Issue 38

R. Fincato et alii, Frattura ed Integrità Strutturale, 38 (2016) 231-236; DOI: 10.3221/IGF-ESIS.38.31 236 Therefore, the crack opens in an intermediate position, closer to the factor that is predominant, which is the deformation in this case. In addition, the graphs show the curves obtained without considering the damage, to highlight the effect of the coupling. Fig. 8 shows the applied forces as a function of the nominal strain measured at the same point on the strain gauge as in Fig. 2. The damage solution agrees well with the solution reported by Bonora et al. [1] showing the complete rupture of the sample at a nominal axial strain of 15%–20%. Between the damage activation and the first crack opening, the red curve shows a small gap for the no damage solution (black line), interpreted as the microvoid formation mechanism. Corresponding to the crack opening, a large decrease is observed, leading to sudden material failure. The damage evolution shows that the fracture starts closer to the surface, although the propagation in the core towards point B is fast, whereas it is slower on the external surface (points C and D). C ONCLUSIONS e performed a monotonic tensile test with a coupled elastoplastic and damage model within the framework of continuum damage mechanics. Following the approach proposed by Lemaitre, the concept of damage as an internal variable was included in an unconventional plasticity model to simulate of the degradation of the mechanical properties in metals during non-linear analyses. The model was used to investigate the behaviour of a notched steel bar undergoing monotonic uniaxial extension. The results showed good agreement with the reference solution in the literature Bonora et al. [1], indicating that the coupled constitutive equations were implemented correctly. To take advantage of the subloading surface model features, the numerical algorithm will be applied to cyclic loading to study the effect of damage on fatigue tests. R EFERENCES [1] Bonora, N., Gentile, D., Pirondi, A., Newaz, G. Ductile damage evolution under triaxial state of stress: theory and experiments, Int. J. Plast., 21(2005) 981-1007. doi:10.1016/j.ijplas.2004.06.003 . [2] De Souza, E.N., Peric, D., Owen, D.J.R., Computational Methods for Plasticity, John Wiley and Sons, Chichester (2008) 471-515. [3] Gurson, A.L., Continuum theory of ductile rapture by void nucleation and growth: part I – yield criteria and flow rules for porous ductile media, J. Eng. Mat. Tech., 99(1977), 2-15. doi:10.1115/1.3443401. [4] Lemaitre, J., A continuous damage mechanics model for ductile fracture, J. Engng. Mat. Tech., 107 (1985) 83-89. doi:10.1115/1.3225775. [5] Lemaitre, J., Coupled elasto-plasticity and damage constitutive equations, Compt. Meth. Appl. M., 51 (1985) 31-49. doi:10.1016/0045-7825(85)90026-X. [6] Needleman, A., Tvergaard, V. An analysis of ductile rapture in notched bars, J. Mech. Phys. Solids., 32(1984) 461. doi:10.1016/0022-5096(84)90031-0. [7] Koplik, J., Needleman, A., Void Growth and coalescence in porous plastic solids, Int. J. Solids Struct., 24(1988), 835- 853. doi:10.1016/0020-7683(88)90051-0. [8] Ohata, M., Toyoda, M., Damage concept for evaluating ductile cracking of steel structure subjected to large-scale cyclic straining, Sci. Technol. Adv. Mat., 5(2004), 241-249. [9] Kachanov, L.M., In: Introduction to Continuum Damage Mechanics, Martinus, Nijhoff Publisher (Ed.), Boston- Dordrecht (1986). [10] Drucker, D.C., Conventional and unconventional plastic response and representation, Appl. Mech. Rev. ASME, 41(1988) 151-167. doi:10.1115/1.3151888. [11] Hashiguchi, K., Subloading surface model in unconventional plasticity, Int. J. Solids Struct., 25(1989), 917-945. doi:10.1016/0020-7683(89)90038-3. [12] Hashiguchi, K., In: Elastoplasticity theory. Lecture notes in applied and computational mechanics, F. Pfeiffer, P. Wriggers (Eds.), Springer, Berlin, Germany, 42 (2009). [13] Lemaitre, J., Chaboche, J.L., In: Mechanics of Solids Materials, Cambdrige University Press (1990). [14] Lemaitre, J., In: A course on damage mechanics, Berlin, Heidelberg, New York: Springer (1996). [15] Benallal, A., Billardon, R., Lemaitre, J., Continuum damage mechanics and local approach to fracture: Numerical procedures, Compt. Meth. Appl. M., 92 (1989) 141-155. doi:10.1016/0045-7825(91)90236-Y. W