Crack Growth Mechanism



The crack tip geometry and averaged stresses over individual elementary material blocks

Figure 1.


A crack and the discrete elementary material blocks

Figure 2.

The idealized crack tip geometry and the discrete structure of a material1.

The following assumptions were applied in this method:
  • The material is assumed to be composed of identical elementary material blocks of a finite dimension ρ* MathType@MTEF@5@5@+=feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacqaHbpGCcaGGQaaaaa@3885@ in Figure 1 and Figure 2
  • The fatigue crack can be analyzed as a sharp notch with a finite tip radius of dimension ρ* MathType@MTEF@5@5@+=feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacqaHbpGCcaGGQaaaaa@3885@
  • The material cyclic and fatigue properties used in the crack growth model are obtained from the Ramberg-Osgood cyclic stress strain curve


    Figure 3.
    and the strain-life(eN) fatigue curve


    Figure 4.
  • The number of cycles N MathType@MTEF@5@5@+=feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacaWGobaaaa@36EA@ to failure of the first elementary material block at the crack tip can be determined from the strain-life fatigue curve (Figure 4) by accounting for the stress-strain history at the crack tip and by using the Smith-Watson-Topper (SWT) fatigue damage parameter and Miner rule. Once accumulated damage reaches 1, N MathType@MTEF@5@5@+=feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacaWGobaaaa@36EA@ is a summation of life ( 1/Di MathType@MTEF@5@5@+=feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacaaIXaGaai4laiaadseacaWGPbaaaa@393C@ ) of found cycles.


    Figure 5.
  • The fatigue crack growth rate can be determined as the average fatigue crack propagation rate over the elementary material block of the size ρ* MathType@MTEF@5@5@+=feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacqaHbpGCcaGGQaaaaa@3885@ .


    Figure 6.
With the above assumptions and average linear stress over the elementary block with the size ρ* MathType@MTEF@5@5@+=feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacqaHbpGCcaGGQaaaaa@3885@ , the following crack growth equations can be derived to calculate crack growth1:


Figure 7.
Where,
Kmax,tot MathType@MTEF@5@5@+=feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacaWGlbWdamaaBaaaleaapeGaamyBaiaadggacaWG4bGaaiilaiaadshacaWGVbGaamiDaaWdaeqaaaaa@3DAC@
Total maximum stress intensity factor
ΔKtot MathType@MTEF@5@5@+=feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacqqHuoarcaWGlbWdamaaBaaaleaapeGaamiDaiaad+gacaWG0baapaqabaaaaa@3B8D@
Total stress intensity range
ΔKth MathType@MTEF@5@5@+=feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacqqHuoarcaWGlbWdamaaBaaaleaapeGaamiDaiaadIgaa8aabeaaaaa@3A8D@
Threshold stress range


Piece-wise linear crack growth equation where total driving force Δκ=Kmax,totp(ΔKtot)1p MathType@MTEF@5@5@+=feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacqqHuoarcqaH6oWAcqGH9aqppaWaaeWaaeaapeGaam4sa8aadaWgaaWcbaWdbiaad2gacaWGHbGaamiEaiaacYcacaWG0bGaam4Baiaadshaa8aabeaaaOGaayjkaiaawMcaamaaCaaaleqabaWdbiaadchaaaGcpaGaaiika8qacqqHuoarcaWGlbWdamaaBaaaleaapeGaamiDaiaad+gacaWG0baapaqabaGccaGGPaWaaWbaaSqabeaapeGaaGymaiabgkHiTiaadchaaaaaaa@4E8A@ .

Figure 8.