Issue 55

K. Fedaoui et alii, Frattura ed Integrità Strutturale, 55 (2021) 36-49; DOI: 10.3221/IGF-ESIS.55.03

    2 1 i

.

i P P

1

P is the surface fraction of the matrix phase, so

Cases

Volume fraction

Shape

 Interphase inclusion V V

 Interphase inclusion V V

 Interphase inclusion V V

/ 2

/ 3

r 1 =0.018mm r 2 =0.023mm r 1 =0.018mm a=0.019mm b=0.032mm r 1 =0.023mm r 2 =0.029mm r 1 =0.023mm a=0.024mm b=0.04mm r 1 =0.033mm r 2 =0.042mm r 1 =0.033mm a=0.035mm b=0.058mm

r 1 =0.018mm r 2 =0.021mm r 1 =0.018mm a=0.019mm b=0.024mm r 1 =0.023mm r 2 =0.026mm r 1 =0.023mm a=0.024mm b=0.03mm r 1 =0.033mm r 2 =0.038mm r 1 =0.033mm a=0.035mm b=0.043mm

r 1 =0.018mm r 2 =0.02mm r 1 =0.018mm a=0.019mm b=0.021mm r 1 =0.023mm r 2 =0.025mm r 1 =0.023mm a=0.024mm b=0.036mm r 1 =0.033mm r 2 =0.036mm r 1 =0.033mm a=0.035mm b=0.038mm

sphere interphase ellipsoid interphase sphere interphase ellipsoid interphase sphere interphase ellipsoid interphase

  1 2 2.5% P P

  1 2 5% P P

  1 2 15% P P

Table 1: Cases of materials.

Elastic properties

Phases



E [MPa]

Matrix

1 5

0.3 0.2 0.3

Inclusion Interphase

0.1 0.2 0.5 0.8 1 3 5 8 10

Table 2: Elastic properties of considered material.

All the components are considered isotropic, this means that their combination in the unit cell generates a composite with isotropic behavior. We note that interphase is located between the matrix and inclusion, its elastic properties are affected by the elastic properties of the two phases. The Young’s modulus and the Poisson’s ratio of all phases are presented in Tab. 2. Governing equations Linear elastic constitutive relationships for an isotropic material are given by [39].

ij ijkl kl C   

(3)

where     , , , 1, 2,3 i j k l  For homogeneous and isotropic materials the tensor C is given by:             ijkl ij kl kl jl il jk C

(4)

with  and  are the Lamé parameters and

  denote a Kronecker delta. The elastic moduli are expressed by:



E

E

2 ,







 





K G

G

(5)



   

 2 1





  1 1 2



3

35