Equations of high frequency vibrations of thermopiezoelectric crystal plates

doi:10.1016/0020-7683(74)90047-X

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Equations of high frequency vibrations of thermopiezoelectric crystal plates

Journal Article•DOI•

Equations of high frequency vibrations of thermopiezoelectric crystal plates

Raymond D. Mindlin¹•Institutions (1)

Columbia University¹

01 Jun 1974-International Journal of Solids and Structures (Pergamon)-Vol. 10, Iss: 6, pp 625-637

TL;DR: In this article, a system of two-dimensional equations for high frequency motions of crystal plates accounting for coupling of mechanical, electrical and thermal fields is derived for high-frequency motions.

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About: This article is published in International Journal of Solids and Structures.The article was published on 1974-06-01. It has received 349 citations till now. The article focuses on the topics: Coupling & Crystal.

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Journal Article•DOI•

Three-Dimensional Solutions of Smart Functionally Graded Plates

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J. N. Reddy¹, Zhen-Qiang Cheng¹•Institutions (1)

Texas A&M University¹

01 Mar 2001-Journal of Applied Mechanics

TL;DR: In this paper, the authors considered the bending of a functionally graded rectangular plate with an attached piezoelectric actuator and obtained a three-dimensional asymptotic solution.

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Abstract: A smart functionally graded plate consists of a plate made of a functionally gradient material and actuators made of an active material. The active material, a layer or set of patches, is bonded on the metal-rich surface of the functionally graded plate. When the ceramic-rich surface of the substrate is subjected to thermomechanical loadings, displacements, and stresses may be controlled, and vibration amplitudes may be suppressed by the actuators with supplied electric power. In the attempt towards a basic understanding of the new type of smart structural system, this study considers a benchmark problem, namely, the bending of a functionally graded rectangular plate with an attached piezoelectric actuator. The transfer matrix and asymptotic expansion techniques are employed to obtain a three-dimensional asymptotic solution. In numerical computations, the locally effective material properties of the functionally gradient material are estimated by the Mori-Tanaka scheme, The three-dimensional distributions of displacements and stresses for different volume fractions of the ceramic and metallic constituents could serve as benchmark results to assess approximate theories and numerical methods.

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187 citations

Journal Article•DOI•

Analysis of distributed thermopiezoelectric sensors and actuators inadvanced intelligent structures

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Singiresu S. Rao¹, Mehmet Sunar¹•Institutions (1)

Purdue University¹

01 Jul 1993-AIAA Journal

TL;DR: In this article, the quasistatic equations of piezoelectricity and thermopiezoeetics are used to develop a finite element formulation of distributed PDE and TPE media.

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Abstract: The quasistatic equations of piezoelectricity and thermopiezoelectricity are used to develop a finite element formulation of distributed piezoelectric and thermopiezoelectric media. The formulation is then integrated with the distributed sensing and control of advanced intelligent structure design. The procedure is illustrated with the help of two example problems. The purpose of the first example, which consists of two piezoelectric layers used as a bimorph robotic finger, is to check the accuracy of the finite element solution with the analytical one. As a second example, an aluminum beam is utilized along with two polyvinylidene fluoride layers acting as distributed actuator and sensor to study the distributed control of the beam when thermal effects are present. It is concluded that the thermal effects are important in the precision distributed control of intelligent structures.

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179 citations

Journal Article•DOI•

Non-linear constitutive relations for piezoceramic materials

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Shiv P. Joshi

01 Mar 1992-Smart Materials and Structures

TL;DR: In this article, a concise formulation of relevant non-linear constitutive relations is presented in order to predict the response of a structure with embedded piezoelectric material, which makes them suitable for sensor and transducer applications.

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Abstract: Piezoelectric materials produce electric charges when mechanically deformed and an electric potential causes a mechanical deformation. This property makes them suitable for sensor and transducer applications. Understanding of the electroelastic constitutive behavior is critical for predicting the response of a structure with embedded piezoelectric material. A concise formulation of relevant non-linear constitutive relations is presented in this paper.

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171 citations

Journal Article•DOI•

Three-dimensional solutions for coupled thermoelectroelastic response of multilayered plates

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Kangming Xu¹, Ahmed K. Noor¹, Yvette Y. Tang¹•Institutions (1)

Langley Research Center¹

01 Oct 1995-Computer Methods in Applied Mechanics and Engineering

TL;DR: In this article, an analytical three-dimensional solution for the coupled thermoelectroelastic response of multilayered hybrid composite plates is presented, where the sensitivity coefficients measure the sensitivity of the response to variations in different mechanical, thermal and piezoelectric material properties of the plate.

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150 citations

Finite element modelling of piezoelectric active structures

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Vincent Piefort, André Preumont

01 Jan 2001

143 citations

Cites background or methods or result from "Equations of high frequency vibrati..."

...Suleman & Venkayya (1995a) and Suleman & Gonçalvès (1995) proposed a 4 node plate element using bilinear shape functions and the Mindlin assumption (constant shear angle) to accomodate thick as well as thin shells; each node has 5 degrees of freedom (3 translations and 2 rotations), the element has one additional electrical degree of freedom per piezoelectric layer (voltage across the thickness). It uses a reduced integration scheme for the transverse shear stiffness to avoid the transverse shear locking phenomenon. This element is demonstrated using the plate described by Crawley & Lazarus (1991), a bimorph pointer and panel flutter control; comparison with the work of Ha et al....
[...]
...Suleman & Venkayya (1995a) and Suleman & Gonçalvès (1995) proposed a 4 node plate element using bilinear shape functions and the Mindlin assumption (constant shear angle) to accomodate thick as well as thin shells; each node has 5 degrees of freedom (3 translations and 2 rotations), the element has one additional electrical degree of freedom per piezoelectric layer (voltage across the thickness). It uses a reduced integration scheme for the transverse shear stiffness to avoid the transverse shear locking phenomenon. This element is demonstrated using the plate described by Crawley & Lazarus (1991), a bimorph pointer and panel flutter control; comparison with the work of Ha et al. (1992) is made....
[...]
...Rao & Sunar (1993) established a finite element formulation of thermopiezoelectric problems starting from the linear thermopiezoelectric constitutive equations established by Mindlin (1974) and the Hamilton’s principle....
[...]
...Rao & Sunar (1993) established a finite element formulation of thermopiezoelectric problems starting from the linear thermopiezoelectric constitutive equations established by Mindlin (1974) and the Hamilton’s principle. In (Sunar & Rao, 1996, 1997), they used the quasistatic equations of thermopiezoelectricity to develop heat, sensor and actuator equations; a finite element formulation is presented. A distributed control system consisting in a cantilever beam sandwiched between a piezoelectric sensor/actuator pair is used to evaluate the proposed finite element approach on the static and dynamic behaviour. Tzou & Ye (1996) derived a 12-nodes triangular thin solid plane element with 4 degrees of freedom per node; it uses shape functions quadratic in the two in-plane directions and linear in the transverse direction with the assumption of a layerwise constant shear angle (Mindlin hypothesis). A laminate is composed of laminae which could be either elastic material or piezoelectric material; the laminated structure is obtained by stacking elements together and connecting the corresponding nodes; this element is validated by modelling the actuation of a bimorph pointer. To stress the influence of the piezoelectric coupling on the vibration characteristics, a semicircular ring shell has been modelled using 60 triangular shell elements (20 for each layer and 10 element meshes along the length); the evolution of its eigenfrequencies with a growing number of short-circuited electrodes is examined. The number of short-circuited electrodes varies from 1 to 10 (fully short-circuited). This element has been extended later by Köppe et al. (1998) to isoparametric curved triangular and quadrangular elements with shape functions of different polynomial degree for each layer; the model is applied to a rectangular plate of composite material with surface bonded piezo patches under static voltage load, simply supported on two edges....
[...]
...Suleman & Venkayya (1995a) and Suleman & Gonçalvès (1995) proposed a 4 node plate element using bilinear shape functions and the Mindlin assumption (constant shear angle) to accomodate thick as well as thin shells; each node has 5 degrees of freedom (3 translations and 2 rotations), the element has one additional electrical degree of freedom per piezoelectric layer (voltage across the thickness). It uses a reduced integration scheme for the transverse shear stiffness to avoid the transverse shear locking phenomenon. This element is demonstrated using the plate described by Crawley & Lazarus (1991), a bimorph pointer and panel flutter control; comparison with the work of Ha et al. (1992) is made. Chattopadhyay et al. (1999) developed a quasi-static coupled thermopiezoelectric model for a smart composite plate structure with surface bonded piezoelectric materials using a variational...
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References

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Thermoelasticity and Irreversible Thermodynamics

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Maurice A. Biot¹•Institutions (1)

Columbia University¹

01 Mar 1956-Journal of Applied Physics

TL;DR: In this article, a unified treatment of thermoelasticity by application and further developments of the methods of irreversible thermodynamics is presented, along with a new definition of the dissipation function in terms of the time derivative of an entropy displacement.

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Abstract: A unified treatment is presented of thermoelasticity by application and further developments of the methods of irreversible thermodynamics. The concept of generalized free energy introduced in a previous publication plays the role of a ``thermoelastic potential'' and is used along with a new definition of the dissipation function in terms of the time derivative of an entropy displacement. The general laws of thermoelasticity are formulated in a variational form along with a minimum entropy production principle. This leads to equations of the Lagrangian type, and the concept of thermal force is introduced by means of a virtual work definition. Heat conduction problems can then be formulated by the methods of matrix algebra and mechanics. This also leads to the very general property that the entropy density obeys a diffusion‐type law. General solutions of the equations of thermoelasticity are also given using the Papkovitch‐Boussinesq potentials. Examples are presented and it is shown how the generalized coordinate method may be used to calculate the thermoelastic internal damping of elastic bodies.

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2,287 citations

Journal Article•DOI•

LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars

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Stephen P. Timoshenko

01 May 1921-Philosophical Magazine Series 1

TL;DR: In this article, the correction for shear of the differential equation for transverse vibrations of prismatic bars is discussed, where the correction is based on the correction of the transverse vibration of a prismatic bar.

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Abstract: (1921). LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Vol. 41, No. 245, pp. 744-746.

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2,255 citations

Journal Article•DOI•

Piezoelectric Crystals and Their Applications to Ultrasonics

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Warren P. Mason, Hans Baerwald

01 May 1951-Physics Today

TL;DR: Piezoelectric crystals and their application to ultrasonics were discussed in this paper, where the authors proposed a method for the extraction of the ultrasonic properties of these crystals.

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Abstract: Piezoelectric crystals and their application to ultrasonics , Piezoelectric crystals and their application to ultrasonics , دانشگاه تهران

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1,057 citations

Journal Article•DOI•

High frequency vibrations of piezoelectric crystal plates

[...]

Raymond D. Mindlin¹•Institutions (1)

Columbia University¹

01 Jul 1972-International Journal of Solids and Structures

TL;DR: In this article, two-dimensional equations of motion of piezoelectric crystal plates are obtained by retaining early terms of power series expansions of the mechanical displacement and electric potential in a variational principle for the three-dimensional equation of PDEs.

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304 citations