Paris' Law¶
Relate intermediate-regime fatigue-crack growth per cycle to a fitted power of crack-tip stress-intensity range.
Core Idea¶
Paris' Law is an empirical power-law fit for the stable middle portion of fatigue-crack growth. It relates an existing crack's average growth per loading cycle to the crack-tip stress-intensity range: \(da/dN=C(\Delta K)^m\), with \(\Delta K=K_{\max}-K_{\min}\). Here \(C\) and \(m\) are fitted coefficients for a specified material and set of conditions, not universal constants. On log–log axes, a valid middle-region fit appears approximately straight.[^ref-a06fe3938c92]
Scope of Application¶
The law organizes data from cyclic crack-growth tests, including NASA's direct Paris-form fit to aircraft-relevant titanium-alloy sheet and U.S. DOT Paris parameters derived from a Walker fit to railroad tank-car steel at \(R=0.6\). The same form can be tested in different materials, but the fitted values and usable \(\Delta K\) interval must be checked anew. Near a growth threshold or an approach to unstable fracture, the ordinary two-parameter fit is outside its scope. Equal \(\Delta K\) need not imply equal growth when stress ratio, crack closure, orientation or residual stress changes.[ref-cf4be00d27c7][ref-d38e755d371f][ref-a06fe3938c92][ref-a50eb33fa142]
Clarity¶
Paris' Law concerns growth of an existing crack, not the number of cycles required to initiate one. It also separates a good fit in one test from warranted transfer to a different geometry or loading context. The U.S. DOT derived its TC-128B steel Paris coefficients by matching a Walker fit at \(R=0.6\), while its broader models addressed mean-stress and curve-knee effects that the Paris fit omits. Thus a quoted \(C,m\) pair is incomplete without its test envelope and units.[^ref-a06fe3938c92]
Manages Complexity¶
Crack size, shape and load influence the crack-tip stress intensity. Within a verified middle regime, the law compresses a rate curve to \(C\) and \(m\), making deviations from a common \(da/dN\)–\(\Delta K\) trend visible. This economy is also its limit: NASA observed AA 2024-T3 rates that changed with \(R\) at fixed \(\Delta K\), and shot-peened D6AC steel surface cracks that failed to follow an uncorrected Paris fit.[ref-d38e755d371f][ref-a50eb33fa142]
Abstract Reasoning¶
Ask whether the observations form a stable straight segment on log–log rate-versus-\(\Delta K\) axes and whether the proposed target case matches the calibration's material, loading and crack-scale conditions. If rates differ systematically at equal \(\Delta K\), the one-variable collapse needs qualification or a different model. Integrating a fitted rate across crack sizes is not, by itself, a safe remaining-life or inspection decision: the crack-tip range, initial and limiting crack sizes, toughness boundary and loading history must also be justified.[ref-a06fe3938c92][ref-d38e755d371f]
Knowledge Transfer¶
The testable relation transfers between titanium-sheet and tank-car-steel fatigue data; their numerical coefficients do not automatically transfer. NASA's aluminum-alloy study separately tests the boundary of a one-variable \(\Delta K\) collapse, not an independent fitted Paris example here. Generic power laws in other fields share an algebraic form but lack the cycles, crack-tip stress intensity and growth-regime boundary that make this Paris' Law. Live Allometry and Scaling Law is therefore a mathematical neighbor, not an asserted strict DAG parent; the staged entry remains unparented.
[^ref-a06fe3938c92]: Joseph W. Cardinal, James H. Feiger and Peter C. McKeighan, Fatigue Crack Growth Equations for TC-128B Tank Car Steel, Southwest Research Institute report for U.S. DOT Volpe Center (2006), printed pp.1–5 and Table 2 p.8. [^ref-d38e755d371f]: William T. Riddell and Robert S. Piascik, Stress Ratio Effects on Crack Opening Loads and Crack Growth Rates in Aluminum Alloy 2024, NASA/TM-1998-206929 (1998), printed pp.1–2, 7–14. [^ref-cf4be00d27c7]: C. Michael Hudson, Investigation of Fatigue Crack Growth in Ti-8Al-1Mo-1V (Duplex-Annealed) Specimens Having Various Widths, NASA TN D-3879 (1967), printed pp.1, 8–10. [^ref-a50eb33fa142]: Wolf Elber, The Effects of Shot-Peening Residual Stresses on the Fracture and Crack-Growth Properties of D6AC Steel, NASA TN D-7716 (1974), printed pp.1, 8–9.
Neighborhood in Abstraction Space¶
Paris' Law sits in a sparse region of the domain-specific corpus (91st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Basquin's Law — 0.83
- Material Ratcheting — 0.83
- Dislocation Creep — 0.80
- Size Effect on Structural Strength — 0.80
- Seismic Gap — 0.78
Computed from structural-signature embeddings · 2026-10-08