Issue 53

K. Sadek et alii, Frattura ed Integrità Strutturale, 53 (2020) 51-65; DOI: 10.3221/IGF-ESIS.53.05

The work W can be expressed as:

2 . 1 W F u 

(4)

where F is the force needed to close the crack virtually, u is the crack opening displacement. Eq (3) allows the calculation of the energy release rate for 2D FE models (modes I and II). The application of the VCCT in 3D FE models is commonly called the 3D VCCT. The extension of the method from 2D to 3D requires replacing Eq. (3) by the following expression:

h a 

1

   , . 

 , . . a s ds dr

0 0   



 

G

u r s

r

(5)

h a

2

where s is the distance from the crack tip in the third direction, and h is the element size in the third direction. To apply Eq. (5) to FE models comprising 8-node brick elements, the integrals in (5) are replaced with the sum as follows:

2

1

 



G

. F u

(6)

ki ki

k h a 

2

1

where index i controls direction, and index k controls the node b number. The shear stresses in the adhesive are given by the following relationship:

1 G u u e  ( a





2 )

(7)

a

Where 1 u and 2 u are displacements in layers 1 and 2 (the plate and the patch) respectively. Experimental design was used for the determination of the optimum patch dimensions [13-15]. Indeed, the patch dimensions process is described by a quadratic model as follows:

2

2

0 1 1 2 2 11 1 22 1 12 1 2 y a a x a x a x a x a x x      

(8)

i x is the normalized

where y is the response of the process (i.e., the J integral at the crack front in the plate) and,

centered value for each factor i u :

u u 

*

i

ic





x

u

(9)

i

i



u

i



u

u

2 imax imin



u

(10)

ic



u

u

(

)

2 imax imin

u  

(11)

i

The quadratic model of J integral is expressed as following [16]:

*

*

*

* * a L W a L t  * *

* *

*2   a L a W a t  *2

*2

0 1 J a a L a W a t      2 3



23 a W t

(12)

12

13

11

22

33

* W is the width, and * t is the thickness of the patch.

* L is the length,

where

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