Foundations — Fatigue — S-N curves, Miner's rule

This page builds — from absolute zero — every symbol, word, and picture the parent note Fatigue topic leans on. Read top to bottom; nothing appears before it is earned.

1. Force, area, and stress $\sigma$

Before we can talk about "loading" anything, we need a fair way to say how hard a material is being squeezed or pulled. Pulling a thick rope with 10 N is gentle; pulling a hair with the same 10 N snaps it. So force alone is not enough — we must share the force over the area it acts on.

Look at the figure: the same force $F$ (orange arrows) pulls two bars. The thin bar has a small area, so the force is crammed into few "columns" of material — high stress (red). The thick bar spreads the same force over many columns — low stress (green). Stress is what the material actually feels, independent of how big the part happens to be.

The Greek letter $\sigma$ (sigma) is just the traditional name-tag for stress. Whenever you see it, read "how hard the material is being squeezed, per unit area". Deeper foundations live in Stress and Strain.

2. Tension, compression, and sign

A pull and a push are opposites, so we give stress a sign.

Why does sign matter for fatigue? Because cracks are opened by tension and pressed shut by compression. A cycle that swings into strong tension is far more dangerous than one that only wiggles in compression. Keeping the sign lets us track which way the load pushes the crack.

3. ${\sigma }_{UTS}$ — the ultimate strength (the "one big pull" limit)

The whole surprise of fatigue is captured in one sentence: parts fail from repeated loading at stresses far below ${\sigma }_{UTS}$. So ${\sigma }_{UTS}$ is the benchmark we compare against to feel that surprise. (Design margins against ${\sigma }_{UTS}$ are handled in Safety Factors & Margins of Safety.)

4. A "cycle" — one up-and-down of stress

Launch shakes a part; orbit heats and cools it. In both cases the stress rises and falls, over and over. One full rise-and-fall is a cycle.

Look at the wavy blue line — stress plotted against time. It has a top and a bottom:

From these two we build the two numbers that actually describe a cycle:

WHAT these are, on the figure: the mean ${\sigma }_m$ (gray dashed line) is the centre the wave wobbles around; the amplitude ${\sigma }_a$ (orange arrow) is how far it swings up (or down) from that centre. WHY split it this way? Because a crack is opened by the swing and held ajar by the centre — so the topic needs both numbers separately. Adding the peak and valley and halving gives the middle; subtracting and halving gives the half-swing. That is all the two formulas are: "middle of two numbers" and "half their gap".

5. $S$ — the stress amplitude, renamed

The parent note writes the amplitude as $S$. There is nothing new here:

6. $N$ — number of cycles, and why we count in powers of ten

Here is the scale problem. A part loaded hard might die at $N=1000$ cycles; loaded gently it might survive $N=500,000,000$. Those numbers differ by a factor of half a million — you cannot fit both on one ruler-style axis. So we use a logarithmic scale.

Look at the two number lines. On the ordinary (linear) line, $10$, $100$, $1000$ are almost stacked at the far left — you cannot see them. On the log line they are evenly spaced. WHY the topic needs logs: fatigue lives across a huge range of cycle counts, and — as the next section shows — the data becomes a straight line only in log–log form. Straight lines are easy to fit and easy to read.

7. Power laws and the straight line in log–log

The parent note says "a straight line in log–log means a power law". Let us earn that.

Take logs of $y=c{x}^k$. Using the log rules "log of a product adds" and "log of a power pulls the power out front": $\log y=\log c+k\log x.$ Now rename $Y=\log y$ and $X=\log x$: this reads $Y=\log c+kX$ — the equation of a straight line with slope $k$. So:

8. Fraction, sum ($\sum$), and "the budget"

Miner's rule adds up small pieces. Two bits of notation:

9. Where these foundations feed the topic

Read it as a river: stress feeds the idea of a cycle; the cycle gives mean and amplitude; amplitude plus the log/power-law tools build the S–N curve; the curve hands each level an $N$; fractions of those $N$'s are summed to $1$ in Miner's rule. Neighbouring topics that pour into fatigue are Random Vibration & PSD and Launch Loads & Environments (they supply the cycles), Thermal Cycling on Orbit (slow orbital cycles), and Fracture Mechanics & Crack Growth (Paris' Law) (the physics of the growing crack itself).

Equipment checklist

Test yourself — you are ready for the parent note when you can answer each without peeking.