recall and use EP = 21 Fx = 21 kx2 for a material deformed within its limit of proportionality

Elastic and Plastic Behaviour

Learning Objective

Recall and use the energy‑storage formula

\$E_{\text{P}} = \frac{1}{2}Fx = \frac{1}{2}kx^{2}\$

for a material deformed within its limit of proportionality.

Key Concepts

• Stress (\$\sigma\$) – force per unit area, \$\sigma = \frac{F}{A}\$.
• Strain (\$\varepsilon\$) – relative extension, \$\varepsilon = \frac{x}{L_{0}}\$.
• Hooke’s Law – within the limit of proportionality, \$F = kx\$ or \$\sigma = E\varepsilon\$, where \$E\$ is Young’s modulus.
• Limit of Proportionality – the maximum stress at which stress and strain remain linearly related.
• Elastic Limit – the greatest stress that a material can sustain without permanent deformation.
• Plastic Region – stress beyond the elastic limit where permanent (plastic) deformation occurs.
• Energy Stored (Elastic Potential Energy) – area under the linear portion of the force–extension graph.

Stress–Strain Diagram

Mathematical Development

For a material obeying Hooke’s law (\$F = kx\$), the work done in stretching it from \$x=0\$ to \$x\$ is

\$E{\text{P}} = \int{0}^{x}F\,dx = \int_{0}^{x}kx\,dx = \frac{1}{2}kx^{2}\$

Since \$F = kx\$, the same expression can be written as

\$E_{\text{P}} = \frac{1}{2}Fx\$

This energy is stored as elastic potential energy and is fully recoverable provided the material is not loaded beyond its elastic limit.

Example Calculation

1. Given: a steel wire of original length \$L_{0}=1.00\ \text{m}\$ and cross‑sectional area \$A=2.0\times10^{-6}\ \text{m}^{2}\$ has a Young’s modulus \$E=2.0\times10^{11}\ \text{Pa}\$. It is stretched by \$x=2.0\ \text{mm}\$.
2. Find the force \$F\$, the spring constant \$k\$, and the elastic potential energy stored.

Important Points for Revision

• The linear (elastic) region ends at the limit of proportionality. Beyond this point, stress and strain are no longer proportional, although the material may still behave elastically up to the elastic limit.
• All energy stored in the elastic region is recoverable; any work done beyond the elastic limit is dissipated as heat or used to create permanent deformation.
• For most engineering materials, the limit of proportionality and the elastic limit are very close, but they are not identical concepts.
• When solving problems, always check that the deformation \$x\$ (or strain) lies within the linear region before applying \$E_{\text{P}} = \frac{1}{2}kx^{2}\$.

Common Misconceptions

1. “All deformation is elastic.” – Only deformation within the elastic limit is fully recoverable.
2. “The area under any stress–strain curve gives the stored energy.” – It gives the energy only up to the elastic limit; the area beyond represents energy dissipated.
3. “Hooke’s law applies to any material.” – Hooke’s law is valid only within the limit of proportionality for linear elastic materials.

Summary

Understanding the distinction between elastic and plastic behaviour is essential for predicting how materials respond to forces. Within the limit of proportionality, Hooke’s law (\$F = kx\$) holds, and the work done in deforming the material is stored as elastic potential energy, given by \$E_{\text{P}} = \frac{1}{2}Fx = \frac{1}{2}kx^{2}\$. Once the elastic limit is exceeded, permanent deformation occurs and the simple energy formula no longer applies.