Elasticity
$\displaystyle \small \bullet$ The property of an object to resume its actual shape after being stretched or compressed due to the application of an external force.
$\displaystyle \small \bullet$ Elasticity is measured as the ratio of stress to strain.
$\displaystyle \small Elasticity=\frac{Stress}{Strain}$
$\displaystyle \small \bullet$ Ex: spring
$\displaystyle \small \bullet$ Examples of elastic materials: natural gum, spandex or lycra, butyl rubber (GDP), fluoroelastomer, elastomers, ethylene-propylene rubber (EPR), resilin, styrene-butadiene rubber (SBR), chloroprene, elastin, rubber epichlorohydrin, nylon, terpene, isoprene rubber, polybutadiene, nitrile rubber, vinyl stretch, thermoplastic elastomer, silicon rubber, neoprene etc.
Plasticity
$\displaystyle \small \bullet$ The ability of an object to be deform repeatedly by action of an external force and to remain deformed after the force is removed.
$\displaystyle \small \bullet$ Thermoplastics: material that can be reheated, remolded, cooled as necessary. Ex: polyolefins, vinyl polymers celluloid, linear polysters etc.
$\displaystyle \small \bullet$ Thermosetting plastics: material that can be modified by heating or cooling but cannot be remolded or recycled. Ex: unsaturated polyster, phenol formaldehyde resins, polyepoxides, polyurethane etc.
Stress
$\displaystyle \small \bullet$ Force per unit area.
$\displaystyle \small \bullet$ It is the internal resistance to the applied external force per unit area on an object.
$\displaystyle \small \bullet$ Value of stress depends upon the applied force and cross-sectional area of these object.
$\displaystyle \small Stress=\frac{Force(kg)}{Cross-sectional\; area(cm^{2})}$
$\displaystyle \small \rho =\frac{F}{A}$
$\displaystyle \small \bullet$ Unit of stress is $\displaystyle \small kg/cm^{2}$ or $\displaystyle \small N/m^{2}$
$\displaystyle \small \bullet$ Types of stress
$\displaystyle \small \circ$ Tensile stress: When two forces act in opposite directions on the same axis of an object and object seem to be under tension, then stress is known as tensile stress
$\displaystyle \small \circ$ Compressive stress: When two forces act in opposite directions on the same axis of an object and object is subjected to compression, then stress is known as compressive stress
$\displaystyle \small \circ$ Shear stress: When two or more forces act in opposite directions on different axis of an object and object generates shear in it, then stress is known as shear stress
Strain
$\displaystyle \small \bullet$ Deformation of solid due to stress.
$\displaystyle \small \bullet$ When a body is subjected to external force, there is some change in dimension of the body. The ratio of change in dimension of the body to the original dimension is known as strain.
$\displaystyle \small \bullet$ Denoted by ‘e’
$\displaystyle \small \bullet$ It has no unit
$\displaystyle \small \bullet$ Types of strain
$\displaystyle \small \circ$ Longitudinal strain
$\displaystyle \small e=\frac{change\; in\; length}{original\; length}$
$\displaystyle \small e=\frac{(l_{2}-l_{1})}{l_{1}}$
$\displaystyle \small \circ$ Lateral strain
$\displaystyle \small e=\frac{change\; in\; area}{original\; area}$
$\displaystyle \small e=\frac{(A_{2}-A_{1})}{A_{1}}$
$\displaystyle \small \circ$ Volumetric strain
$\displaystyle \small e=\frac{change\; in\; volume}{original\; volume}$
$\displaystyle \small e=\frac{(V_{2}-V_{1})}{V_{1}}$
$\displaystyle \small \circ$ Shear strain
The deformation takes place along the direction of deforming force.
Denoted by tangent of angle Ñ„
Hooke’s Law
$\displaystyle \small \bullet$ Hooke’s law states that “Within the elastic limit, stress is directly proportional to the strain”.
$\displaystyle \small \bullet$ With increase in stress, strain also increases
$\displaystyle \small \bullet$ With decrease in stress, strain also decreases
$\displaystyle \small stress\propto strain$
$\displaystyle \small stress=E\times strain$
$\displaystyle \small \rho =E\times e$
$\displaystyle \small \bullet$ E is a constant known as Young’s modulus or elasticity coefficient.
Elastic Limit
$\displaystyle \small \bullet$ Elastic substances can regain their original state after the distorting force is removed, but the quality of elasticity is found upto certain limit.
$\displaystyle \small \bullet$ If the stress is continuously increased, there comes a time after which the material would not be able to regain its original shape and size, even after the external force is removed.
$\displaystyle \small \bullet$ Thus, maximum stress or force per unit area within a material that can arise before the onset of the permanent deformation is called elastic limit.
Stress-Strain Curve
$\displaystyle \small \bullet$ Stress-strain curve represents the states of deformation at different magnitudes of stress.
$\displaystyle \small \bullet$ Tensile force is applied to a mild steel bar.
$\displaystyle \small \bullet$ If force is small, the ratio of stress and strain will remain proportional. (O to A) is the limit of proportionality.
$\displaystyle \small \bullet$ If the force is considerably large, the material will experience elastic deformation. (A to B) is the elastic limit.
$\displaystyle \small \bullet$ Beyond this point the material will experience plastic deformation. Point B is the upper yield point and C is the lower yield point.
$\displaystyle \small \bullet$ D is the maximum ultimate stress.
$\displaystyle \small \bullet$ E is the breaking stress.
Young’s modulus or Elasticity coefficient (E)
Within elastic limit, the ratio of stress and strain in an object is known as Young’s modulus or elasticity coefficient. SI unit is N/sq.m
$\displaystyle \small E=\frac{\rho }{e}$
Ultimate Stress
The maximum value of load which overpowers the internal resistance of an object against any strain is called ultimate stress.
$\displaystyle \small Ultimate\; stress=\frac{ultimate\; load}{cross-sectional\; area}$
Working Stress or Safe Stress
Maximum allowable stress that a material can be subjected to in the course of ordinary use is known as working stress.
$\displaystyle \small Safe\; stress=\frac{maximum\; allowable\; stress}{cross-sectional\; area}$
Factor of Safety or Safety Coefficient
$\displaystyle \small \bullet$ Ratio of ultimate stress and working stress is known as safety coefficient or factor of safety.
$\displaystyle \small \bullet$ It is expressed in form of a number and is always greater than 1.
$\displaystyle \small Safety \; coefficient=\frac{ultimate \; stress}{working\; stress}$
$\displaystyle \small Safety \; coefficient=\frac{ultimate \; load}{working\; load}$
Modulus of Rigidity (N)
Ratio of shear stress and shear strain is known as modulus of rigidity.
$\displaystyle \small N=\frac{shear\; stress}{shear\; strain}$
Poisson’s Ratio (μ or 1/m)
Ratio of lateral strain and linear strain within elastic limit is known as Poisson’s ratio.
Value of m ranges from 3 to 4 for majority of metals.
$\displaystyle \small \mu =\frac{1}{m}=\frac{lateral\; strain}{longitudinal\; strain}$
Bulk Modulus (K)
Ratio of volumetric stress and volumetric strain is known as bulk modulus.
$\displaystyle \small K=\frac{volumetric\; stress}{volumetric\; strain}$
Relationship between the three moduli for a given material
$\displaystyle \small E=2N\left ( 1+\frac{1}{m} \right )=3K\left ( 1-\frac{2}{m} \right )$
where,
E=Young’s modulus
K=Bulk modulus
N=Modulus of rigidity
1/m=Poisson’s ratio
$\displaystyle \small \bullet$ The property of an object to resume its actual shape after being stretched or compressed due to the application of an external force.
$\displaystyle \small \bullet$ Elasticity is measured as the ratio of stress to strain.
$\displaystyle \small Elasticity=\frac{Stress}{Strain}$
$\displaystyle \small \bullet$ Ex: spring
$\displaystyle \small \bullet$ Examples of elastic materials: natural gum, spandex or lycra, butyl rubber (GDP), fluoroelastomer, elastomers, ethylene-propylene rubber (EPR), resilin, styrene-butadiene rubber (SBR), chloroprene, elastin, rubber epichlorohydrin, nylon, terpene, isoprene rubber, polybutadiene, nitrile rubber, vinyl stretch, thermoplastic elastomer, silicon rubber, neoprene etc.
$\displaystyle \small \bullet$ The ability of an object to be deform repeatedly by action of an external force and to remain deformed after the force is removed.
$\displaystyle \small \bullet$ Thermoplastics: material that can be reheated, remolded, cooled as necessary. Ex: polyolefins, vinyl polymers celluloid, linear polysters etc.
$\displaystyle \small \bullet$ Thermosetting plastics: material that can be modified by heating or cooling but cannot be remolded or recycled. Ex: unsaturated polyster, phenol formaldehyde resins, polyepoxides, polyurethane etc.
Stress
$\displaystyle \small \bullet$ Force per unit area.
$\displaystyle \small \bullet$ It is the internal resistance to the applied external force per unit area on an object.
$\displaystyle \small \bullet$ Value of stress depends upon the applied force and cross-sectional area of these object.
$\displaystyle \small Stress=\frac{Force(kg)}{Cross-sectional\; area(cm^{2})}$
$\displaystyle \small \rho =\frac{F}{A}$
$\displaystyle \small \bullet$ Unit of stress is $\displaystyle \small kg/cm^{2}$ or $\displaystyle \small N/m^{2}$
$\displaystyle \small \bullet$ Types of stress
$\displaystyle \small \circ$ Tensile stress: When two forces act in opposite directions on the same axis of an object and object seem to be under tension, then stress is known as tensile stress
$\displaystyle \small \circ$ Compressive stress: When two forces act in opposite directions on the same axis of an object and object is subjected to compression, then stress is known as compressive stress
$\displaystyle \small \circ$ Shear stress: When two or more forces act in opposite directions on different axis of an object and object generates shear in it, then stress is known as shear stress
Strain
$\displaystyle \small \bullet$ Deformation of solid due to stress.
$\displaystyle \small \bullet$ When a body is subjected to external force, there is some change in dimension of the body. The ratio of change in dimension of the body to the original dimension is known as strain.
$\displaystyle \small \bullet$ Denoted by ‘e’
$\displaystyle \small \bullet$ It has no unit
$\displaystyle \small \bullet$ Types of strain
$\displaystyle \small \circ$ Longitudinal strain
$\displaystyle \small e=\frac{change\; in\; length}{original\; length}$
$\displaystyle \small e=\frac{(l_{2}-l_{1})}{l_{1}}$
$\displaystyle \small \circ$ Lateral strain
$\displaystyle \small e=\frac{change\; in\; area}{original\; area}$
$\displaystyle \small e=\frac{(A_{2}-A_{1})}{A_{1}}$
$\displaystyle \small \circ$ Volumetric strain
$\displaystyle \small e=\frac{change\; in\; volume}{original\; volume}$
$\displaystyle \small e=\frac{(V_{2}-V_{1})}{V_{1}}$
$\displaystyle \small \circ$ Shear strain
The deformation takes place along the direction of deforming force.
Denoted by tangent of angle Ñ„
Hooke’s Law
$\displaystyle \small \bullet$ Hooke’s law states that “Within the elastic limit, stress is directly proportional to the strain”.
$\displaystyle \small \bullet$ With increase in stress, strain also increases
$\displaystyle \small \bullet$ With decrease in stress, strain also decreases
$\displaystyle \small stress\propto strain$
$\displaystyle \small stress=E\times strain$
$\displaystyle \small \rho =E\times e$
$\displaystyle \small \bullet$ E is a constant known as Young’s modulus or elasticity coefficient.
Elastic Limit
$\displaystyle \small \bullet$ Elastic substances can regain their original state after the distorting force is removed, but the quality of elasticity is found upto certain limit.
$\displaystyle \small \bullet$ If the stress is continuously increased, there comes a time after which the material would not be able to regain its original shape and size, even after the external force is removed.
$\displaystyle \small \bullet$ Thus, maximum stress or force per unit area within a material that can arise before the onset of the permanent deformation is called elastic limit.
Stress-Strain Curve
$\displaystyle \small \bullet$ Stress-strain curve represents the states of deformation at different magnitudes of stress.
$\displaystyle \small \bullet$ Tensile force is applied to a mild steel bar.
$\displaystyle \small \bullet$ If force is small, the ratio of stress and strain will remain proportional. (O to A) is the limit of proportionality.
$\displaystyle \small \bullet$ If the force is considerably large, the material will experience elastic deformation. (A to B) is the elastic limit.
$\displaystyle \small \bullet$ Beyond this point the material will experience plastic deformation. Point B is the upper yield point and C is the lower yield point.
$\displaystyle \small \bullet$ D is the maximum ultimate stress.
$\displaystyle \small \bullet$ E is the breaking stress.
Young’s modulus or Elasticity coefficient (E)
Within elastic limit, the ratio of stress and strain in an object is known as Young’s modulus or elasticity coefficient. SI unit is N/sq.m
$\displaystyle \small E=\frac{\rho }{e}$
Ultimate Stress
The maximum value of load which overpowers the internal resistance of an object against any strain is called ultimate stress.
$\displaystyle \small Ultimate\; stress=\frac{ultimate\; load}{cross-sectional\; area}$
Working Stress or Safe Stress
Maximum allowable stress that a material can be subjected to in the course of ordinary use is known as working stress.
$\displaystyle \small Safe\; stress=\frac{maximum\; allowable\; stress}{cross-sectional\; area}$
Factor of Safety or Safety Coefficient
$\displaystyle \small \bullet$ Ratio of ultimate stress and working stress is known as safety coefficient or factor of safety.
$\displaystyle \small \bullet$ It is expressed in form of a number and is always greater than 1.
$\displaystyle \small Safety \; coefficient=\frac{ultimate \; stress}{working\; stress}$
$\displaystyle \small Safety \; coefficient=\frac{ultimate \; load}{working\; load}$
Modulus of Rigidity (N)
Ratio of shear stress and shear strain is known as modulus of rigidity.
$\displaystyle \small N=\frac{shear\; stress}{shear\; strain}$
Poisson’s Ratio (μ or 1/m)
Ratio of lateral strain and linear strain within elastic limit is known as Poisson’s ratio.
Value of m ranges from 3 to 4 for majority of metals.
$\displaystyle \small \mu =\frac{1}{m}=\frac{lateral\; strain}{longitudinal\; strain}$
Bulk Modulus (K)
Ratio of volumetric stress and volumetric strain is known as bulk modulus.
$\displaystyle \small K=\frac{volumetric\; stress}{volumetric\; strain}$
Relationship between the three moduli for a given material
$\displaystyle \small E=2N\left ( 1+\frac{1}{m} \right )=3K\left ( 1-\frac{2}{m} \right )$
where,
E=Young’s modulus
K=Bulk modulus
N=Modulus of rigidity
1/m=Poisson’s ratio
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