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Advanced Courses on Mathematical Models of Piezoelectric Solids and Related Problems

Date: 23 - 26 September, 2019

Location: Tbilisi International Centre of Mathematics and Informatics at I. Vekua Institute of Applied Mathematics of I. Javakhishvili Tbilisi State University (Tbilisi, Georgia)

Lecturers:

Ayech Benjeddou (Institut Supérieur de Mécanique de Paris (SUPMECA) & Université de Technologie de Compiègne (UTC)/Centre National de la Recherche Scientifique (CNRS) FRE 2012 ROBERVAL, France)

Piezoelectric Material, Effective and Structural Behaviours

Abstract: Piezoelectric materials are popularly used for sensing, actuation and transduction in the framework of smart structures applications such as for structural noise, vibration, shape, control and health monitoring or energy harvesting for autonomous and wireless smart, internet of things or communication devices. Therefore, mastering the behaviours of such smart materials is the key issue for the correct mathematical modelling of piezoelectric solids and related problems. Piezoelectric materials exist in monolithic and composite forms for which the main representatives of the former are piezoelectric ceramics (or piezoceramics) and polymers (or piezo-polymers), while those of the latter are piezoelectric fibre composites. The latter use particles and short or long fibres as reinforcements of polymer-type (usually epoxy) matrices. Besides, the long fibres can have micro circular or macro rectangular cross sections leading, respectively, to the so-called active fibre composites or macro-fibre composites (MFC).
    After a short introduction, this advanced course starts with recalling the material behaviours of monolithic piezoelectric patches. For this purpose, the coupled response modes are given first; then, the dominant ones under electrodes, polarization and electro-mechanical loads configurations are presented. After that, the effective behaviours of piezoelectric composites in transverse (d₃₁) and shear (d₁₅) response modes are discussed for MFC patches. Focus will be made on the properties identification, through analytical and numerical homogenizations of the active core only and multilayer (7 for the concept or 5 for prototypes) stacks. The last part of the course will focus on the recently introduced concept of structural behaviours of piezoelectric patches that are met once integrated (surface-bonded or embedded) onto a support or into a host. Here, the focus will be on surface-bonded (to a support) shear MFC patches numerical (finite element) and experimental global (displacement) response simulations and measurements and their correlations. Depending on the available time, the case of the transverse response of monolithic piezoceramic (PZT PIC255) patches, surface-bonded on or embedded in host composites (Carbon or Glass Fibre Reinforced Polymers), can be presented and discussed.

Bernadette Miara (Université Paris-Est Marne la Vallée, France)

Homogenization and Control of a Piezoelectric Body

Abstract: In this lecture, focused on the modelling of piezoelectric materials we present two examples in the general framework of linearized three-dimensional evolution equations.
- In the first one we consider an heterogeneous composite made of inclusions periodically distributed in a matrix. And we show that when each microstructure presents "strong" heterogeneities, (i.e., the  characteristics, such as mass density,  elasticity, dielectric and coupling  tensors of the inclusions and of the matrix are "strongly" different in terms of the size of the microstructures), then a band-gaps phenomenon may appears, namely in some intervals of frequency where there is no waves propagation. This modelling is obtained through the "homogenization"' technique.
- The second one addresses the exact controllability of an homogeneous piezoelectric body. More precisely, we investigate the existence of boundary controls (such that elastic displacement and / or electric field) which drive the body to rest after a finite time.

Wolfgang H. Müller, Wilhelm Rickert, Felix A. Reich (Institut für Mechanik, Kontinuumsmechanik und Materialtheorie, Technische Universität Berlin, Germany)

An Examination of Elastic Deformation Predictions of Polarizable Media due to Various Electromagnetic Force Models

Abstract: This study investigates the implications for the deformation of electrets due to various electromagnetic force models. This deformation of dielectric materials due to electromagnetic forces is called electrostriction. Analytical solutions for the electrostriction problem of spherical electrets are derived in five different situations. First, an affine linear dielectric sphere with surface charges in an external field is considered. The electric field is computed analytically in a stationary situation. From this solution several model simplifications yield the electric fields of: a linear dielectric in an external field without surfaces charges, a real-charge electret without polarization, an oriented-dipole electret and a real-charge electret with linear dielectric material. With the electric field solutions the electromagnetic forces can be computed for selected force models. Expanding the forces conveniently in terms of Legendre polynomials, the method of Hiramatsu and Oka is applied to obtain the elastic deformation of the spheres.

Gia Avalishvili^*, Mariam Avalishvili^** (^*Faculty of Exact and Natural Sciences, I. Javakhishvili Tbilisi State University, ^**School of Science and Technology, University of Georgia, Tbilisi, Georgia)

On Variational Methods of Investigation of Mathematical Problems for Thermoelastic Piezoelectric Solids

Abstract: In this lecture, we present the results of investigation of boundary and initial-boundary value problems corresponding to mathematical models of thermoelastic piezoelectric solids with regards to magnetic field. We consider three-dimensional static and dynamical models of multi-domain general inhomogeneous anisotropic thermoelastic piezoelectric solids with mixed boundary conditions, when on certain parts of the boundary density of surface force, and normal components of electric displacement, magnetic induction and heat flux are given, and on the remaining parts of the boundary mechanical displacement, temperature, electric and magnetic potentials vanish. We obtain variational formulations of the boundary and initial-boundary value problems in suitable function spaces and present existence, uniqueness and continuous dependence results. Moreover, we construct and investigate hierarchical models of thermoelastic piezoelectric thin structures applying extensions and generalizations of dimensional reduction method, which was suggested by I. Vekua in the classical theory of elasticity for plates with variable thickness.
This work was supported by Shota Rustaveli National Science Foundation (SRNSF) [Grant number 217596, Construction and investigation of hierarchical models for thermoelastic piezoelectric structures].

George Jaiani (I. Vekua Institute of Applied Mathematics & Faculty of Exact and Natural Sciences of I. Javakhishvili Tbilisi State University, Tbilisi, Georgia)

Piezoelectric Cusped Prismatic Shells

Abstract: The present lecture course is devoted to construction of differential hierarchical models for piezoelectric nonhomogeneous porous elastic and viscoelastic Kelvin-Voigt prismatic shells on the basis of linear theories. Using I. Vekua's dimension reduction method, governing systems are derived and in the Nth approximation of hierarchical models boundary value problems (BVPs) and initial boundary value problems (IBVPs) are set. In the N=0 approximation, considering, e.g.,  elastic, plates of a constant thickness, governing systems mathematically coincide with the governing systems of the plane strain corresponding to the basic three-dimensional (3D) linear theory  up to a separate equation for the out of plane component of the displacement vector. The ways of investigation of BVPs and IBVPs, including the case of cusped prismatic shells, are indicated and some preliminary results are presented. Antiplane deformation of piezoelectric nonhomogeneous materials in the three-dimensional formulation and in N=0 approximation is analyzed. Well-posedness of Dirichlet and Keldysh type problems (BVP) are studied in the N=0 order approximation of hierarchical models for cusped prismatic shells. Some BVPs are solved in explicit forms in concrete cases.
This work was supported by Shota Rustaveli National Science Foundation (SRNSF) [Grant number 217596, Construction and investigation of hierarchical models for thermoelastic piezoelectric structures].

David Natroshvili (Georgian Technical University & I. Vekua Institute of Applied Mathematics of I. Javakhishvili Tbilisi State University , Tbilisi, Georgia)

Dynamical Problems of Generalized Thermo-Electro-Magneto-Elasticity theory

Abstract:   The lecture course is dedicated to the theoretical investigation of basic, mixed and crack type three-dimensional initial-boundary value problems of the generalized thermo-electro-magneto-elasticity theory associated with Green-Lindsay's model. The essential feature of the generalized model under consideration is that heat propagation has a finite speed. We analyse dynamical initial-boundary value problems and the corresponding boundary value problems of pseudo-oscillations, which are obtained from the dynamical problems by the Laplace transform.
    The dynamical system of partial differential equations generate a nonstandard six dimensional matrix differential operator of second order, while the system of partial differential equations of pseudo-oscillations generates a second order strongly elliptic formally non-selfadjoint six dimensional matrix differential operator depending on a complex parameter.
    First, we prove uniqueness theorems of dynamical initial-boundary value problems under reasonable restrictions on material parameters and afterwards we apply the Laplace transform technique to investigate the existence of solutions.  This approach reduces the dynamical problems to the corresponding elliptic problems for pseudo-oscillation equations.
    The fundamental matrix of the differential operator of pseudo-oscillations is constructed explicitly by the Fourier transform technique, and its properties near the origin and at infinity are established.
    By the potential method, the corresponding three-dimensional basic, mixed and crack type boundary value problems and the transmission problems for composite elastic structures are reduced to the equivalent systems of boundary pseudodifferential equations.
    The solvability of the resulting boundary pseudodifferential equations are analysed in appropriate Sobolev-Slobodetskii, Bessel potential, and Besov spaces and  the corresponding uniqueness and  existence theorems of solutions to the   boundary value problems under consideration are proved.
    The smoothness properties and singularities of thermo-mechanical and electro-magnetic fields are investigated near the crack edges and the curves where the different types of boundary conditions collide. It is shown that the smoothness and stress singularity exponents essentially depend on the material parameters and an efficient method for their computation is described.
    By the inverse Laplace transform, the solutions of the original dynamical initial-boundary value problems are constructed and their smoothness and asymptotic properties are analysed in detail.

Coordinators: Gia Avalishvili, Mariam Avalishvili

Sponsor:  The Advanced Courses on Mathematical Models of Piezoelectric Solids and Related Problems will be supported by Shota Rustaveli National Science Foundation of Georgia (SRNSF) [217596, Construction and investigation of hierarchical models for thermoelastic piezoelectric structures].

Deadline for registration: August 10, 2019.

Further information:

Address: Tbilisi International Centre of Mathematics and Informatics, I. Vekua Institute of Applied Mathematics of Tbilisi State University, University Str.2, Tbilisi 0143, Georgia

The Advanced Courses can be followed by junior (Masters, PhDs, or Post-Docs) and senior (Assistant, Associate, or Full Professors) academics, as well as researchers and engineers from industry. Only basic mathematics and mechanics are required to follow adequately and fruitfully the Advanced Courses. The participants will also have an opportunity to give 20-minute talks on their own work at the International Conference on Applications of Mathematics and Informatics in Natural Sciences and Engineering (AMINSE 2019), which will be held during the Advanced Courses. Abstracts of the talks will be published and distributed among the lecturers and participants before the Advanced Courses. The registration fee for participants is 650 EUR which includes all local expenses during the Advanced Courses.

The detailed information about AMINSE 2019 can be found on the website:
http://www.viam.science.tsu.ge/aminse2019/