applied fracture mechanics

166 Applied Fracture Mechanics22 Will-be-set-by-IN-TECH2k 2n+ ⋅10x no01−22−4−2 −1 0 1 2Figure 7. The dependence of the shear stress intensity factor k + 2n on the coordinate x0 n for the case of theboundary of half-plane loaded with normal p(x 0 )=π/4 and frictional τ(x 0 )=−λp(x 0 ) stresses,y 0 n = −0.2, α n = 0, q 0 = 0: λ = 0.1 - curve marked with 1, λ = 0.2 - curve marked with 2 (after Kudishand Covitch [1]). Reprinted with permission from CRC Press.4. Stress intensity factors and directions of fatigue crack propagationThe process of fatigue failure is usually subdivided into three major stages: the nucleationperiod, the period of slow pre-critical fatigue crack growth, and the short period of fastunstable crack growth ending in material loosing its integrity. The durations of the firsttwo stages of fatigue failure depend on a number of parameters such as material properties,specific environment, stress state, temperature, etc. Usually, the nucleation period is short[1]. We are interested in the main part of the process of fatigue failure which is due to slowpre-critical crack growth. In these cases max (k + 1 , k− 1 ) < K f , where K f is the material fracturetoughness.During the pre-critical fatigue crack growth cracks remain small. Therefore, they can bemodeled by small straight cuts in the material. For such small subsurface cracks it is sufficientto use the one-term asymptotic approximations k ± 1 = k 1 and k ± 2 = k 2 for the stress intensityfactors from (43) and (44) which in dimensional variables take the formk 1 = √ l[Y r + q 0 sin 2 α]θ[Y r + q 0 sin 2 α], k 2 = √ l[Y i − q02 sin 2α],Y =π1 ∫a H[p(t)D 0 (t)+τ(t)G 0 (t)]dt, τ = −λp,−a H{Y r , Y i } = {Re(Y), Im(Y)},[]D 0 (t) =2i −t−X 1 + 1t−X − e−2iα (X−X),(t−X)[ ]2G 0 (t) = 1 12 t−X + 1−e−2iαt−X− e−2iα (t−X), X = x + iy,(t−X) 2(45)