Longitudinal wave propagation. Part I—Comparison of rod theories

Spectral Methods for Modelling of Wave Propagation in Structures in Terms of Damage Detection—A Review

“…In the group of articles devoted to the application of different rod theories, one should distinguish the works [101][102][103]108], where different models derived with FDSFEM suitable for modelling of wave propagation in elements with various damages have been proposed. As damage, the authors have proposed a transverse, open and not propagating crack and an additional mass.…”

Exact solution of the Mindlin–Herrmann model for longitudinal vibration of an isotropic rod

“…The model is presented in [2] and [7,11] in Cartesian and cylindrical coordinate respectively and used in [13,19] as improvement of the single mode classical, Rayleigh-Love and Rayleigh-Bishop models. According to Mindlin-Herrmann theory of longitudinal stress wave propagation, the axial displacements u and the transverse (lateral) displacements v and w are assumed to be function of the form:…”

“…For example, in the Rayleigh-Love [1][2][3] and Rayleigh-Bishop [4,5] models, the lateral displacements are supposed to be proportional to the product of longitudinal strain of the rod, its Poisson ratio and the distance from the neutral line of the cross-section. Lack of physical clarity in the interpretation of certain higher effects, such as independent shear displacement and radial motion which describe transverse deformation have also been associated with these approaches [2]. The theory of longitudinal stress wave propagation in an elastic rod which couples axial and independent lateral displacements was first established by Mindlin and Herrmann in 1950 [6] and later on, was developed in more details by Graff in his book [7].…”

“…Krishnaswamy and Batra have investigated the same model with various boundary conditions and shown that the analysis of the corresponding frequency spectrum can be considered in three situations, depending on the domain in which the eigenfrequency belongs to [8][9][10]. Recently Krawczuk, Grabowska and Palacz, Zak and Krawczuk, and Anderson [2,11,12] in their work on different theories of longitudinal vibrations of rods have analysed the frequency equation of each theory including the one of Mindlin and Herrmann, also providing a comparison regarding their accuracy and applicability. These authors have based their approach to wave propagation in a elastic continuum medium only on the analysis of the frequency equation, which shows the relationship between the governing factors of the phenomena, these being: time frequency, spacial frequency or wave number and phase velocity.…”

“…Compared to single mode models, this representation increases the num-ber of unknown functions and hence, the model is described by two partial differential equations. Consequently, the model is more accurate as it has wider frequency range in which the effect of longitudinal vibration of the rod can be analysed [2,12,13]. Nevertheless, on needs to be mindful of the fact that, similar to Classical, Rayleigh-Love, Rayleigh-Bishop models, the Mindlin-Herrman model is based on plane cross-section theory.…”