Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

246

where k n designated the bond stiffness. (Compare with (6.8) 1 and (6.26) 2 .) At this point, it is necessary to define an associated unit lattice cell. The derivation of the equations of the connection between the bond (spring) constants and the effective stiffness components is based on equivalence (6.28). A key step in this process is to establish a connection between u and ε , which depends on the specific geometry of the lattice unit cell and the specific model of the interaction between lattice nodes. Triangular lattice with central interactions The only truss lattice to be considered in detail herein is an equilateral triangular lattice with the central-force interactions between the first-nearest neighbors. The lattice, illustrated in Figure 6.14, is based on a hexagonal unit cell and six lattice elements (springs or trusses) of length ℓ , equal to a half the equilibrium distance, r 0 , between lattice nodes which defines the equilibrium lattice geometry (the reference state). The area (volume of unit thickness) of a hexagonal unit cell is V = 2 √ 3 ℓ 2 . Each bond b , belonging to a given unit cell, is characterized by a spring constant α ( b ) and unit vector n ( b ) defining the bond direction, with corresponding angles θ ( b ) = ( b − 1 ) π / 3 ( b = 1 , . . . , 6 ) with respect to the horizontal.

Figure 6.14: (a) Regular triangular lattice with a hexagonal unit cell and central interactions between the first neighbors ( α model); (b) a serial connection associated with a link between two adjacent unit cells.

The deformation energy stored in the unit hexagonal cell consisting of six evenly stretched connections is

6 ∑ b = 1

6 ∑ b = 1

ℓ 2 2

1 2

α ( b ) n ( b )

( b ) β n

( b ) γ n

( b ) δ ε αβ ε γδ .

( α u · u ) ( b ) =

U cell =

α n

(6.33)

Based on equivalence (6.28), the components of the elasticity tensor can be identified in the following form

6 ∑ b = 1

1 2 √ 3

α ( b ) n ( b )

( b ) β n

( b ) γ n

( b ) δ

C αβ γδ =

α n

(6.34)