Lecture 3 Elasticity PDF

Chapter 3 Elasticity  Contents … Introduction 3.1 Hooke’s Law and Coefficient of Elasticity 3.2 Young’s Modulus, Bulk Modulus and Modulus of Rigidity 3.3 Work done during Longitudinal Strain, Volume Strain and Shearing Strain 3.4 Poisson’s Ration 3.5 Determination of Y of material of a Rect...

Chapter 3 Elasticity  Contents … Introduction 3.1 Hooke’s Law and Coefficient of Elasticity 3.2 Young’s Modulus, Bulk Modulus and Modulus of Rigidity 3.3 Work done during Longitudinal Strain, Volume Strain and Shearing Strain 3.4 Poisson’s Ration 3.5 Determination of Y of material of a Rectangular Thin Bar Loaded at the Centre 3.6 Torsional Oscillation  Introduction: elasticity is the ability of a body to resist a distorting influence or deforming force and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate forces are applied on them. If the material is elastic, the object will return to its initial shape and size when these forces are removed. The physical reasons for elastic behavior can be quite different for different materials. In metals, the atomic lattice changes size and shape when forces are applied (energy is added to the system). When forces are removed, the lattice goes back to the original lower energy state. For rubbers and other polymers, elasticity is caused by the stretching of polymer chains when forces are applied.  Stress and Strain : The internal resistive force per unit area of the body is called stress. If a wire of length L increases in length by ‘l’ then the ratio (l/L) is called strain.  Hooke’s Law and Coefficient of Elasticity: Hooke's Law It states that with in elastic limits, stress is proportional to strain. Within elastic limits, tension is proportional to extension. So, Stress ∝ Strain A ∝ l/L  Young’s Modulus: Young's modulus Y It is defined as the ratio between normal stress to the longitudinal strain. Y = normal stress/longitudinal strain = (F/A)/(l/L) Y = (Mg×L)/(πr2×L) Bulk Modulus : It is defined as the ratio of volume stress of volume strain. Bulk modulus (k) = (Volume stress / Volume strain ). (K) = (PV/v) Modulus of Rigidity (ἠ) : It is defined as the ratio of the tangential stress to the shearing strain. η = tangential stress/ shear strain = (F/A)/θ = T/θ Work done during longitudinal strain: The longitudinal strain equation is the extension of the body of the material when the load or stress is applied. Y = stress / strain Y = (F/A)/(xL)……………………………….(1) F = AY/L × (x) , ……………………………..(2) dw = Fdx ……………………………………(3) From equation 2 and 3 we get dw = AY/L x dx………………………………(4) W = ʃᴸ˳ AY/L x dx AY/2L l² ½ (F/A) × (l/L) ½ × stress × strain  Work done during Volume Strain: K = Pv/v P = Kv/V dw = Fdx = PA dx, dw = Pdv Where, A dx = dv is the small change in volume. W = ʃˇ˳ P dv = ½ k/v v² = ½ (k/v ѵ) × (v) = ½ Pv , w = ½ ×(stress) × (volume strain)  Work done during shearing strain: shearing strain = Ө Modulus of rigidity = η (F/A) / Ө F = η AӨ F= ηL²Ө ,Therefore A= L² , Ө = l/L F= ηLl W = ʃˡ˳ η L l dl W = ½ ηLl² W = F/L l W = ½ Fl W = ½ x (tangential force)x (Displacement) W = (F/A) x (Ө) W = ½ x Shearing stress x Shearing strain  Poisson’s Ratio: Poisson’s Ratio – The ratio of lateral strain to longitudinal strain is called poison's ratio(σ). (σ) = Lateral strain / Longitudinal strain. Lateral strain = Change in diameter/ original diameter. = δD/D Longitudinal strain = Change in length / Original length = δL/L Poisson’s Ratio σ = (δD/D)/( δL/L) σ = (L δD)/( DδL)  Experimental Determination of Y by Loading a Rectangular Thin Bar at The Center: Here the given beam(meter scale) is supported symmetrically on two knife edges and loaded at its centre. The maximum depression is produced at its centre. Since the load is applied only one point of the beam, the bending is not uniform through out the beam and the bending of the beam is called non- uniform bending. In non-uniform bending (central loading), the Young's modulus of the material of the bar is given by I is the moment of inertia of the bar. for a rectangular bar, In non uniform bending, the young’s modulus of the material of the bar is given by,  m - Mass loaded for depression. g - Acceleration due to gravity. l - Length between knife edges. b - Breadth of the bar using venire calipers. d - Thickness of the bar using screw gauge. e - Depression of the bar.  Torsional Oscillations : A body suspended by a thread or wire which twists first in one direction and then in the reverse direction, in the horizontal plane is called a torsional pendulum. The first torsion pendulum was developed by Robert Leslie in 1793. A simple schematic representation of a torsion pendulum is given below,  The period of oscillation of torsion pendulum is given as, Where I=moment of inertia of the suspended body; C=couple/unit twist But we have an expression for couple per unit twist C as, Where l =length of the suspension wire; r=radius of the wire; n=rigidity modulus of the suspension wire Substituting (2) in (1) and squaring, we get an expression for rigidity modulus for the suspension wire as,  END THANK YOU