PSI - Issue 47

Daniela Scorza et al. / Procedia Structural Integrity 47 (2023) 30–36 Scorza et al./ Structural Integrity Procedia 00 (2023) 000–000

33

4

1

1

( ) 1

( ) 1

( ) 1

( ) 1

(4) c v L v L v L v L L L − = − (2) (4) (2) 1 1 2 2 2 c 2

(13)

1

1

( ) 1

( ) 1

( ) 1

( ) 1

(5) c v L v L v L v L L L − = − (3) (5) (3) 1 1 2 2 2 c 2

(14)

The stiffness of the nanobeam at the crack location, that is, the spring stiffness, k , is computed by exploiting both the Griffith energy criterion and the conventional Linear Elastic Fracture Mechanics, as is detailed in the research work by Scorza et al. (2023), that is:

) ( ) I BH E   2

k

=

(15)

(

( )

2 F F π ν ξ ξ − +

72 1

 

II

a H ξ = is the relative crack depth, and I F and II F are computed by exploiting the

where ν is the Poisson ratio,

shape functions, I f and II f , for Mode I and II loading , respectively:

ξ

( )

ξ = 

2 z f dz

F

(16)

I

I

0

ξ

( )

ξ = 

2 z f dz

F

(17)

II

II

0

The expressions of I f and II f depend on the angle θ . It is worth noticing that when 0 θ = ° , 0 II F = . 3. Model applications The model outlined in the previous Section is herein applied to a nanobeam made of aluminium, having elastic modulus 70 E GPa = and Poisson ratio 0.33 ν = . The material internal characteristic length is assumed to be equal to 15 c L m µ = , whereas the point load is 10 P N µ = . The nanobeam sizes herein assumed are: 20 L H = , B H = , 1 0.5 L L η = = and θ alternatively equal to 0 .0 ° , 22 .5 ° , and 45 .0 ° , with 30 L m µ = , corresponding to 0.5 c L L λ = = . It is worth noticing that the crack here examined has a length greater than the critical crack size computed by Scorza et al. (2023) and, in favor of safety, greater than 100 nm . Such an assumption ensures to avoid Linear Elastic Fracture Mechanics breakdown. Moreover, note that, for the examined value of θ , I f and II f were provided by Gross and Srawley (1965) and by Wilson (1969). In Figs. 2-4 the dimensionless deflection is plotted against the relative crack depth, ξ , for the three above values of θ . In each Figure, the solution determined by applying the local classical beam theory is also reported. It can be observed that the deflection increases by increasing the crack depth, for all the examined values of crack angle. In particular, the maximum increment, equal to 187% , is obtained when 0 .0 θ = ° , whereas the deflection increases by 98% and 39% when the crack angle is equal to 22 .5 ° and 45 .0 ° , respectively.