Issue 19

M. Paggi, Frattura ed Integrità Strutturale, 19 (2012) 29-36; DOI: 10.3221/IGF-ESIS.19.03 35 The peak stresses for the various simulations are plotted in Fig. 9 vs. 2 1 max max / l l   . These peak stresses are made dimensionless using the Mode I fracture energy of level 1 and the grain size diameter of the polycrystals composing the rods, 1 l d . In this diagram, the response of a standard polycrystalline material without structural hierarchy, as that shown in Fig. 6, is represented by the red dot in correspondence of 2 1 max max / 1 l l    . The results clearly show that the tensile strength of the material can be significantly increased by using a hierarchical microstructure. The interfaces of the level 2 act as crack-arresters for the microcracks propagating into level 1. The main effect of material hierarchy is therefore to increase the ability of a heterogeneous material to tolerate defects. C ONCLUSIONS n this paper it has been shown that functionally designed microstructures can offer enhanced mechanical properties as compared to traditional heterogeneous materials. Tailoring the interfaces properties allows us to enforce crack propagation along desired paths. In this way, self-resharpening effects can be achieved. Structural hierarchy is also particularly important. In this study it has been demonstrated that the interaction of interfaces with different properties at the different hierarchical levels may explain the experimental results in [7]. Further work has to be done in this direction, especially for the 3D simulation of crack propagation in polycrystalline materials. Finite element analyses should also consider coupled thermo-elastic problems, an issue particularly important in cutting technology due to the high temperature conditions. The present study has been limited to a two-level hierarchical composite material. More hierarchical level should be investigated in the future research. However, due to very different length scales involved in the problem, ranging from the size of the sample to the size of the smallest heterogeneity, modelling the mechanical behaviour is a challenging task and multiscale computational methods should be invoked [15]. One possibility is to define representative volume elements (RVE) that provide a homogenized constitutive relationship to be used at the upper level. However, although such an approach is very appealing and has been pursued by several authors [16], some aspects require special attention. For instance, the definition of a RVE is not obvious, especially in case of localized phenomena, like crack nucleation and propagation. Moreover, the condition of scales separation has to be checked with care, otherwise the risk is to exclude coupling effects between length scales that may influence the mechanical response of the material. A CKNOWLEDGEMENTS he support of the Italian Ministry of Education, University and Research (MIUR), Ateneo Italo-Tedesco, and the Deutscher Akademischer Austausch Dienst (DAAD) to the Vigoni Project "3D modelling of crack propagation in polycrystalline materials" is gratefully acknowledged. R EFERENCES [1] K.K. Chawla, Composite Materials: Science and Engineering, Springer-Verlag, Berlin, (1987). [2] I.M. Daniel, E.E. Gdoutos, K.A. Wang, J.L. Abot, Int. J. Dam. Mech. 11 (2002) 309. [3] A. Carpinteri, M. Paggi, G. Zavarise, Int. J. Solids Struct., 45 (2008) 129. [4] R. De Santis, L. Ambrosio, F. Mollica, P. Netti, L. Nicolais, Modeling of Biological Materials (Chapter 6), F. Mollica, L. Preziosi, K.R. Rajagopal Eds., Birkhäuser, Boston, (2007). [5] H.M. Yao, H.J. Gao, J. Mech. Phys. Solids, 54 (2006) 1120. [6] Z. Zhang, Y.-W. Zhang, H. Gao, In: Proc. R. Soc. B, in press, doi:10.1098/rspb.2010.1093 [7] Z.K. Fang et al., Int. J. Refractory Metals & Hard Materials, 19 (2001) 453. [8] A. Carpinteri, M. Paggi, Chaos, Solitons and Fractals, 42 (2009) 2546. [9] Functional design puts the bite into hard and refractory metals. MPR Technical Trends in Metal-Powder, 26 (2003) 20. [10] A. Carpinteri, M. Paggi, Finite Elements in Analysis Design, 43 (2007) 941. [11] S.G. Moseley, K.-P. Bohn, M. Goedickemeier, Int. J. Refractory Metals & Hard Materials, 27 (2009) 394. I T