Design of materials with extreme thermal expansion using a three-phase topology optimization method

O. Sigmund, S. Torquato

Research output: Contribution to journalArticlepeer-review

823 Scopus citations


Composites with extremal or unusual thermal expansion coefficients are designed using a three-phase topology optimization method. The composites are made of two different material phases and a void phase. The topology optimization method consists in finding the distribution of material phases that optimizes an objective function (e.g. thermoelastic properties) subject to certain constraints, such as elastic symmetry or volume fractions of the constituent phases, within a periodic base cell. The effective properties of the material structures are found using the numerical homogenization method based on a finite-element discretization of the base cell. The optimization problem is solved using sequential linear programming. To benchmark the design method we first consider two-phase designs. Our optimal two-phase microstructures are in fine agreement with rigorous bounds and the so-called Vigdergauz microstructures that realize the bounds. For three phases, the optimal microstructures are also compared with new rigorous bounds and again it is shown that the method yields designed materials with thermoelastic properties that are close to the bounds. The three-phase design method is illustrated by designing materials having maximum directional thermal expansion (thermal actuators), zero isotropic thermal expansion, and negative isotropic thermal expansion. It is shown that materials with effective negative thermal expansion coefficients can be obtained by mixing two phases with positive thermal expansion coefficients and void.

Original languageEnglish (US)
Pages (from-to)1037-1067
Number of pages31
JournalJournal of the Mechanics and Physics of Solids
Issue number6
StatePublished - Jun 1997

All Science Journal Classification (ASJC) codes

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering


  • A. microstructures
  • A. thermomechanical processes
  • B. constitutive behavior
  • C. numerical algorithms
  • C. optimization


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