Issue 29

L. Cabras et alii, Frattura ed Integrità Strutturale, 29 (2014) 9-18; DOI: 10.3221/IGF-ESIS.29.02

/2Lk LS*L ,

N*

EJ M*

p

(i=1, 2 and 3),

(10)

 iu



)dξ

 



0

iS

iN

i (M

EA

beam

spring

where M i * , N i * and S i L* (with i=1, 2 and 3) are the internal actions of the statically admissible structures subjected to forces applied in the point B in horizontal and vertical direction, and the internal actions of the statically admissible structure subjected to a vertical forces applied in the point A in vertical direction. The corresponding displacement are: u 1 = A 1 F N + B 1 F T +C 1 F V u 2 = A 2 F N + B 2 F T +C 2 F V u 3 = A 3 F N + B 3 F T +C 3 F V (11) where A 1-3 , B 1-3 , C 1-3 are coefficients arising from the integrations. Previous equations are explicit linear relations between the forces F N , F T and F V associated to macroscopic stresses

3F FN T psin γ 4F 2 3F 2F V N T σ22 2 3psinγ      σ11

(12)

and the displacements of the points A and B associated to macroscopic strains: A F B F F 2u 1 N 1 1 V 1 T ε 2 11 3psin γ 3psin γ A F B F F u 3 N 3 3 V 3 T ε 2 22 psin γ psin γ C C        

(13)

Solving previous relations (13) in term of F N , F T and substituting the result into eqn. (12) leads to the macroscopic constitutive relation between the macroscopic stress σ and macroscopic strain ε . Clearly, appropriate choices of the forces F N , F T , and F V can be considered in order to set to zero some components of the stress. The in-plane mechanical properties of the lattice with extensional springs are:  Poisson's ratio , and F V





    

1α1d

13α-3

2α1α2d 2α2

1α3d 2

2α4d 2

2α7d 1α6d 2α1α5d 2

1α1d 22ε22σ 11ε11σ 22ε11σ 11ε22σ  

(14)

Lν 





  

)2α7d-1α6d-2α1α10d 2

1α8d 3

1α9d 2 2α1α23d 2α2

2α43d 2

where:

d 1 d 2

=-9cos 4 γ

=sin 4 γ

d 3 =9(cos 6 γ -4 cos 4 γ +2cos 2 γ-1) d 4 =sin 4 γ cos 2 γ d 5 =3(6cos 4 γ-7cos 2 γ-2cos 6 γ+1)

(15)

=-9cos 2 γ =-3cos 2 γ

d 6 d 7 d 8

=3(4sin 2 γcos 2 γ+1)

d 9 =9(3cos 6 γ-4cos 4 γ+2cos 2 γ+1) d 10 =3(10cos 4 γ-6cos 6 γ-5cos 2 γ+3)

 Bulk modulus

 

2 1 LK 

22ε 11ε 22σ 11σ 

Lk γ2 sin 3

(16)



)1α γ2 (cos 2

 Young's modulus

14