Overload-Induced Fatigue Crack Retardation

Material

Cyclic Loading

Peak stress σ_max 150 MPa

Stress ratio R = σ_min/σ_max 0.10

Simulation speed (cycles/frame) 150

Overload Event

Overload ratio σ_OL/σ_max 1.80

Wheeler exponent m₁ (retardation strength) 2.00

Live Readouts

Steady-state Paris growth

Crack length a

5.00 mm

Cycles N

ΔK

— MPa√m

da/dN

— µm/cyc

da/dN vs. cycles—

How it works

An edge crack of length a in a plate under cyclic stress σ (peak σ_max, ratio R) grows by the empirical Paris law:

da/dN = C · (ΔK)^m
ΔK = Y·(σ_max − σ_min)·√(π·a)
K_max = Y·σ_max·√(π·a),  Y ≈ 1.12 (edge crack)

Growth is stable until K_max reaches the material's fracture toughness K_IC, at which point the remaining ligament fails catastrophically — a single overload-free run reproduces this "slow creep, then snap" behaviour.

A single high overload cycle (σ_OL > σ_max) leaves behind a plastic zone at the crack tip larger than the one produced by ordinary cycling. Irwin's plane-stress estimate gives the zone radius:

r_p(K) = (1/2π) · (K / σ_y)²

Subsequent (lower) cycles must first re-grow the crack tip through this residual plastic field before returning to their normal rate — this is retardation. The Wheeler model (1972) captures it with a multiplicative factor on the baseline Paris rate:

C_p = [ r_p(a_i) / (a_OL + r_p,OL − a_i) ] ^ m1     while a_i + r_p(a_i) < a_OL + r_p,OL
da/dN (actual) = C_p · C·(ΔK)^m

C_p starts near zero right after the overload (the crack is deep inside the residual plastic field) and rises back to 1 once the growing crack tip escapes that field — producing the visible dip-then-recovery in the da/dN chart. The exponent m1 controls how strong that dip is: larger m1 means sharper retardation.

Material buttons — switch Paris-law constants C, m, yield strength σ_y and toughness K_IC (illustrative textbook-order values; real alloys vary by heat treatment).
σ_max / R sliders — set the baseline cyclic load.
Apply Overload Cycle — fires one cycle at σ_OL = ratio × σ_max at the crack's current position, seeding a retardation zone.
Wheeler exponent — tunes how aggressively the residual plastic zone slows subsequent growth.

Fatigue crack Current plastic zone Overload retardation zone

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