Certain Hygrothermopiezoelectric Multi-Field Variational Principles for Smart Elastic Laminae

Abstract

In this study Hamilton's principle is stated for the motion and deformation of a regular region of hygrothermopiezoelectric materials in the elastic range, and a four-field variational principle is obtained under the quasi-static approximation of the electric field. The variational principle is modified through an involuntary transformation and a generalized variational principle is derived in analogy to the Hu-Washizu variational principle of elasticity. Likewise, a variational principle is formulated for the region with a fixed internal surface of discontinuity, and another one for the region containing any number of perfectly bonded dissimilar materials. The variational principles are shown to generate, as its Euler-Lagrange equations, all the divergence and gradient equations, the constitutive relations and the natural mixed boundary and continuity (jump) conditions for each region of the dissimilar materials. The admissible states of the variational principles have Cauchy's second law of motion and the initial conditions for each region, as its constraint conditions. Certain special cases of the variational principles operating on all the field variables and their reciprocals are indicated. The multi-field variational principles with their well-known features are applicable to an analysis of the physical response of smart laminae under the coupled piezoelectric and hygrothermal effects.

Keywords:

• variational principles
• smart elastic laminae
• piezoelectricity
• thermopiezoelectricity
• hygrothermoelasticity
• hygrothermopiezoelectricity

ACKNOWLEDGMENT

The authors are grateful to Professor J.N. Reddy, Editor-in-Chief of the Journal, for his suggestion of the paper and the referees for their comments and very careful reading of the manuscript, and they acknowledge the financial support of TUBA. The paper was presented at Mindlin's Symposium, 15th USNCTAM, Boulder, CO., June 2006. The second author (Dökmeci) is sincerely indebted to late Professors Raymond D. Mindlin and B. M. Fraeijs De Veubeke and, in particular, to his academic advisor Professor Bruno A. Boley for their enlightening and encouraging motivation.