Glossary term
Paris–Erdogan Law
An empirical relationship describing the rate of fatigue crack growth per load cycle as a function of the stress intensity factor range.
Definition
lawThe Paris–Erdogan law is an empirical power-law relationship that describes the stable crack growth rate per fatigue cycle as a function of the range of the stress intensity factor at the crack tip.
In the fracture mechanics approach to fatigue, a crack is assumed to pre-exist in the material and the analysis focuses on how fast it grows under cyclic loading. The Paris–Erdogan law provides a quantitative link between the cyclic loading characterised by the stress intensity factor range and the crack advance per cycle. It is the foundation of damage tolerance design, where components are designed not to be crack-free but to contain cracks safely for a defined inspection interval.
In the fracture mechanics approach to fatigue, attention shifts from the smooth-specimen stress amplitude of classical fatigue analysis to the crack tip stress field of a component that already contains a crack. The driving force for crack growth is the stress intensity factor K, which characterises the intensity of the elastic stress field at the crack tip. Under cyclic loading, the stress intensity factor oscillates between a minimum value K_\text{min} and a maximum value K_\text{max}. The range is:
Paul C. Paris and Fazil Erdogan showed in 1963 that, over the intermediate range of crack growth rates (the so-called Paris regime), the crack advance per cycle da/dN follows a power law in \Delta K:
where a is the crack half-length (or crack length, depending on the geometry), N is the number of elapsed cycles, and C and m are material constants determined experimentally from fatigue crack growth tests on standard specimens.
Material constants
The exponent m typically lies between 2 and 4 for most metallic alloys. For steels, m \approx 3 is common; for aluminium alloys, m is often closer to 3.5–4. The pre-factor C varies widely with material, environment (air, seawater, vacuum), load ratio R = K_\text{min} / K_\text{max}, and microstructure. Both C and m are tabulated in material databases and design standards such as BS 7910, ASME FFS-1, and the NASGRO database.
The three regimes of fatigue crack growth
The Paris law applies only in the intermediate, stable crack growth regime. A complete fatigue crack growth rate curve — plotted as \log(da/dN) versus \log(\Delta K) — shows three distinct regions:
In Region I, below a threshold value \Delta K_\text{th}, cracks do not propagate measurably. This threshold is material- and environment-dependent and represents the lower bound below which a pre-existing crack is effectively dormant under cyclic loading.
In Region II — the Paris regime — crack growth is stable and the log-log relationship is approximately linear, following the Paris law. This is the region governing most of the fatigue life of damage-tolerant structures.
In Region III, as K_\text{max} approaches the fracture toughness K_{Ic}, crack growth accelerates rapidly and the remaining ligament fails by monotonic fracture. Design in this regime is conservative because failure is imminent.
Integrating the Paris law
To determine the fatigue life N_f — the number of cycles to grow a crack from an initial size a_0 to a critical size a_c — the Paris law is integrated:
The stress intensity factor range \Delta K depends on the crack size a, the applied stress range \Delta \sigma, and a geometric correction factor Y(a) that accounts for the specimen or component geometry:
Substituting and integrating gives N_f as a function of initial crack size, applied stress, and material constants. This calculation is the core of damage tolerance analysis, which determines the maximum permissible initial crack size and the required inspection interval to ensure that any crack is detected before it reaches the critical size.
Limitations
The Paris law is a simplified model. It does not account for crack closure (contact of crack faces during the compressive part of the cycle, which reduces the effective \Delta K), mean stress effects (load ratio R), overload retardation, short crack behaviour below a few grain diameters, or environmental effects in detail. More sophisticated models — the Walker equation, the Forman equation, and the NASGRO equation — extend the Paris law to incorporate these effects, but the Paris law remains the standard starting point for damage tolerance assessment.
Common mistakes
A common mistake is to use the Paris law as if it predicted crack initiation. It does not: it starts from an assumed or detected crack and estimates propagation in a defined growth regime. Another is to integrate the law without updating the geometry factor as the crack grows. A robust assessment states the initial crack assumption, inspection detectability, stress range spectrum, load ratio, environment, material constants, critical crack size, and whether the calculated interval includes uncertainty and safety factors.