We formulate and solve the mathematical problem for antiplane cracks in a pre-stressed and pre-polarized piezoelectric material with static initial fields, assuming the initially deformed configuration of the body is locally stable. Using the boundary conditions of antiplane cracks, we get the Riemann-Hilbert problems. Nonhomogeneous linear complex differential equations having the unknown complex potential are obtained. For constant value of the applied incremental forces can be obtained the complex potentials, incremental displacement and stress fields corresponding to the third mode of the classical fracture. The problem of interaction of two collinear, unequal cracks in a pre-stressed and pre-polarized piezoelectric material, is also studied.

Nonlinear formulation of isotropic piezoelectric Euler-Bernoulli nano-beam is developed based on consistent size-dependent piezoelectricity theory. By considering geometrically nonlinear and axial displacement of the centroid of beam sections, basic nonlinear equations of piezoelectric nanobeam are derived using Hamilton's principle and variational method. Afterwards, in the special case for the formulation derived, hinged-hinged piezoelectric nanobeam is studied, and static deflection as well as free vibrations of the nanobeam under mechanical loads is determined. In this case, results of the linear formulation of the size-dependent theory are compared to those of the linear and nonlinear classical continuum theory.

In this paper, the electro-elastic fields in a functionally gradient piezoelectric strip with an internal semi-infinite electrode are analyzed by using Fourier transform and Wiener-Hopf technique. The exact forms of asymptotic solutions and intensity factor and energy are obtained. The energy density criterion is proposed to study the fracture behavior near the electrode tip. The fracture initiation angle depends on the fracture resistance of the piezoelectric ceramic, bonding strength between piezoelectric and electrode, and the direction of least energy density factor S inside the piezoceramic.

We use a piezoelectric quartz tuning fork to calibrate the displacement of ceramic piezoelectric scanners that are widely employed in scanning probe microscopy. We measure the static piezoelectric response of a quartz tuning fork and find it to be highly linear, nonhysteretic and with negligible creep. These performance characteristics, close to those of an ideal transducer, make quartz transducers superior to ceramic piezoelectric actuators. Furthermore, quartz actuators in the form of a tuning fork have the advantage of yielding static displacements comparable to those of local probe microscope scanners. We use the static displacement of a quartz tuning fork as a reference to calibrate the three axis displacement of a ceramic piezoelectric scanner. Although this calibration technique is a nontraceable method, it can be more versatile than using calibration grids because it enables characterization of the linear and nonlinear response of a piezoelectric scanner in a broad range of displacements, spanning from a fraction of a nanometer to hundreds of nanometers. In addition, the creep and the speed dependent piezoelectric response of ceramic scanners can be studied in detail.

The present work aims at proposing a rigorous analysis of the mathematical and numerical modelling of ultrasonic piezoelectric sensors. This includes the well-posedness of the final model, the rigorous justification of the underlying approximation and the design and analysis of numerical methods. More precisely, we first justify mathematically the classical quasi-static approximation that reduces the electric unknowns to a scalar electric potential. We next justify the reduction of the computation of this electric potential to the piezoelectric domains only. Particular attention is devoted to the different boundary conditions used to model the emission and reception regimes of the sensor. Finally, an energy preserving finite element/finite difference numerical scheme is developed; its stability is analyzed and numerical results are presented.

Cet article présente une méthode de synthèse optimale pour la conception préliminaire de mécanismes flexibles monolithiques piézoélectriques. La construction d'un tel mécanisme, selon cette méthode baptisée FlexIn, est basée sur l'agencement de blocs flexibles piézoélectriques élémentaires qui peuvent être structurels, actionneurs et/ou capteurs, sélectionnés respectivement dans trois bibliothèques. Cet article décrit plus particulièrement l'approche utilisée pour établir le modèle aux éléments finis multi-physiques des blocs capteurs piézoélectriques, de type treillis de poutre, en vue de son intégration dans le logiciel de synthèse optimale. Les résultats obtenus présentent un écart de quelques pourcents avec ceux issus d'un code aux éléments finis commercial. À titre d'exemple, le modèle capteur est appliqué à une micropince monolithique existante; le résultat de mesure est d'un ordre de grandeur satisfaisant. Ces résultats permettent de dresser les perspectives de la méthode de synthèse optimale proposée dans le cadre général de la conception de structures intelligentes pour la microrobotique.

Les besoins accrus d'encapsulation et d'autonomie des dispositifs decontrôle vibratoire ont ouvert le domaine des matériauxpiézoélectriques shuntés. De par ses propriétésphysiques, l'élément piézoélectrique convertit unepartie d'énergie vibratoire en énergie électrique. Lefait de connecter un circuit électrique (un circuit de shunt)aux bornes du patch modifie la dynamique du système complet(structure + patchs piézoélectriques) et offre ainsi despossibilités de le contrôler. C'est ce que nous nommeronscontrôle vibratoire passif dans le cas où le circuit deshunt est constitué de composants passifs (Résistances etinductances physiques). Dans la littérature, les dispositifs employant desstratégies de contrôle se focalisent principalement surl'efficacité obtenue en termes de facteur d'amortissement modal.Les flux d'énergie entre les différents éléments dusystème complet ne sont pas optimisés. Nous proposons dedévelopper un critère d'optimisation structurale qui prenden compte la dissipation induite non seulement en termesd'amortissement modal mais également en termes d'amplituded'énergie dissipée. Ce critère intègre des grandeursfacilement calculables via un code de calcul par éléments-finis. Il est à noter que l'une des principales difficultésde modélisation de ce type de structures est liée à lacondition électrique qui varie. Il est nécessaire d'employerun modèle complet ou suffisamment représentatif pourcalculer les modes de résonance de la structure à tensionnulle (condition de Dirichlet électrique ) et à courant nul(condition de Neumann électrique). Nous appliquerons notrecritère à une poutre élancée sur laquelle estimplantée un patch piézoélectrique pour différentesconditions aux limites et différents types de polarisation. Nousnous focaliserons sur le contrôle du premier mode de flexion de lastructure.

The purpose of this paper is to derive and study a new asymptoticmodel for the equilibrium state of a thin anisotropicpiezoelectric plate in frictional contact with a rigid obstacle.In the asymptotic process, the thickness of the piezoelectricplate is driven to zero and the convergence of the unknowns isstudied. This leads to two-dimensional Kirchhoff-Love plateequations, in which mechanical displacement and electric potentialare partly decoupled. Based on this model numerical examples arepresented that illustrate the mutual interaction between themechanical displacement and the electric potential. We observethat, compared to purely elastic materials, piezoelectric bodiesyield a significantly different contact behavior.

We analyze the propagation of shear horizontal (SH) piezoelectric waves guided by a plate of polarized piezoceramics between two piezoceramic half-spaces with imperfectly bonded interfaces. The interfaces are described by the so-called shear-lag model with an elastic constant characterizing the interface physical behavior. Exact dispersion relations are obtained. It is found that the waves are sensitive to the physical nature of the interfaces. The results are useful for acoustic wave devices.

In this work, we consider the quasistatic frictionless contact problem between aviscoelastic piezoelectric body and a deformable obstacle. The linear electro-viscoelasticconstitutive law is employed to model the piezoelectric material and the normal compliancecondition is used to model the contact. The variational formulation is derived in a formof a coupled system for the displacement and electric potential fields. An existence anduniqueness result is recalled. Then, a fully discrete scheme is introduced based on thefinite element method to approximate the spatial variable and an Euler scheme to discretizethe time derivatives. Error estimates are derived on the approximative solutions and,as a consequence, the linear convergence of the algorithm is deduced under suitableregularity conditions. Finally, some two-dimensional examples are presented to demonstratethe performance of the algorithm.

We rigorously establish the existence of the limithomogeneous constitutive law of a piezoelectric composite made of periodicallyperforated microstructures and whose reference configuration is athin shell with fixed thickness. We deal with an extension of theKoiter shell model in which the three curvilinear coordinates ofthe elastic displacement field and the electric potential arecoupled. By letting the size of themicrostructure going to zero and by using the periodicunfolding method combined with the Korn's inequality in perforateddomains, we obtain the limit model.

Les céramiques piézoélectriques présentent un couplage électromécanique leurdonnant la capacité de développer sous une contrainte mécanique une charge électrique proportionnelle à celle-ci et de se déformer en présence d'un champ électrique.Cette propriété est à la base des nombreuses applications telles que des actionneursou des capteurs. Pour des chargements plus importants, la réponse n'est pluslinéaire. Ce comportement s'explique par la rotation de zones uniformément polariséesqui engendre une polarisation et une déformation irréversibles responsablesdes comportements ferroélectrique et ferroélastique. L'exploitation de ces comportements non-linéairespermet des déplacements plus importants mais elle est freinée par la complexitédu comportement et l'absence d'outils de dimensionnement.Une modélisation phénoménologique décrivant la piézoélectricité, ferroélasticitéet ferroélectricité est présentée. Des cyclages électriques ainsi que le phénomène de dépolarisation mécaniquesont simulés sur la base de ce modèle.

A new electromechanical finite element is implemented in the Abaqus F.E. code. It takes into account the ferroelectric and ferroelastic behavior. It is represented by a developed phenomenological constitutive law using two internal variables (remnant ferroelastic strain and remnant polarization) and two loading surfaces (electrical and mechanical ones). Saturation phenomena and depolarization under compression mechanical loading are taken into account.

In order to lead to a more accurate description of the domain switching mechanisms, a micromechanical single-crystal ferroelectric and ferroelastic constitutive law was also developed.

The self consistent scale transition technique will be adopted to derive the polycristalline behavior. The results that were obtained by the micromechanical and phenomenological approaches will be compared in order to identify material parameters that will be introduced in the F.E. modeling.

The converse piezoelectric response of a thin film constrained by a substrate is analyzed in different geometries under various boundary conditions. We considerthe effects of elastic deformation of the substrate on the total displacement of thefilm surface induced by the electric field. The change of film thickness and the bending curvature of a film/substrate couple are calculated. For a thin film island clamped on a large thick substrate, the theoretical estimation of the piezoresponse, including a local bending in the vicinity of the island/substrate interface, is in agreement with the finite element calculation.

Exact electroelastic analysis of hollow and solid circular cylinders subjected to a uniform temperature change and electromechanical loads that do not vary axially are presented. Generalized plane strain and torsion of a piezoelectric cylindrical body is formulated in a state space framework. Uniformly valid exact solutions are obtained for hollow and solid circular cylinders subjected to extension, torsion, pressuring, shearing, electric loadings and a uniform temperature change.