A mathematical model that includes layerwise finite elements has been developed for use in numerical simulation of the coupled electrical, mechanical, and thermal responses of composite plate structures that incorporate piezoelectric sensors and actuators. Typically, the sensors and actuators in these so-called "smart" structures consist mainly of patches or layers of piezoceramic material. These "smart" structures can be used to sense and/or induce stresses, strains, and/or displacements in themselves or in larger structures of which they are parts. In one important class of potential applications that is particularly relevant to the present mathematical model, the piezoelectric actuators would be used to counteract thermal distortions. Figure 1. This Graphite/Epoxy Plate With Piezoceramic Patches on its upper and lower surfaces was considered to be simply supported at its right and left edges, and subject to a thermal gradient through its thickness.

The mathematical derivation of the model begins with the representation of coupled mechanical, electrical, and thermal responses at the material level by a set of simultaneous equations that include (1) the equation for mechanical equilibrium in the presence of stress; (2) Maxwell's equation for the conservation of electric displacements; and (3) the constitutive equations that express the relationships among strain, electric field, and temperature in a thermopiezoelectric material. The mechanical displacement, electric potential, and temperature are assumed to be fields that are layerwise continuous through the thickness of a given laminate or plate structure. This assumption provides the capability to capture locally induced piezoelectric effects, leading to increased accuracy in prediction of stresses, especially in a laminate that is thick and/or that exhibits strong through-the-thickness thermal and elastic inhomogeneities.

The layerwise formulation leads to a corresponding finite-element formulation for a bilinear plate element. The finite-element equations can be put into a compact matrix form, with the electric potential partitioned into active (applied) and sensory components. The advantage of this form is that the unknown variables (displacements and sensory electric potentials) appear on the left side, while the known quantities (mechanical loads, thermal loads, electric charges, and applied voltages) appear on the right side of the equation. The partitioned equations can be uncoupled into an independent equation for mechanical displacements and another independent equation for sensory electric potentials. Figure 2. Displacements of the Plate of Figure 1 were computed by use of the mathematical model to illustrate the capability of the model and the use of piezoelectric actuation in a "smart" structure to counteract thermal distortion.

In a test case, the model was applied in a simulation of the behavior of a thermally loaded [0°/±45°] graphite/ epoxy laminated plate with a total of 30 piezoceramic patches mounted in symmetrical patterns on the top and bottom surfaces. The plate (see Figure 1) was considered to be simply supported along its y edges, and to be subjected to a thermal gradient from a temperature of 50 °C at its top surface to 50 °C at its bottom surface. The model was used to calculate the distortion caused by the thermal gradient alone, plus the combined effects of the thermal gradient and piezoelectric actuation. The numerical results, plotted in Figure 2, indicate that application of equal potentials of 70 V to the upper and lower piezoelectric patches counteracts the thermal distortion to such an extent as to reduce the center-line deflection to near zero.

This work was done by Ho-Jun Lee of Lewis Research Center and Dimitrios A. Saravanos of Ohio Aerospace Institute.

Inquiries concerning rights for the commercial use of this invention should be addressed to

###### NASA Lewis Research CenterCommercial Technology OfficeAttn: Tech Brief Patent StatusMail Stop 7 - 321000 Brookpark RoadClevelandOhio 44135

Refer to LEW-16537. ##### NASA Tech Briefs Magazine

This article first appeared in the July, 1998 issue of NASA Tech Briefs Magazine.

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