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 a is crack half-length (m), 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 for a center crack in an infinite plate 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 (Y = 1.0 for an infinite plate center crack, Y = 1.12 for a through-crack at a free surface), and a is crack length in meters.

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).

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.