Short Answer:

Modulus of resilience is defined as the maximum amount of strain energy stored per unit volume of a material when it is stressed up to its elastic limit. It represents the ability of a material to absorb energy within the elastic region and release it upon unloading without permanent deformation.

In simple words, modulus of resilience tells how much energy a material can store elastically before yielding starts. Materials with a high modulus of resilience, such as spring steel, are ideal for applications where energy storage and recovery are required, like in springs, shock absorbers, and elastic couplings.

Detailed Explanation :

Modulus of Resilience

When a body is subjected to an external load, it experiences deformation. If the applied stress is within the elastic limit, the body regains its original shape after the load is removed, and the energy absorbed during deformation is completely recovered. This energy stored per unit volume within the elastic limit is known as the modulus of resilience.

The modulus of resilience is an important property in mechanical engineering, especially in the design of components that must store and release energy repeatedly without permanent deformation — for example, springs, elastic couplings, and buffers.

It gives a quantitative measure of how efficiently a material can absorb energy elastically before yielding occurs. The larger the value of the modulus of resilience, the more energy a material can absorb elastically.

Definition

The modulus of resilience can be defined as:

“The maximum strain energy stored per unit volume of a material without causing permanent deformation is known as the modulus of resilience.”

It represents the area under the stress–strain curve up to the elastic limit. Beyond this limit, the material enters the plastic region, and energy absorption is no longer completely recoverable.

Derivation of Expression

Let a material be loaded gradually within its elastic limit.
Let:

•  = stress on the material,
•  = strain in the material,
•  = modulus of elasticity (Young’s modulus).

From Hooke’s Law,

The strain energy per unit volume (u) stored in a material under stress up to the elastic limit is equal to the area under the stress–strain curve within the elastic region, which is a right-angled triangle.

where,

•  = yield stress (stress at the elastic limit),
•  = strain at yield point.

From Hooke’s Law,

Substitute this in the above equation:

Hence,

This equation gives the modulus of resilience in terms of the yield stress and the modulus of elasticity of the material.

Units

• SI Unit: Joules per cubic meter (J/m³)
• CGS Unit: Erg per cubic centimeter (erg/cm³)

Since modulus of resilience represents energy per unit volume, its unit is the same as that of strain energy density.

Physical Meaning

The modulus of resilience indicates how much energy a material can store elastically before permanent deformation starts.

• A high modulus of resilience means the material can absorb more energy per unit volume within the elastic range.
• A low modulus of resilience means the material stores less energy before yielding.

For example, materials like spring steel, bronze, and copper have high resilience, while brittle materials like cast iron and glass have low resilience.

Factors Affecting Modulus of Resilience

1. Yield Stress ():
   The modulus of resilience is directly proportional to the square of the yield stress.
   Higher yield strength means higher resilience.
2. Elastic Modulus (E):
   It is inversely proportional to the modulus of resilience.
   A material with a lower modulus of elasticity will have higher resilience if the yield stress remains constant.
3. Temperature:
   Increasing temperature usually decreases yield strength, which reduces resilience.
4. Material Composition:
   Alloying elements and heat treatment can improve yield strength and hence increase resilience.

Importance of Modulus of Resilience

1. Design of Springs:
   The modulus of resilience helps in selecting materials that can store maximum energy within the elastic limit without permanent deformation.
2. Shock Absorbers and Buffers:
   It determines the energy absorption capacity under impact or sudden loading conditions.
3. Material Selection:
   Engineers select materials with appropriate resilience for components that experience cyclic or impact loading.
4. Elastic Energy Storage:
   Used in components like flywheels, couplings, and suspension systems where energy must be absorbed and released elastically.
5. Prevention of Yielding:
   Ensures that the structure or component operates safely within the elastic range.

Example

Let a mild steel specimen have:

Then,

 

Thus, the material can store 0.156 MPa of strain energy per unit volume within the elastic limit.

Comparison Between Proof Resilience and Modulus of Resilience

• Proof Resilience is the total strain energy stored up to the elastic limit in a body.
• Modulus of Resilience is the strain energy stored per unit volume up to the elastic limit.

Thus,

Applications

• Design of springs, shafts, buffers, and shock absorbers.
• Used in automotive suspension systems and aerospace components.
• Important in dynamic loading applications, where energy must be absorbed elastically.
• Helps in selecting materials for impact-resistant parts like machine mounts and couplings.

Conclusion

The modulus of resilience represents the maximum strain energy per unit volume that a material can absorb within its elastic limit without permanent deformation. It is mathematically expressed as . This property is vital in designing components like springs, shock absorbers, and elastic members, where energy absorption and recovery are essential. Materials with a high modulus of resilience provide better performance under impact or fluctuating loads, ensuring both strength and durability.