applied fracture mechanics

Fracture Mechanics Based Models of Structural and Contact FatigueFracture Mechanics Based Models of Structural and Contact Fatigue 2316743+k ⋅10 2n22102112−1−21 ox n−3−2 −1 0 1 2Figure 8. 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 = π/2, 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.where i is the imaginary unit (i 2 = −1), θ(x) is a step function: θ(x) =0, x ≤ 0 and θ(x) =1, x > 0. It is important to mention that according to (45) for subsurface cracks the quantitiesof k 10 = k 1 l −1/2 and k 20 = k 2 l −1/2 are functions of x and y and are independent from l.Numerous experimental studies have established the fact that at relatively low cyclic loadsmaterials undergo the process of pre-critical failure while the rate of crack growth dl/dN (Nis the number of loading cycles) in the predetermined direction is dependent on k ± 1 and K f .Anumber of such equations of pre-critical crack growth and their analysis are presented in [6].However, what remains to be determined is the direction of fatigue crack growth.Assuming that fatigue cracks growth is driven by the maximum principal tensile stress (seethe section on Three-Dimensional Model of Contact and Structural Fatigue) we immediatelyobtain the equationk ± 2 (N, x, y, l, α± )=0, (46)which determines the orientation angles α ± of a fatigue crack growth at the crack tips. Dueto the fact that a fatigue crack remains small during its pre-critical growth (i.e. practicallyduring its entire life span) and being originally modeled by a straight cut with half-lengthl at the point with coordinates (x, y) the crack remains straight, i.e. the crack direction ischaracterized by one angle α = α + = α − . This angle is practically independent from crackhalf-length l because k ± 2= k ± 20√l, where k±20 is almost independent from l for small l (see(45)). The dependence of k ± 2 on the number of loading cycles N comes only through thedependence of the crack half-length l on N. Therefore, the crack angle α is just a function ofthe crack location (x, y).