Table of Contents
Elasticity, Stress, Strain and Mechanical Properties
1. Introduction
Elasticity and mechanical properties describe how materials respond when forces are applied to them. These concepts are essential in civil, mechanical and structural engineering, material science, construction and manufacturing.
They help engineers select appropriate materials for buildings, bridges, vehicles, machinery and other structures.
The study of elasticity explains how materials deform under applied forces and whether they return to their original dimensions after the forces are removed.
2. Elasticity
2.1 Definition
Elasticity is the property of a material by which it regains its original shape and dimensions after the removal of an external deforming force, provided the material has not exceeded its elastic limit.
Examples:
- A steel spring returns to its original length after a moderate stretching force is removed.
- A rubber band returns approximately to its original shape after being stretched within its elastic range.
- A compressed elastic ball regains its shape when the force is removed.
2.2 Elastic Deformation
Elastic deformation is temporary deformation that disappears when the applied force is removed.
2.3 Plastic Deformation
Plastic deformation is permanent deformation that remains after the external force is removed.
A material may undergo both elastic and plastic deformation as the applied load increases.
2.4 Elastic Limit
The elastic limit is the maximum stress up to which a material can recover its original dimensions after the applied load is removed.
Beyond this limit, permanent deformation may occur.
2.5 Elasticity and Plasticity
Elasticity is the ability to recover the original shape, while plasticity is the ability to undergo permanent deformation without breaking.
These properties are important in selecting materials for different engineering purposes.
3. Stress
3.1 Definition
Stress is the internal resisting force developed per unit area within a material when an external force acts on it.
[
Stress=\frac{F}{A}
]
Where:
- (F) = applied force
- (A) = cross-sectional area
The SI unit of stress is the pascal (Pa), equivalent to N/m².
Stress describes the intensity of internal force within a material.
3.2 Types of Stress
Tensile Stress
Tensile stress develops when a material is pulled and tends to elongate.
Examples include cables supporting a suspended load and steel reinforcement under tension.
Compressive Stress
Compressive stress develops when a material is pushed or compressed.
Examples include columns supporting buildings and structural members under compression.
Shear Stress
Shear stress develops when forces act parallel or tangential to a surface, tending to cause adjacent layers to slide relative to one another.
Examples include rivets, bolts and materials subjected to cutting forces.
Bulk or Volumetric Stress
Bulk stress occurs when pressure acts uniformly from all directions and changes the volume of a material.
4. Strain
4.1 Definition
Strain is the ratio of the change in dimension of a material to its original dimension.
For linear strain:
[
Strain=\frac{\Delta L}{L}
]
Where:
- (\Delta L) = change in length
- (L) = original length
Strain is dimensionless because it is a ratio of two lengths.
4.2 Types of Strain
Longitudinal Strain
The ratio of change in length to original length.
It may be tensile or compressive.
Lateral Strain
The ratio of change in a lateral dimension to the original lateral dimension.
Shear Strain
The angular deformation produced by shear stress. For small angles, shear strain is approximately equal to the angle of deformation in radians.
Volumetric Strain
The ratio of change in volume to original volume.
[
\text{Volumetric strain}=\frac{\Delta V}{V}
]
5. Hooke’s Law
Hooke’s law states that, within the proportional limit, stress is directly proportional to strain.
[
Stress\propto Strain
]
For a material under simple uniaxial loading:
[
Stress=E\times Strain
]
Where (E) is Young’s modulus.
The proportionality applies only within the material’s linear elastic range.
5.1 Spring Form of Hooke’s Law
For an ideal spring:
[
F=-kx
]
Where:
- (F) = restoring force
- (k) = spring constant
- (x) = displacement from equilibrium
The negative sign indicates that the restoring force acts opposite to the displacement.
The SI unit of the spring constant is N/m.
6. Elastic Moduli
Elastic moduli measure a material’s resistance to deformation.
6.1 Young’s Modulus
Young’s modulus is the ratio of longitudinal stress to longitudinal strain within the linear elastic region.
[
E=\frac{\text{Longitudinal stress}}{\text{Longitudinal strain}}
]
It measures resistance to stretching or compression.
A material with a higher Young’s modulus is generally stiffer under the same conditions.
The SI unit is pascal.
6.2 Bulk Modulus
Bulk modulus measures resistance to uniform compression.
[
K=-\frac{\Delta P}{\Delta V/V}
]
Where:
- (K) = bulk modulus
- (\Delta P) = change in pressure
- (\Delta V/V) = volumetric strain
The negative sign indicates that volume decreases when pressure increases.
A high bulk modulus means the material is difficult to compress.
6.3 Shear Modulus
Shear modulus, also called the modulus of rigidity, is the ratio of shear stress to shear strain in the linear elastic region.
[
G=\frac{\text{Shear stress}}{\text{Shear strain}}
]
It measures resistance to changes in shape caused by shear forces.
6.4 Poisson’s Ratio
Poisson’s ratio is the negative ratio of lateral strain to longitudinal strain under uniaxial loading.
[
\nu=-\frac{\text{Lateral strain}}{\text{Longitudinal strain}}
]
When a material is stretched, it generally becomes thinner in the transverse direction.
Poisson’s ratio describes this relationship.
For many ordinary isotropic materials under conventional elastic behaviour, it lies between 0 and 0.5.
7. Stress-Strain Curve
A stress-strain curve shows how a material responds as stress increases.
For a typical ductile metal, the curve may include the following regions:
7.1 Proportional Limit
The region in which stress is directly proportional to strain and Hooke’s law applies.
7.2 Elastic Limit
The highest stress up to which the material can recover its original shape after unloading.
7.3 Yield Point
The stage at which substantial plastic deformation begins, often with a relatively small increase in stress.
Some materials do not show a clear yield point. Engineers may use a proof stress, such as the 0.2% offset yield strength, instead.
7.4 Ultimate Tensile Strength
The maximum engineering tensile stress reached during a tensile test.
7.5 Fracture Point
The point at which the specimen finally breaks.
The exact shape of a stress-strain curve depends on the material and testing conditions.
8. Mechanical Properties of Materials
8.1 Strength
Strength is the ability of a material to withstand applied loads without failure.
Types include tensile strength, compressive strength and shear strength.
8.2 Stiffness
Stiffness is the resistance of a material or structural member to deformation.
Young’s modulus is a measure of material stiffness under uniaxial loading.
8.3 Hardness
Hardness is the resistance of a material to indentation, scratching, abrasion or local plastic deformation.
Common hardness tests include:
- Brinell test
- Rockwell test
- Vickers test
- Mohs scale for mineral scratch hardness
8.4 Toughness
Toughness is the ability of a material to absorb energy and deform plastically before fracturing.
The area under the complete stress-strain curve represents energy absorbed per unit volume up to fracture.
8.5 Resilience
Resilience is the ability of a material to absorb energy within its elastic range and release that energy when unloaded.
The area under the stress-strain curve up to the elastic limit represents modulus of resilience.
8.6 Ductility
Ductility is the ability of a material to undergo significant tensile plastic deformation before fracture.
Ductile materials can be drawn into wires.
Examples include copper and many forms of steel.
8.7 Malleability
Malleability is the ability of a material to undergo plastic deformation under compressive forces, allowing it to be hammered or rolled into thin sheets.
Gold and aluminium are examples of malleable metals.
8.8 Brittleness
Brittleness is the tendency of a material to fracture with little plastic deformation.
Glass and many ceramics are comparatively brittle.
8.9 Plasticity
Plasticity is the ability of a material to undergo permanent deformation without immediate fracture.
It is useful in metal forming and manufacturing.
8.10 Fatigue
Fatigue is the progressive weakening and eventual failure of a material under repeated or fluctuating loading.
A material may fail by fatigue even when the applied stress is below its static strength.
8.11 Creep
Creep is the gradual time-dependent deformation of a material under sustained stress.
It becomes particularly important at elevated temperatures, although some materials can exhibit creep at ordinary temperatures over long periods.
9. Elastic Potential Energy
When an elastic body is deformed, energy may be stored in it as elastic potential energy.
For an ideal spring:
[
U=\frac{1}{2}kx^2
]
Where:
- (U) = elastic potential energy
- (k) = spring constant
- (x) = extension or compression
The energy is measured in joules.
10. Applications in Civil and Mechanical Engineering
Knowledge of elasticity and mechanical properties is essential for:
- Designing buildings and bridges
- Selecting structural steel and concrete
- Designing columns, beams and foundations
- Manufacturing springs and shock absorbers
- Selecting cables and suspension systems
- Designing pressure vessels and pipelines
- Assessing fatigue in rotating machinery
- Evaluating creep in high-temperature components
- Ensuring the safety and durability of engineering structures
Engineers must consider not only strength but also stiffness, ductility, toughness, fatigue resistance and long-term deformation.
11. Worked Examples
Example 1: Stress
A force of 20,000 N acts on a rod with a cross-sectional area of 0.01 m². Calculate the stress.
[
Stress=\frac{F}{A}
]
[
Stress=\frac{20,000}{0.01}
]
[
Stress=2,000,000,Pa
]
Answer: 2 MPa.
Example 2: Strain
A wire originally 2 m long extends by 0.004 m. Calculate the strain.
[
Strain=\frac{\Delta L}{L}
]
[
Strain=\frac{0.004}{2}=0.002
]
Answer: 0.002 or 0.2%.
Example 3: Young’s Modulus
A material experiences a stress of 100 MPa and a strain of 0.0005 within its linear elastic region. Calculate Young’s modulus.
[
E=\frac{Stress}{Strain}
]
[
E=\frac{100\times10^6}{0.0005}
]
[
E=200\times10^9,Pa
]
Answer: 200 GPa.
Example 4: Spring Force
A spring has a spring constant of 200 N/m and is stretched by 0.05 m. Calculate the magnitude of the restoring force.
[
F=kx
]
[
F=200\times0.05=10,N
]
Answer: 10 N, directed opposite to the displacement.
Example 5: Elastic Potential Energy
A spring has a spring constant of 100 N/m and is compressed by 0.2 m. Calculate its stored elastic potential energy.
[
U=\frac{1}{2}kx^2
]
[
U=\frac{1}{2}\times100\times(0.2)^2
]
[
U=2,J
]
Answer: 2 J.
12. Important Examination Distinctions
| Property | Meaning |
|---|---|
| Elasticity | Ability to recover original dimensions |
| Plasticity | Ability to undergo permanent deformation |
| Stress | Internal force per unit area |
| Strain | Ratio of change in dimension to original dimension |
| Young’s modulus | Resistance to longitudinal deformation |
| Bulk modulus | Resistance to volume change |
| Shear modulus | Resistance to shear deformation |
| Poisson’s ratio | Ratio of lateral to longitudinal strain, with a negative sign |
| Strength | Ability to withstand loads without failure |
| Stiffness | Resistance to deformation |
| Hardness | Resistance to indentation or scratching |
| Toughness | Energy absorbed before fracture |
| Resilience | Energy absorbed within elastic range |
| Ductility | Ability to undergo tensile plastic deformation |
| Malleability | Ability to undergo compressive plastic deformation |
| Fatigue | Failure under repeated or fluctuating loading |
| Creep | Time-dependent deformation under sustained stress |
13. 36 Key Takeaways for Exams
- Elasticity is the ability of a material to recover its original dimensions after unloading.
- Elastic deformation is temporary.
- Plastic deformation is permanent.
- The elastic limit is the maximum stress up to which recovery is possible after unloading.
- Stress is internal resisting force per unit area.
- The SI unit of stress is the pascal.
- Tensile stress develops when a material is pulled.
- Compressive stress develops when a material is compressed.
- Shear stress acts parallel or tangential to a surface.
- Bulk stress is associated with uniform pressure acting on a material.
- Strain is the ratio of change in dimension to original dimension.
- Strain is dimensionless.
- Longitudinal strain concerns change in length.
- Lateral strain concerns change in transverse dimensions.
- Volumetric strain is the ratio of change in volume to original volume.
- Hooke’s law states that stress is proportional to strain within the proportional limit.
- Young’s modulus is the ratio of longitudinal stress to longitudinal strain.
- A higher Young’s modulus generally indicates greater material stiffness.
- Bulk modulus measures resistance to volume change.
- Shear modulus measures resistance to shear deformation.
- Poisson’s ratio relates lateral strain to longitudinal strain.
- The spring form of Hooke’s law is (F=-kx).
- The negative sign in Hooke’s law indicates a restoring force.
- Elastic potential energy in an ideal spring is (\frac{1}{2}kx^2).
- Strength is the ability to withstand loads without failure.
- Stiffness is resistance to deformation.
- Hardness is resistance to indentation, scratching or abrasion.
- Toughness is the ability to absorb energy before fracture.
- Resilience is the ability to absorb energy within the elastic range.
- Ductility is the ability to undergo tensile plastic deformation.
- Malleability is the ability to undergo compressive plastic deformation.
- Brittleness is the tendency to fracture with little plastic deformation.
- Plasticity is the ability to undergo permanent deformation.
- Fatigue is failure caused by repeated or fluctuating stresses.
- Creep is gradual deformation under sustained stress.
- Elasticity and mechanical properties are essential in structural design, construction and material selection.