Elasticity Notes

ELASTICITY Page # 1 ELASTICITY 1. DEFINATION Elasticity is that property of the material of a body by virtue of which the body opposes any change in its shape or size when deforming forces are applied to it, and recovers its original state as soon as the deforming forces are removed. On the basis of defination bodies may be classified in two types : (a) Perfectly Elastic (P.E.) : If body regains its original shape ans size completely after removal of force. Nearest approach P.E. : quartz-fibre (b) Perfectly Plastic (P.P.) : If body does not have tendency to recover its original shape and size. Nearest Approach P.P. : Peetty Limit of Elasticity : The maximum deforming force upto which a body retains its property of elasticity is called the limit of elasticity of the material of the body. 2. STRESS When a deforming force is applied to a body, it reacts to the applied force by developing a reaction (or restoring force which, from Newton's third law, is equal in magnitude and opposite in direction to the applied force. Thereaction force per unit area of the body which is called into play due to the action of the applied force is called stress. Stress is measured in units of force per unit area, i.e. Nm–2. Thus. F Stress = A where F is the applied force and A is the area over which it acts. A 10 N Stress = 10/A Unit of stress : N/m2 Dimension of stress : M1L-1T-2 2.1 Types of stress : Three Types of Stress : (A) Tensile Stress : Pulling force per unit area. It is applied parallel to the length It causes increase in length or volume F A F F (B) Compressive Stress : Pushing force per unit area. A F It is applied parallel to the length It causes decrease in length or volume (C) Tangential Stress : Tangential force per unit area. It causes shearing of bodies. Note : 1. If the stress is normal to surface called normal stress. 2. Stress is always normal to surface in case of change in length of a wire or volume of body. 3. When external force compresses the body  Nature of atomic force will be repulsive. 4. When external forces expanses the body  Nature of atomic force will be attractive. Difference between Pressure v/s Stress : S. No. Pressure Stress 1 Pressure is always normal to the Stress can be normal or area. tangential 2 Always compressive in nature May be compressive or tensile in nature. 3 Sc alar Tensor 1 ELASTICITY Page # 2 3. STRAIN When a deforming force is applied to a body, it may suffer a change is size or shape. Strain is defined as the ratio of the change in size or shape to the original size or shape of the body. Strain is a number; it has no units or dimensions. The ratio of the change in length to the original length is called longitudinal strain. The ratio of the change in volume to the original volume is called volume strain. The strain resulting from a change in shape is called shearing strain. Strain = ΔL L0 = final length– original length original length =  T,, Note : Original and final length should be at same temperature. F 3.1Types of strain : Three Types of Strain : (A) Linear Strain : Change in length per unit length is called linear strain Linear Strain = Change in length Original length L L (B) Volume Strain : Change in volume per unit volume is called volume strain. = V  V Volume Strain Volume sirain Change in volume V = = Original volume V x (C) Shear Strain : Angle through which a line originally normal to fixed surface is turned. L  x  = L Note : Strain is unitless. Shear strain 4. WORK DONE IN STRETCHING A WIRE In stretching a wire work is done against internal restoring forces. This work is stored in body as elastic potential energy or strain energy. If L = length of wire & A = Cross-sectional Area. F/A YA Y =  F = x x/L L work done to increase dx length YA dW = Fdx = xdx L L YA 1 YA Total work done = W = xdx = (L)2 0 L 2 L   L  W 1  Work done per unit volume = = Y  V 2  L  2 [ V = AL] W 1 = Y (strain)2 V 2 W 1 = x stress x strain V 2 W 1 = V 2 ( stress)2 Y  1 W = AL [ Y = 1 2 F L × A L Stress ] Strain ELASTICITY W = Page # 3 1 1 F × L = load x elongation 2 2 5. STRESS-STAIN CURVE If we increase the load gradually on a vertical suspended metal wire, stress C D E B A In Region OA : O strain Strain is small (< 2%) Stress  Strain  Hook's law is valid. Slope of line OA gives Young's modulus Y of the material. In Region AB : Stress is not proportional to strain, but wire will still regain its original length after removing of stretching force. In region BC : Wire yields  strain increases rapidly with small change in stress. This behavior is shown up to point C known as yield point. In region CD : Point D correspondes to maximum stress, which is called point of breaking or tensile strength. In region DE : The wire literally flows. The maximum stress corresponding to D after which wire begin to flow. In this region strain increase even if wire is unloaded and rupture at E. 6. HOOKES' LAW Hookes' law states that, within the elastic limit, the stress developed in a bodyis proportional to the strain produced in it. Thus the ratio of stress to strain is a constant. This constant is called the modulus of elasticity. Thus stress Modulus of elasticity = strain Since strain has no unit, the unit of the modulus of elasticity is the same as that of stress, namely, Nm–2 7. YOUNG'S MODULUS Suppose that a rod of length l and a uniform crossectional area a is subjected to a logitudinal pull. In other words, two equal and opposite forces are applied at its ends. F Stress = A The stress in the present case is called linear stress, tensile stress, or extensional stress. If the direction of the force is reversed so that L is negative, we speak of compressional strain and compressional stress. If the elastic limit is not exceeded, then from Hooke's law Stress  strain or Stress = Y × strain stress F L Y  . or ...(1) strain A L where Y, the constant of proportionality, is called the Young's modulus of the material of the rod and may be defined as the ratio of the linear stess to linear strain, provided the elastic limit is not exceeded. Since strain has no unit, the unit of Y is Nm–2. 7.1LINEAR EXPANSION When the rod is heated, its increase in length L is proportional to its original length L0 and change in temperture T where T is in °C or K. L  L 0  L L0 Before heating After heating dL =  L0dT  L =  L0T If a T << 1 L  = L T where  is called the coefficient of linear expansion whose unit is °C–1 or K–1. 0 L = L0(1 +  T) . Where L is the length after heating the rod. ELASTICITY Page # 4 Variation of  with temperatue and distance (a) If  varies with distance,  = ax + b Then total expansion =  (ax  b) Tdx dx x (b) If  varies with temperature,  = f(T) Then L =  L dT 0 Note : • Actually thermal expansion is always 3-D expansion. When other two dimensions of object are negligible with respect to one, then observations are significant only in one dimension and it is known as linear expansion. • Avery linear dimenstions of the object changes in the same fashion 7.2Measurement of length by metallic scale : Case (I) When object is expanded only  2  1 {1   0 ( 2 – 1)} 1 = actual length of object at 1°C = measure length of object at 1°C.  2 = actual length of object at 2°C = measured length of object at 2°C 0 = linear expansion coefficient of object 2 1 , , 0 3 1 2 2 Case (II) When only measureal instrument is expaneded actual length of object will not change but measured value (MV) decreases. 1 MV = 1{1 –  s ( 2 – 1)} s = linear expansion coefficient of measuring instrument. 2 Case (III) 2  C If both expanded simultaneously MV = l1{1 + (0 – s) (2 – 1)} (i) if 0 > s, then measured value is more than actual value at 1°C (ii) If 0 < s, then measured value is less than actual value at 1°C at 1 °C MV = 3.4, 2 °C MV = 4.1 0 1 2 3 4 5 1 C 2  C 0 1 2 3 4 7.3VOLUME OR CUBICAL EXPANSION When a solid is heated and its volume increases, then the expansion is called volume expansion or cubical expansion. Note : Now after increase in temp by T Vf = a bc b = a[1  T ]3 bc = abc [1 + 3  T]  T << 1 vf = vi [1 + 3 T] So 3 =  = coefficient of volume expansion. c a vi=abc a Hollow a Cubical Container of Solid cube same material 1. When temperature changes the volume of the container and volume of the cube change in the same fashion because a changes in the same fashion. 2. In volume expansion of container we use  of the container material.