## 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.

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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.