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Mechanical Properties of Solids Chapter 8
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Mechanical Properties of Solids Chapter 8

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Questions and Answers

What is the property of a body that tends to regain its original size and shape when the applied force is removed?

elasticity

What is the deformation caused by a body that does not tend to regain its previous shape?

plastic deformation

What is the property called when a body is pressed with a hand and pushed horizontally?

  • Tensile stress
  • Compressive stress
  • Shearing stress (correct)
  • Hydraulic stress

    • Hooke's Law states that stress is not proportional to strain for small deformations.

    <p>False</p> Signup and view all the answers

    What is the typical term for a material if the ultimate strength and fracture points are close?

    <p>brittle</p> Signup and view all the answers

    What is the term used to describe a material if the ultimate strength and fracture points are far apart?

    <p>ductile</p> Signup and view all the answers

    What is the ratio defined as Young's Modulus?

    <p>Stress to Strain</p> Signup and view all the answers

    What is the unit of Young's Modulus?

    <p>N m^-2 or Pascal (Pa)</p> Signup and view all the answers

    What is the name given to materials like rubber and tissue of aorta that exhibit large elastic regions?

    <p>elastomers</p> Signup and view all the answers

    What is the ratio of hydraulic stress to the corresponding hydraulic strain called?

    <p>bulk modulus</p> Signup and view all the answers

    What is the symbol used to denote Bulk Modulus?

    <p>B</p> Signup and view all the answers

    The negative sign in the bulk modulus formula B = - p/(∆V/V) indicates that with an increase in pressure, a decrease in _____ occurs.

    <p>volume</p> Signup and view all the answers

    Which of the following materials has the highest bulk modulus?

    <p>Nickel</p> Signup and view all the answers

    Bulk moduli for liquids are generally smaller than for solids.

    <p>True</p> Signup and view all the answers

    What is the formula for the amount a bar sags when loaded at the centre by a load? Write it in terms of the load (W), length (l), breadth (b), depth (d), and Young's modulus (Y).

    <p>$\delta = \frac{Wl^3}{4bd^3Y}$</p> Signup and view all the answers

    What is the elastic limit for a typical rock if the shear modulus is 30 x 10^7 N m^-2?

    <p>30 x 10^7 N m^-2</p> Signup and view all the answers

    What is the relationship between stress and strain under tension or compression based on Hooke's law?

    <p>$F/A = Y\Delta L/L$</p> Signup and view all the answers

    What are the three types of stresses mentioned in the text?

    <p>Hydraulic stress</p> Signup and view all the answers

    Hooke's law is only valid in the linear part of the stress-strain curve.

    <p>True</p> Signup and view all the answers

    Calculate the bulk modulus of water given: Initial volume = 100.0 litre, Pressure increase = 100.0 atm, Final volume = 100.5 litre. Compare the bulk modulus of water with that of air at constant temperature. Explain why the ratio is so large.

    <p>Bulk modulus of water = $2.27 × 10^9$ Pa. The bulk modulus of water is much larger than air due to water being nearly incompressible compared to air.</p> Signup and view all the answers

    What is the density of water at a depth where the pressure is 80.0 atm, given that its density at the surface is 1.03 × 10^3 kg m^3?

    <p>Density of water at 80.0 atm pressure = $1.08 × 10^3$ kg m^3</p> Signup and view all the answers

    Calculate the fractional change in volume of a glass slab when subjected to a hydraulic pressure of 10 atm.

    <p>Fractional change in volume = 0.0194 or 1.94%</p> Signup and view all the answers

    Determine the volume contraction of a solid copper cube with 10 cm edges when subjected to a hydraulic pressure of 7.0 × 10^6 Pa.

    <p>Volume contraction = $7.0 × 10^{-8}$ m^3</p> Signup and view all the answers

    How much should the pressure on a litre of water be changed to compress it by 0.10%?

    <p>Pressure change required = 1.0 atm</p> Signup and view all the answers

    Study Notes

    Mechanical Properties of Solids

    • Mechanical properties of solids are important in engineering design and everyday life.
    • Solids can be stretched, compressed, and bent, and their properties can be changed by applying external forces.

    Stress and Strain

    • When a force is applied to a solid, it deforms, and the body develops a restoring force to counteract the applied force.
    • Stress is the restoring force per unit area, and its SI unit is N m-2 or Pascal (Pa).
    • Strain is the ratio of change in dimension to the original dimension, and it has no units or dimensional formula.

    Types of Stress and Strain

    • There are three types of stress: tensile, compressive, and shear stress.
    • Tensile stress occurs when a force is applied normal to the cross-sectional area of a body, causing it to elongate.
    • Compressive stress occurs when a force is applied parallel to the cross-sectional area of a body, causing it to compress.
    • Shear stress occurs when a force is applied tangentially to the surface of a body, causing it to deform by an angle.

    Hooke's Law

    • Hooke's law states that stress and strain are proportional to each other for small deformations.
    • The law is an empirical law and is valid for most materials.
    • The proportionality constant is known as the modulus of elasticity.

    Stress-Strain Curve

    • The stress-strain curve is a graph that shows the relationship between stress and strain for a given material.
    • The curve can be used to determine the elastic limit, yield strength, and ultimate tensile strength of a material.
    • The elastic region of the curve is where the stress and strain are proportional to each other.

    Elastic Moduli

    • Elastic moduli are measures of the stiffness of a material.
    • Young's modulus is a measure of the stiffness of a material under tensile or compressive stress.
    • Shear modulus is a measure of the stiffness of a material under shear stress.

    Applications of Elastic Behavior

    • The elastic behavior of materials is important in engineering design, such as in the design of buildings, bridges, and machines.
    • The elastic behavior of materials is also important in biological systems, such as in the structure of bones and tissues.

    Examples and Problems

    • A structural steel rod is stretched by a 100 kN force, and its elongation is calculated.
    • A copper wire and a steel wire are stretched by a load, and the net elongation is calculated.
    • The compression of a thighbone under an extra load is calculated.

    Tables

    • Table 8.1 shows the values of Young's moduli and yield strengths of some materials.
    • Table 8.2 shows the values of shear moduli of some common materials.### Mechanical Properties of Solids
    • Stress and Strain:
      • Stress: the restoring force per unit area, measured in N m^-2 (Pa) or GPa.
      • Strain: the fractional change in dimension, a pure number with no dimensions or units.
    • Young's Modulus (Y):
      • The ratio of tensile stress to tensile strain, measured in N m^-2 (Pa) or GPa.
      • Symbol: Y, unit: N m^-2 (Pa) or GPa.
      • Young's modulus is a measure of the stiffness of a material.
    • Compression in Thighbone:
      • The weight of the body is supported by the legs, and the thighbone is compressed.
      • The compression in each thighbone (∆L) can be computed using the formula: ∆L = (F × L)/(Y × A).
      • The fractional decrease in the thighbone is ∆L/L = 0.000091 or 0.0091%.

    Shear Modulus

    • Shear Modulus (G):
      • The ratio of shearing stress to the corresponding shearing strain, measured in N m^-2 (Pa) or GPa.
      • Symbol: G, unit: N m^-2 (Pa) or GPa.
      • The shear modulus is a measure of the rigidity of a material.
      • The shear moduli of a few common materials are given in Table 9.2.

    Bulk Modulus

    • Bulk Modulus (B):
      • The ratio of hydraulic stress to the corresponding hydraulic strain, measured in N m^-2 (Pa) or GPa.
      • Symbol: B, unit: N m^-2 (Pa) or GPa.
      • The bulk modulus is a measure of the compressibility of a material.
      • The bulk moduli of a few common materials are given in Table 8.3.

    Elastic Potential Energy

    • Elastic Potential Energy in a Stretched Wire:
      • When a wire is put under a tensile stress, work is done against the inter-atomic forces.
      • The work is stored in the wire in the form of elastic potential energy.
      • The elastic potential energy per unit volume of the wire (u) is given by the formula: u = (1/2) × σ ε.

    Poisson's Ratio

    • Poisson's Ratio:
      • The ratio of the lateral strain to the longitudinal strain in a stretched wire.
      • A pure number with no dimensions or units.
      • Poisson's ratio is a measure of the lateral strain response to a longitudinal tensile strain.

    Applications of Elastic Behaviour of Materials

    • Crane Design:
      • The design of a crane requires knowledge of the elastic behaviour of materials.
      • The rope should be able to withstand the weight of the load without deforming permanently.
    • Beam Design:
      • The design of a beam requires knowledge of the elastic behaviour of materials.
      • The beam should be able to withstand the load without bending or breaking.
      • The formula for the sagging of a beam under a load is: δ = W l^3/(4bd^3Y).
    • Bridge Design:
      • The design of a bridge requires knowledge of the elastic behaviour of materials.
      • The beam should be able to withstand the load of the flowing traffic, the force of winds, and its own weight.
      • The cross-sectional shape of the beam is critical in reducing the bending.
    • Mountain Height:
      • The maximum height of a mountain on earth is limited by the elastic properties of rocks.
      • The stress due to the weight of the material on top should be less than the critical shearing stress at which the rocks flow.
      • The formula for the maximum height of a mountain is: h = 30 × 10^7 N m^-2/(3 × 10^3 kg m^-3 × 10 m s^-2) = 10 km.### Types of Stresses
    • There are three types of stresses: tensile stress, shearing stress, and hydraulic stress
    • Tensile stress is associated with stretching, while compressive stress is associated with compression
    • Shearing stress occurs when a pair of forces is applied parallel to the upper and lower faces of an object
    • Hydraulic stress occurs when an object undergoes compression due to a surrounding fluid

    Hooke's Law

    • Hooke's law states that for small deformations, stress is directly proportional to strain for many materials
    • The constant of proportionality is called the modulus of elasticity
    • There are three elastic moduli: Young's modulus, shear modulus, and bulk modulus
    • Young's modulus is used to describe the elastic behavior of objects under tension or compression
    • Shear modulus is used to describe the elastic behavior of objects under shearing stress
    • Bulk modulus is used to describe the elastic behavior of objects under hydraulic stress

    Young's Modulus

    • Young's modulus (Y) is used to calculate the stress and strain of an object under tension or compression
    • The formula for Young's modulus is F/A = Y∆L/L, where F is the applied force, A is the cross-sectional area, and ∆L/L is the tensile or compressive strain

    Shear Modulus

    • Shear modulus (G) is used to calculate the stress and strain of an object under shearing stress
    • The formula for shear modulus is F/A = G × ∆L/L, where F is the applied force, A is the cross-sectional area, and ∆L/L is the displacement of one end of the object in the direction of the applied force

    Bulk Modulus

    • Bulk modulus (B) is used to calculate the pressure and volume change of an object under hydraulic stress
    • The formula for bulk modulus is p = B (∆V/V), where p is the pressure, ∆V/V is the volume strain, and B is the bulk modulus

    Points to Ponder

    • Hooke's law is only valid in the linear part of the stress-strain curve
    • Young's modulus and shear modulus are only relevant for solids, as they have lengths and shapes
    • Bulk modulus is relevant for solids, liquids, and gases, and refers to the change in volume under uniform stress
    • Materials with larger values of Young's modulus require a larger force to produce small changes in length
    • Materials that stretch more are not necessarily more elastic; in fact, materials that stretch less are considered more elastic

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    Description

    This quiz covers the mechanical properties of solids, including stress and strain, and the motion of rigid bodies.

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