What is the stress-intensity range ΔK?

ΔK is the range of the crack-tip stress intensity over a load cycle, ΔK = Y·Δσ·√(πa), where Δσ is the stress range, a the crack length and Y a geometry factor. Crack growth correlates with ΔK rather than with stress alone, which is what makes fracture mechanics predictive for fatigue.

What are the three regions of the da/dN curve?

Region I is near the threshold ΔK_th, below which cracks effectively do not grow. Region II is the stable, log-linear Paris regime. Region III is rapid acceleration as the peak K approaches the fracture toughness and final failure. The Paris law describes Region II.

How do I get remaining life from it?

Integrate the growth law from the initial (detectable or assumed) crack size to the critical size at which K reaches the toughness. Because ΔK depends on crack length, the integral is usually done numerically — the result is the number of cycles, which divided by the cyclic rate gives remaining life.

Fatigue · guide

Paris law and fatigue crack growth

Damage-tolerant design assumes cracks exist and asks how long until one becomes critical. The Paris law is the engine of that calculation: it ties the crack-growth rate to the stress-intensity range, so a measured or assumed flaw can be grown forward to failure.

Growth is governed by ΔK, not stress

The key insight of fracture mechanics is that fatigue crack growth correlates with the cyclic stress-intensity range at the crack tip:

ΔK = Y · Δσ · √(π·a)

where Δσ is the applied stress range, a the crack length and Y a dimensionless factor for the geometry and loading. Two very different components with the same ΔK grow their cracks at the same rate.

The three regions and the Paris law

Plotting growth rate da/dN against ΔK on log–log axes gives a characteristic S-shape. Below the threshold ΔK_th (Region I) cracks barely advance. In the central, stable Region II the data are log-linear — the Paris law:

da/dN = C · (ΔK)^m

with C and m material constants (m is typically 2–4 for metals). Region III is the steep run-up to failure as K_max approaches the fracture toughness.

For loading with a mean stress, the load ratio R = σ_min/σ_max shifts the curve; models like Walker or Forman extend Paris to capture the R-effect and the Region III acceleration.

From growth law to remaining cycles

Because ΔK grows as the crack lengthens, life is the integral of the growth law from the initial flaw a_i to the critical size a_c (where K_max = K_IC):

N = ∫ from a_i to a_c of da / [ C·(Y·Δσ·√(π·a))^m ]

Usually evaluated numerically. The initial size comes from inspection capability (the largest flaw that could be missed) and the critical size from the toughness.

Open the calculatorFatigue crack-growth calculator →Integrate Paris/Walker/Forman growth from an initial flaw to critical size, with the da/dN curve, threshold, and a Monte-Carlo life band.

Where the uncertainty lives

Crack-growth predictions are sensitive to the assumed initial flaw size, the geometry factor Y, and the scatter in C and m — which can span an order of magnitude across heats. Treat a single deterministic life as indicative and prefer a probabilistic crack-growth analysis that carries those uncertainties through to a probability of failure and an inspection interval.

Frequently asked

What is the stress-intensity range ΔK?: ΔK is the range of the crack-tip stress intensity over a load cycle, ΔK = Y·Δσ·√(πa), where Δσ is the stress range, a the crack length and Y a geometry factor. Crack growth correlates with ΔK rather than with stress alone, which is what makes fracture mechanics predictive for fatigue.
What are the three regions of the da/dN curve?: Region I is near the threshold ΔK_th, below which cracks effectively do not grow. Region II is the stable, log-linear Paris regime. Region III is rapid acceleration as the peak K approaches the fracture toughness and final failure. The Paris law describes Region II.
How do I get remaining life from it?: Integrate the growth law from the initial (detectable or assumed) crack size to the critical size at which K reaches the toughness. Because ΔK depends on crack length, the integral is usually done numerically — the result is the number of cycles, which divided by the cyclic rate gives remaining life.

References

P.C. Paris, F. Erdogan, "A Critical Analysis of Crack Propagation Laws," Journal of Basic Engineering, 1963.
ASTM E647, "Standard Test Method for Measurement of Fatigue Crack Growth Rates."
S. Suresh, "Fatigue of Materials," Cambridge University Press.
BS 7910, "Guide to methods for assessing the acceptability of flaws in metallic structures."

Related guides

Browse all engineering guides →