applied fracture mechanics

156 Applied Fracture Mechanics12 Will-be-set-by-IN-TECHof the beam is fixed. Then it can be shown (see Lurye [10]) thatσ x = σ z = 0, σ y = − P xI zx(L − y),τ xz = 0, τ xy =P x 2(1+ν)a 2 +b 2{a2(1+ν)I z 3a 2 +b 2 − x 2 − (1−2ν)a2 z 2},2 2(1+ν)a 2 +b 2(27)τ zy = −P x (1+ν)a 2 +νb 2(1+ν)I z 3a 2 +b 2where I z is the moment of inertia of the beam cross section about the z-axis. By introducingthe principal stresses that satisfy the equation σ 3 − σ y σ 2 − (τ 2 xy + τ 2 zy)σ = 0, one can find thatxz,√σ 1 = 1 2 [σ y − σy 2 + 4(τxy 2 + τzy)], 2 σ 2 = 0,√σ 3 = 1 2 [σ y + σy 2 + 4(τxy 2 + τzy)].2The tensile principal stress σ 1 reaches its maximum 4|P x|Lat the surface of the beam at oneπa 2 bof the points (±a,0,0) (depending on the sign of load P x ) and is acting along the y-axis - theaxis of the beam. Based on equations (28) and the solution for the surface crack inclined tothe surface of the half-space at angle of π/2 (see Hasebe and Inohara [11] and Isida [12]), oneobtains k 10 = 4.484 √|P x |Lπ. Therefore, bending fatigue life of a beam follows from substitutinga 2 bthe expression for k 10 into equation (19)where function g(μ, σ) is determined by equation (26).(28)N =2(n−2)g 0{ 0.395a2 b|P x |L }n g(μ, σ), (29)In both cases of torsion and bending, fatigue life is independent of the elastic characteristic ofthe beam material (see formulas (25), (29), and (26)), and it is dependent on fatigue parametersof the beam material (n and g 0 ), the initial defect distribution (i.e. on μ and σ), the geometryof the beam cross section (a and b), and its length L and the applied loading (P x or M y ).In a similar fashion the model can be applied to contact fatigue if the stress field is known. Amore detailed analysis of contact fatigue is presented below for a two-dimensional case.3. Contact problem for an elastic half-plane weakened by straight cracksA general theory of a stress state in an elastic plane with multiple cracks was proposed in [13].In this section this theory is extended to the case of an elastic half-plane loaded by contactand residual stresses [1]. A study of lubricant-surface crack interaction, a discussion of thedifference between contact fatigue lives of drivers and followers, the surface and subsurfaceinitiated fatigue as well as fatigue of rough surfaces can be found in [1].The main purpose of the section is to present formulations for the contact and fracturemechanics problems for an elastic half-plane weakened by subsurface cracks. The problemsfor surface cracks in an elastic lubricated half-plane are formulated and analyzed in [1]. The