applied fracture mechanics

Fracture Mechanics Based Models of Structural and Contact FatigueFracture Mechanics Based Models of Structural and Contact Fatigue 13157problems are reduced to systems of integro-differential equations with nonlinear boundaryconditions in the form of alternating equations and inequalities. An asymptotic (perturbation)method for the case of small cracks is applied to solution of the problem and some numericalexamples for small cracks are presented.Let us introduce a global coordinate system with the x 0 -axis directed along the half-planeboundary and the y 0 -axis perpendicular to the half-plane boundary and pointed in thedirection outside the material. The half-plane occupies the area of y 0 ≤ 0. Let us considera contact problem for a rigid indenter with the bottom of shape y 0 = f (x 0 ) pressed intothe elastic half-plane (see Fig. 2). The elastic half-plane with effective elastic modulusE (E = E/(1 − ν 2 ), E and ν are the half-plane Young’s modulus and Poisson’s ratio) isweakened by N straight cracks. The crack faces are frictionless. Besides the global coordinatesystem we will introduce local orthogonal coordinate systems for each straight crack ofhalf-length l k in such a way that their origins are located at the crack centers with complexcoordinates z 0 k = x0 k + iy0 k , k = 1,...,N, the x k-axes are directed along the crack faces and they k -axes are directed perpendicular to them. The cracks are inclined to the positive directionof the x 0 -axis at the angles α k , k = 1,...,N. All cracks are considered to be subsurface.The faces of every crack may be in partial or full contact with each other. The indenter isloaded by a normal force P and may be in direct contact with the half-plane or separatedfrom it by a layer of lubricant. The indenter creates a pressure p(x 0 ) and frictional stressτ(x 0 ) distributions. The frictional stress τ(x 0 ) between the indenter and the boundary of thehalf-plane is determined by the contact pressure p(x 0 ) through a certain relationship. Thecases of dry and fluid frictional stress τ(x 0 ) are considered in [1]. At infinity the half-planeis loaded by a tensile or compressive (residual) stress σx ∞ = q 0 which is directed along the0x 0 -axis. In this formulation the problem is considered in [1].Then the problem is reduced to determining of the cracks behavior.dimensionless variablesTherefore, in(x 0 n , y 0 n )=(x0 n , y 0 n)/˜b, (p 0 n , τ 0 n , p n )=(p 0 n , τ0 n , p n)/˜q,(x n, t )=(x n , t)/l n , (v n, u n)=(v n , u n )/ṽ n , ṽ n = 4˜ql nE ,(k ± 1n , k±2n )=(k ±1n , k ± 2n )/(˜q√ l n )(30)the equations of the latter problem for an elastic half-plane weakened by cracks and loadedby contact and residual stresses have the following form [1]∫ 1−1∫ 1−1v k (t)dtt−x k+ ∑N 1∫δ m [v m(t)A r km (t, x k) − u m(t)Bkm r (t, x k)]dtm=1 −1= πp nk (x k )+πp 0 k (x k), v k (±1) =0,u k (t)dtt−x kp 0 k − iτ0 k = − 1 π+ N ∑m=1∫1δ m [v m(t)A i km (t, x k) − u m(t)Bkm i (t, x k)]dt−1= πτ 0 k (x k), u k (±1) =0,∫ b[p(t)D k (t, x k )+τ(t)G k (t, x k )]dt − 1 2 q0 (1 − e −2iα k),a(31)(32)p nk (x k )=0, v k (x k ) > 0; p nk (x k ) ≤ 0, v k (x k )=0, k = 1,...,N,