Issue 58

K. Benyahi et alii, Frattura ed Integrità Strutturale, 58 (2021) 319-343; DOI: 10.3221/IGF-ESIS.58.24

)

(

2 12 33 13

C .C -C



21

(26)

12 S =-

=-

)

(

2

2 2

2

E

33 11 C .C -2.C .C -C .C +2.C .C 11 13 12 33 12 13

22



C

13

31

(27)

13 S =-

=-

)

(

2

E

11 33 C .C -2.C +C .C 13 12 33

33

(

)

C +C

1

11 12

33 S =

=

(28)

)

(

2

E

11 33 C .C -2.C +C .C 13 12 33

33

1 1

S = =

(29)

44

C G

44

12

1 1

S = =

(30)

55

C G

55

13

The determination of the effective elastic properties depends on the stress-state, and also on the property of the composite material (isotropic or orthotropic case) of the RVE. For an isotropic case and a uniaxial stress-state according to the direction X-X, the effective elastic properties are determined:   eff eff 11 11 11 E =E = (31)    eff eff eff 12 13 + = 2 (32) with:   eff 22 =-

     

12

 

11

(33)

eff

33



=-

13



11

For an orthotropic/isotropic case transversely and a uniaxial stress-state along the direction X-X, the effective elastic properties are determined:   eff 11 11 11 E = (34)    eff 22 12 11 =- (35)

G ENERAL IMPLEMENTATION FLOWCHART ON THE A BAQUS CALCULATION CODE

he proposed periodic homogenization method has been implemented numerically using a Python script, to invoke the solution in the Abaqus finite element computational code. The iterative algorithm is presented in Fig. 3. At each incremental step of the analysis, the material properties subroutine (Umat), where the constitutive law was used to carry out the communication between the Python code, and the Abaqus solver. The Python code modifies the boundary conditions to an Abaqus input file, applied to the RVE. T

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