**Modulus of Resilience**

The modulus of resilience is a property that measures the ability of a material to store energy without undergoing permanent deformation. It is an important concept in materials science and engineering, especially in the study of mechanical behavior of materials under loading conditions.

**Definition and Calculation**

The modulus of resilience is defined as the maximum amount of energy that a material can absorb without permanently deforming. It represents the area under the stress-strain curve up to the elastic limit of the material. In other words, it is the amount of energy that can be stored in a material before it reaches its yield point and undergoes permanent deformation.

The modulus of resilience is calculated using the formula:

Resilience = 0.5 × σ × ε

Where:
- Resilience is the modulus of resilience
- σ is the yield strength of the material
- ε is the strain at the yield point

**Interpretation**

The modulus of resilience provides information about the elastic behavior and energy storage capacity of a material. It indicates the amount of energy that can be absorbed by a material without causing permanent damage or deformation.

**Significance**

The modulus of resilience is an important property in materials selection and design. It helps engineers determine the suitability of a material for specific applications where energy absorption is critical. Materials with high resilience are preferred for applications where impact or shock loading is expected, as they can absorb and store energy without undergoing permanent deformation.

**Comparison with Other Properties**

- Property to resist shocks: While the modulus of resilience is related to the ability to resist shocks, it specifically quantifies the energy absorption capacity of a material, rather than its ability to withstand sudden impacts.
- Index of elasticity: The modulus of resilience is not an index of elasticity. Elasticity refers to the ability of a material to return to its original shape after deformation, whereas resilience measures the ability to store energy.
- Index of compressibility: The modulus of resilience is not an index of compressibility. Compressibility refers to the change in volume of a material under applied pressure, whereas resilience measures the energy storage capacity.

Therefore, the correct answer is option B: the property to store energy without undergoing permanent deformation.