PSI - Issue 41

Roberta Massabò et al. / Procedia Structural Integrity 41 (2022) 461–469 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

466

6

( ) ( 1 4 3 + −

)

( 3 1

)

 

+



g

g

a

= −

−

) (

)

(

63

33

34

2 4 3 )  −

(

E

2

2 1 4 3 E − −

 

1

1

• Root rotations in DCB/ADCB specimens:

( P a h h

( P a h h

(9)

),

),

22 a a +

52 a a +

1 

= −



= −

23 2

53

(

)

(

)

   

   

6 3 1

2     + + − 1 12 4 3 E

6 3 1 1

  +



2 2 5 6  

3

with:

,

a

=

2 6 3  + − − − +

1  (1 )

a

=

22

23

E

E

2

4 3 −



1

1

(

)

(

)

( (

) )

6 3 1

2       + + − − − 1 12 1 1 E 4 3

   

   

6 3 1

1 1

  +

+ 



 

2    − + + 3

1

6 5

a

=

,

2 6 3  +

(1 ) 

a

=

−

 

 

52

53

2

2

E

−

E

2

4 3 −



1

1

For the homogeneous DCB specimen, the analytic functions assume values

0.673  = ,

, 33 2.271 g =

0.209

P  =

and 34 0 g = , and the compliance coefficients simplify yielding:



u u

P

= =

1

2

2

E

6

5 1

Pa

P



  

  

(10)

2

0.6

73

2.271

v v

= − =

− +

−

+

1

2

4

2

Eh

E

12 0 7 P 

  

2

1

0.67

3

a h



.

6

3

+

−

2  = − = −

−

1 

 

1

0

2 10

Eh

The calculated root rotations and displacements may be used as boundary conditions to accurately define displacements in the detached layers when using beam or plate theories. In accordance to Timoshenko beam theory, for instance, the load-point displacement in the homogeneous DCB may be obtained by superposing the contributions due to the flexural deflection of a built-in beam, the deflection due to shear deformations along the detached layer and the effects due to root rotation and root displacement at the crack tip given above. This yields:

3 5 Pa Pa Eh Eh + 3 12

  

  

4

(11)

( v x a v x a ) (

) 2 = − − = − =

(1 )

a v

  + − +

(1)

(2)

1

1

Comparing this result with the exact 2-D elasticity result in Ustinov and Massabò (2020) shows that the displacement in (11) is missing the term 3 5 Pa Eh  , which is due to shear deformations near the crack tip which are not included in the second term in the brackets. The rotations of the neutral lines could be used instead of the rotations of the cross sections as kinematics variables to define the compliance matrix. This has been done in Ustinov and Massabò (2020) and the compliance matrix associated to this choice is provided in the paper. Using the compliance coefficients associated to the rotations of the neutral axes in combination with Euler Bernoulli beam theory, which provides the flexural deflection of the detached arm, yields a solution which coincides with the 2-D result. 3.2. Application to homogeneous ELS specimen In the homogeneous ELS specimen, shown in Fig. 4, with 1 2 h h h = = ( 1  = ) and 0   = = , the crack tip field is more complex and described, through equations (7-10), by the five elementary loadings: symmetric bending,