Page 1 Material Science Prof. Satish V. Kailas Associate Professor Dept. of Mechanical Engineering, Indian Institute of Science, Bangalore – 560012 India Chapter 4. Mechanical Properties of Metals Most of the materials used in engineering are metallic in nature. The prime reason simply is the versatile nature of their properties those spread over a very broad range compared with other kinds of materials. Many engineering materials are subjected to forces both during processing/fabrication and in service. When a force is applied on a solid material, it may result in translation, rotation, or deformation of that material. Aspects of material translation and rotation are dealt by engineering dynamics. We restrict ourselves here to the subject of material deformation under forces. Deformation constitutes both change in shape, distortion, and change in size/volume, dilatation. Solid material are defined such that change in their volume under applied forces in very small, thus deformation is used as synonymous to distortion. The ability of material to with stand the applied force without any deformation is expressed in two ways, i.e. strength and hardness. Strength is defined in many ways as per the design requirements, while the hardness may be defined as resistance to indentation of scratch. Material deformation can be permanent or temporary. Permanent deformation is irreversible i.e. stays even after removal of the applied forces, while the temporary deformation disappears after removal of the applied forces i.e. the deformation is recoverable. Both kinds of deformation can be function of time, or independent of time. Temporary deformation is called elastic deformation, while the permanent deformation is called plastic deformation. Time dependent recoverable deformation under load is called anelastic deformation, while the characteristic recovery of temporary deformation after removal of load as a function of time is called elastic aftereffect. Time dependent i.e. progressive permanent deformation under constant load/stress is called creep. For visco- elastic materials, both recoverable and permanent deformations occur together which are time dependent. When a material is subjected to applied forces, first the material experiences elastic deformation followed by plastic deformation. Extent of elastic- and plastic- deformations will primarily depend on the kind of material, rate of load application, ambient temperature, among other factors. Change over from elastic state to plastic state is characterized by the yield strength ( s 0 ) of the material. Forces applied act on a surface of the material, and thus the force intensity, force per unit area, is used in analysis. Analogous to this, deformation is characterized by percentage Page 2 Material Science Prof. Satish V. Kailas Associate Professor Dept. of Mechanical Engineering, Indian Institute of Science, Bangalore – 560012 India Chapter 4. Mechanical Properties of Metals Most of the materials used in engineering are metallic in nature. The prime reason simply is the versatile nature of their properties those spread over a very broad range compared with other kinds of materials. Many engineering materials are subjected to forces both during processing/fabrication and in service. When a force is applied on a solid material, it may result in translation, rotation, or deformation of that material. Aspects of material translation and rotation are dealt by engineering dynamics. We restrict ourselves here to the subject of material deformation under forces. Deformation constitutes both change in shape, distortion, and change in size/volume, dilatation. Solid material are defined such that change in their volume under applied forces in very small, thus deformation is used as synonymous to distortion. The ability of material to with stand the applied force without any deformation is expressed in two ways, i.e. strength and hardness. Strength is defined in many ways as per the design requirements, while the hardness may be defined as resistance to indentation of scratch. Material deformation can be permanent or temporary. Permanent deformation is irreversible i.e. stays even after removal of the applied forces, while the temporary deformation disappears after removal of the applied forces i.e. the deformation is recoverable. Both kinds of deformation can be function of time, or independent of time. Temporary deformation is called elastic deformation, while the permanent deformation is called plastic deformation. Time dependent recoverable deformation under load is called anelastic deformation, while the characteristic recovery of temporary deformation after removal of load as a function of time is called elastic aftereffect. Time dependent i.e. progressive permanent deformation under constant load/stress is called creep. For visco- elastic materials, both recoverable and permanent deformations occur together which are time dependent. When a material is subjected to applied forces, first the material experiences elastic deformation followed by plastic deformation. Extent of elastic- and plastic- deformations will primarily depend on the kind of material, rate of load application, ambient temperature, among other factors. Change over from elastic state to plastic state is characterized by the yield strength ( s 0 ) of the material. Forces applied act on a surface of the material, and thus the force intensity, force per unit area, is used in analysis. Analogous to this, deformation is characterized by percentage change in length per unit length in three distinct directions. Force intensity is also called engineering stress (or simply stress, s), is given by force divided by area on which the force is acting. Engineering strain (or simply strain, e) is given by change in length divided by original length. Engineering strain actually indicates an average change in length in a particular direction. According to definition, s and e are given as 0 0 0 , L L L e A P s - = = where P is the load applied over area A, and as a consequence of it material attains the final length L from its original length of L 0 . Because material dimensions changes under application of the load continuously, engineering stress and strain values are not the true indication of material deformation characteristics. Thus the need for measures of stress and strain based on instantaneous dimensions arises. Ludwik first proposed the concept of, and defined the true strain or natural strain ( e) as follows: ? + - + - + - = ... 2 2 3 1 1 2 0 0 1 L L L L L L L L L e 0 ln 0 L L L dL L L = = ? e As material volume is expected to be constant i.e. A 0 L 0 =AL, and thus ) 1 ln( ln ln 0 0 + = = = e A A L L e There are certain advantages of using true strain over conventional strain or engineering strain. These include (i) equivalent absolute numerical value for true strains in cases of tensile and compressive for same intuitive deformation and (ii) total true strain is equal to the sum of the incremental strains. As shown in figure-4.1, if L 1 =2 L 0 and L 2 =1/2 L 1 =L 0 , absolute numerical value of engineering strain during tensile deformation (1.0) is different from that during compressive deformation (0.5). However, in both cases true strain values are equal (ln [2]). True stress ( s) is given as load divided by cross-sectional area over which it acts at an instant. ) 1 ( 0 0 + = = = e s A A A P A P s Page 3 Material Science Prof. Satish V. Kailas Associate Professor Dept. of Mechanical Engineering, Indian Institute of Science, Bangalore – 560012 India Chapter 4. Mechanical Properties of Metals Most of the materials used in engineering are metallic in nature. The prime reason simply is the versatile nature of their properties those spread over a very broad range compared with other kinds of materials. Many engineering materials are subjected to forces both during processing/fabrication and in service. When a force is applied on a solid material, it may result in translation, rotation, or deformation of that material. Aspects of material translation and rotation are dealt by engineering dynamics. We restrict ourselves here to the subject of material deformation under forces. Deformation constitutes both change in shape, distortion, and change in size/volume, dilatation. Solid material are defined such that change in their volume under applied forces in very small, thus deformation is used as synonymous to distortion. The ability of material to with stand the applied force without any deformation is expressed in two ways, i.e. strength and hardness. Strength is defined in many ways as per the design requirements, while the hardness may be defined as resistance to indentation of scratch. Material deformation can be permanent or temporary. Permanent deformation is irreversible i.e. stays even after removal of the applied forces, while the temporary deformation disappears after removal of the applied forces i.e. the deformation is recoverable. Both kinds of deformation can be function of time, or independent of time. Temporary deformation is called elastic deformation, while the permanent deformation is called plastic deformation. Time dependent recoverable deformation under load is called anelastic deformation, while the characteristic recovery of temporary deformation after removal of load as a function of time is called elastic aftereffect. Time dependent i.e. progressive permanent deformation under constant load/stress is called creep. For visco- elastic materials, both recoverable and permanent deformations occur together which are time dependent. When a material is subjected to applied forces, first the material experiences elastic deformation followed by plastic deformation. Extent of elastic- and plastic- deformations will primarily depend on the kind of material, rate of load application, ambient temperature, among other factors. Change over from elastic state to plastic state is characterized by the yield strength ( s 0 ) of the material. Forces applied act on a surface of the material, and thus the force intensity, force per unit area, is used in analysis. Analogous to this, deformation is characterized by percentage change in length per unit length in three distinct directions. Force intensity is also called engineering stress (or simply stress, s), is given by force divided by area on which the force is acting. Engineering strain (or simply strain, e) is given by change in length divided by original length. Engineering strain actually indicates an average change in length in a particular direction. According to definition, s and e are given as 0 0 0 , L L L e A P s - = = where P is the load applied over area A, and as a consequence of it material attains the final length L from its original length of L 0 . Because material dimensions changes under application of the load continuously, engineering stress and strain values are not the true indication of material deformation characteristics. Thus the need for measures of stress and strain based on instantaneous dimensions arises. Ludwik first proposed the concept of, and defined the true strain or natural strain ( e) as follows: ? + - + - + - = ... 2 2 3 1 1 2 0 0 1 L L L L L L L L L e 0 ln 0 L L L dL L L = = ? e As material volume is expected to be constant i.e. A 0 L 0 =AL, and thus ) 1 ln( ln ln 0 0 + = = = e A A L L e There are certain advantages of using true strain over conventional strain or engineering strain. These include (i) equivalent absolute numerical value for true strains in cases of tensile and compressive for same intuitive deformation and (ii) total true strain is equal to the sum of the incremental strains. As shown in figure-4.1, if L 1 =2 L 0 and L 2 =1/2 L 1 =L 0 , absolute numerical value of engineering strain during tensile deformation (1.0) is different from that during compressive deformation (0.5). However, in both cases true strain values are equal (ln [2]). True stress ( s) is given as load divided by cross-sectional area over which it acts at an instant. ) 1 ( 0 0 + = = = e s A A A P A P s It is to be noted that engineering stress is equal to true stress up to the elastic limit of the material. The same applies to the strains. After the elastic limit i.e. once material starts deforming plastically, engineering values and true values of stresses and strains differ. The above equation relating engineering and true stress-strains are valid only up to the limit of uniform deformation i.e. up to the onset of necking in tension test. This is because the relations are developed by assuming both constancy of volume and homogeneous distribution of strain along the length of the tension specimen. Basics of both elastic and plastic deformations along with their characterization will be detailed in this chapter. 4.1 Elastic deformation and Plastic deformation 4.1.1 Elastic deformation Elastic deformation is reversible i.e. recoverable. Up to a certain limit of the applied stress, strain experienced by the material will be the kind of recoverable i.e. elastic in nature. This elastic strain is proportional to the stress applied. The proportional relation between the stress and the elastic strain is given by Hooke’s law, which can be written as follows: e s ? e s E = where the constant E is the modulus of elasticity or Young’s modulus, Though Hooke’s law is applicable to most of the engineering materials up to their elastic limit, defined by the critical value of stress beyond which plastic deformation occurs, some materials won’t obey the law. E.g.: Rubber, it has nonlinear stress-strain relationship and still satisfies the definition of an elastic material. For materials without linear elastic portion, either tangent modulus or secant modulus is used in design calculations. The tangent modulus is taken as the slope of stress-strain curve at some specified level, while secant module represents the slope of secant drawn from the origin to some given point of the s-e curve, as shown in figure-4.1. Page 4 Material Science Prof. Satish V. Kailas Associate Professor Dept. of Mechanical Engineering, Indian Institute of Science, Bangalore – 560012 India Chapter 4. Mechanical Properties of Metals Most of the materials used in engineering are metallic in nature. The prime reason simply is the versatile nature of their properties those spread over a very broad range compared with other kinds of materials. Many engineering materials are subjected to forces both during processing/fabrication and in service. When a force is applied on a solid material, it may result in translation, rotation, or deformation of that material. Aspects of material translation and rotation are dealt by engineering dynamics. We restrict ourselves here to the subject of material deformation under forces. Deformation constitutes both change in shape, distortion, and change in size/volume, dilatation. Solid material are defined such that change in their volume under applied forces in very small, thus deformation is used as synonymous to distortion. The ability of material to with stand the applied force without any deformation is expressed in two ways, i.e. strength and hardness. Strength is defined in many ways as per the design requirements, while the hardness may be defined as resistance to indentation of scratch. Material deformation can be permanent or temporary. Permanent deformation is irreversible i.e. stays even after removal of the applied forces, while the temporary deformation disappears after removal of the applied forces i.e. the deformation is recoverable. Both kinds of deformation can be function of time, or independent of time. Temporary deformation is called elastic deformation, while the permanent deformation is called plastic deformation. Time dependent recoverable deformation under load is called anelastic deformation, while the characteristic recovery of temporary deformation after removal of load as a function of time is called elastic aftereffect. Time dependent i.e. progressive permanent deformation under constant load/stress is called creep. For visco- elastic materials, both recoverable and permanent deformations occur together which are time dependent. When a material is subjected to applied forces, first the material experiences elastic deformation followed by plastic deformation. Extent of elastic- and plastic- deformations will primarily depend on the kind of material, rate of load application, ambient temperature, among other factors. Change over from elastic state to plastic state is characterized by the yield strength ( s 0 ) of the material. Forces applied act on a surface of the material, and thus the force intensity, force per unit area, is used in analysis. Analogous to this, deformation is characterized by percentage change in length per unit length in three distinct directions. Force intensity is also called engineering stress (or simply stress, s), is given by force divided by area on which the force is acting. Engineering strain (or simply strain, e) is given by change in length divided by original length. Engineering strain actually indicates an average change in length in a particular direction. According to definition, s and e are given as 0 0 0 , L L L e A P s - = = where P is the load applied over area A, and as a consequence of it material attains the final length L from its original length of L 0 . Because material dimensions changes under application of the load continuously, engineering stress and strain values are not the true indication of material deformation characteristics. Thus the need for measures of stress and strain based on instantaneous dimensions arises. Ludwik first proposed the concept of, and defined the true strain or natural strain ( e) as follows: ? + - + - + - = ... 2 2 3 1 1 2 0 0 1 L L L L L L L L L e 0 ln 0 L L L dL L L = = ? e As material volume is expected to be constant i.e. A 0 L 0 =AL, and thus ) 1 ln( ln ln 0 0 + = = = e A A L L e There are certain advantages of using true strain over conventional strain or engineering strain. These include (i) equivalent absolute numerical value for true strains in cases of tensile and compressive for same intuitive deformation and (ii) total true strain is equal to the sum of the incremental strains. As shown in figure-4.1, if L 1 =2 L 0 and L 2 =1/2 L 1 =L 0 , absolute numerical value of engineering strain during tensile deformation (1.0) is different from that during compressive deformation (0.5). However, in both cases true strain values are equal (ln [2]). True stress ( s) is given as load divided by cross-sectional area over which it acts at an instant. ) 1 ( 0 0 + = = = e s A A A P A P s It is to be noted that engineering stress is equal to true stress up to the elastic limit of the material. The same applies to the strains. After the elastic limit i.e. once material starts deforming plastically, engineering values and true values of stresses and strains differ. The above equation relating engineering and true stress-strains are valid only up to the limit of uniform deformation i.e. up to the onset of necking in tension test. This is because the relations are developed by assuming both constancy of volume and homogeneous distribution of strain along the length of the tension specimen. Basics of both elastic and plastic deformations along with their characterization will be detailed in this chapter. 4.1 Elastic deformation and Plastic deformation 4.1.1 Elastic deformation Elastic deformation is reversible i.e. recoverable. Up to a certain limit of the applied stress, strain experienced by the material will be the kind of recoverable i.e. elastic in nature. This elastic strain is proportional to the stress applied. The proportional relation between the stress and the elastic strain is given by Hooke’s law, which can be written as follows: e s ? e s E = where the constant E is the modulus of elasticity or Young’s modulus, Though Hooke’s law is applicable to most of the engineering materials up to their elastic limit, defined by the critical value of stress beyond which plastic deformation occurs, some materials won’t obey the law. E.g.: Rubber, it has nonlinear stress-strain relationship and still satisfies the definition of an elastic material. For materials without linear elastic portion, either tangent modulus or secant modulus is used in design calculations. The tangent modulus is taken as the slope of stress-strain curve at some specified level, while secant module represents the slope of secant drawn from the origin to some given point of the s-e curve, as shown in figure-4.1. Figure-4.1: Tangent and Secant moduli for non-linear stress-strain relation. If one dimension of the material changed, other dimensions of the material need to be changed to keep the volume constant. This lateral/transverse strain is related to the applied longitudinal strain by empirical means, and the ratio of transverse strain to longitudinal strain is known as Poisson’s ratio ( ?). Transverse strain can be expected to be opposite in nature to longitudinal strain, and both longitudinal and transverse strains are linear strains. For most metals the values of ? are close to 0.33, for polymers it is between 0.4 – 0.5, and for ionic solids it is around 0.2. Stresses applied on a material can be of two kinds – normal stresses, and shear stresses. Normal stresses cause linear strains, while the shear stresses cause shear strains. If the material is subjected to torsion, it results in torsional strain. Different stresses and corresponding strains are shown in figure-4.2. Page 5 Material Science Prof. Satish V. Kailas Associate Professor Dept. of Mechanical Engineering, Indian Institute of Science, Bangalore – 560012 India Chapter 4. Mechanical Properties of Metals Most of the materials used in engineering are metallic in nature. The prime reason simply is the versatile nature of their properties those spread over a very broad range compared with other kinds of materials. Many engineering materials are subjected to forces both during processing/fabrication and in service. When a force is applied on a solid material, it may result in translation, rotation, or deformation of that material. Aspects of material translation and rotation are dealt by engineering dynamics. We restrict ourselves here to the subject of material deformation under forces. Deformation constitutes both change in shape, distortion, and change in size/volume, dilatation. Solid material are defined such that change in their volume under applied forces in very small, thus deformation is used as synonymous to distortion. The ability of material to with stand the applied force without any deformation is expressed in two ways, i.e. strength and hardness. Strength is defined in many ways as per the design requirements, while the hardness may be defined as resistance to indentation of scratch. Material deformation can be permanent or temporary. Permanent deformation is irreversible i.e. stays even after removal of the applied forces, while the temporary deformation disappears after removal of the applied forces i.e. the deformation is recoverable. Both kinds of deformation can be function of time, or independent of time. Temporary deformation is called elastic deformation, while the permanent deformation is called plastic deformation. Time dependent recoverable deformation under load is called anelastic deformation, while the characteristic recovery of temporary deformation after removal of load as a function of time is called elastic aftereffect. Time dependent i.e. progressive permanent deformation under constant load/stress is called creep. For visco- elastic materials, both recoverable and permanent deformations occur together which are time dependent. When a material is subjected to applied forces, first the material experiences elastic deformation followed by plastic deformation. Extent of elastic- and plastic- deformations will primarily depend on the kind of material, rate of load application, ambient temperature, among other factors. Change over from elastic state to plastic state is characterized by the yield strength ( s 0 ) of the material. Forces applied act on a surface of the material, and thus the force intensity, force per unit area, is used in analysis. Analogous to this, deformation is characterized by percentage change in length per unit length in three distinct directions. Force intensity is also called engineering stress (or simply stress, s), is given by force divided by area on which the force is acting. Engineering strain (or simply strain, e) is given by change in length divided by original length. Engineering strain actually indicates an average change in length in a particular direction. According to definition, s and e are given as 0 0 0 , L L L e A P s - = = where P is the load applied over area A, and as a consequence of it material attains the final length L from its original length of L 0 . Because material dimensions changes under application of the load continuously, engineering stress and strain values are not the true indication of material deformation characteristics. Thus the need for measures of stress and strain based on instantaneous dimensions arises. Ludwik first proposed the concept of, and defined the true strain or natural strain ( e) as follows: ? + - + - + - = ... 2 2 3 1 1 2 0 0 1 L L L L L L L L L e 0 ln 0 L L L dL L L = = ? e As material volume is expected to be constant i.e. A 0 L 0 =AL, and thus ) 1 ln( ln ln 0 0 + = = = e A A L L e There are certain advantages of using true strain over conventional strain or engineering strain. These include (i) equivalent absolute numerical value for true strains in cases of tensile and compressive for same intuitive deformation and (ii) total true strain is equal to the sum of the incremental strains. As shown in figure-4.1, if L 1 =2 L 0 and L 2 =1/2 L 1 =L 0 , absolute numerical value of engineering strain during tensile deformation (1.0) is different from that during compressive deformation (0.5). However, in both cases true strain values are equal (ln [2]). True stress ( s) is given as load divided by cross-sectional area over which it acts at an instant. ) 1 ( 0 0 + = = = e s A A A P A P s It is to be noted that engineering stress is equal to true stress up to the elastic limit of the material. The same applies to the strains. After the elastic limit i.e. once material starts deforming plastically, engineering values and true values of stresses and strains differ. The above equation relating engineering and true stress-strains are valid only up to the limit of uniform deformation i.e. up to the onset of necking in tension test. This is because the relations are developed by assuming both constancy of volume and homogeneous distribution of strain along the length of the tension specimen. Basics of both elastic and plastic deformations along with their characterization will be detailed in this chapter. 4.1 Elastic deformation and Plastic deformation 4.1.1 Elastic deformation Elastic deformation is reversible i.e. recoverable. Up to a certain limit of the applied stress, strain experienced by the material will be the kind of recoverable i.e. elastic in nature. This elastic strain is proportional to the stress applied. The proportional relation between the stress and the elastic strain is given by Hooke’s law, which can be written as follows: e s ? e s E = where the constant E is the modulus of elasticity or Young’s modulus, Though Hooke’s law is applicable to most of the engineering materials up to their elastic limit, defined by the critical value of stress beyond which plastic deformation occurs, some materials won’t obey the law. E.g.: Rubber, it has nonlinear stress-strain relationship and still satisfies the definition of an elastic material. For materials without linear elastic portion, either tangent modulus or secant modulus is used in design calculations. The tangent modulus is taken as the slope of stress-strain curve at some specified level, while secant module represents the slope of secant drawn from the origin to some given point of the s-e curve, as shown in figure-4.1. Figure-4.1: Tangent and Secant moduli for non-linear stress-strain relation. If one dimension of the material changed, other dimensions of the material need to be changed to keep the volume constant. This lateral/transverse strain is related to the applied longitudinal strain by empirical means, and the ratio of transverse strain to longitudinal strain is known as Poisson’s ratio ( ?). Transverse strain can be expected to be opposite in nature to longitudinal strain, and both longitudinal and transverse strains are linear strains. For most metals the values of ? are close to 0.33, for polymers it is between 0.4 – 0.5, and for ionic solids it is around 0.2. Stresses applied on a material can be of two kinds – normal stresses, and shear stresses. Normal stresses cause linear strains, while the shear stresses cause shear strains. If the material is subjected to torsion, it results in torsional strain. Different stresses and corresponding strains are shown in figure-4.2. Figure 4.2: Schematic description of different kinds of deformations/strains. Analogous to the relation between normal stress and linear strain defined earlier, shear stress ( t) and shear strain ( ?) in elastic range are related as follows: ? t G = where G is known as Shear modulus of the material. It is also known as modulus of elasticity in shear. It is related with Young’s modulus, E, through Poisson’s ratio, ?, as ) 1 ( 2 ? + = E G Similarly, the Bulk modulus or volumetric modulus of elasticity K, of a material is defined as the ratio of hydrostatic or mean stress ( s m ) to the volumetric strain ( ?). The relation between E and K is given by ) 2 1 ( 3 ? s - = ? = E K m Let s x , s y and s z are linear stresses and e x , e y and e z are corresponding strains in X-, Y- and Z- directions, then 3 z y x m s s s s + + = Volumetric strain or cubical dilatation is defined as the change in volume per unit volume. z y x z y x e e e e e e + + ˜ - + + + = ? 1 ) 1 )( 1 )( 1 ( , m e 3 = ? where e m is mean strain or hydrostatic (spherical) strain defined as 3 z y x m e e e e + + = An engineering material is usually subjected to stresses in multiple directions than in just one direction. If a cubic element of a material is subjected to normal stresses s x , s y , and s z , strains in corresponding directions are given by [] ) ( 1 z y x x E s s ? s e + - = , [ ] ) ( 1 x x y y E s s ? s e + - = , and [ ] ) ( 1 y x z z E s s ? s e + - =Read More

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