Fatigue — S-N curves, Miner's rule — Physics notes

WHAT is fatigue?

WHY it matters for spacecraft: Launch = a few minutes of intense random vibration + acoustic loading + thousands of stress cycles. On orbit = thermal cycling every ~90 min (day/night), each cycle straining joints. Nothing is loaded once — everything is loaded repeatedly.

Key quantities describing one cycle:

$\sigma_m = \frac{\sigma_{max}+\sigma_{min}}{2}\quad(\text{mean}),\qquad \sigma_a = \frac{\sigma_{max}-\sigma_{min}}{2}\quad(\text{amplitude})$

The S–N Curve (WHAT / HOW)

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

HOW it's built: Take many identical specimens. Load each at a fixed amplitude $S_i$ until it breaks, record $N_i$ . Plot the points. Higher stress → fewer cycles survived.

Deriving the Basquin power law (from scratch)

Empirically, plotting $\log S$ vs $\log N$ gives a straight line in the high-cycle regime. A straight line in log–log means a power law. Let us build it:

$\log S = \log \sigma_f' - b\,\log N$

Why? A line $y = c - b x$ with $y=\log S$ , $x=\log N$ . Exponentiating both sides:

$\boxed{S = \sigma_f'\,N^{-b}}\qquad\Longleftrightarrow\qquad N = \left(\frac{S}{\sigma_f'}\right)^{-1/b}$

$\sigma_f'$ = fatigue strength coefficient (stress at $N=1$ ).
$b$ = Basquin exponent (slope magnitude, small positive number ~0.05–0.12).

Endurance limit

Miner's Rule (cumulative damage)

The real problem: Real loading is a mix of amplitudes ( $n_1$ cycles at $S_1$ , $n_2$ at $S_2$ , …). The S–N curve only tells us $N_i$ = life if that amplitude acted alone. How do we combine them?

Derivation from first principles

Assume damage accumulates linearly and each cycle at level $i$ contributes an equal share $\dfrac{1}{N_i}$ of a total unit of damage $D$ . For $n_i$ cycles at level $i$ :

$D_i = \frac{n_i}{N_i}$

Total damage is the sum (independent contributions):

$D=\sum_i D_i = \sum_i \frac{n_i}{N_i}$

Failure criterion: damage reaches full budget.

Worked Examples

Worked example Example 1 — Read life off Basquin

A Ti strut has $\sigma_f'=1800$ MPa, $b=0.09$ . Cyclic amplitude $S=300$ MPa. Find cycles to failure $N$ .

Step 1: Use $S=\sigma_f' N^{-b}\Rightarrow N=(S/\sigma_f')^{-1/b}$ . Why this step? We know $S$ ; invert Basquin to get $N$ .

Step 2: $S/\sigma_f' = 300/1800 = 0.1667$ . $N = 0.1667^{-1/0.09}=0.1667^{-11.11}$ . Why? $-1/b = -11.11$ .

Step 3: $\ln N = -11.11\ln(0.1667)=-11.11(-1.7918)=19.9\Rightarrow N=e^{19.9}\approx 4.4\times10^{8}$ cycles. Why? Log makes the power tractable; exponentiate back.

Common Mistakes (Steel-manned)

Active Recall

Recall Explain to a 12-year-old (Feynman)

Bend a paperclip back and forth. One bend = fine. But each wiggle makes a tiny crack a little bigger, and after enough wiggles — snap! — even though you never pulled it super hard. The S–N curve is a chart saying "wiggle THIS hard → it lasts THIS many wiggles." Miner's rule is like a phone battery: each hard wiggle drains more battery, each soft wiggle drains less. When the battery hits empty (100 %), the metal breaks.

Flashcards

What is fatigue failure?

Progressive damage & failure from cyclic loading at stresses below the static ultimate strength.

What are the two axes of an S–N (Wöhler) curve?

Stress amplitude

S

(y) vs cycles-to-failure

N

on a log scale (x).

State Basquin's law.

S = \sigma_f' N^{-b}

, i.e. a straight line in log–log;

b

is the Basquin exponent.

What is the endurance limit and which materials have a true one?

A stress below which life is effectively infinite; a well-defined limit is seen mainly in plain-carbon/low-alloy steels. Aluminium and most common Ti alloys (Ti-6Al-4V) do NOT have one.

State Palmgren–Miner's rule.

Failure when

\sum_i n_i/N_i = 1

; each cycle uses fraction

1/N_i

of total life.

In Miner's rule, what does

n_i/N_i

represent?

The fraction of total fatigue life consumed by

n_i

cycles at stress level

i

Mean vs amplitude stress formulas?

\sigma_m=(\sigma_{max}+\sigma_{min})/2

\sigma_a=(\sigma_{max}-\sigma_{min})/2

Why is Miner's rule only approximate?

It ignores load sequence/interaction; real failure damage sum ranges ~0.3–3, so safety factors are used.

Given

\sigma_f'=1800

MPa,

b=0.09

S=300

MPa, find

N

N=(300/1800)^{-1/0.09}\approx4.4\times10^8

cycles.

Two sources of cyclic load on a spacecraft?

Launch vibration/acoustics and on-orbit thermal cycling (~90-min day/night).

Connections

Stress and Strain — cyclic stress amplitude is the driver
Random Vibration & PSD — launch loads feed the S–N spectrum
Thermal Cycling on Orbit — main long-term fatigue source
Fracture Mechanics & Crack Growth (Paris' Law) — physical mechanism behind S–N
Safety Factors & Margins of Safety — why we design to $D<1$
Launch Loads & Environments

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, fatigue ka matlab hai ki metal ek hi baar ke load se nahi, balki baar-baar (cyclic) load se toot jaata hai — chahe stress yield/ultimate se kaafi neeche ho. Jaise paperclip ko baar-baar mod-mod ke tod dete ho. Spacecraft mein yeh bahut important hai kyunki launch ke time vibration aur acoustic loads thousands of cycles deti hain, aur orbit mein har 90 minute mein din-raat ka thermal cycle joints ko strain karta hai.

S–N curve (Wöhler curve) basically ek graph hai: stress amplitude $S$ vs number of cycles $N$ jitne mein part fail hoga (log scale pe). Jitna zyada stress, utne kam cycles survive. Log–log pe yeh seedhi line banti hai, isliye Basquin's law: $S = \sigma_f' N^{-b}$ . Ek endurance limit — jiske neeche infinite life — sirf mainly plain-carbon aur low-alloy steels mein achhe se dikhta hai. Aluminium aur zyadatar common titanium alloys (jaise Ti-6Al-4V) mein aisa true limit nahi hota — curve $10^7$ cycles ke aage bhi neeche girta rehta hai, isliye inke liye ek specified high-cycle strength ke liye design karte hain, "infinite life" ke liye nahi.

Real life mein load ek fixed amplitude ka nahi hota — mixed hota hai. Yahan aata hai Miner's rule. Idea simple: agar kisi amplitude pe part $N_i$ cycles mein failega, to ek cycle $1/N_i$ life kha gaya. $n_i$ cycles ne $n_i/N_i$ fraction use kiya. Sabhi fractions add karo: $\sum n_i/N_i$ . Jab yeh 1 (100%) ho jaaye, tab failure. Phone battery ki tarah socho — har cycle thodi battery drain karta hai.

Important cheez: Miner's rule ek approximation hai. Loading ka order aur overload effects ignore karta hai, isliye real mein damage sum kabhi 0.3 to 3 tak jaata hai. Isliye engineers safety factor use karte hain aur $D=1$ tak kabhi design nahi karte — margin rakhte hain.

Fatigue — S-N curves, Miner's rule

WHAT is fatigue?

The S–N Curve (WHAT / HOW)

Deriving the Basquin power law (from scratch)

Endurance limit

Miner's Rule (cumulative damage)

Derivation from first principles

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Spacecraft Structures & Systems Engineering

Connections