understand and use the terms load, extension, compression and limit of proportionality

Stress and Strain

Learning Objective

By the end of this lesson you should be able to:

• Define the terms load, extension, compression and limit of proportionality.
• Distinguish between tensile and compressive loading.
• Apply Hooke’s law to calculate stress and strain within the elastic region.
• Interpret a stress‑strain graph and identify the limit of proportionality.

Key Definitions

Load (Force): The external force $F$ applied to an object, measured in newtons (N). It can be tensile (pulling) or compressive (pushing).

Extension: The increase in length of a material when a tensile load is applied. If the original length is $L_0$ and the new length is $L$, the extension $\Delta L = L - L_0$.

Compression: The decrease in length of a material when a compressive load is applied. The change in length is also expressed as $\Delta L$, but $\Delta L$ is negative.

Limit of Proportionality (LoP): The maximum stress at which stress is directly proportional to strain. Beyond this point the material no longer obeys Hooke’s law.

Stress and Strain

Stress ($\sigma$) is the force applied per unit original cross‑sectional area $A$:

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

Strain ($\varepsilon$) is the relative change in length:

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

Both stress and strain are dimensionless in the SI system, though stress is expressed in pascals (Pa) and strain is often expressed as a percentage.

Hooke’s Law

Within the elastic region (up to the limit of proportionality) stress and strain are related by a constant called Young’s modulus $E$:

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

Re‑arranging gives the familiar form for a tensile or compressive spring:

$$F = k\,\Delta L \quad\text{where}\quad k = \frac{EA}{L_0}$$

Identifying the Limit of Proportionality

1. Plot stress ($\sigma$) against strain ($\varepsilon$) as the load is increased.
2. Observe the initial straight‑line region where the graph passes through the origin.
3. The point where the curve first deviates from this straight line marks the limit of proportionality.

Example Calculation

Given a steel rod with:

• Original length $L_0 = 1.20\ \text{m}$
• Cross‑sectional area $A = 2.5 \times 10^{-4}\ \text{m}^2$
• Young’s modulus $E = 2.0 \times 10^{11}\ \text{Pa}$

If a tensile load of $F = 5.0 \times 10^{4}\ \text{N}$ is applied, find the extension $\Delta L$ (assuming the load is within the elastic region).

1. Calculate stress: $\displaystyle \sigma = \frac{F}{A} = \frac{5.0 \times 10^{4}}{2.5 \times 10^{-4}} = 2.0 \times 10^{8}\ \text{Pa}$.
2. Find strain using Hooke’s law: $\displaystyle \varepsilon = \frac{\sigma}{E} = \frac{2.0 \times 10^{8}}{2.0 \times 10^{11}} = 1.0 \times 10^{-3}$.
3. Determine extension: $\displaystyle \Delta L = \varepsilon L_0 = (1.0 \times 10^{-3})(1.20) = 1.2 \times 10^{-3}\ \text{m} = 1.2\ \text{mm}$.

Common Misconceptions

• Load vs. Stress: Load is the total force applied; stress is that force distributed over an area.
• Extension vs. Strain: Extension is an absolute change in length; strain is the relative change (dimensionless).
• Limit of Proportionality vs. Yield Point: The LoP is where linearity ends; the yield point is where permanent deformation begins, which may occur at a higher stress.

Symbol Summary

Symbol	Name	Unit	Expression
$F$	Load (Force)	newton (N)	Given or measured
$A$	Cross‑sectional area	square metre (m²)	Measured or calculated
$\sigma$	Stress	pascal (Pa)	$\sigma = \dfrac{F}{A}$
$\Delta L$	Extension (or compression)	metre (m)	$\Delta L = L - L_0$
$\varepsilon$	Strain	dimensionless (often %)	$\varepsilon = \dfrac{\Delta L}{L_0}$
$E$	Young’s modulus	pascal (Pa)	$E = \dfrac{\sigma}{\varepsilon}$
LoP	Limit of Proportionality	Pa (stress) or dimensionless (strain)	Maximum stress where $\sigma \propto \varepsilon$