Two-phase equilibrium microstructures against optimal composite microstructures

  • Alexander B. FreidinEmail author
  • Leah L. Sharipova


We relate two problems which arise from different branches of mechanics of materials: construction of limiting phase transformation surfaces in strain space and stress–strain diagrams for stress-induced phase transitions and optimal design of two-phase 3D composites in the sense of minimizing its energy. In Antimonov et al. (Int J Eng Sci 98:153–182, 2015) for the case of isotropic phases, it was shown that given a new phase volume fraction and depending on average strain, the strain energy of a two-phase linear-elastic composite is minimized by either direct or inclined simple laminates, direct or skew second-rank laminates or third-rank laminates. Then these results were applied for the construction of direct and reverse transformations limiting surfaces in strain space for elastic solids undergoing phase transformations by additional minimization with respect to the new phase volume fraction and finding the strains at which minimizing volume fraction equals zero or one. In the present paper we construct stress–strain diagrams on various straining paths at which a material undergoes the phase transformation. We demonstrate that an additional degree of freedom—new phase volume fraction—may crucially result in instability of two-phase microstructures even if the microstructures are energy minimizers for composites with given volume fractions of phases. This in turn may lead to incompleteness of monotonic phase transformations and broken stress–strain diagrams. We study how such a behavior depends on a loading path and chemical energies of the phases.


Stress-induced phase transformations Optimal composite microstructure Limit transformation surfaces Stress–strain behavior 



This paper became possible thanks to fruitful collaboration with Andrej Cherkaev who gave Alexander Freidin inspiring lessons on optimal microstructures, which, in turn, arise to Prof. Konstantin Lurie, and for invaluable comments on the draft of the paper. The authors also greatly appreciate financial support of Russian Foundation for Basic Research (Grant No. 16-01-00815)


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Authors and Affiliations

  1. 1.Institute for Problems in Mechanical Engineering of Russian Academy of SciencesSaint PetersburgRussia
  2. 2.Peter the Great St. Petersburg Polytechnic UniversitySaint PetersburgRussia

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