Issue 47

P. Gallo et alii, Frattura ed Integrità Strutturale, 47 (2019) 408-415; DOI: 10.3221/IGF-ESIS.47.31 412 ρ (nm) 2 α (deg) K tn δ f (nm) P f (μN) Specimen 1 6.3 68 4.9 83.74 30.84 Specimen 2 20.2 59 2.9 115.59 65.11 Table 2 : Geometry of the notches, deflection at failure δ f and load at failure P f . D ISCUSSION : NANOCRACKING IN SILICON s is well-known, tensile strength and fracture toughness are fundamental mechanical properties. While tensile strength has been investigated in several works/materials and its scale dependence observed [26,27], the fracture toughness has not been investigated in detail up to the present. The reasons are the well-known difficulties in the fabrication of very small specimens and in the realization of an effective pre-crack at the nanometer level. At very small scale, indeed, even a small notch radius can drastically change the stress singularity and the fracture toughness [16,17]. Ando et al. [16], for example, found that the K IC changes with the crystal orientation, with maximum and minimum values of approximately 2 and 1.2 MPa·m 0.5 , respectively. Li et al. [17] found a value of the fracture toughness of approximately 1.6 MPa·m 0.5 , in apparent disagreement with bulk silicon value. The results presented here show that, if an effective pre- crack is realized, the nanoscale Si fracture toughness agrees with the macro counterpart. However, the realization of pre- crack is still very challenging and time-consuming. Alternatively, it is shown that by employing the TCD, the demanding accurate control on the specimens geometry can be avoided since the fracture toughness is evaluated using at least two generic notches of different sharpnesses , leaving the user free to employ any obtained geometry. Fig. 3(b) shows a fracture toughness in excellent agreement with macro counterpart value and the previous experiments on pre-cracked samples. Additional experiments conducted by [10,11,15,25] further confirm the results presented here. Useful information on crack propagation can be obtained by the experiments conducted on pre-cracked samples. Fig. 4(a) refers to the sample 2 in Tab. 1 and shows that although the not perfect specimen orientation, a stable crack propagation occurred along the cleavage plane (011). Once the critical crack opening displacement was reached (see Tab. 1), unstable crack propagation and subsequent final failure were observed. The propagation once started, is very fast and brings to the final failure rapidly. As is well-known, the fracture is usually studied on the assumption that fracture takes place from pre- existing defects of flaws. Defects, however, vary in number and severity depending on the volume and size of the sample. Ideally, at the nanoscale, the inherent material strength σ 0 can be considered to be the upper limit of fracture stress of components having no defect, i.e., ideal fracture stress [28]. The micro-mechanisms governing the crack propagation in such small components should be located therefore at a smaller level, i.e., atomic structure level. A more in-depth investigation can only be conducted if the discrete nature of atoms is considered, by using numerical experiments. Indeed, recent density functional theory (DFT) calculations showed that the nanocracking in silicon is due to the breaking of a covalent bond at the crack tip. When the applied stress intensity factor K I reaches the critical value of 1 MPa·m 0.5 the bond is broken and the crack propagates instantaneously along the cleavage plane (011). This result was further confirmed with additional DFT simulations considering different models and molecular dynamics (MD) simulations, where the stress is calculated atom-by-atom using the Virial theorem. Regardless of size, geometry and loading conditions, the simulations confirmed that the nanoscale crack always starts to propagate when the atomic stress at the tip reaches its ideal (bond) strength [9,15]. Fig. 4(b) represents the fracture mode displacements at the onset of nanocracking and schematically shows the crack advancement by atomic bond breaking. In the light of crack ultimately governed by atomic bond breaking and therefore by atomic level, it is natural to wonder whether the continuum theories should be applied at such small scales, if there is any limitation to the continuum theories and how eventually evaluate this limit. Theory of critical distances lies between continuum mechanics theories and micro-mechanistic approaches. The material characteristic length should be considered therefore a representative length scale parameter that changes as the mechanisms of fracture do [29,30]. L , therefore, can somehow be representative of the low limit of continuum fracture mechanics. Validation of this hypothesis can be done by comparison with a recent relevant work by Shimada et al. [9]. Those authors proposed an interesting study based on sophisticated DFT and MD calculations on cracked specimens. Several samples were studied by varying the size. It was shown that the length of the singular stress field becomes smaller as the width of the specimen does, while the fracture process zone (defined as the zone where the discrete motion of atoms is concentrated) remains constant. When approaching a critical singular stress field length, continuum theory breaks down. This limit was quantified in 3-6 times the fracture process zone, and therefore in the case of silicon is 1.5-3 nm approximately, with a fracture process zone of 0.4-0.6 nm on average. Interestingly, L falls within the range of that limit, being 1.8 nm, while L /2 is close to the fracture A