13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- Figure 7. Characteristic force-displacement curve of the normal (RF3) and the lateral force (RF1) under load of a single tungsten crystal with {011}<01̅1> - crack system, exemplary for the ratio RF1/RF3 = 0 (CF0.0) and RF1/RF3 = 0.4 (CF0.4). Figure 8. Course of the J-integral along the crack front, from the symmetry plane up to surface., exemplary for the ratio RF1/RF3 = 0 (CF0.0) and RF1/RF3 = 0.4 (CF0.4) at 10 µm reference point displacement. We obtained the J-integral in two ways: first with the analytical relations of the standard ASTM E1820 [15] using the force-displacement curve (commonly applied in fracture experiments) and secondly numerically. J consists of an elastic Jel and plastic Jpl contribution. According to [15] the J-integral is calculated with: 2 2 (1 ) pl el pl o A K J J J E B b , (2) where Apl defines the plastic work of the applied load, B the specimen thickness and bo is defined by the difference between the width W and the initial crack length ao. The non-dimensional η describes the effect of plastic work normalized by the ligament area. For deep notch specimens, standard [7] and [15] prescribe η by the following definition: 2 ,when 0.45 0.55 a W (3) With an investigated a/W-relation of 0.27, the standard approach of η is just an approximation. Based on a numerical fit to a number of different FEM solutions, Nevalainen and Wallin [16] obtained a crack length dependent correlation of η: 2 13.818 25.124 ,when 0 0.274 0.03 1.859 , when 0.274 0.9 1 a a a W W W a W a W (4) According to [16] and with a geometry relation of a/W = 0.27, η can be determined from Eq. (4) to be 1.9 leading to a J of 0.80 mJ/mm2 at an reference point displacement of 10 µm when no lateral force is present. In case of RF1/RF3 = 0.4 the J increases to 0.86 mJ/mm2. In the numerical approach the J-integral can be determined along the crack front. Its course is given in Fig. 8. As expected, the maximum J value occurs at the symmetry plane in the state of plane strain, the so-called critical condition. Here, it is also expected that crack initiation starts with subsequent crack propagations. 0 10 20 30 40 50 60 70 80 90 0 2 4 6 8 10 Reaction Force RF [mN] RP Displacement Δu [µm] RF3 (CF0.0) RF1 (CF0.0) RF3 (CF0.4) RF1 (CF0.4) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 5 10 15 J-Integral [mJ/mm²] Crack Front Distance y [µm] Numerical ABQ (CF 0.0) Analytical RF-U (CF 0.0) Numerical ABQ (CF 0.4) Analytical RF-U (CF 0.4) Symmetry Plane Surface
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