Coffin-Manson Strain-Life Relation

Classical fatigue analysis is written in stress, and it works well while the part stays elastic. It stops working the moment the part yields every cycle — because past yield the stress barely changes as the strain grows, so stress is no longer telling you anything about how hard the material is being worked. That is the low-cycle regime: thermal cycling, seismic demand, forming operations, pressure vessels cycling through start-up and shutdown, and the solder joints under every chip in every device you own.

S. S. Manson at the NACA Lewis laboratory and L. F. Coffin at General Electric's Knolls laboratory arrived at the same answer within about a year of each other in the early 1950s, from different directions — Manson from thermal shock in turbine components, Coffin from thermally cycled stainless steel. Plot plastic strain amplitude against reversals to failure on log-log axes and the result is a straight line: $\Delta\varepsilon_p/2 = \varepsilon_f'(2N_f)^c$.

Count reversals, not cycles. The bracket is $2N_f$, because one cycle contains two reversals — one loading, one unloading. This page takes cycles and doubles them for you, but a constant $c$ fitted against cycles rather than reversals belongs to a slightly different equation, and the two are quietly off by $2^c$, roughly 45% at $c = -0.55$. It is the single most common bookkeeping error on this relation. The other one is entering a strain RANGE where the equation wants an AMPLITUDE — half the range — which is a clean factor of two.

The constants are unusually well behaved. The fatigue ductility exponent $c$ falls between about $-0.5$ and $-0.7$ for essentially every metal, which is a remarkably narrow band for a property that spans aluminium, steel, nickel superalloys and solder. Coffin and Manson both landed near $-0.5$, and that near-universality is what made the relation stick. The fatigue ductility coefficient $\varepsilon_f'$ is approximately the TRUE fracture strain from an ordinary monotonic tensile test — $\ln(A_0/A_f)$ from the reduction of area — and for many steels the two agree within a factor of two. That is a genuinely useful check: the fatigue line extrapolated back to half a cycle should land near where the material simply breaks in tension, and if your fit comes back at ten times the tensile ductility, suspect the reversal convention before you suspect the material.

Because $1/c$ is around $-1.8$, life is brutally sensitive to strain. A 20% increase in plastic strain amplitude cuts the life to roughly 60% of what it was. That sensitivity is why low-cycle fatigue is solved by taking strain out of the part rather than by making anything stronger — an expansion loop in a pipe, a compliant lead on a component, a slip joint, a longer flexible section. Adding stiffness usually makes a strain-controlled problem worse, because the same imposed movement then produces more strain in whatever resists it. This is genuinely counterintuitive to anyone trained on stress-based design, and it is the mistake that gets made.

The same reasoning settles a material choice. This relation is the PLASTIC half of the strain-life curve; Basquin's relation is the elastic half, and adding them gives the full Morrow four-constant expression that covers the whole range. The plastic term dominates below roughly $10^4$ cycles and the elastic term above. The crossing point is the transition life, and it is the most useful single number in strain-life analysis: below it ductility wins, and a soft tough alloy lasts longer; above it strength wins, and a hard one does. Specifying a high-strength alloy for a low-cycle thermal problem is a classic and expensive error — the strong alloy has less ductility, so $\varepsilon_f'$ is smaller, so the plastic line sits lower, so it fails sooner under the same imposed strain.

Two caveats before this is used in anger. The relation counts cycles to INITIATION of a crack of some small defined size — often a fraction of a millimetre — not to separation of the part. In a large component the remaining propagation life can be a substantial addition, and it is Paris law's territory rather than this one's. And at high temperature, where creep and oxidation act between the cycles, the plain Coffin-Manson line overpredicts life; frequency-modified and strain-range-partitioning versions exist precisely because time at temperature does damage that a cycle count cannot see.