PSI - Issue 41

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

467

7

3 / 8 S M Pa = − , axial forces, / 2 D P = , single shear S P = − , uniform bending 0 M Pa = − , while the void loading 0 0 P = . Root rotation and displacements depend on the elastic constants and the analytic functions 0.673  = , 0.209 P  = , 33 2.271 g = and 34 0 g = . 3 / (4 ) S P Pa h = , double shear

• Longitudinal root displacements in homogeneous isotropic ELS specimen:

13 2 a a

3

3

(12)

( u Pa a =

),

,

a − + −

u u = −

14

1

11

12

2

1

4

8

h

h

a a

2

2 E  =

P E  =

3 3 E E E   − − 3 8 8 P

, 12 0 a = , 13 a

with: 11 a

,

a

= −

14

5

2 3 3 3 8 5 4 P  + +

5 8

P E

a h

  

 

Simple substitution yields:

u u

= − =

+





1

2

P

 • Transverse root displacements in homogeneous isotropic ELS specimen:

33 2 a a a a a h

3

3

(13)

( v Pa a =

),

,

a

v v =

34 + − −

36

−

1

31

32

2

1

4

8

h

h

1

3 5

2 1 6 6 3 5 2 E   





  

  

  

3 16 E 

2

2

3

3 P 

a

a

,

,

3 3 2 E a g = , 33

33 2 a a = , 36 a 34

with:

−



= − +

−

= − + − 

=

32

31

2

E

1

Simple substitution shows that  = = − • Root rotations in homogeneous isotropic ELS specimen: 1 2 9 4 P v v Pa Eh 2

23 2 a a

3 Pa a

3

(14)

(

),

a

1 

= −

−

+ −

  =

24

21

22

2

1

4 h h

8

h

a a

(

)

6

12

1 E   −



1 6 

6 5

1 3

9

5 3

  

2  

  

P

with:

,

,

,

a

=

2

2

(1 ) 

(1 ) 

a

a

+





=

− +

=

− +

a

=

 

 

21

22

23

24

4 P

E

E

E

1

1

1

9 4

2 , a 

P

  

Simple substitution shows that

2

=

+

   =

1 



 

2

1

P

P

Eh

h

4. Comparison with 2-D finite element results Root rotations and root displacements in Eqs. (7-14) are compared in Table 1 with the FE results in Qiao and Wang (2004) and in Ustinov and Massabò (2022) for specimens with: 1 16 a h = and 1 2 / 1 h h  = = . The layers in the homogeneous DCB specimen have Poisson ratio 0.3  = . In the ADCB specimen, bimaterial layers with 0.667 and 0.23   = − = − (plane stress; 2 1 1 2 5 , 0.3 E E   = = = ) and 0.667 and 0   = − = (plane stress; 2 1 1 2 5 , 0.995, 0.975 E E   = = = ) are examined to investigate the effects of the simplifying assumption 0  = , which is usually made to avoid near tip oscillating fields. The comparison highlights that the classical assumption 0  = is indeed acceptable when calculating root rotation coefficients while may yield important under/over-estimations of the root displacement coefficients. 5. Conclusions The steps required to define root rotations and root displacements in bimaterial layers subjected to arbitrary end loadings using compliance coefficients associated to six elementary loadings and recently derived through an elasticity technique have been presented with reference to two classical fracture mechanics specimens, the Double Cantilever