applied fracture mechanics

162 Applied Fracture Mechanics18 Will-be-set-by-IN-TECHk 1±4k 2±x 10 −312.5 x 10−3 −42201.53−11−20.51−3x o02 4 6 8 10Figure 3. Comparison of the two-term asymptotic expansions k ± 1a and k± 2a with the numericallycalculated stress intensity factors k ± 1n and k± 2n obtained for y0 = −0.4, δ 0 = 0.1, α = π/2, λ = 0.1, andq 0 = −0.005. Solid curves are numerical results while dashed curves are asymptotical results. (k − 1n group1, k + 1n group 2, k− 2n group 3, k+ 2n group 4) (after Kudish [15]). Reprinted with permission of the STLE.3.3. Stress intensity factors k ± 1n and k± 2nbehavior for subsurface cracksSome examples of the behavior of the stress intensity factors k ± 1n and k± 2n for subsurface cracksare presented below.It is important to keep in mind that for the cases of no friction (λ = 0) and compressive or zeroresidual stress (q 0 ≤ 0) all subsurface cracks are closed and, therefore, at their tips k ± 1n = 0.Let us consider the case when the residual stress q 0 is different from zero. The residual stressinfluence on k + 1n results in increase of k+ 1n for a tensile residual stress q0 > 0 or its decreasefor a compressive residual stress q 0 < 0 of the material region with tensile stresses. Fromformulas (43), (44), and Fig. 4 (obtained for y 0 n = −0.2, α n = π/2, and δ n = 0.1) followsthat for all x 0 n and for increasing residual stress q 0 (see the curves marked with 3 and 5 thatcorrespond to λ = 0.1, q 0 = 0.04, and λ = 0.2, q 0 = 0.02, respectively) the stress intensityfactor k + 1n is a non-decreasing function of q0 . Moreover, if at some material point k + 1n (q0 1 ) > 0for some residual stress q 0 1 , then k+ 1n (q0 2 ) > k+ 1n (q0 1 ) for q0 2> q0 1 (compare curves markedwith 1 and 2 with curves marked with 3 and 4 as well as with curves marked with 5 and6, respectively). Similarly, for all x 0 n when the magnitude of the compressive residual stress(q 0 < 0) increases (see curves marked with 4 and 6 that correspond to λ = 0.1, q 0 = −0.01 and