Reflection from and transmission through a periodically stratified ideal fluid half‐space

The problem of the propagation of quasi-Lamb waves in a pre-deformed incompressible elastic layer that interacts with the half-space of an ideal compressible fluid is considered. The study is conducted on the basis of the three-dimensional linearized equations of elasticity theory of finite deformations for the incompressible elastic layer and on the basis of the three-dimensional linearized Euler equations for the half-space of an ideal compressible fluid. The problem is formulated, and the approach based on the utilization of representations of the general solutions of the linearized equations for an elastic solid and a fluid is developed. Applying the Fourier method, we arrive at two eigenvalue problems for the equations of motion of the elastic body and the fluid. Solving them, we find the eigenfunctions. Substituting the general solutions into the boundary conditions, we obtain a homogeneous system of linear algebraic equations for the arbitrary constants. From the condition for the existence of a nontrivial solution, we derive the dispersion equation. A dispersion equation, which describes the propagation of normal waves in the hydroelastic system, is obtained. The dispersion curves for quasi-Lamb waves over a wide range of frequencies are constructed. The effect of the finite initial deformations in an elastic layer, the thickness of the elastic layer, and the half-space of an ideal compressible fluid on the phase velocities and dispersion of quasi-Lamb modes are analyzed. It follows from the graphical material presented above that, in the case of compression with 0.54, i.e., with a 46 percent’s reduction in the length of the highly elastic incompressible body, the phase velocities of the surface waves (Stoneley waves and Rayleigh waves) vanish. This indicates that surface instability develops at 0.54 for a highly elastic incompressible non-Hookean body initially in a plane stress-strain state. We should point out that these figures agree with results obtained earlier in the theory of stability and correspond to the critical value of the contraction parameter. In the case of highly elastic incompressible bodies, the linearized wave theory makes it possible to study not only general and several specific wave processes, but also the conditions under which the surface instability begins in elastic bodies and hydroelastic systems. It also follows from the graphs that the ideal fluid slightly affects the surface instability of hydroelastic systems. The numerical results are presented in the form of graphs, and their analysis is given.

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On a technique for deriving explicit transfer matrices of orthotropic layers

Vietnam Journal of Mechanics ◽

10.15625/0866-7136/37/4/6534 ◽

2015 ◽

Vol 37 (4) ◽

pp. 303-315 ◽

Cited By ~ 1

Author(s):

Pham Chi Vinh ◽

Nguyen Thi Khanh Linh ◽

Vu Thi Ngoc Anh

Keyword(s):

Wave Propagation ◽

Explicit Form ◽

Half Space ◽

Transfer Matrix ◽

Rayleigh Waves ◽

Secular Equation ◽

Transfer Matrices ◽

Orthotropic Layer ◽

Wave Propagation Problem ◽

Arbitrary Thickness

This paper presents a technique by which the transfer matrix in explicit form of an orthotropic layer can be easily obtained. This transfer matrix is applicable for both the wave propagation problem and the reflection/transmission problem. The obtained transfer matrix is then employed to derive the explicit secular equation of Rayleigh waves propagating in an orthotropic half-space coated by an orthotropic layer of arbitrary thickness.

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