Mechanical Properties of Solids

Hooke's law and Elastic Moduli

  • For a given material longitudinal strain : Shear strain : Bulk strain = 1 : 2 : 3
  • Elastic limit is the maximum value of stress with in which a body can regain its original size and shape.
  • Hooke's law states that with in the Elastic limit stress is directly proportional to strain.
  • Modulus of Elasticity E is defined as the ratio of stress to strain produced in a body
  • With in Elastic Limit stress strain graph is a straight line passing through the origin.
  • Slope of stress strain graph given modules of Elasticity.
  • The point where Elasticity ends and plasticity begins is called yield point.
  • The permanent increase in length of the wire after removing the load called permanent set.
  • The stress required to break a wire called breaking stress.
  • Breaking stress in mathematically breaking force per unit area.
  • Breaking stress depends upon nature of the material but it is independent of dimensions.
  • Breaking force is independent of length of the wire but it depends up on nature of material and area of cross section.
  • Poisson's ratio is defined as the ratio of lateral contraction strain to the longitudinal elongation strain.
  • Poisson's ratio has theoretical limits −1 to 0.5 and practical limits 0 to 0.5.
  • Poisson's ratio has no units and dimension
  • Elastic fatigue is the state of temporary loss of Elastic nature of material.
  • The delay in regaining the original state on removal of the deforming force on a body called Elastic after Effect.
  • For a perfectly plastic body the Elastic after effect is infinity.

View the Topic in this video From 0:40 To 59:37

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1. Hooke's Law: Within the limit of elasticity, the stress is proportional to the strain.
                  Stress ∝ Strain
                  Stress= E × Strain

2. Increment in the length of wire l=\frac{FL}{\pi r^{2}Y}

3. Breaking force = P × A

4. Young's Modulus of Elasticity: It is defined as the ratio of normal stress to the longitudinal strain within the elastic limit.
                                             \tt Y=\frac{Normal\ stress}{Longitudinal\ strain}
                                             \tt Y=\frac{F\Delta l}{Al}=\frac{Mg\ \Delta l}{\pi\ r^{2}l}

5. Bulk Modulus of Elasticity: It is defined as the ratio of normal stress to the volumetric strain within the elastic limit.
                                             \tt K=\frac{Normal\ stress}{Volumetric\ strain}
                                             \tt K=\frac{FV}{A\Delta V}=\frac{\Delta p V}{\Delta V}

6. Modulus of Rigidity (η) (Shear Modulus)
It is defined as the ratio of tangential stress to the shearing strain, within the elastic limit.
                                             \tt \eta=\frac{Tangential\ stress}{Shearing\ strain}
                                             \tt \eta=\frac{F}{A \theta}

7. Compressibility: Compressibility of a material is the reciprocal of its bulk modulus of elasticity.
                               Compressibility (C) = \tt \frac{1}{K}

8. Breaking stress: Breaking stress is fixed for a material but breaking force varies with area of cross-section of the wire.
                              Safety factor = \tt \frac{Breaking\ stress}{Working\ stress}

9. Thermal stress: When temperature of a rod fixed at its both ends is changed, then the produced stress is called thermal stress.
                             Thermal stress \tt \frac{F}{A}=Y\alpha\ \Delta \theta

10. Poisson's ratio (σ)= \tt \frac{Lateral\ strain}{Longitudinal\ strain}=\frac{-\Delta R/R}{\Delta l/l}

11. Relation between Y, K, η and σ
(i) Y = 3K (1 − 2σ)
(ii) Y = 2η (1 + σ)
(iii) \tt \sigma=\frac{3K-2\eta}{2\eta+6K}
(iv) \tt \frac{9}{Y}=\frac{1}{K}+\frac{3}{\eta}\ or\ Y=\frac{9K \eta}{\eta+3K}

12. Potential energy in a stretched wire
U = Average force × Increase in length = \frac{1}{2}\ F\Delta l

13. Elastic potential energy of a stretched spring = \frac{1}{2}\ kx^{2}
where, k = force constant of spring and x = Change in length.