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

Disclaimer: Compete.etutor.co may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. Compete.etutor.co makes no representations whatsoever concerning the content of these sites and the fact that compete.etutor.co has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by compete.etutor.co with respect to such sites, its services, the products displayed, its owners, or its providers.

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.