Guillermo Ramirez

Frictionless contact in a layered piezoelectric half-space

A matrix formulation is presented for the solution of frictionless contact problems on arbitrarily multilayered piezoelectric half-planes. Different arrangements of elastic and piezoelectric materials with hexagonal symmetry within the layered medium are considered. A generalized plane deformation is used to obtain the governing equilibrium equations for each individual layer. These equations are solved using the infinite-Fourier-transform technique. The problem is then reformulated using the local/global stiffness method, in which a local stiffness matrix relating the stresses and electric displacements to the mechanical displacements and electrical potentials in the transformed domain is formulated for each layer. Then it is assembled into a global stiffness matrix for the entire half-plane by enforcing interfacial continuity of traction forces and displacements. This local/global approach not only eliminates the necessity of explicitly finding the unknown Fourier coefficients, but also allows the use of efficient numerical algorithms, many of which have been developed for finite-element analysis. Unlike finite-element methods, the local/global stiffness approach requires minimal input. Application of the mixed boundary conditions reduces the problem to a singular integral equation. This integral equation is then numerically solved for the unknown contact pressure using a collocation technique. Knowledge of the contact pressure and electrostatic distributions is very important for applications where piezoelectric layers are used as sensors and/or actuators. One example includes the active deformation and shape control of support surfaces.

Frictionless contact in a layered piezoelectric medium characterized by complex eigenvalues

A local/global stiffness matrix formulation is presented to study the response of an arbitrarily multilayered piezoelectric half-plane indented by a rigid frictionless parabolic punch. The methodology is extended to accommodate the presence of piezoelectric layers that are characterized by complex eigenvalues. Different arrangements of elastic and transversely orthotropic piezoelectric materials within the multilayered medium are considered. A generalized plane deformation is used to obtain the governing equilibrium equations for each individual layer. These equations are solved using the infinite Fourier transform technique. The problem is then reformulated using the local/global stiffness method, in which a local stiffness matrix relating the stresses and electric displacement to the mechanical displacements and electric potential in the transformed domain is formulated for each layer. Then it is assembled into a global stiffness matrix for the entire half-plane by enforcing continuity conditions along the interface. Application of the mixed boundary conditions reduces the problem to an integral equation for the unknown pressure in the contact area. This integral has a divergent kernel that is decomposed into a Cauchy-type and regular parts using the asymptotic properties of the local stiffness matrix. The resulting equation is numerically solved for the unknown contact pressure using a technique based on the Chebyshev polynomials.

FREE VIBRATIONS OF HOMOGENEOUS AND LAYERED PIEZOELECTRIC HOLLOW SPHERES

A finite element formulation in spherical coordinates is presented for the study of the vibrations of piezoelectric homogeneous and layered hollow spheres. The finite element model is based on nine-node Lagrangian interpolation functions. Representative cases are considered including solid elastic spheres and hollow homogeneous and laminated piezoelectric spheres. The accuracy of the proposed formulation is verified by comparison with existing solutions showing excellent agreement. Several new results are presented for different piezoelectric materials in both tabular and graphical format.

Comment on “Free vibration analysis of laminated piezoceramic hollow spheres” [J. Acoust. Soc. Am. 109, 41 (2001)]

In a recent article, Chen [J. Acoust. Soc. Am. 109, 41–50 (2001)] presented a three-dimensional analysis for piezoelectric hollow thick spheres. Results were presented for several configurations for shells of layered piezoelectric materials. The elastic material constants for PZT-4 that were used in the paper were in error and this letter is an attempt to justify and offer suggestions that can preserve the value of the analysis.