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Role of structure in predicting Thuy Nguyen, Daniel Bonamy

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Thuy Nguyen, Daniel Bonamy. Role of crystal lattice structure in predicting . Phys- ical Review Letters, American Physical Society, 2019, 123 (20), ￿10.1103/PhysRevLett.123.205503￿. ￿hal-02320548v2￿

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Thuy Nguyen1, 2 and Daniel Bonamy1, ∗ 1Service de Physique de l’Etat Condensée, CEA, CNRS, Université Paris-Saclay, CEA Saclay 91191 Gif-sur-Yvette Cedex, France 2EISTI, UniversitÃľ Paris-Seine, Avenue du Parc, 95000 Cergy Pontoise Cedex, France We examine the atomistic scale dependence of material’s resistance-to-failure by numerical simula- tions and analytical analysis in electrical analogs of brittle . We show that fracture toughness depends on the lattice geometry in a way incompatible with Griffith’s relationship between fracture and free surface . Its value finds its origin in the matching between the continuum displace- ment field at the engineering scale, and the discrete nature of at the atomic scale. The generic asymptotic form taken by this field near the crack tip provides a solution for this matching, and subsequently a way to predict toughness from the atomistic parameters with application to .

Predicting when failure occurs is central to many in- The factor two, here, comes from the fact that breaking dustrial, societal and geophysical fields. For brittle solids one bond creates two surface . This formula sig- under tension, the problem reduces to the destabilization nificantly underestimates Γ [5–8]. A major hurdle is to of a preexisting crack: Remotely applied stresses are dra- conciliate the continuum LEFM description with the dis- matically amplified at the crack tip, and all the dissipa- crete nature of solids at the atomic scale: Lattice trap- tion processes involved in the problem are confined in a ping effect has e.g. been invoked [9–14] to explain the small fracture process zone (FPZ) around this tip. Linear observed shift in Griffith threshold. Analytical analysis elastic (LEFM) provides the frame- of crack dynamics in lattices has also evidenced [15, 16] work to describe this concentration; the main re- novel high frequency , nonexistent in the elastic sult is that the near-tip stress field exhibits a square-root continuum, which may also explain the anomalously high singularity fully characterized by a single parameter, the value of Γ [17]. Still, till now, predicting fracture energy stress intensity factor, K [1, 2]. Crack destabilization is remain elusive. governed by the balance between the flux of the mechan- Here, we examined how the material’s resistance to ical energy released from the surrounding material into crack growth is selected in two-dimensional (2D) fuse lat- the FPZ and the dissipation rate in this zone. The for- tices of different geometries. These model digital mate- mer is computable via elasticity and connects to K. The rials, indeed, both possess perfectly prescribed and tun- latter is quantified by the fracture energy, Γ, required able "atomic bonds" and satisfy isotropic linear elastic- to expose a new unit area of cracked surface [3]. Equiva- ity under antiplane deformation at the continuum scale lently, it can be quantified by the fracture toughness, Kc, [18–21]. Fracture energy is observed to be significantly which is the K value above which the crack starts prop- larger than Griffith’s prediction, by a factor depending agating. Both quantities are related via Irwin’s formula on the lattice geometry. It is the fine positioning of con- [4]: tinuum stress/displacement fields onto the discrete lat- tice which sets the value of Kc, and consequently that of √ Γ. We demonstrate how the singular form of these fields Kc = EΓ (1) near the crack tip constrains this positioning, and subse- where E is Young’s modulus. quently permits to predict fracture toughness not only in simplified antiplanar elasticity, but also in genuine elastic Within LEFM theory, both Γ and Kc are material con- stants, to be determined experimentally. Still, their value problems, with an application to graphene. is governed by the various dissipation processes (viscous Numerics – Rectangular fuse lattices of size 2L × L or deformations, crazing, etc) at play in the FPZ. in the x and y directions were created and a horizontal straight crack of initial length 4L/5 is introduced from As such, it should be possible to infer Γ and Kc from the knowledge of the atomistic parameters, at least for per- the left-handed side in the middle of the strip, by with- fectly brittle crystals in which cleavage occurs via succes- drawing the corresponding bonds [Fig. 1(a)]. This fuse sive bond breaking, without involving further elements of network is an electrical analog of an elastic plate under dissipation. In Griffith’s seminal work [3], he proposed antiplanar loading, where voltage field, u(x, y), formally maps to out-of-plane displacement, u(x, y) = u(x, y)e . to relate Γ to the free per unit area, γs, z which is the bond energy times the number of bonds per Triangular, square and honeycomb geometries were con- unit surface: sidered; they to different values for E and γs (Tab. 1, see Supplementary Material [22] for derivation details). The length, conductance and current breakdown thresh- Γ = 2γs (2) old for each fuse were assigned to unity: ` = 1, g = 1 and 2 ic = 1 and all quantities thereafter are given in reduced 2 units: ` unit for length, g √for stresses, ic /g` for surface or fracture energy, and ic/ ` for stress intensity factor or fracture toughness.

Two different loading schemes were imposed: Thin strip (TS) configuration where the voltage difference, 2Uext, is applied between the top and bottom nodes of the lattice, and compact tension (CT) configuration in which the voltage difference is applied between the upper (above crack) and lower (below crack) part of the left-handed side [Fig. 1(a)]. Voltages at each node and currents in each fuse were determined so that Kirchhoff’s and Ohm’s law are satisfied everywhere. For both schemes, Uext was increased until the current flowing through the fuse right at the crack tip reached ic; this point corresponds to the crack destabilization. This fuse was withdrawn, the FIG. 1. (a) Sketch and notation of the simulated fuse lattice. The external voltage difference 2U is imposed between the crack advanced one step, and so on. The procedure was ext blue electrodes in the TS configuration, and between the red repeated until the crack has grown an additional length ones in the CT configuration. In response, a total current of 2L/5; at each step, the tip abscissa, xc, was located at Iext flows through the network. Uext and Iext are the ana- the center of mass of the next fuse about to burn. log of loading displacement and loading force in experimental mechanics. (b) Uext value at breaking onset as a function Γ(xc) was computed using two procedures: (i) Virtual of crack tip position xc, in a 200 × 100 square lattice. (c) work method, where Γ was computed from the loss of to- Γ vs xc in the same system. Note that y-axis ranges from P 2 2.39 to 2.46. Γ is nearly independent of x , no how tal energy stored in the lattice (Etotal = i /2g, where c p p it is measured and the loading configuration. (d) Size de- the current ip flows through fuse p) as the crack fuse is pendency of Γ averaged over xc. For small sizes, Γ depends burnt keeping Uext constant; (ii) Compliance method, on both the measurement method and loading configuration. 2 where Γ(xc) = (1/2)(2Uext(xc)) dGglob/dxc is computed However, the differences vanishes as L → ∞. All curves were p from the xc variation of the global lattice conductance, fitted by Γ = a/L + Γ∞, and the fitted value Γ∞ is a mate- (Gglob = Iext/2Uext where Iext is the total current flow- rial constant, independent of both loading configuration and ing through the lattice, see Fig. 1(a)). The latter is measurement method. Panel (d) concerns the square geom- widely used in experimental mechanics [23] since it re- etry but the same features were observed in triangular and lates fracture energy to continuum-level scale quantities honeycomb lattices. Table 1 (third line) provides the values Γ measured in the three lattice geometries. only.

Results – Figure 1(b) displays examples of the profiles U (x ) obtained in both TS and CT loading configu- ext c a series of elementary solutions Φn: rations: Uext is independent of xc in the first case, and increases with x in the second case. Conversely, Γ is c X nearly constant, within less than 0.5%. In-depth exami- u(r, θ) = anΦn(r, θ), (3) nation actually reveals slight dependencies with the mea- n≥0 surement method and loading conditions [Fig. 1(c)]. The Here, r is the distance to the crack tip and θ is the angle differences are significant in small systems, but they de- with respect to the crack line. In the antiplane (mode III) crease with increasing L [Fig. 1(d)]. In the continuum crack problem examined here, Φ write (Supplementary limit (L → ∞), all values converge toward the same ma- n Materials [22]): terial constant limit, Γ∞, as expected in LEFM. From now on, the ∞ subscript is dropped for sake of simplic- ity. Table 1 (third line) reports the continuum-level scale  rn/2 sin nθ if n odd ΦIII (r, θ) = 2 (4) values in the three lattice geometries considered here. In n 0 otherwise all cases, they are larger than Griffith’s prediction (Eq. 2), by a factor ranging between 1.75 to 3.67 depending The term n = 1 is the usual square√ root singular term and on the geometry. a1 relates to K via K = Ea1 2π/4 (see e.g. [1]). Equa- tions 3 and 4, truncated to the first six terms (n ≤ 11), To understand the selection of Γ, we turned to the were used to fit u(r, θ) obtained in the numerical experi- analysis of the spatial distribution of voltage [Fig. 2(a)]. ments. A difficulty here is to place properly the crack tip Williams showed [24] that, in a continuum elastic body in the lattice; as a first guess, this tip is located in the embedding a slit crack, the displacement field writes as center of the next bond to break [Fig. 2(b)]. The fitted 3

Lattice geometry Square Triangular Honeycomb √ √ E (g) 2 2 3 2/ 3 √ 2 2γs (ic /g`) 1 2 1/ 3 Γ(sim) (i2/g`) 2.414 ± 0.002 3.512 ± 0.012 2.152 ± 0.012 c √ K (sim) (i / `) 2.198 ± 0.011 3.489 ± 0.007 1.576 ± 0.008 c c √ K (approx. theo.) (i / `) 2.19 3.55 1.40 c c√ Kc (exact theo.) (ic/ `) 2.24734 3.57137 1.59697

TABLE I. Synthesis of the different elastic and fracture pa- rameters at play in the three lattice geometries examined here.

FIG. 2. Spatial distribution of voltage field and effect of crack tip mispositioning. (a) Map of voltage field, u(x, y) in a 200× 100 square lattice under TS loading. The crack lies along x in FIG. 3. Fracture toughness K plotted as a function of the the middle of the strip, so that the next bond about to break is c system size L, in both TS and CT loading configurations, for at the map center (red arrow) and the external applied voltage the three lattice geometries: square, honeycomb and triangu- difference, 2U , is at the value required to break this bond. ext lar. K is obtained by fitting the voltage field u(x, y) using Equations 3 and 4 are expected to describe this field. (b) c Eqs. 3 and 4 after having corrected the tip mispositioning us- Zoom on the tip vicinity emphasizing the lattice discreetness. ing the iterative procedure depicted in the main text. For all The difficulty is to place the continuum-scale crack tip in this curves, the continuous line is a power-law fit K = a/Lp+K∞. discrete lattice. At first, this tip is placed in the middle, O, c c The material constant K∞ obtained in the thermodynamic of the bond to break. (c) Absolute difference |u − u|(x, y) c fit limit (L → ∞) is independent of the loading conditions (dot- between the voltage field of panel (a) and the one, u (x, y), fit ted horizontal lines). These values are reported in the fourth fitted using Eqs. 3 and 4, with 1 ≤ n ≤ 11 (6 terms). The line of Tab. 1. fit is very good everywhere, except in the very vicinity of the crack tip (red arrow). (d) Absolute difference |ufit − u|(x, y) after correcting for the mispositioning and placing the continuum-scale tip at the proper position, C, using the iterative procedure depicted in the text. The fit is now very super-singular (n = −1) term has been added; (iii) Cor- good everywhere. rect the origin position by shifting it by d = 2a−1/a1; and (iv) Repeat this process (steps (ii) and (iii)) till the shifting distance d is less than a prescribed value (0.01` in our case). Herein, this convergence occurred in less field is in good agreement with the measured one, except than 3 iterations, irrespectively of the lattice geometry, in the very vicinity of the crack tip [Fig. 2(c)]. Unfor- lattice size and loading scheme. The obtained displace- tunately, this near-tip zone is precisely the one setting ment fields fit now that measured numerically extremely whether or not the next bond breaks. well, even in the very vicinity of the crack tip [Fig. 2(d)]. The near-tip discrepancy above results from the mis- The value of Kc arises from the fitting parameter a1. Kc match problem between the continuum fields of LEFM depends on L for small sizes, in a way depending whether and the lattice discreteness at small scale, yielding a CT or TS configurations are applied. However, as Γ, Kc crack tip mispositioning. A similar problem has been converges toward a constant, loading independent value faced in experimental mechanics to find out the crack as L → ∞ [Fig. 3]. The convergence is faster in lattices tip on samples, knowing the displacement field at a with higher . It also depends on loading condi- set of discrete points via digital image correlation [25]. tions in the case with the highest symmetry (triangular This problem was solved by noting that the elementary lattice). Table 1 (fourth line) provides the asymptotic solutions Φn in the Williams expansion obey [26, 27] values obtained in the square, honeycomb and triangular ∂Φn/∂x = nΦn−2/2. Then, a slight mispositioning of geometries. In all cases, Irwin formula (Eq. 1) relating the tip position x by a small distance d to an addi- Kc and Γ is properly satisfied, within 0.8%. tional, super-singular (n=-1), term in Eq. 3, with a pref- Is it now possible to predict fracture toughness from actor a−1 ' a1d/2 [25]. The crack tip can then be located scratch in the different lattices, without resorting to sim- as follow: (i) Start with an arbitrary mispositioned ori- ulations? To do so, we use the singular nature of the gin (in the center of the next bond to break in our case); near-tip field and make L → ∞. Consider now the very (ii) Fit the displacement field with Eq. 3 to which the vicinity of the next bond about to break, call A and B 4 the two edge atoms, and position, as a first guess, the Nevertheless, 2D crystals include an additional compli- crack tip at the center O of this bond [Fig. 2(b)]. All cating element with respect to the fuse networks consid- the n > 1 terms vanish in the Williams expansion and ered till now: the deformation are accommodated at the the voltage field, there, is fully characterized by two un- atomistic scale by two distinct modes: bond stretching known only: the singular term prefactor, a1, and the and bond bending, characterized by the bond stretching super-singular one, a−1. Their value are then deter- stiffness ks and bond bending stiffness kb [both can be mined by the Kirchhoff laws at A (or equivalently at B): measured experimentally via infrared spectroscopy [28]]. P i(u(Ai) − u(A)) = 0, where Ai are the nodes them- The procedure then follows the same route as for the elec- selves in contact with A. This equation gives a−1/a1, and trical network; it is detailed in Supplementary Material subsequently an approximation of the distance d between [22]. It quantitatively relates Kc to E, ks, kb, `, the bond O and the true position C of the effective continuum-level strength Fc, the Poisson ratio ν and the crystal geome- scale crack-tip [Fig. 2(b)]: try. This procedure has been applied in graphene, which has a planar honeycomb geometry with ` = 0.142 nm and E = 340 N.m−1 [29], ν = 0.18 [30], k ' 688−740 N.m−1 P III III  s Φ (rA , θA ) − Φ (rA, θA) −18 −2 d = −2 i 1 i i 1 (5) [31], kb = 0.769 − 0.776 × 10 N m rad [31], and P III III  −3 −1/2 i Φ−1 (rAi , θAi ) − Φ−1 (rA, θA) Fc = 13.6 nN [32]. It yields Kc = 1.25 × 10 N.m (Supplementary Material [22]). Fracture energy is sub- Shifting the reference frame origin from O to C leads sequently deduced from Irwin formula [Eq. 1]: Γ = to the disappearance of the super-singular term. Fur- −9 −1 4.6 × 10 J.m . The predicted values for Kc and thermore, the prefactor of the singular term is equal to √ Γ are very close to the experimental ones [33]: Kc = a = 4K /E 2π when the force i = u(B) − u(A) 1 c AB 1.4×10−3 N.m−1/2 and Γ = 5.4×10−9 J.m−1. Note that applying to the bond A − B is equal to the breaking Griffith’s√ seminal prediction [Eq. 2] would have yielded threshold ic = 1. This yields: 2 −10 −1 Γ = (1/2 3)(Fc /ks`) ' 5.08−5.47×10 J.m ; hence it underestimates the measured value by a factor of ten! √ E π Concluding discussion – This work demonstrates the Kc =   (6) III ˜ III ˜ key role played by the crystal lattice geometry at the 2 Φ1 (˜rA, θA) − Φ1 (˜rB, θB) atomistic scale in the selection of the resistance to failure at the macroscopic scale. It challenges Griffith’s seminal where (˜r, θ˜) are the polar coordinate of the considered interpretation [3] directly linking Γ to γ . It also ques- point in the new reference frame positioned at C. Equa- s tions the argument proposed in [15–17] to explain Grif- tions 5 and 6 provide an approximation for K in the c fith’s failure: Slepyan et al examined the structure of the three lattice geometries considered here. The values are waves emitted when cracks propagate in discrete lattices reported in Tab. 1 (fifth line) and coincide with that at constant speed, v and, from this, they determined the measured on the numerical experiments within 1%, 11%, fracture energy and its variation with v, Γ(v). They ob- and 2% for square, honeycomb and triangular lattices, served that in the limit of vanishing speed, Γ(v → 0+) is respectively. larger than 2γ . They attributed the difference to the en- Note that the mispositioning d obtained by Eq. 5 is an s ergy of ( lattice-induced high-frequency waves) approximation only since it involves Kirchhoff’s law at a that would survive when v = 0. Inertia is absent in single point, A, and a single iteration in the shifting pro- our simulations and, hence, phonons cannot be activated. cedure. Actually, both assumptions can be released by Still, we observe the same anomalously high value for imposing Kirchhoff laws to an arbitrary number of nodes fracture energy, as that obtained in dynamically crack- and by applying more iterations. These refinements are ing lattices at vanishing speeds; it is worth mentioning described in Supplementary Material [22]. They provides that the value 2.414 determined here for a square lattice ab-initio predictions for the fracture toughness, up to [Tab. 1] coincides exactly with that reported in [15], in any prescribed accuracy! Table 1 (sixth line) presents the v → 0 limit. This means that the fracture energy in the obtained values. They coincide with the numerically excess does not find its origin in the presence of phonons measured ones within 2%, 1.3%, and 2% for square, hon- as argued in [15–17]. The interpretation we propose is eycomb and triangular lattices, respectively. that fracture toughness is fixed by the fine positioning The analytical procedure derived here for model elec- of the near tip singularity of the continuum stress field tric/scalar elastic crack problems can be extended to on the crystal lattice, then by its adjustment so that the genuine 2D elastic (plane stress) crack problems. This force exerted on a is sufficient to break it. comes about via several correspondences: (i) In such Then in a second step, fracture energy is deduced from cases, the (vectorial) displacement field u(x, y) also fol- Irwin formula [Eq. 1]. lows Williams’s asymptotic form [Eq. 3]; and (ii) the I I relation ∂Φn/∂x = nΦn/2 remains valid [25]. 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