Research ArticleMATERIALS SCIENCE

Computational discovery of extremal microstructure families

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Science Advances  19 Jan 2018:
Vol. 4, no. 1, eaao7005
DOI: 10.1126/sciadv.aao7005
  • Fig. 1 A computational process for the discovery of extremal microstructure families.

    Given a set of physical properties and design constraints, we estimate the material property gamut using stochastic sampling and topology optimization. Structures near the gamut boundary are grouped into families using NLDR. A representative from each family is fitted with a template represented as a skeleton. Beams are placed on the skeleton edges with optimized parameters to fit the original structure. Structure variations with the same topology can be generated by varying the beam parameters. Finally, reduced template parameters are computed to reveal domain-specific design principles.

  • Fig. 2 Five auxetic microstructure families identified by our software.

    (A) Structures with similar properties in the gamut are selected to study their commonalities. Our software embeds the structures using NLDR. (B) The auxetic families are plotted in the embedding space numbered from 1 to 5. Families with similar topologies are located closer in the embedding space. Three example structures from family 5 show the underlying connection between seemingly distinct structures through gradual morphing of shape.

  • Fig. 3 Sampled coverage of microstructure templates in the gamut.

    (A) Extracting a skeleton (middle) from a representative structure (top). The skeleton represents the topology of the structure. A beam network is derived from the skeleton by placing a cuboid on each edge of the skeleton. Because we enforce cubic symmetry, the beams in a single tetrahedron determine the entire beam network. A template can generate a new structure (bottom) that approximates the original structure. (B) Coverage of each template in the material property space. (C) Reducing template parameter dimensions with PCR. The first two reduced parameters approximately correspond to varying the Young’s modulus and Poisson’s ratio of a structure.

  • Fig. 4 Discovered auxetic mechanisms.

    Two mechanisms capable of producing auxetic behavior are discovered from our microstructure families. (A) The slanted column transforms vertical stress into horizontal displacement. (B) The rotating triangle mechanism pulls the outer tip of the joint toward the center of the structure, reducing the macroscopic volume. (C) Relationship between vertical strain and rotation of the triangle joint. (D) The rotation is observed in printed samples under vertical load. (E) Stress is concentrated at the lower end of the triangle joint.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/1/eaao7005/DC1

    Supplementary Text

    fig. S1. Smoothing for geometric difference metric.

    fig. S2. Examples of auxetic structures in addition to the five selected families in the main paper.

    fig. S3. Computing a microstructure template from a representative structure.

    fig. S4. Structures with large Poisson’s ratios (ν > 0.3).

    fig. S5. Microstructures that resemble designs from previous works.

    fig. S6. Reduced parameters for family 4.

    fig. S7. Test apparatus for measuring Young’s modulus and Poisson’s ratio.

    table S1. Auxetic families and templates.

    table S2. Simulated and measured Poisson’s ratios of example structures.

    movie S1. Continuous search of a microstructure using topology optimization.

    movie S2. Shape variation of structures from family 4 along principal directions.

    movie S3. Compression testing and simulation of example from families 1, 3, and 5.

  • Supplementary Materials

    This PDF file includes:

    • Supplementary Text    
    • fig. S1. Smoothing for geometric difference metric.   
    • fig. S2. Examples of auxetic structures in addition to the five selected families in the main paper.    
    • fig. S3. Computing a microstructure template from a representative structure.    
    • fig. S4. Structures with large Poisson’s ratios (ν > 0.3).   
    • fig. S5. Microstructures that resemble designs from previous works.    
    • fig. S6. Reduced parameters for family 4.    
    • fig. S7. Test apparatus for measuring Young’s modulus and Poisson’s ratio.    
    • table S1. Auxetic families and templates.    
    • table S2. Simulated and measured Poisson’s ratios of example structures. 
    • Legends for movies S1 to S3
    • Download PDF
    • Movie S1 -

      (.mp4 format). Continuous search of a microstructure using topology optimization.

    • Movie S2 -

      (.mp4 format). Shape variation of structures from family 4 along principal directions.

    • Movie S3 -

      (.mp4 format). Compression testing and simulation of example from families 1, 3, and 5.

    • Matlab files

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