Issue 17

M. Paggi et alii, Frattura ed Integrità Strutturale, 17 (2011) 5-14; DOI: 10.3221/IGF-ESIS.17.01 6 I NTRODUCTION olycrystalline materials present heterogeneous microstructures where polyhedral grains are separated by interfaces. A typical scanning electron microscope (SEM) image of these microstructures is shown in Fig. 1(a). The different colours of the crystals indicate their different crystallographic orientations. The mechanical behaviour is strongly affected by the grain boundaries which govern the strength, the toughness and the ductility of the material. These properties are of paramount importance in forming processes and for the design of super-hard materials. As a general trend, the smaller the grain, the higher the material strength and the hardness. Interfaces are also important in conduction processes. In this case, however, they should be viewed as defects and the material conductivity of single crystalline materials is usually much higher than that of their polycrystalline counterparts. Fracture in polycrystalline materials can be schematically classified according to two different types: intergranular and transgranular . The former mode of fracture corresponds to the decohesion of the grains along the interfaces. Correspondingly, the material response is quite brittle. The latter is characterized by a propagation of cracks into the grains, with the occurrence of high plastic deformations leading to a much more ductile response. In spite of the fact that failure is often the result of a combination of these two modes of fracture, it is instructive to investigate each mode separately and understand the underlying mechanisms. In the present study, attention is paid to intergranular fracture, which is typically observed in brittle polycrystals. During a tensile test, microcracks develop at the grain boundaries (see Fig. 1(b) taken from [1]). At a certain deformation level, the microcracks coalescence and lead to the final failure with the propagation of a single main crack. Correspondingly, the stress-strain curve reaches a maximum and a sudden loss of load carrying capacity takes place. In order to investigate the effect of interfaces on the mechanical response of polycrystalline materials, a nonlinear fracture mechanics model is herein proposed. Intergranular fracture is depicted as a phenomenon of progressive separation at the grain boundaries governed by a nonlinear traction-separation law, or cohesive zone model (CZM). In this context, the finite thickness properties of interfaces are suitably taken into account by using the nonlocal CZM recently proposed in [2, 3] and briefly summarized in the next section. Virtual tensile tests of representative volume elements (RVEs) of material microstructures are carried out in order to simulate the phenomena of crack nucleation, coalescence and strain localization. Finally, grain size effects are investigated by changing the average grain size of the polycrystalline material and computing the peak stress of the simulated stress-strain curves. As it will be shown, numerical results are in agreement with the trend expected by the Hall-Petch law. This result confirms that fracture mechanics of interfaces is one of the most important factors governing the strength of polycrystalline materials. (a) SEM image of the microstructure (b) Microcracks observed during a tensile test Figure 1 : SEM images of brittle polycrystalline materials showing microcracks (courtesy of Dr. Ing. M. Schaper [1]). A NONLOCAL COHESIVE ZONE MODEL FOR INTERFACE FRACTURE olycrystals are an important example of a material with finite thickness interfaces. In particular, examining the materials science literature [4], a power-law dependency of the interface thickness on the grain diameter has been noticed. Figure 2 shows the relation proposed in [4], where the grain diameter is computed as the mean diameter P P