UNIT V PLASTIC DEFORMATION AND FRACTURE MECHANICS DEFORMATION OF METALS Introduction  When force is applied on a metal piece, then the size and/or shape will be altered. Any change in the size and/or shape of the metal is called as deformation of the metal.  Deformation is caused either by the mechanical action of external forces or by various physical and physiochemical processes. For example, changes in volume of separate crystallites in phase transformations are as a result of temperature gradient.  The deformation can be either permanent or temporary. Permanent deformation remains after the removal of the deforming force or applied load, while temporary deformation disappears on removal of the load.  The knowledge of the mechanical deformation of metals is essential to understand and obtain the desired shape/size of the end products through the various mechanical operations such as rolling, forging, drawing, etc. Classification of Metal Deformation Based on the nature of strain produced during deformation, the deformation of metals can be classified into ; 1. 1. Elastic deformation, and 2. Plastic deformation. Elastic Deformation  Elastic deformation is the deformation of a body which completely disappears as soon as the external load is removed from the body.  In other words the deformation which is fully recoverable and virtually time independent is called elastic deformation.  This deformation type obeys Hook’s law.  Besides, there are elastomeric deformations and anelastic deformations which exhibit somewhat elastic deformations but do not obey the Hook’s law. 2. Plastic Deformation  Plastic deformation is the deformation of a body which remains even after removing the external load from the body.  The plastic deformation in crystalline materials occurs at temperatures lower than 0.4 Tm, where Tm is the melting temperature of the material.  In contrast to elastic deformation, which depends primarily on stress in the simplest cases, plastic deformation is a function of stress, temperature, and the rate of straining.  The plastic deformation may occur under the tensile, compressive or torsional stresses.  This property of plastic deformation makes the metal suitable for various forming processes such as rolling, forging, pressing, drawing, spinning, extrusion and stamping. Elastic Vs Plastic Deformations The comparison between elastic and plastic deformation is presented in Table3.4. Table 3.4 Elastic Vs Plastic Deformation S.No. Elastic deformation Plastic deformation 1. It is the deformation of a body which It is the deformation of a body which completely disappears as soon as the remains even after removing the external load is removed from the body. external load from the body. 2. It obeys Hook’s law 3. The elastic deformation is beginning of the progress deformation 4. In elastic deformation, the strain In plastic deformation, the strain occurs reaches its maximum value after the simultaneously with the application of stress has reached its maximum value stress. It does not obey Hook’s law the The plastic deformation takes place of after the elastic deformation has stopped. MECHANISM OF PLASTIC DEFORMATION (Mode of Plastic Deformation) The two modes of plastic deformation are : 1. Slip, and 2. Twinning Deformation by Slip  Slip may be defined as the sliding of blocks of the crystal over one another along definite crystallographic planes called slip planes.  In other words slip represents a displacement of one part of the crystal relative to another along particular crystallographic planes and in certain crystallographic directions. The particular crystallographic planes are called slip planes and the preferable direction is called the slip direction, as shown in Fig. 3.1  There are one or more slip planes and one or more slip directions in each crystal. Fig. 3.1 Slip in a single crystal  The combination of a slip plane and slip direction is known as slip system. Fig.3.2 illustrates a slip system for FCC lattice.  The concept of slip can be considered analogous to the distortion produced in a deck of cards when its is pushed from one end. Fig. 3.2 A slip plane for crystal Mechanism of Slip  The slip occurs by translatory motion along sliding planes and rotation of the specimen.  Fig. 3.3 illustrates the mechanism of slip.  Fig. 3.3 (a) depicts the layers of planes of an ideal crystal. Let us consider the shear stress acting on the crystal, as indicated by the arrows.  When the shear stress exceeds the critical value, then the slip occurs, as shown in Fig. 3.3. (b) Fig. 3.3 Mechanism of slip  It is experimentally proved that the mechanism of slip is actually due to the movement of dislocations in the crystal lattice.  The energy of a dislocation line is given by the relation E  G. l. b2 …. (1) Where G = Shear modulus, l = Length of dislocation line, and b = Unit slip vector also known as Burgers vector.  Thus it can be noted form the above relation that the dislocations having the shortest slip vector (b) are the easiest dislocations to generate and to expand for plastic deformation. Critical Resolved Shear Stress for Slip  The extent of slip in a single crystal depends on the magnitude of the shearing stress produced by external loads, the geometry of the crystal structure, and the orientation of the active slip planes with respect to the shearing stresses.  Slip begins when the shearing stress on the slip plane in the slip direction reaches a threshold value called the critical resolved shear stress. Calculation of critical resolved shear stress Consider a cylindrical single crystal subjected to an axial load as shown in Fig.3.4 Let P = Axial load applied along the crystal axis, A = Area of cross section of the crystal  = Angle between the normal to the slip plane and the tensile axis, and  = Angle which the slip direction makes with the tensile axis Fig. 3.4 The area of the slip plane inclined at the angle  =  A cos The component of the axial load acting in the slip plane in the slip direction = P cos    Critical resolved shear stress = Or  Load Area cr = P cos   P cos  cos  A A cos cr =  cos  cos  Where  = Applied tensile stress = ….. (2) P A  The stress required at a given temperature to initiate slip in a pure and perfect single crystal, for a material is constant. This is known as Schmid’s law.  In equation (2), the term cos  cos  is known as the Schmid’s factor. Deformation by Twinning  The second important mechanism by which metals deform is the process known as twinning.  Twinning is the process in which the atoms in a part of a crystal subjected to stress, rearrange themselves so that one part of the crystal becomes a major image of the other part.  Like slip, twinning almost takes place in special planes called twinning planes.  In most plastic deformation, twinning is relatively insignificant, but it may have considerable influence on the total amount of deformation.  It should be noted that twinning differs from slip in that every plane of atoms suffers some movement, and the crystallographic orientations of many unit cells are altered. Mechanism of Twinning  Fig. 3.6 illustrates the mechanism of twinning.  Fig. 3.6. (a) depicts the arrangement of atoms before twinning.  Fig. 3.6. (b) depicts the arrangement of atoms after twinning. In fig. 3.6. (b), the planes AB and CD are known as twinning planes.  The crystals twin about the twinning planes. The twinning occurs due to the growth and movement of dislocations in the crystal lattice.  It can be noted from the Fig. 3.6. (b) that the arrangement of atoms on either side of the twinning planes remain unaffected. Fig. 3.5 Mechanism of Twinning Types of Twins (Causes of Twins) Two types of twins are : 1. Mechanical twins :  Twins that are produced by mechanical deformation are called mechanical twins.  Mechanical twins are produced in BCC or HCP metals, under the conditions of decreased temperature and shock loading. 2. Annealing twins  Twins that are produced by annealing are called annealing twins.  Most of FCC metals form annealing twins. Slip Vs Twinning The differences between slip and twinning are presented in Table 3.6. Table 3.6 Slip Vs Twinning S.No. Slip Twinning 1. In slipping, the deformation takes place In twinning, the deformation is due to due to the sliding of atomic planes over orientation of one part of the crystal the others. with respect to the other. 2. It occurs along individual slip planes It occurs over general crystallographic planes 3. The atomic movements are over large distances The atomic movements are over a fraction of atomic spacing 4. There is no change in the orientation of the atoms after slip has occurred. Twinned atoms undergo a change in Slip takes place when shear stress reaches resolved critical shear stress There is no role for resolved critical shear stress 5. their orientation and become mirror of the untwined atom. COLD AND HOT WORKING OF METALS Cold Working: Plastic deformation of metals below the recrystallization temperature is known as cold working. It is generally performed at room temperature. In some cases, slightly elevated temperatures may be used to provide increased ductility and reduced strength. Cold working offers a number of distinct advantages, and for this reason various cold-working processes have become extremely important. Significant advances in recent years have extended the use of cold forming, and the trend appears likely to continue. In comparison with hot working, the advantages of cold working are  No heating is required  Bettter surface finish is obtained  Better dimensional control is achieved; therefore no secondary machining is generally needed.  Products possess better reproducibility and interchangeablity.  Better strength, fatigue, and wear properties of material.  Directional properties can be imparted.  Contamination problems are almost negligible. Some disadvantages associated with cold-working processes are:        Higher forces are required for deformation. Heavier and more powerful equipment is required. Less ductility is available. Metal surfaces must be clean and scale-free. Strain hardening occurs ( may require intermediate annealing ). Undesirable residual stresses may be produced Cold forming processes, in general, are better suited to large-scale production of parts because of the cost of the required equipment and tooling. Hot Working: Plastic deformation of metal carried out at temperature above the recrystallization temperature, is called hot working. Under the action of heat and force, when the atoms of metal reach a certain higher energy level, the new crystals start forming. This is called recrystallization. When this happens, the old grain structure deformed by previously carried out mechanical working no longer exist, instead new crystals which are strain-free are formed. In hot working, the temperature at which the working is completed is critical since any extra heat left in the material after working will promote grain growth, leading to poor mechanical properties of material. In comparison with cold working, the advantages of hot working are       No strain hardening Lesser forces are required for deformation Greater ductility of material is available, and therefore more deformation is possible. Favorable grain size is obtained leading to better mechanical properties of material Equipment of lesser power is needed No residual stresses in the material. Some disadvantages associated in the hot-working of metals are:       Heat energy is needed Poor surface finish of material due to scaling of surface Poor accuracy and dimensional control of parts Poor reproducibility and interchangeability of parts Handling and maintaining of hot metal is difficult and troublesome Lower life of tooling and equipment. FRACTURE AND ITS PREVENTION FRACTURE AND ITS PREVENTION What is Meant by Fracture?  Fracture is the mechanical failure of the material which will produce the separation or fragmentation of a solid into two or more parts under the action of stresses.  The understanding of various phenomenons of fracture is necessary to minimize and prevent the fracture. Cause of Fracture The fracture is caused due to the presence of submicroscopic defects known as cracks. Then the initial crack is propagated to cause complete failure. Types of Fractures The four important types of fractures are : 1. Brittle fracture 2. Ductile fracture 3. Fatigue fracture, and 4. Creep fracture. Now we shall discuss the above types, in detail, in the following sections. BRITTLE FRACTURE What is Meant by Brittle Fracture?  Brittle fracture defined : A brittle fracture may be defined as a fracture which takes place by the rapid propagation of crack with a negligible deformation.  It may be noted that in amorphous materials such as glass, the fracture is completely brittle whereas in crystalline materials, the fracture occurs after a small deformation.  In crystalline materials, the fracture takes place normal to the specific crystallographic planes, called cleavage planes.  In polycrystalline materials, the fracture takes place along the grain boundaries.  The tendency of the brittle fracture is increased with decreasing temperature and increasing strain rate. Mechanism of Brittle Fracture : Griffith’s Theory (Griffith Crack Theory) Introduction It is provided that the stress at which a material fractures is far below the lower value of the ideal breaking strength calculated from the atomic strength. In other words, the fracture strength of real materials is far lower than (about 10 4 MPa) the theoretical minimum value for an ideal solid. According to Griffith, the discrepancy between the strengths of real and ideal materials is due to many fine cracks which act to concentrate the stress at their tips or ends. Griffith Theory Griffith’s postulation is as follows : 1. In a brittle material, there are many fine cracks. These cracks concentrate the applied stress at their tips or ends. 2. When the stress at the tips of a crack exceeds the theoretical stress values, the crack expands and fracture occurs. Explanation of Mechanism of Brittle Fracture Let us consider a crack of elliptical cross-section in a rectangular specimen as shown in Fig.3.6. Fig. 3.6 Model for the Griffith crack theory Let  = Tensile stress applied to the specimen m = Maximum stress at the tip of the crack c = Half length of the crack, and  = Radius of curvature at the ends of the ellipse. It is observed that when a tensile stress is applied to the specimen, then the applied stress is distributed about the crack in such a way that the maximum stress occurs at its tips. The maximum stress s(m) at the tip of the crack is given by c m = 2    It is understood that when an elastic material is stressed, potential energy is stored in the material before crack occurs. This stored energy is known as elastic strain energy. When a crack begins propagating, elastic energy is released. It is also understood that as the crack propagates, new surfaces are created and a certain amount of energy, called surface energy, must be provided to create them. Derivation for Fracture Strength Griffith supposed that the crack propagates when the released strain energy is just sufficient to provide the surface energy necessary for the creation of the new surface. According to elastic theory, 2 The strain energy per unit volume =  2E E = Young’s modulus of the material. Where Then the elastic strain energy released by the spreading of a crack of unit width is given by, 2 2 UE =   Area  Width =   c2 1 2E 2E =  2c 2 …. (i) 2E A more detailed calculation shows that the strain-free volume is larger and the elastic energy released will be twice the value given in equation (i). Then equation (i) becomes, UE =   2c 2 E … (ii) The negative sign indicates that the elastic energy stored in the material is released as the crack forms. If  = Surface energy per unit area in joules /m2, then the surface energy (Us) for a crack of length 2 c with unit width is given by Us = 4  c …. (iii) Since there are two surfaces, the equation (iii) is multiplied by two. The total change in potential energy, resulting from the creation of the crack, is given by,   2c 2 U = UE + US = +4c … (iv) E According to Griffith, the crack will propagate and produce brittle fracture when an incremental increase in its length does not change the total energy of the system. Mathematically, dU 0 dc  d    2c 2     4c  = 0 dc dc  E  dU  Or 2 2c 2 E Or +4c=0  2E c … (3) The above equation is called Griffith’s fracture equation. Observations From equation (3) the following observations can be equation. (i) The stress necessary to cause brittle fracture varies inversely as the square root of the crack length. (ii) The critical tensile stress required to propagate the crack spontaneously is smaller than the Young’s modulus of the material. (iii) The Griffith theory is valid only for a perfect brittle material like glass. DUCTILE FRACTURE What is Meant by Ductile Fracture?  Ductile fracture defined : Ductile fracture may be defined as the fracture which takes place by a slow propagation of crack with appreciable plastic deformation.  When a ductile specimen is subjected to tensile stress the stress-strain curve can be obtained as shown in Fig.3.7.  In Fig. 3.7 at point F the fracture takes place. Fig. 3.7 Ductile fracture Mechanism of Ductile fracture  The various stages in the development of a ductile fracture, also called cup-and-cone fracture is shown in Fig.3.8.  Fig. 3.8 (a) indicates the formation of a neck, when a ductile specimen is subjected to tensile stress. When the tensile stress is increased beyond the ultimate tensile stress (i.e. at point E in Fig.3.7) a neck is formed in the specimen.  The continuation of the plastic deformation produces many fine cavities in the specimen, as shown in Fig.3.8 (b).  Under continued straining, these cavities grow and form a central crack, as shown in Fig. 3.8 (c).  The crack grows in a direction perpendicular to the axis of the specimen until it approaches the surface of the specimen. It then propagates to the surface of the specimen to form the cone part of the fracture, as shown in Fig. 3.8 (d)  The central ‘cup’ region of the fracture has a very fibrous appearance, as shown in Fig.3.8. Fig. 3.8 Stages in the formation of a cup-and-cone fracture Observations The following observations can be made during the ductile fracture : 1. A ductile fracture occurs by a slow tearing of the metal with the expenditure of considerable energy. 2. Unlike brittle fracture, a ductile fracture presents a rough dirty-grey surface. Brittle Fracture Vs Ductile Fracture The comparison between brittle and ductile fractures is presented in Table 3.6. Table 3.6 comparison between brittle and ductile fractures S.No. 1. Brittle fracture It occurs with negligible Ductile fracture plastic It occurs with large plastic deformation. deformation 2. It occurs at the point where micro crack It occurs in some localized region is more where the deformation is very large. 3. The rate of crack propagation is rapid The rate of crack propagation is slow 4. Failure is due to the direct stress Failure is due to the shear stress 5. It is characterized by separation of It is characterized by the formation of normal to tensile stress cup and cone Fractured surface shows a s harp planar Fractured surface is a rough dirty-grey facet contour 6. 7. The brittle fracture can be increased by The ductile fracture can be increased by decreasing temperature, increasing dislocations and other defects in metals. strain rate and work hardening Prevention of Ductile Fracture In order to prevent the ductile fracture, the material should have the following characteristics :  The material should have fine grains.  It should have higher hardness value.  It should have higher Young’s modulus and cohesive energy.  It should not have any defects/dislocations. FATIGUE FRACTURE What is Meant by Fatigue Fracture?  Fatigue fracture defined : The fatigue fracture is defined as the fracture which takes place under repeatedly applied fatigue stresses.  In other words, the behaviour of the materials subjected to fluctuating or repeated loads is called fatigue.  Three main features that distinguishes the fatigue fracture are : o Loss of Strength o Loss of ductility and o Increased uncertainty in strength and service life.  The fatigue fracture occurs at stresses well below the tensile stress of the materials.  Fatigue must be considered in machine components subjected to continuously cyclic loading such as motor shafts, bolts, springs, gear teeth, valves, turbine blades, airplanes, automobile and gas-engine parts, suspension bridges, wire ropes and many others.  Since 80 to 90% of the total failures of high-speed machine parts is due to fatigue, therefore the study of fatigue failure is very important. Stress Cycles Fig. 3.9 Illustrates the different arrangements of fatigue loadings.  Fig. 3.9 (a) shows the arrangements of fatigue loadings.  Fig. 3.9 (b) shows the arrangement of the fluctuating stress.  Fig. 3.9 © shows the arrangement of the irregular stress. Fig. 3.9 Stress cycles for fatigue loading Mechanism of Fatigue Fracture  Fatigue fracture begins at irregularities on the surface imperfections such as machine marking and slip steps. The nucleation of microcrack is due to the slip movements. The slip movements starts within few cycles of loading. The microcracks act as the points of stress concentration.  As the cycles of loading continue, the microcrack propagates and grows in its size. In brittle materials, the crack grows to a critical size very fastly. But in duct ile materials, the crack keeps growing until the remaining area cannot support the load. So the fatigue fracture in ductile materials occur suddenly.  The high temperature increases the mobility of atoms, facilitating slip and hence the fatigue fracture. S-N Diagram  The S-N diagram can be obtained by plotting the number of cycles of stress reversals (N) required to cause fracture against the applied stress level (S), as shown in Fig.3.10. Fig. 3.10 The S-N curves for different materials  It can be seen from the Fig.3.10 that the fatigue strength is more for steels than for nonferrous metals (such as aluminium) and their alloys.  Fatigue stress (or fatigue strength) : The stress at which a metal fails by fatigue is termed as fatigue strength.  Fatigue limit (or endurance limit) : It is defined as the value of stress below which the material will not fail even when it is loaded for infinite number of cycles.  Fatigue life : It is the total number of cycles required to bring about final fracture under a given conditions of use. Factors Affecting Fatigue Strength 1. Fatigue strength is influenced by many factors such as chemical composition, grain size, and amount of cold working. 2. Fatigue strength is high at low temperatures and gradually decreases with rise in temperature. 3. Environmental effects such as corrosion of the product by moisture decreases the fatigue strength. 4. The design of the product also influences the fatigue strength. Because changes in shape/size may change the stress gradients and residual stresses of the materials. Prevention of Fatigue Failure The following methods can be adopted to prevent the fatigue failure. 1. Use of good design to avoid stress concentration by eliminating sharp recesses and severe stress raisers. 2. Control of the surface finish by avoiding damage to surface machining, punching, stamping, shearing, etc. 3. Reduction of corrosion environmental effects by surface heat treatments like polishing, coatings, carburizing, nitriding, etc. 4. The material should have fine grain structure and also it should be free from residual stresses and dislocations. CREEP What is Meant by Creep?  Creep defined : The creep is defined as the property of a material by virtue of which it deforms continuously under a steady load.  In other words, creep can be defined as the permanent deformation of a material under a steady load as a function of time, usually at higher temperatures.  The property of creep is important in : o The soft metals used at about room temperature such as lead coverings on telephone cables and white metal bearings. o The metals used at high temperature such as furnace parts, turbine blades, pressure vessel parts, rocket and missiles, supersonic jets, etc.  When a material is subjected to a constant loading, then the time-dependent strain occurring under the constant stress is known as creep.  Some materials such as zinc, lead and tin creep more at room temperature. Some other materials such as iron, nickel, and copper creep more at elevated temperatures only. Creep Curve A creep curve shows the variation of the extension of a metal with time under different stresses. A typical creep curve under constant nominal stress and constant temperature, is shown in Fig.3.11. Fig. 3.11 Typical creep curve for a long time, and high temperature creep test Different Stages in Creep Curve The three distinct stages of a creep curve (Fig. 3.11) are : 1. Primary Creep  In this stage, the creep is mainly due to dislocation movement.  During this stage, the recovery effect is less than the workhardening effect. Hence the creep rate decreases logarithmically. 2. Secondary Creep  During this stage, the rates of work hardening and recovery are equal, so the material creeps at steady rate.  For the above reason, secondary creep is usually termed as steady-state creep.  Steady-state creep may be viscous or plastic in character, depending upon the state level and temperature.  It is the important part of the creep curve which is used to estimate the service life of the alloy. 3. Tertiary Creep  In this stage, creep rate increases with time until fracture occurs.  Generally the tertiary creep occurs due to necking of the specimen or grain boundary sliding. Important Creep Related Properties 1. Creep Strength : It is the constant nominal stress that will cause a specified creep extension in a given time at a constant temperature. 2. Creep limit : It is defined as the maximum static stress that will result in creep at a rate lower than some assigned rate at a given temperature. 3. Creep life : It is the time required for occurrence of creep fracture under a given static load. 4. Creep resistance : It is the resistance offered by the material for its continuous deformation under steady load. Factors Affecting Creep 1. Grain size : Grain size influences the creep resistance. Generally, coarse – grained materials exhibit better creep resistance than fine-grained ones. 2. Thermal stability of the micro-structure : The material having greater thermal stabilities possess greater creep resistance than the material with poor thermal stability. 3. Chemical reactions : Chemical reactions in materials seriously affect the creep rate. 4. Prior strain : Prior cold working or work-hardening has also a strong effect on creep. It increases the creep resistance. Mechanism of Creep Fracture The creep fracture is due t o any of the following three mechanisms : 1. 1. Dislocation climb, 2. Vacancy diffusion, and 3. Grain boundary sliding Dislocation climb : At high temperatures, atomic movements permit the dislocation to climb. Because of this dislocation climb, the diffusion rate of vacancies may produce a motion in response to the applied stress. The mechanism of dislocation climb is illustrated in Fig. 3.12 (a).s Fig. 3.12 Mechanism of creep 2. Vacancy diffusion : In this mechanism, the diffusion of vacancies controls the creep rate but does not involve the climb of edge dislocations. Vacancies move in response to the applied stress in the directions shown in Fig.3.12 (b). This movement finally will result in creep fracture. 3. Grain boundary sliding : At lower temperatures, the creep fracture takes place due to sliding of grain boundaries, as shown in Fig.3.12 ©. Prevention of Creep Fracture The following methods can be adopted to prevent the creep failure. 1. Use of coarse grained materials will avoid creep fracture. 2. Strain hardening can be done to avoid creep fracture. 3. The material should be free from any residual stresses and dislocations. 4. Precipitation – hardened alloys can be used to avoid creep fracture. 5. Heat treatment reduces the occurrence of creep fracture.