Triangular lattice/network

Fig. 3.17. Molecular dynamic simulation results for the onset of fracture growth instablity in a triangular lattice network with Lennard-Jones potential, having an initial crack at the left-side boundary, (a) Initial stages of growth, and (b) late stage unstable growth with large propagation velocities (Abraham et al 1994).

Fig. 3.20. Computer simulation results for the ratio C j x versus the fraction of unbroken springs in a triangular lattice network with different bond-bending forces (/3 = 0, 0.01, 0.3 and 1), having uniform distribution of bond-breaking thresholds. The ratio Cn/ji seems to converge to an universal value (c 1.25) as the complete fracture point is approached (Sahimi and Arbabi 1992).

Fig. 4 Elastic moduli of two types of cable network, (a) A triangular lattice network and designated test region (green) in the middle, (b) Network in (a) is stretched horizontally, (c), (d) Unstretched and stretched square lattice network, (e) Young s modulus E determined from the stretching experiment for both types of network, plotted as a function of number of nodes in the test region. Note that, a prestressed network has a larger Young s modulus, (d) Poisson s ratio a for triangular and square lattice networks. Clearly, prestress reduces the Poisson s ration for both types of network.

Fig. 3.11. Molecular dynamic simulation results for the average fracture stress CTf for various disorder concenrations on triangular lattices, (a) For site dilute Lennard-Jones system (Chakrabarti et al 1986), and (b) for bond dilute spring network (Beale and Srolovitz 1988).

Sahimi and Arbabi (1993) also studied the fracture strength distribution of a two-dimensional (triangular lattice) randomly diluted network with both central and bond-bending forces (with Hamiltonian given by (1.11) in Section 1.2.1 (f)). The results showed that, although the Weibull distribution fits the data initially for small disorder (p near unity), the data fits the Gumbel distribution considerably better and much more accurately as disorder increases (p Pc)- Iii fact, one can define a quantity A as... 

For polygons having four or more sides, there are several ways of placing the broken bonds, which are shown in Fig. 72. Requiring the bond network to have the topology of a triangular lattice forces a... 

Figure 6.9 pictures the example of the triangular-square network taken from [6.38]. Some 4-coordinated sites are seen comprising inner boundaries of this LRC. It is easy to notice that there are two 5-coordinated atoms in the first coordination sphere of each 5-coordinated atom in LRC. This circumstance follows from the fact that, in P-polyhedra, we have an even number of squares leading to the formation of the MRO, which manifests itself in the formation of chains of 5-coordinated atoms. Collins and Kawamura studied the thermodynamic properties of triangular-square lattices. Kawamura established the existence of the first-order phase transition connected with the transformation of the crystalline structure into a topologically disordered one. 

To proceed, consider two finite planar networks, a regular Euclidean triangular lattice (interior valence v — 6, but with boundary defect sites of valence v = A and v = 2) of dimension d — 2, and a fractal lattice... 

Figure 3.1 shows a non-reconstructed ideal (lll)-(l x 1) surface of diamond with the three lattice vectors (211), (112) and (121) that form a triangular network with a unit mesh size of 0.252 x 0.252 nm" Shown also is the unit cell (shaded) with side lengths and ( /3/V ao- Superposed on this... 

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