Short Answer:
The modulus of resilience is the maximum amount of energy per unit volume that a material can absorb without permanent deformation. It represents the material’s ability to store elastic energy when stressed within its elastic limit.
In simple terms, the modulus of resilience measures how much energy a material can absorb and recover when the load is removed. It is given by the area under the stress-strain curve up to the elastic limit. The formula for modulus of resilience is:
where = yield stress and = modulus of elasticity.
Detailed Explanation :
Modulus of Resilience
The modulus of resilience is an important property of materials that helps engineers understand how much elastic energy a material can store before yielding or undergoing permanent deformation. It is particularly significant in applications where materials are required to absorb energy and return to their original shape, such as in springs, shock absorbers, and suspension systems.
When a load is applied to a material, it deforms and stores energy within its elastic range. If the load is removed before the yield point, this energy is released, and the material returns to its original shape. The capacity of a material to store this recoverable energy is what defines its modulus of resilience.
Definition
Modulus of resilience is defined as:
“The maximum strain energy per unit volume that a material can absorb without permanent deformation when loaded within its elastic limit.”
It represents the elastic strain energy density in the material. This energy is recoverable and does not cause any residual deformation or damage.
Mathematical Expression
For a material following Hooke’s Law, stress (σ) and strain (ε) are related as:
The strain energy per unit volume (U) stored in the material up to a certain stress level is given by:
Substituting , we get:
But, from Hooke’s law,
Substitute this value of :
Hence,
Where:
- = Modulus of resilience (J/m³)
- = Yield stress (Pa or N/m²)
- = Modulus of elasticity (Pa or N/m²)
Explanation of Terms
- Stress (σ): The internal resistance developed within a material due to applied load.
- Strain (ε): The change in dimension per unit original dimension.
- Elastic Limit: The maximum limit up to which a material can be stressed without permanent deformation.
- Resilience: The ability of a material to absorb and recover energy in the elastic region.
The modulus of resilience, therefore, is a measure of the capacity of a material to absorb energy before it starts yielding.
Graphical Representation
On a stress-strain diagram, the area under the curve up to the yield point represents the modulus of resilience.
- The curve starts from the origin (O) and increases linearly up to the elastic limit (point A).
- Since the relationship between stress and strain is linear, the area under the line OA forms a triangle.
- Hence, the modulus of resilience is the area of this triangle:
This confirms that modulus of resilience represents the recoverable elastic energy stored in the material before plastic deformation begins.
Units of Modulus of Resilience
- SI Unit: Joules per cubic meter (J/m³)
- FPS Unit: Inch-pound per cubic inch (in·lb/in³)
Since it is energy per unit volume, its dimensional formula is:
Significance of Modulus of Resilience
- Elastic Energy Absorption:
Materials with higher modulus of resilience can absorb more energy before yielding, making them suitable for shock and impact applications. - Spring Design:
Spring materials must have a high modulus of resilience to store large amounts of energy within the elastic limit. - Safety and Efficiency:
Components like couplings, fasteners, and machine elements are designed using this property to avoid permanent deformation during service. - Material Selection:
Helps engineers choose materials that can safely absorb vibrations, shocks, or impact loads. - Fatigue Resistance:
Materials with higher resilience are generally better at withstanding cyclic or fluctuating loads.
Factors Affecting Modulus of Resilience
- Yield Strength (σy):
The modulus of resilience increases directly with the square of yield strength. Thus, stronger materials store more energy. - Elastic Modulus (E):
It varies inversely with E. A lower modulus of elasticity allows more strain before yielding, resulting in higher resilience. - Heat Treatment:
Processes such as quenching and tempering can increase yield strength, improving resilience. - Material Type:
Ductile materials like steel and copper generally have higher modulus of resilience than brittle materials like cast iron. - Temperature:
Increasing temperature may lower yield strength, reducing resilience.
Typical Values of Modulus of Resilience
| Material | Yield Stress (MPa) | Elastic Modulus (GPa) | Modulus of Resilience (kJ/m³) |
| Mild Steel | 250 | 200 | 0.16 |
| High Carbon Steel | 400 | 210 | 0.38 |
| Aluminum | 100 | 70 | 0.07 |
| Copper | 70 | 110 | 0.02 |
From the values, it is clear that high-strength steels have a higher modulus of resilience compared to softer metals.
Applications of Modulus of Resilience
- Springs:
To ensure that the spring can absorb and release energy without permanent deformation. - Shock Absorbers:
Materials with high resilience are used to absorb impact energy efficiently. - Automotive Components:
Used in car suspensions and engine parts subjected to cyclic loading. - Machine Tools:
Components such as couplings and shafts are designed to resist shock loads using materials with suitable resilience. - Sporting Goods:
Bows, golf clubs, and helmets use materials with high resilience for better performance and safety.
Difference Between Resilience and Toughness
- Resilience: Energy absorbed per unit volume within the elastic limit.
- Toughness: Energy absorbed up to fracture, including both elastic and plastic regions.
Thus, resilience deals with recoverable energy, while toughness deals with total energy before failure.
Conclusion
The modulus of resilience is a measure of the elastic energy a material can absorb per unit volume without undergoing permanent deformation. It is mathematically expressed as and represents the area under the stress-strain curve up to the yield point. Materials with high modulus of resilience, such as spring steel, are ideal for applications involving shocks, impacts, and cyclic loads. This property helps engineers design components that can store and release energy safely and efficiently within the elastic range.