Analysis and homogenization of functionally graded viscoelastic porous structures with a higher order plate theory and statistical based model of cellular distribution

Farooq Al Jahwari, Hani E. Naguib*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

31 Citations (Scopus)


Functionally graded materials (FGMs) have the advantage of continuous microstructure over conventional composite materials. This continuity eliminates many undesired problems like delamination and abrupt change of material properties as in conventional composites. It is also favoured for numerical analysis when homogenizing the properties with a continuous function. Most of the focus has been on particulate-based functionally graded composites and metallic-ceramic FGMs which performed very well in thermal-barrier system and many other applications. Little attention has been on functionally graded porous structures. In this work, we study experimental and numerical procedures of this class of material using a constrained foaming process and homogenization based finite element analysis with plate theory. Plate-like structures of polylactic acid (PLA) are fabricated with a constrained foaming process. The fabricated structures have a gradual porosity with variable cellular size throughout the thickness but homogeneous on-average at parallel planar sections. SEM characterization of the microstructure is statistically analysed with Burr distribution and used as an input to a new homogenization model that accounts for cellular distribution. A higher order plate theory is developed for the analysis that satisfies the free traction condition a priori to the consistency of transverse shear strain energy which had been rarely considered by similar theories in the area. Implementing the theory in finite element procedure necessitates the curvature continuity across elements. This was resolved by implementing the penalty enforcement technique and the use of conforming/nonconforming finite elements with derivative based nodal degrees of freedom. The conforming element showed a better convergence and thus selected for further analysis. The linear viscoelastic behaviour of PLA is assumed to obey Boltzmann superposition principle with hereditary integrals. The stress relaxation functions are determined experimentally for property characterization and from the homogenization model. The proposed plate theory and homogenization model agree very well with the work done by other researchers and experimental data. The numerical tool proved its validity for conducting accurate computer experiments with the minimal input about material properties and microstructure.

Original languageEnglish
Pages (from-to)2190-2205
Number of pages16
JournalApplied Mathematical Modelling
Issue number3
Publication statusPublished - Feb 1 2016
Externally publishedYes


  • Functionally graded
  • Homogenization
  • Plate theory
  • Porous structure
  • Statistical
  • Viscoelastic

ASJC Scopus subject areas

  • Modelling and Simulation
  • Applied Mathematics


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