Properties of Solids: Elasticity & Stress-Strain

Elasticity and Deformation

When an external force is applied to a solid body, it undergoes deformation. If the body regains its original shape after the removal of the deforming force, it is called elastic; otherwise it is plastic. The study of elasticity deals with the relationship between stress and strain within the elastic limit.

Stress

Stress is defined as the restoring force per unit area developed inside the body when subjected to a deforming force.

$\sigma = \frac{F}{A}$

• $\sigma$: stress (Pa or N m$^{-2}$)
• $F$: applied force (N)
• $A$: cross-sectional area (m$^2$)
• Dimensional formula: [M L$^{-1}$ T$^{-2}$]

Types of stress:

1. Tensile stress — force pulls the body along the length
2. Compressive stress — force pushes/compresses the body
3. Shear stress — force acts tangentially; $\tau = F/A$
4. Volumetric (hydraulic) stress — uniform pressure applied from all sides; equals the change in pressure $\Delta P$

Strain

Strain is the ratio of change in dimension to the original dimension. It is dimensionless.

1. Longitudinal strain: $\varepsilon_L = \Delta L / L$
2. Shear strain: $\gamma = \Delta x / L = \tan\theta \approx \theta$ (for small angles)
3. Volumetric strain: $\varepsilon_V = \Delta V / V$

Hooke's Law

Within the elastic limit, stress is directly proportional to strain:

$\sigma = E \cdot \varepsilon$

where $E$ is the modulus of elasticity (Pa).

Elastic Moduli

1. Young's Modulus ($Y$): resistance to longitudinal deformation $Y = \frac{\sigma}{\varepsilon_L} = \frac{F \cdot L}{A \cdot \Delta L}$

2. Shear Modulus / Modulus of Rigidity ($G$ or $\eta$): $G = \frac{\text{Shear stress}}{\text{Shear strain}} = \frac{F/A}{\theta}$

3. Bulk Modulus ($B$ or $K$): $B = \frac{-\Delta P}{\Delta V / V} = \frac{-\Delta P \cdot V}{\Delta V}$

The negative sign ensures $B > 0$ since volume decreases when pressure increases.

4. Compressibility: $k = 1/B$ (Pa$^{-1}$)

Poisson's Ratio

$\nu = \frac{\text{Lateral strain}}{\text{Longitudinal strain}} = -\frac{\Delta d / d}{\Delta L / L}$

• Dimensionless; theoretical range: $-1 \leq \nu \leq 0.5$
• Most materials: $0.2 \leq \nu \leq 0.5$
• Rubber: $\nu \approx 0.5$ (incompressible)
• Cork: $\nu \approx 0$ (ideal for bottle stoppers)

Relations Among Elastic Constants

$Y = 3B(1 - 2\nu) = 2G(1 + \nu)$ $Y = \frac{9BG}{3B + G}$

Stress-Strain Curve (Ductile Metal like Mild Steel)

Key points on the curve:

1. Proportional limit (A) — Hooke's law valid; linear region
2. Elastic limit (B) — beyond this, permanent deformation begins
3. Upper yield point (C) — stress at which yielding starts
4. Lower yield point (D) — stress during plastic flow
5. Ultimate tensile strength (E) — maximum stress the material can withstand
6. Fracture point (F) — material breaks

Elastic Potential Energy

Energy stored per unit volume (energy density): $u = \frac{1}{2} \times \text{stress} \times \text{strain} = \frac{\sigma^2}{2E} = \frac{1}{2} E \varepsilon^2$

Total elastic potential energy in a wire: $U = \frac{1}{2} \frac{Y A (\Delta L)^2}{L} = \frac{1}{2} F \cdot \Delta L$

Thermal Stress

When a rod is clamped at both ends and temperature changes by $\Delta T$: $\sigma_{\text{thermal}} = Y \alpha \Delta T$ $F = Y A \alpha \Delta T$

where $\alpha$ is the coefficient of linear expansion (K$^{-1}$).

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