13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- Another fracture parameter is the J-integral which characterizes the strain energy release rate and which can be calculated in the linear elastic range for mode I with 2 2 Ic Ic Ic 1 J G K E (1) from the fracture toughness KIc. GIc corresponds to the critical strain energy release rate, ν is the Poisson's ratio of 0.28 and E is the elastic modulus of tungsten with 410 GPa. Applying the purely elastic FE model the stress intensity factor for each mode (I, II or III) as well as the J-integral was calculated along the crack front for various indenter tip geometries. Additionally a parameter study with a friction coefficient varying from 0 to 0.4 was performed. The results are shown in Fig. 4. The J-integral (at an indenter tip displacement of 11.1 µm) is plotted as a function of the ratio of the lateral force to the normal force (RF1/RF3). First it shows that independently of the indenter tip geometry the J-integral decreases linearly with increasing lateral force RF1. Furthermore, it is apparent that even without friction (μ = 0) a lateral force occurs where the RF1/RF3 values vary from 0.25 to 0.44. The lateral force itself increases with increasing friction coefficient. The diagram also reveals the influence of the indenter tip geometry: the more localized the penetration of the indenter, the smaller is the overall bending of the cantilever leading to smaller J values (see Indenter R2 and R10). Instead of using the indenter tip as reference point for the displacement a different reference point (RP) outside the process zone of the indenter has been selected at the sample surface of the beam with the coordinates of 10 µm in negative x-direction at the crack tip. This time the J-integrals are compared at 10 µm displacement of this newly selected reference point (Fig. 3 a)). Fig. 5 shows the computed J-integrals as a function of the lateral to normal load for various indenters. In this way the dependence of the J-integral on the indenter geometry is eliminated and the j-integral is only determined by the ratio of the lateral to normal force. The deviation of the J values for the same indenter tip displacement can reach up to 12%. In a mesh study the local mesh refinement at the contact surfaces from 0.5 to 0.2 µm indicates no effect on the J-integral. Figure 4. J-integral as a function of the ratio of lateral to normal load at 11.1 µm indenter tip displacement revealing an influence of the indenter geometry. Figure 5. J-integral as a function of the ratio of lateral to normal load at 10 µm displacement of a chosen reference point (RP). 3.4 3.6 3.8 4.0 4.2 4.4 4.6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 J-Integral [mJ/mm²] Reaction Force RF1/RF3 [ – ] Indenter R2 (0.2µm) Indenter R10 (0.2µm) Indenter R10 (0.5µm) Wedge R10 (0.5µm) Concentrated Force Pressure µ = 0 µ = 0.4 µ = 0 µ = 0.4 µ = 0 µ = 0.4 3.4 3.6 3.8 4.0 4.2 4.4 4.6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 J-Integral [mJ/mm²] Reaction Force RF1/RF3 [ – ] Indenter R2 (0.2µm) Indenter R10 (0.2µm) Indenter R10 (0.5µm) Wedge R10 (0.5µm) Concentrated Force Pressure
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