Non-linear axisymmetric transient analysis of orthotropic thin annular plates with a rigid central mass

The geometrically non-linear, axisymmetric transient elastic response is determined of cylindrically orthotropic thin annular plates with a rigid central mass subjected to a uniformly distributed load on the plate as well as a central load on the rigid mass. The dynamic analogue of the von Kármán equations in terms of the normal displacement w and the stress function Ψ are employed. The displacement w and stress function Ψ are expanded in finite power series and the orthogonal point collocation method in the space domain and the Newmark β scheme in the time domain are used. The response of isotropic and orthotropic, clamped as well as simply supported, annular plates with a rigid central mass, subjected to step function and sinusoidal pulse loads, has been calculated for two values of the annular ratio. The influence of the mass ratio and the magnitude of the step load on the deflection response has been determined. The effect of mass ratio, amplitude and duration of sinusoidal pulse on the deflection response has also been studied.     

On the non-linear vibrations of rectangular plates

A solution, based on a one-term mode shape, for the large amplitude vibrations of a rectangular orthotropic plate, simply supported on all edges or clamped on all edges for movable and immovable in-plane conditions, is found by using an averaging technique that helps to satisfy the in-plane boundary conditions. This averaging technique for satisfying the immovable in-plane conditions can be used to resolve many anisotropic and skew plate problems where otherwise, when a stress function is used, the integration of the u and v equations becomes difficult, if not impossible. The results obtained herein are compared with those available in the literature for the isotropic case and excellent agreement is found. Results available for the one-term mode shape solutions of these problems are compared and the non-linear effect is presented as functions of aspect ratio and of the orthotropic elastic constants of the plate. The results are further compared with those based on the Berger method and the detailed comparative studies show that the use of the Berger approximation for large deflection static and dynamic problems and its extension to anisotropic plates, skew plates, etc., can lead to quite inaccurate results.     

Effects of large amplitude and transverse shear on vibrations of triangular plates

In this paper an analytical investigation of large amplitude free flexural vibrations of isotropic and orthotropic moderately thick triangular plates is carried out. The governing equations are expressed in terms of the lateral displacement, w, and the stress function, F, and are based on an improved non-linear vibration theory which accounts for the effects of transverse shear deformation and rotatory inertia. Solutions to the governing equations are obtained by using a single-mode approximation for w, Galerkin's method and a numerical integration procedure. Numerical results are presented in terms of variations of non-linear frequency ratios with amplitudes of vibrations. The effects of transverse shear, rotatory inertia, material properties, aspect ratios, and thickness parameters are studied and compared with available solutions wherever possible. Present results are in close agreement with those reported for thin plates. It is believed that all of the results reported here that are applicable for moderately thick plates are new and therefore, no comparison is possible.     

Small-scale effect on the vibration of thin nanoplates subjected to a moving nanoparticle via nonlocal continuum theory

The vibration of elastic thin nanoplates traversed by a moving nanoparticle involving Coulomb friction is investigated using the nonlocal continuum theory of Eringen. The eigen function technique and the Laplace transform method are employed to solve the governing equations of the nanoplate. The explicit expressions of the in-plane and transverse displacements are obtained when the moving nanoparticle traverses the nanoplate on an arbitrary straight line. In a special case, the obtained results are also compared with those of other researchers and a reasonably good agreement is achieved. The effects of small-scale parameters and velocity of the moving nanoparticle on the dynamic response as well as the dynamic amplitude factors (DAFs) of the in-plane and transverse displacements are then explored in some detail. The results indicate that the magnitude of DAF of the transverse displacement of the nanoplate (i.e., DAF_w) increases with the first small-scale effect parameter, irrespective of the values of the second small-scale effect parameter and the velocity of the moving nanoparticle. As the first small-scale effect parameter grows, the maximum values of DAF_w as a function of the moving nanoparticle velocity increase and generally occur in the lower levels of the moving nanoparticle velocity.     

Non-linear axisymmetric vibration of orthotropic thin circular plates on elastic foundations

This study deals with the large amplitude axisymmetric free vibrations of cylindrically orthotropic thin circular plates resting on elastic foundations. Geometric non-linearity due to moderately large deflections has been included. Movable and immovable simply supported plates and immovable clamped plates resting on Winkler, Pasternak and non-linear Winkler foundations have been considered. The von Kármán type governing equations have been employed. Harmonic vibrations are assumed and the time t is eliminated by the Kantorovich averaging method. An orthogonal point collocation method is used for spatial discretization. Numerical results are presented for the linear natural frequency of the first axisymmetric mode and for the ratio of the non-linear period to the linear period of natural vibration. The effects of foundation parameters, the orthotropic parameter and the edge conditions on the non-linear vibration behaviour have been investigated.     

Influence of domain boundary width on the statics of 90° domains in epitaxial ferroelectric thin films

The influence of the domain boundary width on the statics of single 90° elastic domains (twins) in epitaxial ferroelectric tetragonal films grown on a cubic substrate is theoretically investigated. The inhomogeneous internal stresses arising in polydomain epitaxial systems are calculated by the effective dislocation method. The elastic energy stored in the heterostructure is determined. The equilibrium domain size is found and the stability diagram for single domains at different wall widths is constructed by minimizing the total internal energy of the system. It is demonstrated that, as the domain boundary width 2w increases, the stability region of 90° domains increases and qualitatively changes for ultrathin films when the parameter 2w exceeds the specific critical value 2w _cr. The equilibrium width 2w* of domain walls in thin films is predicted to be larger compared to the width 2w ₀ of domain boundaries in a macroscopic crystal.     

A note on the vibrations of rectangular plates of variable thickness with two opposite simply supported edges and very general boundary conditions on the other two

An approximate solution for the title problem is obtained by making use of the Galerkin method. The plate displacement function is approximated by means of a sinusoid multiplied by a polynomial. Translational and rotational flexibilities are taken into account at x = ±a/2. It is shown that the free edge situation (Kirchoff's boundary condition) can be treated as a special case by means of the approach developed herein. A simple algorithm which allows evaluation of the fundamental frequency of vibration is derived and rough estimates of amplitudes and stress resultants are also given when the plate is subjected to a p₀cosωt-type excitation.     

Analyzing in-plane magnetic anisotropy from surface morphology for amorphous films

A method is developed to analyze the in-plane magnetic anisotropy from surface morphology for amorphous films. The lateral sizes along radial direction (R_RD) and tangent direction (R_TD) of rotational substrate, which are extracted from the surface morphology of Co_66.3Zr_33.7 amorphous films, are used to calculate stress anisotropy energy E_σ. It is found that E_σ is consistent with the magnetic anisotropy energy K_μ for the samples deposited on Si (1 0 0) substrate and then a relationship K_μ ∝ 1/R_RD − 1/R_TD can be obtained. This method is sensitive to the initial state of substrate so its application range is discussed.     

Large amplitude flexural vibration of eccentrically stiffened plates

The large amplitude free flexural vibrations of thin, orthotropic, eccentrically and lightly stiffened elastic rectangular plates are investigated. Clamped boundary conditions with movable in-plane edge conditions are assumed. A simple modal form of one-term transverse displacement is used and in-plane displacements are made to satisfy the in-plane equilibrium equations. By using Lagrange's equation, the modal equations for the nonlinear free vibration of stiffened plates are obtained for the cases when the stiffeners are assumed to be smeared out over the entire surface of the plate, and when the stiffeners are located at finite intervals. Numerical results are obtained for various possibilities of stiffening and for different aspect ratios of the plate. By particularizing the problem to different known cases, the results obtained here are compared with available analytical and experimental results, and the agreement is good.     

The effect of an oblique magnetic field on the superparamagnetic relaxation time

The effect of a constant magnetic field, applied at an angle ψ to the easy axis of magnetization, on the Néel relaxation time τ of a single domain ferromagnetic particle with uniaxial anisotropy is investigated by calculating the smallest non-vanishing eigenvalue of the appropriate Fokker-Planck equation as a function of ψ. The curve of λ₁ versus ψ is symmetric about ψ = 90°. Thus the maximum decrease in τ occurs at ψ = 45° with maximum increase to a value exceeding that at ψ = 0, at ψ = 90°, the ψ = 0 value being again attained at ψ = 180°. The results are shown to be consistent with the behaviour predicted by the Kramers theory of the rate of escape of particles over potential barriers. This theory when applied to the potential barriers for the ψ = 90° orientation for rotation in space yields a simple approximate formula for the escape rate which is in reasonable agreement with the exact λ₁ calculated from the Fokker-Planck equation.     

Combined effects of the in-plane orientation angle and the loading angle on the dynamic enhancement of honeycombs under mixed shear-compression loading

The combined effect of the loading angle (ψ) and the in-plane orientation angle (β) on the dynamic enhancement of aluminium alloy honeycombs is investigated. Experimental results are analysed on the crushing surfaces (initial peak and average crushing forces). A significant effect of the loading angle is reported. The dynamic enhancement rate depends on the loading angle until a critical loading angle (ψ_critical). Beyond, a negative dynamic enhancement rate is observed. Concerning the in-plane orientation angle β effect, it depends on the loading angle ψ under quasi-static conditions. Under dynamic conditions, a significant effect is reported independently of the loading angle ψ. Therefore, the dynamic enhancement rate depends on the combined effects of ψ and β angles. A global analysis of the buckling mechanisms allowed us to explain the combined effect of ψ and β angles on the initial peak force. The collapse mechanisms analysis explain the negative dynamic enhancement rate for large loading angles.     

Properties of amorphous CoZr(RE) (RE=Gd,Sm, Dy) films with various uniaxial anisotropies,prepared by a new process

If a DC magnetic field is applied parallel to the plane of amorphous CoZr(RE) thin films during sputter depositing, a uniaxial anisotropy is formed the direction of which depends upon the choice of RE substituted and its concentration. When RE=Gd a perpendicular anisotropy K_p forms over a large concentration range, a spin reorientation process being at the origin of the process. A well-defined K_p is developed also in CoFeZrGd and CoZrGdSm films. CoZrGdDy films exhibit simultaneously a perpendicular and an in-plane uniaxial anisotropy. The related magnetization process and domain structures are quite peculiar.     

Optimum designs for multi-layered film structures based on the knowledge on residual stresses

Residual stresses are inevitably generated within the multi-layered film structures due to the mismatches of material properties between the adjacent layers. Using the force and moment equilibrium conditions and beam bending theory, the residual stresses in each layer can be predicted and expressed as σ_i(z) = E_i[?′ + K(z + δ)], where E_i is the elastic modulus of the layer, ?′ the strain due to the in-plane force resulting from the misfit strain, K(z + δ) characterizes the bending contribution. For a bilayer system, the expression of the residual stress in the film is relatively simple. If the each layer thickness is much less than the substrate thickness, Stoney's equation will be derived. The assumption of a constant elastic modulus throughout the system is only applicable when the film and the substrate thickness ratio is less than 0.1. Specific analyses are performed for the thermal stresses in ZrO₂/NiCoCrAlY thermal barrier coatings (TBCs) to illustrate the implementation of the analytical model. Moreover, the effects of single interlayer and graded interlayer inserted between the metallic layer and the ceramic layer on the residual stress distributions in TBCs are investigated. Additionally, the zero-deflection design is also discussed for typically duplex-layer TBC system.     

Observation and theory of strong stripe domains in amorphous sputtered FeB films

Domain structures in thin sputtered amorphous FeB films are studied by means of the longitudinal Kerr effect. In addition to the irregular domain structure characteristic of soft magnetic materials, we observe in certain regions a fine equilibrium domain structure with periodicity of a few micrometers. The Kerr contrast indicates that the magnetization at the surface of the film lies partially along the stripe direction. These characteristics and the behavior in applied fields suggests that the domains are similar to type II “strong stripe domains” observed earlier in permalloy films. Extending an earlier theory by Hara, we use a stray-field-free model with tilted orthorhombic anisotropy to show that there are at least two qualitatively different strong stripe structures: type IIa with surface magnetization perpendicular to the stripes and type IIb with surface magnetization at least partially parallel to the stripes. Type IIb is favored when K_p/K₀<cos 2θ ₀ where K₀ is the anisotropy component with axis tilted by θ₀ out of the film plane, and K_p is an in-plane anisotropy perpendicular to K₀. Strong stripes in amorphous FeB appear to be type IIb while those in permalloy are usually type IIa.     

A new procedure for investigating three-dimensional stress fields in a thin plate with a through-the-thickness crack

In the paper, a new procedure is proposed to investigate three-dimensional fracture problems of a thin elastic plate with a long through-the-thickness crack under remote uniform tensile loading. The new procedure includes a new analytical method and high accurate finite element simulations. In the part of theoretical analysis, three-dimensional Maxwell stress functions are employed in order to derive three-dimensional crack tip fields. Based on the theoretical analysis, an equation which can describe the relationship among the three-dimensional J-integral J(z), the stress intensity factor K(z) and the tri-axial stress constraint level T _z (z) is derived first. In the part of finite element simulations, a fine mesh including 153360 elements is constructed to compute the stress field near the crack front, J(z) and T _z (z). Numerical results show that in the plane very close to the free surface, the K field solution is still valid for in-plane stresses. Comparison with the numerical results shows that the analytical results are valid.     

Large amplitude flexural vibration of stiffened plates

The large amplitude free flexural vibration of thin, elastic orthotropic stiffened plates is studied. The boundary conditions considered are either simply supported on all edges or clamped on all edges and the in-plane edge conditions are either immovable or movable. The governing dynamic equations are derived in terms of non-dimensional parameters describing the stiffening achieved, and the solutions are obtained on the basis of an assumed one-term vibration mode shape for various stiffener combinations. In all cases, the non-linearity is found to be of the hardening type (i.e., the period of non-linear vibration decreases with increasing amplitude). Some interesting conclusions are drawn as to the effect of the stiffening parameters on the non-linear behaviour. A simple method of predicting the postbuckling and static large deflection behaviour from the results obtained in this analysis is included.     

Electromechanical properties of Co0.6 Zn0.4 Fe2O4 ferrite transducer

Co_0.6Zn_0.4Fe₂O₄ ferrite has been prepared by the general ceramic method. The resonance frequency of the disk for the radial and thickness mode was investigated as a function of mechanical stress. The increase of the resonance frequency of the transducer is attributed to domain wall motion inhibition. The increase of ultrasonic velocity with increasing mechanical stress can be explained as due to retarding the oscillation of the ferrimagnetic domains under applied high-frequency electric field. The induced strain increases the resonance frequency. The high values of the electromechanical coupling factor for the radial mode K_p, and thickness mode, K_t, indicate that this composition is useful for producing high-frequency ultrasonic waves that can be used in modern technology.     

SUPER AND COMBINATORIAL HARMONIC RESPONSE OF FLEXIBLE ELASTIC CABLES WITH SMALL SAG

The paper deals with the analysis of cables in stayed bridges and TV-towers, where the excitation is caused by harmonically varying in-plane motions of the upper support point with the amplitude U. Such cables are characterized by a sag-to-chord-length ratio below &0uml;02, which means that the lowest circular eigenfrequencies for in-plane and out-of-plane eigenvibrations, ω₁and ω₂, are closely separated. The dynamic analysis is performed by a two-degree-of-freedom modal decomposition in the lowest in-plane and out-of-plane eigenmodes. Modal parameters are evaluated based on the eigenmodes for the parabolic approximation to the equilibrium suspension. Superharmonic components of the ordern , supported by the parametric terms of the excitation and the non-linear coupling terms, are registered in the response for circular frequency ω?ω₁/n. At moderate U, the cable response takes place entirely in the static equilibrium plane. At larger amplitudes the in-plane response becomes unstable and a coupled whirling superharmonic component occurs. In the paper a first order perturbation solution to the superharmonic response is performed based on the averaging method. For ω?(m/n)ω₁, m<n, the geometrical non-linear restoring forces gives rise to a substantial combinatorial harmonic component with the circular frequency (n/m)ω. Both entirely in-plane and coupled in-plane and out-of-plane responses occur. Based on an initial frequency analysis of the response, an analytical model for these vibrations is formulated with emphasis on superharmonics of the order n=3 and combinatorial harmonics of the order (n, m)=(3,2). All analytical solutions have been verified by direct numerical integration of the modal equations of motion.