Fatigue fracture surfaces of broken high strength materials exhibit rough conoidal cracks at the vertex of which are located inclusions or heterogeneities
Experimental: The observations refer to Sakai et al. (2002), Abdesselam et al. (2018), Stinville et al. (2018) ... These cracks have been named “fish-eye marks” by two former authors and their formations have been divided into three stages: (i) formation of the characteristic area as a fine granular area (FGA); (ii) crack propagation to form the fish-eye (i.e. according to us “rough conoidal crack”); (iii) rapid crack propagation to cause the catastrophic fracture.
Modelling: The rough conoidal crack is represented by a continuous distribution of dislocations with infinitesimal Burgers vectors. This is a smooth cone on average, and for definiteness, with a vertex O at the origin, vertical symmetrical axis, horizontal circular basis with radius R= a at a height x2= h(a). Its surface consists of profound striations running radially to the vertex. The rough conoidal crack dislocation at the elevation x2= h(R) (R≤ a) is obtained as the intersection of a vertical cylinder of radius R (axis x2) and the rough crack. The dislocation distribution consists of three families (m) (distribution function Dm; m= 1, 2 and 3) with burgers vectors bm directed along the positive xm- directions. The stresses are applied uniformly at infinity with tension S22a along x2 and shears S21a and S23a along x1 and x3. Poisson’s normal stresses -νS22a (ν is Poisson’s ratio) acting in the x1and x3 directions are incorporated into the analysis. Plastic distortions associated with these dislocations are first given. Expressions of the elastic fields (displacement and stress) are also provided. Then distribution function Dm0 of circular horizontal dislocations (with the identical Burgers vectors bm) covering the smooth conoidal crack are considered and three singular integral equations that determine Dm0 are written down. Dm0 depends on angle θ, complementary to the half angle at the vertex of the cone. Dm0 may be used to approximate Dm. The crack-tip stresses have been given as functions of Dm0. Explicit expressions of the rough circular crack extension force GRC per unit length of the crack front are given that are associated with Dm0 ( θ=0). Expressions < GRCs> of GRC (crack extension force) averaged over the length of the oscillatory crack-front are plotted for the special case of sinusoidal fronts. These plots reveal that the rough circular cracks can expand under small shearing stresses. Agreements are found with high cycle fatigue experiments.
General expressions of the rough conoidal crack extension force GRCo as a function of the average crack cone angle θ have been given equally.
These works lead naturally to the characterisation of open fracture surfaces in the form of caps inside materials.
Please refer to our publications of this year 2022.
With this work at hand (i.e. "ROUGH CONOIDAL CRACK GROWING UNIFORMLY UNDER GENERAL LOADING"), it becomes possible to follow the evolution (propagation) of highest complexity cracks that nucleate from defects (such as heterogeneities, inclusions ...) located inside materials. The provided G (the crack extension force per unit length of the crack front) is function of highest number of variables and parameters.
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