PSI - Issue 42

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

1679

4

For a smooth hourglass specimen, during an ultrasonic fatigue test, the system machine + specimen vibrates at a longitudinal resonance mode, in which the displacement is symmetric and maximum at the ends of the specimen. On the other hand, the stress is maximum at the middle of the specimen, which is also a node of vibration (a section where the displacement is zero). Another important aspect to highlight is the stress and displacement distribution along the machine, which are zero at threaded connections to avoid problems of tightening and high stresses due to regions of stress concentration. An example of this, it is the thread connection between the specimen and the machine where stress is zero, since it would be a local of stress concentration (Anes et al., 2011).

3. Analytical Formulation

Bathias and Paris (2004) developed an analytical methodology to define the geometry of a smooth specimen, to be tested in an ultrasonic system, and which is based on the theory of elastic wave propagation. Thus, considering a three-dimensional isotropic elastic body and assuming a cartesian coordinate system, the di ff erential equations of motion can be written as:

+ ∇ 2 u + ∇ 2 v + ∇ 2 w

E 1 + ν E 1 + ν E 1 + ν

∂ 2 u ∂ t 2 ∂ 2 v ∂ t 2 ∂ 2 w ∂ t 2

1 1 − 2 ν 1 1 − 2 ν

∂ e ∂ x ∂ e ∂ y

(1)

ρ

=

(2)

ρ

=

1 1 − 2 ν

∂ e ∂ z

(3)

ρ

=

where u , v , and w are displacements along x , y , and z ; E is the Young’s modulus, ν is the Poisson’s ratio, ρ is the density, ∇ 2 is the Laplacian and e is the volume dilatation. The elastic wave theory identifies two di ff erent types of waves: longitudinal and transverse, which velocity can be defined, respectively, by the following equations: c = E (1 − ν ) (1 + ν )(1 − 2 ν ) ρ (4) c = E 2(1 + ν ) ρ (5) In order to study an ultrasonic fatigue testing, a longitudinal wave should be considered to solve the di ff erential equation problem. Furthermore, the wave theory applied to the fatigue smooth specimens, which geometry is repre sented in Fig.2, can be approximated to a one-dimensional problem and the Poisson ratio assumed as zero, which allows the simplification of the longitudinal wave into the equation bellow: c = E ρ (6)

Fig. 2. Geometry of ultrasonic smooth specimen (adapted from (Bathias and Paris, 2004) )