Foundations — Stress and strain — σ = F - A, ε = ΔL - L, Young's modulus E

This page assumes you know nothing. Every letter, every slash, every unit that the parent note Stress and Strain throws at you is built here from the ground up. Read it top to bottom — each idea is a brick for the next.

0. The picture we keep returning to

Everything in this topic is one experiment: a bar being pulled at both ends. Fix that image now.

That bar has a length (how long, left to right), a cross-section (the flat face you'd see if you sliced it), and it feels a force (the pull). Those three plain things become our three symbols. Let's earn each one.

1. Force — the symbol $F$

The picture: the two arrows tugging the ends of the bar in the figure above. One newton is roughly the weight of a small apple resting in your hand — the pull gravity gives it.

Why the topic needs it: the whole point of a strut is to carry a load. The load is a force. But — and this is the parent note's opening warning — force alone cannot tell you if something breaks. A pull of $1000\text{N}$ snaps a hair and does nothing to a girder. We need to compare force to how much material shares it. That's the next symbol.

2. Area — the symbol $A$

The picture: imagine cutting the bar with a knife and looking at the cut face — a little rectangle or circle. Its size is $A$.

Why "cross-sectional" and not just "area"? Because a bar has many faces (the long sides, the ends). Only the face facing the pull matters — that is the surface the internal force must cross. Look at the figure: the fat bar's cut face is big, the thin bar's is small, even though the pull is identical.

3. Length and change of length — $L$ and $\Delta L$

The picture: the bar was $L$ long; after pulling it is $L+\Delta L$ long. The extra sliver on the end is $\Delta L$.

Reading $\Delta$ out loud: "delta L" means "the little bit of L that changed." It is not $\Delta$ multiplied by $L$ — $\Delta$ is not a number, it's an instruction: take the difference.

Why the topic needs it: stretching $1\text{mm}$ is a huge deal for a $2\text{mm}$ sample and nothing for a $2\text{m}$ beam. So raw stretch $\Delta L$ is as misleading as raw force $F$. We must compare it to the original $L$ — that ratio is strain, section 6.

4. Ratios, and why dividing is the whole trick

Both key moves in this topic are divisions:

• crowding $=\frac{F}{A}$ (force compared to area)
• proportion of stretch $=\frac{\Delta L}{L}$ (stretch compared to length)

The slash $/$ just means "divided by." $\text{N}/{\text{m}}^2$ reads "newtons divided by square metres." Keep that in mind — units divide exactly like numbers.

5. Stress — the symbol $\sigma$

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

The picture: the density of the little internal-force arrows spread across the cut face. Many arrows squeezed into a small face = high $\sigma$.

Reading the units: $1\text{Pa}$ is one newton on one square metre — a tiny crowding. Real metals live at millions of pascals, so we use the megapascal: $1\text{MPa}=10^6\text{Pa}$. The little superscript $10^6$ means "1 followed by 6 zeros," i.e. a million.

6. Strain — the symbol $\epsilon$

$\epsilon =\frac{\Delta L}{L}$

The picture: the sliver $\Delta L$ measured as a fraction of the whole bar $L$. Half a bar longer? $\epsilon =0.5$. A thousandth longer? $\epsilon =0.001$.

Why no units? It's metres divided by metres — the units cancel, leaving a pure number (like a percentage). Because real strains are tiny, we often say microstrain: $1\mu \epsilon =10^{-6}$, i.e. one part in a million. The $\mu$ ("mu") means "millionth."

7. Proportional to — the symbol $\propto$

The picture: plot $\sigma$ up, $\epsilon$ across — for small pulls the points fall on a straight line starting at the origin. That straight-line region is where all our formulas live. (You'll meet the full curved graph in Stress-Strain Curve.)

Why "proportional" and not just "related"? Because "related" could be any wiggly connection. Proportional is the special, clean, straight-line relationship — and only a straight line has a single constant slope. That slope is our final symbol.

8. Young's modulus — the symbol $E$

The picture: the steepness of the straight line in section 7. A steep line = large $E$ = the material barely strains for a big stress = stiff. A gentle line = small $E$ = stretches easily.

This last one — the "engineer's stretch formula" — is the whole topic packed into five symbols you now understand. This proportional law is Hooke's Law applied to a material.

9. Reading Greek letters aloud (so notation never scares you)

How the foundations feed the topic

Every arrow is a division or a slope you built above. Nothing entered from outside.

Where these foundations go next

• The straight line becomes a full curve in Stress-Strain Curve.
• The proportional law itself is Hooke's Law.
• Pull one way, it thins the other way — Poisson's Ratio.
• Heat also causes $\Delta L$ — Thermal Stress.
• Real struts route these forces — Spacecraft Load Paths and Struts.
• Staying safely below the ceiling — Factor of Safety and Yield Strength and Plastic Deformation.

Equipment checklist

Test yourself — cover the right side.