Optimization of material distribution in functionally graded structures with stress constraints

Fernando V. Stump, Emílio C.N. Silva, Glaucio H. Paulino

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31 Scopus citations


This work describes a topology optimization framework to design the material distribution of functionally graded structures considering mechanical stress constraints. The problem of interest consists in minimizing the volumetric density of a material phase subjected to a global stress constraint. Due to the existence of microstructure, the micro-level stress is considered, which is computed by means of a mechanical concentration factor using a p-norm of the Von Mises stress criterium (applied to the micro-level stress). Because a 0-1 (void-solid) material distribution is not being sought, the singularity phenomenon of stress constraint does not occur as long as the material at any point of the medium does not vanish and it varies smoothly between material 1 and material 2. To design a smoothly graded material distribution, a material model based on a non-linear interpolation of the Hashin-Strikhman upper and lower bounds is considered. Consistently with the framework adopted here in, the so-called 'continuous approximation of material distribution' approach is employed, which considers a continuous distribution of the design variable inside the finite element. As examples, the designs of functionally graded disks subjected to centrifugal body force are presented. The method generates smooth material distributions, which are able to satisfy the stress constraint.

Original languageEnglish (US)
Pages (from-to)535-551
Number of pages17
JournalCommunications in Numerical Methods in Engineering
Issue number6
StatePublished - Jun 2007
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Software
  • Modeling and Simulation
  • General Engineering
  • Computational Theory and Mathematics
  • Applied Mathematics


  • Functionally graded materials
  • Stress constraint
  • Topology optimization


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