Issue 29

A. Bacigalupo et alii, Frattura ed Integrità Strutturale, 29 (2014) 1-8; DOI: 10.3221/IGF-ESIS.29.01

ˆ





                  11,1 12,2 1 1 21,1 22,2 2 2 ˆ d d k v u k v u

1 1 u 

 ˆ k

1 2  u







          

                   1,1 m m k u v k u v 2,2 21 1 1 2 2 2 ˆ ˆ ˆ d d k I       

I



12

1

(2)

2 1  v

2 2  v

where the stiffnesses ˆ d

d cell k k A  and ˆ

cell A being the area of the periodic cell, together with

k k A   

are introduced,

cell

M

M

J

J

 

 

1 A  and



and

, and the micro-inertia terms

, respectively.

the mass densities

I

I

1

2

2

A

A

A

1

2

1

2

cell

cell

cell

cell

i m are the overall stress components, namely the asymmetric stress components and the micro

ij  and

In Eq. (2)

cell A 

2 2 2 3 cos l

 and the following inertial parameters

couples, respectively. In case of hexachiral lattice, one obtains

are obtained

        3 sin 2

s

1

24

2       2 3 tan sin l 2       2 3 sin 24 i A 1 24 M s I 2

cell

2     3 J

2

2 sin tan

2  

I

l

192 i

2

A

cell

  24 s       , 1 sin 2

cell A 

 and the resulting inertia parameters are

For the tetrachiral lattice one obtains

2

2 cos

l

48 s    

2       , M

2     J

2

2

1   . To obtain the displacement formulation of the equations of motion, the compatibility equations involving the macrostrain components 11 1,1 u   , 22 2,2 u   , 12 1,2 u     ,     21 2,1 u and the curvatures 1 ,1    and 2 ,2    have to be considered together with the constitutive equation. sin I l  , 2 2 sin 48 i cell A 2 2 sin tan 2 2 192 i cell I l A

M ICROPOLAR CONSTITUTIVE EQUATION FOR HEXA - AND TETRA - CHIRAL LATTICES

T

he constitutive equation of hexachiral honeycomb corresponds to that obtained in [13, 15] and is written as follows

   

    

11                            22 12 21 1 2      

A A A A

2

0 0 0 0 S 0

0 0 0 0 0 S

11             22

2      A A k A A k 



       

k k

(3)

12



21       1 m m  

   

0 0

0 0

0 0

0 0

2   

in which the five elastic moduli

4