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¹¹ Note that the criterion for fracture used here is based on exceeding a critical maximum stress within a discrete element, with no consideration of the subsequent elastic energy gained upon the relaxation of the remaining network. Fracture does not appear to be based, therefore, on the equilibrium energy approach of Griffith. However, the Griffith criterion could be preserved on a smaller scale since the hypothetical tensile test performed oneach element does not actually specify how failure of the element occurred. Failure may have initiated by the Griffith mechanism, with essentially all of the elastic energy released within the elementitself (which is valid if the initial defect is much smallerthan the element size), and then the network relaxation energy is simply the excess energy released on further unstable propagation of the initial defect up to the element size.Google Scholar

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¹³ The elastic properties of the triangular network of nonfailing springs with distributed moduli a have been studied extensively in recent years in connection with the elastic percolation problem [see S. Feng and P. B. Sen, Phys. Rev. Lett. 52, 216 (1984), for example].If a fraction 1-p of the springs are assigned modulus a = 0 and placed at random in the network, with the remaining fraction p of the springs having a nonzero α (usually all identical), the elastic moduli of these highly defective systems goes to zero at a critical value of p, estimated ai p_c = 0.65, despite the fact that the network is entirely connected. Including additional spring-angle dependent forces into the network shifts this percolation point down to p_c = 0.33, where the network does become disconnected. Thus, highly defective triangular networks with spring forcesonly do have some unusual mechanical properties, but they will not generally be manifested in the fracture problems studied here. There are other lattice effects appearing in low-defect networks and we discuss them briefly in the accompanying paper.Google Scholar

¹⁴ The algorithm used here for evolution of the material with increasing strain is entirely deterministic. That is, the distributed disorder present at the outset of the simulation (ε = 0) determines the subsequent behavior (at ε > 0) completely. The choice of which spring to break at any point is clear and unique, with no probabilistic aspect to it. Such a deterministic algorithm is appropriate in situations where the distributed disorder dominates any possible effects due to thermal fluctuations. This is most often the case, since the springs represent material regions large compared to atomic dimensions with failure occurring because of local high stress, stiffness, or weakness. A probabilistic algorithm [Y. Termonia and P. Meakin, Nature 320, 429 (1986)] should be used only when the mechanical properties of every region are essentially identical so that only fluctuation effects distinguish the different regions.+0)+completely.+The+choice+of+which+spring+to+break+at+any+point+is+clear+and+unique,+with+no+probabilistic+aspect+to+it.+Such+a+deterministic+algorithm+is+appropriate+in+situations+where+the+distributed+disorder+dominates+any+possible+effects+due+to+thermal+fluctuations.+This+is+most+often+the+case,+since+the+springs+represent+material+regions+large+compared+to+atomic+dimensions+with+failure+occurring+because+of+local+high+stress,+stiffness,+or+weakness.+A+probabilistic+algorithm+[Y.+Termonia+and+P.+Meakin,+Nature+320,+429+(1986)]+should+be+used+only+when+the+mechanical+properties+of+every+region+are+essentially+identical+so+that+only+fluctuation+effects+distinguish+the+different+regions.>Google Scholar

¹⁵ In the absence of inhomogeneities, the elements may represent any length scale. But once failure occurs at one element, completely brittle macroscopic failure usually follows since each subsequent element at the crack tip will exceed the failure stress required for cracking. This scale independence is also basically relevant for the case where only “cracks” (i.e., missing elements or elements with zero modulus) are present as the inhomogeneities, with brittle failure initiating at that element under the higheststress due to stress concentration from some most-critical defect configuration. Thus, the nice work of Srolovitz and Beale,¹⁶ based on an electrical analog studied by Duxbury et al.,¹⁷ using a network model of springs with a distribution of α = 0 defects only, need not be motivated by microstructural considerations. In contrast, the recent work of Kahng et al. on the electrical analog of a distribution of failure strains (a distribution of failure currents in a network of resistive fuse elements) is motivated incorrectly.¹⁸ Kahng et al. motivate a distribution of failure currents by conceptually performing a coarse-graining of the Duxbury et al. system of randomly blown fuses: blocks of the fuse network, containing within them various configurations of blown fuses, are regarded as single effective fuses, and block to block variations generate a distribution of critical currents in a “renormalized” or rescaled network of effective fuses. This hypothetical procedure misses a major point of the Duxbury et al. work; namely that the macroscopic failure is dependent on one most critical defect configuration no matter where it appears in the network nor which effective fuse contains it upon coarse-graining. The essentially arbitrary boundaries of the coarse-graining procedure do not arrest the most-critical defect once it begins to propagate. Stress concentration on a much smaller length scale than the arbitrary coarse-graining scale is the crucial feature, and it cannot be accounted for properly for coarse-graining. Kahng et al.'s results are, of course, interesting and potentially useful for considering the mechanical analog of a distribution of failure strains on an appropriately considered length scale. But to reiterate, in brittle failure the retention of the intrinsic disorder on the smallest relevant length scale is the key to the problem.Google Scholar

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¹⁹ This behavior may be compared to that which prevails in the case of distributed breakdown strengths. There, all nonlinearity is due to accumulated damage with no release of residual strain and so at any point on the macroscopic σ-ε curve, the effective modulus is precisely σ/ε. Therefore, extensive nonlinear regimes in the distributed ε^c case must be accompanied by considerable modulus reductions.Google Scholar

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