Stress Strain Elasticity Modulus Young s Modulus Shear Modulus Bulk Modulus. Case study

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Juniper Hawkins
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2 Stress Strain Elasticity Modulus Young s Modulus Shear Modulus Bulk Modulus Case study 2

3 In field of Physics, it explains how an object deforms under an applied force Real rigid bodies are elastic we can slightly change their dimensions by pulling, pushing, twisting or compressing them. 3

4 Stress: Deforming force per unit area Strain: Unit deformation Stress = Elasticity Modulus x Strain Elastic modulus: describes the elastic behavior (deformations) of objects as they respond to forces that act on them Tensile stress: associated with stretching 4

5 Shearing stress Hydraulic stress Stress = Elasticity Modulus x Strain Elastic modulus: describes the elastic behavior (deformations) of objects as they respond to forces that act on them 5

6 (1) Stress = cte x Strain Recovers original dimensions when stress removed. (2) Stress > yield strength S y specimen becomes permanently deformed. (3) Stress > ultimate strength S u specimen breaks. Stress = F A L Strain = L F L = E A L ( F= force applied perpendicular to the area A of the object) ( fractional change in length of the specimen) Stress = (Young s modulus) x Strain Units of Young modulus: N/m 2 6

7 Elasticity in Length Everything is a spring L F = Y A L until it breaks! 7

8 The phenomenon of yielding occurs at the onset of plastic or permanent deformation Yield strength is indicative of stress at which plastic deformation begins Tensile strength is taken as the stress level at the maximum point on the engineering stress-strain curve; it represents the maximum tensile stress that may be sustained by a specimen 8

$Ductility is a measure of the degree to which a material will plastically deform by the time fracture occurs Quantitatively, ductility is measured in terms of percent$

9 Ductility is a measure of the degree to which a material will plastically deform by the time fracture occurs Quantitatively, ductility is measured in terms of percent elongation and reduction area Percent elongation (%EL) is a measure of the plastic strain at fracture Percent reduction in area (%RA) may be calculated according to equation 9

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11 One end of steel rod of radius R = 9.5 mm and length L =81 cm is held in a vise. A force of magnitude F = 62 kn is then applied perpendicularly to the end face (uniformly across the area) at the other end. What are the stress on the rod and the elongation L and strain of the rod? 11

$SHEAR MODULUS Elasticity of Shape F Stress = ( F= force in the plane of the area A) A Strain = x ( fractional change in length of the$

12 SHEAR MODULUS Elasticity of Shape F Stress = ( F= force in the plane of the area A) A Strain = x ( fractional change in length of the specimen) L X F = S A L0 Shear moduli and Young moduli are comparable in magnitude for most materials (Liquids have zero shear modulus) 12

13 Twisted bar In the field of solid mechanics, torsion is the twisting of an object due to an applied torque. Torsion is expressed in newtons per square metre (Pa) or pounds per square inch (psi) while torque is expressed in newton metres (N m) or foot-pound force (ft lbf). In sections perpendicular to the torque axis, the resultant shear stress in this section is perpendicular to the radius. 13

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$pound per square inch Bulk modulus B relates fractional volume change to pressure change V P= B V 0 Hydraulic Stress = (Bulk modulus) x Hydraulic compression$

15 Volume Elasticity Pressure is defined as force per unit area P F A Unit of pressure is N/m 2 = Pascal (Pa) Atmospheric pressure ~ 10 5 Pa= 15psi 1 psi = 1 pound per square inch Bulk modulus B relates fractional volume change to pressure change V P= B V 0 Hydraulic Stress = (Bulk modulus) x Hydraulic compression 15

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17 If the tension in the cable was 940 N. What diameter should a 10-m-long steel wire have if we do not want it to stretch more than 0.5 cm under these conditions? 17

18 From the definition of Young s modulus, we can solve for the required cross-sectional area. Assuming that the cross section is circular, we can determine the diameter of the wire. 18

19 A solid brass sphere is initially surrounded by air, and the air pressure exerted on it is 1.0 x 10 5 N/m 2 (normal atmospheric pressure). The sphere is lowered into the ocean to a depth at which the pressure is 2.0 x 10 7 N/m 2. The volume of the sphere in air is 0.50 m 3. By how much does this volume change once the sphere is submerged? 19

20 From the definition of bulk modulus, we have 20

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22 STRESS, STRAIN AND HOOKE S LAW L F / A = Y L X F / A = S L0 V P= B V 0 F/A or pressure are known as stress Fractional changes in length or volume are known as strain Above relations between stress and strain account for elastic behavior and Hooke s law 22

23 A steel wire has diameter of 2 mm and length 4 m. The wire used to hang an object with a mass of 5.0 kg. The Young s modulus of the wire is 200 x 10 9 N/m 2. Based on those information, determine the elongation and spring constant of the wire 23

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25 or or If kef is a substitute constant for the arrangement of the two wires above, then apply From the equation of total length or We get 25

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27 If kef is a substitute constant for the arrangement of the two wires above, then apply Because the downward force and the amount of upward force at the load must be equal then or 27

28 What is the period of oscillation of the object hung on two springs arranged in parallel In serial if each of the spring constant 250 N/m and 550 N/m. Suppose the weight of object is 600 N (g = 10 N/kg). The oscillation frequency follow this equation 28

29 A Table has three legs that are 1.00 m in length and a fourth leg that is longer by d = 0.50 mm, so that the table wobbles slightly. A steel cylinder with the mass M= 290 kg is placed on the table (which has mass less than M) so that four legs are compressed but unbuckled and the table is level but no longer wobbles. The legs are wooden cylinders with crosssectional area A= 1.0 cm 2 ; Young modulus is E = 1.3 x N/m 2. What are the magnitudes of the forces on the legs from the floor? 29

Objectives: After completion of this module, you should be able to:

Chapter 12 Objectives: After completion of this module, you should be able to: Demonstrate your understanding of elasticity, elastic limit, stress, strain, and ultimate strength. Write and apply formulas