Short Answer:
Proof resilience is the maximum amount of strain energy that a material can absorb per unit volume without causing permanent deformation. It is the energy stored in a material when it is stressed up to its elastic limit. Beyond this limit, the material will not return to its original shape after removing the load.
In simple words, proof resilience measures the capacity of a material to store elastic energy safely before yielding begins. It is an important property for materials used in components like springs, shafts, and suspension systems, where elasticity and energy absorption are required.
Detailed Explanation :
Proof Resilience
When a material is subjected to an external load, it undergoes deformation. If the applied stress remains within the elastic limit, the deformation is reversible, and the energy absorbed during loading is released completely when the load is removed. This recoverable energy is known as strain energy, and the maximum strain energy that can be stored within the elastic limit is called proof resilience.
Thus, proof resilience is a measure of the elastic energy absorption capacity of a material. It helps engineers understand how much energy a material can store safely without permanent deformation or damage. It is one of the key properties considered in designing springs, beams, and shafts, where elastic recovery and energy storage are important.
Definition
The proof resilience can be defined as:
“The total strain energy stored in a body at the elastic limit is called proof resilience.”
When this energy is expressed per unit volume, it is termed the modulus of resilience.
Hence,
- Proof resilience → Total strain energy stored at elastic limit.
- Modulus of resilience → Strain energy per unit volume at elastic limit.
Proof resilience is expressed in Joules (J), while modulus of resilience is expressed in Joules per cubic meter (J/m³).
Derivation of Expression
Let us derive the expression for proof resilience of a material.
Consider a material subjected to a gradually applied tensile load .
Let,
- = Original length of the specimen
- = Cross-sectional area
- = Stress in the material
- = Strain in the material
- = Modulus of elasticity
From Hooke’s Law,
The strain energy stored per unit volume (up to elastic limit) is the area under the stress-strain diagram within the elastic range.
where,
= Yield stress or stress at elastic limit
= Strain corresponding to yield point
Substituting Hooke’s law (), we get
Hence,
Now, to find proof resilience, multiply the modulus of resilience by the volume of the material (V):
Thus, proof resilience depends on the material properties (yield stress and modulus of elasticity) and the size of the component.
Units
- In SI system: Joules (J)
- In CGS system: Erg (1 Joule = 10⁷ ergs)
For modulus of resilience:
- SI unit: Joules per cubic meter (J/m³)
Physical Meaning
Proof resilience represents the maximum recoverable elastic energy that a material can store before the yield point is reached. If the material is loaded beyond this limit, permanent deformation occurs, and some energy is lost as heat or plastic work.
For example:
- In a spring, proof resilience defines how much energy it can store elastically before it takes a permanent set.
- In a beam or shaft, proof resilience shows how much energy can be absorbed safely during bending or twisting.
Therefore, materials used in these components must have high proof resilience to ensure safety and performance.
Factors Affecting Proof Resilience
- Yield Strength:
Proof resilience increases with higher yield strength since the material can withstand greater stress before yielding. - Elastic Modulus (E):
A material with a lower modulus of elasticity (more flexible) has higher proof resilience for the same yield stress. - Volume of Material:
The total proof resilience increases with the volume of the specimen. - Temperature:
Higher temperature may reduce yield strength and, hence, the proof resilience. - Material Type:
Ductile materials like steel and copper have higher proof resilience compared to brittle materials like cast iron or glass.
Significance of Proof Resilience
- Design of Springs:
Springs are designed to store maximum energy without exceeding the elastic limit. Proof resilience determines the energy absorption capacity. - Energy Absorbing Devices:
Used in shock absorbers, buffers, and vibration dampers to avoid permanent deformation. - Selection of Materials:
Helps in selecting suitable materials for dynamic or impact load applications. - Structural Safety:
Ensures that structures can sustain temporary overloads or impact forces within safe limits. - Performance under Impact Loads:
Components subjected to sudden or fluctuating loads must have high proof resilience to avoid failure.
Example Calculation
Let a steel bar have:
Then,
Hence, the proof resilience of the steel bar is 0.156 Joules.
Difference Between Proof Resilience and Modulus of Resilience
| Property | Proof Resilience | Modulus of Resilience |
| Definition | Total strain energy stored up to elastic limit | Strain energy per unit volume up to elastic limit |
| Unit | Joule (J) | Joule per cubic meter (J/m³) |
| Depends on | Volume of material | Material property only |
Practical Applications
- Design of springs, shock absorbers, and buffers.
- Used in flywheels to store energy temporarily.
- Design of machine components under fluctuating or impact loading.
- In sports equipment like bows and rackets for energy absorption and return.
Conclusion
The proof resilience of a material is the total strain energy stored within the elastic limit. It defines the maximum elastic energy a body can absorb without undergoing permanent deformation. It is expressed as . Materials with high yield strength and flexibility have greater proof resilience. Hence, proof resilience is an important property in designing springs, shock absorbers, and energy-storing devices, ensuring that materials perform safely and elastically under load.