On graded elements for multiphysics applications

Emílio C.N. Silva, Ronny C. Carbonari, Glaucio H. Paulino

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


Recently, the functionally graded material (FGM) concept has been explored in piezoelectric materials to improve properties and to increase the lifetime of bimorph piezoelectric actuators. For instance, elastic, piezoelectric, and/or dielectric properties may be graded along the thickness of a piezoceramic. Thus, the gradation of piezoceramic properties influences the performance of piezoactuators. The usual FGM modelling using traditional finite element formulation and discretization into layers gives a highly discontinuous stress distribution, which is undesirable. In this work, we focus on nonhomogeneous piezoelectric materials using a generalized isoparametric formulation based on the graded finite element concept, in which the properties change smoothly inside the element. This approach provides a continuum material distribution, which is appropriate to model FGMs. Both four-node quadrilaterals and eight-node quadrilaterals for piezoelectric FGMs were implemented using the graded finite element concept. A closed form two-dimensional analytical model of piezoelectric FGMs is also developed to check the accuracy of these finite elements and to assess the influence of material property gradation on the behavior of piezoelectric FGMs. The paper discusses and compares the behavior of piezoelectric graded elements under four loading conditions with respect to the analytical solutions (derived in this work) considering exponential variation of elastic, piezoelectric, and dielectric properties separately. The analytical solutions provide benchmark problems to verify numerical procedures (such as the finite element method and the boundary element method).

Original languageEnglish (US)
Pages (from-to)2408-2428
Number of pages21
JournalSmart Materials and Structures
Issue number6
StatePublished - Dec 1 2007
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Signal Processing
  • Civil and Structural Engineering
  • Atomic and Molecular Physics, and Optics
  • General Materials Science
  • Condensed Matter Physics
  • Mechanics of Materials
  • Electrical and Electronic Engineering


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