Definition: Elasticity is the property of a material by virtue of which it regains its original shape and size after the removal of a deforming force. It is the fundamental physical characteristic that describes how materials respond to external stresses, bridging the gap between rigid body mechanics and fluid dynamics.
Understanding Stress and Strain
When you apply an external force to a body, it tends to change its shape or volume. This is known as a deforming force. Because the body is composed of internal particles held together by intermolecular forces, it develops an internal restoring force that opposes this deformation. Stress is defined as this internal restoring force per unit area of cross-section.
Stress = Force / Area (Units: N/m² or Pascal)
While stress describes the effort to deform, strain measures the degree of deformation. It is a dimensionless quantity because it is simply the ratio of change in dimension to the original dimension. Depending on how the force is applied, we categorize strain into longitudinal strain (change in length), volumetric strain (change in volume), and shearing strain (change in shape/angle).
Hooke’s Law and the Elastic Limit
In 1676, Robert Hooke formulated the empirical law that governs the behavior of elastic materials. Hooke’s Law states that within the elastic limit, the stress applied to a body is directly proportional to the strain produced. This relationship is linear, meaning if you double the stress, the strain also doubles.
However, this proportionality does not hold indefinitely. If you continue to increase the stress, the material reaches its proportionality limit, followed by the elastic limit (or yield point). Beyond this point, the material undergoes permanent, or plastic deformation, meaning it will not return to its original dimensions even after the force is removed.
Young’s Modulus and Elastic Constants
The proportionality constant in Hooke’s Law is known as the Modulus of Elasticity. For longitudinal deformation, this is specifically called Young’s Modulus (Y). It provides a measure of a material’s stiffness. A higher Young’s Modulus indicates that the material is harder to stretch or compress.
- Young’s Modulus (Y): Longitudinal Stress / Longitudinal Strain
- Bulk Modulus (B): Volumetric Stress / Volumetric Strain
- Modulus of Rigidity (η): Shearing Stress / Shearing Strain
These constants are intrinsic properties of the material, not the shape of the object. For instance, steel has a very high Young’s Modulus compared to rubber, which explains why steel is considered more “elastic” in terms of its resistance to deformation, despite common misconceptions.
Elastic Potential Energy
When a wire is stretched, work is done against the internal restoring forces. This work is stored within the material as elastic potential energy. As you pull a wire, the force required increases linearly with extension, similar to the behavior of a spring.
The energy density, or potential energy per unit volume, is given by the area under the stress-strain curve. For a material obeying Hooke’s Law, the energy stored is:
U = 1/2 × Stress × Strain × Volume
This energy is released when the deforming force is removed, allowing the body to return to its original state. In real-world materials, some energy is lost as heat during the loading and unloading cycle, a phenomenon known as elastic hysteresis.
Important Facts and Formulas
| Concept | Formula | Unit |
|---|---|---|
| Stress | σ = F / A | N/m² |
| Young’s Modulus | Y = (F/A) / (ΔL/L) | N/m² |
| Energy Density | u = 1/2 × Stress × Strain | J/m³ |
| Poisson’s Ratio | σ = – (Lateral Strain) / (Longitudinal Strain) | Dimensionless |
Previous Year Question Hints
- Stiffness Comparison: Be prepared to compare the Young’s Modulus of two different materials based on a given stress-strain graph. Remember: the steeper the slope, the higher the Modulus.
- Temperature Effects: Frequently, questions ask how elasticity changes with temperature. For most metals, elasticity decreases as temperature increases because intermolecular bonds weaken.
- Wire Stretching: When a wire is cut into two equal halves, Young’s Modulus remains unchanged, but the extension for a given load will be halved.
Quick Revision Summary
- Stress is defined as restoring force per unit area; it is not a vector, but a tensor.
- Strain is a ratio of changes in dimensions and is always dimensionless.
- Hooke’s Law is valid only up to the limit of proportionality.
- Young’s Modulus measures resistance to longitudinal deformation.
- Plasticity is the opposite of elasticity; the material retains its deformed shape.
- Elastic Hysteresis explains the energy loss in materials during cyclic loading.
- Bulk Modulus applies to changes in volume under uniform pressure.
- Poisson’s Ratio relates lateral contraction to longitudinal elongation.
