Mathematical Physics - Volume II - Numerical Methods

6.4 Lattices

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approach innovatively introduces the post-critical behavior as a part of the micro-scale constitutive model. Ever since, various approaches were devised to deal with the softening of lattice elements [97]. This model has been immensely influential in development of numerical techniques for these classes of materials (e.g., [70], [82]-[85]). It cannot be overemphasized that Bažant and co-authors [67], [68] used the post critical softening regime (i.e., the progressive degradation of stiffness) in the constitutive stress-strain relations for the bonding matrix and the interface despite the fact that it is not considered an inherent material property. The softening, thus introduced into the model, is then defined indirectly by the fracture energy of the interparticle bond. More than a decade after this model gained prominence, the assignment of the softening properties to the micro/meso scale has not been not fully defined despite great efforts in that direction (e.g., [86]-[88]). The standard test to determine softening parameters is still elusive due to the unavoidable experimental material-structural interaction. Consequently, the introduction of softening in constitutive relations on the meso-scale was criticized by van Mier [87], on the grounds that there is a danger that the basic mechanisms may be missed due to the fact that the desired results are achieved by introducing additional model parameters. At the same time, many authors advocated the use of softening in constitutive modeling as indispensable (e.g., [89]) since, in their opinion, the meso-scale heterogeneity alone cannot account fully for the experimentally-observed dissipative response at the macro-scale. This difference of opinion is important to point out since the introduction of softening on micro and meso-scales (based on macroscopic observations) is a common practice as well as an active research topic nowadays due to the computational benefits it provides. 6.4.3 Lattices with Central and Angular Interactions ( α − β Models) The lattice with a central-angular interaction (also known as the α − β model) was created by upgrading the α model by adding an angular spring between adjacent connections that meet at the same node [90]. Consequently, the energy is necessary to overcome the bending angle resistance reminiscent of (6.8) 2 . The triangular α − β model was considered in detail by Kale and Ostoja-Starzewski [91] in the context of studying the material damage.

Triangular lattice with central and angular interactions

A triangular lattice with central and angular interactions was obtained by introducing angu lar springs in the manner illustrated in Figure 6.15. The stiffness of these angular springs is defined by the spring constant β ( b ) . According to the conditions of symmetry of the unit cell with respect to the corresponding lattice node, the elastic properties of a hexagonal unit cell are completely determined by six independent spring constants { α ( b ) , β ( b ) , b = 1 , 2 , 3 } : α ( b ) = α ( b + 3 ) and β ( b ) = β ( b + 3 ) for b = 1 , 2 , 3 (isotropic Kirkwood model).