Elasticity is the mechanical property of a material by virtue of which it resists any change in its size or shape when an external deforming force is applied, and completely recovers its original dimensions once the deforming force is removed.
Microscopic Origin of Elasticity
In a solid material, constituent atoms or molecules are held together in a stable equilibrium position by inter-atomic or inter-molecular forces. When an external force deforms the solid, these atoms are displaced from their equilibrium positions, altering the inter-atomic distances. This displacement triggers an internal restoring force that attempts to pull the atoms back to their original equilibrium configuration.
Key Structural Definitions
- Deforming Force: An external force applied to a body that changes its shape, length, or volume.
- Restoring Force: An internal force that develops within a deformed body, equal in magnitude and opposite in direction to the applied deforming force, aimed at restoring the body to its original state.
- Perfect Elastic Body: A body that recovers its original shape and size instantaneously and completely after the removal of the deforming force (e.g., Quartz and Phosphor Bronze approach perfect elasticity).
- Perfect Plastic Body: A body that shows no tendency to recover its original shape and size after the deforming force is removed; it remains permanently deformed (e.g., Putty, paraffin wax, and wet clay).
Concepts of Stress and Strain
To quantify elasticity, physics introduces two primary variables: stress (the internal resistance intensity) and strain (the deformation measure).
Stress
Stress is defined as the internal restoring force developed per unit cross-sectional area of a deformed body.
- SI Unit: Newton per square meter (N/m2) or Pascal (Pa).
- Dimensional Formula: [M1L-1T-2] (identical to pressure, though stress is a tensor quantity, not a simple scalar).
Strain
Strain is the fractional change produced in the dimensions of a body due to the application of a deforming force. It is a ratio of two identical dimensions.
- Physical Nature: Since it is a ratio of identical physical quantities, strain is a dimensionless and unitless quantity.
Classification Matrix of Stress and Strain
| Type of Deformation | Stress Type | Strain Type | Dimensional Formula / Expression |
| Linear (Length Change) | Tensile / Compressive Stress: Force applied perpendicular to the cross-section to elongate or compress. | Longitudinal Strain: Ratio of change in length (Δ L) to original length (L). | Strain = Δ L/L |
| Volume (Size Change) | Volume / Hydraulic Stress: Uniform perpendicular force applied across all faces of a body. | Volume Strain: Ratio of change in volume (Δ V) to original volume (V). | Strain = Δ V/V |
| Shape (Shear Change) | Tangential / Shearing Stress: Force applied parallel to the surface, shifting layers tangentially. | Shearing Strain: Angle (θ) in radians through which a line originally perpendicular to the fixed face is turned. | Strain = θ ≈ Δ x/L |
Hooke’s Law and Modulus of Elasticity
Formulated by the English physicist Robert Hooke in 1676, this law establishes the linear relationship between stress and strain for elastic materials.
Statement of the Law
Hooke’s Law states that, within the elastic limit of a material, the stress developed in a body is directly proportional to the strain produced in it.
- E is a constant of proportionality known as the Modulus of Elasticity or Elastic Modulus.
- The unit and dimensions of the Modulus of Elasticity are identical to those of stress (N/m2 or Pa; [M1L-1T-2]), because strain is dimensionless.
The Three Moduli of Elasticity
- Young’s Modulus (Y): The ratio of longitudinal stress to longitudinal strain. It applies exclusively to solids (as liquids and gases do not possess a fixed length). Y = F · L/A · Δ L.
- Bulk Modulus (B or K): The ratio of hydraulic stress to volume strain. It applies to solids, liquids, and gases. The reciprocal of the Bulk Modulus is called Compressibility (k = 1/B).
- Shear Modulus / Modulus of Rigidity (η): The ratio of tangential stress to shearing strain. It determines a material’s structural resistance to twisting or shape distortion.
The Stress-Strain Curve Analysis
The mechanical behavior of a material under an increasing tensile load is mapped via a standard stress-strain graph, which reveals distinct physical thresholds.
Crucial Threshold Points on the Curve
- Proportional Limit (Point A): The region from the origin to Point A follows Hooke’s Law strictly. Stress is perfectly linear to strain.
- Elastic Limit / Yield Point (Point B): Beyond point A up to point B, stress and strain are not strictly linear, but the material remains elastic. If the load is removed at B, the body completely recovers its original dimensions. Point B represents the maximum stress a material can withstand without permanent deformation.
- Permanent Set: If the material is stressed beyond point B into the plastic region and the load is removed, it does not regain its original size. It retains a permanent residual deformation called a permanent set.
- Ultimate Tensile Strength (Point C): The maximum stress value that the material can handle before structural necking or thinning begins. Beyond this point, additional strain occurs even with reduced stress.
- Fracture / Breaking Point (Point D): The point of ultimate structural failure where the material physically snaps or breaks.
Material Classification Based on the Curve
- Ductile Materials: Materials that have a wide plastic deformation region between the elastic limit (B) and the fracture point (D). They can be drawn into thin wires (e.g., Copper, Gold, Iron, Aluminum).
- Brittle Materials: Materials that fracture almost immediately after passing the elastic limit. Their points B and D are extremely close (e.g., Glass, Ceramic, Cast Iron).
- Elastomers: Materials that can be stretched to cause large strains but do not obey Hooke’s law, and recover their shape completely without a distinct yield point (e.g., Tissue of aorta, rubber vulcanizates).
UPSC High-Yield Scientific Trivia
The Steel vs. Rubber Elasticity Paradox
In common parlance, rubber is considered highly elastic because it stretches easily. However, scientifically, steel is far more elastic than rubber. Elasticity measures a material’s internal resistance to deformation and its restoring capability. For a given amount of stress, steel undergoes a much smaller strain than rubber, meaning it requires a vastly higher force to deform and generates a powerful internal restoring force. Therefore, Young’s Modulus for steel is significantly larger than that for rubber.
Structural Engineering Selection (I-Shaped Beams)
Iron and steel beams utilized in the construction of bridges and high-rise buildings are engineered with an “I-shaped” cross-section. The bending or sagging (δ) of a beam loaded at the center is inversely proportional to its Young’s Modulus (Y), its width (b), and the cube of its depth (d3):
Elastic Fatigue
When a material is subjected to repeated, alternating deforming forces over a long duration, it loses its elastic strength temporarily. This condition is known as elastic fatigue. Because of elastic fatigue, bridges are declared unsafe after long periods of active service; the constant structural loading and unloading make the metal components brittle, risking a sudden collapse under stress.
Temperature Influence on Elasticity
The modulus of elasticity is inversely proportional to temperature. As the temperature of a substance rises, the inter-molecular thermal vibrations increase, causing expansion. This widens the equilibrium distance between atoms, weakening the inter-molecular restoring forces and rendering the material more plastic and less elastic. (Exception: Invar, an iron-nickel alloy, maintains a near-constant modulus of elasticity across normal temperature ranges, making it ideal for precision clocks and scientific instruments).
Last Modified: May 27, 2026