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**PH0101 UNIT 1 LECTURE 1 Elasticity and Plasticity Stress and Strain**

Hooke’s Law Twisting couple on a cylinder Worked and Exercise Problems PH0101 UNIT 1 LECTURE 1

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**Elasticity and Plasticity**

The property of the body by virtue of which it tends to regain its original shape or size on the removal of deforming force is called elasticity. The property of the body by virtue of which it tends to retain the altered size and shape on removal of deforming forces is called plasticity. PH0101 UNIT 1 LECTURE 1

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Stress and Strain Stress is a quantity that characterizes the strength of the forces causing the deformation, on a “force per unit area” basis. The deforming force per unite area of the body is called stress. The SI unit of stress is the Pascal (abbreviated Pa, and named for the 17th century French scientist and philosopher Blaise Pascal). PH0101 UNIT 1 LECTURE 1

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**One Pascal equals one Newton per square meter.1 Pascal = 1Pa = 1N/m2.**

Strain is a quantity which describes the resulting deformation. Strain is the fractional deformation produced in a body when it is subjected to a set of deforming forces. Strain being ratio has no units PH0101 UNIT 1 LECTURE 1

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**There are following three types of stress and strain **

Tensile stress and strain Bulk stress and strain Shear stress and strain PH0101 UNIT 1 LECTURE 1

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**Tensile Stress and Strain**

Initial state lo PH0101 UNIT 1 LECTURE 1

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Tensile stress at the cross section is defined as the ratio of the force to the cross – sectional area A. Tensile stress = The tensile strain of the object is equal to the fractional change in length, which is the ratio of the elongation to the original length . PH0101 UNIT 1 LECTURE 1

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**Tensile strain is stretch per unit length**

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Bulk Stress and Strain Initial state PH0101 UNIT 1 LECTURE 1

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**Pressure plays the role of stress in a volume deformation.**

The force per unit area is called the pressure PH0101 UNIT 1 LECTURE 1

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**Bulk stress = , an increase in pressure. **

The fractional change in volume that is the ratio of the volume change to the original volume Vo, is called as bulk strain. Bulk (volume) strain = Volume strain is the change in volume per unit volume. PH0101 UNIT 1 LECTURE 1

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**Shear Stress and Strain**

Initial state PH0101 UNIT 1 LECTURE 1

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**Shear strain is defined as the ratio of the displacement x to the transverse dimension h**

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Hooke’s Law Hooke’s law states that within the elastic limit, the stress developed is directly proportional to the strain. The constant of proportionality is the elastic modulus ((or modulus of elasticity). = elastic modulus (Hooke’s law) PH0101 UNIT 1 LECTURE 1

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Strain Plastic range Stress Elastic limit Elastic range Permanent set The plot between stress and strain is called stress - strain diagram. It is clear from the graph that Hooke’s law holds good only for the straight line portion of the curve PH0101 UNIT 1 LECTURE 1

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**Three kinds of elastic moduli**

Elastic Modulus Definition Nature of strain Young’s modulus (Y) Tensile strain Change of shape and size Bulk modulus (B) Bulk strain Change of size but not shape Shear modulus or Rigidity modulus (S) Shear strain Change of shape but not size Bulk stress Tensile stress Shear stress PH0101 UNIT 1 LECTURE 1

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**Approximate elastic moduli**

Material Young’s Modulus Y(Pa) Bulk Modulus B(Pa) Shear Modulus S(Pa) Aluminium 7.0x1010 7.5x1010 2.5x1010 Brass 9.0 x1010 6.0 x1010 3.5 x1010 Copper 11 x1010 14 x1010 4.4 x1010 Crown Glass 5.0 x1010 2.5 x1010 Iron 21 x1010 16 x1010 7.7 x1010 Lead 1.6 x1010 4.1 x1010 0.6 x1010 Nickel 17 x1010 7.8 x1010 Steel 20 x1010 7.5 x1010 PH0101 UNIT 1 LECTURE 1

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**Compressibility Poisson’s ratio**

The reciprocal of bulk modulus is called compressibility.The unit of compressibility is same as that of reciprocal pressure, Pa-1 Poisson’s ratio Within the elastic limits the ratio of the lateral strain to the longitudinal strain is constant for the material of the body and is known as Poisson’s ratio and is denoted by . PH0101 UNIT 1 LECTURE 1

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Worked Example 1 A steel rod 2.0m long has a cross sectional area of 0.30cm2. The rod is now hung by one end from a support structure and a 550kg milling machine is hung from the rod’s lower end. The Young’s modulus of steel is 20x1010Pa. Determine the stress, the strain and the elongation of the rod. PH0101 UNIT 1 LECTURE 1

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**Elongation = = (strain) x o = (9.0x10-4) (2.0m) = 0.0018m = 1.8mm**

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Worked Example 2 A box – shaped piece of gelatin dessert has a top area of 15cm2 and a height of 3cm. When a shearing force of 0.50N is applied to the upper surface, the upper surface displaces 4mm relative to the bottom surface. What are the shearing stress, the shearing strain, and the shear modulus for the gelatin? PH0101 UNIT 1 LECTURE 1

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**Twisting Couple on a Cylinder (or Wire)**

The twisting of a structural member about its longitudinal axis by two equal and opposite torques is expressed through a certain angle. The stress seen in this situation is not tensile or compressive, it is said to be shearing or shear stress. The strain in this case is measured by an angle in unit of radians PH0101 UNIT 1 LECTURE 1

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**Let us consider a cylindrical rod of length l and radius r with its upper end fixed.**

Let a twisting couple-be applied to the lower end of the rod in a plane perpendicular to its length and let the rod twist through an angle θ (radians). While the rod is twisted restoring couple acts in the opposite direction and in the position of equilibrium, the twisting couple is equal and opposite to the restoring couple. PH0101 UNIT 1 LECTURE 1

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To calculate this couple, let us consider the solid cylinder to be made up of a larger number of concentric thin walled cylinders. Let us consider one such hollow cylinder of radius x, and radial thickness dx. When the rod is twisted through an angle θ, the angle through which the rim of the cylinder is sheared is ф. PH0101 UNIT 1 LECTURE 1

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i.e. BB’=l ф Also BB’=x θ :. ф = From this it is clear that with x, ф varies. ф has the maximum value when x is the greatest. i.e., the strain is maximum on the outermost part of the cylinder and minimum on the innermost. In other words, the shearing stress is not uniform through out the material. PH0101 UNIT 1 LECTURE 1

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**If N is the rigidity modulus,**

Hence F =N ф= PH0101 UNIT 1 LECTURE 1

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**Now the face area of the hollow cylinder = 2πxdx **

:. Total shearing force on this area = 2πxdx. Nx θ / l = 2πN . θ. x2.dx l PH0101 UNIT 1 LECTURE 1

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**Therefore, moment of this force about the axis 00’ = 2πN θ x2dx . x l **

Total twisting couple of the cylinder can be obtained by integrating this expression between limits x = 0 and x = r. PH0101 UNIT 1 LECTURE 1

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**:. Total twisting couple on the cylinder =**

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**Twisting couple per unit twist C= πNr4/2l **

If θ = l radian, we have Twisting couple per unit twist C= πNr4/2l This twisting couple per unit twist of the wire is called the torsional rigidity or modulus of torsion of the cylinder of wire. It is evident form this relation that the couple required is proportional to the fourth power of the radius. PH0101 UNIT 1 LECTURE 1

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**Twisting couple of the cylinder**

Note: Hollow Cylinder For a hollow cylinder of the same length l and of inner radius r1 and outer radius r2, Twisting couple of the cylinder PH0101 UNIT 1 LECTURE 1

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**If θ = 1 radian, twisting couple per unit twist C= (r24 – r14)**

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**Worked Example 3 Angle twisted by wire θ = radians**

A wire of length 1 meter and diameter 1 mm is fixed at one end and a couple is applied at the other end so that the wire twists by π/2 radians. Calculate the moment of the couple required if rigidity modulus of the material = 2.8 ×1010 N/m2. Rigidity modulus of the material N = 2.8x1010 N/m2 Angle twisted by wire θ = radians PH0101 UNIT 1 LECTURE 1

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Required couple C = 4.3 x10-3 N-m. PH0101 UNIT 1 LECTURE 1

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Worked Example 4 A wire of length l m and diameter 1mm is clamped at one of its ends. Calculate the couple required to twist the other end by 90o. Given N = 2.8 × 1010 N/m2. The torque required to twist the free end of a clamped wire of length through θ radian will be PH0101 UNIT 1 LECTURE 1

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**For θ = 90o = π/2 radian , C = N= 2.8x10-10 N/m2, l =1m**

r = 5mm = m :.C PH0101 UNIT 1 LECTURE 1

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Exercise Problem A wire of length 1 metre and diameter 1mm is clamped at one of its ends. Calculate the couple required to twist the other end by 90o. Given the modulus of rigidity is 298GPa. Hint : Twisting couple = PH0101 UNIT 1 LECTURE 1

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THANK YOU THANK YOU PH0101 UNIT 1 LECTURE 1

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