Issue 65
S. M. J. Tabatabee et alii, Frattura ed Integrità Strutturale, 65 (2023) 208-223; DOI: 10.3221/IGF-ESIS.65.14
With this result, generalized elastic moduli will have the following form [28]:
1 2
n
1
1
12
(19)
E
n
(2
I
1
21 12
E E
n
2
2
x y
6
1 2
n
1
1
12
(20)
E
n
(2
II
1
21 12
2
n
2
E
2
6
x
Several studies have been made to calculate the effect of the pores and micro-cracks on the material’s overall properties. The porous materials are load-induced anisotropic, i.e., mechanical properties depend on the load pass. However, if all the pores are all open and have a random distribution, they can assume as isotropic [4]. Budiansky and O'Connell have obtained the effective elastic moduli of a body containing open micro-crack. For a solid containing uncorrelated pores, we have [8]:
3
15 N a
2
E E
1
3 ( ) 2 ( , ) f g
(21)
3
( )
3(1 2 ) N a f
2
K K
1
(22)
3 1 1 1 0 3 G G E G K G E E K K
(23)
where , , , E K G v are effective elastic modulus bulk modulus, shear modulus, Poisson’s ratio respectively, f and , g represent the shape coefficient that defined based on pores geometry, N is the number of pores per unit volume and the symbol . calculates the mean value. For narrow elliptical pores, these shape coefficients are defined as follows [8]:
2 2 4 1 3 ( ) b a E k
( )
f
(24)
2
b a
4
2
2
2
(25)
( , )
g
R k
Q k v
1
( , )cos
( , )sin
3
2 2 1/2 (1 / ) k b a
(26)
where a and b are major and minor axes of the considered ellipse. Q and R are complete elliptic integrals of the first kind with argument k. effective elastic modules for material extracted as follow [8]:
E E G G
16 1 (1 )(5 4 ) 45
(27)
8 1 (10 7 ) 45
(28)
where is micro-crack density parameter and calculate as follows [8]:
219
Made with FlippingBook - Share PDF online