Pham Chi Vinh

In this paper, based on the Hermitian property of surface impedance matrices, it has been proven that for Rayleigh waves propagating in traction-free anisotropic elastic half-spaces, the traction vector at any plane parallel to the surface of half-spaces is orthogonal to the displacement vector at the surface. This result generalizes the previous ones discovered by Hassan and Nagy (J. Acoust. Soc. Am. 104 (1998) 3107–3110) for isotropic half-spaces and Royer (J. Acoust. Soc. Am. 123 (2008) 599-601) for orthotropic materials. The orthogonal property may supply us with new approaches to study Rayleigh waves. As will be shown, it provides a new derivation of the explicit secular equation, polarization vector, and Stroh eigenvalues with positive imaginary parts of two-component Rayleigh waves without solving their characteristic equations.

The question of existence of surface waves is of great importance in both theoretical aspect and practical application. In this paper, we investigate the existence and uniqueness of electromagnetoacoustic shear horizontal waves propagating along the interface between two piezoelectric half-spaces (hereafter referred to as interface EMA-SH waves), without using the quasi-static approximation, by employing the complex function method. The results recover the necessary and sufficient conditions for the existence of interface EMA-SH waves under the quasi-static approximation, known as Maerfeld–Tournois waves. Explicit formulas for the slowness (the reciprocal of velocity) of interface EMA-SH waves are also obtained.

The present study addresses the propagation of Rayleigh waves in pre-stressed elastic half-spaces subjected to an isotropic internal constraint. The primary objective is to establish exact explicit expressions for the H/V ratio, which defines the amplitude relationship between horizontal and vertical displacements at the surface of half-spaces. By utilizing the surface impedance matrix methodology, a critical link is formed between the velocity and the H/V ratio of the Rayleigh wave. From this relationship and the secular equation, a cubic equation for the squared H/V ratio is derived. By applying the theory of cubic equations, we have formulated precise and explicit expressions for the H/V ratio, which are valid for an arbitrary strain energy function. Subsequently, these general formulas are specified for various strain energy functions and internal constraints. As the obtained formulas for the H/V ratio are totally explicit, they are a convenient tool to evaluate pre-stresses appearing in structures from measured values of the H/V ratio. To prove this point, two examples are considered and they show that for determining the pre-stresses we have only to solve nonlinear (algebraic) equations.

In this paper, we investigate the propagation of Stoneley waves along the unbonded interface between two micropolar elastic half-spaces. Using the complex function method, we derive explicit formulas for the wave velocity. These formulas have applications in various scientific fields, particularly in non-destructive evaluation. Two numerical examples illustrate how the obtained wave velocity formulas can be used to evaluate material parameters non-destructively.

In this paper, we study the propagation of Rayleigh waves in nonlocal orthotropic half-spaces coated by a nonlocal orthotropic layer with sliding contact using the weakly nonlocal elasticity model. This model was recently introduced and different from other existing nonlocal models it has been proven to be well-posed for all harmonic plane wave problems. The transfer matrix method and the effective boundary condition method are employed for deriving the explicit dispersion equation of Rayleigh waves. Using the obtained dispersion equation, the effect of the nonlocality and the thickness of the layer on the velocity of Rayleigh waves is considered through some numerical examples. It is shown that the nonlocality and the thickness of the layer strongly affect the velocity of Rayleigh waves and they make it decreasing. Since the dispersion equation of Rayleigh waves is totally explicit, it will be a powerful tool for monitoring the health of the layer/half-space structures during loading.

The existence of Rayleigh waves (propagating in isotropic elastic half-spaces) with the tangential and normal impedance boundary conditions was investigated. It has been shown that for the tangential impedance boundary condition (TIBC), there always exists a unique Rayleigh wave, while for the normal impedance boundary condition (NIBC), there exists a domain (of impedance and material parameters) in which exactly one Rayleigh wave is possible and outside this domain a Rayleigh wave is impossible. In this paper, we consider the existence of Rayleigh waves with the full impedance boundary condition (FIBC) that contains both TIBC and NIBC. It is shown that the existence picture of Rayleigh waves for this general case is more complicated. It contains domain for which exactly one Rayleigh wave exists, domain where a Rayleigh wave is impossible, and domain for which all three possibilities may occur: two Rayleigh waves exist, one Rayleigh wave exists, and no Rayleigh wave exists at all. The obtained existence results recover the existence results established previously for the cases of TIBC and NIBC. The formulas for the Rayleigh wave velocity are derived. As these formulas are totally explicit, they are very useful in various practical applications, especially in the non-destructive evaluation of the mechanical properties of structures. In order to establish the existence results and derive formulas for the Rayleigh wave velocity, the complex function method, which is based on the Cauchy-type integrals, is employed.

In this paper, we establish the transfer matrix for an orthotropic elastic layer made of nanomaterial that is modeled by the weakly nonlocal elasticity model. This model is proposed recently and different from other existing nonlocal models of elasticity it has been proved to be well-posed for any problem of plane waves. Since the established transfer matrix is totally explicit, it is a convenient tool in investigating various problems of plane waves propagating in layered nonlocal elastic media. To prove this point, we use it to derive the dispersion equation of Lamb waves propagating in a composite plate consisting of two weakly nonlocal orthotropic layers. The dispersion equation in explicit form of Lamb waves has been derived, and it is shown that the derivation of this equation is much more simple than the one using the traditional technique. The obtained explicit dispersion equation will be used to establish techniques monitoring the health of layered nanostructures.

In this paper, we establish a criterion for well-posedness of Eringen’s non-local elasticity theory for problems of harmonic plane waves in domains with non-empty boundaries, and introduce a novel method for solving well-posed problems. The criterion for well-posedness says that for problems of harmonic plane waves, Eringen’s non-local elasticity theory is well-posed when the constitutive boundary conditions contain all equilibrium boundary conditions, otherwise it is ill-posed in the sense of no solutions. With this well-posedness criterion, it is easy to check whether a non-local harmonic plane wave problem is well-posed or not. If it is a well-posed problem, its solution will be found by employing the novel method. It has been shown that Eringen’s method, which has been used widely to solve problems of non-local harmonic plane waves, does not give their correct solutions. Therefore, it must be replaced by the novel method. As an application of the criterion for well-posedness and the novel method, two well-posed problems of harmonic plane waves are considered including Rayleigh waves and SH waves propagating in traction-free non-local isotropic elastic half-spaces. Exact solutions of these problems have been obtained including explicit expressions of displacements, local and non-local stresses and dispersion equations.