# Solids

Mechanical Properties of Solids

Deforming Force – A force which when applied changes the shape and size of the body.

Elasticity – The Property due to which a body regains it’s original shape and size when deforming force is removed.

Plasticity – the property due to which a body does not regain it’s shape and size when deforming force is removed.

Stress – The restoring force setup per unit area of the deformed body is called its stress.

Stress = Restoring Force / Area = F / A

SI Unit of stress is N/m2
CGS unit is dyne/cm2
Dimensional Formula is [M1L-1T-2]

Types of Stress –
1) Normal Stress – It is of three types —
a) Tensile Stree
b) Compressive Stree
c) Hydraulic Stress
2) Tangential / Shearing Stress

Strain – The ratio of change in dimension produced in the body to the original dimension is called strain.

Strain = Change in dimension / original dimension

The strain is pure number hence it does not have any units and dimensions.

Types of Strain –
1) Longitudinal Strain
= change in length (ΔL) / Original Length (L)

2) Volumetric Strain
= change in Volume (ΔV) / Original Volume (V)

3) Shearing Strain
= θ = ΔL / L

Elastic Limit

The maximum stress up to which stress is proportional to strain is called elastic limit.

Q. What is Young’s Modulus of Elasticity for a perfectly rigid body?

Young’s Modulus (Y) = Longitudinal Stress / Longitudinal Strain
Y = F L/A ΔL

For a perfect Rigid Body, ΔL = 0
∴ Y = ∞
Hence, Young’s Modulus for a perfect rigid body is infinite ( ∞ ).

Hooke’s Law

According to Hooke’s Law, within the elastic limit, the stress developed is directly proportional to the strain produced in the body.

i.e. Stress ∝ Strain
Stress = E x Strain

where E is a constant of proportionality called ‘coefficient of elasticity’ or ‘Modulus of Elasticity’ of the material.

Note : Slope of Stress vs Strain curve gives ‘E’.

‘Coefficient of Elasticity’ or ‘Modulus of Elasticity’

E = Stress / Strain

It is defined as the ratio of the stress to the corresponding strain, within the elastic limit.

SI Unit of E is N/m2
CGS unit of E dyne/cm2
Dimensional Formula of E is [M1L-1T-2]

It is of three types :

1) Young’s Modulus of Elasticity (Y)

Y = Longitudinal Stress / Longitudinal Strain

Y = (F /A) / (ΔL/L)
Y = F L/A ΔL

SI Unit is N/m2
CGS unit is dyne/cm2
Dimensional Formula is [M1L-1T-2]

It is only valid fo solid objects.

2) Bulk Modulus of Elasticity (B)

B = Volumetric Stress / Volumetric Strain

Y = (F /A) / (-ΔV/V)
Y = – F V/A ΔV
-ve sign shows that volume is decreasing when pressure is applied.
SI Unit is N/m2
CGS unit is dyne/cm2
Dimensional Formula is [M1L-1T-2]

It is valid for solid, liquid as well as for gases.

Compressibility (k) – It is defined as the reciprocal of Bilk Modulus of elasticity (B).

Bulk Modulus of solids are much larger than the bulk modulus of for liquids which are again much higher than the bulk modulus for gaseous.

2) Modulus of Rigidity /
Shear Modulus of elasticity
(G or η )

B = Tangential Stress / Tangential Strain
G = (F /A) / ( θ )
G = F /A θ or FL/ A Δ L [ ∴ θ = Δ L / L]

SI Unit is N/m2
CGS unit is dyne/cm2
Dimensional Formula is [M1L-1T-2]
It is valid for solid only.

Stress-Strain curve for a Metallic Wire

Explanation –

1) The initial part OA of the graph is a straight line indicating that stress is directly proportional to strain. Up to the point A, Hooke’s law is obeyed. The Point A is called Proportional Limit. In this region, the wire is perfectly elastic.

2) After the point A, the stress is not
proportional to strain and a
curved portion AB i obtained. However, if the load is removed at any point between O and B, the curve is retraced along BAO and wire attains its original length. The Portion OB of the graph is called the elastic region and the point B is called the elastic limit or yield point.

3) Beyond point B, the graph is represented by BC, where the strain increases much more rapidly with the strain. If the load is removed at any point C, the wire does not come back to its original length but traces dotted line. Hence when stress is zero then strain is not equal to zero. In this region, the material is said to have acquired a permanent set.

4) If the load is increased beyond the point C, there is a large increase in the strain (length of the wire). In this region necks and waist develop at few points along the length of the wire and the wire ultimately breaks at point E called fracture point.

Summary
A – Proportional limit
B– Elastic Limit or Yield Point
The Stress corresponding to point B is called Yield Strength.
The Stress corresponding to the point D is called Tensile Strength.
E – Fracture Point.

OB – Elastic Behaviour
BE – Plastic Behaviour

Elastomers – The material which can be elastically stretched to large values of strain are called elastomers. They don’t follow Hooke’s law.
For Example – Rubber and Tissues of Aorta.

Elastic After Effect – The delay in regaining the original state by a body on the removal of deforming force is called elastic after effect.

Elastic Fatigue – It is the property of the elastic body due to which its behaviour becomes less elastic under the action of repeated alternating deforming forces.

Poisson’s Ratio
Within the elastic limit, the ratio of lateral strain to the longitudinal strain is called Poisson’s Ratio.

Suppose the length of a loaded wire increases from L to L+ΔL and its diameter decreases from D to D+ΔD then

Longitudinal Strain = Δ L / L
Lateral strain = – Δ D/D

Poisson’s Ratio, σ  = Lateral strain / Longitudinal Strain

σ  = [- Δ D/D]/[ Δ L/ L]
σ  = -LΔD/DΔL

The -ve sign shows that longitudinal and lateral strain in opposite directions.

Poisson’s Ratio is a pure number. So it don’t have any units and dimensions.E