recall and use Hooke’s law

Stress and Strain

Key Definitions

• Stress (\$\sigma\$): Force applied per unit area.

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

  Units: pascals (Pa) or newtons per square metre (N·m⁻²).

• Strain (\$\varepsilon\$): Relative deformation of a material.

  \$\varepsilon = \frac{\Delta L}{L_0}\$

  It is dimensionless (ratio of lengths).

Hooke’s Law

For many solids the stress is directly proportional to the strain provided the material is not permanently deformed. This linear relationship is expressed by Hooke’s law:

\$\sigma = E\,\varepsilon\$

where \$E\$ is the Young’s modulus (elastic modulus) of the material.

Young’s Modulus (\$E\$)

Young’s modulus quantifies the stiffness of a material. It is the gradient of the initial linear portion of a stress‑strain curve.

Using Hooke’s Law – Example Problem

1. Given: A steel rod of original length \$L_0 = 1.20\ \text{m}\$ and cross‑sectional area \$A = 2.0\times10^{-4}\ \text{m}^2\$ is subjected to a tensile force \$F = 12.0\ \text{kN}\$. Calculate the stress, strain, and the extension \$\Delta L\$.
2. Solution:

   • Stress:

     \$\sigma = \frac{F}{A} = \frac{12.0\times10^{3}\ \text{N}}{2.0\times10^{-4}\ \text{m}^2}=6.0\times10^{7}\ \text{Pa}\$

   • Young’s modulus for steel \$E = 200\ \text{GPa}=2.0\times10^{11}\ \text{Pa}\$.
   • Strain from Hooke’s law:

     \$\varepsilon = \frac{\sigma}{E}= \frac{6.0\times10^{7}}{2.0\times10^{11}} = 3.0\times10^{-4}\$

   • Extension:

     \$\Delta L = \varepsilon L_0 = (3.0\times10^{-4})(1.20\ \text{m}) = 3.6\times10^{-4}\ \text{m}=0.36\ \text{mm}\$

Limits of Hooke’s Law

Hooke’s law is valid only within the elastic region of the material. Beyond the proportional limit:

• The stress‑strain curve becomes non‑linear.
• Permanent (plastic) deformation occurs.
• Materials may reach a yield point, after which they deform significantly under little increase in stress.

Typical Stress‑Strain Curve

Quick Revision Checklist

• Know the definitions and units of stress and strain.
• Remember Hooke’s law: \$\sigma = E\varepsilon\$.
• Be able to rearrange the formula to solve for any of the three variables.
• Identify the elastic region on a stress‑strain graph.
• Recall typical values of Young’s modulus for common materials.