PSI - Issue 66

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

408

3

i k . Under the Euler-Bernoulli assumption, beam segment according to the SDM:

1 n + differential governing equations can be written for each i th −

( )

M x

1

( ) ( ) 2 x

( ) x





i

−

=−

(1)

i

i

2 c

2 c L EI

L

being ( ) i x  the curvature of the i th − beam segment,

( ) ( ) 2 i x



its second derivative with respect to x and c L the

internal characteristic length of the material (Scorza et al., 2022). Moreover, two constitutive boundary conditions at the nanobeam ends and 2 n constitutive continuity conditions at each internal crack are added to the above equations: ( ) ( ) ( ) 1 1 1 1 0 0 0 c L − =   (2a) ( ) ( ) ( ) 1 1 1 1 0 n n c L L L + + + =   (2b) ( ) ( ) ( ) ( ) 1 1 1 2 n i i i i i, j i c c j i x x x L L + = + =     (2c)

2 i

1

 

( ) ( ) 1 1 i i x +

( ) i x

( ) i x





−

=−

(2d)

1

1 i , j +

i

+

L

L

c

c j

1

=

being

1

if 1 if

q q

i j i j  

=

x

  

( )

(

)

j

M t

q x t   − 

1



with

( ) x

j



exp

dt

=

 

 

=−

i, j

2

L

L

EI

c

c

x

1

j

−

2.2. Model formulation for a double cracked cantilever nanobeam Let us consider a cantilever nanobeam, shown in Figure 1, containing two edge-cracks with same depth and orientation, for which three beam segments can be identified.

(b)

B

(a)

B

y

y

k 1

k 2

d

O

H

O

H

x

a

a

x

x 1

x 1

F

F

x 2

x 2

L

L

Fig. 1. (a) cantilever nanobeam with two edge-cracks and (b) its schematic representation with two massless elastic rotational springs. The governing equations, Eq. (1), can be rewritten in terms of transverse (vertical) displacement, ( ) i v x , by exploiting the Euler-Bernoulli assumption ( ( ) ( ) ( ) 2 i i x v x =  ) and by differentiating Eq. (1) twice with respect to x . In this way, the following relationship can be obtained: ( ) ( ) ( ) ( ) ( ) ( ) 2 6 4 2 2 1 with 123 i i i c c M x x x i , , L L EI v v − =− = (3) Analogously, Eqs (2) are also rewritten as here reported: