Fundamental Concept

Hooke’s Law describes the linear relationship between stress and strain within the elastic limit of a material. It forms the basis of constitutive relations for linear-elastic (Hookean) materials: $$ \varepsilon \propto \sigma $$ This implies that deformation is directly proportional to the applied force, provided the material remains within its elastic range.

Elastic Constants

Hooke’s Law yields three primary elastic constants that quantify a material’s resistance to deformation:

1. Modulus of Elasticity (Young’s Modulus)

For axial loading: $$ E = \frac{\sigma}{\varepsilon} \quad \text{(Eq. 2.3)} $$ - \( \sigma \): Normal stress - \( \varepsilon \): Longitudinal strain - \( E \): Young’s modulus

Higher \( E \) implies greater stiffness. Hoop Stress in a ring or cylinder: $$ \sigma_h = \frac{d_f - d_i}{d_i} \cdot E $$ where \( d_i \) and \( d_f \) are initial and final diameters.

2. Modulus of Rigidity (Shear Modulus)

For shear loading: $$ G = \frac{\tau}{\gamma} \quad \text{(Eq. 2.4)} $$ - \( \tau \): Shear stress - \( \gamma \): Shear strain - \( G \): Modulus of rigidity

3. Bulk Modulus of Elasticity

For volumetric loading: $$ K = \frac{\sigma}{\varepsilon_v} \quad \text{(Eq. 2.5)} $$ - \( \sigma \): Volumetric stress (pressure) - \( \varepsilon_v \): Volumetric strain - \( K \): Bulk modulus

Speed of sound in a material: $$ a = \sqrt{\frac{K}{\rho}} $$ where \( \rho \) is the material density.

Longitudinal Deformation Formulas