PSI - Issue 41

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

463

3

1 1 − − = −  12 I M

S M

M

2) symmetrical bending moments:

(2)

1

3

(3)

3) symmetrical shear forces (double shear):

2 D V =

(4)

4) asymmetrical shear forces (single shear):

3 S V =

where:

2 h      2 3 1 3 h

2 h    +  + + − +    =  1 3 h 

2    −  + 

1 1 2 1

I

,

(5)

 =

h

The elastic mismatch 1 2 / E E  = in isotropic layers under plane stress conditions, with i E the Young’s modulus in layer i and i  the Poisson coefficient. Extension to plane strain problems is straightforward. 11(2) 11(1) /    = , with 11( ) i  a constant of compliance in layer i , becomes

(1)

(2)

(3)

(4)

(5)

(6)

Fig. 2. Elementary loads 1-6 acting at the crack tip cross sections of the bimaterial layer

The last two elementary loadings, (5) and (6) in Fig. 2, are void loadings which do not produce singularity at the crack tip but affect the near tip deformations. The first void loading is uniform compression, with 0 3 P P = so that ( ) ( ) 1 0 2 0 / 1 , / 1 P P P P    =  +  = − +  for equilibrium. The second is uniform bending with 0 3 M M = so that ( ) ( ) ( ) 3 1 2 1 0 2 0 1 2 0 1 1 / , 1 / , 6 1 / M M M M P P M h         − =  +  = − +  = =  + to satisfy equilibrium, with 4 2 3 2 4 6 4 1      = + + +     + . The kinematics of the layer is described by longitudinal and transverse displacements, ( ) ( ) , i i u v with 1, 2 i = for the upper and lower sub-layers. Different kinematic variables have been used in Ustinov and Massabò (2022) to define the boundary conditions for the detached layers, when they are studied as beams or plates. The original derivation has been based on the relative displacements and rotations of the neutral lines of the detached layers and the interface. The solutions have then been modified to describe different kinematic variables. Here solutions are presented for the kinematic variables which are used in Timoshenko type beam theory. They are defined as: crack tip relative longitudinal displacements, 1 (1) 1 (2) 1 ( 0 , / 2) ( 0 , ) h u u x y h u x y h − + = = = − = = −  and 1 v and 2 v (having similar forms), between the neutral axes of the detached and intact parts of the layer (formulas presented for the case 1 0 h h   ) ; and relative rotations, 1  and 2  , of the cross sections between the two parts. They are shown in the schematic in Fig. 3. 2 (2) u u x − + = = = − − = = −  2 (2) u x 1 ( 0 , / 2) ( 0 , ) h y h y h and relative transverse displacements,