Mathematical Physics - Volume II - Numerical Methods

6.4 Lattices

245

6.4.1 Basic idea of Lattice Models When formulating the lattice model, it is crucial to establish the relationship between the lattice parameters and the mechanical properties of the solid material being simulated. The gist is that under the uniformly applied load, the lattice should reproduce the linear elastic behavior of the corresponding equivalent continuum and its uniform deformation. Different approaches have been proposed in this regard (e.g., [67]-[71]). The methodological approach employed herein is based on the deformation energy equivalence and follows closely the original works of Ostoja-Starzewski and co-workers [3], [8], [72], [73]. The basic idea is to ensure the equivalence of the deformation energy contained in the deformed unit cell of the lattice (e.g., the hexagonal unit cell in Figure 6.14a) with that in the associated continuum structure (of the same volume V )

U cell = U continuum .

(6.28)

The deformation energy is defined in continuum mechanics by expression

1 2 Z V

1 2 Z V

σ : ε d V =

σ αβ ε αβ d V .

U continuum =

(6.29)

If we restrict ourselves to a uniform strain field ε , Equation (6.29) becomes

V 2

V 2

ε : C : ε =

C αβ γδ ε αβ ε γδ

U continuum =

(6.30)

In order to establish relationships between the macroscopic material properties (e.g., the effective stiffness components C αβ γδ ) and the lattice parameters, it is necessary to define a lattice unit cell based on specific periodic arrangements of the associated nodal points and their mutual interactions. For brevity, only the regular triangular lattice with the hexagonal unit cell will be used in the following deliberations (Figures 6.14-6.16). 6.4.2 Lattices with Central Interactions ( α Models) The lattice with central-force interactions (also known as the spring-network or α model) is the basic model in the sense that each bond represents a truss or a spring that transmits only an axial force: f = f n n directed along the bond direction defined by the unit vector n (Figure 6.14). The deformation energy contained in a unit lattice cell with central interactions is the sum of deformation energies of the constituent lattice elements (bonds) U cell = ∑ b E ( b ) = 1 2 N b ∑ b ( f · u ) ( b ) . (6.31) In Equation (6.31), u = u i j = u i − u j designates the resulting change of length of the lattice element that connects the lattice nodes (material points) i and j , b is bond index ( b -th truss/spring), and N b is their total number. If we restrict ourselves to linearly elastic interactions, Equation (6.31) can be written as

N b ∑ b

1 2

( k n u · u ) ( b )

U cell =

(6.32)