FINITE ELEMENT ANALYSIS OF THE CONTACT DEFORMATION OF PIEZOELECTRIC MATERIALS

Piezoelectric materials in the forms of both bulk and thin-film have been widely used as actuators and sensors due to their electromechanical coupling. The characterization of piezoelectric materials plays an important role in determining device performance and reliability. Instrumented indentation is a promising method for probing mechanical as well as electrical properties of piezoelectric materials. The use of instrumented indentation to characterize the properties of piezoelectric materials requires analytical relations. Finite element methods are used to analyze the indentation of piezoelectric materials under different mechanical and electrical boundary conditions. For indentation of a piezoelectric half space, a three-dimensional finite element model is used due to the anisotropy and geometric nonlinearity. The analysis is focused on the effect of angle between poling direction and indentation-loading direction on indentation responses. For the indentation by a flat-ended cylindrical indenter, both insulating indenter and conducting indenter without a prescribed electric potential are considered. The results reveal that both the indentation load and the magnitude of the indentation-induced potential at the contact center increase linearly with the indentation depth. For the indentation by an insulating Berkovich indenter, both frictionless and frictional contact between the indenter and indented surface are considered. The results show the indentation load is proportional to the square of the indentation depth, while the indentation-induced potential at the contact center is proportional to the indentation depth. Spherical indentation of piezoelectric thin films is analyzed in an axisymmetric finite element model, in which the poling direction is anti-parallel to the indentation-loading direction. Six different combinations of electrical boundary conditions are considered for a thin film perfectly bonded to a rigid substrate under the condition of the contact radius being much larger than the film thickness. The indentation load is found to be proportional to the square of the indentation depth. To analyze the decohesion problem between a piezoelectric film and an elastic substrate, a traction-separation law is used to control the interfacial behavior between a thin film and an electrically grounded elastic substrate. The discontinuous responses at the initiation of interfacial decohesion are found to depend on interface and substrate properties

Analytical model of wave propagation in piezo thermo elastic multilayered PZT5A/LEMV/SWCNT/LEMV/PZT5A circular cylinder

In this study we revised the axisymmetric vibration of an infinite thermo piezoelectric composite circular hollow cylinder made of inner and outer thermo piezoelectric layer bonded together by a Linear Elastic Material with Voids (LEMV) and Single Walled carbon Nano Tube (SWCNT) is studied. The frequency equations are obtained for the traction free outer surface with continuity conditions at the interfaces. The equations of motion, heat and electric conduction also exactly solved. Numerical results are carried out for the inner and outer hollow piezoelectric layers bonded by LEMV and SWCNT layers. The dispersion curves are compared with core/LEMV/core, core cylinders

Elastic and piezoelectric fields due to polyhedral inclusions

AbstractWe derive explicit, closed-form expressions describing elastic and piezoelectric deformations due to polyhedral inclusions in uniform half-space and bi-materials. Our analysis is based on the linear elasticity theory and Green’s function method. The method involves evaluation of volume and surface integrals of harmonic and bi-harmonic potentials. In case of polyhedra, such integrals are expressed through algebraic functions. Our results generalize numerous studies on this subject, and they allow to obtain fully analytical solutions for a number of physical and engineering problems. In the limiting case of an infinite space, our relations have an essentially more compact form, than relations obtained by other authors. We present solutions to classical Mindlin and Cherruti problems. We describe the elastic relaxation of a misfitting polygonal quantum dot in bi-materials assuming isotropic and vertically isotropic properties. It is explained how to analyze non-hydrostatic and non-uniform inclusions. We also study piezoelectric fields induced by inclusions in materials with cubic and hexagonal lattices. Among other results, we have found that a cubic inclusion in an isotropic material reproduces fields of quantum dots in GaAs (0,0,1) and GaAs (1,1,1) depending on the orientation of the cube. This suggests that one can qualitatively model crystals with different lattices by choosing an appropriate inclusion shape

Elastodynamics of radially inhomogeneous spherically anisotropic elastic materials in the Stroh formalism

A method is presented for solving elastodynamic problems in radially inhomogeneous elastic materials with spherical anisotropy, i.e.\ materials such that $c_{ijkl}= c_{ijkl}(r)$ in a spherical coordinate system ${r,\theta,\phi}$. The time harmonic displacement field $\mathbf{u}(r,\theta ,\phi)$ is expanded in a separation of variables form with dependence on $\theta,\phi$ described by vector spherical harmonics with $r$-dependent amplitudes. It is proved that such separation of variables solution is generally possible only if the spherical anisotropy is restricted to transverse isotropy with the principal axis in the radial direction, in which case the amplitudes are determined by a first-order ordinary differential system. Restricted forms of the displacement field, such as $\mathbf{u}(r,\theta)$, admit this type of separation of variables solutions for certain lower material symmetries. These results extend the Stroh formalism of elastodynamics in rectangular and cylindrical systems to spherical coordinates.Comment: 15 page

Wave Propagation in Materials for Modern Applications

In the recent decades, there has been a growing interest in micro- and nanotechnology. The advances in nanotechnology give rise to new applications and new types of materials with unique electromagnetic and mechanical properties. This book is devoted to the modern methods in electrodynamics and acoustics, which have been developed to describe wave propagation in these modern materials and nanodevices. The book consists of original works of leading scientists in the field of wave propagation who produced new theoretical and experimental methods in the research field and obtained new and important results. The first part of the book consists of chapters with general mathematical methods and approaches to the problem of wave propagation. A special attention is attracted to the advanced numerical methods fruitfully applied in the field of wave propagation. The second part of the book is devoted to the problems of wave propagation in newly developed metamaterials, micro- and nanostructures and porous media. In this part the interested reader will find important and fundamental results on electromagnetic wave propagation in media with negative refraction index and electromagnetic imaging in devices based on the materials. The third part of the book is devoted to the problems of wave propagation in elastic and piezoelectric media. In the fourth part, the works on the problems of wave propagation in plasma are collected. The fifth, sixth and seventh parts are devoted to the problems of wave propagation in media with chemical reactions, in nonlinear and disperse media, respectively. And finally, in the eighth part of the book some experimental methods in wave propagations are considered. It is necessary to emphasize that this book is not a textbook. It is important that the results combined in it are taken “from the desks of researchers“. Therefore, I am sure that in this book the interested and actively working readers (scientists, engineers and students) will find many interesting results and new ideas

Static and Free Vibration Analyses of Composite Shells Based on Different Shell Theories

Equations of motion with required boundary conditions for doubly curved deep and thick composite shells are shown using two formulations. The first is based upon the formulation that was presented initially by Rath and Das (1973, J. Sound and Vib.) and followed by Reddy (1984, J. Engng. Mech. ASCE). In this formulation, plate stiffness parameters are used for thick shells, which reduced the equations to those applicable for shallow shells. This formulation is widely used but its accuracy has not been completely tested. The second formulation is based upon that of Qatu (1995, Compos. Press. Vessl. Indust.; 1999, Int. J. Solids Struct.). In this formulation, the stiffness parameters are calculated by using exact integration of the stress resultant equations. In addition, Qatu considered the radius of twist in his formulation. In both formulations, first order polynomials for in-plane displacements in the z-direction are utilized allowing for the inclusion of shear deformation and rotary inertia effects (first order shear deformation theory or FSDT). Also, FSDTQ has been modified in this dissertation using the radii of each laminate instead of using the radii of mid-plane in the moment of inertias and stress resultants equations. Exact static and free vibration solutions for isotropic and symmetric and anti-symmetric cross-ply cylindrical shells for different length-to-thickness and length-to-radius ratios are obtained using the above theories. Finally, the equations of motion are put together with the equations of stress resultants to arrive at a system of seventeen first-order differential equations. These equations are solved numerically with the aid of General Differential Quadrature (GDQ) method for isotropic, cross-ply, angle-ply and general lay-up cylindrical shells with different boundary conditions using the above mentioned theories. Results obtained using all three theories (FSDT, FSDTQ and modified FSDTQ) are compared with the results available in literature and those obtained using a three-dimensional (3D) analysis. The latter (3D) is used here mainly to test the accuracy of the shell theories presented here

Static and Free Vibration Analyses of Composite Shells Based on Different Shell Theories

Equations of motion with required boundary conditions for doubly curved deep and thick composite shells are shown using two formulations. The first is based upon the formulation that was presented initially by Rath and Das (1973, J. Sound and Vib.) and followed by Reddy (1984, J. Engng. Mech. ASCE). In this formulation, plate stiffness parameters are used for thick shells, which reduced the equations to those applicable for shallow shells. This formulation is widely used but its accuracy has not been completely tested. The second formulation is based upon that of Qatu (1995, Compos. Press. Vessl. Indust.; 1999, Int. J. Solids Struct.). In this formulation, the stiffness parameters are calculated by using exact integration of the stress resultant equations. In addition, Qatu considered the radius of twist in his formulation. In both formulations, first order polynomials for in-plane displacements in the z-direction are utilized allowing for the inclusion of shear deformation and rotary inertia effects (first order shear deformation theory or FSDT). Also, FSDTQ has been modified in this dissertation using the radii of each laminate instead of using the radii of mid-plane in the moment of inertias and stress resultants equations. Exact static and free vibration solutions for isotropic and symmetric and anti-symmetric cross-ply cylindrical shells for different length-to-thickness and length-to-radius ratios are obtained using the above theories. Finally, the equations of motion are put together with the equations of stress resultants to arrive at a system of seventeen first-order differential equations. These equations are solved numerically with the aid of General Differential Quadrature (GDQ) method for isotropic, cross-ply, angle-ply and general lay-up cylindrical shells with different boundary conditions using the above mentioned theories. Results obtained using all three theories (FSDT, FSDTQ and modified FSDTQ) are compared with the results available in literature and those obtained using a three-dimensional (3D) analysis. The latter (3D) is used here mainly to test the accuracy of the shell theories presented here

Linear interface crack under harmonic shear : Effects of crack's faces closure and friction

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