Effective Anisotropic Elastic and Plastic Yield Properties of Periodic Foams Derived from Triply Periodic Schoen's I-WP Minimal Surface

Rashid K. Abu Al-Rub, Dong Wook Lee, Kamran A. Khan, Dong Wook Lee, Rashid K. Abu Al-Rub, Anthony N. Palazotto

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42 Scopus citations


Recently, there has been a growing interest in studying cellular materials made of unit cells that take the topology of the mathematically known triply periodic minimal surfaces (TPMS). Architected foams based on the shell-based Schoen's I-graph-wrapped package (I-WP) TPMS (i.e., IWP-foam) seems a promising mechanical metamaterial and might yield better mechanical properties than stochastic or strut-based foams. The IWP-foam has a cubic symmetry and is interconnected and intertwined in the three-dimensional (3D) space. Changing the thickness of the I-WP minimal surface can change the relative density of the IWP-foam. In this paper, the elastic properties (uniaxial, shear and bulk moduli, and Poisson's ratio), elastic anisotropy, plastic properties (tensile, shear, and hydrostatic yield strengths), and anisotropic yield surfaces under different load combinations of IWP-foam, as a function of the relative density, are computed through conducting microscopic finite-element simulations. The anisotropic elastic and plastic properties of IWP-foam are compared with that of other common types of periodic cellular solids. Furthermore, computationally inexpensive closed-form macroscopic isotropic and anisotropic yield functions are used to predict the microscopic yield behavior of IWP-foam. It is found that the uniaxial elastic modulus and yield strength, as well as the shear elastic modulus and shear yield strength, scale with the Gibson-Ashby power-law, indicating a mixed stretching-dominated and bending-dominated deformation behavior. On the other hand, the bulk modulus and hydrostatic yield strength scale linearly with relative density, indicating a stretching-dominated behavior. It is shown that a modified version of Hill's anisotropic yield function can better describe the macroscopic yield behavior of the IWP-foam under several different stress states than the isotropic Deshpande-Fleck model, especially for combinations of shear stress states. Finally, it is noticed from comparing the elastic and plasticity yield properties of the IWP-foam with several other common foams and lattices that, for a given relative density, the IWP-foam has the highest uniaxial stiffness and strength.

Original languageBritish English
Article number04020030
JournalJournal of Engineering Mechanics
Issue number5
StatePublished - 1 May 2020


  • Elastic properties
  • Finite-element analysis
  • Lattices
  • Plastic properties
  • Schoen's I-WP surface


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