Failure strength of brittle materials containing nanovoids Mariella Ippolito, 1,2 Alessandro Mattoni, 2 Nicola Pugno, 3 and Luciano Colombo 1,2, * 1 Department of Physics, University of Cagliari, Cittadella Universitaria, I-09042 Monserrato, Cagliari, Italy 2 Sardinian Laboratory for Computational Materials Science (SLACS, CNR-INFM), c/o Department of Physics, University of Cagliari, Cittadella Universitaria, I-09042 Monserrato (Ca), Italy 3 Department of Structural Engineering and Geotechnics, Politecnico di Torino, I-10138 Torino, Italy Received 4 August 2006; revised manuscript received 30 March 2007; published 15 June 2007 By means of atomistic simulations, we investigate the failure strength in plane strain conditions of a brittle solid containing nanosized stress concentrators, i.e., a straight crack, a cylindrical hole, or a spherical hole. We find that the failure strength of the defected solid strongly depends on the defect size, in contrast with the predictions of standard elasticity theory. A high strength reduction due to voids as large as few atoms is observed. Such results have been included in two analytical failure criteria, namely, the average stress criterion and the point stress criterion. Both models introduce a length scale typical of the system, tailored at describing the process zone near the nanovoids. We provide a numerical estimate for this length scale, which is found to be specific for any defect, and we reconcile atomistic results to continuum into a coherent picture. DOI: 10.1103/PhysRevB.75.224110 PACS numbers: 62.25.+g, 62.20.Mk, 81.40.Np I. INTRODUCTION Defects such as cracks and voids affect the mechanical behavior of brittle solids since they modify the overall strength of the material. Sometimes such defects are un- avoidable because they form during materials synthesis and processing such as, e.g., ceramic sintering. On the other side, voids may be introduced into the material by design in order to obtain specific properties. This is the case of porous ma- terials where pores at a suitable concentration are used to control the thermal or acoustic isolation, the impact energy absorption, and many other properties. 1 In any case, such inhomogeneities are of great relevance on the mechanical response of the system, since they enhance the local stress and they possibly may initiate failure. In addition, as the technological demand for extremely high strength materials increases as well as the development of nanoscale devices or machines, defects as small as a few nanometers cannot be neglected. As an example, it has been recently found that even one- or two-atom vacancy defect can reduce the failure strength of carbon nanotubes by an amount of 26%. 2,3 We will show that sizable strength reduction due to voids as large as few atoms are observed in bulk -SiC as well. The strength of materials containing cracks and voids is traditionally described according to stress intensification or stress concentration arguments, respectively. 4,5 The need of different approaches is motivated, according to linear elastic fracture mechanics LEFM, by the mathematical divergence of the stress field near the crack tip. Following LEFM, load- ing produces a 1/  x singularity at the crack tip where x is the distance from the crack tip along the plane of the crack and a critical stress equal to zero is expected. As a conse- quence, a straightforward prediction of failure stress as uniquely based on local stress criteria cannot be applied. The critical stress of the cracked body is therefore calculated by analyzing the stress singularity at the crack tip: the failure takes place when the stress intensity factor K is equal to the material fracture toughness K c . 6,7 This criterion relies on the energy balance of the Griffith theory. 8 In contrast, elasticity theory predicts that the failure from a void as it is the case of cylindrical or spherical holes takes place when the maxi- mum local stress equals the ideal material strength  th . 5 Both alternative continuum approaches for cracks and voids are based on linear elasticity and they unlikely work at the nanoscale. Their possible weaknesses could, in principle, be due to the failure of at least one of the three underlying constitutive hypotheses they rely on: either continuum me- chanics, elasticity, or linearity. In order to improve classical continuum models, modern theories of fracture are generally formulated so as to incor- porate into their formalism a suitable material length scale : this key quantity is aimed at describing a process zone close to the crack tip where at least one of the above constitutive hypotheses fails. The characteristic length scale is typically given by   2K c 2  th 2 . 1 The interpretation of the length  is not unique and it could be related to the existence of either a plastic zone i.e., the mechanical response is beyond pure elasticity, a cohesive zone linearity is lost, or a discrete unit for crack advance- ment continuum hypothesis is no longer applicable. In the framework of brittle fracture formalism, the char- acteristic length  has been incorporated in four different models. The point stress criterion 9,10 PSC assumes that the failure occurs if the stress becomes equal to  th at a suitable distance l from the notch, corresponding to l   /4. An al- ternative approach is the average stress criterion 9,10 ASC according to which the failure occurs if the mean value of the stress along a line or a surface, or a volume starting at the notch root is equal to  th ; the length l of such a line is in this case as large as . Furthermore, the equivalent linear elastic fracture mechanics equivalent LEFM 4 assumes the exis- tence of a crack at the root of the notch i.e., the effective crack length is longer than its original size: the failure is predicted to occur when this effective crack reaches the criti- PHYSICAL REVIEW B 75, 224110 2007 1098-0121/2007/7522/2241107 ©2007 The American Physical Society 224110-1