#round 3 "Fatigue Crack Growth Analysis - Paris Law 'This calculator estimates fatigue crack growth rate and remaining fatigue life using the Paris-Erdogan law for linear elastic fracture mechanics. ' 'All inputs are dimensionless. Assumed consistent unit system: stress in MPa, length in mm, toughness in MPa sqr mm. "References 'Ref 1: Paris, P.C. and Erdogan, F., 1963, A Critical Analysis of Crack Propagation Laws, Journal of Basic Engineering, Vol. 85, pp. 528-534 'Ref 2: Anderson, T.L., 2005, Fracture Mechanics Fundamentals and Applications, 3rd Edition, CRC Press "Inputs '

Stress Parameters

' Applied cyclic stress range in MPa -'d_sigma = ? {100} ' Maximum applied stress in MPa -'σ_max = ? {200} '

Crack Geometry

' Initial crack length in mm -'a_0 = ? {2} ' Geometry correction factor -'Y = ? {1.12} 'Y = 1.12 for edge crack in semi-infinite plate per Ref 2 Table 2.2 '

Material Fracture Toughness

' Fracture toughness K_Ic in MPa sqr mm -'K_Ic = ? {3000} 'Equivalent to approximately 95 MPa sqr m '

Paris Law Coefficients

' Paris coefficient C multiplier -'C_val = ? {2} ' Paris coefficient C negative power of 10 -'C_exp = ? {13} 'C = C_val times 10 to the minus C_exp in units of mm/cycle/(MPa sqr mm)^m ' Paris law exponent, must not equal 2 -'m_p = ? {3} 'Typical range for metals: 2 to 4. Must not equal 2. "Calculations '

'Click to expand/collapse detailed calculations '

Paris Law Coefficient

C = C_val*10^(-C_exp) ' Paris law coefficient -'C '

[Ref 1]

Stress Intensity Factor Range

'At the initial crack length a_0: d_K = Y*d_sigma*sqr(π*a_0) ' Stress intensity factor range in MPa sqr mm -'d_K '

[Ref 2, Eq. 2.41]

Crack Growth Rate

'Per the Paris-Erdogan law da/dN = C times DK^m: da_dN = C*d_K^m_p ' Crack growth rate in mm per cycle -'da_dN '

[Ref 1, Paris-Erdogan Law]

#hide da_dN_E6 = da_dN*10^6 #show '

Critical Crack Length

'The crack length at which K reaches K_Ic under maximum stress: a_cr = (K_Ic/(Y*σ_max))^2/π ' Critical crack length in mm -'a_cr '

[Ref 2, Eq. 2.43]

Remaining Fatigue Life Estimation

'Closed form integration of Paris law from a_0 to a_cr. 'Valid for Paris exponent m not equal to 2. n_exp = 1 - m_p/2 ' Integration exponent -'n_exp Q_fac = (Y*d_sigma*sqr(π))^m_p ' Stress-geometry parameter -'Q_fac #if a_0 ≥ a_cr N_life = 0 'Initial crack already at or above critical size. #else N_life = (a_cr^n_exp - a_0^n_exp)/(n_exp*C*Q_fac) #end if ' Estimated remaining fatigue life in cycles -'N_life '

[Ref 2, Eq. 12.16]

"Results 'Stress Intensity Factor Range (MPa sqr mm): d_K 'Crack Growth Rate: 'da/dN = 'da_dN_E6' x 10^-6 mm per cycle da_dN 'Critical Crack Length (mm): a_cr 'Remaining Fatigue Life (cycles): N_life '

Assessment - Crack Size

#if a_0 < a_cr '

PASS - Crack Below Critical Size

Initial crack a_0 = 'a_0' mm is less than critical crack a_cr = 'a_cr' mm

Estimated remaining fatigue life: 'N_life' cycles

#else '

FAIL - Crack At or Above Critical Size

Initial crack a_0 = 'a_0' mm equals or exceeds critical crack a_cr = 'a_cr' mm

Immediate fracture risk. Remove component from service.

#end if '

Assessment - Fatigue Life

#if a_0 ≥ a_cr '

CRITICAL - No Remaining Fatigue Life

Crack already at or above critical size. Immediate action required.

#else if N_life < 10000 '

WARNING - Low Remaining Fatigue Life

Remaining life of 'N_life' cycles is less than 10000 cycles

Recommend immediate inspection and repair planning.

#else '

OK - Adequate Remaining Fatigue Life

Remaining life of 'N_life' cycles exceeds 10000 cycles

Schedule routine inspection per maintenance program.

#end if "Conclusion 'The fatigue crack growth analysis per the Paris-Erdogan law has been completed following Ref 1 and Ref 2. For an initial crack of 'a_0' mm with geometry factor Y = 'Y' under a stress range of 'd_sigma' MPa the stress intensity factor range is 'd_K' MPa sqr mm. The crack growth rate at the initial crack length is 'da_dN_E6' x 10^-6 mm per cycle. The critical crack length for fast fracture is 'a_cr' mm giving an estimated remaining fatigue life of 'N_life' cycles.

Fatigue Crack Growth Analysis

About This Calculator