A simple crack system (Figure 1) can be readily studied to estimate the Hertzian conoidal crack angle and length, and also the stress intensity factor.
This is a 3-D brittle elastic half-space on the flat boundary Ox1x3 of which a rectilinear contact pressure along Ox3 is exerted by a cylinder whose axis is parallel to x3; the cylinder lies along Ox3 on the flat boundary. A planar straight-front crack inclined by an angle θ with respect to x1x3 is present under the action of the load along x2 due to the cylinder. The relevance of this modelling may be understood as follows. A slab of cylinder with thickness dx3 at spatial position O’ (0, 0, x3) exerts elastic fields (displacement and stress) proportional to those of a point load at O’ (proportionality coefficient dx3). Physically, this corresponds to the action of a spherical indenter to which is associated a conoidal fracture surface for sufficiently large load (Roesler (1956) as quoted by Frank and Lawn (1967)). The coalescence of conoidal cracks from different slabs of cylinder along Ox3 would produce planar fracture surface envelops parallel to x3 at large crack lengths. Therefore, we expect the modelling in Figure 1 to provide the experimentally observed fracture surface inclination angle θ and crack length l as a function of critical load P by both a spherical indenter and cylinder. This is the essence of the modelling depicted schematically in Figure 1.
In a recent work intitled “Fracture Mechanics in a three-dimensional elastic half-space under the rectilinear contact pressure of a cylinder” (see our contribution in ResearchGate), this crack system has been investigated. Expressions of the relative displacement of the faces of the crack, crack-tip stresses and crack extension force G per unit length of the crack front are given. G displays a maximum at an angle θ that is confronted to experiment. dG / dθ = 0, the condition that determines the crack angle, is seen to depend on Poisson’s ratio only.
Second observation:
An exact derivation of the crack extension force (or stress intensity factor) is not available in the case of the Hertzian conoidal crack produced by a spherical indenter. Our simple planar crack system provides a derivation of the crack extension force that must agree with that of a conoidal crack; this allows to get information on the dependence of the crack extension force upon the elastic constants.
- Badges
- Science topic
- Similar topics
- Physical Sciences
- Materials Science
- Material Characterization