mechanical-properties-and-working-of-metals-and-alloys_compress

1.5 Elements of Plastic Deformation 31 fewer number of more widely separated, coarser particles. In Disloction line this state, since the widely dispersed precipitates become ineffective barriers to the motion of dislocations, the alloy Particle Sheared particle loses its strength and becomes softer. The alloy is then said to be in the overaged condition. As the precipitate particles Movement of dislocation grow, coherency between the precipitates and the matrix is lost and there is a further decrease in strength of the alloy Fig. 1.19 A dislocation cutting through a particle (schematic) due to loss of coherency. the dislocation bow3 around the precipitates particles by the On the other hand, dispersion-hardened alloys can be Orowan’s mechanism of dispersion hardening. obtained by closely dispersing an almost insoluble and thermally stable fine second-phase particles in a metallic The ease with which shearing of particles can occur is matrix mechanically or by chemical reaction, where these affected by six properties of the particles. These lead to the particles strengthen the alloy by effectively hindering the following six ways in which the dislocation and particles dislocation motion. Generally, there is no coherency between interact to affect the strength of alloys: the lattices of the second-phase particles and the matrix. The condition of a decreasing solid solubility with decreasing • Coherency strains: Mott and Nabarro (Smallman 1970) temperature is also not required for dispersion-hardened suggested that the strain resulting from a slight mismatch alloys. The advantage of a dispersion-hardened alloy over a between the matrix and a particle gives rise to internal precipitation-hardened alloy is that the second-phase parti- stress fields which prevent the movement of gliding cles in the former are thermally stable at very high temper- dislocation and results in coherency hardening. Assum- atures because of very small solubility of second-phase ing particles to be spherical, the increase in flow stress constituents in the matrix and, thus, the particles in the former required to pass the dislocation through the fields of resist coarsening to a greater extent than in the latter case. internal stress is: Further, if cold worked, a dispersion-hardened material will resist recovery and recrystallization better than a Dr ¼ 2Gecs precipitation-hardened material. In dispersion-hardened alloys, inert oxides particles, such as thoria (ThO2), alu- where G is the shear modulus, e is the strain due to lattice mina (Al2O3) or silica (SiO2), are mixed with powders of mismatch between particles and the matrix, and cs is the base metal and the mixture is compacted and sintered into a atomic concentration of solute. solid mass by powder metallurgy techniques. Examples of • Stacking fault energy difference: If the stacking fault such type of alloy are thoria-dispersed nickel, known as TD nickel, and sintered aluminium powder, called SAP. Car- energy of the particles differs significantly from that of bides, nitrides, borides, etc., apart from oxides, can also be the matrix, the dislocations will have a different width used as second-phase particles in dispersion-hardened alloys. inside the particle than in the matrix, so that extra work is At least theoretically, it is possible to produce an almost done when the glide dislocation enters the particles. The infinite number of dispersion-hardened systems in contrast to magnitude of strengthening effect will vary directly with a limited number of age-hardening alloys. the size of the precipitate and inversely with the dislo- cation width in the matrix. According to Hirsch and The movement of dislocations is retarded in two distinct Kelly (1965), the increase in flow stress is directly pro- ways by second-phase particles distributed in the matrix. portional to the stacking fault energy difference between The dislocations may either cut through the particles, or if the matrix and the particle. cutting of particles becomes very difficult, the dislocations will bend around and bypass them. When the particles are 3Bowing a dislocation means bending a dislocation line like a bow soft and/or too small, such as clusters of solute atoms around the obstacles to bypass them. forming coherent zones (e.g., coherent G.P. zones in Al— 4.5 wt% Cu alloy with interzone spacing of about 150–250 Å), dislocation can cut and deform the particle as shown in Fig. 1.19. In precipitation-strengthened alloys, this mecha- nism operates when the separation of the centres of the precipitates in zones is too small (about 150–250 Å) for dislocations to bend and curve around the zones. However, when the stage of intermediate transition precipitate or/and normal equilibrium precipitate is reached, the separation of the centres of the precipitates is large (1000–10,000 Å) and

32 1 Tension • Chemical hardening: When a dislocation cuts though a qD ¼ Gb=2s zone, a change in the number of solvent–solute near neighbours occurs across the slip plane. This tends to where G is the shear modulus and b is the Burgers vector. If reverse the clustering process and, hence, needs addi- the separation distance between particles is k ; and k ffi qD; tional energy, which must be supplied by the applied then to avoid obstacles the dislocation would take a form like stress, resulting in an increase in flow stress. This process the one shown in Fig. 1.20a. For widely separated particles, is known as chemical hardening. where k [ qD; the dislocation line moves between the par- In a similar way, when dislocations cut through ordered ticles as illustrated in Fig. 1.20b based on Orowan’s mech- structure of particles then antiphase domain boundaries anism (Orowan 1947) of dispersion hardening proposed for are introduced leading to strengthening effect, which the overaged non-coherent precipitates. Figure 1.20b shows depends on the details of spacing and size of the parti- that in stage (1), a dislocation line approaches from left side cles. Good high-temperature strength of many superal- towards two particles separated by a distance k; and at stage loys is due to strengthening from ordered precipitate. (2), the line bends and reaches the critical radius of curvature after which it moves forward without further reduction in its • Modulus difference: If the modulus of the particles differs radius of curvature. Assuming that the particles do not significantly from that of the matrix, the energy of a deform with the matrix and k ¼ 2qD; the shear stress needed dislocation will be lowered or raised as it passes through to force the dislocation between the particles is: the particles, because the energy of a dislocation depends linearly on the local modulus. In most alloys, the mod- s ¼ ðGbÞ=k ulus difference between the particle and the matrix is not sufficient to cause a strong strengthening effect. The yield stress is determined by the shear stress s required to bow a dislocation. It is theoretically estimated • Interfacial energy and morphology of particle: When a that in age-hardened and dispersion-hardened alloys, the dislocation cuts through a particle, a step is created at the most effective dispersion of particles for optimum strength particle–matrix interface. This increase in interface area should have an interparticle spacing k of about 10À4 m. between particle and matrix caused by the shearing Overageing increases the interparticle spacing k and process is accompanied by an increase in surface energy, decreases the strength. which must be provided by the external stress. The increase in flow stress is proportional to the particle– (a) Stress field of matrix surface energy (Kelly and Nicholson 1963) and precipitate depends on the morphology of the particle. Strengthening Dislocation increases with increasing the surface-to-volume ratio of line the particle. So, finer spherical particles will give higher strength than coarser ones and greater strengthening is (b) Dislocation produced from thin plate-shaped precipitates, like the loop left G.P. zones or h00 precipitates in the Al–Cu system, than Moving behind from the spherical particles. For example, at equal vol- dislocation ume fraction of second-phase particles, strengthening produced by rod- and plate-shaped particles is about two line times of that by spherical particles (Kelly 1972). λ Precipitate • Lattice friction stress: Strength increase due to the Peierls stress in the particle and matrix (Gleiter and Hornbogen 1967) is proportional to the strength difference between the particle and the matrix. If there is an interface between the matrix and the particle (1) (2) (3) (4) or the orientation changes abruptly at the matrix–particle interface or the particles are strong enough to resist cutting, Fig. 1.20 Schematic diagrams showing a curling of a dislocation round the stress fields from precipitates, and b stages in the passage of a cutting through the particle will not be possible. Under such dislocation between widely spaced precipitates, based on Orowan’s mechanism of dispersion hardening circumstances, the dislocations bend around and bypass the particles. If qD is the radius of curvature to which the dis- location is bent under the action of a shear stress s; it has been shown (Smallman 1970) that

1.5 Elements of Plastic Deformation 33 Since the portions of dislocation meeting on the right side of single crystal was found (Maddin and Cotrell 1955) to the particle are of opposite sign, they can cancel each other over increase from 5 to 50 kPa by the presence of such part of their length and leave behind a dislocation loop around quenched-in vacancies, because excess vacancies migrate to each particle, as shown in stage (3) in Fig. 1.20b. Finally, the dislocations and pin them in a way similar to solute atoms. original dislocation is free to glide as shown in stage (4). A back Vacancies and interstitials can also be created by the stress is exerted by these loops, which must be overcome by movement of jogs produced by intersection of dislocations increasing the shear stress so that further slip can occur. As a and by high energy irradiation, where fast-moving (high result, the bowing of dislocations between the dispersed non- energy) atomic particles collide with a solid metal, such as coherent particles strain hardens the matrix rapidly, whereas the neutron irradiation. Neutron irradiation of an annealed strain hardening of matrix produced by deformation that occurs metal increases its yield stress in the tensile stress–strain by cutting through the particles is small. curve by a factor of 2–4. Neutron irradiation increases the ductile-to-brittle transition temperature in structural steels. The strength limit in overaged alloys or dispersion- After irradiation, a sharp yield point is developed in FCC hardened alloys is the yielding, or rupture, of the particles, or metals, such as copper and aluminium, but the yield point is the tearing of the matrix away from the particles. However, often eliminated in BCC metals, such as steel and since the particles are usually very fine intermetallic com- molybdenum. pounds, their strengths are high. 7. Fibre Strengthening 9. Martensite Strengthening The reinforcement of high strength fine fibres with high elastic Martensite is a phase produced by martensitic transforma- modulus in a ductile matrix, which is nonreactive with the tion, which involves the coherent formation of one phases fibres, can produce high strength materials, especially with from another of the same composition by a diffusionless, high strength-to-weight ratio. These fibre-reinforced materials lattice shear process. Several metallurgical systems show are generally known as composite materials. Graphite, boron martensitic transformation, but a pronounced strengthening or metal wire such as tungsten is used as fibres in most effect is observed in the alloys based on Fe–C system fibre-strengthened materials, although whiskers of materials because of the presence of interstitial carbon atoms. Thus, such as Al2O3, due to their very high strength, have been used the transformation of austenite to martensite in steel causes as fibres with good results. Long and continuous or discon- its yield strength and hardness to increase with increase in tinuous fibres may be used. The matrix of materials is gener- carbon content (Bain and Paxton 1961). The reasons of ally made of metals or polymers. The most common fibre- strengthening in Fe–C martensite are: strengthened materials are glass-fibre-reinforced polymers. • Solid solution strengthening by interstitial carbon atoms, In these materials, the entire load is essentially carried by which is considered to be a major cause of high hardness the high elastic modulus fibres. The functions of the ductile and strength of martensite. Figure 1.21 (Krauss 1999) matrix are: shows the creation of distortion dipole by displacements of iron atoms by excess amount of carbon atoms residing • Separation of the individual fibres. in sites of octahedral voids of BCT lattice in martensite. • Protection of fibres from surface damage. This distortion of iron lattice creates stress field around it • Load transfer to the fibres. and makes the movement of dislocation though this stress • Prevention of crack propagation by blunting its tips by field very difficult. This strengthening follows a square-root dependency with carbponffifficffiffiffioffiffinffiffitffiffieffiffinffiffitffiffiffioffiffifffiffiffiaffiffiuffi stenite, plastic deformation, if cracks develop due to breakage of i.e. 0:2% offset yield strength4 / weight% of C; which fibres. implies that the rise of strength with increasing carbon is initially rapid, but subsequently slows down at higher A unidirectional array of fibres in a matrix will result in a carbon. In case of alloy steel, solid solution strengthening highly anisotropic material where properties will vary with due to substitutional alloying elements (which is not so change in the orientation of the body. Note that strong great) is added to that due to carbon. composite materials can also be produced by reinforcement of particles in a matrix, in which the properties are expected to be isotropic; i.e., the properties are equal in all directions. 8. Strengthening Due to Vacancies and Interstitials An excess of vacancies can be retained by quenching a metal 4See Sect. 1.9.1 in this chapter for discussion on 0:2% offset yield from near its melting point, and the CRSS of aluminium strength.

34 1 Tension martensite, with sometimes improvement in ductility and/or toughness. Austenite can be deformed: Fe atoms I. Prior to its transformation to martensite: C atoms (a) Above the upper critical temperature of steel with Range of or without recrystallization, by a process known Fe-atom respectively as controlled hot rolling or hot-cold displacements work. (b) Within upper and lower critical temperature of c steel, by a process known as intercritical TMT. A typical example of the product after quenching a is dual-phase steel consisting of martensite fibres in ferrite. a (c) Below the lower critical temperature of steel without allowing transformation to pearlite or Fig. 1.21 Displacements of iron atoms due to carbon atoms in bainite, by a process known as ausforming or martensite (Krauss 1999) ausrolling or ausworking. • Martensitic substructure consists of high dislocation II. During its transformation to martensite, by a process density or fine twin structure. Low-carbon martensite known as isoforming or zerolling, where martensite laths, which are aligned parallel to form a packet, usually transformation can be due to a stress-assisted or contain a high dislocation density of 109–1010 mm−2 and stress-induced transformation. For example, this treat- high carbon martensite plates, which are oriented ran- ment can be applied to 18/8 austenitic stainless steels. domly, usually contain very fine parallel transformation twins, each about 0.1 lm thick. This martensitic sub- III. After its transformation to martensite, by a process structure acts as effective barriers to dislocation motions known as marforming. For example, marforming and aids in increasing the strength of martensite. This treatment with a high degree of cold working can be substructure strengthening is relatively constant as a applied to soft (about RC 30) martensite in maraging function of carbon content and, except at low-carbon steel. concentrations, does not make nearly as great a contri- bution as does the carbon solid solution strengthening. The most important TMT is ausforming, in which steel in a metastable austenitic condition is deformed (usually rolled) • Carbon atom segregation to the dislocation fine structure in excess of 50% (usually 80–90%) below the lower critical and/or lath and packet boundaries during autotempering temperature in the bay of time–temperature–transformation of low-carbon martensite having high Ms (Ms is the curve of the steel (usually between 400 and 600 °C) fol- temperature, where martensite reaction starts). lowed by quenching below the Ms to transform to marten- site. This treatment can increase strength up to 50% with no • Fine particle size of martensite crystal. significant deterioration in ductility and impact toughness which, in some cases, are even higher than those obtained in Thermomechanical treatments (TMT) applied to steel to a conventionally formed martensite. The yield and tensile form martensite from plastically deformed strain-hardened strength of ausformed steel is proportional to the strain austenite containing higher dislocation density generally imparted to the metastable austenite. Strengthening is usu- produce higher strength than the conventionally formed ally in the range of 4–9 MPa for each 1% deformation and greater for higher initial yield strength of steel. Strength properties of ausformed martensite increase on lowering the deformation temperature of the metastable austenite because of its increased strain hardening. Hence for a given alloy, highest strength is obtained by imparting greatest possible deformation to the metastable austenite at the lowest tem- perature. The presence of strong carbide forming elements in

1.5 Elements of Plastic Deformation 35 steel promotes increased strength. The main reasons where r1; r2 and r3 represent the principal stresses in responsible for the increased strength in ausforming are: the principal directions ‘1’, ‘2’ and ‘3’. • Strain-induced precipitation of carbides that occurs in (2) The other component of stress is the stress deviator, austenite prior to its transformation, particularly in the usually denoted by r0; which involves the shear stresses presence of strong carbide formers such as Mo. The martensite is dispersion strengthened as soon as it is and is responsible for producing plastic deformation. If formed and the pre-existing fine carbides also serve to refine the structure of martensite. the stress deviators in the principal directions ‘1’, ‘2’ and ‘3’ are, respectively, r10 ; r02 and r30 ; they can be • Inheritance of dislocation structure of the deformed determined by subtracting the mean stress rm; from austenite by the ausformed martensite. Dislocations with their corresponding principal stresses r1; r2 and r3; a very high density (1011 mmÀ2), which are distributed uniformly in martensite, are locked by carbon atoms and which are shown below: carbon precipitates. r10 ¼ r1 À rm ¼ r1 À r1 þ r2 þ r3 • Development of texture in martensite. 3 • Refinement or reduction of structural units in a direction ! ð1:60aÞ 1 2 1 normal to the rolling plane because of reduction in the ¼ 3 ½2r1 À r2 À r3Š ¼ 3 r1 À 2 ðr2 þ r3Þ size of austenite grains in that direction. 2 1 ! As a result of these strengthening mechanisms, the yield 3 2 strengths of ausformed steels can reach to a very high value of Similarly; r20 ¼ r2 À rm ¼ r2 À ðr1 þ r3Þ about 2–3 GPa with reductions of area varying from 40 to 20%. ð1:60bÞ 2 1 ! 3 2 And; r30 ¼ r3 À rm ¼ r3 À ðr1 þ r2Þ ð1:60cÞ 1.5.8 Spherical and Deviator Components 1.5.9 Yielding Criteria for Ductile Metals of Stress In the theory of plasticity, the part of the total stress, which is For a material subjected to any possible combination of capable of producing the plastic deformation, must be indi- stresses, it is important to predict the conditions at which cated. Hence, the total stress can be divided into two com- plastic flow of the material begins. In uniaxial loading, as in ponents as follows: a uniaxial tension test, the onset of macroscopic plastic deformation occurs at the yield stress, r0; also known as (1) Hydrostatic or spherical component of stress, which flow stress. includes only pure tension or compression. Materials can bear very large hydrostatic or spherical state of The yielding criteria are basically empirical relationships, stress, without experiencing any plastic deformation. which correlate the initiation of plastic yielding under a However, hydrostatic stress can cause elastic change in situation of combined stresses with some particular combi- volume. For example, when equal compressive stresses nation of principal stresses. To predict the onset of yielding act from all directions on a material subjected to plastic in ductile metals, two yielding criteria based on a number of deformation, the stresses will push the atoms closer experimental observations are generally accepted, which are together in all directions, but the atomic structure is not discussed below: distorted. Hence, there is no plastic deformation of the material and only its volume is reduced. Under an ideal (I) Von Mises’ or Distortion Energy Yielding Criterion state of hydrostatic compression, even the failure of the or Theory material becomes impossible. Bridgman (1945) has confirmed experimentally that the yield stress of metals Von Mises (1913) proposed that yielding would occur when is practically independent of hydrostatic pressure. The the following relation is satisfied spherical, or the hydrostatic component of stress, usu- ally represented by rm; is the mean stress and given by 1 ½ðr1 À r2Þ2 þ ðr2 À r3Þ2 þ ðr3 À r1Þ2Š ¼ k2 ð1:61Þ 6 rm ¼ r1 þ r2 þ r3 ð1:59Þ where the constant k in (1.61) can be identified as follows by 3 considering the state of stress in pure shear that is produced

36 1 Tension in biaxial torsion. If r1 is the algebraically largest principal Advantages of Von Mises’ yielding criterion: normal stress and r2 is the algebraically smallest principal normal stress, the principal stresses in biaxial torsion are: • Since the criterion involves squared terms, the result is independent of the sign of individual stresses. r1 ¼ rmax ¼ Àr2 ¼ smax; and r3 ¼ 0; • In order to use this criterion, it is not necessary to know Because from (1.48), which are the largest and the smallest principal stresses. maximum shear stress; smax ¼ r1 À r2 ¼ r1 ¼ rmax: Why the Name Distortion Energy Yielding Criterion? 2 Now, from (1.61) at yielding, 1 ½4r21 þ r21 þ r12Š ¼ k2; The part of total strain energy per unit volume that is 6 involved in the change of shape as opposed to a change in ) r1 ¼ rmax ¼ Àr2 ¼ smax ¼ k ð1:62Þ volume is called the distortion energy per unit volume. Let Therefore, k represents the yield stress in pure shear ðUV Þdistortion represents the distortion energy per unit volume. (biaxial torsion) or the shearing yield stress. Henky (1924) showed that yielding will occur when the The constant k in (1.61) can be evaluated and related to distortion energy ðUV Þdistortion reaches a critical value which the yielding in the uniaxial tension test as follows. At is equal to the distortion energy for a uniaxial state of stress. yielding, the principal stresses in uniaxial tension are: ðUV Þdistortion is given by the following equation: r1 ¼ r0; and r2 ¼ r3 ¼ 0; ðUV Þdistortion ¼ 1 h À r2Þ2 þ ðr2 À r3Þ2 12G ðr1 i ð1:65Þ where r0 is the flow stress or yield stress in uniaxial tension. þ ðr3 À r1Þ2 Putting these values in the above (1.61) of the Von where G ¼ shear modulus or modulus of rigidity in shear. Mises’ yielding criterion, we get In a uniaxial state of stress like a uniaxial tension test, if 1 Âr20 þ r02à ¼ k2; or; r02 ¼ 3 k2; r0 is the yield stress or flow stress, the principal stresses at 6 pffiffi yielding will be: r1 ¼ r0; and r2 ¼ r3 ¼ 0: Or, r0 ¼ 3k; ) From (1.65) .pffiffi ) k ¼ r0 3 ¼ 0:577 r0 ð1:63Þ ðUV Þdistortion stress¼ 1 2r20 ð1:66Þ for uniaxial state of 12G Therefore from (1.63), it is evident that according to the According to Henky (1924), yielding occurs when Von Mises’ yielding criterion the yield stress in pure shear or biaxial torsion, k; will be less than the yield stress or flow ðUV Þdistortion ¼ ð UV Þdistortion for uniaxial state of stress: stress in uniaxial tension, r0. Substituting the above value of k from (1.63) in the original (1.61) of Von Mises’ yielding Therefore, from (1.65) and (1.66), the final form of the criterion, it takes the following final form, given by (1.64). distortion energy criterion will be: 1 h r2Þ2 r3Þ2 i r0 ¼ p1ffiffi h À r2Þ2 þ ðr2 À r3Þ2 þ ðr3 À r1Þ2i1=2 6 ðr1 r1Þ2 2 ðr1 À þ ðr2 À þ ðr3 À ¼ k2 ¼ r20 ; ð1:67Þ ) r0 ¼ p1ffiffi h3 À r2Þ2 þ ðr2 À r3Þ2 Since the yielding criterion based on the distortion energy 2 ðr1 given by (1.67) is identical with that proposed by Von Mises shown by (1.64), so the ‘Von Mises’ yielding criterion’ is þ ðr3 À r1Þ2i1=2 also called the ‘distortion energy yielding criterion’. ð1:64Þ Equation (1.64) predicts that plastic yielding under a (II) Maximum Shear Stress or Tresca Yielding Crite- situation of combined stresses will begin when the yield rion or Theory stress in uniaxial tension, r0; is exceeded by the differences of principal stresses given by the right-hand side of (1.64). The above yielding criterion, suggested by Tresca (1864), states that yielding will occur in any state of stress when the

1.5 Elements of Plastic Deformation 37 maximum shear stress reaches a critical value equal to the Now, substituting the value of r0 from (1.69) in (1.68a), maximum shear stress in a uniaxial tension test. Thus, we get the maximum shear stress yielding criterion in the yielding occurs when smax ¼ s0; where s0 the maximum following form: shear stress in a uniaxial tension test. r1 À r2 ¼ r0 ¼ 2k ð1:68bÞ From (1.48), it can be recalled that maximum shear stress; smax ¼ r1 À r2 : Thus, two expressions for the maximum shear stress 2 yielding criterion are: Since in uniaxial tension, ðiÞ r1 À r2 ¼ r0: ðiiÞ r1 À r2 ¼ 2k: r1 ¼ r0; ½r0 is the flow stress or yield stress where r1 is the algebraically largest principal normal stress and r2 is the algebraically smallest principal normal stress. in uniaxial tensionŠ; and r2 ¼ r3 ¼ 0: Note that according to the maximum shear stress yielding criterion, yielding is independent of the magnitude of the So, s0 ¼ r0 ; ) r1 À r2 ¼ r0 ; intermediate principal stress r3; which is not really true. 2 22 The advantage of the maximum shear stress yielding Or, r1 À r2 ¼ r0 ð1:68aÞ criterion is its mathematical simplicity compared to the Von Mises’ yielding criterion. Its difficulty is that one needs to The maximum shear stress or Tresca yielding criterion is know in advance which are the largest and the smallest expressed mathematically by (1.68a). principal stresses. The yield stress in pure shear shown by (1.62) and rep- It has been experimentally shown (Taylor and Quinney resented by k; can be evaluated and related to the yielding in 1931) that practical results are closely approximated by the the uniaxial tension test according to the above theory, as von-Mises’ yielding criterion than by the Tresca yielding follows: criterion. Hence, von-Mises’ yielding criterion is generally accepted and applied in most analysis throughout this book. For a state of pure shear the principal stresses are: r1 ¼ Àr2 ¼ k; and r3 ¼ 0: From (1.48), the maximum shear stress for a state of pure shear is: smax ¼ r1 À r2 ¼ 2 r1 ¼ r1 1.5.10 Octahedral Shear Stress and Shear Strain 2 2 ¼ k ¼ maximum shear stress in uniaxial tension The octahedral planes are the faces of a three-dimensional octahedron. They make equal angles with each of the three ¼ s0 ¼ r0 ; principal stress axes; i.e., their direction cosines are same. 2 The angle between the normal to one of the planespaffinffi d the r0 nearest principal axes is 54°44′, whose cosine is 1 3: The ) k ¼ 2 ¼ 0:5 r0 or; r0 ¼ 2k family of {111} planes in an FCC crystal lattice are octa- hedral planes. ð1:69Þ The stress acting on each face of such octahedron is oc- The ratio of yield stress in pure shear, k; to the yield stress tahedral stress, which can be resolved into two components in uniaxial tenpsioffiffin, r0; according to Von Mises’ yielding (Nadai 1950): criterion is 1 3; as evident from (1.63), while the ratio according to maximum shear stress yielding criterion is 1=2; 1. Normal octahedral stress. It is the hydrostatic component as shown by (1.69). of the total stress and cannot cause plastic deformation in solid materials. It is denoted by roct; and given by Hence; kVon Mises0 yielding criterion pkmffiffiaximum shearÀstress yielding criterion r0= 3 p2ffiffi r1 þ r2 þ r3 ¼ r0=2 ¼ 3 ¼ 1:155; roct ¼ 3 ¼ rm ð1:70Þ Therefore, the value of shear stress for yielding, i.e. ‘k’, as 2. Octahedral shear stress acting along the octahedral plane, given by the more accurate Von Mises’ yielding criterion, is represented by soct; and it is responsible for yielding. So, 15.5% higher than that given by the maximum shear stress its function resembles to that of the stress deviator and yielding criterion. given by

38 1 Tension soct ¼ 1 h À r2Þ2 þ ðr2 À r3Þ2 þ ðr3 À r1Þ2i1=2 ð1:71Þ Nadai (1937) has shown that the octahedral shear stress 3 ðr1 and shear strain are invariant functions of stress and strain Hence, it is assumed that yielding will occur in any state of that describe plastic deformation regardless of the state of stress when the octahedral shear stress soct reaches a critical value equal to the octahedral shear stress in a uniaxial tension stress. Most frequently used invariant stress and strain test ðsoctÞuniaxial tension; i.e., when soct ¼ ðsoctÞuniaxial tension: Hence for uniaxial tension, substituting r1 ¼ r0 into (1.71), functions, which describe the flow curve independent of the we get state of stress, are effective or significant stress, usually denoted by r; and effective or significant strain, normally represented by e: These are shown respectively by (1.76) and pffiffi (1.77a). ðsoctÞuniaxial tension¼ 2 r0 ¼ 0:471r0 ð1:72Þ Significant or effective stress is: 3 pffiffi h r1Þ2i1=2 Equating (1.71) and (1.72), the yielding criterion can be r ¼ 2 ðr1 À r2Þ2 þ ðr2 À r3Þ2 þ ðr3 À ð1:76Þ written as 2 h r1Þ2i1=2 Significant or effective strain is: ðr1 soct ¼ 1 À r2Þ2 þ ðr2 À r3Þ2 þ ðr3 À e ¼ pffiffi h À e2Þ2 þ ðe2 À e3Þ2 þ ðe3 À e1Þ2i1=2 ð1:77aÞ 3 pffiffi 2 ðe1 3 2 ¼ ðsoct Þuniaxial tension¼ 3 r0 ; To express the effective or significant strain, another form ¼ ) r0 p1ffiffi h À r2Þ2 þ ðr2 À r3Þ2 þ ðr3 À r1Þ2i1=2 of (1.77a) is as follows: 2 ðr1 pffiffi ð1:73Þ de ¼ 2 h À de2Þ2 þ ðde2 À de3Þ2 þ ðde3 À de1Þ2i1=2 3 ðde1 Since (1.73) is same as (1.67) in distortion energy theory, ð1:77bÞ so the octahedral shear stress yield theory and the distortion energy yield theory will produce the identical result. The strain considered in (1.77) is the plastic strain, where plastic strain ¼ total strain À elastic strain: Elastic strains Similar to octahedral normal stress, the octahedral linear can be ignored in comparison with enormous amount of strain eoct acting on each face of the three-dimensional plastic strains involved during metal working processes, but octahedron, which is the mean of the total strain, is given by in many plasticity problems the elastic strains cannot be neglected. eoct ¼ e1 þ e2 þ e3 ¼ em ð1:74Þ 3 Octahedral shear strain is given by 1.5.12 Levy–Mises Equations for Ideal Plastic Solid coct ¼ 2 h À e2Þ2 þ ðe2 À e3Þ2 þ ðe3 À e1Þ2i1=2 ð1:75Þ 3 ðe1 1.5.11 Invariants of Stress and Strain The relation between stresses and the differentials or incre- ments of plastic strain for an ideal rigid plastic solid, where Invariant functions of stress and strain are those which the elastic strains are negligible, is called flow rules or the describe the flow curve independent of state of stress (the Levy–Mises equations. The Levy–Mises equations neglect type of test). When the plastic stress–strain curve, i.e. the elastic strains and are only applicable to problems of large flow curve is plotted with invariant stress function as the plastic deformation, such as in metal working processes. ordinate and invariant strain function as the abscissa, approximately the same curve will be obtained irrespective Let us consider yielding under uniaxial tension where the of the state of stress. So, a complex state of stress and strain principal stresses are: r1 ¼6 0; but r2 ¼ r3 ¼ 0, and the can be simplified by means of invariant functions of stress respective principal plastic strains are e1; e2 and e3: The and strain. For example, when the flow curves for a uniaxial hydrostatic component of total stress in uniaxial tension will tension test and a biaxial torsion test are plotted in terms of be: rm ¼ r1=3: The stress deviators, which are only respon- the invariant functions of stress and strain, both the curves sible for yielding in uniaxial tension, are shown below: will be identical. r01 ¼ r1 À rm ¼ 2r1 ; and r02 ¼ r30 ¼ r2 À rm ¼ r3 À rm 3 ¼ 0 À r1 ¼ À r1 33

1.5 Elements of Plastic Deformation 39 From above, it is found that pffiffi \"  3r22 4  3r32 2 4 dk2 3r1 dk2 3r2 de ¼ À þ À r01 ¼ À2r02 ¼ À2r03 39 2 2 9 2 2 ð1:78Þ Since the volume remains constant during plastic defor- þ 4  À 3r12#1=2 mation, the value of Poisson’s ratio is m ¼ 1=2: Hence, one dk2 3r3 can write for an ideal plastic solid, 9 22 Poisson’s ratio; m ¼ transvere strain ¼ pffiffi h À r2Þ2 þ dk2ðr2 À r3Þ2 longitudinal strain 2 dk2ðr1 3 ¼ transvere strain increment ¼ de2 þ dk2ðr3 À r1Þ2i1=2 longitudinal strain increment de1 ¼ þp3ð2ffiffird3kÀp2rffi2ffi1&Þ2pi21ffi2=ffi2h)ðr1 À r2Þ2 þ ðr2 À r3Þ2 ¼ de3 ¼ À 1 de1 2 ) de1 ¼ À2de2 ¼ À2de3 ð1:79Þ Now from (1.78) and (1.79), one can relate the stress ¼ 2 dkr: ½with the help of ð1:76ފ deviators with the plastic strain increments as follows: 3 r10 r01 ) dk ¼ 3 de r02 r03 2 r de1 de1 ¼ À2 ¼ de2 ; or; ¼ À2 ¼ de3 ð1:80Þ ð1:83Þ Equation (1.80) can be generalized in following form of Putting the value of dk from (1.83) in (1.82a), (1.82b) and the Levy–Mises equation: (1.82c), the final form of Levy–Mises equation is obtained as follows: ! de1 ¼ de2 ¼ de3 ¼ constant ¼ dk ðsay) ð1:81Þ de1 ¼ de r1 À 1 ðr2 þ r3 Þ ð1:84aÞ r01 r20 r30 r 2 Levy and Von Mises proposed that the increments of de 1 ! plastic strain are related to stress deviators by the above r 2 relation given by (1.81). It shows that at the moment of de2 ¼ r2 À ðr1 þ r3 Þ ð1:84bÞ plastic deformation the ratio of the plastic strain increments to the instantaneous stress deviators is constant. With the de 1 ! help of (1.60a), (1.60b) and (1.60c), (1.81) can be expressed r 2 in terms of the actual applied stresses as follows: de3 ¼ r3 À ðr1 þ r2Þ ð1:84cÞ 2 1 ! Comparison between (1.32) developed for an elastic solid 3 2 de1 ¼ dk r1 À ðr2 þ r3Þ ð1:82aÞ and (1.84) developed for an ideal rigid plastic solid shows they are very similar, except the pre-bracketed part 1=E of the 2 1 ! former equation has been replaced by a ratio of de=r in the 3 2 latter equation. Further, in the flow rule (1.84), the value of m de2 ¼ dk r2 À ðr3 þ r1Þ ð1:82bÞ has been taken 1=2 due to plastic deformation, while no value of m has been substituted in (1.32) because of variation of 2 1 ! m-values from 1=4 to less than 1=2 during elastic deforma- 3 2 tion of nonporous solids. It is to be noted that 1=E always de3 ¼ dk r3 À ðr1 þ r2Þ ð1:82cÞ remains constant but the ratio de=r in the equation of flow rule changes throughout the period of plastic flow and this ratio By substituting de1; de2; de3; from (1.82a), (1.82b) and can be estimated by considering the plastic strain increment de (1.82c) in the effective strain (1.77b), the constant dk; can be in the plot of effective stress versus effective strain, by the evaluated as follows: method shown in Fig. 1.22, where de=r ¼ cot h:

40 1 Tension (I) Maximum Shear Stress Yielding Criterion Since r3 is a principal stress intermediate between r1 and r2; the above criterion (1.68b) is given by r1 À r2 ¼ r0 ¼ 2k ¼ r00 ð1:86aÞ where σ σ r00 flow stress under plane strain condition, r0 flow stress under homogeneous strain condition, and k ¼ yield stress in pure shear ¼ r00  ð1:87Þ 2 θ cot θ = dε The above (1.86a) can also be expressed in terms of the dε σ deviatoric stresses r01 and r02; as follows: Prior strain ε r10 À r02 ¼ r0 ¼ 2 k ¼ r00 ð1:86bÞ increments Fig. 1.22 Method of evaluating de=r in flow rule (1.84a) (II) Von Mises’ Yielding Criterion 1.5.13 Yielding Criteria Under Plane Strain By substituting the value of r3 from (1.85) into the above criterion given by (1.64), it changes to (1.88) as shown If there is no strain in one of the primary directions, the below: situation is called plane strain condition. To produce such condition, plastic deformation must be physically restrained p1ffiffi Þ2 nÀr1 À r2o2 2 2 along one direction. Such constraint can be developed with r0 ¼ ðr1 À r2 þ the help of an external barrier like a die wall, or the middle þ nÀr1 À r2o2!1=2 part of a material can be plastically deformed which can be constrained by the surrounding rigid part (the region may be h 2 i 4ðr1 r2Þ2 elastically deformed) of the material. For example, during 1 À r2Þ2 þ ðr1 À r2Þ2 þ ðr1 À ¼ 2r02 4 rolling, the width of the stock remains usually more or less constant and, hence, rolling is normally considered to be a Or, 6 ðr1 À r2Þ2 ¼ 2r20; 4 plane strain deformation. 8 Let the three mutually perpendicular principal axes are Or, ðr1 À r2Þ2 ¼ 6 r20; ‘1’, ‘2’ and ‘3’, which are respectively normal to the prin- rffiffi cipal planes of ‘23’, ‘31’ and ‘12’. Plane strain condition 86r0 p2ffiffi ) r1 À r2 ¼ ¼ 3 r0 ð1:88aÞ exists if the plastic flow occurs everywhere only parallel to a given plane; i.e., all plastic deformations are restricted to a Since, on the basis of von Mises’ yielding criterion, the particular plane, say, the plane ‘12’, and the flow does not uniaxial tensile yiepldffiffi stress r0 is related to the shearing yield depend on the direction ‘3’. In such case, the principal stress k; as Àr0¼pffiffiÁ3k; [see (1.63)], from (1.88a), it can be written as 2 3 r0 ¼ 2 k ¼ r00; where r00 ¼ flow stress plastic strain in direction ‘3’ is practically absent, i.e. e3 ¼ 0; or the increment of principal plastic strain in the direction ‘3’ under plane strain condition, also known as constrained is: de3 ¼ 0: Hence from Levy–Mises (1.84c), we get yield stress. Hence according to Von Mises’ yielding crite- de 1 ! rion, r00 ¼ 1:155r0; i.e. r00 is 15% greater than the homo- r 2 geneous yield stress r0; and r00 must always be used for de3 ¼ 0 ¼ r3 À ðr1 þ r2Þ : plane-strain deformation. ) Stress in the principal direction ‘3’ is: Hence under plane strain condition, the von Mises’ r3 ¼ 1 ðr1 þ r2Þ ð1:85Þ yielding criterion finally reduces to 2 Therefore, even though the strain in a direction is zero, a ) r1 À r2 ¼ 2 k ¼ r00 ð1:88bÞ constraining stress acts in that direction.

1.5 Elements of Plastic Deformation 41 It must be noted that the above both yielding criteria are 1.6.1 Type I: Elastic Behaviour equivalent under a plane strain condition. It is also to be Elastic deformation is reversible where the object returns to noted that in both (1.86a) and (1.88b), r1 is the algebraically its original shape and size on release of the applied force. greatest principal normal stress and r2 is the algebraically Material under this category will show elastic deformation smallest principal normal stress. right up to the point of fracture. The resulting stress–strain curve for this kind of material is usually linear, as shown in 1.6 Types of Tensile Stress–Strain Curve Fig. 1.24a, although it may be sometimes nonlinear as shown in Fig. 1.25. Brittle materials, such as glasses, rocks, In a uniaxial tensile test, a suitably designed specimen, after concrete, many ceramics, heavily cross-linked polymers, measuring its lateral dimensions and marking its gage cast iron and some other B.C.C. metals at low temperature, length, is subjected to increasing axial load until it fractures. will show linear elastic behaviour, although some brittle Schematic diagram of one model of a tensile testing machine materials such as cast iron and some other B.C.C. metals at is displayed in Fig. 1.23. During the test, the simultaneous low temperature usually exhibit a little amount of plastic readings of load, P, and deformation or extension, DL; are range. A completely brittle material, such as glass, exhibits recorded at frequent intervals. These above recorded read- no plastic deformation as shown in Fig. 1.24a, but a brittle ings are converted to the engineering stress, S; and the metal-like white cast iron will show a slight amount of engineering strain, e, according to (1.24) and (1.5), respec- plasticity before fracture, as shown in Fig. 1.24b which has tively, and the values are plotted as S À e diagram. The also been included in the same category of stress–strain shape of S À e diagram will be the same as that of the curve. Generally, brittle materials are stronger in compres- load-deformation, i.e., P–DL diagram. Different types of sion than in tension, sometimes by a large factor. Naturally, tensile stress–strain curve reflecting different deformation applications of these materials under tensile loading are characteristics have been discussed below in this section. restricted, but they may be applied successfully in services Hooke’s law stated in Sect. 1.4 is applicable to the linear under compressive loads as compression tends to close up elastic part of stress–strain curves discussed below. cracks present in brittle materials causing much greater resistance to fracture. For example, concrete is used widely Cross yoke in compression but not in tension, and if required, the con- crete is reinforced by the addition of steel bars that bear the Load cell tensile loads. Upper limit Testing machine In Fig. 1.25, nonlinear elastic stress–strain diagram has switch knob main unit been shown for elastomer or rubber which is characterized by very large elastic strain, often in excess of 100%. Shape Sensor Movable memory alloys such as Nitinol also exhibit large and non- switch crosshead linear elastic deformation range. The tensile stress–strain Control/display curve of elastomer or rubber has some similarity with that of Load fixture panel crystalline polymer under the category of Type V behaviour, discussed later, though there are notable differences as fol- Lower limit Control unit lows. Elastomers or rubbers show no drop of stress at switch knob intermediate strains; i.e., the slope of the stress–strain curve for them is always positive. Further, most strains in them are Fixed fully reversed, though nonlinearly. In the unstressed condi- crosshead tion, the structure of rubber is vey disorganized with the chains having a random coiled configuration. On tensile Emergency switch loading, the chains of the rubber become straightened and aligned. As a result, the extent of elastic deformation is very Fig. 1.23 Schematic diagram of one model of a universal tensile high for rubber. The straightening of the chains in rubber is testing machine responsible for the rapid rise of stress at large strains, as shown in Fig. 1.25. However, on the release of applied load, the chains of rubber again return to the random curled configuration, indicating that this configuration is the pre- ferred one for rubber.

42 1 Tension Fig. 1.24 Type I: schematic (a) Proportional limit (b) S1 S2 engineering stress–strain curves e1 e2 showing linear elastic behaviour and fracture strength Young Modulus, E = = for a a completely brittle material, such as glass and b a brittle Fracture material with a very little Proportional limit ductility, such as white cast iron Loading path Fracture Engineering stress, S Ultimate and Engineering stress, S fracture strength Unloading path S1 Type I S2 Type I e1 e2 O Engineering strain, e O Engineering strain, e The essential feature is that both the linear and nonlinear A elastic curves will return to the origin, if the applied load is removed from the tensile specimen before the point of Engineering stress Type I fracture. The reversible nature of strain is a basic element of Loading curve elastic strain in any material, irrespective of its capability to undergo much larger total strain or not. In case of a linear 1 elastic plot, the unloading curve will flow the same path but 2 in reverse direction as compared to the loading curve, as shown by the arrow marks in Fig. 1.24a. But the paths fol- Unloading curve lowed by a nonlinear elastic curve during loading and unloading are different. The nonlinear unloading curve is usually somewhat lower than the nonlinear loading curve as shown in Fig. 1.25. 1.6.2 Type II: Elastic–Homogeneous Plastic O Engineering strain Behaviour Fig. 1.25 Type I: schematic engineering stress–strain diagram show- Material under this category will show linear elastic defor- ing nonlinear elastic behaviour for elastomer or rubber during loading mation up to small strain and thereafter a gradual transition and unloading, represented respectively by curves ‘1’ and ‘2’ from elastic to homogeneous plastic deformation, i.e., irre- versible flow that continues up to the point of fracture as 1.6.2.1 Engineering Stress–Strain Diagram shown in Fig. 1.26. Beyond the elastic strain, the stress– A real ductile material, which undergoes strain hardening, strain curve in Fig. 1.26 shows a smooth parabolic region will cause an increase in load as well as engineering stress which is associated with homogeneous plastic deformation required to cause plastic deformation as long as the decrease process, such as irreversible movement of dislocation in in cross-sectional area along the gage length of the specimen metals, ceramics and crystalline polymers. Generally, this is uniform. Because the strain hardening increases the load type of curve is very common for face-centred-cubic needed to deform the material plastically, this increase is (FCC) metals. Body-centred-cubic (BCC) metals at ele- greater than the decrease in load caused by the reduction in vated temperature also show this kind of curve. instantaneous cross-sectional area of specimen. Hence dur- ing uniform elongation, the higher rate of strain hardening

1.6 Types of Tensile Stress–Strain Curve 43 Strain to fracture, ef with the progress of plastic deformation due to localized Uniform strain, eu reduction in cross-sectional area of the specimen. Engineering stress, S Offset Fracture 1.6.2.2 True Stress–Strain Diagram yield Actually, the strain hardening of the tensile specimen con- strength, S0 tinues throughout till the point of fracture so that the stress required for further progress of plastic deformation should Ultimate tensile Fracture also increase. The above can easily be understood from the strength, Su strength, Sf true stress–true strain diagram (r À e curve) that increases continuously up to the point of fracture unlike the S À e Type II curve, because the true stress r is based on the actual cross-sectional area of specimen at the moment of observa- Engineering strain, e tion, which causes r to increase. Fig. 1.26 Type II: schematic engineering stress–strain curve showing The true stress r can be determined from the value of linear elastic behaviour followed by a region of homogeneous plastic conventional or engineering stress S; according to (1.28) or deformation. Different tensile properties are also shown in the diagram (1.29), which is applicable only during uniform plastic deformation because both constancy of volume and homo- makes the material stronger and increases the applied load. geneous distribution of strain along the gage length of tensile But as soon as ‘necking’ of the specimen of a ductile specimen are assumed in the derivation of (1.28) or (1.29). material begins, the condition is reversed causing the applied But once the necking begins, the true stress, r, must be load to drop. Thus, the point of maximum load in the load– determined from measurements of the actual load and the deformation curve indicates the point of initiation of necking instantaneous minimum cross-sectional area in the necked in the tensile specimen. Necking is the localized plastic region of the tensile specimen, according to (1.25). Thus, a deformation at some weak region in the tensile ductile true stress–true strain diagram (r À e curve) can be con- specimen. Once the necking starts, all further plastic defor- structed from the engineering stress versus the engineering mations are concentrated in that weak region causing this strain diagram up to the point of maximum load beyond region to thin down locally. Finally, it results in the localized which measurements of load at the moment of observation reduction of cross-sectional area at that weak region in the and instantaneous minimum cross-sectional area at the neck tensile specimen. At this stage of the tensile test, the load are required to generate the r À e curve. Similarly beyond needed to continue plastic deformation drops due to rapid the point of the maximum load, the true strain e cannot be decrease in the cross-sectional area in the necked region of calculated from the measured value of conventional or the tensile ductile specimen. So the engineering stress S engineering strain e, with the help of (1.10), which is valid based on the original cross-sectional area of specimen for a homogeneous distribution of strain along the gage decreases causing the engineering stress versus engineering length of the tensile specimen. Hence, the natural or true strain curve (S À e curve) to decline beyond the point of strain, which is based on the measurement of actual maximum load, i.e. the onset of necking for a ductile cross-sectional area or diameter of the tensile specimen, material (Fig. 1.26). This falling-off continues till the point must be determined according to (1.89a) or (1.89b) beyond of fracture and fracture usually takes place at the necked the point of the onset of necking. region of the tensile specimen. But an ideal plastic material in which no strain hardening occurs would start to neck as Let us assume that L0, A0 and D0 represent, respectively, soon as the material yields and the applied load as well as original gage length and initial average cross-sectional area engineering stress (proportional to load) would decrease and initial average diameter of tensile specimen and that L; A; and D are, respectively, instantaneous values of gage length, smallest cross-sectional area and minimum diameter during tensile deformation at any point between the onset of necking and the fracture of tensile specimen. Due to the constancy of volume, V; during plastic deformation, one can write

44 1 Tension V ¼ LA ¼ constant; ) dV ¼ AdL þ LdA ¼ 0; of maximum load, since the true strain is higher and the true Or; dL ¼ À dA ; stress calculated from (1.25) is always higher than the engineering stress calculated from (1.24), the true curve LA overtakes the engineering curve; i.e., the points on the true curve are to the right and above of those on the engineering Hence from (1.7), curve beyond the point of maximum load. From the point of maximum load to fracture, the true curve increases contin- true strain uously upward and is frequently linear, whereas the engi- neering curve continuously bends downward. The fracture Ze ZL ZA ð1:89aÞ point on the true curve is far to the right and above of that on e ¼ de ¼ dL ¼ À dA ¼ ln A0 the engineering curve. L AA A true-stress–true-strain curve is also known as a flow 0 L0 A0 curve, and the true stress is called flow stress, because the curve represents the basic plastic-flow characteristics of the Or, for cylindrical tensile specimen, true strain material and any point on this curve can be considered as the  yield stress or flow stress of a material that has been plas- A0 pD02 4 D0 ð1:89bÞ tically deformed in tension up to that point. Thus, if the load e ¼ ln A ¼ ln pD2=4 ¼ 2 ln D is released at that point and then reapplied, the material will show elastic behaviour over almost total span of the Figure 1.27 shows the true-stress–true-strain curve with reloading. This has been shown with the help of the its corresponding engineering stress–strain curve for the true-stress–true-strain curve drawn in Fig. 1.28, where the purpose of comparison. In this figure, the elastic regions of material is unloaded after being loaded into the plastic range both the curves have been compressed into the stress axis and then reloaded. In this context, it is relevant to discuss on because of the relatively large plastic strains. On the two loading of a member into the plastic range and unloading it curves in Fig. 1.27, the corresponding points are joined to from there. show their relations one to the other. In agreement with (1.10) and (1.28) or (1.29), the points on the true-stress– 1.6.2.3 Loading and Unloading true-strain curve are always to the left and above of those on Figure 1.29 shows the relative amounts of elastic, anelastic the engineering curve till the point of maximum load, i.e. the and plastic strain in tensile specimens of the same material onset of necking. Beyond this point of maximum load, because of high local necking strain, the values of true strain calculated from (1.89a) or (1.89b) becomes higher than those of engineering strain calculated from (1.5). Beyond the point Fig. 1.27 Comparison of Applied true engineering and true stress–strain fracture strength, σfapp curves for a ductile material (schematic) True fracture strain, εf True Tensile strength, σu Unifrom True local necking strain, εn true strain, εu Engineering and true stress True curve Corrected true fracture strength, σf true Corrected for necking Engineering curve Fracture strength, Sf Unifrom Maximum load engineering strain, eu Fracture Tensile strength, Su Engineering and true strain

1.6 Types of Tensile Stress–Strain Curve 45 Continued loading Loading True stress Reloading Unloading Unloading Loading Stress Parallel lines Loading O True strain Fig. 1.28 Repeated loading (schematic) when they are unloaded from different points beyond the O Strain linear elastic part of stress–strain curve. Suppose the speci- mens are plastically strained in tension up to two points Fig. 1.30 Hysteresis loop in unloading and reloading (schematic) A and B on this curve, where OA and OAB are, respectively, loading paths and AA′ and BB′ are, respectively, unloading where elastic strain from D to B′, eD!B0 ; is greater than paths of the specimens, and the respective total strains, elastic strain from C to A′, eC!A0 ; as proved below. If rA and consisting of elastic, anelastic and plastic parts, are OC and rB are respective tensile stresses at points A and B on the OD. Neglecting slight curvature of the unloading curves, as stress–strain curve, then shown in Fig. 1.30, it is assumed that the unloading curves AA′ and BB′ follow paths nearly parallel to the linear elastic eC!A0 ¼ rA=E and eD!B0 ¼ rB=E; part of the true stress–true strain curve; i.e., the slope of * rB [ rA; ) eD!B0 [ eC!A0 : unloading curve is equal to the elastic modulus, E of the material. The immediate recoverable elastic strains on Therefore, it is clear from above that the higher the tensile unloading are respectively from points C to A′ and D to B′, stress applied to deform the material plastically is, the higher the immediate recoverable elastic strain on unloading will Young’s modulus, E = σ B be. This implies that the stronger or harder the material, the ε higher the immediate elastic recovery is. A However, on unloading, the remaining strains from points O to A′ and O to B′ are not fully permanent plastic strain, but Unloading a small part of the remaining strain is elastic in nature, which will disappear with time. This time-dependent recoverable Loading elastic strain is called anelasticity. The extent of anelasticity depends on material and test temperature. Anelastic effects at Stress room temperature are usually very small in metals but can be large for polymeric materials. Figure 1.29 shows that the σσ σ elastic strains that disappear with time are from points A′ to εε ε A″ and B′ to B″. So, in Fig. 1.29, the permanent plastic strains will be from points O to A″ and O to B″. O A\" A' B\" B' D C Usually, the unloading path from the plastic range of the stress–strain curve is not exactly parallel to the linear elastic Strain part of the curve and will be slightly curved as shown in Fig. 1.30. Further on reloading, the stress–strain curve will Fig. 1.29 Loading and unloading in the plastic range, showing generally bend over as the reloading stress approaches permanent plastic strain and recoverable elastic and anelastic strains towards that value of stress from which the tensile specimen (schematic) was unloaded earlier. After a little additional plastic strain, the reloading curve becomes a continuation of the prior

46 1 Tension stress–strain curve that would have been generated if no True stress σ, on log scale n= a unloading had taken place. Thus, in fact, the unloading and b reloading curves create a hysteresis loop as shown in Fig. 1.30. a b Generally, the anelastic strain as well as the hysteresis behaviour arising due to unloading and reloading from a K plastic strain is neglected in the theory of plasticity. So it may be concluded that subsequent to the unloading of the 0.001 0.01 0.1 1.0 tensile specimen from the plastic range, the reloading curve True strain ε will start from point A′ or B′ in Fig. 1.29. Moreover, it is assumed that yielding will start respectively from point A or Fig. 1.31 Log–log plot of true stress–strain curve from the onset of B on the stress–strain curve up to which the specimens were yielding to the point of maximum load. The slope of this linear plot is plastically deformed earlier in tension, as in Fig. 1.29. This the strain hardening coefficient, n ¼ a=b ; as shown. The strength means that when the material is reloaded in tension after coefficient, K ¼ r ; at a true plastic strain, e ¼ 1:0 unloading from its plastic range, the tensile stress required for yielding increases, i.e., its tensile yield strength, is raised to a higher value. 1.6.2.4 Empirical Relationship for Flow Curve where e0 ¼ the amount of strain hardening imparted to the The flow curve of many materials in the region of uniform material prior to the tension test. plastic deformation, i.e. from the onset of yielding to the point of maximum load (initiation of necking), can be The fallacy of (1.90a) lies in the fact that when the true described empirically by the following parabolic true-stress– plastic strain is zero, i.e. e ¼ 0; the true stress at the onset of true-strain relationship proposed by Hollomon (1945): yielding of the material is also zero, i.e. r ¼ 0: So, the origin of the true stress–true strain curve will be at the yield stress r ¼ Ken ð1:90aÞ value of zero, but this is not the case in reality. This dis- crepancy has been overcome by the following relationship Or; log r ¼ log K þ n log e ð1:90bÞ given by (1.90d), which is proposed by Ludwik: where r ¼ true stress, K ¼ a material constant, known as r ¼ r0 þ Ken ð1:90dÞ ‘strength coefficient’, e ¼ true plastic strain, and n ¼ strain-hardening coefficient or exponent. Since (1.90b) rep- where r0 ¼ true stress at the yield point, i.e. true yield stress; resents a straight line, the plot of log r versus log e from the i.e., when the true plastic strain e ¼ 0; the true stress r ¼ r0; onset of yielding to the point of maximum load will be a and K and n are the same parameters as in (1.90a). The value straight line as shown in Fig. 1.31, if (1.90a) is obeyed by of r0 in (1.90d) can be obtained from the intersection of the the experimental data. The value of the strain-hardening parabolic curve expressed by (1.90a) and the linear elastic coefficient, n, is the slope of this linear plot, whose intercept region expressed by (1.31b) at the point of yielding (Mor- at log e ¼ 0; is log K: It means the strength coefficient, K, is rison 1966). Most materials show a gradual transition from the value of the true stress, r, at a true plastic strain, e ¼ 1:0; elastic to plastic behaviour, and it is very difficult to define (corresponding to reduction of area, r ¼ 0:63), with the precisely the point of the onset of plastic deformation or dimension of K being the same as that of r. To obtain the yielding because it depends on the sensitivity of the strain value of K, the above straight line has to be extended to the measuring instruments. So, it is assumed that the Hooke’s right side of the above plot, i.e. to a higher value of true law holds good up to the point of yielding that produces a plastic strain till e ¼ 1:0: If a nonlinear log-log plot is very small measurable amount of plastic deformation, but observed for a given material, often at low strain values of theoretically the Hooke’s law is obeyed up to the point of 10À3 or high strain values of 1, the strain-hardening coeffi- ‘proportional limit’, to be discussed subsequently in this cient, n, will be often defined at a particular strain value. chapter. Sometimes, the experimental data that deviates from (1.90a) will generate a linear plot according to the following rela- At the yield point, (1.31b) can be written as tionship proposed by Datsko (1966): r ¼ Kðe0 þ eÞn ð1:90cÞ r0 ¼ E e0 ð1:91aÞ

1.6 Types of Tensile Stress–Strain Curve 47 Again at the yield point, (1.90a) can be written as σ r0 ¼ K en0 ð1:91bÞ where r0 and e0 are, respectively, the true stress and true n= 1 strain at the yield point. 2 By solving (1.91a) and (1.91b), the value of r0 in (1.90d) n=0 can be obtained as given below: From (1.91a), e0 ¼ ðr0=EÞ and from (1.91b), e0 ¼ ðr0=KÞ1=n K n=1 ) r0 ¼ r0 1=n or, rð01nÀ1Þ ¼ K 1=n ; ; E EK  n  1 ð1:92Þ K 1=n 1Àn K 1Àn En ) r0 ¼ E ¼ 1.0 ε 1.6.2.5 Strain-Hardening Coefficient Fig. 1.32 Different forms of true stress–strain curve expressed by r ¼ Ken for various values of n (strain-hardening coefficient) The magnitude of the strain-hardening coefficient, n, indi- 1.6.2.6 Tensile Instability cates the ability of the material to resist further deformation. On reaching a particular value of applied stress, plastic flow The values of strain-hardening coefficient may vary from will start at the weakest region within the gage length of the n ¼ 0 for an ideal plastic material to n ¼ 1 in case of a tensile specimen. The strain hardening caused by this perfectly elastic solid, as shown in Fig. 1.32. For most localized plastic flow in a real material increases its local materials, the values of n lie between 0.10 and 0.50. To strength and prevents further plastic deformation in that know techniques for measuring n; the reader is referred to region. Then, the applied stress must be increased so that the discussion by Duncan (1967) and ASTM Standard further plastic flow occurs in the next weakest region within (ASTM E646 2016a). It is important to note that the values the gage length. The material is again strain hardened in this of n are sensitive to thermal as well as mechanical treatment region, and in this way, the above action of shifting the area and normally higher for materials in the annealed condition of deformation continues. Prior to the onset of necking, and lower in the cold-worked state. In general, n of a macroscopically there is a uniform extension of the gage material decreases as the strength level of that material length that in turn causes a uniform reduction in the increases and vice versa. Further, n of a material depends on instantaneous cross-sectional area A along the gage length to the cross-slip of screw dislocations in that material, which in maintain the constancy in volume during plastic deforma- turn is determined by its stacking fault energy. Since the tion. The reduction in A requires lower load for plastic extent of strain hardening is greater in a material with lower deformation, while strain hardening requires higher load for stacking fault energy than with higher stacking fault energy, plastic deformation. The net effect causes the applied load to the value of n will increase with decreasing stacking fault increase with the progress of deformation in the uniform energy and vice versa. plastic deformation range, i.e. prior to the onset of necking. The rate of strain hardening, dr=de; at any given strain, Prior to the onset of necking, since the increase in sometimes called the modulus of strain hardening, is equal load-bearing capability of the material caused by strain to the slope of the r À e curve at that point and is not the hardening is greater than the increase in stress r due to same as the strain-hardening coefficient n, but directly rela- decrease in the instantaneous cross-sectional area, necking is ted to n, as shown below: prevented. Hence, the higher rate of strain hardening results in a uniform elongation and as long as ðdr=deÞ [ r; necking As n is the slope of the linear plot of log r against log e, is prevented. With increase of load, the rate of strain hard- so ening gradually decreases due to the increase in strength of material, although strain hardening continues till the point of n ¼ dðlog rÞ ¼ dr=r ¼ e dr ; or; dr ¼ n r ð1:93Þ fracture. At the point of maximum load, the strength of the dðlog eÞ de=e r de de e material due to strain hardening is equal to the applied stress r: But once the necking of specimen begins, the load The rate of strain hardening is an important property in the analysis of tensile instability and working operations.

48 1 Tension required for continuation of deformation drops due to rapid the strain axis at a linear distance of unit strain, i.e. at a value decrease of the cross-sectional area, A; in the necked region of true strain, e ¼ 1; when measured linearly along the strain of the tensile specimen. The increase in r due to the rapid axis from the point of tangent, because according to (1.95) decrease in A becomes greater than the increase in the slope at this point is ru=1: This point of tangent on the load-bearing capability of the material caused by strain flow curve is the true tensile strength and can be estimated hardening so that r [ ðdr=deÞ: Therefore at the point of from the horizontal projection of the tangent point on the maximum load, necking begins causing instability in ten- stress axis. The linear distance along the strain axis between sion, which is expressed by the condition dP ¼ 0; because at the stress axis and the vertical drop from the point of tangent this point the decrease in load due to the reduction in A is is a measure of true strain at maximum load or uniform true equal to the increase in load due to strain hardening. strain and has been denoted by eu: As P ¼ rA; ) dP ¼ Adr þ rdA ¼ 0; The tangent on each point of the true stress–true strain curve is a measure of the rate of strain hardening, dr=de; and Or, Adr ¼ ÀrdA; it can be estimated against each value of the true strain of the flow curve. In Fig. 1.33b, the rate of strain hardening versus ) dr ¼ À dA ð1:94aÞ the true strain curve, i.e. the dr=de versus e curve, is r A superimposed on the flow curve, i.e. on r versus e curve. As per (1.95), at the point of maximum load, the rate of strain From the constancy of volume, V; during plastic defor- hardening equals the stress. So the point at which r versus e mation, one can write curve and dr=de versus e curve intersects each other is the point of maximum load on the flow curve. When this point V ¼ LA ¼ constant; ) dV ¼ AdL þ LdA ¼ 0; ð1:94bÞ of intersection is horizontally projected on the stress axis, it Or; dL ¼ À dA ; gives a measure of the true tensile strength, ru: The uniform true strain, eu; can be estimated from the vertical drop of this LA point of intersection on the strain axis. Since from ð1:7Þ; dL ¼ de; ) À dA ¼ de For obtaining the point of maximum load, the former LA method shown in Fig. 1.33a requires searching of the point of tangent on the flow curve and, in the latter method shown From (1.94a) and (1.94b), it follows that in Fig. 1.33b, the construction of dr=de versus e curve is not only laborious but also time-consuming. To determine the dr ¼ de or; dr ¼ r ¼ ru ð1:95Þ point of maximum load avoiding the above-mentioned dif- r de ficulties, a geometrical construction known as “Considère’s construction” (Considère 1885), involving true stress versus where ru ¼ true stress at the maximum load or true tensile engineering strain diagram, i.e., r versus e curve, can be strength. Hence with the help of (1.95), the point of maxi- used as discussed below. mum load where necking begins can be obtained from the flow curve as explained in Fig. 1.33a, b. On the true stress–true strain curve in Fig. 1.33a, such a point is determined that the tangent at that point intersects Fig. 1.33 Graphical (a) (b) σ, dσ dε determination of true tensile strength, ru; and uniform true σ dσ vs. ε curve strain, eu; from necking criterion σu dε σ vs. ε curve σu dσ = σ dε εu εu ε ε 0 1 0

1.6 Types of Tensile Stress–Strain Curve 49 Starting with (1.95), the necking criterion can be Therefore, the amount of uniform true plastic strain is expressed using the engineering strain, e, as follows: numerically equal to the value of strain-hardening coeffi- cient, if (1.90a) is obeyed. dr ¼ dr de ¼ dr dL=Lo ¼ dr L ¼ dr ð1 þ eÞ ¼ r de de de de dL=L de Lo de 1.6.2.7 Stress Field at the Neck and Bridgman Analysis ) dr ¼ r ¼ ru de 1þe 1 þ eu The development of a neck in the tensile specimen intro- duces a triaxial state of tensile stress in that region, in ð1:96aÞ addition to the necking strains. This triaxial stress field raises the value of axial tensile stress required to cause plastic And using (1.28), deformation in the longitudinal direction. The necked sec- tion of the sample, having a lower cross-sectional area, dr ¼ S ¼ Su ð1:96bÞ experiences a higher true stress than the unnecked regions. de The more highly stressed material within the neck, under- going large local extensions in the longitudinal direction, where seeks to contract laterally because conservation of volume must be maintained during the plastic deformation process. Su engineering stress at maximum load or ultimate tensile But the unnecked regions of the sample experiencing a much strength and lower stress level induce radial, rr; and transverse, rh; true tensile stresses that constrain such lateral contractions and eu engineering strain at maximum load or uniform engi- resist deepening of the neck. Thus, the induced constraining neering strain. tensile stresses developed in the radial and transverse directions along with the applied longitudinal tensile stress The stress–strain curve plotted in terms of true stress constitute the triaxial stress field, which acts to constrain the material from plastically deforming in the necked region. versus engineering strain to determine the point of maximum Consequently, the applied axial tensile stress required to cause plastic deformation at the neck must be higher than the load is illustrated in Fig. 1.34. Suppose point A on the axis stress which would be required to cause plastic deformation if uniaxial tensile stress condition prevailed. So, after the of engineering strain at the left side of the origin of the above onset of necking, the higher values of axial stresses, caused by the development of the plastic constraint in the necked plot represents a negative engineering strain value of 1.0, i.e. region, are recorded on the true stress–strain curve (Fig. 1.27). Based on the following assumptions, Bridgman e ¼ À1: If a straight line is drawn from the point A so that it (1944) corrected the measured axial true tensile stress, rapp; applied under the triaxial stress condition in the presence of becomes tangent at any point on the stress–strain curve, then neck to determine the uniaxial true tensile stress, rtrue; which would be necessary for plastic deformation if necking had that point of tangent will establish the point of maximum not introduced triaxial stresses. The assumptions are: load, because according to (1.96a), the slope of the line • The neck contour is represented by the arc of a circle. • The cross-section of the necking zone remains circular joining the point A and the tangent point is ru=ð1 þ euÞ: This point of tangent on the r versus e curve is the true stress at during the test. • The Von Mises’ yielding criterion is applicable. maximum load, ru; whose value can be determined from the • The strains remain uniform over the cross-section of the horizontal projection of the tangent point on the true stress neck. axis. When this tangent point is vertically dropped on the The geometry of the necked region and the triaxial engineering strain axis, it gives a measure of the uniform stresses developed within the necked region are illustrated in Fig. 1.35. The true stress–strain curve has been corrected for engineering strain. The point of intersection between the necking with the following Bridgman relation, and the resulting curve is shown in Fig. 1.27. above tangent drawn from the point A at e ¼ À1 and the stress axis gives a measure of the ultimate tensile strength, because according to (1.96b), the slope of the line joining the point A and the point of intersection is Su=1: If the condition of tensile instability expressed by (1.95) is applied to the Hollomon parabolic true-stress–true-strain relation given by (1.90a) at the point of maximum load, the following simple relationship between the uniform true strain, eu; and the strain-hardening coefficient, n, is developed. Differentiating (1.90a) with respect to e; we get: dr ¼ KnenÀ1: de Substituting the above in (1.95), KnenÀ1 ¼ r ¼ Ken; [From (1.90a)]. ) At the point of maximum load; n ¼ eu ð1:97Þ

50 1 Tension Fig. 1.34 Considère’s True stress σ construction, to establish the point of maximum load and determine true tensile strength, ru; uniform engineering strain, eu; and ultimate tensile strength, Su σu Su –1 +1 A 0 Engineering 1 eu strain e 1 + eu rtrue ¼ rapp ð1:98Þ (a) (b) 2R=aÞ½lnð1 þ ð1 þ a=2Rފ where σapp rapp measured true tensile stress at the neck, applied in the axial direction under the triaxial stress condition in the presence of neck, i.e. rapp ¼ axial tensile load ; minimum cross-sectional area of the specimen at the neck rtrue uniaxial true tensile stress that would be necessary for R σr plastic deformation if necking had not introduced a triaxial stresses; σθ R radius of curvature of the neck contour; and a radius of the minimum cross-sectional area at the neck It is obvious from the Bridgman relation that the stress Fig. 1.35 a Geometry of necked region. b Development of triaxial necessary to produce a given level of plastic deformation tensile stresses within the necked region which acts to constrain will increase with increasing sharpness of the neck contour, additional deformation in the neck i.e. with decreasing the value of R for a given depth of the neck contour. assumption, the incremental transverse true tensile strain ¼ deh ¼ de3 ¼ de2 ¼ der ¼ the incremental radial true tensile The fact that rapp is higher than rtrue; has been proved strain. Hence, according to the Levy–Mises equations below. With reference to Von Mises’ yielding criterion (1.84b) and (1.84c), it yields (1.99). given by (1.64), we can write r0 ¼ rtrue; r1 ¼ rapp; r2 ¼ rr and r3 ¼ rh; in the present context. As the necked region of the sample contracts symmetrically in the lateral directions to maintain a uniform strain, so according to this above

1.6 Types of Tensile Stress–Strain Curve 51 deh ¼ de rh À 1 À þ Á! ¼ der ¼ de rr À 1 À þ Á! r 2 rapp rr r 2 rapp rh Or, rh À 1 rr ¼ rr À 1 rh; or, 3 rh ¼ 3 rr ; 2 2 2 2 ) Transverse true tensile stress rh ¼ Radial true tensile stress rr Engineering stress ð1:99Þ With the help of Von Mises’ yielding criterion given by (1.64), the required proof is shown by (1.100): rtrue ¼ p1ffiffi h À rr Þ2 þ ðrr À rr Þ2 þ ðrr À rappÞ2i1=2 Type III 2 ðrapp Engineering strain ¼ p1ffiffi h À rr Þ2i1=2; 2 2ðrapp Fig. 1.36 Type III: schematic engineering stress–strain curve showing linear elastic behaviour followed by a region of heterogeneous plastic Or, rtrue ¼ rapp À rr; or, rapp ¼ rtrue þ rr; deformation. This serrated stress–strain curve is caused by twinning controlled deformation or interactions of solute atom or vacancy with i:e:; rapp [ rtrue lattice dislocations ð1:100Þ body-centred-cubic (BCC) or face-centred-cubic (FCC) metals. However, twinning can occur in metals of cubic The values of a=R required for the Bridgman correction crystal system when the twinning stress is lower than the (1.98) can be obtained (Trozera 1963) by direct measure- stress required for slip because of increase in the magnitude ments of ‘a’ and ‘R’ when a tensile specimen is unloaded of CRSS for slip at low deformation temperatures or at high after it has been subjected to a given amount of strain strain rates. So, a similar stress–strain curve may also be beyond necking. Otherwise, these parameters can also be observed in BCC and FCC metals due to occurrence of measured continuously beyond the point of necking using twinning, when they are tested at low deformation temper- photography or a tapered ring gage. The applied true tensile atures or at high strain rates. stress rapp and the corrected true tensile stress rtrue beyond the point of maximum load (the onset of necking) are shown During twinning, extension of the gage length proceeds in respectively by solid line and dashed line in Fig. 1.27. For a discrete bursts, often with an emission of audible click [tin flat tensile specimen, Aronofsky (1951) has considered a cry], that are associated with twin band nucleation and correction for the triaxial stresses in the neck. growth. This causes the instantaneous strain rate of the specimen to increase remarkably. Whenever this instanta- 1.6.3 Type III: Elastic–Heterogeneous Plastic neous strain rate in the specimen exceeds the cross-head Behaviour velocity of the tensile testing machine, a drop of load with a corresponding drop in engineering stress will occur in order Material under this category will show a non-uniform or to decrease the instantaneous strain rate in the specimen so heterogeneous plastic deformation after the normal range of that a balance is maintained between the two rates. The linear elastic deformation as shown in Fig. 1.36. When a deformation stress increases due to strain hardening of the series of serrations are superimposed on the parabolic por- material and causes further twinning. Every time the speci- tion in Fig. 1.26, a serrated stress–strain curve will result. men twins, each jog of the irregular stress–strain curve is Beyond the elastic strain, such serrated stress–strain curve in created due to the drop of stress, and thus, serrations in Fig. 1.36 is exhibited by this category of material either due stress–strain curve in Fig. 1.36 are produced. to occurrence of twinning or interactions of solute atom or vacancy with lattice dislocations. Serrated stress–strain curves are also observed in plain carbon steels tested at moderately elevated temperature of Von Mises showed that the operation of five independent approximately 200°C, depending on the strain rate used in slip systems is required for plastic deformation of a crystal the test. The occurrence of serrations in the stress–strain by slip. As this requirement is not fulfilled by the HCP curve is associated with the dynamic strain ageing beha- metals, twinning occurs during the tension test of hexagonal viour, called the ‘Portevin-Le Chatelier effect’, which is due close-packed (HCP) metals at orientations that are unfa- to solute atom or vacancy interactions with lattice vourable for slip on basal plane. Therefore, this type of curve is generally the most common for HCP metals. Twinning is not a dominant deformation mechanism in metals of cubic crystal system having many possible slip systems, such as in

52 1 Tension dislocations (Cotrell 1958). Due to the raised test tempera- Unyielded Lüders band ture, the solute atoms of carbon and/or nitrogen are able to metal diffuse in the specimen at a rate equal to or slightly faster than the speed of dislocations so as to catch and lock them. Engineering stress Upper A C D Therefore, the applied stress must increase in order to unlock yield B Reloading dislocations. When dislocations will break free from their point Lower yield point solute clusters, there will be a drop of load and the corre- sponding stress. If the solute atoms can diffuse quickly Yield enough to re-trap these dislocations, then more stress must elongation be applied to enable dislocations to again break free from their solute atmosphere. This process occurs many times, Unloading Type IV producing the serrations, i.e. discontinuous or repeated yielding in the stress–strain curve in Fig. 1.36. 1.6.4 Type IV: Elastic–Heterogeneous– Engineering strain Homogeneous Plastic Behaviour Fig. 1.37 Type IV: schematic engineering stress–strain curve showing The stress–strain curve under this category consists of three a narrow heterogeneous plastic deformation region between initial segments—a linear elastic region is separated from a linear elastic and final homogeneous parabolic plastic deformation homogeneous plastic flow region with a relatively narrow regions. Yielding initiates locally at upper yield point A followed by a segment of heterogeneous plastic deformation that normally sudden drop of stress to lower yield point B, at which yield-point ranges from approximately 1–3% strain, but strain over 10% elongation (heterogeneous deformation) and formation and spreading can be obtained with low-carbon steel under proper condi- Lüders bands over the entire gage length occur from B to C. Homo- tions. Figure 1.37 shows a steady increase of stress with geneous plastic deformation starts at point C, beyond which if the elastic strain followed by a sudden drop of stress from a specimen is unloaded from any point, say point D, and immediately point A, defined as upper yield point, to a point B, defined as reloaded, then the subsequent stress–strain curve will not display any lower yield point. At point B, the stress–strain curve fluc- yield point tuates about some nearly constant value of stress, at which the engineering strain or elongation that occurs by hetero- Type II behaviour, where the material continues to deform geneous deformation is called yield-point elongation. It plastically in a homogeneous manner. extends from B to C as shown in Fig. 1.37. After loading to the upper yield point A, a discrete deformed band, often Note that if the tensile specimen is unloaded from any visible with the naked eye, appears at a point of stress point beyond the Lüders strain region BC, say, from point concentration such as a fillet and its development causes the D (Fig. 1.37), and reloaded immediately, then the subse- initial drop of stress from point A to the lower yield point at quent stress–strain curve will not display any yield point. B. The band then spreads along the entire gage length of the This type of curve was found originally in low-carbon steel specimen, causing the heterogeneous segment of yield-point and is generally observed in many BCC iron based alloys elongation. During this period of yield-point elongation, and some nonferrous alloys. In addition to iron and steel, occasionally more than one band may form at several points yield points have been observed in polycrystalline molyb- of stress concentration. These bands are generally named as denum, titanium, and aluminium alloys and in single crystals Lüders bands, or stretcher strains, or Hartmann lines. For- of iron, cadmium, zinc, alpha and beta brass, and aluminium mation of several Lüders bands makes the flow curve during and also in ionic and covalent materials. the yield-point elongation irregular; i.e., each jog of the irregular flow curve is created with the formation of a new A pronounced upper yield point is achieved when the Lüders band. The Lüders bands form normally at an angle of testing uses an elastically rigid or hard machine, a specimen approximately 45° with the tensile axis. After all Lüders free from stress concentrations, a very careful axial align- bands have spread simultaneously to cover the entire gage ment of the specimen in the test grips, a high strain rate, and, length of the specimen, the segment of yield-point elonga- a subambient temperature. The value of stress at the lower tion will end. Finally, the increase in dislocation density yield point is usually reported as the yield strength of causes an enhanced strain hardening through interactions of material under this category because measured value of dislocations, and the stress starts to rise gradually with fur- stress at the upper yield point suffers from considerable ther strain in the usual manner, similar to that described for scatter due to its extreme sensitiveness to the above-mentioned several experimental factors. Usually, the upper yield point is 10–20% higher than the lower yield point, but it can be roughly double of the lower yield point if the effect of stress concentration is avoided during the for- mation of first Lüders band. It is to be noted that according to

1.6 Types of Tensile Stress–Strain Curve 53 the Hall-Petch relationship given by (1.57a), the upper and Dislocations multiply and q increases with strain; b ¼ lower yield points and the corresponding stresses will Burgers vector; v ¼ average velocity of mobile dislocations; decrease with increasing grain size. Further, the yield drop which according to Johnston and Gilman (1959) and Stein tends to become less pronounced and ultimately may vanish and Low (1960), depends on the resolved shear stress as as the grain size becomes larger. given by the following (1.102). 1.6.4.1 Explanation for Yield-Point Behaviour  s m0 ð1:102Þ v ¼ The yield-point phenomenon was found originally in low-carbon steel having a small amount, with a minimum of B only about 0.001%, of carbon and/or nitrogen atoms. Either of these interstitial elements in iron readily diffuses to the posi- where s ¼ applied resolved shear stress; B ¼ material tion of minimum energy in dislocations and thus a solute property; m0 ¼ dislocation-velocity stress sensitivity, i.e., “atmosphere”, known as ‘Cottrell atmosphere’, is formed around each dislocation core. Since these dislocations are exponent describing the stress dependence of dislocation “pinned” by such solute atmospheres, dislocation motion is severely restricted. Apart from the solute-dislocation inter- velocity and it is a material property. action, pinning can also arise by precipitation of fine carbides or nitrides along the dislocation. When a sufficiently high For materials with a low initial density of mobile dislo- stress corresponding to the upper yield point is applied, dis- locations will break free from their solute clusters causing a cations or with strongly pinned dislocations, as in steel, the drop of stress to the lower yield point and then slip can occur at a lower yield stress. Alternatively, in case of strongly pinned strain rate, given by qbv; would be less than the rate of dislocations new dislocations must be created at the points of stress concentrations to allow the flow stress to drop. This movement of the test machine cross-head unless average explains the origin of the upper yield stress and the drop in stress after yielding has begun. The applied stress combined velocity of mobile dislocations, v; becomes high to maintain with the stress concentration produced due to the pile-up of dislocations at grain boundaries will either unlock disloca- a balance between the above two rates. But a higher velocity tions from the influence of the solute atoms or generate new dislocations in the next grain and in this way a Lüders band of mobile dislocations can only be achieved at higher propagates over the specimen. The development and propa- gation of Lüders band in the region of yield-point elongation applied stress level, according to (1.102). On the other hand, continue until essentially all dislocations have broken free from their respective solute atom clusters. Thus, irregular once some dislocations begin to move they also begin to segment of yield-point elongation in the curve is caused by a dislocation-solute atom interaction. multiply rapidly and so the density of mobile dislocations, q; In other materials such as silicon, germanium, lithium increases rapidly. Under such condition, the strain rate in the fluoride and copper whiskers the dislocation density is quite low and dislocation locking by interstitial atoms cannot material would exceed the cross-head velocity of the test explain similar yield-point behaviour. Dislocations pinning by solute atmospheres thus becomes a special case of machine. To match the two rates, the dislocation velocity yield-point behaviour. For all materials showing a yield drop, i.e., a drop in yield stress after yielding has begun, must be decreased and to accomplish this, the stress needed Johnston (1962) and Hahn (1962) have proposed a more generalized theory, which is discussed below. to move the dislocations must drop according to (1.102). The relation between the strain rate in the material and the Note that the decrease in the stress due to the drop in dis- average velocity of mobile dislocation is given by, location velocity is higher than the increase in the stress due to some strain hardening introduced by the increase in dis- location density Thus, once yielding begins there will be a drop in yield stress, the magnitude of which depends on the dislocation-velocity stress sensitivity parameter, m0: According to this model proposed by Johnston and Hahn, yield-point behaviour is controlled by two parameters that are the initial mobile dislocation density, q; and the dislocation-velocity stress sensitivity, m0: The lower is the value of these parameters the higher will be the magnitude of the yield drop, as explained below. We obtain the following relation by applying (1.101) and (1.102) at the upper and lower yield points, which are denoted respectively by the subscripts ‘U’ and ‘L’. sU ¼ ðvU Þ1 B ¼ e_ =ðqU bÞ!1=m0 ¼  qL 1=m0 ð1:103Þ sL m0 e_=ðqLbÞ qU ðvL Þ1 B m0 e_ ¼ qbv ð1:101Þ If values of m0 is very small, for example, less than 20 as in the case of covalent- and ionic-bonded materials as well as where e_ ¼ strain rate; q ¼ density of mobile dislocations, i.e., number of mobile dislocations per unit area. in some BCC metals and alloys and the initial mobile dis- location density of the material qU is also very less from the beginning of the test up to the upper yield point, then the ratio sU=sL will be large and a strong yield drop will result.

54 1 Tension For steel having m0 ¼ 35; there will be a substantial yield stress-drop shows a stage of minimum stress level extending drop provided qU is less than about 10 mmÀ2: This is only over a range of plastic strain, the reason of which has been possible if most of the dislocations are pinned by solute explained by Meinel and Peterlin (1971) considering the atmosphere because the mobile dislocation density of competition of two events. In the region of minimum stress annealed steel is usually at least 104 mmÀ2: By contrast, for level, these events occur side by side within the polymer most FCC metals and alloys values of m0 is large, normally sample and they are—(1) the breakdown of the original greater than 100–200, and an initial mobile dislocation crystalline structure and (2) its subsequent reorganization density qU is also high. These factors cause sU % sL and into a highly oriented strong material. The former event make a yield drop an unlikely event in most FCC metals and leads to softening due to broken structure and the latter one alloys. From the above discussion, a pronounced yield point causes hardening due to orientation strengthening. The is observed in crystalline materials that balance between these two events results in the minimum in the curve. As more and more of the fresh reoriented strong • have few mobile dislocations at the start of the test, structures are created with progress of plastic deformation, • possess the capability to multiply dislocations rapidly the polymer sample experiences more and more resistance to deformation due to strain hardening, which is caused by with progress of plastic deformation so that the disloca- molecular alignment. Enhanced molecular alignment results tion density at the lower yield point i.e., qL; is quite high in the development of more highly oriented structure that and generates greater optical transparency in the gage section of • show relatively low dislocation-velocity stress the polymer specimen and the milky white regions, produced sensitivity. by the initial break-down of original crystal structure, gradually become clear once again. Finally beyond the Since these above characteristics are possessed by many minimum stress region, the stress–strain curve begins to rise ionic- and covalent-bonded crystals (Tegart 1966) they once again and continues up to the point of fracture stress exhibit yield points. because of the strain hardening effect. 1.6.5 Type V: Elastic–Heterogeneous– Homogeneous Plastic Behaviour for Some Crystalline Polymers Type V stress–strain response, shown in Fig. 1.38, is gen- Orientation erally observed during the tensile deformations of some strengthening crystalline polymers, such as polypropylene. This type of stress–strain curve at the end of linear elastic region exhibits Engineering stress Yield an upper yield point and subsequent drop of stress similar to Cold drawing that observed in Type IV behaviour. But the plastic defor- Type V mation region of this curve undoubtedly differs from that exhibited by the Type IV curve. O Engineering strain As soon as yielding of the material begins at the upper yield point there is a breakdown of the initial crystalline Fig. 1.38 Type V: schematic engineering stress–strain curve usually structure within the polymer, which results the initial drop of observed in some crystalline polymers exhibiting linear elastic region load and corresponding stress. While the stress drops, the followed by upper yield point, yield drop, a stage of minimum stress region of greatest deformation within the gage section of the level extending over a range of plastic strain, called cold drawing, and tensile specimen becomes milky white and the necking of ultimately, rise in stress level with further strain, called orientation the specimen begins. The necking process may or may not strengthening, that continues up to the point of fracture produce failure in the specimen within a short time. If fracture of the specimen does not take place shortly after the beginning of the neck formation, continuing plastic defor- mation will bring about a reorganization of the broken-down crystallite into fresh and highly oriented strong structures. The plastic zone of the stress–strain curve after the

1.7 Linear Elastic Properties 55 1.7 Linear Elastic Properties The modulus of elasticity is an example of structure-insensitive mechanical property. It is only slightly The linear elastic properties like stiffness or modulus of affected by changes in microstructure caused by alloying elasticity, proportional limit, elastic strength or limit, and additions, heat treatment, or cold-work (Mack 1946). For resilience of materials that follow Hooke’s law are generally example, while heat treatment and minor alloying additions determined by the static tensile test. The resilience of a linear may increase the magnitude of strength of steel from 210 to elastic material is usually measured by modulus of resilience. 2400 MPa, the value of the elastic modulus remains rela- The Poisson’s ratio, m, can also be measured by observing tively unchanged with a rise of about 200–210 GPa. How- the lateral contraction and longitudinal extension of the ever, the values of modulus of elasticity for metals and specimen in tension using (1.13). ceramics decrease with increasing temperature, while those for rubbers or elastomers increase with increasing 1.7.1 Modulus of Elasticity temperature. If the material obeys Hooke’s law, i.e., has a linear stress– The reason behind the relative insensitiveness of the strain curve in the elastic region, its stiffness or relative elastic modulus to micro-structural changes is that the resistance to separation of the two adjacent atoms or ions is modulus of elasticity depends on the strength of the inter- measured by the modulus of elasticity or Young’s modulus, atomic binding forces between adjacent atoms or ions that E. The modulus of elasticity is found geometrically by vary with the type of bonding found in a given material. measuring the slope of the initial linear portion of the stress– These interatomic forces cannot be altered unless the basic strain curve, since E ¼ elastic stress=elastic strain: In this nature of the material is changed. Let us consider the curve current section, these elastic stress and strain will be in Fig. 1.39, where interatomic bond stress, rðrÞ; between respectively denoted by r and e; since in the elastic range it two adjacent atoms or ions is plotted against interatomic is immaterial whether we take the stress and strain as engi- spacing, r, to illustrate the relation between this curve and neering or true. It is to be noted that the higher the value of the elastic modulus, E. This curve is similar to the curve of the elastic modulus, the lesser is the elastic strain produced interatomic bonding force, F(r), between two adjacent atoms by a particular applied stress. As the value of modulus of or ions versus interatomic spacing, r, as shown in Fig. 9.1, elasticity is required to compute deflections of beams and only the bonding force in Fig. 9.1 has been replaced by other structural members, it is an important design intensity of that force, i.e., the bond stress in Fig. 1.39. On parameter. this diagram in Fig. 1.39, the value of E can be shown graphically, as follows. On this diagram at point B, which is the point of equilibrium interatomic spacing, r0, Fig. 1.39 Variation of bond Bond stress σ (r) σc = Cohesive strength stress with interatomic spacing to Atomic spacing r establish graphically the modulus of elasticity (schematic) rx r0 B σx O x rx – r0 A

56 1 Tension corresponding to zero value of the bond stress, a tangent BA 1.7.2 Proportional and Elastic Limit is drawn so that it intersects the stress axis at point A at a linear distance of OA from the origin O of the curve. The Proportional limit is the highest engineering stress up to which stress is directly proportional to strain according to slope of the tangent BA to this curve at the point r ¼ r0 ¼ (1.31) and Hooke’s law is valid. The proportional limit is OB; is given by ½drðrÞ=drŠr¼r0: For small atomic displace- shown in Fig. 1.24. The proportional limit, Spl; is obtained ment, which usually occurs in case of elastic strain, it may be by noting the value of stress from the engineering stress– strain curve at which the curve deviates from its linearity and assumed that this tangent line BA practically coincides with is given by the curve up to the point of small displacement, which is shown by point x in Fig. 1.39. The theoretical cohesive strength, rc; represented by the maximum ordinate of the Spl ¼ Ppl ð1:105Þ curve in Fig. 1.39, is usually 100–1000 times as great as the A0 maximum elastic stress at which elastic action in most materials terminates by yielding or rupture. The elastic range where Ppl is the load at which the load–deformation curve obtained from tensile test deviates from the linearity and A0 is therefore restricted to a very minute part of the curve in is the average initial cross-sectional area in the gage section of the tensile specimen. The slope of the stress–strain curve Fig. 1.39 and the assumption that the curve coincides with in this region is the modulus of elasticity or Young’s mod- ulus, E. its tangent in this range is quite justified. Elastic limit or Elastic strength is the highest engineering With the above assumption, the slope of tangent BA, can stress that the material can withstand without showing any measurable permanent strain remaining on the complete be expressed approximately by rx=ðrx À r0Þ; where rx and withdrawal of applied load. Most materials exhibit a gradual transition from elastic to plastic behaviour and the stress at rx are respectively the corresponding values of elastic stress which plastic deformation or yielding begins depends on the and atomic spacing at point x: Since the elastic strain ex sensitivity of the displacement transducer that measures the corresponding to elastic stress rx is represented by strain. The more sensitive the strain gage, the lower is the ðrx À r0Þ=r0; it follows that the elastic modulus is: stress level at which plastic flow begins. Usually the elastic limit is higher than the proportional limit because the sen- rx  sitivity of strain measurement generally performed in engi- elastic stress; rx À r0Þ=r0 rx neering applications is on the order of 10−4. But the value of E ¼ elastic strain; ex ¼ ðrx ¼ r0 rx À r0 the elastic strength decreases gradually with increase in the sensitivity of strain measurement and becomes lower than ¼ r0  slope of tangent BA ¼ OB  OA ¼ OA the proportional limit. When special capacitance strain gages OB for micro-strain measurements with sensitivity of strain on the order of 10-6 are employed, this elastic limit ultimately ð1:104Þ arrives at a very low stress value, called true elastic limit, which is associated with the movement of a few hundred Or, the distance OA is equal in magnitude to the modulus dislocations. The elastic limit or strength is determined by of elasticity. noting the value of applied load from the load–deformation curve up to which the deformation strains are fully reversible The modulus of elasticity related to differences in the and dividing this noted value by the average original strength of the interatomic forces shows a wide range of cross-sectional area in the gage section of the tensile speci- values for different materials, because the equilibrium men. But to obtain the elastic limit, a laborious incremental spacing r0 normally varies between 1 and 4 Å with different loading unloading test procedure is required. bonds and the slope of the curve of Fig. 1.39 at equilibrium spacing, r0, differs with change of the basic nature of the 1.7.3 Resilience and Modulus of Resilience material. Elastic resilience or simply, resilience in tension is the The above-mentioned fact that the decrease in values of capacity of a material to absorb energy when deformed modulus of elasticity, E, with increasing temperature for under elastic loading condition and to release it when the metals and ceramics depends on the equilibrium spacing applied load is removed completely. Thus the term between two adjacent atoms or ions in their unstrained condition, i.e., r0. It has been seen that the elastic stiffness or the modulus of elasticity increases with the decrease in value of r0 and vice versa. As materials expand on heating so the equilibrium distance of atom or ion separation, r0, increases with temperature and results in a decrease in the value of the modulus of elasticity, E. With increasing temperature, the loss of material stiffness or the decrease of elastic modulus is gradual, showing only a small percent decrease with an increase in temperature by 100 °C. In contrast, an increase in temperature results in contraction in length of a rubber or an elastomer and causes to increase its modulus of elasticity.

1.7 Linear Elastic Properties 57 ‘resilience’ in tension, denoted by UR, can be expressed as As long as the deformation is both elastic and linear the follows: unloading path will follow the same path as the loading and UR ¼ Recoverable energy on complete release of applied tensile load Energy absorbed originally by a material during the elastic stretching the strain energy stored in the material will become com- ð1:106aÞ pletely recoverable. Therefore, both the above areas obtained The amount of energy absorbed by the material comes on loading and unloading will be equal to each other. Thus directly from the work done by the applied tensile forces and for all linear elastic materials that follow Hooke’s law, the this energy is stored in the material in the form of strain value of resilience in tension, UR ¼ 1: To distinguish among energy. This strain energy for uniaxial tension is the area different linear elastic materials, this property is usually under the linear elastic portion of the load–deformation diagram obtained from tensile test, which is nothing but the measured by the modulus of resilience, which has been area of a right angled triangle with base and aÀltitude bÁeing denoted by UMR : The modulus of resilience in tension is respectively DLpl and Ppl: This is given by PplDLpl 2; defined as the elastic strain energy per unit volume required where Ppl is the load applied at the proportional limit and to stress the material from zero stress to the proportional DLpl is the extension or deformation of the gage section in limit Spl; which is as follows according to (1.107) and the tensile specimen at the point of proportional limit. Hooke’s law (1.31a): Hence, the elastic strain energy per unit volume for uniaxial tension is the area between the linear elastic portion of the Modulus of resilience in tension; UMR ¼ Splepl engineering stress–strain curve and the strain axis, which is 2 given by ¼ Spl Spl ¼ Sp2l 2 E 2E ð1:108aÞ PplDLpl For convenience in engineering application, the propor- 2 Â ðinitial volume in the gage section of the tensile specimenÞ tional limit Spl in (1.108a) can be replaced by the material’s yield strength, S0 ; (discussed subsequently in this chapter). ¼ PplDLpl ¼ 1 Ppl DLpl ; If e0 is the engineering strain at the yield point, then 2ðA0L0Þ 2 A0 L0 assuming the Hooke’s law to be valid up to the point of yield stress, (1.108a) can be represented as follows: Or, elastic strain energy per unit volume for uniaxial tension Modulus of resilience in tension; UMR ¼ S0 e0 ¼ S02 2 2E ¼ Splepl 2 ð1:108bÞ ð1:107Þ If an engineering design requires a material that allows elastic behaviour with large energy absorption and must not where undergo permanent distortion, such as mechanical spring, then according to (1.108b), the material possessing high Spl proportional limit, defined by (1.105),  yield strength but a low modulus of elasticity will be ideal. epl engineering strain at the proportional limit ¼ DLpl L0; The moduli of resilience for high carbon spring steel and medium carbon structural steel are shown by the cross- in which L0 is the initial gage length of the tensile hatched regions in Fig. 1.40, where each of the steels has the same value of elastic modulus. The modulus of resilience in specimen and DLpl is the extension at proportional tension for high carbon spring steel is higher than that of the limit. Hence, from (1.106a), the resilience in tension, UR, can be expressed by the ratio of the areas as follows: UR ¼ area under elastic engineering stressÀstrain curve on full release of applied load ð1:106bÞ area under engineering stressÀstrain curve when material is deformed elastically

58 1 Tension High carbon spring steel X Structural steel SX Engineering stress SY Y Engineering Stress S Engineering strain Fig. 1.40 Schematic engineering stress–strain curves for high carbon O eY eX spring steel and structural steel, showing that the former has a higher Engineering strain e modulus of resilience than the latter because of higher yield strength of the former, although the elastic moduli of both steels are same. The Fig. 1.41 Schematic nonlinear elastic stress–strain curve, illustrating moduli of resilience for both steels are represented by the crosshatched secant modulus and tangent modulus regions medium carbon structural steel because of higher yield with the value obtained by (1.109a) for the secant modulus strength of the former than the latter. ESec: If stiffness related to an infinitesimally small change or increment in stress is desired, instantaneous stiffness or in- 1.8 Nonlinear Elastic Properties cremental stiffness of material is determined instead of the The properties in the elastic range of a nonlinear elastic secant modulus. Instantaneous stiffness at a particular stress material determined from the static tensile test are normally average stiffness or secant modulus, instantaneous stiffness value can be measured from the slope of the tangent to the or tangent modulus and elastic resilience or simply resilience. stress–strain curve at the point of that particular stress. This slope is called the tangent modulus, which is illustrated by a tangent drawn at point Y on the stress–strain curve as shown in Fig. 1.41 and given by ! dS 1.8.1 Secant and Tangent Modulus ET ¼ de S¼SY ð1:109bÞ If the material does not follow Hooke’s law, i.e., has a As for the secant modulus, the stress at the point Y on the stress–strain curve, i.e., the stress SY must be specified with nonlinear stress–strain curve like Fig. 1.41, its stiffness does the value obtained by (1.109b), for the tangent modulus ET. not remain constant but varies with stress. Sometimes av- erage stiffness of such material at a particular stress value is found from the average slope of the stress–strain curve with 1.8.2 Elastomer or Rubber Elasticity a secant drawn from the point of that particular stress to the In addition to very large nonlinear elastic strains, often in excess of 100%, the other distinguished characteristic of origin of the curve. The average stiffness at that particular elastomer or rubber is that its elastic modulus increases with increasing temperature, which is opposite to that observed stress is thus measured by the slope of the secant, shown by in other materials. In the unstressed condition the structure of rubber is very disorganized with the chains preferring to line OX in Fig. 1.41 and this quantity is represented by the remain in random curled positions. Rubber elasticity is associated mainly with the straightening of the chains from secant modulus, ESec, as follows: their random coiled arrangements into partially extended configurations. ESec ¼ ! ð1:109aÞ S e S¼SX Since the value of the secant modulus depends on the location of the point X on the stress–strain curve. i.e., on the magnitude of the stress SX; so this stress must be specified

1.8 Nonlinear Elastic Properties 59 The fact that the elastic modulus of rubber increases with is stretched out under a tensile load, the entropy decreases as increase of temperature can be explained by using the first and second laws of thermodynamics. From the first law of the chains become straightened and aligned causing a thermodynamics dÀecrease in Á their degree of disorder. As a result @ÀSentropy @L TÁ is negative. Thus, the second term dU ¼ dQ À dðW:D:Þ ð1:110aÞ T @Sentropy @L T becomes additive with the first term ð@U=@LÞT in (1.111) and increases the value of the applied where load P at constant temperature. Hence, with increase of temperature T; the load P required to extend the elastomeric dU the change in internal energy of the system; rod by Àan amount dLÁincreases due to increase in the additive dQ the change in heat absorbed or released between term T @Sentropy @L T : As the load and corresponding stress the system and the surroundings; applied on the rubber to produce a particular elastic strain dðW :D:Þ work done by the system. increases with increasing temperature, its elastic modulus If work is done on the system, (1.110a) can be re-written also increases with increasing temperature. In contrast to as rubber, the initial degree of order of atoms or ions in metals dU ¼ dQ þ dðW:D:Þ ð1:110bÞ and ceramics are very high and so their entropy term is If an elastomeric rod of length L is stretched by an initially negligible in comparison with rubber. But on amount dL under the application of a tensile load of P, the work done on the system, i.e., on the rod is dðW:D:Þ ¼ PdL; stretching under a tensile load, the entropy of metals and assuming the change in cross-sectional area of the rod to be ceramics increases due to increase in the degree of disorder insignificant. Hence, we get from (1.110b): Àof their atoÁmic or ionic arrangements. As a result dU ¼ dQ þ PdL ð1:110cÞ @Sentropy @L T is positive in (1.111), which causes to decrease the modulus of elasticity of metals and ceramics with rise of temperature. For a reversible process, the second law of thermody- 1.8.3 Elastic Resilience or Resilience namics gives dQ ¼ T dSentropy ð1:110dÞ Similar to that mentioned in the linear case in Sect. 1.7.3, the elastic strain energy per unit volume stored in non linear where T ¼ temperature; and dSentropy ¼ change in entropy. elastic material under uniaxial tensile loading condition is Combining both (1.110c) and (1.110d), the following is equal to the area under the elastic portion of the engineering stress–strain diagram, and is given by obtained. dU ¼ TdSentropy þ PdL ð1:110eÞ Stored elastic strain energy per unit volume; Zee If the temperature T remains constant, (1.110e) can be ð1:112Þ ðUV Þelastic strain¼ Sde rearranged as follows: 0    @U @ Sentropy At constant temperature; P ¼ @L ÀT @L where ee ¼ the extent of elastic engineering strain experi- enced by the material. The modulus of resilience, however, T T is no longer applicable to the case of a non linear elastic material because it does not possess a proportional limit. ð1:111Þ Even though the material is elastic, the elastic strain where change in the strain energy due to elastic energy stored in the material will not be always completely À@UÁ straining under an applied tensile load; and recoverable if the unloading path in tensile test does not change in the entropy or order of the elastomer follow the same path as the loading. This has been explained @L T with its elastic deformation on tensile loading. in Fig. 1.25, which is a typical engineering stress–strain diagram for rubber, a non linear elastic material. In  Fig. 1.25, the rubber is loaded from the origin, ‘O’ to a point ‘A’ following the curve ‘1’ but when it is unloaded from ‘A’ @Sentropy to ‘O’, it follows the path represented by curve ‘2’. For @L T rubber, the unloading curve lies usually somewhat below the loading curve and thus the area under the unloading curve The entropy of a system is measured by the degree of disorder, i.e., higher is the degree of disorder higher will be the entropy. Since the chains of rubbers prefer random coiled configurations prior to loading, their degree of disorder and corresponding entropy is initially high. But when the rubber

60 1 Tension ‘2’ is lower than that under the loading curve ‘1’. Now 1.9.1 Yield Strength according to (1.106b), the resilience in tension, UR, for rubber is given by The yield strength is a stress associated with the onset of irreversible plastic deformation and an important structure UR ¼ area under unloading stress-strain curve 2, in Fig.1.25 \\1: sensitive mechanical property of material. Actually the area under loading stress-strain curve 1; in Fig. 1:25 conventionally determined engineering yield strength is many times higher than the stress level at which plastic Hence, the elastic strain energy recovered on complete deformation begins as revealed by the micro-strain mea- release of applied load is less than that supplied originally to surement. However in practical engineering applications, the the material when it is stretched, because some stored strain yield strength is the most commonly used parameter to mark energy is expended for internal friction in the rubber. This the beginning of yielding and the end of the elastic range. loss of energy is equivalent to the mechanical hysteresis loop formed by the area between the two curves ‘1’ and ‘2’ in For materials showing discontinuous yielding, the initia- Fig. 1.25. If the value of resilience for a material is low, it tion of plastic strain occurs abruptly at certain point, known will possess high damping capacity whereas a high value of as yield point, which can easily be detected by the sudden resilience is desired for low internal heat generation. Rubber stoppage of increment in the tensile load. The engineering has good damping capacity due to its low resilience and so, stress at the yield point where yielding; i.e., plastic defor- used as a base for machinery to absorb vibration and mini- mation begins is called the yield strength S0 of the material, mize noise. as shown in Fig. 1.42. Sometimes with the initiation of yielding, the load may drop sharply from upper yield point 1.9 Inelastic Properties Engineering stressto lower yield point with a more-or-less steady value of load at which further plastic deformation proceeds, as discussed Beyond the elastic range, the inelastic action starts where in Sect. 1.6.4 and shown in Fig. 1.37. Since the stress at irrecoverable changes of structure take place in the material. upper yield point ðS0ÞU; suffers from considerable scatter Two possible ways by which the elastic range can end, are due to its extreme sensitiveness to several experimental by yielding or by fracture. Plastic range often refers to the factors and the stress at lower yield point ðS0ÞL; is relatively range of mechanical behaviour in which yielding and stable, the value of engineering stress at the lower yield point strain-hardening take place. Inelastic behaviour of material is usually reported as the yield strength S0 of material. The thus consists of plastic range and / or fracture. In engineering yield strength S0; is given by applications, most commonly used inelastic tensile proper- ties that are evaluated in terms of engineering stress and S0 strain include yield strength, ultimate tensile strength (UTS) or simply tensile strength, ductility, toughness or 0 Engineering strain modulus of toughness and breaking or fracture strength. The secant modulus and the tangent modulus in the nonlinear Fig. 1.42 Yield strength for materials showing discontinuous yielding plastic region can also be measured in terms of engineering (schematic) stress and strain respectively by (1.109a) and (1.109b), which is identical with that described under nonlinear properties in Sect. 1.8.1. The inelastic tensile properties, measured in terms of true stress and strain, are true tensile strength, true fracture strength, true fracture strain com- prising of true uniform strain and true local necking strain. In relation to the flow curve, other inelastic properties that can also be determined from the tension test are strength coefficient, strain-hardening coefficient or exponent and rate of strain-hardening or modulus of strain-hardening, which have been discussed in Sect. 1.6.2.

1.9 Inelastic Properties 61 S0 ¼ load at yield point of load-deformation curve obtained from tensile test ¼ Py ð1:113Þ average initial cross-sectional area in gage section of tensile specimen A0 As most materials exhibit a gradual transition from elastic strain is generally 0.2 or 0.1%, i.e., e0 ¼ 0:002 or to plastic behaviour instead of abrupt change as cited above, e0 ¼ 0:001: In Great Britain the offset yield strength is it is very difficult to define precisely the stress at which often called as the proof stress, where the offset value of plastic deformation or yielding begins because it depends on strain is usually taken as 0.1 or 0.5% i.e., e0 ¼ 0:001 or the sensitivity of the strain gage. For such materials, the e0 ¼ 0:005: engineering stress required for producing a small specified amount of conventional plastic strain or permanent set is The offset method cannot be used for some materials that called the yield strength, which is usually defined as the show essentially no linear part in their stress–strain curve, offset yield strength. A test procedure specified by ASTM Standard (ASTM E8/E8M 2016b) is adopted to maintain for example, soft copper or gray cast iron. In such case the uniformity in the determination of the offset yield strength. The value of offset yield strength is obtained in the following usual practice is to determine the engineering stress required manner from the engineering stress–strain diagram, such as to produce some total strain, eT ; comprising of elastic and the one shown in Fig. 1.43: plastic strain, for example, eT ¼ ee þ e0 ¼ 0:005 and to report this stress as the yield strength of the material. • A specified plastic strain is measured from the origin on the strain axis and at this offset a line is constructed If it is desired that a material must remain elastic under parallel to the linear elastic portion of the engineering stress–strain diagram. The usually accepted offset is load, the yield strength is commonly used for design and 0.002 or 0.2% strain unless otherwise specified. The specification purposes or used as the basis for working stress offset strain is nothing but the yield strain (engineering plastic strain at the yield point), and will be denoted by in a majority of engineering applications, because the prac- e0: tical difficulties to measure the elastic limit or proportional limit can thus be avoided. The working stress, ðS0ÞW ; is • The engineering stress at which this offset line intersects obtained as follows by reducing the yield strength, S0; or the engineering stress–strain curve is the required yield 0.2% offset yield strength ½S0Še0¼0:002; with a suitable factor stress. This value of engineering stress is usually defined of safety, ny; which depends mostly upon the previous as the offset yield strength. If offset strain value is 0.002 experience. or 0.2% as cited above, this value is called the 0.2% offset yield strength, which is expressed as follows: ðS0ÞW ¼ S0 or ½S0 Še0 ¼0:002 ð1:115Þ ny At the yield point, since the difference between the original and the instantaneous cross-sectional area is insignificant, i.e., A0 % A; so, S0 or ½S0Še0¼0:002% the true ½S0 Še0 ¼0:002 ¼ load required to produce engineering plastic strain; e0 ¼ 0:0002 average initial cross-sectional area in gage section of tensile specimen ð1:114Þ ¼ Pengineering strain offset; e0¼0:002 A0 One way to understand the offset yield strength is that the stress at the yield point, i.e., the flow stress, r0: Hence, tensile specimen will become 0.2% longer than its original S0 or ½S0Še0¼0:002 can be replaced by r0 during yielding and length when unloaded after it has been loaded to its 0.2% for engineering purposes, the yield stress or 0.2% offset yield offset yield strength. In the United States the specified offset strength is designated by r0:

62 1 Tension Offset Yield strength strength. Sometimes, UTS (ultimate tensile strength) may be [S0]e = 0.002 used as a basis to calculate working stress in a ductile material where some permanent deformation may be Engineering stress, S 0.2% Offset line allowed but actual fracture must be prevented. For example, the tensile strengths of ductile materials used to make 0 Engineering strain, e pressure vessel and piping system may be used for their e0 Offset (0.2%) design because some increase of diameter in them is often acceptable but actual rupture must not take place. However, = 0.002 because of the prolong use of the tensile strength of a ductile material to determine its service-strength in earlier days, it Fig. 1.43 Determination of offset yield strength for a ductile metal or has presently become an important mechanical property used alloy from its engineering stress–strain diagram (schematic). Tensile for identification of a material similar to that served by its yield strength is defined at the intersection of stress–strain curve and chemical composition. Moreover, the tensile strength being 0.2% offset line easily measurable and a quite reproducible property, is used for the purposes of specifications and for quality control of a 1.9.2 Ultimate and True Tensile Strength product. Further, there are quite useful empirical relation- ships between the tensile strength and other mechanical The ultimate tensile strength (UTS) or simply, the tensile properties like hardness and fatigue strength. strength and the true tensile strength are important structure sensitive mechanical properties of material. The tensile The tensile strength is a valid design criterion for a brittle strength (Fig. 1.26) is the maximum engineering stress that a material, where the working stress needed to protect against material can withstand under condition of uniaxial tensile fracture, ðSuÞW ; is determined as follows from the ultimate loading. It is defined as the maximum load, Pmax; divided by tensile strength, or the fracture strength reduced by a suitable the average original cross-sectional area, A0, within the gage factor of safety, nu. length of the tensile specimen and is given as follows: For brittle materials; ðSuÞW ¼ Su or Sf ð1:118Þ nu UTS; Su ¼ average maximum load area The factor of safety, nu; must be chosen higher than that original cross-sectional used with the yield strength, because of the scatter in the ð1:116Þ ¼ Pmax tensile strengths or fracture strengths of brittle materials and A0 the need to protect against complete failure of the material. The maximum load is measured by the peak point on the load–deformation curve, obtained from the tensile test. For The true tensile strength (Fig. 1.27), usually denoted by ductile materials, the applied load drops beyond the point of ru; is the true stress at the point of maximum load measured the ultimate load because of necking in the tensile specimen, by the highest point on the load–deformation curve. It is but brittle material showing no neck formation breaks at the defined as the maximum load, Pmax; divided by the instan- point of the maximum load. Thus for brittle materials, the taneous cross-sectional area, Au; within the gage length of tensile strength, Su; is the same as the breaking or fracture the tensile specimen at the point of maximum load and is strength, Sf ; as shown in Fig. 1.24b. given as follows: True tensile strength; ru ¼ maximum load ¼ Pmax ð1:119Þ instantaneous cross-sectional Au area For brittle materials, UTS; Let eu and eu denote respectively the engineering strain and the true strain at the point of maximum load. Then from Su ¼ Pmax ¼ load at fracture, Pf ¼ fracture strength; Sf A0 A0 (1.28) and (1.29), one can write (see Sect. 1.3.1) the fol- lowing relationship between the true tensile strength, ru; and the ultimate tensile strength, Su: ð1:117Þ ru ¼ Suð1 þ euÞ ð1:120Þ For ductile materials, the tensile strength is rarely used for ru ¼ Su expðeuÞ ð1:121Þ design or as a basis to determine working stress. Rather, the design of a ductile material is currently based on its yield Sometimes the point of maximum load on the load– deformation curve becomes difficult to establish for the

1.9 Inelastic Properties 63 material having usually a lower value of strain-hardening • To indicate the additional margin of safety beyond the exponent, n, which shows normally a lower rate of strain elastic range that may prevent complete mechanical hardening and makes the curve more flat around its highest failure of the material if some part of it is momentarily point. This above difficulty may arise when the value of overstressed. In such case, if the material is ductile, it will n becomes normally lower for stronger materials or for the yield locally without fracture in spite of the wrong stress same material in the cold-worked state than in the annealed calculation or overlooking of the prediction of momen- condition or for materials having higher stacking fault tary large overload by the designer. energies, as discussed earlier in Sect. 1.6.2.5. For example, steel with a lower value of n generally produces a load– • The “quality” level of material that depends on changes deformation curve flatter than that exhibited by annealed in impurity level or processing conditions may be copper having a higher value of n. In such case, the deter- assessed by ductility measurement. mination of the point of maximum load has been explained by Fig. 1.33a, b and by Considère’s construction in 1.9.3.1 Tensile Specimen Fig. 1.34. Out of these three methods Considère’s con- Prior to the discussion on the measurement of ductility, it is struction is the best one because it is not only easy to con- necessary to know the geometry of tensile specimen. The struct but also provides the values of ultimate tensile strength cross-section of a tensile specimen is usually circular, square and true tensile strength, whereas the former two methods or rectangular. A round specimen is the most common but a (Fig. 1.33a, b) only provide one strength-value that is the flat specimen is normally used when the specimen has to be true tensile strength, as discussed earlier in Sect. 1.6.2.6. prepared by machining of sheet and plate stock. The mid-section of the length of the cylindrical or prismatic 1.9.3 Ductility specimen has usually a smaller cross-section than the end sections of the specimen that are placed in the gripping Ductility is one of the most desirable structure sensitive device of the testing machine. The end portions are inten- mechanical properties of material because ductility mea- tionally enlarged to provide additional strength so that the surement in tension test provides the information regarding failure of the specimen can be avoided at or near the grips. the ability of the material to flow plastically before fracture The ends of cylindrical specimen may be plain, shouldered and is useful in the following three ways (Dieter 1968): or threaded and those of prismatic specimen are generally plain, although they may be sometimes shouldered or have a • To indicate the amount of plastic deformation that can be hole for bearing a pin. Various shapes of specimen-ends are imparted to a material without causing its fracture for shown in Fig. 1.44. The central part is not only reduced to formation of unlimited variety of useful shapes in get fracture in that region but its cross-sectional area is mechanical working operations such as rolling, drawing, maintained largely constant for uniform distribution of stress extrusion etc. across the cross-section. Care must be taken during speci- men preparation so that the centre lines of the reduced portion and the ends must coincide and both sides of the reduced portion are symmetrical with respect to the Fig. 1.44 Typical specimen and Distance between shoulders various shapes of specimen-ends “Reduced” section for tension test May have parallel sides or Plain end slight taper to mid length Diameter, D0 Fillet Gage length, L0 Over all length Shouldered end Threaded end Pin end

64 1 Tension Table 1.4 Dimensional ratios of tensile specimens standardized by different countries Germany 11.3 Type of tensile specimens Dimensional ratio United States Great Britain India 10.0 Standardized by ASTM Recommended by I.S.O. 4.5 Flat pffiffiffiffiffi 4.0 5.65 5.65 Round L0 A0 L0=D0 5.0 5.0 longitudinal axis over the whole length in order to prevent elongation, denoted by ef ; and (2) the reduction of bending during application of tensile load. Further, both cross-sectional area at fracture, represented by rf ; both of sides of the reduced section are made parallel with each which are usually expressed as percentage and shown below. other but they are often slightly tapered from 0.076 to 0.127 mm (0.003–0.005 in.) towards the centre so that the According to (1.5) probability of fracture becomes more near the centre. Sometimes, brittle specimen is curved throughout the length Elongation; ef ¼ DLf ¼ Lf À L0 ð1:122aÞ of its reduced section in order to avoid its fracture at or near L0 L0 the grips. In such case, the stress distribution is not uniform across the cross-section and hence to obtain comparable Or, % elongation; % ef ¼ DLf  100 ¼ Lf À L0  100 results, all dimensions of the specimen have to be stan- L0 L0 dardized. In accordance with the standards mentioned in Table 1.4 discussed subsequently in Sect. 1.9.3.2, the gage ð1:122bÞ length is fixed usually by punching two gage marks on the surface of the reduced section of the undeformed specimen, where which is slightly more in length than the gage length. All experimental measurements are made over this gage length. L0 original gage length of the tensile specimen before The shift from the larger ends to the reduced section must be loading; made by fillets of suitable large radius so that the stress concentration caused by the abrupt changes in the Lf gage length of the tensile specimen after its fracture, cross-section is reduced. Typical tensile specimen with its which is measured by joining the fractured parts of different sections has been illustrated schematically in the specimen together; and Fig. 1.44. As an example, dimensions of different sections of a 5 in. (127 mm) long cylindrical tensile specimen for DLf total extension or deformation up to the point of ductile metals as per ASTM Standard are given below fracture, i.e., increase in length from original gage (Davis et al. 1964; Richards 1961). length, L0; to gage length at fracture, Lf : End sections: Diameter ¼ 0:75 in. ð19:05 mmÞ; and Reduction of area at fracture; rf ¼ A0 À Af ð1:123aÞ Length ¼ 1:375 in. ð34:925 mmÞ; A0 Fillet: Usual radius ¼ 0:5 in. ð12:7 mmÞ; and Or; Minimum recommended radius = 0.375 in. (9.525 mm) Reduced section: % reduction of area at fracture; % rf ¼ A0 À Af  100 Diameter ¼ 0:505 þ 0:003 to 0:005 in. A0 ð12:827 þ 0:0762 to 0:127 mmÞ (for taper), And length ¼ 2:25 in. ð57:15 mmÞ; ð1:123bÞ Gage length: Diameter ¼ 0:505 in. ð12:827 mmÞ; and where length ¼ 2 in. ð50:8 mmÞ: A0 average original cross-sectional area within the gage 1.9.3.2 Ductility Measurement length of the tensile specimen before loading; and The ductility of a material is conventionally measured from the tension test by the following two standard parameters— Af minimum cross-sectional area of the tensile specimen (1) the engineering strain at fracture, usually called the after fracture, measured from the fractured parts of the necked segment The magnitude of total elongation, ef ; reported for a given tension test depends on the value of original gage length of the specimen used in the test, L0; because the largest part of the plastic strain is concentrated in the necked region of the tensile specimen with relatively little in other parts of it. The shorter is the initial gage length the greater

1.9 Inelastic Properties 65 Onset of necking B C E A D D Fracture C Engineering stress E Elongation B A Engineering strain Gage length Fig. 1.45 Engineering stress–strain curve for a ductile material that Fig. 1.46 Schematic illustration of distribution of elongation along undergoes necking followed by fracture. Different arbitrary stress levels specimen gage length. Uniform elongation occurs up to the point of are indicated on the curve by points A, B, C, D, and E, which are maximum load (point C in Fig. 1.45), at which the necking process discussed in Fig. 1.46 initiates (curve C) and further elongations up to the point of fracture are localized in necked region (curve D and E). A, B, C, D, and E are different arbitrary stress levels as indicated in Fig. 1.45 will be the fraction of strain contributed to the total elon- gation from the necked region and the higher will be the The tensile elongation, ef ; can now be expressed by value of elongation, ef : For this reason, it is essential that the ef ¼ DLf ¼ euL0 þ DLn ¼eu þ DLn ð1:125Þ original gage length, L0; must always be specified when L0 L0 L0 L0 reporting values of percentage elongation, % ef : The distri- bution of total extension up to the point of fracture along the The above (1.125) clearly shows that the larger is the original gage length, L0; the smaller will be the value of gage length of the tensile specimen is represented schemat- DLn=L0; and the lower will be the difference between the values of ef and eu: Therefore, for a tensile specimen with ically in Fig. 1.46 for different arbitrary stress levels as longer gage length, the total elongation i.e., engineering fracture strain, ef ; is mainly affected by uniform elongation, indicated by various points on the engineering stress–strain which increases usually with the value of strain-hardening exponent, n; and thus, the total elongation depends on the curve in Fig. 1.45. From Fig. 1.46, it is clear that the total strain-hardening capacity of the material. extension includes two components, the uniform extension, From (1.125), it is clear that the smaller is the original gage length, L0; the higher will be the total elongation up to DLu; up to the point of maximum load at which the necking the point of fracture. This is illustrated with the following process initiates (point C in Fig. 1.45 that corresponds to example. Suppose two tensile specimens of the same mate- curve C in Fig. 1.46) and the local necking extension, DLn; rial and the same cross-sectional area and shape, having from the onset of necking to the point of fracture. The original gage lengths of 50 and 200 mm, are stretched with amount of uniform extension, DLu; increases usually with the value of strain-hardening exponent, n; i.e., normally with equal uniform elongation of 10% till the point of maximum load at which neck begins, i.e., eu ¼ 0:1 for both specimens. the decrease in strength or stacking fault energy of material, Let the necked segments for both specimens were originally (see Sect. 1.6.2.5). The size and shape of specimen also 12.5 mm long at the point of maximum load, but their affect the extent of uniform extension. The uniform engi- lengths after fracture become 25 mm, showing a 100% localized elongation only in the necked regions, i.e., DLn ¼ neering strain that occurs till the point of maximum load, 25 À 12:5 ¼ 12:5 mm: According to (1.125), the per cent elongation, % ef ; of the above 50 and 200 mm long speci- i.e., up to the onset of necking is known as uniform elon- mens are respectively given below: gation, which will be denoted by eu: The total deformation up to the point of fracture, DLf ; is given by ÀÁ DLf ¼ Lf À L0 ¼ ðLu À L0Þ þ Lf À Lu ¼ DLu þ DLn ¼ euL0 þ DLn ð1:124Þ where eu ¼ DLu=L0:

66 1 Tension  pffiffiffiffiffi DLn various values of L0 A0 for flat test pieces, or, L0=D0 for When; L0 ¼ 50 mm; %ef50 ¼ eu þ L050 Â 100 round specimens, which is given in Table 1.4.  ¼ 0:1 þ 12:5 Â 100¼35% Now to obtain approximately the same elongation-value, 50 ef ; of two different specimens, 1 and 2, having the same shape made from the same material, the following is ð1:126aÞ expected from (1.128):  DLn When, L0 ¼ 200 mm; %ef200 ¼ eu þ L0200 Â 100 pLffi0ffiffi1ffiffiffi ¼ pLffi0ffiffi2ffiffiffi ð1:129aÞ  A01 A02 ¼ 0:1 þ 12:5 Â 100 ¼ 16:25% 200 where original gage length and cross-sectional area of ð1:126bÞ specimen 1 are respectively L01 and A01 and those of spec- imen 2 are respectively L02 and A02 : The elongation of the From (1.126) it can be readily seen that since L050 \\L0200 ; specimen 2 can be predicted by performing the actual tensile so, % ef50 [ % ef200 : To minimize such variations in the test on the specimen 1 with the cross-sectional area of A01; values of % elongations resulting from the uses of different where its gage length is required to be adjusted according to original gage lengths, J. Barba in 1880 first stated that (1.129a) so that its gage length becomes geometrical similarities among test pieces are required to sffiffiffiffiffiffi compare the elongation-values of different-sized test speci- L01 ¼ L02 A01 ð1:129bÞ A02 mens. Numerous investigations since then have been made to rationalize the strain distribution (Hsu et al. 1965) in the tension test. The most general conclusion that has been The application of (1.129b) is explained with the fol- lowing example. Suppose the elongation of a flat specimen confirmed is that geometrically similar specimens develop of a material with 40 mm gage length, 12 mm width and 2 mm thickness is desired by the customer. But in produc- geometrically similar necked regions. The total elongation, tion shop, 12 mm wide and 8 mm thick flat product of the identical material is available and the reduction of the ef ; of the same material remains practically constant (Moore thickness is restricted probably due to involvement of cost 1918; Nichols et al. 1927) when the Barba’s law (Barba and / or infrastructural facility etc. In such case to predict the required elongation, a test specimen has to be made from the 1880) is followed. 8 mm-thick product but the only requirement is to adjust its gage length, L0; so that pffiffiffiffiffi ð1:127Þ Barba’s law: DLn ¼ C A0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L0 ¼ 40 Â ð8 Â 12Þ=ð2 Â 12Þ mm ¼ 80 mm: where This above procedure has been experimentally verified by DLn local necking extension, Kula and Fahey (1961). C a constant that varies with the type of specimen used On the other hand, the prediction of ductility from the in the test, and tension test by the measurement of the reduction of area at A0 original cross-sectional area within the gage length of fracture does not suffer from the difficulty encountered by elongation measurement, mentioned above, i.e., the value of tensile specimen reduction of area at fracture does not depend on the original gage length-value of tensile specimen. Reduction of area at By substituting (1.127) in (1.125), the elongation equa- fracture is chiefly influenced by the necking process as it is mainly a measurement of the deformation required for tion becomes fracture. Since the development of neck is controlled by the specimen geometry and the triaxial state of stress in the pffiffiffiffiffi neck, the value of reduction of area at fracture depends on A0 the specimen geometry and deformation behaviour at the ef ¼ eu þ C L0 ð1:128Þ neck. Therefore to obtainpthffiffieffiffiffisame elongation-value for a given Both the elongation and reduction of area increase gen- material, the value of A0 L0 depending on the type of test erally with the test temperature. Further, they vary usually with the heat treatment, composition or alloy content, etc., specimen mpuffiffisffiffitffi be kept constant, as predicted by (1.128). As pnoffirffiffimffiffially A0\\L0; so to avoid the fractional values of A0 L0; the cpritffiifficffiffiaffi l geometrical factor that is maintained constant is L0 A0 for flat test pieces, or, L0=D0 for round pffiffiffiffiffi pspeffifficffiffiiffimffiffiffiffieffiffinffiffiffisffi,ffiffi because for round specimens, A0 ¼ ðp=4ÞD02 pffiffiffi pffiffiffiffiffi to D0 / A0: Hence, ¼ ð p=2ÞD0 and thus, minimize the variations in the elongation-data, the stan- dardizing organizations in different countries have specified

1.9 Inelastic Properties 67 but the most structure-sensitive ductility parameter is the Only the ratio of A0 to Af is used to calculate the value of ef ; as shown by the following (1.132). reduction of area at fracture. ZAf À dA ¼ ln A0 The ductility of a material can also beÀ pÁredicted by A Af zero-gage-length elongation, represented by ef 0; which is True fracture strain; ef ¼ ð1:132Þ obtained from an equivalent value of reduction of area at fracture, rf ; as follows: A0 Due to the constancy of volume, V; during plastic ¼ true reduction of area at fracture; rf0 deformation, one can write V ¼ A0L0 ¼ Af Lf ; or, A0 ¼ Lf Note that the value of true fracture strain is the same as Af L0 that of the true reduction of area at fracture, represented by Hence from (1.123a) rf0 ; as shown by (1.132). The true fracture strain, ef ; can also be related to the engineering reduction of area at fracture, rf ; rf ¼ 1 À Af ; or; A0 ¼ 1 ð1:130Þ by the following (1.133). A0 Af À As from (1.130) 1 rf A0 1 Af À À FÁrom (1.122a) and (1.130), zero-gage-length elongation ¼ 1 rf ; ef 0 is given by ln A0 1 ð1:133Þ Af À rf ) ef ¼ ¼ ln 1 ÀÁ Lf A0 1 rf ef 0¼ L0 À 1 ¼ Af À 1 ¼ 1 À rf À 1 ¼ 1 À rf ð1:131Þ The true uniform strain, eu; is the true strain required to deform the specimen from the start of loading in tension to The zero-gage-length elongation represents the elonga- tion when the gage length at the point of fracture approaches the point of maximum load. Since the strain distribution zero. As it is based on the shortest possible gage length near the fracture, it gives the highest value of elongation for a along the gage length of the tensile specimen is homoge- given material and is independent of the original gage neous up to this point, so the true uniform strain, eu; may be length-value of tensile specimen. calculated from the measured value of eu; with the help of (1.10) as well as by the ratio of either gage length at the 1.9.3.3 Ductility in Terms of True Strain point of maximum load, Lu, to original gage length, L0; or A0 to Au; as shown respectively by the following (1.134) and The measurement of true strain from the tension test is often (1.135). required to assess the ductility of material. Specification of ductility in terms of true strain is more significant than in True uniform strain; eu ¼ lnð1 þ euÞ ð1:134Þ terms of engineering strain. The strains, shown in the true ð1:135aÞ stress–strain curve in Fig. 1.27, are true fracture strain, where eu ¼ Lu ÀL0 : represented by ef ; true uniform strain, denoted by eu; and L0 true local necking strain, denoted by en; which are discussed below. Let ‘A’ denotes average cross-sectional area within ZLu dL ¼ ln Lu the gage length of the tensile specimen and its subscripts ‘0’, L L0 ‘u’ and ‘f’ represent respectively original ‘A’ before loading, eu ¼ ‘A’ at the point of the maximum load and the minimum ‘A’ after fracture, for the following (1.132)–(1.136). L0 The maximum true strain that can be tolerated by the ZAu À dA ¼ ln A0 A Au material prior to its fracture is given by the value of true Or; eu ¼ ð1:135bÞ fracture strain, ef ; and this value is higher than the value of total fracture strain in the engineering stress–strain curve, A0 i.e., ef ; for a ductile material which shows necking. Since the strain distribution along the gage length of tensile specimen From the standpoint of the use of material in stretch forming operations or in any other engineering applications, is inhomogeneous beyond the onset of necking, so the value the maximum possible amount of uniform strain prior to the of ef ; can neither be calculated from the measured value of onset of localized necking is always desired. It has been ef ; with the help of (1.10), nor be calculated by the ratio of shown in (1.97) that the extent of uniform true strain, eu; is gage length after the fracture, Lf ; to original gage length, L0: numerically equal to the magnitude of the strain-hardening exponent, n; where the material obeys Hollomon true stress– strain relation given by (1.90a). So, the higher the magnitude of n the more will be the value of eu and the greater will be the formability or longevity of the material. For this reason,

68 1 Tension the material selected must have higher value of n that can fracture of a ductile material, its ultimate tensile strength, Su; is more important than its fracture strength, Sf : easily be judged from the results of tension test by the measurement of uniform true strain, eu : Moreover to avoid It must be noted that the fracture strength observed for a the complication from necking, ductility measurement is brittle material is strongly influenced by the stress concen- based on the uniform strain, i.e., the determination of eu is trations due to presence of some imperfections in the material, often used to estimate the ductility of the material. such as a microscopic crack. The values of stress concentra- tions again depend on the size, sharpness and orientation of The true local necking strain, en; is the true strain needed for the imperfections. But this flaw-geometry will vary among the deformation of material from the onset of necking to the different samples of a brittle material resulting different observed fracture strengths for the same material. Therefore, point of fracture and calculated with the following (1.136). the fracture strength of a brittle material will not have a single value rather the values will be scattered significantly. The True local necking strain; arithmetic mean or the median of a set of observed values is usually reported as the fracture strength for a brittle material. ZAf À dA ¼ ln Au But it must be kept in mind that this reported value of fracture A Af strength is higher than roughly 50% of the observed fracture en ¼ ð1:136Þ strengths. Further, the fracture strength of a brittle material will vary with size of specimens made from it. The chance of Au having larger flaws is more in a bigger sized sample than in a smaller sized sample of the same material and as a result, the The minimum cross-sectional area after fracture of the larger sample containing bigger flaws will show a lower tensile specimen, Af ; is measured from the fractured parts of average fracture strength than the smaller one. So, this fact the necked segment. From the above three (1.132), (1.135b) must not be forgotten when the fracture strengths of large and (1.136), it is clear that members of brittle materials are predicted from the results of laboratory tests on small specimens. True fracture strain ¼ true uniform strain þ true local necking strain: The applied true fracture strength, rfapp ; is an applied true stress required to cause fracture, which is shown in Owing to the variation of elongation, ef ; with the gage Fig. 1.27. It is defined as the applied load at fracture, Pf ; length, L0; of tensile specimen, researchers occasionally divided by area of the smallest cross-section of the tensile report either both of total engineering fracture strain, ef ; and specimen after its fracture, Af ; as given below: uniform engineering strain, eu; or both of total true fracture strain, ef ; and uniform true strain, eu: 1.9.4 Fracture Strength and True Fracture Applied true fracture strength, Strength Both fracture strength and true fracture strength are structure rfapp ¼ applied load at fracture fracture ¼ Pf sensitive properties. The breaking or fracture strength, Sf ; is minimum cross-sectional area at Af shown in Fig. 1.26 for ductile material and Fig. 1.24 for ð1:138Þ brittle material. The fracture strength Sf is an engineering stress required to cause fracture. It is defined as the load at where Af is measured from the fractured parts of the tensile fracture, Pf, divided by the average original cross-sectional specimen. The value of rfapp is always higher than the tensile area within the gage length of the tensile specimen, A0, as strength, Su; irrespective of whether the material is ductile or given below: brittle since Af \\A0 and it gives a measure of the strain hardening capacity of the material. The measured value of Fracture strength; ð1:137Þ true tensile fracture strength applied in the axial direction Sf ¼ average load at fracture area ¼ Pf rfapp for a ductile material is frequently in error since the original cross-sectional A0 formation of neck in tensile specimen produces a condition of triaxial stress prior to the fracture,. The value of rfapp is The breaking or fracture strength, Sf ; for a brittle material higher than uniaxial true fracture strength, say rftrue ; that is given by its tensile strength, Su; as shown in (1.117). But would be necessary for fracture if necking had not intro- for a ductile material showing necking the fracture strength, duced triaxial stress condition. If rfapp is corrected for Sf ; is lower than the tensile strength, Su; as the fracture load necking, probably by Bridgman relation discussed in is lower than the maximum load. So in the process of Sect. 1.6.2.7, it is possible to obtain the value of rftrue :

1.9 Inelastic Properties 69 1.9.5 Toughness Toughness is an important structure sensitive property of the Engineering stress, S A material. It is defined as the capacity of the material to C absorb mechanical energy in the plastic range up to the point of fracture. The higher the energy absorbed by the B material the higher will be its toughness and the greater will be the resistance offered by the material to its fracture. So, a Engineering strain, e large amount of energy is expended to cause fracture of a tough material, whereas a material is said to be brittle if it Fig. 1.47 Schematic engineering stress–strain curves for three mate- absorbs little energy for its fracture. High toughness is rials of different toughness. Material A has a high strength with little desired particularly for components like coupler joints, ability for plastic deformation. Material B has a poor strength but a high chains, crane hooks, gears etc. which must be capable to ductility. Material C has an optimum combination of strength and bear incidental rise of stress above the yield strength and ductility resulting in a maximum toughness resist fracture. total area under the engineering stress–strain curve is the The amount of energy absorbed by a unit volume of a material till the point of its fracture comes directly from the maximum for material C and so it is the toughest material. amount of work done by the applied tensile force on a unit volume of the material without causing it to break. This Therefore toughness is a property which includes both energy stored in the material in the form of strain energy per unit volume is the area under the tensile stress–strain curve. strength and ductility. Neither a highly strong nor an Hence, toughness is considered to be equal to the area measured under the engineering plastic stress–strain curve. exceptionally ductile material alone can provide good Toughness is sometimes termed as the modulus of toughness because it is measured by the strain energy per unit volume toughness. The optimum combination of strength and duc- like the modulus of resilience but the difference is that the energy absorption is in the plastic range up to the point of tility will be required to achieve the best toughness. fracture in the former one and in the elastic range up to the point of proportional limit or yield strength in the latter one. Sometimes the total area under the engineering stress– Since the fracture strain includes the elastic as well as the strain curve is mathematically approximated by the follow- plastic strain, the energy to break, i.e., the toughness is normally measured by the total area under the engineering ing (1.140) and (1.141), where Su ¼ ultimate tensile stress–strain curve from beginning to end. For an un-notched strength, S0 ¼ yield strength and ef ¼ total elongation. For a tensile specimen, the amount of energy absorbed per unit volume of the specimen up to the point of its fracture is ductile material that shows necking prior to its fracture, therefore À Á Zef ð1:139aÞ either of the following equations may be used to calculate its UToughness Engg:¼ Sde toughness: 0 ÀÁ where ef ¼ engineering strain at fracture: UToughness Engg:% Suef ð1:140aÞ To compute the modulus of toughness, the above area can Or, ÀÁ Su þ S0 ef ð1:140bÞ be measured by the use of a planimeter. UToughness Engg:% 2 Figure 1.47 schematically shows engineering stress– Since the highly localized strain is involved after necking strain curves for three materials of different toughness. for a ductile material, the reporting of its toughness should Material A has a high yield and tensile strength with little include the gage length of the specimen used in testing. ability for plastic deformation, material B is of poor strength Assuming the engineering stress–strain curve to be a para- but shows a greater total elongation and so has a high bola for a brittle material that shows no neck formation, the ductility, whereas both the strength and ductility levels of toughness may be given by the following equation: material C are optimum. Out of these three materials, the

70 1 Tension ÀÁ 2 Suef ð1:141Þ increase in ductility and the easier is the plastic deformation UToughness Engg:% 3 and mechanical working processes. But if changes in micro-structure like precipitate formation, strain ageing etc. It is to be noted that the engineering stress–strain curve occur at elevated temperature, the above mentioned general based on the original cross-sectional area of the specimen behaviour may be changed. does not represent the true behaviour of the material in the plastic range. For this reason, the total area under the true The magnitude of flow stress at the time of its measure- stress–strain diagram up to the point of rupture is used to ment is controlled by the dislocation structure prevailing at measure the toughness in the most severely strained region that time. The dislocation structure again varies with strain, of the specimen. This area represents the energy per unit strain rate and temperature. Therefore, the flow stress r is a volume actually put into the material at the smallest part of function of strain e; strain rate e_ and temperature T; i.e., the neck. The total area under the true stress–strain curve up mathematically r ¼ f ðe; e_; TÞ: At constant strain e and strain to the point of fracture is given by rate e_; the following general relationship exists between the flow stress r of a material and the test or deformation tem- À Á Zef perature T: UToughness True¼ rde ð1:139bÞ Q ð1:142Þ 0 r ¼ C1eRÁT je;e_ where ef ¼ true strain at fracture: where If there would not be any neck formation and the material Q an activation energy for plastic deformation, in J mol−1; would follow Hollomon true-stress–true-strain relation given R molar gas constant = 8.314 J mol−1 K−1; by (1.90a) up to the point of fracture, the area under both the T deformation or test temperature in K engineering and true plastic stress–strain curve would be equal to each other, i.e., the value of true toughness would The slope of the linear plot of ln r versus 1=T at constant be equal to that of engineering toughness—it has been strain and strain rate can provide the value of Q=R; from proved subsequently in the section of solved problems. where the activation energy for plastic deformation Q can be determined, and the value of C1 is obtained from the inter- Strain rate, i.e., rate of loading during tensile testing, cept ln C1 in this plot. At constant dislocation structure, the deformation or test temperature and presence of notch can activation energy for plastic deformation Q can be evaluated affect the toughness of a material. As the strain rate increases by a temperature change test (Fig. 1.48). In this test per- or the deformation or test temperature is lowered, ductility of formed at a constant true strain rate e_; the temperature is a material will decrease resulting in a decrease in toughness. increased from T1 to T2 at a given value of true strain e; so A material having good toughness under uniaxially applied that the flow stress decreases from r1 to r2: Q is given by the tensile stress condition tends to be brittle and shows poor following (1.143). toughness under multi-axial state of stress that is produced due to the presence of a notch. A notched thick tensile T1 specimen having a triaxial state of tensile stress is more prone to brittle fracture than a notched thin specimen that True stress, σ T2 T2 creates a biaxial state of tensile stress. Apart from the tensile σ1 toughness discussed in this section, there are impact T1 < T2 toughness (also called notch toughness) and fracture ε = constant toughness, which will be discussed respectively in Chaps. 6 and 9. 1.10 Influence of Temperature on Tensile σ2 Properties The stress–strain diagram and the strength and ductility Time properties obtained from the tensile test strongly depend on the test temperature. Generally, the higher the deformation Fig. 1.48 Temperature change test to determine activation energy for temperature the more is the reduction in strength and the plastic deformation Q; at constant dislocation structure

1.10 Influence of Temperature on Tensile Properties 71   covalent and BCC crystals are highly directional which Q Q results in a small width of dislocation and hence a high r1 ¼ C1 exp RT1 and r2 ¼ C1 exp RT2 ; Peierls–Nabarro stress is observed in the above crystals. The decrease in the test or deformation temperature limits the Or, r1 ¼ exp½Q=ðR T1ފ ; motion of dislocations and increases the Peierls stress. r2 exp½Q=ðR T2ފ However with the decrease in temperature, there is a minor rise in the Peierls stress resulting in a slight increase in the Or, ln r1 ¼ Q À Q ; yield strength for FCC and ideal HCP crystals having large r2 RT1 RT2  dislocation width, because the initial Peierls stress was very small. On the contrary, similar decrease in temperature will ) Q ¼ R ln r1 T1T2 ð1:143Þ cause a rapid increase in the Peierls stress for ionic, covalent r2 T2 À T1 and BCC crystals having small width of dislocation and as a result, their yield strengths will rise sharply. The temperature Note that a given temperature may be quite high for a low sensitivity of yield strength for different materials is com- pared in the following Table 1.5. melting point material but the same temperature may be quite From Table 1.5, it is evident that the yield strength in low for a high melting point material. So the comparison of FCC metals is much less sensitive to temperature. On the other hand, with increasing temperature the rate of strain mechanical properties of various materials at different tem- hardening in a material decreases, which makes the stress– strain curve quite flat beyond its yield point. Thus, the tensile peratures must be based on their melting points. The ratio of strength of the material decreases and comes closer to its yield strength as the temperature is increased. This results in deformation or test temperature, in Kelvin ¼ Tt ðin KÞ the tensile strength of FCC metal to be more dependent on melting point of individual material, in Kelvin Tm ðin KÞ temperature than the yield strength, which is weakly dependent on temperature. is known as the homologous temperature and the The influence of temperature on the ductility expressed in flow stresses of different materials must be compared at term of percentage reduction in area is also dependent on the type of material. With decreasing temperature the behaviour the same value of the homologous temperature. More- of BCC metal and ceramic material changes from ductile to brittle with a drastic decrease in the percentage reduction in over, instead of comparing only the flow stresses of area while the ideal HCP and FCC metals always show slightly lower ductility measured by a slight decrease in the different materials, it is better to compare the ratio of percentage reduction in area. For example (Bechtold 1955), r0 for a drop in temperature of 200 K, the reduction in area flow stress of individual material ¼ E for different decreases for iron (BCC) from about 70%, for molybdenum elastic modulus of individual material (BCC) from 80%, and for tungsten (BCC) from 20% to nearly 0%, showing complete brittleness respectively at 48, materials to rectify for the influence of temperature on 173, and 373 K. In contrast, the reduction in area for nickel (FCC) decreases from about 95–80% for a drop in temper- modulus of elasticity. ature from 1070 to 70 K. The limited ductility in BCC metals and ceramic materials at low temperature are partly The temperature sensitivity of yield strength depends on because of their large Peierls stress. But with an increase in the type of material. When the test or deformation temper- ature is decreased the yield strengths of BCC metals and ceramic materials increase sharply while the yield strengths of ideal HCP and FCC metals increase very slightly, showing meagre dependency on the temperature. This can be explained with respect to the temperature sensitivity of Peierls–Nabarro stress that is related to the yield strength by (1.57b). Again from (1.55), it can be inferred that the larger the dislocation width W; the lower is the Peierls stress spÀn: Furthermore, the dislocation width is determined by the type of binding forces between atoms. When the interatomic binding forces are spherically distributed and acting along the centre-line of the atoms as found in close-packed crystal structures, the width of dislocation is large. So, a low Peierls–Nabarro stress is observed in FCC and ideal HCP crystals. By contrast, the interatomic binding forces in ionic, Table 1.5 Comparison of temperature sensitivity of yield strength for different materials Materials Dislocation Peierls Peierls Stress temperature Temperature sensitivity of width stress sensitivity yield-strength Insignificant FCC and ideal HCP Very large Very small Negligible metal Sharp Small Large Strong Sharp BCC metal Small Large Strong Sharp Very small Very large Strong Ionic-bonded ceramic Covalent-bonded ceramic

72 1 Tension temperature, there is a rapid decrease in the Peierls stress, –200°C ( = 73K ) which improves the ductility of these materials at elevated Room temperature ≈25°C ( = 298K ) temperatures. Engineering stress 200°C ( = 473K ) 1.10.1 Effect of Temperature on Stress–Strain Curve of Mild Steel The influence of temperature on the flow curve of a material 400°C ( = 673K ) at constant strain rate, as shown schematically in Fig. 1.49, indicates that the increase of temperature causes mainly the Strain rate, ε = constant following three effects: Flow stress Temperature increasing 0 1. Lowering of the yield stress; Engineering strain 2. Decrease in the rate of strain hardening and 3. Increase in the ductility. Fig. 1.50 Schematically showing variations in engineering stress– strain curves with temperature for mid steel, at a constant strain rate However, Fig. 1.50 shows schematically the variations in engineering stress–strain curves with temperature for mid very low test temperature of the order of À200 C: Further, steel, where it is assumed that the strain rate remains con- the lower the temperature the steeper is the linear elastic stant at all temperatures. The most common engineering modulus line of the stress–strain curve because the modulus stress–strain curve of mid steel tested at room temperature of of elasticity decreases with the increase of temperature. around 25 °C is the Type IV curve showing elastic– heterogeneous–homogeneous plastic response with The Type I curve can be explained in terms of the vari- yield-point phenomenon. ations in yield strength and fracture or cleavage strength of mild steel with temperature. At room temperature, since the For mild steel, it is possible to obtain Type II curve yield strength of mild steel is lower than its cleavage showing elastic–homogeneous plastic response with lower strength, plastic deformation occurs resulting in a ductile strength properties and higher percentage elongation at fracture. The yield strength of mild steel is much more higher test temperature, on the order of 400 °C, which of sensitive to temperature than its cleavage strength. When the course depends on the value of strain rate. But as the tem- test or deformation temperature is decreased below the room perature decreases the curve shifts gradually to higher temperature, the yield strength of mild steel (BCC) increases strength levels with lower percentage elongation and shows sharply as discussed earlier, while the fracture strength of sequentially elastic–heterogeneous plastic response i.e., mild steel increases very slightly, showing weak dependency serrated stress–strain curve of Type III, then Type IV curve on temperature. At around a temperature of À200 C the and finally, Type I curve showing elastic response with little yield strength of mild steel becomes nearly equal to its plasticity prior to fracture i.e., brittle nature that occurs at cleavage strength and the mild steel undergoes brittle frac- ture with a little amount of plasticity prior to fracture, thus Constant strain rate exhibiting the Type I curve. Strain Curves of Types II to IV can be explained with respect to the competition between the speed of dislocations and that of Fig. 1.49 Schematically showing the effect of temperature on flow the interstitial solute atoms in mild steel i.e., carbon and stress versus strain curve at constant strain rate nitrogen atoms. Since the strain rate during the test is assumed to be constant, the variation in the speed of dislo- cation motion is only dependent on the change in tempera- ture. When the test-temperature is higher on the order of 400 °C, the solute atoms move faster than the dislocations because the interstitial solute atoms are smaller in size than the dislocations. So, pinning of slow moving dislocations by atmosphere of rapid moving solute atoms cannot occur. Thus, the unhindered movement of dislocations generates the smooth parabolic curve of Type II. When the temperature is decreased to about 200 °C, the speed of the solute atoms of carbon and/or nitrogen also decreases and becomes nearly equal to the speed of

1.10 Influence of Temperature on Tensile Properties 73 dislocations. As a result, the dislocations are pinned by such the testing machine, the extension, dL; of the specimen per solute atmospheres and dislocation motion is severely unit time is equal to the cross-head velocity. Hence, restricted. This requires a higher applied stress to unlock the pinned dislocations from their solute atmosphere and Crosshead velocity, v ¼ dL ð1:144cÞ unpinning of dislocations causes a drop of stress. Since the dt velocities of solute atoms and dislocations are nearly the same, this process of pinning and unpinning of dislocations The strain rate is related to the cross-head velocity, v, by occurs several times producing the serrated stress–strain the following (1.145a) and (1.145b). From (1.144a) and curve of Type III, at test temperature on the order of 200 °C, (1.144c): which of course depends on the value of strain rate. Engineeng strain rate, e_ ¼ dL=L0 ¼ dL=dt ð1:145aÞ When the temperature is decreased to room temperature, dt L0 the velocity of the solute atoms decreases further and causes an initial locking of most of the dislocations. But even then ¼ v ¼ crosshead velocity there will be some unpinned mobile dislocations which will L0 original gage length move with deformation at higher velocity to maintain a constant strain rate (1.101), resulting in an increase in stress It is clear from (1.145a) that e_ / v; since L0 ¼ constant: (1.102). These dislocations multiply rapidly to increase the Hence, the tension test can easily be carried out at a constant mobile dislocation density. The high mobile dislocation engineering strain rate e_; if the cross-head velocity, v; is kept density will cause a drop in dislocation velocity for main- constant. From (1.144b) and (1.144c): taining a constant strain rate, and a corresponding drop of stress from upper to lower yield point. The next phase of True strain rate, e_ ¼ dL=L ¼ dL=dt ð1:145bÞ deformation is the yield-point elongation—all these have dt L been discussed in detail in Sect. 1.6.4. Finally, when the velocity of all unpinned dislocations exceeds that of the ¼ v ¼ crosshead velocity solute atoms then only homogeneous dislocation flow will L instataneous gage length take place producing the smooth parabolic portion of Type IV curve. As the tensile specimen is stretched its instantaneous gage length, L; increases. Therefore, the true strain rate e_ 1.11 Strain Rate decreases when the cross-head velocity v is maintained constant, as evident from (1.145b). To achieve a constant The rate at which strain is applied to a specimen can be true strain rate e_; during uniform extension of the tensile expressed in terms of engineering or conventional strain specimen, the cross-head velocity must be increased in rate, denoted by e_; and in terms of true strain rate, repre- sented by e_; which are defined as follows: proportion to the increase in the instantaneous gage length of Engineeng strain rate, e_ ¼ de ¼ dL=L0 ¼ 1 dL ð1:144aÞ the specimen (Lautenschlager and Brittain 1968). At the dt dt L0 dt beginning of the test, i.e., at initial time t ¼ 0; the original gage length of the specimen is L ¼ L0: Let the gage length of the specimen after test duration of t ¼ tn; is L ¼ Ln: At time t ¼ tn; the cross-head velocity vn must increase according to the following (1.146b) to maintain a constant true strain rate e_ during uniform deformation of the tensile specimen. True strain rate, e_¼ de ¼ dL=L ¼ 1 dL ð1:144bÞ Since e_ ¼ dL=L and e_ ¼ constant, dt dt L dt dt where L0 and L are respectively the original and instanta- ZLn Ztn ð1:146aÞ neous gage length of the tensile specimen and dL is the so, dL ¼ e_ dt; extension of the specimen in the time interval dt: The con- ventional unit of strain rate is s−1, i.e., “per second” and the L strain rate for static tension test operated with hydraulic or L0 0 screw-driven machines varies normally from 10−5 to 10−1 s−1. The rate of movement of the cross-head of the Or; ln Ln ¼ e_ tn; or, Ln ¼ L0 expðe_tnÞ L0 testing machine is called the cross-head velocity, which will ) vn ¼ e_ Ln ¼ e_ L0 expðe_ tnÞ ð1:146bÞ be denoted by v. As one end of the tensile specimen remains When the localized plastic deformation occurs along the fixed and its other end is attached to a movable cross-head of gage length to form ‘neck’ in the specimen the true strain for the constant volume plastic deformation process is given by

74 1 Tension e ¼ ÀðdA=AÞ; according to (1.89a). Under such condition a Flow stress at true strain ε = 0.002 (log scale) Temperature, T1 > T2 >T3 > T4 > 25°C constant true strain rate e_ can be achieved, if the initial Room temperature ≈ 25°C ( = 298K ) cross-sectional area A0 within the gage length of the tensile specimen at time t ¼ 0; changes to the cross-sectional area An after test duration of t ¼ tn; according to the following (1.147): ZAn Ztn T4 À dA ¼ e_ dt; ½since, e_ ¼ constantŠ T3 A T2 A0 0 Or; ln An ¼ Àe_tn; A0 ) An ¼ A0 expðÀe_tnÞ ð1:147Þ T1 Further, the relation between the true strain rate, e_; and 0 10–1 the conventional strain rate, e_; during uniform extension of 10–5 the tensile specimen is given by the following (1.148): Strain rate, s–1 (log scale) Since engineering strain; e ¼ L À 1; Fig. 1.51 Flow stress at true strain e ¼ 0:002 versus strain rate at L0 various temperatures (schematic) Or, L0 ¼ 1 ; C2 ¼ proportionality constant in (1.149), and it is also L þ 1 e dependent on temperature. ) e_ ¼ v ¼ v L0 ¼ e_ ð1:148Þ The slope of the linear plot of log r versus log e_ at L L0 L þ 1 e constant strain and temperature can provide the value of 1.11.1 Relation Between Flow Stress and Strain strain-rate sensitivity m; and the value of C2 is obtained from Rate the intercept of this plot. However, a more precise method to determine the exponent m is by means of the strain-rate change test (Fig. 1.52). In this test performed at a constant Flow stress of a material usually increases with increase of temperature T; the strain rate is increased from e_1 to e_2 at a strain rate e_; as revealed from Fig. 1.51. The effect of strain given value of strain e; so that the flow stress increases from rate is more on the flow stresses at lower plastic strains and on the yield strength than on the tensile strength of the r1 to r2: The strain-rate sensitivity m is given as follows: material. Further, the increase in strength properties with strain rate is more with the increase in temperature, which    means the difference between flow stresses at low and high @ log r D log r strain rate is insignificant at room temperature but becomes m¼ @ log e_ ¼ D log e_ appreciable at elevated temperatures, i.e., at above half of the  melting point of the material expressed in Kelvin. So, the  e; T e;T ! ð1:150Þ stress–strain curve of a material at room temperature remains practically independent on the strain rate and it becomes ¼ log r2 À log r1 ¼ logðr2=r1Þ immaterial to specify the strain rate in tension test at room log e_2 À log e_1 logðe_2=e_1Þ e;T temperature. Low-carbon steel that does not show a yield e;T point under ordinary strain rates will exhibit the yield point at high strain rates. The value of strain-rate sensitivity m is usually lower than 0.1 for metals at room temperature but with increase of Figure 1.51 shows that at constant strain e; and constant temperature this value increases, particularly when the temperature T; the following general relationship exists temperature is above half of the melting point of the mate- between the flow stress, r; of a material and the strain rate, e_; rial, expressed in Kelvin. During hot working the values of applied during the tensile test: m generally vary from 0.1 to 0.2. The variation of flow stress with strain rate for steels at constant strain e and temperature T is better described by the following semi-logarithmic relationship than by the above (1.149). r ¼ k1 þ k2 log e_je;T ð1:151Þ r ¼ C2ðe_Þmje;T ð1:149Þ In the above (1.151), k1 and k2 are constants which can be evaluated respectively from the intercept and the slope of the where the exponent m ¼ strain-rate sensitivity, the value of linear plot of r versus log e_ at constant strain and which depends on temperature; temperature.

1.11 Strain Rate 75 True Stress, σ ε2 • An extremely fine and uniform grain size or interphase σ1 spacing of the order of 1À10 lm; most commonly 1À3 lm; in the material. To make the alloy superplastic σ2 the grain refining treatment is highly needed. But the alloy with refined grain size is not suitable for applica- ε1 tions at high temperature under load, as in creep defor- mation (described in Chap. 7). This unsuitability is due Strain rate, ε2 > ε1 to the lower strength of grain boundary than the grain at constant temperature body at an elevated temperature above the equicohesive temperature at which the strengths of grain boundary and True Strain, ε grain body are equal. Therefore, after the grain-refining treatment required for superplastic formation of the Fig. 1.52 Strain-rate change test at a constant temperature to deter- desired component the alloy is subjected to mine strain-rate sensitivity grain-coarsening treatment for its application at high temperature under load. 1.11.2 Superplasticity • A high deformation temperature, usually above 0:4 Á Tm; Superplasticity (Johnson 1970; Edington et al. 1976; Taplin (where Tm is the melting point of the superplastic et al. 1979) is the behaviour of a material which can undergo material in Kelvin); an extensive total engineering strain usually between 100 and 1000% or sometimes in excess of 1000% without • A high (Backofen et al. 1964; Avery and Backofen 1965; necking during tensile deformation. A superplastic material Johnson 1970) strain-rate sensitivity factor m [ 0:3; of exhibits pronounced resistance to necking primarily due to a the material, due to which the material exhibit pro- high value of strain-rate sensitivity m ð0:3\\m\\1:0Þ: nounced resistance to necking; Superplastic behaviours have been observed in a number of metals and alloys that are mostly of eutectic or eutectoid • Application of a low strain rate, generally strain-rates compositions, for example, in some stainless steels, alu- lower than 0:01 sÀ1 are used during superplastic forming. minium, nickel and titanium alloys. Superplasticity has also There is a limiting strain-rate above which no super- been observed in ceramics (Johnson 1970). To avoid internal plastic material will remain superplastic. Generally, cavity formation to a large extent, there must be similarity in superplastic behaviour is observed over a range of strain strengths between the matrix and the second phase in a rate. This strain-rate range usually increases with superplastic material. decrease in grain size of the material and increase in the deformation temperature, as illustrated schematically in Superplasticity occurs in a material under the following log–log plot of normalized stress, r=G; versus strain rate conditions, which are: in Fig. 1.53. One characteristic of superplastic materials is that their grains remain mainly equiaxed even after large amount of deformation probably due to occurrence of grain-boundary migration. The main advantage of a superplastic material is Fig. 1.53 Schematic log–log Log σ Increasing temperature plot of normalized stress, r=G ; G Decreasing grain size versus strain rate, illustrating the increase in the strain-rate range associated with superplastic behaviour with increasing temperature and decreasing grains size Superplastic range Log strain rate

76 1 Tension that a low flow stress, of the order of 5–30 MPa, is needed Now, substituting e_ from (1.152a) into (1.152b) we get during its deformation due to high deformation temperature and / or its structural condition. This advantage is useful dA P1=m 1 1=m  P 1=m  ð1=mÞÀ1 during mechanical working operations of superalloys that are A ¼ 1 difficult to work and for embossing finer details in several À ¼ applications. This low flow stress gives a low rate of strain dt A C2 C2 A hardening dr=de for a superplastic material, but in a ¼  P 1=m ! non-superplastic material, as long as the rate of strain hard- C2 1 ening is higher than the increase in the axially applied flow Að1ÀmÞ=m stress r; caused by the reduction in the cross-sectional area, i.e., as long as ðdr=deÞ [ r; necking will not occur. Then the ð1:153Þ question is how necking is being resisted in a superplastic material. The reason behind it is that as soon as some region in In the extreme case, the material exhibits Newtonian a superplastic material begins to neck locally, the strain rate in viscous flow where the flow stress is given by that region increases due to this localized plastic flow. Hence the flow stress r required for further deformation in that r ¼ ge_ ð1:154Þ region increases rapidly according to (1.149), because of the local rise of strain rate e_ in that region and a high value of where g ¼ constant related to coefficient of viscocity: strain-rate sensitivity m of the superplastic material. This local Comparing (1.154) with (1.149), it is found that the rise in the flow stress prevents further deformation and incipient neck to propagate inward in that region and hence, strain-rate sensitivity m ¼ 1 for a Newtonian viscous solid. the hardening due to the high value of strain-rate sensitivity Under such condition, i.e., when m ¼ 1; (1.153) reduces to exponent exceeds the axially applied flow stress r: At this moment, the deformation shifts to another region where no À dA ¼ P ð1:155aÞ neck has initiated. The material is again deformed in this dt C2 region and the flow stress r increases locally according to (1.149). In this way the above action of shifting the area of Further for any value of m; when A ¼ P=C2; ÀdA=dt deformation continues producing uniform large elongation becomes independent of m; which is evident from (1.155b): with the suppression of necking. Hence, the presence of strain-rate hardening (Avery and Backofen 1965; Hart 1967; À dA ¼  P 1=m\" 1 # ¼ P ð1:155bÞ Al-Naib and Duncan 1970) prevents necking. dt C2 ðP=C2Þ1=mðP=C2ÞÀ1 C2 The fact that a high value of strain-rate sensitivity m of a It is clear from (1.155a) that when m ¼ 1; the rate of superplastic material suppresses necking during tensile decrease in cross-sectional area ÀdA=dt is only dependent deformation and results in an extensive elongation can be on the applied load P and does not depend on any geo- established from the following analysis. Let a specimen metrical irregularities in the cross-sectional area of specimen made of superplastic material is subjected to an axial tensile like incipient neck and machining mark etc., which are load of P applied normal to the cross-sectional area A of the simply retained and do not spread inward. In such case, the specimen. From (1.149), it can be written as material exhibits extreme extensibility without the formation of neck. For example, a very long fibre can be drawn without r ¼ P ¼ C2ðe_Þm; or, e_m ¼ P ; necking from the glass in molten condition having the A AC2 strain-rate sensitivity m % 1: But for the exponent m\\1; the  1 1 lesser the cross-sectional area A of specimen the higher is the ) e_ ¼ P m 1 m rate of reduction in cross-sectional area, i.e., the faster is the A C2 ð1:152aÞ decrease in the cross-sectional area. The dependence of the rate of decrease in cross-sectional area ÀdA=dt on A for Since the true strain for the constant volume plastic various values of m has been shown in Fig. 1.54. As the deformation process is given by e ¼ ÀðdA=AÞ; according to value of m comes near to 1, the rate of inward propagation of (1.89a), so the true strain rate will be expressed by the incipient neck is severely decreased. In accordance with the above explanation, Fig. 1.55 shows schematically the e_ ¼ de ¼ ÀðdA=AÞ ¼ À 1 dA ; increase in percentage tensile elongation of superplastic dt dt A dt materials with m: Or, À dA ¼ Ae_ ð1:152bÞ The mechanisms that cause the superplastic flow have not dt been well accepted. After considerable debate, it is believed that the mechanisms responsible for superplasticity are climb of edge dislocation at high stress level and grain-boundary- sliding with diffusion controlled accommodation (Alden 1967, 1968; Ashby and Verall 1973) at low stress level. The

1.11 Strain Rate 77 1 103 m= 4 m= 1 Tensile elongation, % 102 2 – dA dt 3 10 m= 0 0.2 0.4 0.6 0.8 4 Strain rate sensitivity, m P m=1 1 Fig. 1.55 Schematically showing the increase in percentage tensile C2 3/4 elongation of superplastic materials with strain rate sensitivity m 1/2 1/4 A mild steel with increasing strain rate are similar to those with decreasing temperatures, which has been described in P/C2 Sect. 1.10.1. Thus at a lower strain rate, the Type II curve with lower strength properties and higher percentage elon- Fig. 1.54 Graphical illustration of (1.153), showing the variation of gation is observed. But at constant temperature, with the rate of decrease in cross-sectional area À dA=dt with increasing strain rate the curve shifts gradually to higher cross-sectional area A for different values of strain-rate sensitivity m strength levels with lower percentage elongation and shows sequentially Type III, then Type IV curve and finally Type I reader is referred to Hubert and Kay (1973) for commercial curve. However, there is no effect of strain rate on the linear applications of superplasticity. elastic modulus line of the stress–strain curve unlike the effect of temperature on the linear elastic modulus line. 1.11.3 Effect of Strain Rate on Stress–Strain Curve of Mild Steel The Type I curve can be explained in terms of the vari- ations of yield strength and fracture or cleavage strength of The influence of strain rate on the flow curve of a material at mild steel with strain rate. The yield strength of mild steel is constant temperature, as shown schematically in Fig. 1.56 much more dependent on strain rate (as on temperature) than indicates that increasing strain rate has mainly the following its cleavage strength. The yield strength of mild steel, which three effects: was lower than its cleavage strength at low strain rate, increases according to (1.151) at higher strain rate (which is 1. Increase in the flow stress; e_1 in Fig. 1.57), and becomes nearly equal to its fracture 2. Increase in the rate of strain hardening and strength, exhibiting brittle fracture with a little amount of 3. Decrease in the ductility. plasticity prior to fracture, as shown in the Type I curve. Thus, the effect of increasing strain rate on the flow curve Similar to the effect of temperature, curves of Types II to of a material is opposite to that caused by the increase of IV can be explained with respect to the competition between temperature. the speed of dislocations and that of the interstitial carbon and nitrogen atoms in mild steel. Since the interstitial solute However, Fig. 1.57 shows schematically the variation of atoms are smaller in size than the dislocations, a given engineering stress–strain curve with strain rate for mild steel, constant test-temperature causes the solute atoms to move where it is assumed that the temperature remains constant at faster than dislocations. At a low strain rate, e_4 in Fig. 1.57, all strain rates. The variations in the stress–strain curves of the speed of dislocations is quite lower (1.101) than that of the solute atoms. So, pinning of slow moving dislocations by

78 1 Tension Strain rate increasing strain rate e_2 in Fig. 1.57 has been given in relation to the room temperature effect on stress–strain curve of mild steel Flow stress (see Sect. 1.10.1). 1.12 Testing Machine Constant temperature The machine which is equipped to test the specimen under Strain compressive loading or bending load apart from testing Fig. 1.56 Schematically showing the effect of strain rate on flow various types of specimens under tensile loading is called stress versus strain curve at a constant temperature universal testing machine, one model of which has been atmosphere of rapid moving solute atoms cannot occur, resulting in the smooth parabolic curve of Type II. shown in Fig. 1.23. A tensile testing machine is character- When the strain rate is increased to e_3; as shown in (1.57), ized by its spring constant or stiffness. This characteristic the speed of dislocation increases and becomes nearly equal strongly influences the shape of stress–strain curve. A rigid to the speed of the solute atoms of carbon and/or nitrogen, testing machine having a high stiffness Mm; is called a “hard As a result, the dislocations are pinned by such solute machine” and a machine with a low stiffness Mm; is known atmospheres requiring a higher applied stress to unlock the as a “soft machine”. During the test, a soft machine itself pinned dislocations. Unpinning of dislocations from their undergoes more elastic deformation and stores more elastic solute atmosphere causes a drop of stress. This process of pinning and unpinning of dislocations occurs several times strain energy than a hard one. The elastic strain energy producing the serrated stress–strain curve of Type III. stored in each machine is givenÀ by thÁe area under the load Although further increase in strain rate to a value of e_2; as ðPÞ versus elastic deformation DLem diagram and a com- shown in (1.57), increases the dislocation velocity but there parison of this energy at a given load between hard and soft will be an initial locking of most of the dislocations by testing machines is shown in Fig. 1.58. Since P ¼ MmDLme ; atmosphere of slow moving solute atoms. Even at this stage, so the elastic strain energy stored at a given load P1 with there will be some unpinned mobile dislocations that move reference to Fig. 1.58 is: at a rapid rate. Further explanation for the Type IV curve at For¼haðrMd mmÞaHc2ÀhDinLeem;Á21ðU¼mÞ2HðM¼Pm21 12ÞHP1ÀDLemÁ1 ð1:156aÞ For¼soðfMt mmaÞcShÀ2DinLeme; Á22ðU¼mÞ2S ð¼MP21m21ÞPS1ÀDLemÁ2 ð1:156bÞ Engineering stress ε•1 ÀwhereÁ ðÀUMÞHÁ; ðUMÞS are stored elastic strain energy, ε•2 ε•3 DLem 1; DLme 2 are elastic deformation and ðMmÞH; ðMmÞS are stiffness for hard and soft machines respectively. Since Strain rate, ε•1> ε•2 > ε•3 > ε•4 ðMmÞH [ ðMmÞS; therefore from (1.56) it is clear that at contsant temperature ðUM ÞH \\ðUM ÞS : A hard machine is often important to use in instability experiments, such as for determination of the discontinuous yielding of mild steel. The reason is the reduction in the ε•4 elastic deformation of such machine, which in turn reduces the energy absorbed by the machine during the test. In tensile testing two types of machines are generally used: 0 (1) Hydraulic machine, which is load controlled, i.e., the Engineering strain load is adjusted by the operator and the displacement depends on the regulated load. It is usually less Fig. 1.57 Schematically showing variations in engineering stress– expensive and capable of applying a load of about 2300 strain curves with strain rate for mid steel, at a constant temperature

1.12 Testing Machine P Soft Machine 79 P1 Low Mm Fig. 1.58 Schematically High Mm comparing hard and soft testing machines with respect to their stored elastic strain energy, represented by shaded areas at a given load P1 Hard machine 0 (ΔLem)2 ΔLem (ΔLem)1 metric ton ð $ 22:55 MNÞ and more. It is a soft v t ¼ DLes þ DLps þ DLem ð1:157Þ machine, by which the yield-point phenomenon of mild steel is difficult to detect. where (2) Mechanical type or screw-driven machine, which is displacement controlled, i.e., a constant velocity pres- DLse elastic displacement of the specimen, elected for the movement of the machine cross-head D Lsp plastic displacement of the specimen, and controls the displacement and the load accommodates DLme elastic displacement of the testing machine itself with that displacement. It is usually more expen- sive than the hydraulic machine but its load applying If L0 is the original gage length of the tensile specimen, capacity is less, which seldom exceeds 200 metric ton E is its modulus of elasticity and S is the applied engineering ð $ 1:96 MNÞ: It is usually a hard machine, by which stress, then according to Hooke’s law, D Les ¼ ðSL0Þ=E: If ep the upper and lower yield point of mild steel can be is the engineering plastic strain in the specimen, then DLps ¼ precisely recorded. epL0: If P is the tensile force applied on the specimen mounted in the machine and Mm is the stiffness of the testing A versatile machine that serves to control anyone of both machine then according to (1.30), DLem ¼ P=Mm; where P ¼ load and displacement is the servo hydraulic testing A0S and A0 is the original cross-sectional area within gage machine, where it is possible to carry out the test by com- length of the specimen. Hence, from (1.157), puter control. vt ¼ SL0 þ epL0 þ P ð1:158Þ E Mm 1.12.1 Influence of Testing Machine on Strain ) ep ¼ vt À S À A0S ð1:159Þ and Strain Rate L0 E MmL0 All testing machines undergo elastic deformation when the Thus, elastic deformations of both specimen and testing specimen mounted in the machine is subjected to load. Let machine must be considered to obtain the correct value of us assume that the cross-head velocity v of the testing the plastic engineering strain in the specimen according to machine is constant and it produces a total displacement of the above (1.159). v t at a particular time t. This above total displacement is the summation of the following three components. When the cross-head velocity v of the testing machine is kept constant, we obtain a constant engineering strain rate e_; according to (1.145a). This total strain rate e_; applied by a

80 1 Tension testing machine is the summation of the following three Equation (1.163) shows that engineering strain rate e_ in components, which are the specimen will not be proportional to the prefixed cross-head velocity v of the testing machine, which was • The elastic strain rate in the tensile specimen, shown by (1.145a). Rather, e_ will depend on the engineering • The plastic strain rate in the tensile specimen and plastic strain rate e_p; elastic modulus E of the specimen and • The strain rate caused by the elastic deformation of stiffness or spring constant Mm of the testing machine. The spring constants, Mm; of testing machines have experimen- testing machine. tally been found to have values in the range of 7 to 32 MPa m-1, which are usually much lower than the spring Now, to obtain the correct engineering strain rate in the constants of the specimens except for wire specimens specimen e_; the following method is adopted. In above (Hockett and Gillis 1971). (1.158), the time ‘t’ is replaced by an infinitesimally small period ‘dt‘ at which there are infinitesimally small changes 1.13 Notch Tensile Test in the engineering plastic strain of the specimen, the applied Metallurgical transformations or environmental attacks may engineering stress and the applied load, i.e., ‘ep’, ‘S’, ‘P’ of cause the reduction in local ductility of a material and make (1.158) are respectively replaced by ‘dep’; dS and ‘dP’: it susceptible to brittle fracture but this fact is not always Further, dP ¼ A0dS; is substituted in above (1.158). Hence, detected by the measurement of ductility on a smooth nor- (1.158) is expressed as mal tensile specimen. The susceptibility to brittle fracture in the presence of a notch is known as notch sensitivity, which v dt ¼ L0dS þ L0dep þ A0dS ð1:160Þ can be assessed by employing a notched specimen in a E Mm uniaxial tension test. The notch sensitivity can also be examined with notched-bar impact test, which is described Since v; L0; and A0 are constants, differentiating (1.160) in Chap. 6. The state of stress is better defined in the notch with respect to dt; we get tensile test than the notched-impact test, although the latter has the advantage of testing over a broad span of tempera- v¼ L0 dS þ L0 dep þ A0 dS ; ture and ease in making specimens. The notch tensile test has Or, E dt  dt Mmdt e_ p ; been widely applied to study hydrogen embrittlement in v¼ S_ steels and titanium, to evaluate the notch sensitivity of L0 E 1 þ A0E þ high-strength steels and high-temperature alloys and to MmL0 detect metallurgical embrittlement, such as tempered martensite embrittlement (TME) in steels and temper ) S_ ¼ ðv=L0Þ À e_p ð1:161Þ embrittlement (TE) in alloy steels and for measurements E 1 þ ½ðA0EÞ=ðMmL0ފ involving fracture mechanics. where S_ ¼ engineering stress rate ¼ dS=dt; and For the notch tensile specimen (Kuhn and Medlin 2000), e_p ¼ engineering plastic strain rate ¼ dep=dt: a 60° V-shaped notch with a tip radius of 0.025 mm or less is the most common that is created in the mid-length of Engineering strain rate in the specimen is given by specimen along the circumference for a round specimen, called circumferential notch or at two opposite faces for a e_ ¼ e_ e þ e_p ¼ S_ þ e_p ð1:162Þ flat specimen, called double-edge notch. Generally the notch E penetrates into the specimen up to a depth such that the area of cross-section at the tip of the notch is one-half of the Substituting S_ E from (1.161), into (1.162), we get cross-sectional area in the smooth region of the specimen. e_ ¼ 1 þ ðv=L0Þ À e_p þ e_ p The notched specimen after it has been properly aligned ½ðA0EÞ=ðMmL0ފ is subjected to uniaxial tensile load till it fractures and the maximum load that the notched specimen can withstand is ¼ ðv=L0Þ þ ½ðA0EÞ=ðMmL0ފe_p recorded. The notch strength, Snet; and the notch strength 1 þ ½ðA0EÞ=ðÂMmL0ފ ratio, NSR, are defined as follows: ½ðA0EÞ=ðMmL0ފ fðv MmÞ=ðA0EÞg þ e_ p à 1 þ ½ðA0EÞ=ðMmL0ފ ¼ ¼ ½ðvMmÞ=ðA0Eފ þ e_p ð1:163Þ ½ðMmL0Þ=ðA0Eފ þ 1