Chapter - 8

MECHANICAL PROPERTIES OF SOLIDS

Deforming Force: A force which changes the size or shape of a body.

Restoring Force: A force which bring the body back to its size or shape.
ELASTICITY
If a body regains its original shape and size after the removal of deforming force, it is said to be
elastic body and this property is called elasticity.
Perfectly elastic body: If a body regains its original shape and size completely and immediately after
the removal of deforming force, it is said to be perfectly elastic body.

PLASTICITY
If a body does not regain its original shape and size even after the removal of deforming force,
it is said to be plastic body and this property is called plasticity.
Perfectly plastic body: If a body does not show any tendency to regain its original shape and size
even after the removal of deforming force, it is said to be a perfectly plastic body.
Cause of Elasticity - Interatomic forces are the cause of elasticity.

F – interatomic force, r – interatomic distance & u – elastic potential energy

STRESS
The internal restoring force set up per unit area of cross section of the deformed body is called
stress.
𝐴𝑝𝑝𝑙𝑖𝑒𝑑 𝐹𝑜𝑟𝑐𝑒 𝐹
Stress = =
𝐴𝑟𝑒𝑎 𝐴
SI unit – N/m2 or Pa

Types of Stress
1. Tensile stress/Longitudinal stress
2. Hydrostatic stress/volumetric stress
3. Shear/Tangential stress

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STRAIN
The ratio of change of any dimension produced in the body to the original dimension is called
strain.
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛
Strain =
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛
It is a dimensionless quantity
Types of strain
1. Longitudinal strain :- It is defined as the increase in length per unit original length.
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑙𝑒𝑛𝑔𝑡ℎ ∆𝑙
Longitudinal strain = =
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 𝑙

2. Volumetric strain :- It is defined as the change in volume per unit original volume
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑜𝑙𝑢𝑚𝑒 ∆𝑉
Volumetric strain = =
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑉

3. Shear strain :- it is defined as the angle θ (in radian), through which the face originally
perpendicular to the fixed face gets turned on applying tangential deforming force.
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑝𝑙𝑎𝑛𝑒𝑠
Shear strain = θ = tan θ =
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑝𝑙𝑎𝑛𝑒𝑠

Elastic limit: The maximum stress within which the body regains its original size and shape after the
removal of deforming force is called elastic limit.

HOOKE’S LAW
The extension produced in a wire is directly proportional to the load applied.
OR
Within the elastic limit, the stress is directly proportional to strain
Stress α Strain
or Stress = Constant x Strain
𝑆𝑡𝑟𝑒𝑠𝑠
or = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑆𝑡𝑟𝑎𝑖𝑛

The constant of proportionality is called modulus of elasticity or coefficient of elasticity of material.

MODULUS OF ELASTICITY
The modulus of elasticity or coefficient of elasticity of a body is defined as the ratio of stress to
the corresponding strain, within the elastic limit.
SI unit – N/m2 or Pa
Types of moduli of elasticity:
1. Young’s modulus (Y), i.e., the modulus of elasticity of length.
2. Bulk modulus (k), i.e., the modulus of elasticity of volume.
3. Modulus of rigidity or shear modulus (η), i.e., modulus of elasticity of shape.

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YOUNG’S MODULUS OF ELASTICITY (Y)
Within the elastic limit the ratio of longitudinal stress to the longitudinal strain is called
Young’s modulus of the material of the wire.
𝐹 𝑙
Y= .
𝐴 ∆𝑙

BULK MODULUS OF ELASTICITY (k)

Within the elastic limit, the ratio of normal stress to the volumetric strain is called bulk
modulus of elasticity.
𝐹 𝑉 𝑝𝑉
k=− . = −
𝐴 ∆𝑉 ∆𝑉

Compressibility: The reciprocal of the bulk modulus of a material is called its compressibility.
1
Compressibility =
𝑘
-1 2
SI unit of compressibility = N m .

MODULUS OF RIGIDITY OR SHEAR MODULUS (Qualitative idea Only)

Within the elastic limit, the ratio of tangential stress to shear strain is called modulus of
rigidity.
𝑇𝑎𝑛𝑔𝑒𝑛𝑡𝑖𝑎𝑙 𝑆𝑡𝑟𝑒𝑠𝑠 𝐹⁄
𝜂= = 𝐴 = 𝐹 = 𝐹. 𝑙
𝑆ℎ𝑒𝑎𝑟 𝑆𝑡𝑟𝑎𝑖𝑛 𝜃 𝐴𝜃 𝐴 ∆𝑙

STRESS – STRAIN CURVE FOR A METALLIC WIRE

OB – Elastic region
BD – Plastic region
A – Proportional limit (Hooke’s law is valid till A)
B – Elastic limit/Yield point (Wire is elastic till B)
Sy – Yield strength (Stress corresponding to Yield point)
OE – Permanent set (Stress is zero but some strain is left, ie; wire is permanently changed its shape)

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CE – Stress-Strain curve is not retraced on reversing the stress
D – Breaking point/Fracture point
Su – Ultimate strength/Tensile strength (Stress corresponding to the breaking point)

Classification of materials on the basis of stress- strain curve:

1. Ductile materials: The materials which have large plastic range of extension are called ductile
materials. For example, copper, silver, iron, aluminum, etc.

2. Brittle materials: The materials which have very small range of plastic extension are called
brittle materials. For example, cast iron, glass, ceramics, etc.

3. Malleable materials: The yield point (B’) obtained under compression is called crushing
point. For example, gold, silver, lead, etc.

4. Elastomers: The materials which can be elastically stretched to large values of strain are called
elastomers. There is no well-defined plastic region. The material does not obey Hooke’s law.
For example, Aorta and rubber.

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Elastic after effect - The delay in regaining the original state by a body on the removal of the
deforming force is called elastic after effect.
Quartz or phosphor-bronze alloy has small elastic after effect. But a glass fibre takes hour to regain its
original state.
Elastic fatigue - It is defined as the loss in strength of a material caused due to repeated alternating
strains to which the material is subjected.

ELASTIC POTENTIAL ENERGY OF A STRETCHED WIRE (Refer Note book for the
derivation)
1
𝑈 = 𝐹 × ∆𝑙
2
1
u = 2 𝑠𝑡𝑟𝑒𝑠𝑠 × 𝑠𝑡𝑟𝑎𝑖𝑛
1
u = 2 𝑌𝑜𝑢𝑛𝑔′ 𝑠 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 × 𝑠𝑡𝑟𝑎𝑖𝑛2

POISSON’S RATIO
Within the elastic limit, the ratio of lateral strain to the longitudinal strain is called Poisson’s ratio.
∆𝑙
Longitudinal strain =
𝑙
∆𝐷
Lateral Strain =−
𝐷

∆𝐷 𝑙
σ = − .
∆𝑙 𝐷
The negative sign indicates that the longitudinal and lateral strains are in opposite sense.

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