Mathematical Physics - Volume II - Numerical Methods

6.4 Lattices

247

which, in the case of equal spring constants α ( b ) = α ( b = 1 , . . . , 6 ) , reduces to

9 8 √ 3 3 8 √ 3 3 8 √ 3

E 1 − ν 2 E ν 1 − ν 2

α =

C 1111 = C 2222 =

,

α =

C 1122 = C 2211 =

(6.35)

,

E 2 ( 1 + ν )

α =

C 1212 = =

.

It should be noted that expressions (6.35) satisfy the isotropy condition C 1212 = ( C 1111 − C 1122 ) / 2 . (6.36) In addition, since the value of the apparent plane-strain Poisson’s ratio 2 is fixed, the spring constant defines only the plane modulus of elasticity of the unit lattice cell

α √ 3

1 3

C 1122 C 1111

ν =

E =

(6.37)

=

,

.

Importantly, for all types of lattices, the upper limit of the plane Poisson’s ratio is defined by the value (6.37) 1 . This value could be modified in various ways. For example, it is possible to define the so-called “triple honeycomb network” (sometimes the term α − β − γ model is also used [8]) in which the base of the unit cell is a regular hexagon but, in the notation used in Figure 6.14, α ( 1 ) = α ( 4 )̸ = α ( 2 ) = α ( 5 )̸ = α ( 3 ) = α ( 6 ) . This selection of spring stiffnesses results in the following expression ν = 1 − 2 h 1 + 2 9 α ( 1 ) + α ( 2 ) + α ( 3 ) 1 α ( 1 ) + 1 α ( 2 ) + 1 α ( 3 ) i for the plane Poisson’s ratio, which can reproduce values in the range 1 / 3 to 1. When α ( 1 ) = α ( 4 ) = α ( 2 ) = α ( 5 ) = α ( 3 ) = α ( 6 ) = α , the value (6.37) 1 is recovered. The plane coefficients E and ν , defined by (6.37) in terms of the spring stiffness α , are not material properties but simply their 2D counterparts, mere parameters, obtained by combining the elastic properties under the plane-strain (or, generally, the plane-stress) conditions. Since, the 2D triangular lattice (Figure 6.14) is equivalent to three-dimensional continuum under the plane strain conditions [74], the corresponding relationships between the real (3D) and the apparent (2D) material properties are

ν ( 3 D ) 1 − ν ( 3 D )

E ( 3 D ) 1 − [ ν ( 3 D ) ] 2 .

ν =

E =

(6.38)

,

Consequently, Poisson’s ratio and the modulus of elasticity corresponding to Equations (6.37) are ν ( 3 D ) = , E ( 3 D ) = α . (6.39)

5 √ 3 16

1 4

2 The apparent plane-strain Poisson’s ratio and the apparent plane-strain modulus od elasticity are henceforth, for brevity, referred to as the plane Poisson’s ratio and the plane modulus od elasticity.