Ab initio aided strain gradient elasticity theory in prediction of nanocomponent fracture
Introduction
In the analysis of brittle fracture of nanostructured solids (e.g. nanocomponents of advanced devices, with the thickness ranging from few nanometers to few hundred nanometers) two major questions arise in an attempt to predict the fracture toughness: (i) whether continuum fracture mechanics works at this scale and (ii) whether the stress intensity factor is sufficient to predict fracture toughness. The two questions are closely interconnected when considering the basic hypothesis of fracture mechanics that the K-dominant region must be geometrically much larger than the fracture process zone which incorporates inelastic deformation near the crack-tip. It was shown that beyond a certain limit, the size of the K-dominance zone starts to decrease with decreasing size (width) of the nanocomponent (Shimada et al., 2015). When the size of the K-dominance zone becomes smaller than 2–3 nm it starts to be comparable to the fracture process zone (0.4–0.6 nm) that embraces a highly concentrated discrete motion of atoms near the crack-tip in atomistic models. The fracture process starts to be dominated by far-stress field terms and, therefore, the critical stress intensity factor can no more represent the total fracture driving force. Indeed, it becomes size dependent and inadequate to characterize fracture toughness of a nanostructure. The Griffith energy release rate (ERR) exhibits similar behavior. This critical case corresponds to the specimen width of approximately 10 nm below which the classical continuum approach breaks down. Such a failure comes up because, in the critically small specimens, the actual strain energy distribution near the crack-tip is no longer described by the classical continuum approach. In this respect cohesive zone models with a traction-separation law determined from atomistic simulations combined with the assumption that the material behavior near the crack tip is controlled by the K-field may fail beyond a certain critical size of a nanocomponent.
There are attempts in literature to capture the size dependency of the fracture toughness Kc using a two parameter model based on Williams's expansion, however the size dependency of Kc and/or critical ERR contradicts the constant critical ERR obtained using atomistic simulations (Sun and Qian, 2009). By analyzing the stability of a crack in the silicon nano-panel (Kotoul and Skalka, 2017) we proved that the continuum assumption can be extended beyond the limit of the classical fracture mechanics using the strain gradient elasticity theory (SGET) proposed by Mindlin (1964) which removes the presence of singular strain field at the crack tip consistently with atomistic simulations based on either the density functional theory (DFT) or molecular statics/dynamics (MS/D). This approach allows addressing deformation and fracture problems at micron and nano scales in an effective and computationally robust manner thus helping to bridge the gap between classical continuum theories and atomic-lattice theories.
The most simplified form of SGET employs only one internal length scale parameter in addition to the two classical Lamé constants. In this context it should be noted that even in case of linear, isotropic, gradient-dependent elastic material, the most general form of the strain energy depends (except of the two Lamé constants) on five additional gradient coefficients which appear in Mindlin's strain gradient theory (Mindlin, 1964). Besides the difficulties to find these coefficients, the resulting equations are quite formidable and hard to solve. Thus, a physical simplification could be useful. As already suggested in Aifantis (1984), the higher-order elastic parameters might be assumed proportional to the conventional elastic stiffness coefficients by the internal length material parameter. Persuasive arguments for such assumption are offered by Lazar and Maugin (2005). One of the arguments follows the approach suggested by Feynman in linear gravity (Feynman, 1995). The second one is based on the suggestion that the elastic energy should exhibit the same symmetry in gradient terms like the stress-strain symmetry of the elastic energy.
Thus, the crucial point is to identify the above mentioned internal length scale parameter. In the present paper, two approaches are employed for its identification by fitting the solutions for (i) phonon dispersion curves and (ii) displacement fields near the screw dislocation as obtained by DFT approaches. Since the simplified form of SGET should particularly work for isotropic materials, the tungsten single crystal was selected for determination of its length scale parameter and the critical ERR by combined SGET/DFT and DFT/MS approaches, respectively. Moreover, experimental values of the critical ERR are also known for this material. The silicon single crystal with a rather small anisotropy and widely experimentally explored fracture properties was selected as reference material. Both the critical ERR and the critical crack tip opening displacement (CTOD) for the Si crystal were already calculated using the atomistic approaches (Shimada et al., 2015; Perez and Gumbsch, 2000) and, therefore, these time-consuming computations could be omitted here. Let us note that both selected crystals are convenient for the present study also because they are, at least at low temperatures, intrinsically brittle (Pokluda et al., 2015).
Due to higher-order terms the classical linear relationship between the frequency and the wave vector becomes nonlinear for longer wave-vectors thus manifesting dispersive effects, the first phenomenon suitable for a determination of the length scale parameter. The acoustic part of the dynamical matrices is simplified by requiring that the stress-strain symmetry of the elastic energy should also be valid for the gradient terms, which leads to Laplacian version of SGET. Thus, the sixth-order tensor of higher order elastic constants of the SGET is replaced by a fourth-order tensor of classical elasticity multiplied by the second power of the yet unknown material length scale parameter. So instead of calculating six independent components of the sixth-order tensor by numerical fitting of the acoustic part from ab-initio calculations (based on the density functional theory (DFT)) one needs to obtain just a single material length scale parameter. Apparently, it would be sufficient to consider the dynamical matrix along a single high-symmetry direction, e.g. [1,0,0]. However, also other high-symmetry directions such as [1,1,0] or [1,1,1] should be considered to determine whether the material length scale parameter changes or not. Additionally, as shown by Mindlin (1964), one also needs to consider higher order terms in the kinetic energy which lead to acceleration gradients in the equation of motion.
In the second method for a determination of the length scale parameter, the analytical SGET-solution for the displacement field near the screw dislocation in the gradient elasticity is compared with the DFT calculations of the developing displacement field near the screw dislocation in cubic crystals of W and Si. We started with the screw dislocation with the Burgers vector ½ a [111] in tungsten (expressed in terms of the lattice constant a) and consecutively, the screw dislocation ½ a 〈110〉 was analyzed in silicon.
The stability of cracks in the tungsten nanocrystal was then studied using molecular statics (MS) simulations and both the critical CTOD and ERR were determined as the critical fracture parameters. For the silicon crystal these data were taken over from the literature (Shimada et al., 2015; Perez and Gumbsch, 2000). Simultaneously, the critical fracture parameters and the identified length scale parameters were used in simulations of the crack stability in W and Si nanocrystals in terms of the SGET.
Section snippets
Basic equations of strain gradient elasticity
A detailed presentation of the Form II of Mindlin's theory of SGET can be found, e.g., in Gourgiotis and Georgiadis (2009), Mindlin and Eshel (1968), Grentzelou and Georgiadis (2008), Georgiadis and Grentzelou (2006) and Aravas and Giannakopoulos (2009). The strain energy density w is a function of the linear strain tensor and of its gradient ∂kɛij,. The monopolar stress tensor τij and the so-called dipolar (or double) stress tensor mkij are then defined as
Determination of the length scale parameters for tungsten and silicon
We have used two approaches to determine length scale parameters as briefly mentioned in Introduction. The first one is based on fitting acoustic phonon dispersions generated by ab-initio calculations along high-symmetry directions. Similar procedure was also used in the work (Maranganti and Sharma, 2007), where however more higher order coefficients were considered, but on the other hand, the acceleration gradients were not incorporated.
Considering a plane-wave solution of the form
Analysis of nano-cracks in silicon and tungsten single crystals
It was well documented experimentally and by atomistic simulations (Gumbsch, 2001, Gumbsch, 2003) that the brittle fracture process in both silicon and tungsten single crystals exhibits an anisotropy with respect to the direction of crack propagation on a specific cleavage plane due to lattice trapping nanopanels effects. This anisotropy is easier to observe in silicon which can be produced as a virtually dislocation-free single crystal and a crack can propagate without a dislocation activity
Summary
The material length scale parameters l1 of the simplified form of the SGET was determined by fitting the ab-initio DFT calculations of the phonon-dispersion relations for crystals of silicon and tungsten and, for comparison, by adjusting the analytical SGET solution for the displacement field near the screw dislocation with the DFT calculations of this field. Both adjusting methods led to the values l1 = 0.28 nm for the Si-crystal and l1 = 0.22 nm for W crystal. These values well corresponded
Acknowledgements
The authors gratefully acknowledge a financial support of the Czech Science Foundation under the Project No. 17-18566S. Computational resources were provided by the Ministry of Education, Youths and Sports of the Czech Republic under the Project IT4Innovations National Supercomputer Center (Project No. LM2015070) within the program Projects of Large Research, Development and Innovations Infrastructures. Fruitful discussions with Dr. Jan Fikar from the Institute of Physics of Materials, Academy
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