PSI - Issue 41

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

464

4

The way to define the rotations is not unique. In Ustinov and Massabò (2022) the coefficients associated to the rotations of the cross sections have been defined so that the averaged squares of the difference between the real horizontal displacements and the displacements due to rotation by this angle over the layer thickness be minimal.

Fig. 3. (a) Bimaterial layer, undeformed configuration; (b) root rotations and root displacements defined as relative displacements between the neutral axes of the detached and intact parts (solid-thick lines: deformed configuration; dashed lines: undeformed configuration).

The matrix of compliances, which relate the kinematic variables to the elementary loadings is:

0 0 0 0

u

a a a a

P

        

              =

                   

         

1

11

12

13

14

S

1

h a a a a 

−

1 S M h

1 1  −  

21

22

23

24

v

a a a a a a

D

(6)

1

31

32

33

34

35

36

0 0 0 0

41 a a a a a a a a 42 43

S

u

44

2

P

2 1 h



−

51

52

53

54

0

 

 

1

−

a a a a a a M h

v

61

62

63

64

65

66

0 1

2

where the coefficients ij a have been related in Ustinov and Massabò (2002) to the terms of the energy release rate associated to the elementary loads by using exact solutions for the displacement fields far away from the crack tip and the reciprocity theorem of 2-D elasticity. Explicit expressions for the coefficients have been obtained for: bimaterial isotropic layers with mid-thickness cracks (26 out of the 28 coefficients derived along with a relation for the last two, 34 64 , a a ); homogeneous orthotropic symmetric layers (all coefficients but 33 63 a a = − and 34 64 a a = − ); and relevant coefficients describing thin films on half-planes. For isotropic layers with second Dundurs’ parameter 0  = , the coefficients depend on the first Dundur’s parameter ( 1) / ( 1)  =  −  + and some analytic functions, 33 34 , , , P g g   given in Ustinov and Massabò (2022). 3. Application to fracture mechanics specimens with known crack tip forces The compliance coefficients in the matrix in Eq. 6 can be used to calculate root rotations and root displacements in fracture specimens and layered structures. In this section some examples will be presented with reference to the specimens analyzed by Qiao and Wang (2004). The analytical results will then be compared with finite elements solutions from the same paper or obtained by Ustinov and Massabò (2022.)