Issue 44

P.S. Valvo, Frattura ed Integrità Strutturale, 44 (2018) 123-139; DOI: 10.3221/IGF-ESIS.44.10



f

f

f

f

  

  

1

1 1

uN M uM N  

Valvo

2

2

Valvo

2



C wQ C M f Q 



uN uM C f N f M  C

(22)

G

G

and

(

)

I

II

B

f

B f

2

2

uN

uN

where, for orthotropic specimens,

   

   

4 1 1 E B H H   

  

6 1 1 G B H H   

  

6 1 1



f  





f

f

f

,

,

, and

uM N 

uN

wQ

2

2

x E B

5

H H





x

zx

1

2

1

2

1

2

(23)

  

  

12 1 1 .



  

f



M

3

3

E B H H

x

1

2

Comparison of Eqs. (22) and (23) with (20) and (21) shows that Valvo’s method [24] reduces to Wang and Qiao’s method [20], if the crack-tip bending moment, M C , is disregarded. In general, however, M C gives a contribution to both modes I and II. Instead, shear forces and shear deformation contribute only to mode I. By substituting Eqs. (23) into (22) and expressing the crack-tip forces in terms of the internal forces on the delaminated sublaminates, the expressions for the modal contributions to the energy release rate become

2

2

H H N N 

  

1 2 1 H H Q Q H H H   

  

3

3

2 1 2





vVal o

1

2

2





H H 





G

 

I

1

2

2 B E

3

2 B G

H H

H

8

5



1

2

1

2

x

zx

2

   

   

1 2 H H H H H H M M    1 2 1 2 3 3

3



and

(24)

1

2

2

3

2

2

2 B E H H

H

x

1

2

2

2

  

  

1 2 H H N N H H H    1

  

1 2 H H M M

1

9

Valv

o

2

1 2

2 2





  

G

II

2 B E

2 B E

H

H

H

8

2



1

2

x

x

2

1

According to Eqs. (24), axial forces may contribute also to mode I. This contribution – neglected by both Williams’ [16] and Wang and Qiao’s [20] methods – passes through the crack-tip bending moment, M C , and depends on the contribution of axial forces on the moment balance on the laminate cross section [24]. Elasticity-theory models As opposite to Williams’ global method [16], Suo and Hutchinson [17] developed a local method based on the analysis of the singular stress field at the delamination crack tip. They analysed the problem of a semi-infinite crack between two homogeneous and isotropic elastic layers subjected to axial forces and bending moments. First, they computed the energy release rate based on simple beam theory. Then, within linear elastic fracture mechanics, they solved numerically a plane elasticity problem to obtain the mode I and II stress intensity factors, K I and K II [48]. Li et al. [19] extended the local method to include the effects of shear forces. Andrews and Massabò [21] further extended the method to include the effects of root rotations. They computed the total energy release rate based on the J -integral and then determined the stress intensity factors. Their analytical expressions depend on numerical coefficients obtained through finite element analyses and given in a tabular form. If no axial forces and bending moment are present, but only shear forces, their expression for the energy release rate takes the form   Andrews,Massabò 2 2 2 2 Shear 2 1 1 1 2 cos V D V S V V D S V V D S D S D S x G f V f V f f V V H E B H             (25) and the stress intensity factors can be written as         1 4 3 8 Andrews,Massabò I,Shear 1 1 4 1 8 Andrews,Massabò II,Shear 1 2 cos cos and 1 2 cos cos 1 V D V V S V D D S S V D V V S V D D S S K f V f V H K f V f V H                                          (26)

131