Issue 19

M. Paggi, Frattura ed Integrità Strutturale, 19 (2012) 29-36; DOI: 10.3221/IGF-ESIS.19.03 33 Each exagonal rod (mesostructure) is composed of a standard polycrystalline material (microstructure). At the microscopic scale (called level 1 in this work), polycrystalline grains are separated by interfaces. Such polycrystals compose the material mesostructure (level 2), which is represented by the exagonal rods. Such rods are also separated from each others by interfaces, much thicker and with different composition with respect to the interfaces of level 1. As proposed in [13,14] interface fracture can be modelled by simplifying the real material microstructure and considering zero-thickness interface elements between the grains. Then, a suitable cohesive zone model (CZM) which takes into account the properties of finite thickness interfaces has to be used. In the present study, we consider the nonlocal CZM recently proposed by Paggi and Wriggers [13,14]. A sketch of the interfaces of a standard polycrystalline material is shown in Fig. 6. Ideal exagonal shapes are considered for the polycrystals. The constitutive model of each interface is described by a Mixed Mode stress-separation relation, given by the nonlocal CZM [13,14]. Figure 6 : 2D model of a polycrystal (CZM interface elements are shown in red with a suitably amplified thickness, for visual representation). To model the present materials processing, we remark that each rod is realized first through sintering of polycrystalline materials as those shown in Fig. 6. Then, the individual roads are joined together using high pressure and temperature conditions, such that the interfaces of level 2 develop. This configuration is sketched in Fig. 7, where yellow interfaces define the boundaries of the rod cells. A direct comparison between Figs. 6 and 7 clearly shows that the two microstructures are not physically similar, if different constitutive laws are used for the interfaces at the two levels. As an example, let us consider interfaces at the second level tougher than those of the first level. In particular, we select 2 1 IC IC / 5 l l G G  . Keeping constant the CZM parameters of the interfaces of level 1, different CZM shapes are considered for the interfaces of level 2, as shown in Fig. 8 in case of pure Mode I deformation. Here, 1 max l  denotes the peak cohesive stress of the interfaces of level 1, and 1 l Nc g is the critical relative opening displacement corresponding to vanishing cohesive stresses for the interfaces of level 1. Considering virtual tensile tests, imposing a monotonic horizontal displacement to the nodes on the vertical right side of the material microstructure, the homogenized response of the representative volume element of the hierarchical material is determined. 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.