Part II Probability, Design and Management in NDE
Probability Distributions The probability that a flaw is between x and x + dx is p( xdx ) x p( x ) is the flaw size is the probability density pxdx ( ) = 1 xa = xp( x) dx x a average flaw size The probability that a flaw is smaller than x is Px ( ) x ( ) = ( ) Px pxdx Px ( ) cumulative probability lim Px ( ) = 1 x Px ( m) =.5 x m median flaw size
Probability of Detection The probability that a flaw of given size x is detected: POD ( x ) POD x lim POD( x) = 1 x The probability of detecting a flaw of size between x and x POD () x p() x dx + dx is The probability of having an undetected flaw of size between x and x [1 POD ( x)] p( x) dx + dx is After NDT, the modified probability density of the reduced ensemble Re-normalization p '( x) = k [1 POD ( x)] p( x) p'( x) dx = 1
Probability of Detection Operator training, alertness and confidence Correct application of the proper technique Environment of the test (laboratory, field) Material homogeneity and isotropy Flaw characteristics Shape of the part, surface roughness Calibration and capability of the system etc. eddy current ultrasonic 1. upper 1% of operators 1. upper 1% of operators Probability of Detection.8.6.4.2 mean 95% confidence Probability of Detection.8.6.4.2 mean 95% confidence.15.3.45.6.75.15.3.45.6.75 Crack Length [in] Crack Length [in] (US Air Force maintenance facilities, round-robin inspection of box-wing)
Probability Distributions 1.4 Probability Density, p (x ) 1.2 1.8.6.4.2 after NDT before NDT 1 2 3 Flaw Size, x Cumulative Probability, P(x ) 1.4 1.2 1.8.6.4.2 after NDT POD before NDT 1 2 3 Flaw Size, x
Noise-Limited POD grain noise in polycrystalline materials Amplitude [a. u.] flaw signal grain noise 2 3 4 5 6 Time [µs] 4 Probability Density 3 2 1 material noise flaw signal 1 2 3 4 Detected Signal
False Alarms versus Missed Flaws 2 material noise Probability Density 1 missed flaws low detection threshold false alarms flaw signal 1 2 3 4 Detected Signal 2 material noise Probability Density 1 missed flaws medium detection threshold false alarms flaw signal 1 2 3 4 Detected Signal 2 material noise Probability Density 1 missed flaws high detection threshold false alarms flaw signal 1 2 3 4 Detected Signal
Misqualified Cases N = N + N D DF NMQ = NMF + NFA N total number of components N D defective components N DF defect-free components N MQ misqualified components N MF missed flaws N FA false alarms Probability Density 2 1 material noise ( p N ) detection threshold ( V D ) missed flaws false alarms flaw signal ( p F ) 1 2 3 4 Detected Signal ( V ) V D NMF ND pf ( V) dv FA DF V N N N p ( V) dv p F probability density of the flaw signal p N probability density of the material noise V amplitude V D detection threshold D
Modified Probabilities after NDE x flaw size p( x ) original probability density of flaw size POD( x ) probability of detection among defective components p dd ( x ) = k d POD ( x ) p ( x ) pdd ( x ) probability density of flaw size among defective components k d re-normalization factor (to assure that pdd ( x ) dx = 1 ) among defect-free components p ( x ) = k [1 POD ( x )] p ( x ) df f pdf ( x ) probability density of flaw size among defect-free components k f re-normalization factor (to assure that p ( x) dx = 1) df Sensitivity V = S( x,...) S sensitivity function (typically nonlinear, multi-variable) for small defects V n x, where n could be as high as 3-5 for large defects V constant
Failure Prevention load environment fracture properties NDT strength available before service Probability Density stress demand sevice degradation failure nominal load peak load safety factor nominal strength
Failure Failure Rate, f burn-in final failure Time, t The probability that a specimen fails between t and t + dt is f () t dt f () t is the failure rate (probability density) The probability that a specimen fails after a given time t is P() t Initial failure (P()<< t 1) P( t + dt) P( t) = [ 1 P( t)] f ( t) dt Pt () is the probability of failure P( t + dt) P() t = [ 1 P()] t f () t dt Constant failure rate dp() t f() t dt dp() t = [1 Pt ( )] dt Pt () = 1 e f t f Ultimate probability of failure lim P() t t = 1
Failure Rate and Probability of Failure Probability Probability Probability 1.8.6.4 probability of failure.2 failure rate [per 1, hours] 1 2 3 4 5 1.8.6.4.2 Time [1, hours] probability of failure failure rate [per 1, hours] 1 2 3 4 5 1.8.6 Time [1, hours] probability of failure.4 failure rate.2 [per 1, hours] 1 2 3 4 5 Time [1, hours]
Nature of Fatigue Damage Material fatigue means changes in properties that can occur due to repeated application of stresses/strains and usually means a process which leads to cracking or failure. mechanical fatigue thermo-mechanical fatigue creep-fatigue corrosion-fatigue, etc. A common feature of these failure processes is that they take place under the influence of loads whose peak values are considerably lower than the safe loads estimated on the basis of static fracture analysis.
Fatigue Life I. crack initiation extends from the commencement of service until the moment when a terminal fatigue crack appears a) before crack nucleation b) after crack nucleation II. crack growth starts with the moment of crack initiation and lasts until the abrupt failure in the remaining cross-sectional area The duration of crack initiation and crack propagation depend on: material properties (strength, toughness, etc.) mechanical load (type, level, load ratio, etc.) environmental effects (temperature, humidity, etc.) surface conditions (roughness, contamination, corrosion, etc.)
Stages of Fatigue and the State of NDE Crack Nucleation Crack Initiation Substructural and Microstructural Damage Microscopic Crack Formation Growth and Coalescence into Dominant Cracks Stable Propagation of Dominant Macrocrack Structural Instability or Fracture Field Inspection Techniques State-of-the-Art Inspection Techniques Experimentary Laboratory Techniques Interdisciplinary Research Techniques
F Fracture Mechanics F F F F F σ σ σ Y σ a x σ a x σ a x' x' x average stress stress concentration stress intensity factor factor Stress singularity lim x ' σ( x ') K x ' Stress intensity factor K = σa π a a is the half-width of the crack (2D slit or Griffith crack)
Critical Crack Size F 2a Critical stress intensity factor, fracture toughness F K c Critical crack size 2 ac 2 2 K = π 2 σc Crack growth rate (Paris Law) da dn m ΔK = AΔ K = B Kc m n number of cycles ΔK = Kmax Kmin stress intensity factor range A, B, m material (temperature T, stress ratio R = σ min /σ max, etc.) variables B = AKm c
Influence of Stress Ratio Paris Law da m max dn = Δ = A K A( K K ) min m Kmax max a a Kmin = σa π a =σ π and min average stress R =.6 σ max a R =.3 R = time σ min a R = -1 Paris-Walker Law da dn = ( Kmax Kmin ) A (1 R) w m R σmin = a σ max a
Design Philosophies Engine Structural Integrity Program (ENSIP, USAF) Retirement for Age Low-Cycle-Fatigue Design Criteria (safe life) based on statistical lower bound 1 in 1, components is predicted to initiate a 1/32 crack Usage (e.g., stress) σ m mean lower bound -3 σ typical distribution N s N m log Life (e.g., cycles)
Retirement for Age versus Cause This photo emphasizes that the USAF retires a great number of turbine engine disks. Disks represent a significant asset, thus discarding them prior to their full useful life represents a significant cost to the Air Force. Technologies are required to more fully use the fatigue lives inherent in turbine engine disks.
Retirement for Cause Damage-Tolerant Design Criteria 1-3 safety inspections during service life a c Crack Size a d a i crack nucleation crack initiation crack limit inspection period terminal crack Cycles (or equivalent) a c Crack Size a d a i 1st inspection 2nd inspection 3rd inspection retirement Cycles (or equivalent)