Mechanical & Aerospace Fracture Mechanics
Paris-Erdogan Fatigue Crack Growth Life Formula
Calculate Critical Fracture Crack Size a_c = (1 / PI()) * (K_IC / (Y * Delta_sigma))^2 and exact Paris-Erdogan Fatigue Cycles to Failure N_f across both power-law (m != 2) and logarithmic (m = 2) exponents.
Integrate subcritical crack propagation from NDT flaw size a_0 to fracture toughness K_IC!
In aerospace damage-tolerance engineering, Paul Paris and Fazil Erdogan (1963) showed that subcritical fatigue crack growth per load cycle follows da/dN = C * (Delta_K)^m, where Stress-Intensity Range is Delta_K = Y * Delta_sigma * SQRT(PI() * a) (for constant geometry factor Y and zero-to-tension cyclic stress Delta_sigma = sigma_max)! Unstable brittle fracture occurs when K_max reaches Plane-Strain Fracture Toughness K_IC at Critical Crack Length a_c = (1 / PI()) * (K_IC / (Y * Delta_sigma))^2. Integrating da / a^(m/2) from initial non-destructive inspection flaw size a_0 < a_c to a_c yields N_f = (a_0^(1 - m/2) - a_c^(1 - m/2)) / (C * (m/2 - 1) * (Y * Delta_sigma * SQRT(PI()))^m) when m != 2, and the exact logarithmic branch N_f = LN(a_c / a_0) / (C * PI() * Y^2 * Delta_sigma^2) at m = 2!
Verify on Initial_Crack_Length a_0 = 0.004 m (4.00 mm in B2), Fracture_Toughness K_IC = 50 MPa*sqrt(m) (B3), Cyclic_Stress_Range Delta_sigma = 100 MPa (B4), Geometry_Factor Y = 1.00 (B5), Paris_Coefficient C = 1e-11 (B6), and Paris_Exponent m = 3.0 (ferritic steel, B7): Critical Crack Size is a_c = (1 / PI()) * (50 / 100)^2 = 0.079577 m = 79.58 mm (E2); Initial Stress Intensity is Delta_K_0 = 1.0 * 100 * SQRT(PI() * 0.004) = 11.21 MPa*sqrt(m) (E3); and Cycles to Failure equal N_f = 440,580 cycles (E4)! Notice the classic spreadsheet #DIV/0! trap: omitting the m = 2 IF branch divides by (m/2 - 1) = 0 for aluminum alloys near m = 2! Test m = 2 with C = 1e-9: our guarded formula switches cleanly to LN(a_c / a_0) and returns 95,189 cycles!
| Damage-Tolerance Fracture Metric (a_0=4mm, K_IC=50, dSigma=100 MPa, Y=1) | Formula | Verified Output |
|---|---|---|
| Critical Crack Length a_c (mm) | 1000 * (1 / PI()) * (K_IC / (Y * dSigma))^2 | 79.58 mm (0.07958 m) |
| Initial Stress Intensity dK_0 | Y * dSigma * SQRT(PI() * a_0) | 11.21 MPa*sqrt(m) (< 50 K_IC) |
| Fatigue Life N_f (Steel m=3, C=1e-11) | (a_0^-0.5 - a_c^-0.5) / (0.5*C*(Y*dSig*SQRT(PI))^3) | 440,580 cycles |
| Logarithmic Branch (m=2, C=1e-9) | LN(a_c / a_0) / (C * PI() * Y^2 * dSigma^2) | 95,189 cycles (No #DIV/0!) |
Singularity-Safe Paris-Erdogan Cycles-to-Failure Formula
Check a_0 >= a_c first: if the detected flaw already meets or exceeds Critical Crack Length a_c, the component fails on the very first load cycle ("Immediate Unstable Fracture (a0 >= ac)")!
=LET(a_in, B2, k_ic, B3, d_sg, B4, y_gm, B5, c_pr, B6, m_ex, B7, IF(OR(a_in <= 0, k_ic <= 0, d_sg <= 0, y_gm <= 0, c_pr <= 0, m_ex <= 0), "Invalid Fracture Input", LET(a_cr, (1 / PI()) * (k_ic / (y_gm * d_sg)) ^ 2, IF(a_in >= a_cr, "Immediate Unstable Fracture (a0 >= ac)", LET(gm_k, y_gm * d_sg * SQRT(PI()), ROUND(IF(m_ex = 2, LN(a_cr / a_in) / (c_pr * gm_k ^ 2), (a_in ^ (1 - m_ex / 2) - a_cr ^ (1 - m_ex / 2)) / (c_pr * (m_ex / 2 - 1) * gm_k ^ m_ex)), 0))))))Critical Crack Size (mm) & Inspection Safety Margin Formula
Aerospace NDT schedules set inspection intervals to N_f / 2 (or 2x detectable flaw size)—note that for m = 3, over half of total fatigue life is spent growing just from 4 mm to 14 mm!
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0, B7 <= 0), "Invalid Fracture Input", ROUND(1000 * (1 / PI()) * (B3 / (B5 * B4)) ^ 2, 2))Try the example
Use the original values or change them to test your understanding.