Issue34

L. E. Kosteski et alii, Frattura ed Integrità Strutturale, 34 (2015) 226-236; DOI: 10.3221/IGF-ESIS.34.24 235 In Fig. 8 (left) the curve denoted by the green dots corresponds to the first elements to become damaged in a given x coordinate. Therefore, this set of points forms a curve linked with the damage propagation velocity at the crack tip and the propagation velocity of the main crack itself, according to the previous assumption made. Thus, 10 representative points were selected to carry out the curve a versus N . The cycle count begins in the normalized time t*=70, when the loading reaches permanent regime (see Fig. 6 right). Knowing the crack propagation velocity (da/dN) during the simulation process, it is possible to plot Fig. 8 (right), where the log(da/dN) versus log(  K) is presented. The tendence line of this curve models the region in crack propagation known as region of stable growth or region II (see Gdoutos [14]). The applied tendence line gives the straight line equation: log(da/dN)  0.22 log(  K) -4.37 , this means m = 0.22 and C = 4.26  10 -5 . Therefore, it is possible to characterize the modeled material according to Paris law as: da/dN = 4.26  10 -5  K 0.22 . Pointing out that calculation procedure to obtain the constants C and m was carried out with stress values in Pascal [Pa] and length in meters [m]. As previously said, the present work focus on exploring the possibilities of the LDEM tool in simulating and characterizing fatigue damage development. Matters regarding real material modeling are topics for future works. C ONCLUSIONS n this work, two applications of the Lattice Discrete Element Method for studying damage development in quasi-brittle materials were presented. In the first example, unstable crack propagation in functional graded material was analyzed. In the second example, subcritical crack propagation due to fatigue damage was studied. In both examples it was possible to verify the great potential of the applied discontinuous mechanics tool in the study of damage development. A CKNOWLEDGEMENT he authors acknowledge the support of CNPq and CAPES (Brazil). R EFERENCES [1] Xu, X., Needleman, A., Numerical simulations of dynamic crack growth along an interface, International Journal of Fracture, 74 (1995) 289-324. [2] Paulino, G.H., Zhang, Z., Cohesive Zone modeling of dynamic crack propagation in functionally graded materials, 5th GRACM International Congress on Computational Mechanics. Limassol, (2005). [3] Xu, Y., Yuan, H., Computational analysis of mixed-mode fatigue crack growth in quasi-brittle materials using extended finite element methods, Engineering Fracture Mechanics, 76 (2009) 165-181. [4] Unger, F. J., Eckardt, S., Könke, C.: Modelling of cohesive crack growth in concrete structures with the extended finite element method, Comput. Methods Appl. Mech. Engrg., 196 (2007) 4087-4100. [5] Yang, Z. J., Chen, J., Finite element modelling of multiple cohesive discrete crack propagation in reinforced concrete beams, Engineering Fracture Mechanics, 72 (2005) 2280-2297. [6] Yamaguchi, T., Okabe, T., Kosaka, T., Fatigue Simulation for Ti/GFRP Laminates using Cohesive Elements, Advanced Composite Materials, 19 (2010) 107-122. [7] Siegmund, T., Cyclic Crack Growth and Length Scales. School of Mechanical Engineering, Purdue University, Indiana, USA, (2007a). [8] Riera, J.D., Local effects in impact problems on concrete structures. Conference on Structural Analysis and Design of Nuclear Power Plants, 3 (1984). [9] Hillerborg, A., A model for fracture analysis. Division of Building Materials, Lund Institute of Technology, TVBM- 3005 (1978) 1-8. [10] Kosteski, L.E., Iturrioz, I., Batista, R.G., Cisilino, A.P., The truss-like discrete element method in fracture and damage mechanics, Engineering Computations, 6 (2011) 765  787. I T