ElasticityElasticity : The property of a body by virtue of which it resists deformation force and regains its original shape wen deformation force is removed is called elasticity. Elastic body : A body which regains its original shape when the deformation force is removed is called an elastic body. Inelastic or Plastic body : A body which cannot regain its original shape, when the deformation force is removed is called a plastic body. Elasticity is molecular property of the matter. Stress : The restoring force per unit area developed inside the body is called stress. ( Or )
Stress = force / area Units : C G S system : dyne cm-2 S I system : newton m-2 or pascal Pascal is also unit of pressure . Its Dimensional formula is ML-1T-2 Strain : The change produced per unit magnitude of a body is called strain. Units : Strain being a ratio , has no units or dimensions. Elastic Limit :The maximum value of stress within which the body completely regains its original condition when deformation force is removed is called elastic limit. Hooke's Law :Stress is proportional to strain provided the strain does not exceed the elastic limit . i.e ., stress µ stain, or stress / strain = E , a constant called the modulus of elasticity. Modulus of elasticity depends on the nature of the material and the past treatment it has gone through . Modulus of elasticity : The modulus of elasticity of a body is the magnitude of stress to be applied to produce unit strain. Three kinds of strain : 1) Longitudinal or linear strain or tensile strain. 2) Shearing strain or tangential strain .3) Bulk strain or volume strain.1) Longitudinal or linear strain or tensile strain :When an external force is applied to a rod along its length the fractional change in its length is called longitudinal strain . If l is the original length and Dl the longitudinal extension .
Longitudinal strain , e = D l / l 2) Shearing strain : When simultaneous compression and extension in mutually perpendicular directions take place in a body , the change of shape it undergoes is called shearing strain . Where Dl is the displacement of the upper surface and l the length of the vertical edge, when q is small, Shearing strain ( q ) = Dl / l 3) Volume or Bulk strain :If V is the original volume DV the change in volume , the bulk strain v = - DV / V The negative sign indicates the decrease in volume . Moduli of Elasticity : 1) Young's modulus ( Y ) : Within the elastic limit of a body , the ratio between the longitudinal stress and the longitudinal strain is called the Young's modulus of elasticity .
Y = longitudinal stress / longitudinal strain = Fl / A Dl Where F is the force acting on the surface of area A Dl is the increase in length in a length l of the wire. ( Or )
Y = Mg / pr 2 * l / Dl Where M is the mass r is the cross sectional radius of the material. Units of Y :dyne / cm2 or N / m2 ( pascal (Pa) ) 1 pascal = 10 dynes cm-2 109 Pascal = 1 Giga Pascal 1 atmosphere = 1.02 * 105 Pa 2) Rigidity modulus :Within the elastic limit of a body , the ratio of tangential stress to the shearing strain is called rigidity modulus of elasticity. The rigidity modulus , n = tangential stress / shearing strain = (F / A ) / q = (F / A ) / (Dl / l) = Fl / A Dl where F / A is the tangential stress, Dl the displacement in a length l in the perpendicular direction . Units of n : N / m2 3) Bulk modulus ( K ) : It is defined as the ratio of stress to volumetric strain . or
K = stress / bulk strain = -PV / DV Where P is the pressure , V is the original volume and v is the change in volume of the material . Unit of K : N / m 2 4) Compressibility :The reciprocal of the bulk modulus is called compressibility . or
Compressibility = 1 / K Breaking stress :It is the maximum stress that a material can withstand and beyond the breaking stress, the material breaks. The breaking stress is also called tensile strength. The breaking stress of a material is constant. Since stress = F / A , it follows that F / A = constant for material or , Breaking force F µ A Elastic Fatigue :The state of temporary loss of elastic nature of a body due to continuous strain is called elastic fatigue. Poisson's Ratio :It is the ratio between the lateral strain and the longitudinal strain . or
s = (Db / b) / (Dl / l) = lateral contraction strain / longitudinal elongation strain Limiting values of s are from -1 to + 0.5 and practical value of s lies between 0 and 0.5. Relation between elastic constants : Let Y, n , k and s be the Young's modulus, rigidity modulus , bulk modulus and Poisson's ratio of the material . A ) Relation between Y , n and s n = Y / 2 ( 1 + s ) ……………….( 1 ) B ) Relation between Y , k and s K = Y / 3 ( 1 - 2s ) ……………….( 2 ) C ) Relation between n, k and s s = 3k - 2n / 2 ( 3k + n ) ………..( 3 ) D ) Relation between Y , k and n 9 / Y = 1 / k = 3 / n or Y = 9nk / 3k + n …………………( 4 ) Inter-atomic force constant ( k ) : The inter- atomic force constant of a solid is equal to the product of the Young's modulus of the material and the spacing between the atoms or , k = Yr0 Where k is the inter-atomic force constant , Y is the Young's modulus and r0 is the distance between two atoms . Thermal stresses :Let a material of coefficient of linear expansion a , Young's modulus Y, length l, cross-sectional area A be heated through a temperature difference of t° C. If e is the expansion due to heating e = a l t ……………………..( 1 ) If equal contraction is to be produced applying a force F ,Y = F / A * l / e or , e = Fl / YA …………………………………….( 2 )\ Fl / YA = a l t or , F = YA a t ……………………………..( 3 ) The stress F / A = Y a t ………………………..( 4 ) Elastic Potential energy : A ) When a wire is stretched , the work done is stored as elastic potential energy. 1 ) The work done per unit volume to stretch a wire. W = ½ * stress * strain ……………………( 1 ) 2 ) The work done is stretching a wire is , W = ½ * F * e ……………………………..( 2 ) Where F is the stretching force and e is the elongation produced. B ) Work done in stretching a spring W = ½ k x2 = ½ Fx = ½ F2 / k Where x is the elongation of the spring , F is the stretching force and k is the spring constant or force constant. k = F / x
Stress = force / area Units : C G S system : dyne cm-2 S I system : newton m-2 or pascal Pascal is also unit of pressure . Its Dimensional formula is ML-1T-2 Strain : The change produced per unit magnitude of a body is called strain. Units : Strain being a ratio , has no units or dimensions. Elastic Limit :The maximum value of stress within which the body completely regains its original condition when deformation force is removed is called elastic limit. Hooke's Law :Stress is proportional to strain provided the strain does not exceed the elastic limit . i.e ., stress µ stain, or stress / strain = E , a constant called the modulus of elasticity. Modulus of elasticity depends on the nature of the material and the past treatment it has gone through . Modulus of elasticity : The modulus of elasticity of a body is the magnitude of stress to be applied to produce unit strain. Three kinds of strain : 1) Longitudinal or linear strain or tensile strain. 2) Shearing strain or tangential strain .3) Bulk strain or volume strain.1) Longitudinal or linear strain or tensile strain :When an external force is applied to a rod along its length the fractional change in its length is called longitudinal strain . If l is the original length and Dl the longitudinal extension .
Longitudinal strain , e = D l / l 2) Shearing strain : When simultaneous compression and extension in mutually perpendicular directions take place in a body , the change of shape it undergoes is called shearing strain . Where Dl is the displacement of the upper surface and l the length of the vertical edge, when q is small, Shearing strain ( q ) = Dl / l 3) Volume or Bulk strain :If V is the original volume DV the change in volume , the bulk strain v = - DV / V The negative sign indicates the decrease in volume . Moduli of Elasticity : 1) Young's modulus ( Y ) : Within the elastic limit of a body , the ratio between the longitudinal stress and the longitudinal strain is called the Young's modulus of elasticity .
Y = longitudinal stress / longitudinal strain = Fl / A Dl Where F is the force acting on the surface of area A Dl is the increase in length in a length l of the wire. ( Or )
Y = Mg / pr 2 * l / Dl Where M is the mass r is the cross sectional radius of the material. Units of Y :dyne / cm2 or N / m2 ( pascal (Pa) ) 1 pascal = 10 dynes cm-2 109 Pascal = 1 Giga Pascal 1 atmosphere = 1.02 * 105 Pa 2) Rigidity modulus :Within the elastic limit of a body , the ratio of tangential stress to the shearing strain is called rigidity modulus of elasticity. The rigidity modulus , n = tangential stress / shearing strain = (F / A ) / q = (F / A ) / (Dl / l) = Fl / A Dl where F / A is the tangential stress, Dl the displacement in a length l in the perpendicular direction . Units of n : N / m2 3) Bulk modulus ( K ) : It is defined as the ratio of stress to volumetric strain . or
K = stress / bulk strain = -PV / DV Where P is the pressure , V is the original volume and v is the change in volume of the material . Unit of K : N / m 2 4) Compressibility :The reciprocal of the bulk modulus is called compressibility . or
Compressibility = 1 / K Breaking stress :It is the maximum stress that a material can withstand and beyond the breaking stress, the material breaks. The breaking stress is also called tensile strength. The breaking stress of a material is constant. Since stress = F / A , it follows that F / A = constant for material or , Breaking force F µ A Elastic Fatigue :The state of temporary loss of elastic nature of a body due to continuous strain is called elastic fatigue. Poisson's Ratio :It is the ratio between the lateral strain and the longitudinal strain . or
s = (Db / b) / (Dl / l) = lateral contraction strain / longitudinal elongation strain Limiting values of s are from -1 to + 0.5 and practical value of s lies between 0 and 0.5. Relation between elastic constants : Let Y, n , k and s be the Young's modulus, rigidity modulus , bulk modulus and Poisson's ratio of the material . A ) Relation between Y , n and s n = Y / 2 ( 1 + s ) ……………….( 1 ) B ) Relation between Y , k and s K = Y / 3 ( 1 - 2s ) ……………….( 2 ) C ) Relation between n, k and s s = 3k - 2n / 2 ( 3k + n ) ………..( 3 ) D ) Relation between Y , k and n 9 / Y = 1 / k = 3 / n or Y = 9nk / 3k + n …………………( 4 ) Inter-atomic force constant ( k ) : The inter- atomic force constant of a solid is equal to the product of the Young's modulus of the material and the spacing between the atoms or , k = Yr0 Where k is the inter-atomic force constant , Y is the Young's modulus and r0 is the distance between two atoms . Thermal stresses :Let a material of coefficient of linear expansion a , Young's modulus Y, length l, cross-sectional area A be heated through a temperature difference of t° C. If e is the expansion due to heating e = a l t ……………………..( 1 ) If equal contraction is to be produced applying a force F ,Y = F / A * l / e or , e = Fl / YA …………………………………….( 2 )\ Fl / YA = a l t or , F = YA a t ……………………………..( 3 ) The stress F / A = Y a t ………………………..( 4 ) Elastic Potential energy : A ) When a wire is stretched , the work done is stored as elastic potential energy. 1 ) The work done per unit volume to stretch a wire. W = ½ * stress * strain ……………………( 1 ) 2 ) The work done is stretching a wire is , W = ½ * F * e ……………………………..( 2 ) Where F is the stretching force and e is the elongation produced. B ) Work done in stretching a spring W = ½ k x2 = ½ Fx = ½ F2 / k Where x is the elongation of the spring , F is the stretching force and k is the spring constant or force constant. k = F / x
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