Issue 29

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

2 2 σ σ 11 22 







E

2K (1 - ν)

L

L

σ ε σ ε 11 11 22 22 

(17)

4 2 4 2 2 2 2 2 3 (9cos γ α sin γ α 3(2cos γ sin γ 1)α α 9α 3α )sin γ k 1 2 1 2 1 2 L 3 2 2 2 2 d α d α α 3d α α d α 3d α d α α - d α - d α 1 1 8 1 2 2 1 2 9 1 4 2 10 1 2 6 1 7 2           



where the constants d 1,

d 2

, d 4

, d 6-10

are given in Eq. (15).

 Shear modulus

σ σ 1 

1 ν 1 ν  

11 22 11 22  ε







μ

K

L

L

ε

2

(18)

4 2 4 2 2 2 2 3 (9cos γ α sin γ α 3(2cos γ sin γ 1)α α 9α 3α )sin γ k 1 2 1 2 1 2 L 2 2 2 2 2 2 2 2 2 12cos γα α 4sin γα α 9sin 2γ 6(2 sin 2 )α α sin 2γ 1 2 1 2 1 1 2 2             



When rotational springs are considered the effective constants are as follows:  Poisson's ratio 2 2 e α e α e α α e α e α 1 3 2 4 3 3 4 4 3 45 ν R 2 2 e α e α e α α e α - e α 6 3 7 4 8 3 4 4 3 45        

(19)

where:

e 1 =9 (2cos 4 γ-3cos 2 γ+1) e 2 =(2cos 4 γ-cos 2 γ) e 3 =3(4cos 2 γ sin 2 γ-1) e 4 =27cos 2 γ e 6 =9(2cos 4 γ-cos 2 γ-1) e 7 =2cos 4 γ-3cos 2 γ e 8 =12cos 2 γ sin 2 γ-9 e 5 =9cos 2 γ

(20)

 Bulk modulus

2 3 3 (k / p ) R

(20)

2 2 2 2 (3sin γ α cos γ α 9cos γ) 3 4   

K R

 Young's modulus

2 6 3 (3α α )(k / p ) 3 4 R E R 2 2 e α e α e α α - e α - e α 6 3 7 4 8 3 3 4 3 35    

(21)

 Shear modulus

2 3 3 (3α α )(k / p ) 3 4 R 

(22)



μ R

2 2 2 2 sin 2γ   

2

 

3 4   

9sin 2γ

6(2 sin 2 )

3

4

A NALYSIS OF EFFECTIVE PROPERTIES

W

e analyze now the effective properties of the microstuctured media. We consider the case of vanishing stiffness of the springs k L , k R →0. We have that:

15