Module 28 Metal working: Deformation processing II Lecture 28 Metal working: Deformation processing II 1
Keywords : Difference between cold & hot working, effect of macroscopic variables on deformation processing, annealing, recovery, re crystallization & grain growth, grain boundary energy, origin of texture Introduction Metal is known for its ductility. It can be formed into any shape by subjecting it to stress. The last module was devoted to various methods of plastic deformation by which a metal or an alloy can be given its final shape. This occurs without any change in its volume. In most of these, the dimension of the work piece increases along one or two directions at the cost of the other. During rolling the length increases at the cost of its thickness whereas in wire drawing its length increases at the cost of its cross section. The ability of a metal to deform depends on its flow stress which is a function of strain rate, temperature and its microstructure. It is always easier to shape the metal at higher temperatures because it can flow easily at low stresses. The flow stress remains unchanged with deformation. However at lower temperatures the flow stress is high. It needs higher stresses to deform and it keeps increasing with strain because of strain hardening. The temperature at which there is such a change in its deformation behavior is known as its re crystallization temperature. Deformation above this temperature is known as hot working whereas that below it is called cold working. This module looks at the difference between the two, a little more critically and tells about their effects on the structure and properties of the metal. Difference between cold & hot working: Plastic deformation below the re crystallization temperature is known as cold working whereas that above it is known as hot working. A rough estimate of the re crystallization temperature is around 0.5T m where T m is the melting point of metal in Kelvin. Slide 1 gives a summary of the difference between the two. 2
Hot vs. cold work Cold work Hot work Temp < 0.5Tm > 0.5Tm Load High Low Structure Elongated grain Equiaxed grains Texture Preferred Random Strength High Low Ductility Poor High Conductivity Poor Good Slide 1 Slide 2 gives a summary of the effect of cold working on the structure and properties of metals. There is a substantial change in the shape of the grains. They get elongated along the direction of metal flow. The density of dislocation increases. It leads to strain hardening and loss of ductility. Deformation takes place by slip within the grains. In order to maintain continuity apart from the change in its shape there is a change in the orientation of the grains. This results in preferred orientation. Effect of cold working 1. Change in grain shape / size 2. Development of texture 3. Increase in dislocation density 4. Loss of ductility Slide 2 5. Increase in strength 6. Increase in residual stress / stored energy 7. Crystal structure remains unaltered 8. Corrosion resistance decreases: stress corrosion cracking 3
Effect of hot working 1. Change in product shape / size with little change in grain shape & morphology 2. Absence of texture (isotropic) 3. Little change in dislocation density 4. No loss of ductility / improves ductility of cast structure Slide 3 5. No significant increase in strength 6. Little residual stress / stored energy 7. Crystal structure remains unaltered Slide 3 gives a summary of the effect of hot working on the structure and properties of metals. If the initial structure is homogeneous consisting of equiaxed grains. There would be hardly any noticeable change in the structure. The plastic deformation is accompanied by recrystallization. Deformed grains get continuously replaced by strain free grains. The new grains are usually randomly oriented. There is no preferred orientation or texture. One of the main objectives of hot working is to remove casting defects in metals make them homogeneous. In such an event there is a significant change in the structure of the metal and its properties. T A + L Structure of cast alloy Structure after hot work + Fig 1 % B 4 Figure 1 shows the phase diagram and the microstructure of a terminal solid solution. Under equilibrium cooling you do not expect this to have eutectic. Under normal cooling there is a concentration gradient within the primary grains of. The shading represents variation in composition. The core is relatively pure and the periphery has higher concentration of solutes. In the inter dendritic channels there is eutectic. This is the effect of segregation. As a result of hot working the structure gets refined. It has uniform grain structure. There is no composition gradient or eutectic surrounding the primary grains. You also expect some precipitates of if the alloy is cooled slowly after hot working.
Effect of macroscopic variables on deformation processing: The structure and properties of metals after deformation processing would depend on the processing parameters. These are stress, temperature, strain rate & the magnitude of strain at every stages of deformation. Although in an actual deformation process these may vary with time from point to point it is often helpful to consider material response under constant stress or strain rate to understand the underlying physical processes taking place during deformation. In general the strain rate at point within a metal during deformation is a function of the local stress (), temperature (T) and its microstructure (S). It may be represented as follows:,, (1) There may be several structural parameters that may affect deformation behavior. They include grain size, dislocation density, volume fraction of precipitate, inter particle spacing etc. The subscript i in the structural parameter S indicates the contribution of the i th component. The evaluation of deformation behavior of metals is done by subjecting it to uniaxial tension or compression. The test is done by either load or displacement control mode. Let us first consider the case where the load is increased till it reaches a specified value and thereafter it is held at the same level. During the test both stress and strain are monitored as a function of time. Slide 4 gives the nature of these plots at three different temperatures. The sketch at the top describes how the test was performed. Note that initially the (load) stress increases linearly as long as the stress is below the elastic limit. Thereafter the rate of increase in stress keeps decreasing till it attains the specified value. The response shown by strain depends on the temperature at which the test is performed. Slide 4 shows strain () versus time plots at three different temperatures (0.1T m, 0.5T m & 0.9T m ). 5
Evolution of strain / strain rate with time at constant temp & stress t Slide 4 t T ~ 0.1Tm t T ~ 0.5 Tm t T ~ 0.9 Tm At lower temperatures strain increases linearly as long as the stress is below the elastic limit. Once the yield stress is exceeded plastic deformation takes place. This is accompanied by strain hardening and a corresponding increase in dislocation density. Therefore deformation continues as long as the stress keeps increasing. Since the stress remains constant after some time the strain too remains fixed. The stress () strain () relation beyond the yield point ( 0 ) is given by the following expression: (2) Where k is a measure of the yield strength and n is known as strain hardening exponent. At low temperatures dislocations can only glide. In metals having high stacking fault energy there may be some amount strain softening because of cross slip. Therefore the extent of strain hardening is more in case of metals having low stacking fault energy such as copper whereas it is low in the case of metals like aluminum having high stacking fault energy. Alloy addition lowers stacking fault energy. Therefore alloys like alpha brass or austenitic steel (we would about it later) have very high strain hardening coefficients. 6 During plastic deformation new dislocations are generated. The subsequent glide motion encounters more dislocation dislocation interactions. This leads to the formation of dislocation locks and pile ups that act as obstacles to the further movement of dislocations. This is the reason for work hardening. At a higher temperature where vacancies are more mobile (self diffusion coefficient is high) the edge dislocations can could climb as well. The additional mobility helps the dislocations to climb over the obstacle and move over to a new slip plane where it may get annihilated by interaction with dislocations having opposite character or get
rearranged in the form of a more stable network having lower stored energy. This process is known as recovery. This is responsible for softening. Thus during deformation at relatively higher temperatures there are two concurrent processes occurring within the metal. These are strain hardening and recovery. At a given temperature the former depends only on the stress and it is independent of time whereas the latter being associated with diffusion depends on time. As a consequence after some time the process tends to reach a steady state when the increase in work hardening is balanced by the loss of strength due to recovery. When this happens the test piece continues to deform at a constant rate. Therefore the strain time plot may look like the one at 0.5T m in slide 4. If the temperature is raised further the deformed grains become unstable. They get replaced by a set of new strain free grains. The process is called re crystallization. It needs a certain amount of stored energy to initiate. This is why you get an oscillating strain time plot. The strain time plot at 0.9Tm in slide 4 shows such a behavior. Evolution of flow stress with time at constant temp & strain rate dot n k 0 t Slide 5 t T ~ 0.1Tm t T ~ 0.5 Tm t T ~ 0.9 Tm 7 Slide 5 shows how the flow stress evolves with time if a metal is deformed at a constant temperature under strain (or displacement) control mode. The sketch at the top shows strain rate as a function of time. After an initial increase in strain rate the metal is allowed to deform at a constant rate. The micro mechanism of deformation is still the same. The slide includes a set of 3 stress time plots at 3 different temperatures. At lower temperatures the deformation is dominated by strain hardening. This is reflected in the shape of the stress time plot at 0.1Tm in slide 5. The flow stress of the metal continues to rise with time. The stress strain relation can
be divided into elastic and plastic parts. 0 in the expression given in slide 5 is the yield stress. The subsequent part indicates the increase in yield stress with strain or work hardening. Look at the stress time plot at 0.5Tm. This is where the effect of thermally activated motion of dislocation shows up. The initial part is dominated by strain hardening. The flow stress continues to rise. However the rate of increase in flow stress keeps decreasing due to strain softening. When the increase in strength due to strain hardening is balanced by the decrease in strength due to recovery the flow stress attains a limiting value. This represents the steady state deformation stage. During this the flow stress remains constant. The initial part of the stress time plot at 0.9Tm is similar to that at 0.5Tm. It needs a certain amount of stored strain energy for re crystallization to start. Once it sets in the flow stress keeps oscillating around a mean stress. In short thermally activated dislocation glide, recovery by annihilation & rearrangement of dislocations, formation of sub grains, and re crystallization are the main mechanisms of hot working. The stacking fault energy plays an important on the evolution of a stable dislocation network during the process. Al & ferritic steel have high stacking fault energy. The dislocations can cross slip. The recovery or the rearrangement of dislocation is easier. The deformation process is also supported by thermal activation. Austenitic steel & Ni have low stacking fault. This makes cross slip difficult. The effect of strain hardening is more. With strain the stored energy increases until re crystallization sets in and new strain free grains are formed. The new grains too undergo deformation. This gives oscillating flow stress. It is also known as dynamic re crystallization. Recovery: 8
Recovery Slide 6 Formation of sub grain Dislocation annihilation Tilt boundary Stored elastic energy is minimized through dislocation annihilation & rearrangement 9 Slide 6 illustrates how thermal activation helps in the rearrangement of dislocation. The sketch on top left shows the direction of forces that acts on a dislocation due to the elastic stress field of the one located at the origin. The firm lines are the reference axes. The dotted lines are at 45 to the reference axes. If the location of the dislocation is below the dotted line as shown there will be a force along the vertical axis and repulsive force along the horizontal axis. The vertical force helps the dislocation climb up. Once it crosses the dotted line the nature of the force along the horizontal axis changes from repulsion to one of attraction. As a result it comes to occupy a position just above the dislocation at the centre. This happens to be a position of stable equilibrium. Even if it is displaced by a small distance there will be restoring force acting on it to bring it back. If there are several dislocations and the temperature is high enough to provide thermal activation to support the process of climb though diffusion of vacancies an array of dislocations similar to the one shown on the right in slide 6 would develop. The two sides of the array get tilted because of the stress field. The line separating the two is known as the tilt boundary. The formation of such an array is associated with reduction in elastic stored energy. Apart from this the climb may also help in annihilation of dislocations. This is schematically shown in the sketch at the bottom of slide 6. If a dislocation climbs to a plane where there is another dislocation of opposite character the force of attraction between the two would bring them closer leading to complete annihilation.
Re crystallization: Re-crystallization Original structure Deformed structure Nucleation of strain free grain 90 % of the work is dissipated as heat & 10% is stored as elastic energy during working. Cold worked structure is thermodynamically unstable. 4 3 2 G r Gv 4 r 3 G v < 0: it increases with cold work but > 0 Slide 7 10 The bulk of the work done on the metal during plastic deformation is dissipated as heat. However some of it remains within the metal in the form of stored elastic energy. This makes it thermodynamically unstable. The sketch on the left shows the initial structure of a homogeneous alloy. It consists of equi axed grains having random orientations. The sketch at the center shows its structure after plastic deformation. The grains are elongated. This is accompanied by a corresponding increase in the area of the boundary. The dislocation density within the grain also increases. These two account for the increase in the elastic stored energy. Let the stored energy per unit volume be equal to. This is less than zero. In thermodynamics negative energy is an indicator of instability. It acts as the driving force for transformation to a more stable state. Even though the cold worked state has enough driving force for transformation it could remain in this state indefinitely. This is because the formation of new strain free grains is associated with the formation of new grain boundaries. The elastic stored energy may not be enough to provide the same. It needs additional thermal activation to support it. Let the energy of the new boundary be equal to. This is greater than zero. If the radius of the new grains be r, the total energy of transformation is given by the following expression. 4 (3)
A reaction can occur spontaneously only if the total free energy (G) is negative. Equation 3 shows that it is a function of the size of the new strain free grains. It has a maximum at particular value of r. The slide 7 shows the steps involved in the derivation of the same. The maximum value of G is the magnitude of activation barrier which must be crossed for new grains to nucleate. Critical nucleus size & activation energy 4 3 2 G r Gv 4r 3 G 2 4r Gv 8r 0 r 3 * 2 * 16 r G G 3 G v v 2 G r Slide 7 G v stored energy As cold work increases activation hill decreases. Lower thermal activation is required for re-crystallization. Grain size is finer. Note that is a measure of stored energy. It increases with the increasing amount of cold work. It is always less than zero. is the magnitude of the activation barrier (or the height of the activation hill) which has to be overcome for re crystallization to occur. As cold work increases the size of the critical nucleus decreases, and the thermal activation needed for recrystallization too decreases. 11 The thermal activation for re crystallization is provided by keeping the metal at a higher temperature for a certain length of time. The process is called annealing. The transformation of the cold worked structure occurs by nucleation and growth. The nucleation rate depends on the activation barrier and temperature. It follows Boltzmann statistics. Slide 8 shows how it can be represented as a function of the activation barrier and temperature. Free energy is given by G = H TS where H = change in enthalpy and S = change in entropy. It has two parts. One of these is a function of temperature. The temperature independent part is called the activation energy of nucleation or the transformation. Slide 8 gives the expression for the nucleation rate. Note that it increases with increasing temperature and decreases with increasing activation energy. The slide 8 also gives a plot of fraction transformed (f) as a function of time. This
represents the extent of re crystallization or the fraction of the total number of elongated grains that has been replaced by new strain free grains. Like all processes controlled by nucleation and growth this has a typical S shape. At a given temperature the nucleation rate is constant. The contribution of growth to the transformation process increases with the increase in the number of stable nuclei. Therefore initially it is less (because the number of stable nuclei is less) and it keeps increasing with the increase in the number of stable nuclei until it reaches a peak. Thereafter the rate of growth deceases when some of the new strain free grains touch each other. This is often called the impingement factor. As a consequence the fraction transformed can be represented as follows: 1 (4) Where; k and n are constants. The kinetics of the process is best followed by monitoring the strength or the hardness (or any other physical properties) of the cold worked metal as a function of time at various temperatures. The process of restoring the properties of cold worked metal by recovery & re crystallization is known as annealing. Re-crystallization Nucleation & growth Gv H N exp exp RT RT G H TS Slide 8 f f n 1expkt t 12 Slide 9 gives a plot of the strength of a cold worked metal as function of annealing temperature if the hold times for each of the samples are identical. The hold time for annealing depends on the size of the sample. It may be of the order of an hour. Note the shape of the plot. It is a mirror image of the shape of the f versus t plot in slide 8. This indicates that the fraction transformed can be represented as the loss strength (). It may be represented as follows:
(4) Note that by using the equation 4 the strength versus temperature or time plots can be converted into fraction transformed versus temperature or time plots. Annealing strength Internal strain I II III GS Slide 9 Annealing temp Three stages of structural change during annealing: recovery, re-crystallization & grain growth Annealing: Slide 9 shows the effect of annealing temperature on the strength or hardness of cold worked metal as a function of temperature for a given hold time (say 1hr). Initially the rate of drop in strength is less, but it keeps increasing with the increase in temperature reaches a peak and thereafter it keeps decreasing. The plot can be divided into three stages. The stage I is characterized by very small change in strength. The stage II has the maximum drop in strength. During the stage III the strength is characterized by an ever decreasing rate of drop in strength. These three stages of annealing are more commonly known as recovery, re crystallization and grain growth. 13 Recovery stage is characterized by reduction of micro & macro residual stresses. Note that the sketch in slide 9 includes a plot of internal strain as a function of temperature. The nature of the residual stress is always elastic. The stress is directly proportional to the strain. If the temperature of the metal is increased its yield strength decreases. Therefore the residual stress gets relieved by local plastic deformation. There is also a reduction in the stored internal energy by rearrangement of dislocations. This is responsible for the formation of sub grains.
During stage II re crystallization takes place. The elongated grains are replaced by a new set of strain free grains. This is associated with the maximum drop in the strength of the metal. There is further reduction in the residual macro stresses. The properties of the metal are restored to their initial state. The stage III is characterized by grain growth. The sketch in slide 9 includes a plot of grain size (GS) as a function of temperature. Like every free surface there is an energy associate with the grain boundary. The total energy stored in the grain boundary is proportional to its area. The area of the grain boundary in per unit volume in a fine grain structure is more than that in a coarse grain structure. The reduction in the total grain boundary energy is therefore the main driving force for grain growth. Grain boundary: We are familiar with low angle grain boundary. It develops during the recovery stage of annealing. This is an array of edge dislocations arranged one over the other. The distance between the two dislocations gives a measure of the angle of mis orientation between the two sub grains. The energy of such a boundary depends on this angle. This is usually less than 5. The orientation of grains on the two sides of the boundary is random. There is no relation between the orientations of the two. Such boundaries are known as high angle boundaries. Their energies are higher than that of a low angle grain boundary. Slide 10 shows the sketch of a microstructure of an annealed polycrystalline metal. In 2D grain boundaries are represented as lines. Grain boundary energy is expressed as energy per unit area (J/m 2 ). It can also be visualized as line tension having the dimension of force / unit length (N/m). The points at which more than two grains meet may be considered as a node. Look at the case where three grains meet at a point. The grain boundary energies are denoted as 12, 23, and 31. They also denote line tension acting along the 3 directions. In order to maintain equilibrium the ratio of the line tension and the sin of the opposite angles should be equal. The relation is shown in slide 10. 14
Stability of grain boundary Grain 3 Grain 2 Grain 1 Slide 10 23 31 12 sin sin sin 1 2 3 If the three line tensions are equal then the angles between these must be equal to 120. However energy of every boundary may not be the same. Nevertheless we do find the angles at triple points (where 3 boundaries meet) are around this. Is there a way we could estimate the energy of a grain boundary? 15
Estimation of GB energy vapor Solid sv sv Solid Slide 11 ss 2 svcos 2 ss Origin of surface energy lies in the number of free bonds / atom Bond energy is proportional to latent heat of vaporization Slide 11 illustrates a method of estimating the grain boundary energy. The origin of surface energy lies in existence of free bonds at the surface. Look at the arrangement of atoms in a 2D lattice shown in slide 11. Atoms are denoted by filled circles. The lines are the bonds. We may visualize that a free surface is created by removing or evaporating a layer of atoms. Therefore the energy of free surface denoted as sv can be estimated from the latent of evaporation. Take a piece of metal having a polished surface. Heat it in vacuum for some time so that some of atoms from the top face evaporate. This is known as heat etching. Atoms near the grain boundaries are in a relatively more excited state than those of within the grains. They would evaporate more easily leaving behind grooves along the boundaries. This is shown in the form of a sketch at top left in slide 11. The angle of the groove is a function of the grain boundary energy. The sketch in slide 11 shows the 3 forces acting at the bottom of the groove. An expression for the grain boundary energy ( ss ) is obtained by equating the vertical components of the forces. This is given in slide 11. 16 The grain boundaries in the microstructure given in slide 10 are shown as straight lines. What would happen if grain boundaries were curved? This is illustrated in slide 12. Look at the sketch given on the top left of slide 12. Consider a small segment of the grain boundary subtending an angle at the centre as shown. The grain boundary line tension is denoted as. Note that the horizontal components cancel each other. However the vertical components are along the same direction. Therefore there is a certain amount of force acting on the grain boundary. It is directed towards the centre of curvature of the line representing the grain boundary. The derivation is given in slide 12. It shows that the force is inversely proportional to the radius of
curvature of the boundary. The force approaches zero or negligible if grain boundaries are straight. It suggests curved grain boundaries are unstable. Note that l is the length of a small segment of the grain boundary and r is its radius of curvature. This explains why during annealing grains tend to grow until the boundaries are straight. Grain growth l F 2 sin 2 r D D F=0 if r approaches 2 2 infinity D D Kt 0 Slide Having shown that curved boundaries are unstable it is possible to guess how grains would grow during annealing. Let the average grain diameter be denoted as and the rate of change of grain diameter is. Since the main driving force is inversely proportional to the radius of curvature (which is half of the grain diameter) we may assume that the rate of change of grain diameter is also inversely proportional to its diameter. Therefore grain diameter at any time is given by the following expression: (5) Where; k is a rate constant. It is a function of grain boundary energy. Re crystallization: effect of variables 17 Re crystallization takes place during annealing of cold worked structure. It occurs above a critical temperature called re crystallization temperature. It takes place by nucleation and growth. Therefore the extent of re crystallization depends on both time and temperature. If we increase annealing time re crystallization may occur at a lower temperature. However it needs a minimum thermal activation. On an average it is above 0.3T m where T m is the melting
temperature K. The amount of cold work also has a profound effect on the kinetics of recrystallization. The size of critical nucleus is inversely proportional to the amount of cold work. Higher the amount of cold work smaller is the size of critical nucleus. This is why higher cold work gives fine grain structure after annealing. The re crystallization temperature too depends on the amount of cold work. The expression derived readily shows that the activation hill decreases with increasing cold work. Therefore re crystallization would occur at a lower temperature if the amount of cold work is high. Apart from these the kinetics of recrystallization is also affected by the initial structure of the metal and the presence of alloying elements or second phase. In fact the presence of second phase particle can significantly raise or suppress re crystallization and grain growth. They may pin (block) the boundaries and thus resist grain growth. It is exploited in making extremely fine grain steel by controlled thermomechanical processing. We shall talk about it in detail in one of the subsequent modules. Origin of texture: During the deformation of single crystal, the tensile axis tends to rotate towards its slip direction. If the stress is compressive the loading axis moves towards its slip plane normal. In polycrystalline material grains cannot rotate freely. The presence of neighboring grains is a major constraint. However strain compatibility at every point has to be maintained. The net strain is the same in every grain. This is possible only if the crystals have 5 independent slip systems. Most common metals are either FCC or BCC. Both of these have sufficient numbers of independent slip system to choose from. Therefore they have excellent ductility. The operative slip system in a neighboring grain may be different. During deformation the orientation of the grains would keep changing. It may result in lattice bending and fragmentation. This results in cold work texture. For example during wire drawing all grains tend to have a common crystal direction aligned along the axis of the wire. Likewise in a cold rolled plate a particular plane in each grain tend to be parallel to the plane of deformation. Annealing texture develops if cold work is more than a critical level. Presence of inclusions on which new grains nucleate often determines the annealing texture. Aluminum killed steel has AlN as inclusions. This promotes a texture which exhibits good deep drawing property. Summary: 18 In this lecture we looked at the effect of cold work and annealing on the evolution of structures in metal. The structural change that occurs during annealing can be classified into three stages. Stage I is the recovery stage where micro & macros stresses get relieved. Re crystallization takes place during stage II. The deformed grains are replaced by new strain free grains. Usually they have random orientations. Even after the process is complete the grains continue to grow during annealing. The main driving force is the grain boundary energy. The curved grain
boundaries are unstable. During the third stage of annealing grain boundary tends to become straight and its size keeps growing. During cold work the grains tend to get reoriented. This leaves behind a cold work texture. The texture disappears on annealing. However If the amount of cold work is beyond a critical amount, it is difficult to remove texture completely. Annealing in the presence of second phase precipitates may develop a special texture called annealing texture. The presence of texture makes the material anisotropic. It may be beneficial for certain types of applications. Exercise: 1. While designing rolling process discuss whether large reduction in a single pass is prefrerable to smaller reductions in successive passes? 2. The following figure gives a typical microstructure of a pure metal. Magnification is 100X. Find out its grain size in terms of ASTM number (N) which is given by: 2. Where n represents the number of grains per unit area 4 cm 4 cm 3. Count the number of grains (F), number of edges (E) and number of points (P) in the microstructure given above. State the type of relationship these must follow. 4. Describe how by cold work & annealing the following microstuctures can be developed in a single phase alloy (a) fine grain structure (b) coarse grain structure 5. Suggest at least two methods to know if a sheet of steel has preferred orientation. Answer: 19 1. During rolling contact length with roll surface is a function of reduction per pass. Let this be l and plate thichness be t. The ratio = l/t gives an idea about the deformation zone. If >> 1 deformation will be inhomogeneous. This would result in residual stresses and may lead to cracking. On the other hand if D << 1 there will be too much of friction between work piece and the roll. For a given plate thickness the former represents large deformation where as the latter represents small deformation. The optimum rolling schedule is some where in between.
2. Note that a large number indicates fine grain structure where as a small number denotes coarse grain structure. The above stucture appears to be coarse. Measurement shows following result. Count the number of grains assuming contribution of corner grain as ¼ & grains at edge as ½. Thus n= (6+3/4+9/2)/A = 11.25/A A = 16/2.54/2.54 = 2.48 in 2 n = 2 = 11.25/2.48 = 4.54 N = 3.2 3. Number of grains F = 18, E = 39 & P = 22: It follows the relation P+F = E+1. This is a 2 D form of Euler rule in Topology (There is a striking similarity with phase rule). If you count the number of corners, faces & edges of the following 3D shapes you get the Euler s rule (a) Tetrahedron (b) Cube (c) Octahedron P = 4, F = 4, E=6 P+F = E + 2 P = 8, F =6, E=12 P+F = E + 2 P =6, F =8, E=12 P+F = E + 2 4. Critical neuclus is inversely proportional to amount of cold work. If before annealing if the metal is given small amount of cold wok and then annealed at a relatively higher temperature it is likely to have coarse grain structure. On the other hand high amount of cold followed by annealing just above its recrystallisation temperature would give fine grain structure. 5. Preferred orientation would give different properties along different direction. Measurement of elastic modulus along diffent direction is a common method to determine anisotropy. It can also be established by direct measurement of crystal orientation by X Ray diffraction technique or SEM using EBSD (Electron Back Scattered Diffraction). 20