Issue 17

M. Paggi et alii, Frattura ed Integrità Strutturale, 17 (2011) 5-14; DOI: 10.3221/IGF-ESIS.17.01 13 mechanics through the introduction of a damage variable that reduces the elastic modulus of the interface material. This approach allows us to perform an upscaling of the complex nonlinear mechanisms occurring in the interface region. The parameters entering the damage formulation can be tuned according to simple axial and shear tests to be performed on RVEs of the interface microstructure. This strategy has the great advantage of avoiding computationally expensive multiscale simulations based on the FE 2 method which requires, for each Gauss point, a micromechanical computation of the response of a lower scale RVE with a complete description of its constitutive nonlinearities [20-22]. Possible applications to fiber-reinforced interfaces and polymeric interfaces are therefore envisaged, with a tuning of the damage evolution law depending on the actual forms of nonlinearities present in those materials. Finally, regarding the grain size effects on the tensile strength of polycrystals, it has to be remarked that an inversion of the trend suggested by Hall and Petch has been observed at the nanoscale. Although a different description of the material has probably to be invoked, using molecular dynamics simulations instead of continuum mechanics, our proposed model may provide an insight into this debated problem. The results obtained in the present study show that the thinner the interface, the higher its fracture energy. As a result, the tensile strength of the material increases by refining the material microstructure. This suggests that an inversion of the Hall-Petch law may occur in case of thicker interfaces at the nanoscale. Experimental results seem to confirm this predicted trend. In fact, data in [23] show that the grain boundary thickness tends to vanish at the nanoscale. However, the percentage of atoms at the grain vertices, the so called triple junctions , drastically increases. As a result, the volumetric content of the all interface atoms (the sum of the atoms belonging to triple junctions and those belonging to grain boundaries), is much higher than that suggested by the scaling law holding at the microscale and shown in Fig. 2. A CKNOWLEDGEMENTS he support of AIT, MIUR and DAAD to the Vigoni 2011-2012 project "3D Modelling of fracture in polycrystalline materials" is gratefully acknowledged. MP would also like to thank the Alexander von Humboldt Foundation for supporting his research fellowship at the Institut für Kontinuumsmechanik, Leibniz Universität Hannover (Hannover, Germany) from February 1, 2010, to January 31, 2011. R EFERENCES [1] P. Kustra, A. Milenin, M. Schaper, A. Gridin, Computer Methods in Materials Science, 9 (2009) 207. [2] M. Paggi, P. Wriggers, Computational Materials Science, 50 (2011) 1625. [3] M. Paggi, P. Wriggers, Computational Materials Science, 50 (2011) 1634. [4] D.J. Benson, H.-H. Fu, M.A. Meyers, Materials Science and Engineering A, 319–321 (2001) 854. [5] Yan-Qing Wu, Hui-Ji Shi, Ke-Shi Zhang, Hsien-Yang Yeh, International Journal of Solids and Structures, 43 (2006) 4546. [6] T. Luther, C. Könke, Engineering Fracture Mechanics, 76 (2009) 2332. [7] P.D. Zavattieri, P.V. Raghuram, H.D. Espinosa, Journal of the Mechanics and Physics of Solids, 49 (2001) 27. [8] G. Beer, International Journal for Numerical Methods in Engineering, 21 (1985) 585. [9] V. Tvergaard, Material Science and Engineering A, 107 (1990) 23. [10] N. Point, E. Sacco, International Journal of Fracture, 79 (1996) 225. [11] M. Ortiz, A. Pandolfi, International Journal for Numerical Methods in Engineering, 44 (1999) 1267. [12] J. Segurado, J. Llorca, International Journal of Solids and Structures, 41 (2005) 2977. [13] S. Li, M.D. Thouless, A.M. Waas, J.A. Schroeder, P.D. Zavattieri, Composites Science and Technology, 65 (2005) 281. [14] C. Leppin, P. Wriggers, Computers & Structures, 61 (1996) 1169. [15] J.C.J. Schellekens, R. de Borst, International Journal for Numerical Methods in Engineering, 36 (1993) 43. [16] H.D. Espinosa, P.D. Zavattieri, Mechanics of Materials, 35 (2003) 365. [17] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method, 5th ed., Butterworth-Heinemann, Oxford and Boston, (2000). [18] E.O. Hall, Proceedings of the Physical Society of London B, 64 (1951) 747. [19] N.J. Petch, Journal of the Iron Steel Institute of London, 173 (1953) 25. [20] C.B. Hirschberger, S. Ricker, P. Steinmann, N. Sukumar, Engineering Fracture Mechanics, 76 (2009) 793. T