Issue 39

M.A. Tashkinov, Frattura ed Integrità Strutturale, 39 (2017) 248-262; DOI: 10.3221/IGF-ESIS.39.23

2





2

2

2







23 22 33    







  cr 22 33 2 22  

12 13









  

when

0, 

0

22 33

  cr 23 

  cr 12 

2

2

2

   



2







2

2

2

cr

   

   

  

   











23 22 33    













22 33

22 33

12 13

22 







(9)

1

0, 

  cr 23 

  cr 23 

  cr 12 



cr

cr

2

2

2

 



2

4

23

22



when 22 

33   

0

cr 22   or

cr 22   is the laminate transversal tension and compression strength, cr 23  and cr 12

 are transverse and

where

longitudinal shear strength constants. The multicontinuum theory (MCT) criterion of composite materials, proposed in [28], also allows to split degradation of matrix and inclusions. It is assumed that fiber and matrix have transversely isotropic properties. Within multicontinuum theory for composite materials, the following relationships between stress and strain fields in the material with effective properties and the individual constituents of the composite are introduced [28]:

С f p p    σ σ σ C ε f m m c c

(10)

С f f p p   ε ε ε m m

(11)

σ and f

ε are stress and strain in fibers; m σ ,

σ and С

ε are stress and strain fields for homogenized material; f

where С

f C ,

m ε are stress and strain in matrix;

c C ,

m C are tensors of

f p and m

p is volume fraction of fiber and matrix;

structural elasticity moduli of homogenized composites, fibers and matrix, respectively. Stresses and strains fields of in the matrix m and the fiber f can be expressed as follows:

f f f  σ C ε , m m m  σ C ε

(12)

p p



  1 



m

ε

 C C C C ε 

 

(13)

f

c

f

c

m m

f





1 

m m   ε I p

A ε

p

(14)

f

c

p p



  1 



m

where C C C C , I is identity tensor. Thus, the strain field in inclusions and matrix may be expressed from the effective strains using relations (7), (9) and (10). This representation is used for finite element modeling for verification of fulfillment of failure criteria of composites. Fiber failure criterion is set with the following equation:   f f f f a J a J 2 1 1 4 4 1   (15) c f c m f     A

where ,     f f f J 2 2 4 12 13     . Matrix failure occurs if the following identities are true:     m m m m m m m m m m m a J a J a J a J a J J 2 2 1 1 2 2 3 3 4 4 5 1 2 1      f f J 1 11  

(16)

256