Issue 74

P. Zuliani et alii, Fracture and Structural Integrity, 74 (2025) 385-414; DOI: 10.3221/IGF-ESIS.74.24

E

= ) and p(x) the ratio of the derivate of the section (S’(x)) to the section

being c the velocity of wave propagation (

c

ρ

(S(x)). Defining k as the ratio of the frequency of vibration ( ω ) to the velocity c and ( )

( ) ( ) u x,t U X sin ωt = , the following equation

is obtained:

( ) ( ) ( ) ( ) 2 0 U x p x U x k U x ′′ + + = ′

(10)

To solve the equation, the authors defined three different functions to describe the section: ( ) 1 D x , 2 = if s l x L < ≤ for the circumferential part

(11)

x l −

D

s

1 = +

if

3 s L x l < ≤ for the conical part

(12)

x

,

θ 2      

2

tan

( ) 2 2 2 D x R x 2 =− − + +

if

3 0 x L < ≤ for the cylindrical part

(13)

R,

Since the analytical solution of Eqn. (13) may be difficult, the function y(x) has been simplified with the following function using the catenary approach.

D

(

)

( )

2 cosh

b x α

=

if

(14)

y x

3 0 x L < ≤

2

2 R R D   + −     sin 2 θ

D

2

1

2

=

b α

(15)

2       θ

Rcos

2

Finally, the equations have been solved using the proper boundary conditions and final geometry has been verified using FEA.

Figure 24: Geometry of notched specimens for the analytical procedure proposed by Dantas et al. [4]. Finally, although Dantas et al. [4] demonstrated the validity of their analytical approach for blunt notch geometries, further analytical methods need to be developed for designing notched specimens with different geometries in ultrasonic fatigue testing of steel.

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