PSI - Issue 42

Rita Dantas et al. / Procedia Structural Integrity 42 (2022) 1676–1683 Rita Dantas / Structural Integrity Procedia 00 (2019) 000–000

1681

6

U ( x ) + 2 α tanh ( α x ) U ( x ) + k 2 U ( x ) = 0 , | x | ≤ L 2

(21)

By solving these di ff erential equations and establishing the boundary conditions, the specimen’s resonance length is defined by:

1 k ( β coth ( β L 2 ) − α tanh ( α L 2 ))

arctan

1 k

L 1 =

(22)

where:

β = √ α 2

− k 2

(23)

On the other hand, the solutions of Eqs. 20 and 21 are:

cos ( kL 1 ) cosh ( α L 2 ) sinh ( β L 2 )

sinh ( β x ) cosh ( α x )

U ( x ) = A 0

(24)

, | x | ≤ L 2

U ( x ) = A 0 cos ( k ( L − x )) , L 2 ≤ | x | ≤ L

(25)

Finally, the strain and stress can be determined from the Eqs. 26-29, which are defined bellow:

• For the variable section part ( | x | ≤ L 2 )

cos ( kL 1 ) cosh ( α L 2 ) sinh ( β L 2 )

β cosh ( β x ) cosh ( α x ) − α sinh ( β x ) sinh( α x ) cosh 2 ( α x ) β cosh ( β x ) cosh ( α x ) − α sinh ( β x ) sinh( α x ) cosh 2 ( α x )

ε ( x ) = A 0

(26)

cos ( kL 1 ) cosh ( α L 2 ) sinh ( β L 2 )

σ ( x ) = E d A 0

(27)

• For the constant part ( L 2 ≤ | x | ≤ L ) ε ( x ) = kA 0 sin ( k ( L − x ))

(28)

σ ( x ) = EkA 0 sin ( k ( L − x ))

(29)

4. Experimental Work

As explained in the previous section, a specimen can only be tested in an ultrasonic machine, when its first longi tudinal natural vibration is approximately equal to 20 kHz (which is the standard working frequency of an ultrasonic machine). In order to define the specimen’s geometry, Young’s modulus ( E ) and density ( ρ ) of the material to test need to be known. Thus, the main mechanical properties of S690 structural steel were experimentally determined and are listed in Tab. 1.

Table 1. S690 steel properties (mechanical and physical) ρ [ kg / m 3 ] E [ MPa ]

y [ MPa ]

σ u [ MPa ]

σ

7815

211593

781

819

Moreover, the specimens design should be also defined taking into account the range of stress which is matter of analysis. Thus, the critical cross section as well as the di ff erent lengths ( L 1 , L 2 and L 3 ) can be adjusted to achieve higher or lower stresses during the fatigue testing, and at the same time ensure the resonance frequency of 20 kHz . The target stress range was defined based on the conventional definition of fatigue limit for R = − 1 under the loading