“…(ii) The non-locality of Eringen’s model decreases the Rayleigh wave velocity more strongly than the one of the weakly non-local model [46]. Figure 1The dependence of Rayleigh wave velocity on the non-locality parameter e∈false[0,0.9false] with γ=0.25 that corresponds to Eringen’s non-local model with the different constitutive boundary conditions for t11: (4.8) (solid line), (4.30) (dashed line) and to the weakly non-local model (dash-dot line) (eqn (24) in [48]).…”
Section: Rayleigh Waves In Traction-free Non-local Isotropic Elastic ...mentioning
confidence: 99%
“…The dependence of Rayleigh wave velocity on the non-locality parameter e∈false[0,0.9false] with γ=0.25 that corresponds to Eringen’s non-local model with the different constitutive boundary conditions for t11: (4.8) (solid line), (4.30) (dashed line) and to the weakly non-local model (dash-dot line) (eqn (24) in [48]).…”
Section: Rayleigh Waves In Traction-free Non-local Isotropic Elastic ...mentioning
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.
“…(ii) The non-locality of Eringen’s model decreases the Rayleigh wave velocity more strongly than the one of the weakly non-local model [46]. Figure 1The dependence of Rayleigh wave velocity on the non-locality parameter e∈false[0,0.9false] with γ=0.25 that corresponds to Eringen’s non-local model with the different constitutive boundary conditions for t11: (4.8) (solid line), (4.30) (dashed line) and to the weakly non-local model (dash-dot line) (eqn (24) in [48]).…”
Section: Rayleigh Waves In Traction-free Non-local Isotropic Elastic ...mentioning
confidence: 99%
“…The dependence of Rayleigh wave velocity on the non-locality parameter e∈false[0,0.9false] with γ=0.25 that corresponds to Eringen’s non-local model with the different constitutive boundary conditions for t11: (4.8) (solid line), (4.30) (dashed line) and to the weakly non-local model (dash-dot line) (eqn (24) in [48]).…”
Section: Rayleigh Waves In Traction-free Non-local Isotropic Elastic ...mentioning
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.
Wave propagation in slender beams is addressed in the framework of nonlocal continuum mechanics. The elastodynamic problem is formulated exploiting consistent methodologies of pure integral, mixture and nonlocal strain gradient elasticity. Relevant wave solutions are analytically provided, with peculiar attention to reflection and near field phenomena occurring in presence of boundaries. Notably, the solution field is got as superimposition of incident, reflected, primary near field and secondary near field waves. The latter contribution represents a further effect due to the size dependent mechanical behaviour. Limit responses for vanishing nonlocal parameter are analytically evaluated, consistently showing a zero amplitude of the secondary near field wave. Parametric analyses are carried out to show how length scale parameter, amplitude of incident wave and geometric and elastic properties of the beam affect the amplitudes of reflected, primary near field and secondary near field waves. The results obtained exploiting different nonlocal integral elasticity approaches are compared and discussed.
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.
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