Hertz crack

Fig. 18. Adhesive contact of elastic spheres. pH(r) and pa(r) are the Hertz pressure and adhesive tension distributions, (a) JKR model uses a Griffith crack with a stress singularity at the edge of contact (r = a) (b) Maugis model uses a Dugdale crack with a constant tension aa in a < r < c [1111.

Cracks caused by contact of an elastic ball of indenter with an elastic brittle surface are given by Hertz s model (Fig. 6.2.7a, b). 

Fig. 6.2.7 (a). Diagram of parameters of chevron cracks associated with spherical indenter action and (b) crack mechanism seen in cross-section of Hertz s cone (C—C crack plane). Crack growth tends to enlarge in (r, 0) in relation to the initiation of the crack peak, which optimizes the conditions for energy liberation. (After Lawn et at., 1975)... 

The Hertz method of determining Kc involves the use of an optically flat surface and a blunt— i.e., spherical— indenter. The sphere is pressed with a series of loads P until a critical load P ris found that gives a 50% chance of Hertz ring crack formation. In practice this is found by plotting the fracture probability against applied load for any given indenter radius a typical plot would look like Figure 5.18. 

In case of pressurized plasma spray specimen, different kinds of damage were observed, such as Hertz cone crack caused by impact loading in the spray coating film, lateral cracks caused by unloading due to fracture of sphere, and debonding (Fig. 8) from the bond coat extending radialy from the tip of Hertz cone crack ,... 

The cross-sections of impression for different combination of specimen and indenter size are shown in from Figs. 13 to Fig. 16. For the observation of plastic zone immediately below the impression" . While there was little distinction in the fracture of spray film and the formation of plastic zone in different specimens, they presented clear differences depending on the indenter size. In case of indenter 1 mminterfacial debonding" . As the indenter size increased, the fracture tended to turn closer to the interfacial debonding. With smaller indenter, the indentation pressure is greater, to produce plastic deformation in the substrate. With 1 mmc[) indenter, plastic deformation occurred in proportion to the size of damage area, while with 4mmc[) indenter, the plastic... 

When an atmospheric spray specimen was subjected to a quasi-static test with an indenter of 1 mmcj), both the substrate and the bond coat were affected with marked plastic deformation, and in particular, the substrate underwent so drastically that the limit of elastic-plastic deformation was stepped over to present a porous zone with concentrated cracks. As the indenter size grew, the area of plastic deformation in the substrate and the bond coat was reduced, to present Hertz cone cracks in the cross-section. In addition to Hertz cone cracks, a number of cracks were seen in parallel to the substrate, which were thought to have occurred when the load was eliminated. With an indenter of 4 mmcj), the area of plastic deformation was reduced further, and lateral cracks were observed as the load was eliminated. Debonding between the bond coat and the substrate was observed only with the 4-mm indenter. Comparing to impact tests results(Fig. 7), qusi-static tests(Fig. 14) can reproduce similar fracture behavior. 

The subj t of adhesive contact mechanics may be said to have started when Kendall (//), solving the problem of the adhesive contact of a rigid flat cylinder punch indenting the smooth plane surface of an elastic medium, demonstrated that the border of the contact area can be considered as a crack tip. The more complex problem of a spherical punch was solved in 1971 by Johnson, KendaU and Roberts (72). The JKR theory predicts the existence of contact area greater then that ven by the elastic contact Hertz s theory. The molecular attractive forces are responsible for this increase and, even in the absence of external compressive loading, the contact area has a finite size. Separating the two solids requires the application of an adherence force despite the existence of infinite normal stresses in the border of the contact area. 

In chapter 1, the properties of the viscoelastic functions are explored in some detail. Also the boundary value problems of interest are stated. In chapter 2, the Classical Correspondence Principle and its generalizations are discussed. Then, general techniques, based on these, are developed for solving non-inertial isothermal problems. A method for handling non-isothermal problems is also discussed and in chapter 6 an illustrative example of its application is given. Chapter 3 and 4 are devoted to plane isothermal contact and crack problems, respectively. They utilize the general techniques of chapter 2. The viscoelastic Hertz problem and its application to impact problems are discussed in chapter 5. Finally in chapter 7, inertial problems are considered. 

Connelly GM, Rimnac CM, Wright TM, Hertz-berg RW, Manson JA. Fatigue crack propagation behavior of ultrahigh molecular weight polyethylene. J Orthop Res 1984 2(2) 119-25. 

A typical Hertz fracture obtained after a blunt impact is shown in Figure 8.8 on a laminated (annealed) windscreen. Even small cracks develop imder humid... 

Fig. 3.22. Schematic illustration of the formation of longitudinal (1), lateral (2) and radial (4) cracks in the surrounding of a Hertz pressure half sphere (3) in glasses...

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