Mathematical Physics - Volume II - Numerical Methods

6.4 Lattices

251

equations, the plane Poisson’s ratio and the plane modulus of elasticity can be expressed as functions of the axial ( α ) and angular ( β ) spring constants

1 3

1 − 3 β / ( α ℓ 2 ) 1 + β / ( α ℓ 2 ) ,

α √ 3

1 + 3 β / ( α ℓ 2 ) 1 + β / ( α ℓ 2 )

C 1122 C 1111

ν =

E =

(6.45)

=

Importantly, the two 2D parameters (6.45) are also dependent upon the model resolu tion defined by ℓ . Substitution of β = 0 in Equations (6.45) recovers the plane moduli of the α model defined by Equations (6.37). Complete range of definition of the plane-strain Poisson’s ratio − 1 < ν < 1 / 3 can be obtained from expression (6.45) 1 for two boundary cases: β / α → ∞ and β / α → 0 ( α model). The plane moduli of compression and shear are defined as functions of the axial ( α ) and angular ( β ) springs constants by the following expressions β α ℓ 2 which demonstrates that angular springs have no effect on volume change. 6.4.4 Lattices with Beam Interactions The lattice with beam interactions (beam lattice) is the result of upgrading the α model (truss lattice, spring network) by replacing structural elements capable of transmitting only axial force ( F ) with beam elements that can also transmit shear forces and moments ( Q , M ; Figure 6.16). The beam lattice represents a micro-polar continuum with independent nodal displacement and nodal microrotations fields. As a result there are six (three) degrees of freedom per lattice node in 3D (2D, Figure 6.16c) models. The beam lattice presentation that follows is based mainly on articles of Ostoja-Starzewski [8] and Karihaloo and co-authors [89]. K = √ 3 4 α , µ = √ 3 8 α 1 + 3

Figure 6.16: (a) Triangular beam lattice with (b) two adjacent hexagonal unit cells; (c) Degrees of freedom per 2D beam lattice node: two translational and one rotational.