Wave propagation and dynamic analysis of smoothly graded heterogeneous continua using graded finite elements

Zhengyu (Jenny) Zhang, Glaucio H. Paulino

Research output: Contribution to journalArticlepeer-review


The dynamic behavior of smoothly graded heterogeneous materials is investigated using the finite element method. The global variation of material properties (e.g., Young's modulus, Poisson's ratio and mass density) is treated at the element level using a generalized isoparametric formulation. Three classes of examples are presented to illustrate this approach and to investigate the influence of material inhomogeneity on the characteristics of wave propagation pattern and stress redistribution. First, a cantilever beam example is presented for verification purposes. Emphasis is placed on the comparison of numerical results with analytical ones, as well as modal analysis for beams with different material gradation profiles. Second, wave propagation patterns are explored for a fixed-free slender bar considering homogeneous, bi-material, tri-layered and smoothly graded materials (steel/alumina), which also provide further verification of the numerical procedures. Comparison of stress histories in these samples indicates that the smooth transition of material gradation considerably alleviates the stress discontinuity in the bi-material system (with sharp interface). Third, a three-point-bending epoxy/glass graded beam specimen is investigated for validation purposes. The beam is graded along the height direction. Stress evolution history at a location of interest is analyzed in detail, which not only reveals the dependence of stress evolution on material gradation direction, but also provides information predictive of potential material failure time for graded beams with different material gradation profiles. Jointly, these three classes of examples provide proper verification and validation for the present numerical techniques.

Original languageEnglish (US)
Pages (from-to)3601-3626
Number of pages26
JournalInternational Journal of Solids and Structures
Issue number11-12
StatePublished - Jun 1 2007
Externally publishedYes


  • Dynamic analysis
  • Finite element method
  • Generalized isoparametric formulation
  • Graded finite element
  • Graded material
  • Heterogeneous materials
  • Wave propagation

ASJC Scopus subject areas

  • Modeling and Simulation
  • General Materials Science
  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics


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