Mathematical Physics - Volume II - Numerical Methods

6.4 Lattices

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Note that identical geometry of beams is assumed in derivation of Equation (6.60) and the subscript ( b ) is, consequently, omitted from geometric parameters for brevity. By comparing expressions (6.55) and (6.60), it follows that the stiffness components and elastic constants of Timoshenko beam theory are obtained from Bernoulli-Euler beam theory when ¯ h 2 is replaced by ¯ h 2 / ( 1 + ς T ) . Therefore, the plane Poisson’s ratio and the modulus of elasticity expressions

¯ h

1 + ¯ h 2 / ( 1 + ς T ) 3 + ¯ h 2 / ( 1 + ς T )

1 − ¯ h 2 / ( 1 + ς T ) 3 + ¯ h 2 / ( 1 + ς T ) ,

t ( b ) t

E = 2 √ 3

E ( b )

ν =

(6.61)

should be used in lieu of (6.58). In general, Tymoshenko beams are by definition more appropriate for use than Bernoulli-Euler beams when lattice elements are not very slender but rather stocky. As an example, in order to directly include the interface layers in the model (Figure 6.3b), the network resolution (i.e., the beam span) must be limited by the interface thickness. Consequently, there are practical difficulties and extremely large computational efforts. This problem has led to the development of alternative lattice models based on generalized beams [96]. 6.4.5 Various Aspects of Lattice Modeling The selection of lattice elements is crucial for simulation of complex cracking. Schlangen and Garboczi [93], [94] performed a paramount, and later very influential, comparative analysis of simulation techniques using lattice modeling of heterogeneous materials with random micro structure. Experimental crack propagation patterns obtained using a con crete double-edge-notched specimen were compared with those obtained by computer simulations using lattice models with different types of interactions and spatial lattice orientations. The selected results are illustrated in Figure 6.17. It is important to note that the geometric disorder was not used in these lattice models to emphasize the ability of a particular type of lattice element to capture the fracture pattern. With respect to Figure 6.17, it is obvious that—in the absence of geometric disorder—the beam elements are superior in the reproduction of experimentally observed complex crack patterns compared to the two truss lattices. Nevertheless, it is noticeable that the shape of the cracks even in that case (Figure 6.17d) reveals the bias inevitable in geometrically regular lattices. Thus, another important selection for correct prediction of complex crack patterns is related to the lattice geometry, especially its regularity. The cracking directions of regular lattices are strongly predetermined (as illustrated in Figure 6.17) but it is easy to achieve the uniform deformation. On the other hand, the irregular lattices are characterized by less biased cracking patterns but, in general, do not behave homogeneously under uniform loading. Schlangen and Garboczi [93] demonstrated importance of the lattice geometric disorder for realistic simulation of crack propagation by comparing crack patterns obtained by computer simulations using four different lattices (based on a square grid, two differently oriented triangular grids, and irregular triangular grids). They also proposed an approach (based on iterative adjustment of the beam properties) to obtain an irregular lattice with the elastically uniform deformation.