Elasticity, Stress, Strain, and Mechanical Properties of Materials

Definition

Elasticity is the fundamental property of a material by virtue of which it regains its original shape and size after the removal of an external deforming force. Stress is the internal resistance offered by a body per unit area against deformation, while strain is the ratio of change in dimension to the original dimension.

Main Content

1. Elasticity, Stress, and Strain

Elasticity: If a material returns to its original state after unloading, it is elastic. If it retains permanent deformation, it is plastic.
Stress ($\sigma$): Defined as Force ($P$) divided by Area ($A$). Measured in $N/m^2$ or Pascals.
Strain ($\epsilon$): A dimensionless quantity representing deformation, expressed as $\Delta L / L$.

2. Elastic Limit and Elastic Constants

Elastic Limit: The maximum stress a material can withstand without permanent deformation.
Hooke’s Law: Within the elastic limit, stress is directly proportional to strain ($\sigma = E \cdot \epsilon$).
Constants: Young’s Modulus ($E$), Rigidity Modulus ($G$), and Bulk Modulus ($K$).

3. Lateral Strain and Poisson’s Ratio

Lateral Strain: Deformation occurring perpendicular to the direction of the applied load.
Poisson’s Ratio ($\nu$): The negative ratio of lateral strain to longitudinal strain.

4. Composite Sections and Temperature Stresses

Composite Sections: Structural members made of two or more materials acting as a single unit. They share the same total strain.
Temperature Stresses: Stresses induced in a body due to temperature change when expansion or contraction is restricted.

5. Volumetric Strain and Resilience

Volumetric Strain: The change in volume per unit original volume of a body subjected to stress.
Resilience: The capacity of a material to absorb energy when it is deformed elastically.

Working / Process

1. Calculating Stress and Strain

Identify the cross-sectional area and the applied axial load.
Divide Load by Area to find Stress.
Divide change in length by original length to find Strain.

2. Solving Composite Section Problems

Set the strain in both materials to be equal (e.g., $\epsilon_1 = \epsilon_2$).
Write force equilibrium equations: Total Load = $P_1 + P_2$.
Solve the resulting system of equations to find the distribution of load.

3. Calculating Thermal Stress

Calculate free expansion: $\Delta L = L \alpha \Delta T$.
Calculate stress: $\sigma = E \cdot \alpha \cdot \Delta T$.
If the bar is constrained, the internal force opposes this expansion.

[Load P applied to a rod]
   |          |
   |   Area A |
   |--------->| <--- Deformation (dL)
   |          |
   |          |

Advantages / Applications

Structural engineering uses elasticity principles to ensure buildings remain stable under wind and dead loads.
Thermal stress analysis is critical in railway track design to prevent buckling during high temperatures.
Composite materials (like Reinforced Concrete) utilize the high compressive strength of concrete and tensile strength of steel simultaneously.

Summary

This unit covers the mechanical behavior of materials, focusing on how solids deform under force and temperature changes. Key concepts include Hooke’s law, the relationship between different elastic moduli, and the calculation of internal energy stored during deformation.

Important terms to remember: - Young’s Modulus ($E$): Measures stiffness. - Poisson’s Ratio ($\nu$): Measures lateral contraction. - Thermal Stress: Stress induced by restricted expansion. - Strain Energy: Energy stored in an elastic body.