Harmonic waves in three-dimensional elastic composites

For a periodic elastic composite which consists of a matrix and fibers with finite dimensions (i.e. a three-dimensional problem), here are given estimates for eigenfrequencies and eigenfunctions. Calculations are based on a new quotient which has been proposed by Nemat-Nasser. The periodic character of the eigenfrequencies is pointed out, and illustrative examples are given.     

Plane infinitesimal waves in homogeneous elastic plates

Using the theory of elastic Cosserat plates, we obtain conditions for reality of the frequencies of vibration.

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Résumé En utilisant la théorie des plaques élastiques de Cosserat, on obtient les conditions pour réalité des fréquences de vibration.

     

Further studies on edge waves in anisotropic elastic plates

A modified Stroh-type formalism for edge waves in unsymmetrical anisotropic plates is derived. Explicit expressions of the fundamental matrices for the formalism are presented. The existence conditions for one or two subsonic edge waves in the unsymmetrical anisotropic plates are discussed based on the formalism, and a procedure for finding an explicit secular equation for the edge-wave speed is proposed.     

Vibration reduction of floating elastic plates in surface waves

Based on dynamical theories of water waves and dynamics of Mindlin thick plates,the investigation of the wave-induced responses and the vibration reduction of an elastic floating plate are presented using the Wiener-Hopf technique.Without regard to the case of elastic connector,the calculated results obtained by the present method are in good agreement with those from the literature and the experiment.It can be shown that the present method is valid.Relations between the spring stiffness to be used to connect the sea bottom and the floating plate and the parameters of wave-induced responses of floating plates are investigated using the present method.Therefore, these results can be used as theoretical bases for the design stage of super floating platform systems.     

Thermoplastic strain of circular sandwich plates on an elastic base

We consider thermomechanical bending of an elastoplastic circular (solid or annular) light-filler sandwich plate resting on an elastic base. The hypotheses of broken normal are used to describe the kinematics of the plate stack nonsymmetric along the thickness. The base reaction is described by the Winkler model. We obtain the system of equilibrium equations and its exact solution in terms of displacements. We also present numerical results for a sandwich annular metal-polymer plate.     

Diffraction of surface waves by floating elastic plates

Based on the dynamical theories of water waves and Mindlin thick plates, the diffraction of surface waves by a floating elastic plate is presented by using the Wiener–Hopf technique. Firstly, the problem is related to a wave guide in water of finite depth, which is analysed to determine the poles. The resulting hybrid boundary value problem is reduced to solving an infinite system of linear algebraic equations. The results obtained are compared with those calculated by an alternative analysis, and with experimental data. Finally, the effects of the geometric and physical parameters on the distribution of deflection and bending moments in plates are analysed and discussed.     

Resonant effects of local loads on circular sandwich plates on an elastic foundation

The axisymmetric forced vibrations of a circular sandwich plate on an elastic foundation are studied. The plate is subjected to axisymmetric surface and mechanical loads with frequency equal to one of the natural frequencies of the plate. The foundation reaction is described by the Winkler model. To describe the kinematics of an asymmetric sandwich, the hypothesis of broken normal is used. The core is assumed to be light. The analytical solution of the problem is obtained and numerical results are analyzed     

Dispersion of elastic guided waves in piezoelectric infinite plates with inversion layers

In this work, we study the dispersion of elastic waves in piezoelectric infinite plates with ferroelectric inversion layers. The motivation is to analyze the effect of ferroelectric inversion layers on wave dispersion and resonant behavior under impulsive line loads. A semi-analytical finite-element (SAFE) method has been adopted to analyze the problem. Two model problems are considered for analysis. In one, the plate is composed of a layer of 36° rotated y-cut LiNbO₃ with a ferroelectric inversion layer. In the other, material is PZT-4 with a ferroelectric inversion layer. Comparison with experimental results, reported in the literature for isotropic materials, shows a very good agreement with theoretical predictions obtained using SAFE method. Furthermore, comparison of the resonance frequencies of the S₁ modes, calculated using KLM approximation (f₀ = C_d/2h) and SAFE method, are illustrated for each problem. The frequency spectra of the surface displacements show that resonant peaks occur at frequencies where the group velocity vanishes and the phase velocity remains finite, i.e., a minimum in the dispersion curve below the cut-off frequency. The effect of the ratio of the thicknesses of the inversion layer (IL) and the plate on the frequencies and strength of the resonant peaks is examined. It is observed that for PZT-4 with 50% IL to plate thickness ratio the frequency for the second resonant peak is about twice that for the first one. Results are presented showing the dependence of resonant frequencies on the material properties and anisotropy. Materials selection for single-element harmonic ultrasound transducers is a very important factor for optimum design of transducers with multiple thickness-mode resonant frequencies. The theoretical analysis presented in this study should provide a means for optimum ultrasound transducer design for harmonic imaging in medical applications.     

Surface adapted partial waves for the description of elastic vibrations in bilayered plates

An approach is proposed in which the form of the partial waves in the decomposition of the displacement field is chosen in such a way that boundary conditions on the outer surfaces of a layered structure are matched exactly. So found basis functions are further used to solve for the remaining boundary conditions. Explicit analytic expressions for the vibration spectrum and normalized amplitudes of a free plate in the long wavelength regime are derived and their behavior in the full space of material parameters is discussed. Thus, it is found that properties of the fundamental flexural mode change in a strongly non-monotonous way. In particular, by increasing the thickness of added layer one may observe the same propagation velocity for up to three different values of layer thicknesses ratio. Existence of an important criterion based on the relative signs taken by the amplitudes on the outer surfaces in this regime is revealed. It allows to establish a clear correspondence between symmetric and antisymmetric solutions for a single layer and those for a bilayered plate. It is suggested that the criterion is not limited to case of two layers. Branches of the spectrum are classified accordingly and their evolution from long to short wavelength is discussed. Analysis of a specific example which includes volume and surface modes (Rayleigh and Stoneley) is presented. The results demonstrate the efficiency of the proposed scheme and possible extensions are suggested.     

Three-dimensional modeling of elastic guided waves excited by arbitrary sources in viscoelastic multilayered plates

This paper provides a modal solution for the three-dimensional modeling of Lamb and SH waves excited by sources of arbitrary shape. This solution is applicable to elastic and viscoelastic plates, in the far-field as well as in the near-field regions, under the assumption of transverse isotropy about the thickness direction. The theoretical developments are conducted based on a semi-analytical finite element formulation. This formulation yields a one-dimensional modal problem, fast from a computational point of view, and allows to readily handle heterogeneous materials having depth-varying properties (multilayered, piecewise or continuously varying, functionally graded). The modal solution is shown to be expressed in terms of Hankel functions of multiple order thanks to a proper application of inverse transforms and Cauchy residue calculus. The link between the proposed formulation and a fully analytical approach is discussed. The solution of this paper is then successfully compared to literature results and degenerates to the point source case. Formulas are presented to calculate point source excitabilities from lines sources. These formulas remain valid for non-propagating modes, viscoelastic materials and account for the near-field contribution. Finally, the example of a viscoelastic bilayer waveguide excited by a rectangular source is considered in order to check the theoretical results.     

Thermal buckling and post-buckling response of imperfect temperature-dependent sandwich FGM plates resting on elastic foundation

Post-buckling behaviour of sandwich plates with functionally graded material (FGM) face sheets under uniform temperature rise loading is considered. It is assumed that the plate is in contact with a Pasternak-type elastic foundation during deformation, which acts in both compression and tension. The derivation of equations is based on the first-order shear deformation plate theory. Thermomechanical non-homogeneous properties of FGM layers vary smoothly by the distribution of power law across the thickness, and temperature dependency of material constituents is taken into account. Using the non-linear von-Karman strain-displacement relations, the equilibrium and compatibility equations of imperfect sandwich plates with FGM face sheets are derived. The boundary conditions for the plate are assumed to be simply supported in all edges. The governing equations are reduced to two coupled equation in terms of stress function and lateral deflection. Employing the single mode approach combined with Galerkin technique, an approximate closed-form solution is presented to calculate the critical buckling temperature and post-buckling equilibrium path of the plate. Presented numerical examples contain the influences of power law index, sandwich plate geometry, geometrical imperfection, temperature dependency, and the elastic foundation coefficients.     

PMN-PT夹层弹性半空间结构中SH波的传播Propagation of SH waves in sandwich structures consisting of PMN-PT single crystal layer and elastic half-spaces

分析了弹性上下半空间和PMN‐PT单晶层组成的夹层结构中SH波的传播性质,PMN‐PT单晶沿[011]c方向极化,宏观上呈mm2对称,且晶体沿角度θ方向切割。基于正交各向异性压电材料和各向同性弹性材料的基本方程,得到了夹层结构中SH波传播时行列式形式的频散方程。通过对数值算例进行分析可以看出,PMN‐PT单晶的切割角度和弹性材料属性对结构中的相速度有很大影响,因此波的某些传播性能可以通过材料的设计以及晶体切割的方向来实现,这些结论为声表面波器件的开发和应用提供了理论依据。     

Numerical modeling of elastic waves in micropolar plates and shells taking into account inertial characteristics

Mathematical models of micropolar plates and shells are considered within the framework of the approximation approach. The governing equations of the theories are written in a thermodynamically consistent form of the conservation laws. This ensures hyperbolicity and correctness of the initial boundary value problems. For numerical solution, we propose parallel algorithms for supercomputers with graphics processing units. The algorithms are based on the splitting method with respect to spatial variables. We present the results of numerical computations of wave propagation in micropolar rectangular plates and cylindrical panels for media with different types of microstructure particles.     

Wave propagation in elastic plates

The problem of wave propagation within the framework of a complete linear theory of elastic plates is treated using the method of wave curves. A complete classification of the various extensional and bending waves is obtained, along with the corresponding speeds of propagation. These are shown to correspond to the phase velocities of harmonic waves for infinite wave number. The decay and coupling equations are found, and the problem of waves due to a punch applied to the plate surface is treated.

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Résumé On a étudié le problème de la propagation des ondes en utilisant la méthode des courbes ondiales et assumant la théorie linéaire et complète des plaques minces. On obtient une classification totale des ondes diverses, de l'extension et de la flexion, avec des vitesses de la propagation. On prouve aussi, que celles-ci correspondent avec des vitesses des phases de la propagation, en cas de nombre infinite des ondes. Les équations de la décadence et de l'accouplement sont derivées, et on a étudié le problème d'une onde, qui est produite par une emporte-pièce appliquée sur la surface d'une plaque.

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This work was supported in part by funds from the National Research Council of Canada, under Grant Number A3805.     

Nonlinear analysis of skewed sandwich plates

Nonlinear static and dynamic behaviours of freely supported Rhombic sandwich plates have been studied following Banerjee's hypothesis. Numerical results for 0° skew angle are compared with other known results. Results for other skew angles are believed to be new.

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Sommario Si studia, seguendo l'ipotesi di Banerjee, il comportamento nonlineare statico e dinamico di piastre rombiche semplicemente appoggiate. Si presentano risultati numerici relativi a piastre rombiche e rettangolari: questi ultimi vengono paragonati a risultati già noti, mentre i primi si ritengono nuovi.

     

Nonlinear bending of circular sandwich plates

In this paper, fundamental equations and boundary conditions of nonlinear axisymmetrical bending theory for the circular sandwich plates with a soft core are derived by means of the method of calculus of variations. Especially in the case of very thin faces, the preceding fundamental epuations and boundary conditions simplity considerably. For example, a circular sandwich plate with edge clamped but free to siip under the action of uniform lateral load is considered. A more accurate solution of this problem has been obtained by means of the modified iteration method.Notation r, ,z system of cylindrical coordinates - a radius of plate boundary - t thickness of the face - h thickness of the core - h ₀ distance from middle of thickness of lower face to middle of thickness of upper face - E Young's modulus of the face - Poisson's ratio of the face - G ₂ shear modulus of the core - D _f flexural rigidity of the face - D flexural rigidity of the plate - C shear rigidity of the plate - q uniform lateral load - u _i ,v _i ,w _{i(i=1, 2, 3)} radial, tangential and normal displacement of upper face, core and lower face, respectively - u radial displacement of the middle plane of the plate - w deflection of the middle plane of the plate - rotation of connecting line of corresponding points in middle planes of two faces - _1i , _i , _zi , _ri , _zi , _{rzi (i=1,2,3)} strains at a point of upper face, core and lower face - _ri , _i , _zi , _ri , _zi , _rzi(i=1,2,3) stresses at a point of upper face, core and lower face - _r , ₀ radial and tangential stress of the middle plane of the plate, respectively - U _i(i =1, 2, 3) strain energy of upper face, core and lower face, respectively - V work done by the external force - U total potential energy of the plate - M _r radial moment of the plate - Q _r shearing force of the plate - m radial moment of the face - stress function - dimensionless radial coordinate - k dimensionless characteristic parameter - W dimensionless deflection - W ₀ dimensionless center deflection - S _r ,S ₀ dimensionless radial and tangential stress, respectively - S _r (0),S ₀ (0) dimensionless radial and tangential stress at center, respectively - S ₀(1) dimensionless tangential stress at edge - P dimensionless uniform lateral load - A ₂,A ₃,B ₂,B ₃,a ₁,.....,a ₂, ₁, ₂,l _1,1,...l _{1 1,3},m ₁,...,m ₃₃,n ₀,₂,...n ₂₂,₆,R ₁,,...R ₃₃ auxiliary quantity - L differential operator     

Large deflection of rectangular sandwich plates

The governing differential equations and the boundary conditions for the large deflection of rectangular sandwich plates are derived using the principle of the complementary energy. The governing differential equations are transformed into systems of nonlinear algebraic equations using the finite difference method, and solved by successive iteration. For the purpose of illustration, deflection behavior of simply supported rectangular plates under uniform load is presented. The deflection behavior of plates with various values of shear rigidities and intensity of applied loads is studied. The change in the stress patterns of the face layers of the plate is also discussed.