Journal Club for November 2016: 3D Fracture Mechanics at the Atomic Scale

Introduction

The study of fracture is as old as mankind: without the creation of sharp stone tools by controlled fracture of stones, human evolution might have been radically different [1]. One could speculate that this might be an evolutionary reason for the visceral joy toddlers show when smashing things -- and why I still like to break stuff, however, now only virtually on the computer. With fracture being omnipresent, one could also think that all fracture-related problems should be solved by now. And indeed, the advent of linear elastic fracture mechanics (LEFM) and elasto-plastic fracture mechanics (EPFM) in the last 60 years has significantly reduced the number of catastrophic fracture-related failures. Nevertheless, failure due to fracture still nowadays has a huge economic impact, with estimates of around 4% of the Gross National Product of advanced industrialized nations [2]. 

The scientific and engineering challenges in understanding, modelling and controlling fracture can be traced back to its inherent multiscale nature. Whether and how a structure will fracture, depends on processes taking place at vastly different time and length scales, which in turn depend on the class of the material. Consequently, the study of fracture encompasses different fields, with physicists e.g. analyzing the rupture of atomic bonds and atomic vibrations, chemists researching the influence of the chemical environment on the breaking bond and the created surfaces, materials scientist focusing on the influence of the material’s microstructure, and mechanical engineers developing continuum scale theories of fracture. Addressing the “hard problems” in engineering fracture mechanics, like stress corrosion fatigue or creep-fatigue interaction will, however, require an integrative approach, which brings together experts from different fields [3]. 

With this journal club, I would like to stimulate discussions between physicists, material scientists and mechanicians on if and how one can apply continuum mechanical concepts to fracture at the atomic scale, and how atomistic simulations can in turn contribute in extending continuum theories to account for atomic scale aspects.

Since most textbooks treat fracture as an essentially 2D problem, with the crack front reduced to a point rather than a line, and much work has been devoted on the continuum and atomistic scale to 2D models, I suggest to focus here on 3D aspects. 

In the following I’ll present a short literature overview on some 3D crack problems in continuum fracture mechanics and 3D atomistic fracture simulations. I encourage readers to suggest and discuss additional literature, as this is intended only as a starting point for our journal club, which lives through the participation of the iMechanica community. In the tradition of the journal club, I will end by putting out two “challenges” that might be of interest to our community. 

3D Crack Problems in Continuum Fracture Mechanics

 Curved crack fronts like in the case of penny-shaped cracks or half-elliptical surface cracks require a three-dimensional treatment. Assuming isotropic linear elasticity, it can be shown that the local crack-tip stress field is of the same type than in the 2D case [4]. Taking a point P along the crack plane as origin (see Fig.1), the near field stresses can then be described by the usual expression:

 The stress intensity factors (SIF) K can however vary as function of the position (s) along the crack front: K=K(s).

 

Fig. 1: Local coordinate system for a curved crack front (from [4]).

 

To determine the SIFs and whether an arbitrarily curved crack is stable under a given loading condition or how it would propagate requires, however, a more elaborate treatment. 

 The problem of slightly curved planar cracks was studied by Rice and Gao [5,6]. For someone who is used to the concept of “line tension”, e.g., from dislocations, it is important to note that cracks cannot be described by such a string-like model. The same authors also applied their methodology to “somewhat circular cracks” [7] and to the trapping of cracks by an array of obstacles [8] . The recent advances in such perturbation approaches in linear elastic fracture mechanics have been reviewed by Véronique Lazarus [9]. 

It is now possible to determine in the framework of LEFM under which loading conditions and in which direction and along which distance a quasistatically loaded crack will propagate [9]. The predictions of perturbation approaches to the pinning of cracks by heterogeneities can now be tested in controlled experiments for interfacial cracks in peeling tests of elastomers [10,11]. There it was shown that in order to reproduce the experimental crack front geometry of the pinned crack required a second-order expansion [11]. 

Interestingly, it seems that although atomistic simulations would in principle allow for full control of the relevant parameters, no quasistatic simulations of cracks interacting with obstacles have yet been performed and compared to continuum theories.

 A related, inherently 3D, question is how the interaction of dynamically propagating cracks with obstacles/heterogeneities influences the crack propagation, leading, e.g., to crack front waves [10,12,13], which in turn are expected to lead to crack surface roughness [14] or crack deflection [15]. 

The current state regarding continuum theories of dynamic fracture was recently reviewed by Bouchbinder, Goldman and Fineberg [16]. There, the authors show that, in agreement with conclusions from experimental observations [10], dynamically propagating cracks require a nonlinear theory of fracture [16]. Although they present a weakly nonlinear theory, it is still a 2D theory. It is now generally seen as crucial next step to extend dynamic fracture mechanics to 3D  [16,17]. Atomistic simulations could serve as important tool to develop and test such theories. 

3D Aspects in Atomistic Fracture Simulations

We have recently reviewed the current state of the field of atomistic fracture simulations [18]. Even though the computational resources have constantly improved over the years, most simulations are still performed on straight crack front using quasi-2D setups with periodic boundary conditions (PBC) along the crack front direction. Such simulations have provided important insights in atomistic aspects of fracture like lattice- or bond trapping [19–21]. However, the importance of using samples with larger extension along the crack front direction rather than the often used minimal periodicity length becomes obvious when considering e.g. dislocation nucleation. Such essentially 2D setups suppress fundamental mechanisms  like the emission of dislocations on oblique glide planes [22–24]. Furthermore, the energy barrier to nucleate a dislocation half loop on an inclined glide plane is significantly lower than the energy barrier for the nucleation of a straight dislocation in quasi-2D simulations [25]. It is now commonly accepted that dislocation nucleation does not take place homogenously along the crack front but at crack front defects [26]. The study of dislocation nucleation from crack tip defects like jogs or ledges requires however the simulation of long crack fronts, and only few such studies exist [27]. Furthermore, the study of crack propagation by the formation and migration of kinks [28,29] is also only possible using long crack fronts.

 

Fig. 2: Plastic events around a penny-shaped crack on the (010)-plane in alpha-Fe. Dislocations emitted from one part of the crack front interact with dislocations or twins nucleated at other parts of the crack as well as with the crack itself (color according to potential energy, only atoms with increased energy are shown). From [18].

 Atomistic simulations of penny-shaped crack have so far only been performed by two groups [30,31]. In general, it is found that penny shaped cracks show behaviors not shown by through thickness cracks on the same crack plane, even when simulating long crack fronts. Nanoscale penny shaped cracks show a strong propensity for crack tip plasticity, which is attributed to the increased probability of penny-shaped cracks to find favorably oriented crack front segments for dislocation emission [31]. Furthermore, dislocations nucleated at one part of the crack front interacted with other parts of the crack front or dislocations nucleated from others sites at the crack front.  

Similarly, only few simulations of propagating crack fronts with localized obstacles exist [18,32]. Uhnáková et al. studied for example the interaction] crack in alpha-Fe with a rectangular bcc-Cu precipitate. Compared to the situation without precipitate, the precipitate retards the crack propagation. After breaking through the precipitate, crack front waves are visible and the resulting fracture surfaces are rough, in agreement with experiments [13,14]. In our studies of a propagating crack interacting with a void, the interaction locally pinned the crack front, leading to a local reorientation of the crack front close to the void. Once the crack front locally attained a certain orientation, new slip planes become available on which perfectly blunting dislocations were nucleated, effectively locally inhibiting further crack advance (see Fig. 3). This mechanism leads to a characteristic ‘V’-shape of the crack front, which can be found e.g. in experiments on cracks propagating in a temperature gradient in an Si-crystal with the same orientation as used in the simulations [33]. As this mechanism of dislocation emission is only available to propagating cracks, the simulations could explain the differences in the dislocation source configurations observed at crack tips in dislocation-free single-crystal Si [33,34].

Fig. 3: a-c) snapshots of a propagating crack front interacting with a void[18]. The void traps the crack causing the local reorientation of the crack front which enables the emission of dislocations on previously not accessible glide planes. d) etch pits on a fracture surface in Si, showing the V-shaped dislocation sources caused by this mechanism[33].

 

 All these observations clearly show the need for full 3D simulations of cracks.  

 

Challenges: Using 3D Atomistic Simulations to Inform Continuum Theories for 3D Crack Problems

 Fully 3D atomistic simulations of fracture will surely help in the development, parametrization and testing of 3D continuum fracture mechanics theories. 

Possible test cases / “challenges” could be

-  the prediction of the expansion of an initially circular crack in a purely brittle crystalline material with anisotropic elasticity and anisotropic lattice trapping in the slow crack propagation regime [28]. 

 - the prediction of crack front waves caused by the interaction of propagating cracks with obstacles of different properties and shapes (see e.g. [32]), including their damping by e.g., interactions with phonons. 

 

Literature:

[1]B. Cotterell, Fracture and Life, Imperial College Press, London, 2010.

[2]R.P. Reed, J.H. Smith, B.W. Christ, The economic effects of fracture in the United States. Part 1 - A Synopsis of the September 30, 1982 Report to NBS by Battelle Columbus Laboratories, National Bureau of Standards Special Publication 647-1, 1983.

[3]R.P. Wei, Fracture mechanics: integration of mechanics, materials science, and chemistry, Cambridge University Press, 2010.

[4]D. Gross, T. Seelig, Fracture Mechanics - With an Introduction to Micromechanics, Springer, Berlin, 2006.

[5]H. Gao, J.R. Rice, Shear Stress Intensity Factors for a Planar Crack With Slightly Curved Front, J. Appl. Mech. 53 (1986) pp. 774.

[6]J.R. Rice, First-Order Variation in Elastic Fields Due to Variation in Location of a Planar Crack Front, J. Appl. Mech. 52 (1985) pp. 571.

[7]H. Gao, J.R. Rice, Somewhat circular tensile cracks, Int. J. Fract. 33 (1987) pp. 155–174.

[8]H. Gao, J.R. Rice, A First-Order Perturbation Analysis of Crack Trapping by Arrays of Obstacles, J. Appl. Mech. 56 (1989) pp. 828–836.

[9]V. Lazarus, Perturbation approaches of a planar crack in linear elastic fracture mechanics: A review, J. Mech. Phys. Solids. 59 (2011) pp. 121–144.

[10]J. Chopin,  a. Prevost,  a. Boudaoud, M. Adda-Bedia, Crack front dynamics across a single heterogeneity, Phys. Rev. Lett. 107 (2011) pp. 2–5.

[11]M. Vasoya, A.B. Unni, J.-B. Leblond, V. Lazarus, L. Ponson, Finite size and geometrical non-linear effects during crack pininning by heterogeneities: An analytical and experimental study, J. Mech. Phys. Solids. 89 (2016) pp. 211–230.

[12]E. Sharon, G. Cohen, J. Fineberg, Propagating solitary waves along a rapidly moving crack front., Nature. 410 (2001) pp. 68–71.

[13]J. Fineberg, E. Sharon, G. Cohen, Crack front waves in dynamic fracture, Int. J. Fract. 121 (2003) pp. 55–69.

[14]E. Bouchaud, J.P. Bouchaud, D.S. Fisher, S. Ramanathan, J.R. Rice, Can crack front waves explain the roughness of cracks?, J. Mech. Phys. Solids. 50 (2002) pp. 1703–1725.

[15]J.R. Kermode, L. Ben-Bashat, F. Atrash, J.J. Cilliers, D. Sherman, A. De Vita, Macroscopic scattering of cracks initiated at single impurity atoms., Nat. Commun. 4 (2013) pp. 2441.

[16]E. Bouchbinder, T. Goldman, J. Fineberg, The dynamics of rapid fracture: instabilities, nonlinearities and length scales., Rep. Prog. Phys. 77 (2014) pp. 46501.

[17]I. Kolvin, G. Cohen, J. Fineberg, Crack Front Dynamics: The Interplay of Singular Geometry and Crack Instabilities, Phys. Rev. Lett. 114 (2015) pp. 175501.

[18]E. Bitzek, J.R. Kermode, P. Gumbsch, Atomistic aspects of fracture, Int. J. Fract. 191 (2015) pp. 13–30.

[19]R.  Perez, P. Gumbsch, R. Perez, P. Gumbsch, Directional anisotropy in the cleavage fracture of silicon, Phys. Rev. Lett. 84 (2000) pp. 5347–5350.

[20]M. Marder, Effects of atoms on brittle fracture, Int. J. Fract. 130 (2004) pp. 517–555.

[21]J.J. Möller, E. Bitzek, Fracture toughness and bond trapping of grain boundary cracks, Acta Mater. 73 (2014) pp. 1–11.

[22]J. Zhang, S. Ghosh, Molecular dynamics based study and characterization of deformation mechanisms near a crack in a crystalline material, J. Mech. Phys. Solids. 61 (2013) pp. 1670–1690.

[23]F.F. Abraham, R. Walkup, H.J. Gao, M. Durchaineau, Tomas, M. Seager, et al., Simulating materials failure by using up to one billion atoms and the world’s fastest computer: Work-hardening, Proc. Nat. Acad. Sci. U.S.A. 99 (2002) pp. 5783–5787.

[24]S.J. Zhou, D.M. Beazley, P.S. Lomdahl, B.L. Holian, Large-scale molecular dynamics simulations of three-dimensional ductile failure, Phys. Rev. Lett. 78 (1997) pp. 479–482.

[25]T. Zhu, J. Li, S. Yip, Atomistic Study of Dislocation Loop Emission from a Crack Tip, Phys. Rev. Lett. 93 (2004) pp. 25503.

[26]A.S. Argon, Mechanics and Physics of Brittle to Ductile Transitions in Fracture, J. Eng. Mater. Technol. Asme. 123 (2001) pp. 1–11.

[27]P.A. Gordon, T. Neeraj, M.J. Luton, The effect of heterogeneities on dislocation nucleation barriers from cracktips in ?-Fe, Model. Simul Mater Sci Eng. 17 (2009) pp. 25005.

[28]J.R. Kermode, A. Gleizer, G. Kovel, L. Pastewka, G. Csányi, D. Sherman, et al., Low Speed Crack Propagation via Kink Formation and Advance on the Silicon (110) Cleavage Plane, Phys. Rev. Lett. 115 (2015) pp. 135501.

[29]T. Zhu, J. Li, S. Yip, Atomistic configurations and energetics of crack extension in silicon, Phys. Rev. Lett. 93 (2004) pp. 25504.

[30]C.H. Ersland, I.R. Vatne, C. Thaulow, Atomistic modeling of penny-shaped and through-thickness cracks in bcc iron, Model. Simul. Mater. Sci. Eng. 20 (2012) pp. 75004.

[31]J.J. Möller, E. Bitzek, On the influence of crack front curvature on the fracture behavior of nanoscale cracks, Eng. Fract. Mech. 150 (2015) pp. 107–208. -- Just email me if you do not have access.

[32]A. Uhnáková, A. Machová, P. Hora, O. ?ervená, Growth of a brittle crack (001) in 3D bcc iron crystal with a Cu nano-particle, Comput. Mater. Sci. 83 (2014) pp. 229–234.

[33]B.J. Gally, A.S. Argon, Brittle-to-ductile transitions in the fracture of silicon single crystals by dynamic crack arrest, Philos. Mag. A. 81 (2001) pp. 699–740.

[34]A.A. George, G.G. Michot, Dislocation loops at crack tips - nucleation and growth - an experimental study in silicon, Mat. Sci. Eng. A. 164 (1993) pp. 118–134.