Issue34

L. E. Kosteski et alii, Frattura ed Integrità Strutturale, 34 (2015) 226-236; DOI: 10.3221/IGF-ESIS.34.24 232 Case 3, Graded E and G c : In case 3, the Elasticity Modulus E and the toughness parameter G c are changed simultaneously in vertical direction. In Paulino and Zhang’s [2] work, a graduation was proposed to the interface properties in the vertical direction, in the same manner of case 2, and the Elasticity Modulus E in the elements was modified. In DEM approach for case 3, the constitutive elemental law was modified as shown in Fig. 3(c). In Fig.4(iii) the comparison between Paulino and Zhang’s [2] implementation (a) and LDEM implementation (b), in terms of final configurations, is shown. In the present case, the similarity between both configurations is not too clear, but both show the same general tendency. (i) (ii) (iii) Figure 4 : (i) Final configuration for the case 1, homogeneous material. Final configuration for the graded materials (ii) for the case 2, (iii) for the case 3, (in all cases (i-iii) (a) the Paulino and Zhang’s [2] config., (b) the LDEM config. (a) (b) (c) d) Figure 5 : a) Energy balance during the simulated process in case 1; b) The elastic energy; c) Kinetic energy; d) Damage dissipated energy for the three analyzed cases. Comparison in terms of Energy Balance: Here the results in terms of Energy Balance during the whole fracture process are discussed. In Fig. 5(a) the complete balance of energy during the whole process for case 1 is shown. In Fig. 5 (b), (c), and (d), the elastic, kinetic and dissipated energy through all the damage process are shown for the three cases. It can be verified in Fig. 5(d) that case 1 shows high level of dissipated energy due to the greater fracture area generated during the simulation. In Fig. 5(c), the abrupt change in kinetic energy in cases 2 and 3 shows a sensitive change in the crack propagation velocity during the whole process. This is in accordance with the final configuration obtained, where branching and secondary cracks appear more than in case 1. 0.0 2.0x10 -6 4.0x10 -6 6.0x10 -6 8.0x10 -6 1.0x10 -5 1.2x10 -5 1.4x10 -5 0.0 5.0x10 -5 1.0x10 -4 1.5x10 -4 2.0x10 -4 2.5x10 -4 3.0x10 -4 3.5x10 -4 4.0x10 -4 4.5x10 -4 5.0x10 -4 Elastic Energy Kinetic Energy Internal Energy Damage disipated Energy Energy (J) t (s) 0.0 2.0x10 -6 4.0x10 -6 6.0x10 -6 8.0x10 -6 1.0x10 -5 1.2x10 -5 1.4x10 -5 0.0 5.0x10 -5 1.0x10 -4 1.5x10 -4 2.0x10 -4 2.5x10 -4 3.0x10 -4 3.5x10 -4 4.0x10 -4 Case 1 Case 2 Case 3 Elastic Energy (J) t (s) 0.0 2.0x10 -6 4.0x10 -6 6.0x10 -6 8.0x10 -6 1.0x10 -5 1.2x10 -5 1.4x10 -5 0.0 5.0x10 -5 1.0x10 -4 1.5x10 -4 2.0x10 -4 2.5x10 -4 3.0x10 -4 3.5x10 -4 4.0x10 -4 Case 1 Case 2 Case 3 Kinetic Energy (J) t (s) 0.0 2.0x10 -6 4.0x10 -6 6.0x10 -6 8.0x10 -6 1.0x10 -5 1.2x10 -5 1.4x10 -5 0.0 5.0x10 -5 1.0x10 -4 1.5x10 -4 2.0x10 -4 2.5x10 -4 3.0x10 -4 3.5x10 -4 4.0x10 -4 Case 1 Case 2 Case 3 Damage disipated Energy (J) t (s)