Nonlinear behaviour of piezoceramics under weak electric fields: Part-I: 3-D finite element formulation

Piezoceramic materials exhibit different types of nonlinearities under different combinations of electric and mechanical fields. When excited near resonance in the presence of weak electric fields, they exhibit typical nonlinearities similar to a Duffing oscillator such as jump phenomena and presence of superharmonics in the response spectra. In order to model such nonlinearities, a nonlinear electric enthalpy density function (using quadratic and cubic terms) valid for a general 3-D piezoelectric continuum has been proposed in this work. Linear (i.e. proportional) and nonlinear damping models have also been proposed. The coupled nonlinear finite element equations have been derived using variational formulation. The classical linearization technique has been used to derive the linearized stiffness and damping matrices which helps in assembling the nonlinear matrices and solution of resulting nonlinear equation. The general 3-D finite element formulation is discussed in this paper. In a companion paper by Samal et al., numerical results on various typical examples are shown to match very well with the experimental observations.     

Non-linear longitudinal vibrations of piezoceramics excited by weak electric fields

Typical non-linear effects, e.g. dependence of the resonance frequency on the amplitude, superharmonics in spectra and a non-linear relationship between excitation voltage and vibration amplitude as well as jump phenomena are observed in experiments with piezoceramics excited at resonance by weak electric fields. These non-linear effects can be observed for both the piezoelectric 31- and the 33-effect. In contrast to the well-known non-linear effects exhibited by piezoceramics in the presence of strong electric fields, these effects are not described in detail in the literature.In this paper, we attempt to model these phenomena using an electric enthalpy density to capture the cubic-like effects observed in the experiments. The equations of motion for the system under consideration are derived via the Ritz method using Hamilton's principle. The ‘non-linear’ parameters are identified and the numerical results are compared to those obtained experimentally. The effects described herein may have a significant influence in structures excited close to resonance frequencies via piezoelectric elements.     

An analytical formulation in 3D domain for the nonlinear response of piezoelectric slabs under weak electric fields

Piezoceramic materials exhibit different types of nonlinearities depending upon the magnitude of the mechanical and electric field strength within the body. Some of the nonlinear phenomena observed under weak electric fields near resonance frequency excitation are the presence of superharmonics in the response spectra and the jump phenomena etc. In this work, an analytical solution for the nonlinear response of rectangular piezoceramic slabs have been obtained by Rayleigh–Ritz method and perturbation technique in the 3-D domain using a generalized nonlinear electric enthalpy density function. Forced vibration experiments (excitation with electric field) have been conducted on a rectangular piezoceramic slab at varying electric field amplitudes and the analytical solutions have been shown to compare very well with the experimental results.     

Elastic behaviour of shrink-fitted forming die assemblies. Part II: Numerical application and results

Summary A study is presented on the elastic behaviour of a set of assemblies consisting of a cylinder prestressed by a shrink-fitted ring of different length as is customary in extrusion toolsets. An accurate analysis is carried out of the individual and interactive effects pertaining both to the bulk support provided by the relatively unstressed parts of the die, and the external compressive stresses induced by radial support of the shrinking ring.Stresses and displacements in the two cylindrical components are evaluated by a numerical application of the three-dimensional elastic theory and the point-matching method.Again a discussion is presented of the limitations involved by the numerical approach.

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Sommario L'articolo riferisce sul comportamento elastico di accoppiamenti realizzati per forzamento di un anello di grosso spessore su un cilindro cavo di lunghezza finita e diversa da quello dell'elemento forzante, del tipo di quelli impiegati nelle attrezzature di estrusione.Si analizzano individualmente e nella loro interazione le influenze che su tale comportamento hanno l'irrigidimento fornito da quelle porzioni dell'anello interno che sono relativamente scariche di tensioni e la pressurizzazione esterna indotta dall'elemento forzante.Il calcolo delle tensioni e degli spostamenti nei due componenti è operato mediante un'applicazione numerica della teoria dell'elasticità tridimensionale e del metodo delle collocazioni.Si discutono inoltre le limitazioni connaturate con l'approccio di tipo numerico.

     

Calculation of some magnetic and electric fields with cylindrical symmetry

Summary It is assumed that the magnetic and electric fields have cylindrical symmetry and at the same time mirror symmetry with respect to a median plane. The field strength in the median plane is given. The problem is to calculate the field at an arbitrary point in space, the equipotential surfaces, and the form of the polepieces of magnets. Solutions are given in the form of power expansions. Analytical solutions are given when the field strength in the median plane as a function of the radius is a polynomial.     

Interaction of weak free-stream disturbance with an oblique shock:validation of the shock-capturing method

Transition prediction is of great importance for the design of long distance flying vehicles. It starts from the problem of receptivity, i.e., how external disturbances trigger instability waves in the boundary layer. For super/hypersonic boundary layers,the external disturbances first interact with the shock ahead of the flying vehicles before entering the boundary layer. Since direct numerical simulation(DNS) is the only available tool for its comprehensive and detailed investigation, an important problem arises whether the numerical scheme, especially the shock-capturing method, can faithfully reproduce the interaction of the external disturbances with the shock, which is so far unknown.This paper is aimed to provide the answer. The interaction of weak disturbances with an oblique shock is investigated, which has a known theoretical solution. Numerical simulation using the shock-capturing method is conducted, and results are compared with those given by theoretical analysis, which shows that the adopted numerical method can faithfully reproduce the interaction of weak external disturbances with the shock.     

Interaction of a jet of plasma with electric and magnetic fields

Summary A generalized ohm's law is used to predict changes in the direction and magnitude of the momentum of a stream of plasma in a crossed field accelerator in terms of plasma properties and species temperatures and concentrations. Measurements are reported of accelerator thrust, plasma jet deflection angle, current-voltage characteristics, accelerator inlet momentum, cooling rates and cross-arc power input for argon and helium plasmas. Estimates are given of the mean values of some of the parameters of argon and helium plasma obtained under nonequilibrium conditions. These results are compared with other methods requiring the assumption of LTE.This work was supported in part by the U.S. Air Force through the Propulsion Division, Directorate of Engineering Sciences, Office of Aerospace Research, AFOSR, under contract AF 49(638)-1445.     

Nonlinear dynamic analysis of Timoshenko beams by BEM. Part II: applications and validation

In this two-part contribution, a boundary element method is developed for the nonlinear dynamic analysis of beams of arbitrary doubly symmetric simply or multiply connected constant cross section, undergoing moderate large displacements and small deformations under general boundary conditions, taking into account the effects of shear deformation and rotary inertia. In Part I the governing equations of the aforementioned problem have been derived, leading to the formulation of five boundary value problems with respect to the transverse displacements, to the axial displacement and to two stress functions. These problems are numerically solved using the Analog Equation Method, a BEM based method. In this Part II, numerical examples are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. Thus, the results obtained from the proposed method are presented as compared with those from both analytical and numerical research efforts from the literature. More specifically, the shear deformation effect in nonlinear free vibration analysis, the influence of geometric nonlinearities in forced vibration analysis, the shear deformation effect in nonlinear forced vibration analysis, the nonlinear dynamic analysis of Timoshenko beams subjected to arbitrary axial and transverse in both directions loading, the free vibration analysis of Timoshenko beams with very flexible boundary conditions and the stability under axial loading (Mathieu problem) are presented and discussed through examples of practical interest.

     

Dynamics of Geometrically Nonlinear Rods: II. Numerical Methods and Computational Examples

The paper selects and develops appropriate numerical solutionmethods for initial boundary value problems of the equations of motionof geometrically nonlinear extensible Euler–Bernoulli-beams. A finiteelement method that uses first-order Hermitian polynomials as interpolationfunctions for the rod axis position vector is used as discretizationtechnique. An averaging method for the calculation of net forces andmoments is developed that achieves a better approximation than thedirect calculation from the strains. Time integration is done usingan energy and momentum conserving algorithm that is proposed in thispaper and Newmark type methods. The derived algorithms are used tosolve problems from space and marine engineering. The obtained simulationresults are compared with results which have been already publishedin the literature or were calculated by different methods.     

Acousto-mechanical behaviour of ex-vivo skin: Nonlinear and viscoelastic properties

The mechanical behaviour of skin is significant for some applications including dermatology, surgery, and impact biomechanics science. In this work, we have investigated the study of the acousto-mechanical viscoelastic properties of skin. For that, both tensile-relaxation and ultrasonic tests were conducted on porcine tissue samples in fibre directions. To understand the complex skin aging phenomena, we used strength tensile test correlated with the Nonlinear Time Reversal signal processing tool extension “TR-NEWS”. Uniaxial tensile tests were carried out at a strain rate of $5?{10}^{?3}\text{ mm}{\text{s}}^{?1}$ on skin using a load-relaxation-discharge load path with increasing amplitude and offset. This work is also under way to extend the frequency range of ultrasounds to 50 MHz. Digital Image Correlation was used for 2D strain measurement of the dermis. From this analysis, we conclude that fresh porcine skin should be modelled as a nonlinear viscoelastic material with strain-rate dependence. The obtained hysteresis loop shall be taken as significant skin damage.     

Modeling of bubble detachment in reduced gravity under the influence of electric fields and experimental verification

A simple model for predicting bubble volume and shape at detachment in reduced gravity under the influence of electric fields is described in the paper. The model is based on relatively simple thermodynamic arguments and relies on and combines several models described in the literature. It accounts for the level of gravity and the magnitude of the electric field. For certain conditions of bubble development the properties of the bubble source are also considered. Computations were carried out for a uniform unperturbed electric field for a range of model parameters, and the significance of model assumptions and simplifications is discussed for the particular method of bubble formation. Experiments were conducted in terrestrial conditions and reduced gravity (during parabolic flights in NASAs KC-135 aircraft) by injecting air bubbles through an orifice into the electrically insulating working fluid, PF5052. Bubble shapes visualized experimentally were compared with model predictions. Measured data and model predictions show good agreement. The results suggest that the model can provide quick engineering estimates concerning bubble formation for a range of conditions (both for formation at an orifice and boiling) and such a model reduces the need for complex and expensive numerical simulations for certain applications. a Major axis of spheroid (m) - a _m Measured bubble height (m) - b Minor axis of spheroid (m) - b _m Measured bubble width (m) - A, B, C, F Parameters of the Kumar-Kuloor model - a/b Computed aspect ratio - a _m /b _m Measured aspect ratio - D Orifice diameter (m) - E Magnitude of the electric field (V/m) - g Gravitational acceleration (m/s²) - g _t Terrestrial gravity (g _t = 9.81 m/s²) - N _w Electrical Weber number - p Pressure (Pa) - Q Volume flow rate (m³/s) - r Radius of the spherical bubble (m) - R Radius of curvature at the tip of the bubble (m) - t Time (s) - t Time interval (s) - T Temperature (°C) - U Electrical potential (V) - u Velocity (m/s) - V Volume (m³) - x, y Dimensionless coordinates of the Cartesian coordinate system - x, y Scaled coordinates, Cheng-Chaddock model - X, Y Dimensional coordinates of the Cartesian coordinate system - Characteristic wave number (m^–1) - Eötvös number - Absolute dielectric permittivity (F/m) - Contact angle (deg.) - Gibbs free energy (J) - Surface tension (N/m) - Dynamic viscosity (Pa s) - Density (kg/m³) - cr Critical value - d Detachment - eq Equilibrium - g Gas - K Refers to the Kumar-Kuloor model - l Liquid - m Measured value - t Terrestrial     

Energy spectra and bubble velocity distributions in pseudo-turbulence: Numerical simulations vs. experiments

Direct numerical simulations (DNS) are performed to study the behavior of a swarm of rising air bubbles in water, employing the front tracking method, which allows to handle finite-size bubbles. The swarms consist of monodisperse deformable 4 mm bubbles with a gas fraction of 5% and 15%. This paper focuses on the comparison of the liquid energy spectra and bubble velocity probability density functions (PDFs) with experimental data obtained by phase-sensitive constant-temperature anemometry (CTA) and three-dimensional particle tracking velocimetry (PTV), respectively.     

Alterations in peristaltic pumping of Jeffery nanoliquids with electric and magnetic fields

The combined effects of electric and magnetic fields on peristaltic flow of Jeffery nanoliquids are analytically investigated. Double-diffusive convection in the asymmetric microchannel is also carried out. The walls of the microchannel are propagating with a finite phase difference in a sinusoidal manner. Rosseland diffusion flux model is employed to examine the thermal radiation effect. The zeta potential on the walls is considered very low to apply Hückel–Debye approximations. The coupled non-linear governing equations are simplified by using dimensional analysis and lubrication theory. The closed form solutions for potential function, nanoparticle fraction field, solute concentration field, temperature field, stream function, and axial velocity are derived under the appropriate boundary conditions. It is noteworthy that the pumping characteristics strongly depend on the magnetic fields, electric fields, electric double layer thickness, Jeffery parameter, thermal radiation and Grashof number. Furthermore, trapping phenomenon is analyzed under the effects of Hartmann number, Jeffrey parameter, Grashof number and Helmholtz–Smoluchowski velocity. The novelty of the present work is the amalgamation of biomimetics (peristaltic propulsion), electro-magneto-hydrodynamics and nanofluid dynamics to produce a smart pump system model for smart drug delivery systems.     

Shape recovery behaviour of NiTi strips in bending: experiments and modelling

We present a theoretical and experimental investigation of the bending recovery performances for a commercial NiTi shape memory alloy strip. We evaluate the mechanical properties and the shape setting parameters and estimate the evolution of the curvature during heating in an Ethylene Glycol-based water solution. To model the strip bending response, we use a one-dimensional phenomenological constitutive equation for the shape memory material, based on the introduction of (twinned and detwinned) martensite and austenite volume fractions as internal variables. Under the assumption of uniform bending, we calculate a quasi-closed-form solution for the stress and martensite fraction distributions in a shape memory beam during bending and subsequent shape recovery. Using our characterisation data as input parameters of the model, we find that the theoretical curvature evolution is in good agreement with experimental data.     

Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments

This paper presents a modified regularized formulation of the Ambrosio-Tortorelli type to introduce the crack non-interpenetration condition in the variational approach to fracture mechanics proposed by Francfort and Marigo [1998. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (8), 1319-1342]. We focus on the linear elastic case where the contact condition appears as a local unilateral constraint on the displacement jump at the crack surfaces. The regularized model is obtained by splitting the strain energy in a spherical and a deviatoric parts and accounting for the sign of the local volume change. The numerical implementation is based on a standard finite element discretization and on the adaptation of an alternate minimization algorithm used in previous works. The new regularization avoids crack interpenetration and predicts asymmetric results in traction and in compression. Even though we do not exhibit any gamma-convergence proof toward the desired limit behavior, we illustrate through several numerical case studies the pertinence of the new model in comparison to other approaches.     

Nonlinear characteristic of a circular composite plate energy harvester: experiments and simulations

Conservative chaotic systems are rare, especially autonomous smooth dynamical systems. This paper reports two four-dimensional (4D) autonomous conservative systems. The conservation of these two systems has been verified using the trace of Jacobian matrix, perpetual point theory and Hamiltonian energy theory. Numerical analyses, including phase portrait, Poincaré section, Lyapunov exponent spectrum and bifurcation diagram, verify the existence of the chaotic and quasiperiodic flows. Moreover, a electronic circuit in Multisim is built to demonstrate their chaotic dynamics, whose circuit experimental results agree well with the numerical results.     

Numerical investigation of heat and mass transfer processes accompanying crystallization under conditions when body forces produce weak accelerations

A numerical investigation has been made into the development of the flow of a liquid and distribution of an dopant during crystallization under conditions when weak body forces act. It is shown that the development and structure of the flow, and also the distribution of the dopant over the height of the liquid volume and along the radius near the crystallization boundary are influenced by the thermal boundary conditions and the wettability of the ampoule wall by the liquid. The mean (over the height of the ampoule) radial inhomogeneity of the dopant distribution is found as a function of the time and the regime parameters (the Grashof, Marangoni, and Schmidt numbers).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 186–192, January–February, 1981.