Dispersion of elastic waves in a strongly inhomogeneous three-layered plate

Abstract

Elastic wave propagation in a three-layered plate with high-contrast mechanical and geometric properties of the layers is analysed. Four specific types of contrast arising in engineering practice, including the design of stiff and lightweight structures, laminated glass, photovoltaic panels, and electrostatic precipitators in gas filters, are considered. For all of them the cut-off frequency of the first harmonic is close to zero. Two-mode asymptotic polynomial expansions of the Rayleigh-Lamb dispersion relation approximating both the fundamental bending wave and the first harmonic, are derived. It is established that these can be either uniform or composite ones, valid only over non-overlapping vicinities of zero and the lowest cut-off frequencies. The partial differential equations of motion associated with two-mode shortened dispersion relations are also presented.

Asymptotic analysis of an anti-plane dynamic problem for a three-layered strongly inhomogeneous laminate

“…In Figure 2, µ = 0.232, ρ = 3.0, and h = 1.0, while µ = 0.014, ρ = 0.03, and h = 1.0 in Figure 3. In Figure 3, the lowest cut-off frequency Ω = 0.17, and formula (17) gives the same value. As might be expected, the cut-off frequencies are not observed over the low-frequency range in the non-contrast case, see Figure 2.…”

“…In the absence of contrast, the problem of interest does not support the fundamental mode with a zero cut-off frequency. Under the same condition as in [17] on the contrast parameters, a low-frequency mode arises. For this mode, the shortened forms of the dispersion relation demonstrate different asymptotic behaviours within setups (a) and (b).…”

“…In spite of a substantial progress in terms of the qualitative understanding of the effect of contrast on the dynamic response of a layered plate coming from the aforementioned paper [17], asymptotic derivation of the related shortened differential equations of motion from the original 3D vector formulation may still seemingly need a more preliminary insight. This is also due to two distinct kinds of asymptotic behaviour governing bending and shear motions.…”

On thermal strains and residual stresses in the linear theory of anti‐sandwiches

“…[17] Moreover, the propagation of elastic waves within the laminates with a particular focus on high-contrast material properties has also been investigated. [18] Analysing the current literature reveals that the XLWT is mostly used in mechanical analyses and more specifically, it is applied by means of the finite element method. [19] The environmental effects and manufacturing side-effects on anti-sandwiches should be considered to complement the mechanical point of view.…”

“…Recently, the XLWT has been further extended to cover mechanical analysis of curved composite panels and merged into a more general shell theory . Moreover, the propagation of elastic waves within the laminates with a particular focus on high‐contrast material properties has also been investigated …”