Key Points

Elasticity is the property of a body to regain its original shape and size after the removal of deforming forces. Plasticity is the property by which a body undergoes permanent deformation and does not regain its original state.

Stress is the internal restoring force per unit area of a deformed body. It is calculated as $\sigma = \frac{F}{A}$, where F is the applied force and A is the cross-sectional area. Its SI unit is N/m$^2$ or Pascal (Pa).

There are three main types of stress: Longitudinal (tensile or compressive), Shearing (tangential), and Hydraulic (volume) stress, which corresponds to pressure.

Strain is the fractional change in the dimension of a body due to stress. It is a dimensionless quantity. Types include Longitudinal strain ($\frac{\Delta L}{L}$), Shearing strain ($\frac{\Delta x}{L} \approx \theta$), and Volume strain ($\frac{\Delta V}{V}$).

Within the elastic limit, stress is directly proportional to strain. The law is expressed as $\text{Stress} = k \times \text{Strain}$, where k is the modulus of elasticity for the material.

This curve shows a material's behavior under load. Key points include the proportional limit (Hooke's law obeyed), elastic limit (or yield point), ultimate tensile strength (maximum stress), and the fracture point.

Ductile materials (e.g., steel) show a large plastic deformation range before fracturing. Brittle materials (e.g., glass) fracture soon after the elastic limit is crossed, with little to no plastic deformation.

Young's Modulus is the ratio of longitudinal stress to longitudinal strain. It is a measure of stiffness and is given by $Y = \frac{\text{Longitudinal Stress}}{\text{Longitudinal Strain}} = \frac{F/A}{\Delta L/L}$.

Shear Modulus is the ratio of shearing stress to shearing strain. It measures a solid's resistance to shape change and is given by $G = \frac{\text{Shearing Stress}}{\text{Shearing Strain}} = \frac{F/A}{\Delta x/L}$.

Bulk Modulus is the ratio of hydraulic stress (pressure) to the corresponding volume strain. It measures resistance to compression and is given by $B = -\frac{p}{\Delta V/V}$. The negative sign indicates volume decreases as pressure increases.

Compressibility is the reciprocal of the Bulk Modulus and measures how much a material's volume changes under pressure. It is defined as $k = \frac{1}{B}$. Solids are least compressible, while gases are most compressible.

Poisson's ratio is the ratio of lateral strain to the longitudinal strain within the elastic limit. It is a pure number with no units, defined as $\sigma = \frac{\text{Lateral Strain}}{\text{Longitudinal Strain}} = \frac{\Delta d/d}{\Delta L/L}$.

The work done in stretching a wire is stored as elastic potential energy. The elastic potential energy per unit volume (u) is given by $u = \frac{1}{2} \times \text{Stress} \times \text{Strain}$.

Beams used in bridges and buildings are often I-shaped. This shape provides a large depth to minimize bending and a wide load-bearing surface, offering maximum strength with reduced weight and cost.