PSI - Issue 66

Daniela Scorza et al. / Procedia Structural Integrity 66 (2024) 406–411 Author name / Structural Integrity Procedia 00 (2025) 000 – 000

409

4

1

( ) ( ) 3 0

( ) ( ) 2

v

v

0 0 =

−

(4a)

1

1

c L

1

( ) ( ) 3 0

( ) ( ) 2

v

v

0 0 =

−

(4b)

1

1

c L

2

1

2



( ) ( ) 3 1 i x

( ) ( ) 2 1 i x

( ) i x

v

v

with

12 ,

i

+

=



=

(4c)

i, j i +

L

L

c

c j i =

2

1

2



( ) ( ) 3 1 1 i x +

( ) ( ) 2 1 1 i x +

( ) i x

v

v

with

12 ,

i

−

=−



=

1 i , + (4d) The general integral solution obtained from the integration of the three sixth-order differential equations, Eq. (3), is a function of 18 integration constants, which can be determined by employing the above six constitutive conditions, Eqs (4), and by adding other twelve boundary conditions. In particular, by considering the case of the cantilever nanobeam in Figure 1, constrained in 0 x = and subjected to a concentrated force F in x L = , the following kinematic and static conditions can be written: c c j i = L L

1

    

( ) ( ) 4 L

( ) ( ) 2 L

v

v

0

−

=

( ) ( ) ( ) 1 1 1 v

0 0 =

   

3

2 3

L

c

and

(5)

v

0 0 =

1

1

F

( ) ( ) 5 L

( ) ( ) 3 L

v

v

−

=

3



2 3

2 c

EI

L

L

c

Whereas, the following two kinematic and two static continuity conditions can be imposed at each edge crack (that is, at i x x = with 1,2 i = ): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 2 2 4 1 i i i i i i i i i i c i i i x x EI x x x L x k v v v v v v + + =      − = −    and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 4 2 4 2 2 1 1 3 5 3 5 2 2 1 1 i i c i i i i c i i i i c i i i i c i i x L x x L x x L x x L x v v v v v v v v + + + +  − = −   − = −  (6)



being the spring stiffness, i k , determined as is described in the following. 3. Stiffness computation for nanobeams with two cracks

The stiffness i k (or compliance i c ) of a cracked cross-section contained in a nanobeam under Mode I loading is here computed by exploiting both the Griffith’s energy criterion and the conventional Linear Elastic Fracture Mechanics (LEFM) (Scorza et al. 2023a; 2023b), by assuming the singularity of the stress field at the crack tip and by holding the material continuum hypothesis. A detailed description of the procedure to be used in order to compute the stiffness of a single crack contained in a nanobeam is reported in Ref. (Scorza et al. 2023a). Note that, when more than a single edge-crack is taken into account, the compliances of the cracked cross sections depend on both the distance between cracks and their depth, and, if an analytical solution is not available, an approximated procedure can be followed by exploiting numerical simulations. In particular, the compliance i c of the i th − crack with depth i a can be computed through the following equation provided that the geometric shape function * f is defined: ( ) ( ) 2 2 4 0 72 1 i a * i i c y f y / H dy BH E   −   =    (7) being  the Poisson’s ratio. Therefore, the problem of the compliance computation comes down to the calculation