the slip plane, slip direction and the axis of tension for a cylindrical crystal. A consideration of the tensile test in this way shows that it is shear stresses which lead to plastic deformation, and for this reason the mechanical behaviour exhibited by a material will depend, to some extent, on the type of test applied. For example, a ductile material can be fractured without displaying its plastic properties if tested in a state of hydrostatic or triaxial tension, since under these conditions the resolved shear stress on any plane is zero. Conversely, materials which normally exhibit a tendency to brittle behaviour in a tensile test will show ductility if tested under conditions of high shear stresses and low tension stresses. In commercial practice, extrusion approximates closely to a system of hydrostatic pressure, and it is common for normally brittle materials to exhibit some ductility when deformed in this way (e.g. when extruded). 7.3.3 Relation of slip to crystal structure An understanding of the fundamental nature of plastic deformation processes is provided by experiments on single crystals only, because if a polycrystalline sample is used the result obtained is the average behaviour of all the differently oriented grains in the material. Such experiments with single crystals show that, although the resolved shear stress is a maximum along lines of greatest slope in planes at 45 ~ to the tensile axis, slip occurs preferentially along certain crystal planes and directions. Three well-established laws governing the slip behaviour exist, namely: (1) the direction of slip is almost always that along which the atoms are most closely packed, (2)slip usually occurs on the most closely packed plane, and (3) from a given set of slip planes and directions, the crystal operates on that system (plane and direction) for which the resolved shear stress is largest. The slip behaviour observed in fcc metals shows the general applicability of these laws, since slip occurs along (1 1 0) directions in {1 1 1} m planes. In cph metals slip occurs along (1 1 2 0) directions, since these are invariably the closest packed, but the active slip plane depends on the value of the axial ratio. Thus, for the metals cadmium and zinc, c/a is 1.886 and 1.856, respectively, the planes of greatest atomic density are the {000 1} basal planes and slip takes place on these planes. When the axial ratio is appreciably smaller than the ideal value of c/a = 1.633 the basal plane is not so closely packed, nor so widely spaced, as in cadmium and zinc, and other slip planes operate. In zirconium (c/a = 1.589) and titanium (c/a = 1.587), for example, slip takes place on the {1 0 10} prism planes at room temperature and on the {1 0 11} pyramidal planes at higher temperatures. In magnesium the axial ratio (c/a = 1.624) approximates to the ideal value, and although only basal slip occurs at room temperature, at temperatures above 225~ slip on the {1 0 1 1} planes has also been observed. Bcc metals have a single well-defined closepacked (1 1 1) direction, but several planes of equally high density of packing, i.e. {1 1 2}, {1 1 0} and {1 2 3}
Mechanical behaviour of materials 205 The choice of slip plane in these metals is often influ ts basal plane perpendicular to the tensile enced by temperature and a preference is shown for is,ie.φ=0° ejecting [11 2 below Tm/4,(110) from Tm/4 to Tm/2 and contrast to its ter haviour. where it is brittle it [123] at high temperatures, where Tm is the melt- will now appear since the shear stress on the ng point. Iron often slips on all the slip planes at slip plane is only zero for a tensile test and not for a once in a common (11 1> slip direction, so that a bend test. On the other hand, if we take the crystal with slip line (i.e. the line of intersection of a slip plane its basal plane oriented parallel to the tensile axis (i.e surface of a crystal) takes 90 )this specimen will appear brittle whatever appearance. tress system is applied to it. For this crystal, although the shear force is large, owing to the large area of the 7.3.4 Law of critical resolved shear stres slip plane, A/ cos o, the resolved shear stress is always This law states that slip takes place along a given slip sling small and insufficient to cause deformation by stress reaches a slipper value. In most crystals the high symmetry of arrangement provides several crystallographic 7.3.5 Multiple slip equivalent planes and directions for slip (i. e. cph crys- The fact that slip bands, each consisting of many slip als have three systems made up of one plane contain- Ing three directions, fcc crystals have twelve systems lines, are observed on the surface of deformed crystals made up of four planes each with three directions, hows that deformation is inhomogeneous, with exten while bcc crystals have many systems)and in such sive slip occurring on certain planes, while the crystal direction for which the maximum stress acts (aw3 formed. Figures 7. 12a and 7 12b show such a crystal tension a series of zinc single crystals. Then, because slip direction. In a tensile test, however, the ends of zinc is cph in structure only one plane is available for a crystal are not free to move sideways'relative the slip process and the resultant stress-strain curve each other, since they are constrained by the grips of will depend on the inclination of this plane to the the tensile machine. In this case, the central portion of tensile axis. The value of the angle is determined the crystal is altered in orientation and rotation of both by chance during the process of single-crystal growth, the slip plane and slip direction into the axis of ten- and consequently all crystals will have different values sion occurs, as shown in Figure 7. 12c. This behaviour more conveniently demonstrated on a stereographic stress,(e. the stress on the glide plane in the glide ma is shown in the unit triangle m re Te .r than tie en have different values of the flow stress as shown in projection of the crystal by considering the rotation Figure 7. 11a. However, because of the criterion of a the tensile axis relative to the crystal rathe critical resolved shear stress, a plot of resolved shear versa. This is illustrated in Figure 7. 13a for the defor within experimental error, for all the specimens. This P and [101], and P and (11 1) are equal to a and p, plot is shown in Figure 7. 1 1b. respectively. The active slip system is the(11 1) plane The importance of a critical shear stress may be and the [101] direction, and as deformation proceeds demonstrated further by taking the crystal which has the change in orientation is represented by the point, P, shear strain Figure 7.11 Schematic representation of (a) variation of stress versus elongation with orientation of basal plane and (b) constancy of revolved shear stress
The choice of slip plane in these metals is often influenced by temperature and a preference is shown for {l 1 2} below Tm/4, {110} from Tin~4 to Tin~2 and {l 2 3} at high temperatures, where Tm is the melting point. Iron often slips on all the slip planes at once in a common (l 1 1) slip direction, so that a slip line (i.e. the line of intersection of a slip plane with the outer surface of a crystal) takes on a wavy appearance. 7.3.4 Law of critical resolved shear stress This law states that slip takes place along a given slip plane and direction when the shear stress reaches a critical value. In most crystals the high symmetry of atomic arrangement provides several crystallographic equivalent planes and directions for slip (i.e. cph crystals have three systems made up of one plane containing three directions, fcc crystals have twelve systems made up of four planes each with three directions, while bcc crystals have many systems) and in such cases slip occurs first on that plane and along that direction for which the maximum stress acts (law 3 above). This is most easily demonstrated by testing in tension a series of zinc single crystals. Then, because zinc is cph in structure only one plane is available for the slip process and the resultant stress-strain curve will depend on the inclination of this plane to the tensile axis. The value of the angle tp is determined by chance during the process of single-crystal growth, and consequently all crystals will have different values of tp, and the corresponding stress-strain curves will have different values of the flow stress as shown in Figure 7.1 l a. However, because of the criterion of a critical resolved shear stress, a plot of resolved shear stress (i.e. the stress on the glide plane in the glide direction) versus strain should be a common curve, within experimental error, for all the specimens. This plot is shown in Figure 7.1 lb. The importance of a critical shear stress may be demonstrated further by taking the crystal which has Mechanical behaviour of materials 205 its basal plane oriented perpendicular to the tensile axis, i.e. ~- 0 ~ and subjecting it to a bend test. In contrast to its tensile behaviour, where it is brittle it will now appear ductile, since the shear stress on the slip plane is only zero for a tensile test and not for a bend test. On the other hand, if we take the crystal with its basal plane oriented parallel to the tensile axis (i.e. tp = 90 ~ this specimen will appear brittle whatever stress system is applied to it. For this crystal, although the shear force is large, owing to the large area of the slip plane, A~ cos 4~, the resolved shear stress is always very small and insufficient to cause deformation by slipping. 7.3.5 Multiple slip The fact that slip bands, each consisting of many slip lines, are observed on the surface of deformed crystals shows that deformation is inhomogeneous, with extensive slip occurring on certain planes, while the crystal planes lying between them remain practically undeformed. Figures 7.12a and 7.12b show such a crystal in which the set of planes shear over each other in the slip direction. In a tensile test, however, the ends of a crystal are not free to move 'sideways' relative to each other, since they are constrained by the grips of the tensile machine. In this case, the central portion of the crystal is altered in orientation, and rotation of both the slip plane and slip direction into the axis of tension occurs, as shown in Figure 7.12c. This behaviour is more conveniently demonstrated on a stereographic projection of the crystal by considering the rotation of the tensile axis relative to the crystal rather than vice versa. This is illustrated in Figure 7.13a for the deformation of a crystal with fcc structure. The tensile axis, P, is shown in the unit triangle and the angles between P and [ 1 01], and P and (1 1 1) are equal to ~ and ~b, respectively. The active slip system is the (1 1 1) plane and the [ 10 1 ] direction, and as deformation proceeds the change in orientation is represented by the point, P, 15 ~ 30 ~ 60 ~ f i f f f f I ,, ,4,, .= , i elongation shear strain (a) (b) f / , ! Figure 7.11 Schematic representation of (a) variation of stress versus elongation with orientation of basal plane and (b) constancy of revoh,ed shear stress
206 Modern Physical Metallurgy and Materials Engineering observations on virgin crystals of aluminium and cop- per, but not with those made on certain alloys, or pure metal crystals given special treatments(e. g. quenched Lattice from a high temperature or irradiated with neutron Results from the latter show that the crystal continues to slip on the primary system after the orientation has direction reached the symmetry line, causing the orientation to overshoot this line, i.e. to continue moving toward Lattice [101, in the direction of primary slip. A otation mount of this additional primary slip the conjugate system suddenly operates, and further slip shootin that slip on the conjugate system must intersect that on the primary system, and to do this is presumably more difficult than tofit a new slip plane in the relatively undeformed region between those planes on which slip cess Is direction more difficult in materials which have a low stacking fault energy(e. g. a-brass) 7.3.6 Relation between work-hardening and Figure 7.12 (a)and (b) show the slip process in an slip constrained single crystal;(c) illustrates the plastic ending in a crystal gripped at its ends. The curves of Figure 7, I show that following the yield phenomenon a continual rise in stress is required to continue deformation. i.e. the flow stress of a deformed metal increases with the amount of strain. This resis raca tance of the metal to further plastic fiow as the defor mation proceeds is known as work-hardening. The legree of work-hardening varies for metals of different rystal structure, and is low in hexagonal metal crys- als such as zinc or cadmium, which usually slip on one family of planes only. The cubic crystals harden rapidly on working but even in this case when sl icted to one e specimen A, Figure 7, 14) the coefficient of harden may plane efined as slope of the plastic portion of the stress-strain curve, is small. Thus this type of harden- ing, like overshoot, must be associated with the inte Figure 7.13 Stereographic representation of (a)slip systens action which results from slip on intersecting familie in fcc crystals and(b) overshooting of the primary slip of planes. This interaction will be dealt with more fully moving along the zone, shown broken in Figure 7. 13a, towards [101], i.e. A decreasing and increasing Specr As slip occurs on the one system, the primary sys em, the slip plane rotates away from its position of maximum resolved shear stress until the orientation of the crystal reaches the [001]-[11 1] symmetry line Beyond this point, slip should occur equally on both he primary system and a second system(the system)(11 1)[011], since these two syster single slip equal components of shear stress. Subsequer the process of multiple or duplex slip the I rotate so as to keep equal stresses on the two active sys ems, and the tensile axis moves along the symmetry gure 7.14 Stress-stl ilium deformed line towards [1 12]. This behaviour agrees with early by single and multiple slip (after Liicke and Lange, 1950)
206 Modern Physical Metallurgy and Materials Engineering plane.~ (a) ,Slip direction Lattice ,Lattice rotation "- Slip q direction (b) (c) Figure 7.12 (a) and (b) show the slip process in an unconstrained single crystal; (c) illustrates the plastic bending in a crystal gripped at its ends. c,. ,..,~, 1~01] IUQ llt I~III"~ t,v,J Crdl(.ll 1114,111"~ ,o+,, I11 ~r...r__. ~ ,.~I~111 Cross l~,llnl II Pr,mlry pllne (a) I0.1 Figure 7.13 Stereographic representation of (a) slip systems in fcc crystals and (b) overshooting of the primary slip system. observations on virgin crystals of aluminium and copper, but not with those made on certain alloys, or pure metal crystals given special treatments (e.g. quenched from a high temperature or irradiated with neutrons). Results from the latter show that the crystal continues to slip on the primary system after the orientation has reached the symmetry line, causing the orientation to overshoot this line, i.e. to continue moving towards [1 0 1 ], in the direction of primary slip. After a certain amount of this additional primary slip the conjugate system suddenly operates, and further slip concentrates itself on this system, followed by overshooting in the opposite direction. This behaviour, shown in Figure 7.13b, is understandable when it is remembered that slip on the conjugate system must intersect that on the primary system, and to do this is presumably more difficult than to 'fit' a new slip plane in the relatively undeformed region between those planes on which slip has already taken place. This intersection process is more difficult in materials which have a low stacking fault energy (e.g. c~-brass). 7.3.6 Relation between work-hardening and slip The curves of Figure 7.1 show that following the yield phenomenon a continual rise in stress is required to continue deformation, i.e. the flow stress of a deformed metal increases with the amount of strain. This resistance of the metal to further plastic flow as the deformation proceeds is known as work-hardening. The degree of work-hardening varies for metals of different crystal structure, and is low in hexagonal metal crystals such as zinc or cadmium, which usually slip on one family of planes only. The cubic crystals harden rapidly on working but even in this case when slip is restricted to one slip system (see the curve for specimen A, Figure 7.14) the coefficient of hardening, defined as the slope of the plastic portion of the stress-strain curve, is small. Thus this type of hardening, like overshoot, must be associated with the interaction which results from slip on intersecting families of planes. This interaction will be dealt with more fully in Section 7.6.2. moving along the zone, shown broken in Figure 7.13a, towards [ 1 0 1], i.e. ~ decreasing and 4> increasing. As slip occurs on the one system, the primary system, the slip plane rotates away from its position of maximum resolved shear stress until the orientation of the crystal reaches the [0 0 1] - [ 1 1 1] symmetry line. Beyond this point, slip should occur equally on both the primary system and a second system (the conjugate system) (1 1 1) [0 1 1 ], since these two systems receive equal components of shear stress. Subsequently, during the process of multiple or duplex slip the lattice will rotate so as to keep equal stresses on the two active systems, and the tensile axis moves along the symmetry line towards [1 1 2]. This behaviour agrees with early '%~o~ I /;p~c,m+oB. [111] ~100[ st,p)llb""~ ' (s,ngte Soe;me; A, I I o I ~) 3 z. 5 6 G! Dde -----,,- % Figure 7.14 Stress-strain curves for aluminium deformed by single and multiple slip (after Liicke and Lange, 1950)
