Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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Note that the fixed value of Poisson’s ratio, 1 / 4 (6.39) 1 , is a reasonable approximation for many engineering materials. Finally, in lattice modeling it is often useful to designate the spring stiffnesses directly to the bond between two lattice nodes (instead of using the half values associated with the hexagonal unit cell). With regards to Figure 6.14b, since the two springs between the adjacent unit cells A and B are connected in series, their equivalent spring stiffness is

α ( 2 )

( 5 ) B

A α

1 α eq

1 α ( 2 ) A

1 α ( 5 ) B

⇒ α eq =

(6.40)

=

+

.

α ( 2 )

( 5 ) B

A + α

For the case of equal spring stiffnesses, α ( b ) = α ( b = 1 , . . . , 6 ) , for all unit cells, the preceding expression ( α eq = α / 2 ) in combination with (6.39) 2 yields

8 5 √ 3

E ( 3 D ) .

α eq =

(6.41)

Thus, Equation (6.41) provides a direct link between the (meso-scale) spring stiffness α eq of the equilateral triangular lattice with the central interactions and the (macro-scale) material property E ( 3 D ) . (Compare with expression (6.27).) Chronology of development of lattices with central interactions Lattices with central interactions, as the simplest lattices, gained popularity very early in the failure modeling of heterogeneous materials with disordered microstructure. Although they began to be used earlier, they achieved the greatest momentum through the pioneering works of Bažant and his associates [67], [68]. Schlangen and van Mier [75], [76] noticed very early the possibilities of the α model to simulate qualitatively the process of damage and fracture of concrete represented by a network of aggregates (as lattice nodes) bonded by the cement (as lattice elements) of inferior tensile strength. However, it should be borne in mind that these simple models are inherently unable to realistically capture more complex crack propagation patterns and reproduce fracture shapes resulting from a combination of basic failure modes [69], [74], [77], [78]. For example, if α models are calibrated to reproduce cracks due to tension and fracture in the first mode, they will do that reasonably well; however, that same model will significantly exaggerate the compressive strength and will not realistically reproduce the stress-strain curve in the post-critical (softening) regime. Also, α models are not able to accurately predict fracture envelopes at complex stress states due to the oversimplified unit-cell stiffness. Despite all of the above, thanks to their simplicity and computational efficiency, these models are—regardless of the abundance of more sophisticated methods—still popular among researchers for simulations in which these shortcomings are not fully expressed [79], [80]. Bažant and co-authors [67], [68], [81] used a central-force lattice with irregular geometry to model the brittle heterogeneous material. In contrast to lattice models in which the distance between nodes is an arbitrary input parameter, they selected the lattice node initial locations by mapping the actual meso-structure of concrete. In other words, the positions of the lattice nodes coincide with the centers of aggregates; therefore, the lattice topology reflects the actual concrete texture. As it will be discussed later, this modeling