Paris Law Fatigue Crack Growth Calculator

The Paris-Erdogan law (1963) relates the fatigue crack growth rate per cycle to the range of stress intensity factor:

da/dN = C * (delta_K)^m

where N is number of cycles, delta_K is the stress intensity factor range, and C and m are material constants determined from test data.

The stress intensity factor range under remote stress is:

delta_K = delta_sigma * Y * sqrt(pi * a)

where delta_sigma is the stress range (MPa), Y is a geometry correction factor, and a is the crack dimension in metres.

What “a” means, and why it is not always the length you measured. This is the step that costs people a factor of two, so it is worth being explicit. The dimension a is whatever the geometry factor Y was derived for:

So a 4 mm centre crack is entered as a = 2 mm with Y = 1.0, while a 4 mm edge crack is entered as a = 4 mm with Y = 1.12. Get this backwards and delta_K is off by sqrt(2), which after the cube in the Paris exponent means the predicted life is out by nearly a factor of three.

Integrating to find total cycles. For constant amplitude loading with m not equal to 2:

N = (a_f^(1-m/2) - a_0^(1-m/2)) / ((1 - m/2) * C * (delta_sigma * Y * sqrt(pi))^m)

For m = 2 exactly, N = ln(a_f/a_0) / (C * (delta_sigma * Y * sqrt(pi))^2).

Typical Paris constants for steels (SI units, a in m, delta_K in MPa*sqrt(m)): C is approximately 1e-12, m is approximately 3. For aluminum alloys: C is approximately 4e-12, m is approximately 3.

Fracture toughness. The final crack size a_f is typically set to the size at which K_max = K_IC (material fracture toughness). This calculator uses a_f as a direct input. K_IC for high-strength steel is roughly 50-100 MPasqrt(m); for aluminum 25-35 MPasqrt(m).

Worked example. A steel plate with C = 1e-12, m = 3, cycling at a stress range of 100 MPa. Start from a 1 mm centre crack and run to 50 mm, with Y = 1.0.

• Initial delta_K = 100 x 1.0 x sqrt(pi x 0.001) = 5.60 MPa*sqrt(m)
• Initial growth rate = 1e-12 x 5.60^3 = 1.76e-10 m/cycle, about 0.18 microns per cycle
• Cycles to failure = 9.75 million

Two things fall out of that. The growth rate is so small at the start that the crack is nearly invisible for most of its life, and yet the total is finite and countable. That is the whole value of the method: it puts a number on an inspection interval.

Limitations. Paris law applies only in the intermediate da/dN range (Region II). Near threshold (Region I) and at fast fracture (Region III), more complex models are needed. Crack closure and mean stress effects are not included here.

The threshold matters more than people expect. Steels have a delta_K_th of roughly 2 to 5 MPa*sqrt(m) at high load ratio, and below it a crack simply does not propagate. If the initial delta_K reported by this calculator lands under that, the honest answer is not the cycle count on screen but “this crack is dormant at this stress range”. The integration will happily return a huge but meaningless number. The calculator flags that case in the result.